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+{"text":"\\section{Introduction}\n\n\\subsection{Polyfold regularization}\n\nConsider a compactified moduli space arising from the study of $J$-holomorphic curves in symplectic geometry.\nA foundational problem is finding some way to give this moduli space enough structure to define invariants.\nPolyfold theory, as developed by Hofer, Wysocki, and Zehnder, has been successful in providing a general abstract framework in which it is possible to ``regularize'' such a moduli space, yielding a perturbed moduli space which has sufficient additional structure.\n\n\\begin{theorem}[Polyfold regularization theorem, {\\cite[Thm.~15.4, Cor.~15.1]{HWZbook}}]\n\tIn some established cases, we can construct a polyfold ${\\mathcal Z}$ such that the compactified moduli space $\\overline{\\mathcal{M}}$ is equal to the zero set of a $\\text{sc}$-smooth Fredholm section $\\overline{\\partial}$ of a strong polyfold bundle ${\\mathcal W} \\to {\\mathcal Z}$, i.e., $\\overline{\\mathcal{M}} = \\smash{\\overline{\\partial}}\\vphantom{\\partial}^{-1} (0) \\subset {\\mathcal Z}$.\n\t\n\tWe may then ``regularize'' the moduli space $\\overline{\\mathcal{M}}$ by means of an ``abstract perturbation.''\n\tThe perturbed moduli space $\\overline{\\mathcal{M}}(p):=(\\overline{\\partial}+p)^{-1}(0)$ then has the structure of a compact oriented ``weighted branched orbifold.''\n\\end{theorem}\n\nIn the boundaryless case, such an approach has been successful in regularizing the Gromov--Witten moduli spaces (see \\cite{HWZGW}).\nA specialized approach has yielded a proof of the Arnold conjecture (see \\cite{filippenko2018arnold}).\nThis approach is also being used in the pursuit of a well-defined symplectic field theory (see \\cite{fish2018sft}).\n\nFor a suitably constructed abstract perturbation, the perturbed moduli space $\\overline{\\mathcal{M}}(p)$ has the structure of a compact oriented weighted branched orbifold, and therefore possesses sufficient structure to define the ``branched integration'' of differential forms.\n\n\\begin{theorem}[Polyfold invariants, {\\cite[Cor.~15.2]{HWZbook}}]\n\tLet ${\\mathcal O}$ be an orbifold and consider a $\\text{sc}$-smooth map $f: {\\mathcal Z} \\to {\\mathcal O}$.\n\tWe may define the \\textbf{polyfold invariant} as the homomorphism obtained by pulling back a de Rahm cohomology class from the orbifold and taking the ``branched integral'' over a perturbed moduli space:\n\t\\[\n\tH^*_{\\dR} ({\\mathcal O}) \\to \\mathbb{R}, \\qquad \t\\omega \t\t\t\\mapsto \\int_{\\overline{\\mathcal{M}}(p)} f^*\\omega.\n\t\\]\n\tThis homomorphism does not depend on the choice of abstract perturbation.\n\\end{theorem}\n\nIn particular, this is precisely the form for the polyfold Gromov--Witten invariants defined in \\cite[Thm.~1.12]{HWZbook}.\n\n\\subsection{Naturality of polyfold invariants}\n\nGiven a compactified moduli space $\\overline{\\mathcal{M}}$, it is possible to model $\\overline{\\mathcal{M}}$ in subtly different ways.\nThat is, it is possible to construct distinct polyfolds ${\\mathcal Z}$ and ${\\mathcal Z}'$ which contain $\\overline{\\mathcal{M}}$ as a compact subset, $\\overline{\\mathcal{M}} \\subset {\\mathcal Z}$ and $\\overline{\\mathcal{M}} \\subset {\\mathcal Z}'$.\nAfter regularization of the moduli space $\\overline{\\mathcal{M}}$ we obtain perturbed moduli spaces $\\overline{\\mathcal{M}}(p) \\subset {\\mathcal Z}$ and $\\overline{\\mathcal{M}}(p') \\subset {\\mathcal Z}'$ which have the structure of compact oriented weighted branched suborbifolds.\nWe obtain distinct polyfold invariants by taking the branched integral over these perturbed moduli spaces.\nThus, we find ourselves in the following situation: given a moduli space $\\overline{\\mathcal{M}}$ we can define polyfold invariants associated to the distinct polyfolds ${\\mathcal Z}$ and ${\\mathcal Z}'$ and which, a priori, we cannot assume are equivalent.\n\\emph{Therefore the polyfold invariants, which aspire to be agnostic of all possible choices, may depend on the subtle choices made in modeling a given moduli space.}\n\nIn this paper, we provide a general framework for studying and resolving this problem.\nThe first step is to find a third polyfold ${\\mathcal Y}$ which models $\\overline{\\mathcal{M}}$ and which refines the different structures or choices made, in the sense that there are inclusion maps\n\t\\[\n\t{\\mathcal X}' \\hookleftarrow {\\mathcal Y} \\hookrightarrow {\\mathcal X}.\n\t\\]\nThe problem then reduces to showing that the polyfold invariants for ${\\mathcal Y}$ and ${\\mathcal X}$ are equal.\nWe consider a commutative diagram of inclusion maps between polyfolds and between strong polyfold bundles of the form:\n\t\\[\\begin{tikzcd}\n\t{\\mathcal V} \\arrow[r, hook] \\arrow[d, \"\\overline{\\partial}_{\\mathcal Y} \\quad\"'] & {\\mathcal W} \\arrow[d, \"\\quad \\overline{\\partial}_{\\mathcal Z}\"] &  \\\\\n\t{\\mathcal Y} \\arrow[r, hook] \\arrow[u, bend left] & {\\mathcal Z} \\arrow[u, bend right] & \n\t\\end{tikzcd}\\]\nin addition to a commutative diagram with target space the orbifold ${\\mathcal O}$:\n\t\\[\\begin{tikzcd}\n\t&  & {\\mathcal O} \\\\\n\t{\\mathcal Y} \\arrow[r, hook] \\arrow[rru, \"f_{\\mathcal Y}\"] & {\\mathcal Z} \\arrow[ru, \"f_{\\mathcal Z}\"'] & \n\t\\end{tikzcd}\n\t\\]\nAs outlined at the start of \\S~\\ref{subsec:intermediary-subbundles-naturality}, we assume that these inclusion maps satisfy a number of properties.\nAlthough these hypothesis are somewhat lengthy at a glance, they will be natural from the point of view of our applications, and moreover reflect some commonalities of the practical construction of distinct polyfolds which model the same moduli space.\nIn these application, we furthermore note that the bundle ${\\mathcal V}$ is not the same as the pullback bundle of ${\\mathcal W}$, hence we may not use the methods of pulling back abstract perturbations of Theorem~\\ref{thm:pullback-regular-perturbation}.\n\nThe most substantial hypothesis is the existence of an ``intermediary subbundle,'' a subset of the target strong polyfold bundle ${\\mathcal R}\\subset {\\mathcal W}$ whose object space is fiberwise a (not necessarily complete) vector space and which satisfies some additional properties (see Definition~\\ref{def:intermediate-subbundle}).\n\n\\begin{theorem}[Naturality of polyfold invariants]\n\t\\label{thm:naturality-polyfold-invariants}\n\tConsider a compactified moduli space $\\overline{\\mathcal{M}}$ which is modeled by two polyfolds ${\\mathcal Y}$ and ${\\mathcal Z}$, i.e., $\\overline{\\mathcal{M}} \\subset {\\mathcal Y}$ and $\\overline{\\mathcal{M}} \\subset {\\mathcal Z}$.\n\tSuppose there is an inclusion map ${\\mathcal Y} \\hookrightarrow {\\mathcal Z}$.\n\tMoreover, assume we satisfy the hypothesis of the general framework described in \\S~\\ref{subsec:intermediary-subbundles-naturality}.\n\t\n\tSuppose there exists an intermediary subbundle ${\\mathcal R} \\subset {\\mathcal W}$. Then the polyfold invariants for ${\\mathcal Y}$ and ${\\mathcal Z}$ defined via the branched integral are equal.\n\tThis means that, given a de Rahm cohomology class $\\omega\\in H^*_{\\dR} ({\\mathcal O})$ the branched integrals over the perturbed moduli spaces are equal,\n\t\\[\n\t\\int_{\\overline{\\mathcal{M}}(p)} f_{\\mathcal Y}^* \\omega = \\int_{\\overline{\\mathcal{M}}(p')} f_{\\mathcal Z}^* \\omega,\n\t\\]\n\tfor any regular abstract perturbations.\n\\end{theorem}\n\nThe proof ends up being somewhat involved as we encounter some substantial technical difficulties, which we sketch briefly.\nRoughly, the existence of an intermediary subbundle allows the construction of abstract perturbations $p'$ of the strong polyfold bundle ${\\mathcal W} \\to {\\mathcal Z}$ whose restrictions induce a well-defined abstract perturbation $p$ of the strong polyfold bundle ${\\mathcal V} \\to {\\mathcal Y}$.\nThis allows us to consider a well-defined restriction between perturbed moduli spaces, \n\t\\[\\overline{\\mathcal{M}}(p) \\hookrightarrow \\overline{\\mathcal{M}}(p').\\]\nOn the level of topological spaces, this restriction is a continuous bijection.\nWhile we can achieve transversality for both perturbations, the abstract polyfold machinery is only able to ``control the compactness'' of the target perturbed moduli space, hence via usual methods we can only assume that $\\overline{\\mathcal{M}}(p')$ is a compact topological space.\n\nUsing only knowledge of the underlying topologies of both of these spaces, it is impossible to say anything more. \nThe key to resolving this problem is understanding the additional structure that these spaces possess---the branched orbifold structure---and using this structure to prove an invariance of domain result for weighted branched orbifolds (see Lemma~\\ref{lem:invariance-of-domain-branched-orbifolds}).\nThis result will allow us to assert that the above map is a homeomorphism---and therefore, $\\overline{\\mathcal{M}}(p)$ is also compact.\n\nThe second major difficulty comes from the fact that the restricted perturbation $p$ on the source space is not a ``regular'' perturbation (see Definition~\\ref{def:regular-perturbation}).\nThis is problematic due to the fact that the present theory only guarantees the existence of a compact cobordism between abstract perturbations which are both assumed to be ``regular'' (see Theorem~\\ref{thm:cobordism-between-regular-perturbations}).\nIn order to resolve this problem, we must generalize the abstract perturbation theory to allow for perturbation of $\\text{sc}$-smooth proper ``Fredholm multisections'' (see \\S~\\ref{subsec:fredholm-multisections}).\nThis generalization enables us to construct a compact cobordism from the restricted perturbation $p$ to a regular perturbation (see Proposition~\\ref{prop:cobordism-multisection-regular}).\n\n\\begin{comment}\nExplain the problem in simple terms...\nAnd explain the main theorem in simple terms.\nAnd then explain some basic steps and ideas required in the proof of the main theorem.\n\nGiven a compact moduli space $\\overline{\\mathcal{M}}$, it is possible to model the problem with distinct polyfolds which contain $\\overline{\\mathcal{M}}$ as a compact subset.\nAfter regularization, we then have perturbed moduli spaces, which have the structure of compact weighted branched orbifolds.\nThe polyfold invariants are defined by integration over these compact weighted branched orbifolds.\nAlthough they are supposed to model the same moduli space, a priori we cannot assume that these polyfold invariants are equal.\n\nWe provide a general framework for studying this problem.\nFirst step, is to find a polyfold which refines the different structures \/ choices of the two polyfolds xxx, in the sense that there should be inclusion maps\n\nBy symmetry, it is equivalent to show the polyfold invariants for ${\\mathcal Y}$ and ${\\mathcal X}$ are equal.\nAs part of the general framework, we consider commutative diagrams of the form:\n\t\\[\\begin{tikzcd}\n\tDIAGRAM\n\t\\end{tikzcd}\\]\nThere are a long list of expected properties of the smooth inclusion ${\\mathcal Y} \\hookrightarrow {\\mathcal X}$ should naturally satisfy, see the introduction of the section XXX.\n(We should note at the outset that the bundle W is not the same as the pullback bundle)\nThese expected properties don't really help us in solving this problem, but they do give us a foothold in understanding the difficulties.\n\nThe key object of study is an ``intermediary subbundle,'' a subset $R\\subset W$ which is fiberwise a bundle, and satsfies some additional properties.\n\n\\begin{theorem}\n\tSuppose we are in the general framework as outlined in section XXX.\n\tSuppose there exists an intermediary subbundle.\n\tThen the polyfold invariants are equal.\n\\end{theorem}\n\n\nWe give a sketch of some of the ideas involved in the proof of the theorem.\nRoughly, the existence of an intermediary subbundle allows us to construct abstract perturbations which have a well defined restriction.\nThis allows us to get a well-defined restriction between perturbed solution spaces:\n\nHowever, while we can control the compactness of the target space, we have no hope of controlling the compactness of the source space.\nAnd from the point of view of point-set topology, we are totally stuck.\n\nETC continue the argument...\n\\end{comment}\n\n\\subsection{Application: Naturality of the polyfold Gromov--Witten invariants}\n\nThe construction of a Gromov--Witten polyfold structure requires choices, such as the choice of a cut-off function in the gluing constructions, choices of good uniformizing families of stable maps, choice of a locally finite refinement of a cover of M-polyfold charts, as well as the exponential gluing profile.\n\nIn addition to these choices, one must also choose a strictly increasing sequence $(\\delta_i)_{i\\geq 0} \\subset (0,2\\pi)$, i.e.,\n\t\\[\n\t0<\\delta_0 < \\delta_1 < \\cdots < 2\\pi.\n\t\\]\nThis sequence is used to define $\\text{sc}$-Banach spaces which are then used to define the M-polyfold models of the Gromov--Witten polyfold ${\\mathcal Z}_{A,g,k}^{3,\\delta_0}$ (see \\cite[\\S~2.4]{HWZGW}).\n\nThe following theorem states that, having fixed the exponential gluing profile and a strictly increasing sequence $(\\delta_i)_{i\\geq 0}\\subset (0,2\\pi)$, different choices lead to Morita equivalent polyfold structures. Hence the Gromov--Witten polyfold invariants are independent of such choices.\n\n\\begin{theorem}[{\\cite[Thm.~3.37]{HWZGW}}]\n\tHaving fixed the exponential gluing profile and a strictly increasing sequence $(\\delta_i)_{i\\geq 0}\\subset (0,2\\pi)$, the underlying topological space ${\\mathcal Z}_{A,g,k}^{3,\\delta_0}$ possesses a  natural equivalence class of polyfold structures.\n\\end{theorem}\n\nWe can use Theorem~\\ref{thm:naturality-polyfold-invariants} to show that the polyfold Gromov--Witten invariants are also independent of the choice of increasing sequence, and hence are natural in the sense that they do not depend on any choice made in the construction of the Gromov--Witten polyfolds.\n\n\\begin{corollary}[Naturality of the polyfold Gromov--Witten invariants]\n\t\\label{cor:naturality-polyfold-gw-invariants}\n\tThe polyfold Gromov--Witten invariants do not depend on the choice of an increasing sequence $(\\delta_i)_{i\\geq 0} \\allowbreak \\subset (0,2\\pi)$.\n\\end{corollary}\n\nWe now consider the choice of puncture at the marked points\nThe underlying set of the Gromov--Witten polyfolds consist of stables curves.\nAs constructed in \\cite{HWZbook}, these stable curves are required to satisfy exponential decay estimates on punctured neighborhoods of the nodal pairs.\nIn contrast, for these Gromov--Witten polyfolds no such decay is required at the marked points.\n\nHowever, in some situations we would like to treat the marked points in the same way as the nodal points.\nFor example, this is true in the context of the splitting and genus reduction axioms, where we will wish to identify a pair of marked points with the same image with a nodal pair.\nAllowing a puncture with exponential decay at a specified marked point is a global condition on a Gromov--Witten polyfold, and hence different choices of puncture at the marked points yield distinct Gromov--Witten polyfolds.\n\nWe again use Theorem~\\ref{thm:naturality-polyfold-invariants} to show that the polyfold Gromov--Witten invariants are independent of such choice of puncture at the marked points.\n\n\\begin{corollary}\n\t\\label{cor:punctures-equal}\n\tThe polyfold Gromov--Witten invariants do not depend on the choice of puncture at the marked points.\n\\end{corollary}\n\n\\subsection{Pulling back abstract perturbations in polyfold theory}\n\nConsider distinct moduli spaces $\\overline{\\mathcal{M}}$ and $\\overline{\\mathcal{M}}'$ which are modeled by polyfolds ${\\mathcal Y}$ and ${\\mathcal Z}$, respectively.\nConsider a naturally defined $\\text{sc}$-smooth map between polyfolds $f: {\\mathcal Y} \\to {\\mathcal Z}$ which restricts to a map between moduli spaces $f|_{\\overline{\\mathcal{M}}} : \\overline{\\mathcal{M}} \\to \\overline{\\mathcal{M}}'$.\nIn many situations we would like to study the geometry of this map and in order to establish algebraic relationships between the respective polyfold invariants.\n\nHowever, without work, we cannot assume that this map will \\emph{persist} after abstract perturbation.\nAbstract perturbations are constructed using bump functions and choices of vectors in a strong polyfold bundle, which in general we cannot assume will be preserved by the $\\text{sc}$-smooth map $f$.\n\nTo solve this problem, consider a pullback diagram of strong polyfold bundles as follows:\n\t\\[\\begin{tikzcd}\n\tf^* {\\mathcal W} \\arrow[d, \"f^*\\overline{\\partial} \\quad\"'] \\arrow[r, \"\\text{proj}_2\"'] & {\\mathcal W} \\arrow[d, \"\\quad \\overline{\\partial}\"] &  \\\\\n\t{\\mathcal Y} \\arrow[r, \"f\"'] \\arrow[u, bend left] & {\\mathcal Z}. \\arrow[u, bend right] & \n\t\\end{tikzcd}\\]\nThe natural approach for obtaining a well-defined map between the perturbed moduli spaces is to take the pullback an abstract perturbation.\nThe main technical point is ensuring that we can control the compactness of the pullback perturbation.\nThis is achieved by a mild topological hypothesis on the map $f$, called the ``topological pullback condition'' (see Definition~\\ref{topological-pullback-condition}).\n\n\\begin{theorem}\n\t\\label{thm:pullback-regular-perturbation}\n\tConsider a $\\text{sc}$-smooth map between polyfolds, $f: {\\mathcal Y} \\to {\\mathcal Z}$, and consider a pullback diagram of strong polyfold bundles as above.\n\tIf $f$ satisfies the topological pullback condition then there exists a regular perturbation $p$ which pulls back to a regular perturbation $f^*p$.\n\t\n\tIt follows that we can consider a well-defined restriction between perturbed moduli spaces,\n\t\t\\[\n\t\tf|_{\\overline{\\mathcal{M}}(f^*p)} : \\overline{\\mathcal{M}}(f^*p) \\to \\overline{\\mathcal{M}} (p).\n\t\t\\]\n\\end{theorem}\n\nThis theorem follows from the more technically stated Theorem~\\ref{thm:compatible-pullbacks}.\n\n\\subsection{Application: Permutation maps between perturbed Gromov--Witten moduli spaces}\n\nLet $(Q,\\omega)$ be a closed symplectic manifold, and fix a homology class $A \\in H_2 (Q;\\mathbb{Z})$ and integers $g,\\ k\\geq 0$ such that $2g+k \\geq 3$.\nFix a permutation $\\sigma: \\{1,\\ldots, k\\} \\to \\{ 1,\\ldots, k \\}$.\nConsider the natural $\\text{sc}$-diffeomorpism between Gromov--Witten polyfold defined by permuting the marked points,\n\t\\[\n\t\\sigma: {\\mathcal Z}_{A,g,k}\\to {\\mathcal Z}_{A,g,k}.\n\t\\]\nFor a fixed compatible almost complex structure $J$, this map has a well-defined restriction to the unperturbed Gromov--Witten moduli spaces\n\t\\[\n\t\\sigma|_{\\overline{\\mathcal{M}}_{A,g,k}(J)} : \\overline{\\mathcal{M}}_{A,g,k}(J) \\to \\overline{\\mathcal{M}}_{A,g,k}(J).\n\t\\]\n\nAs we have mentioned, abstract perturbations are constructed using bump functions and choic\\-es of vectors in a strong polyfold bundle, which in general will not exhibit symmetry with regards to the labelings of the marked points.\nAs a result, given a stable curve $x\\in {\\mathcal Z}_{A,g,k}$ which satisfies a perturbed equation $(\\overline{\\partial}_J+p)(x)=0$ we cannot expect that $(\\overline{\\partial}_J+p)(\\sigma(x))=0$, as the perturbations are not symmetric with regards to the permutation $\\sigma$.\nTherefore, naively there does not exist a permutation map between perturbed Gromov--Witten moduli spaces.\n\nHowever, since $\\sigma: {\\mathcal Z}_{A,g,k}\\to {\\mathcal Z}_{A,g,k}$ is a homeomorphism on the level of the underlying topological spaces, it is immediate that it satisfies the topological pullback condition, hence we immediately obtain the following corollary.\n\n\\begin{corollary}\n\t\\label{cor:pullback-via-permutation}\n\tThere exists a regular perturbation which pulls back to a regular perturbation via the permutation map $\\sigma:{\\mathcal Z}_{A,g,k}\\to {\\mathcal Z}_{A,g,k}$.\n\tTherefore, we can consider a well-defined permutation map between the perturbed Gromov--Witten moduli spaces,\n\t\t\\[\n\t\t\\sigma|_{\\overline{\\mathcal{M}}_{A,g,k}(\\sigma^*p)} : \\overline{\\mathcal{M}}_{A,g,k}(\\sigma^*p) \\to \\overline{\\mathcal{M}}_{A,g,k} (p).\n\t\t\\]\n\\end{corollary}\n\n\\subsection{Organization of the paper}\n\nWe give a self contained introduction to the basic abstract perturbation machinery of polyfold theory in \\S~\\ref{sec:abstract-perturbations-polyfold-theory}.\nIn \\S~\\ref{subsec:polyfolds-ep-groupoids} we review scale calculus, the definition of a polyfold as an ep-groupoid, and discuss the induced topology on subgroupoids and on branched suborbifolds.\nIn \\S~\\ref{subsec:abstract-perturbations} we discuss strong polyfold bundles, $\\text{sc}$-smooth Fredholm sections and $\\text{sc}^+$-multisection perturbations. In addition, we also discuss transverse perturbations, how to control the compactness of a perturbation, and questions of orientation.\nIn \\S~\\ref{subsec:branched-integral-polyfold-invariants} we consider $\\text{sc}$-smooth differential forms, the definition of the branched integral on a weighted branched suborbifold, and how to define the polyfold invariants.\n\nWe provide a general framework for proving that the polyfold invariants are natural, and do not depend on the construction of a polyfold model for a given moduli space in \\S~\\ref{sec:naturality-polyfold-invariants}.\nIn \\S~\\ref{subsec:invariance-of-domain} we prove an invariance of domain result for branched suborbifolds, Lemma~\\ref{lem:invariance-of-domain-branched-orbifolds}.\nIn \\S~\\ref{subsec:fredholm-multisections} we generalize the polyfold abstract perturbation theory to the case of a $\\text{sc}$-smooth proper Fredholm multisection.\nIn \\S~\\ref{subsec:intermediary-subbundles-naturality} we provide the general framework, introduce the definition of an intermediary subbundle, and prove that the equality of polyfolds invariants in Theorem~\\ref{thm:naturality-polyfold-invariants}.\nIn \\S~\\ref{subsec:independence-sequence} we apply Theorem~\\ref{thm:naturality-polyfold-invariants} to show that the polyfold Gromov--Witten invariants are independent of the choice of increasing sequence.\nIn \\S~\\ref{subsec:independence-punctures} we apply Theorem~\\ref{thm:naturality-polyfold-invariants} to show that the polyfold Gromov--Witten invariants are independent of the choice of puncture at the marked points.\n\nWe discuss how to pull back regular perturbations in \\S~\\ref{sec:pulling-back-abstract-perturbations}.\nIn \\S~\\ref{subsec:pullbacks-strong-polyfold-bundles} we define the pullback of a strong polyfold bundle and of a $\\text{sc}^+$-multisection.\nIn \\S~\\ref{subsec:topological-pullback-condition-controlling-compactness} we introduce the topological pullback condition and show how it allows us to pullback a pair which controls compactness.\nIn \\S~\\ref{subsec:construction-regular-perturbation-which-pullback} we construct regular perturbations which pullback to regular perturbations, proving Theorem~\\ref{thm:compatible-pullbacks}.\nIn \\S~\\ref{subsec:permutation-map} we apply Theorem~\\ref{thm:compatible-pullbacks} to obtain a well-defined permutation map between the perturbed Gromov--Witten moduli spaces.\n\nIn Appendix~\\ref{appx:local-surjectivity} we consider some basic properties of the linearized Cauchy--Riemann operator, which allow us to assert the simple fact that cokernel vectors can be chosen so that they vanish on small neighborhoods of the marked or nodal points.\n\n\n\\section{Abstract perturbations in polyfold theory}\n\t\\label{sec:abstract-perturbations-polyfold-theory}\n\nIn this section we recall and summarize the construction of abstract perturbations in polyfold theory, as developed by Hofer, Wysocki, and Zehnder.\n\n\\subsection{Polyfolds and ep-groupoids}\n\t\\label{subsec:polyfolds-ep-groupoids}\n\nWe use the modern language of \\'etale proper Lie groupoids to define polyfolds.  \nThe notion of orbifold was first introduced by Satake \\cite{satake1956generalization}, with further descriptions in terms of groupoids and categories by Haefliger \\cites{haefliger1971homotopy,haefliger1984groupoide,haefliger2001groupoids}, and Moerdijk \\cites{moerdijk2002orbifolds,moerdijk2003introduction}.  \nWith this perspective, a polyfold may be viewed as a generalization of a (usually infinite-dimensional) orbifold, with additional structure.  This generalization of the \\'etale proper Lie groupoid language to the polyfold context is due to Hofer, Wysocki, and Zehnder \\cite{HWZ3}.\nFor full details in the present context, we will refer the reader to \\cite{HWZbook} for the abstract definitions of ep-groupoids in the polyfold context.\n\n\\subsubsection[sc-Structures, M-polyfolds, and polyfold structures]{$\\text{sc}$-Structures, M-polyfolds, and polyfold structures}\n\n\nWe begin by discussing the basic definitions of ``scale calculus'' in polyfold theory.\nScale calculus is a generalization of classical functional analytic concepts, designed to address the classical failure of reparametrization actions to be differentiable (see \\cite[Ex.~2.1.4]{ffgw2016polyfoldsfirstandsecondlook}).\nThus, scale calculus begins by generalizing notions of Banach spaces and of Fr\\'echet differentiability in order to obtain scale structures where reparametrization will be a smooth action.\n\n\n\\begin{definition}[{\\cite[Def.~1.1]{HWZbook}}]\n\tA \\textbf{$\\text{sc}$-Banach space} consists of a Banach space $E$ together with a decreasing sequence of linear subspaces\n\t\\[\n\tE=E_0\\supset E_1 \\supset \\cdots \\supset E_\\infty := \\cap_{i\\geq 0} E_i\n\t\\]\n\tsuch that the following two conditions are satisfied.\n\t\\begin{enumerate}\n\t\t\\item The inclusion operators $E_{m+1} \\to E_m$ are compact.\n\t\t\\item $E_\\infty$ is dense in every $E_i$.\n\t\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}[{\\cite[Def.~1.9]{HWZbook}}]\n\t\\label{def:ssc-differentiability-ssc-Banach-spaces}\n\tA map $f:U\\rightarrow U'$ between two open subsets of $\\text{sc}$-Banach spaces $E$ and $E'$ is called a \\textbf{$\\text{sc}^0$-map}, if  $f(U_i)\\subset U'_i$  for all $i\\geq 0$ and if the induced maps  $f:U_i\\rightarrow U'_i$ are continuous.  \n\tFurthermore, $f$ is called a \\textbf{$\\text{sc}^1$-map}, or of \\textbf{class $\\text{sc}^1$}, if the following conditions are satisfied.\n\t\\begin{itemize}\n\t\t\\item For every $x\\in U_1$ there exists a bounded \n\t\tlinear  map $Df(x)\\in  \\mathcal{L}(E_0, E'_0)$ satisfying for $h\\in\n\t\tE_1$, with $x+h\\in U_1$,\n\t\t\\[\\frac{1}{\\norm{h}_1}\\norm{f(x+h)-f(x)-Df(x)h}_0\\to 0\\quad\n\t\t\\text{as\\ $\\norm{h}_1\\to 0$.}\\mbox{}\\\\[4pt]\\]\n\t\t\\item  The tangent map  $Tf:TU\\to TU'$,\n\t\tdefined by\n\t\t\\[Tf(x, h)=(f(x), Df(x)h),\n\t\t\\]\n\t\tis a $\\text{sc}^0$-map between the tangent spaces.\n\t\\end{itemize}\n\\end{definition}\n\nIf $Tf:TU\\to TU'$ is of class $\\text{sc}^1$, then $f:U\\to U'$ is called of class $\\text{sc}^2$; inductively, the map $f:U\\to E'$ is called of class $\\text{sc}^k$ if the $\\text{sc}^0$-map $T^{k-1}f:T^{k-1}U\\to T^{k-1}E'$ is of class $\\text{sc}^1$.  A map which is of class $\\text{sc}^k$ for every $k$ is called  \\textbf{$\\text{sc}$-smooth}  or of  \\textbf{class $\\text{sc}^{\\infty}$}.  The basic building block which allows us to check the $\\text{sc}$-differentiability of maps is the chain rule.\n\n\\begin{proposition}[Chain rule, {\\cite[Thm.~1.1]{HWZbook}}]\n\tAssume that $E$, $F$, and $G$ are $\\text{sc}$-smooth Banach spaces and $U\\subset E$ and $V\\subset F$ are open sets.  Assume that $f:E\\to F$, $g:V\\to G$ are of class $\\text{sc}^1$ and $f(U)=V$.  Then the composition $g\\circ f :U\\to G$ is of class $\\text{sc}^1$ and the tangent maps satisfy\n\t\\[\n\tT(g\\circ f) = Tg \\circ Tf.\n\t\\]\n\\end{proposition}\n\n\n\n\\begin{definition}[{\\cite[Defs.~2.1,~2.2]{HWZbook}}]\n\tConsider a $\\text{sc}$-Banach space $E$ and consider an open subset $U\\subset E$.  A $\\text{sc}$-smooth map $r:U\\to U$ is called a \\textbf{$\\text{sc}$-smooth retraction} on $U$ if $r\\circ r = r$.\n\tA \\textbf{local M-polyfold model (without boundary)} is a pair $(O,E)$ consisting of a $\\text{sc}$-Banach space $E$ and a subset $O\\subset E$ such that there exists a $\\text{sc}$-smooth retraction $r:U \\to U$ defined on an open subset $U\\subset E$ such that $r(U)= O$.  We call $O$, equipped with the subspace topology $O\\subset E$, a \\textbf{$\\text{sc}$-retract}.\n\\end{definition}\n\nThese definitions of $\\text{sc}$-differentiability extend to local M-polyfolds models in the following way.\n\\begin{definition}[{\\cite[Def.~2.4]{HWZbook}}]\n\tA map $f:O \\to O'$ between two local M-polyfold models is of \\textbf{class $\\text{sc}^k$} if the composition  $f\\circ r:U\\to E'$ is of class $\\text{sc}^k$ where $U\\subset E$ is an open subset of the $\\text{sc}$-Banach space $E$ and where $r:U\\to U$ is a $\\text{sc}$-smooth retraction onto $r(U)=O$. \n\\end{definition}\n\n\\begin{comment}\n{only include this if i want to elaborate more about reparametrization.\nIn Gromov--Witten theory the compactification phenomena consist of nodal curves; hence our local M-polyfold models are without boundary and the $i$th-noded curves appear as interior points in the local M-polyfold models of codimension $2i$ (see \\cite[Rem.~5.3.2]{ffgw2016polyfoldsfirstandsecondlook}).\nOther cases such as Hamiltonian Floer theory (as first introduced in \\cite{floer1988unregularized}) and Symplectic Field Theory (as first introduced in \\cite{egh2000introduction}) would require the inclusion of partial quadrants in order to deal with boundaries and corners (see \\cite[Defs.~1.6,~2.2]{HWZbook} for details on local M-polyfold models with partial quadrants).}\n\\end{comment}\n\nIn the absence of isotropy, we may consider the following definition of an ``M-polyfold,'' short for a ``polyfold of manifold type.''\n\n\\begin{definition}[{\\cite[Def.~2.8]{HWZbook}}]\n\tWe say that a paracompact Hausdorff topological space $Z$ is an \\textbf{M-polyfold} if every point $z\\in Z$ has an open neighborhood\n\twhich is homeomorphic to a $\\text{sc}$-retract $O$, and such that the induced transition maps between any two $\\text{sc}$-retracts are $\\text{sc}$-smooth.\n\\end{definition}\n\nHowever, in almost all situations that arise isotropy is inevitable, and must be dealt with.\nIn this sense, polyfold behave like infinite-dimensional orbifolds, and so we introduce the language of ep-groupoids.\n\n\\begin{definition}[{\\cite[Defs.~7.1,~7.3]{HWZbook}}]\n\tA \\textbf{groupoid} $(Z,{\\bar\\m}{Z})$ is a small category consisting of a set of objects $Z$, a set of morphisms ${\\bar\\m}{Z}$ which are all invertible, and the five structure maps $(s,t,m,u,i)$ (the source, target, multiplication, unit, and inverse maps).\n\tAn \\textbf{ep-groupoid} is a groupoid $(Z,{\\bar\\m}{Z})$ such that the object set $Z$ and the morphism set ${\\bar\\m}{Z}$ are both M-polyfolds, and such that all the structure maps are $\\text{sc}$-smooth maps which satisfy the following properties.\n\t\\begin{itemize}\n\t\t\\item \\textbf{(\\'etale).}  The source and target maps\n\t\t$s:{\\bar\\m}{Z}\\to Z$ and $t:{\\bar\\m}{Z}\\to Z$ are surjective local sc-diffeomorphisms.\n\t\t\\item \\textbf{(proper).}  For every point $z\\in Z$, there exists an\n\t\topen neighborhood $V(z)$ so that the map\n\t\t$t:s^{-1}(\\overline{V(z)})\\rightarrow Z$ is a proper mapping.\n\t\\end{itemize}\n\\end{definition}\n\nFor a fixed object $z\\in Z$ we denote the \\textbf{isotropy group of $z$} by\n\t\\[\n\t{\\bar\\m}{G}(z) := \\{\t\\phi \\in {\\bar\\m}{Z} \\mid s(\\phi)=t(\\phi = z)\t\\}.\n\t\\]\nBy \\cite[Prop.~7.4]{HWZbook}, the properness condition ensures that this is a finite group.\nThe \\textbf{orbit space} of the ep-groupoid $(Z,{\\bar\\m}{Z})$,\n\t\\[\n\t\\abs{Z} := Z \/ \\sim,\n\t\\]\nis the quotient of the set of objects $Z$ by the equivalence relation given by $z\\sim z'$ if there exists a morphism $\\phi\\in {\\bar\\m}{Z}$ with $s(\\phi)=z$ and $t(\\phi)=z'$.  It is equipped with the quotient topology defined via the map \n\t\\begin{equation}\\label{eq:quotient-map}\n\t\\pi: Z\\to\\abs{Z}, \\qquad z\\mapsto \\abs{z}.\n\t\\end{equation}\n\n\n\n\n\\begin{definition}[{\\cite[Def.~16.1]{HWZbook}}]\n\tLet ${\\mathcal Z}$ be a second countable, paracompact, Hausdorff topological space.  A \\textbf{polyfold structure} on ${\\mathcal Z}$ consists of an ep-groupoid $(Z,{\\bar\\m}{Z})$ and a homeomorphism $\\abs{Z}\\simeq {\\mathcal Z}$.\n\\end{definition}\n\nDefining an ep-groupoid involves making a choice of local structures.  Taking an equivalence class of ep-groupoids makes our differentiable structure choice independent.  The appropriate notion of equivalence in this category-theoretic context is a ``Morita equivalence class'' (see \\cite[Def.~3.2]{HWZ3})\n\n\\begin{definition}[{\\cite[Def.~16.3]{HWZbook}}]\n\tA \\textbf{polyfold} consists of a second countable, paracompact, Hausdorff topological space ${\\mathcal Z}$ together with a Morita equivalence class of polyfold structures $[(Z,{\\bar\\m}{Z})]$ on ${\\mathcal Z}$.\n\\end{definition}\n\nTaking a Morita equivalence class of a given polyfold structure (in the case of polyfolds) is analogous to taking a maximal atlas for a given atlas (in the usual definition of manifolds).\nGiven distinct polyfold structures which define an orbifold or a polyfold, the method of proving they define the same Morita equivalence class is by demonstrating that both polyfold structures possess a common refinement.\n\nThe scales of a $\\text{sc}$-Banach space induce a filtration on the local M-polyfold models, which is moreover preserved by the structure maps $s,t$.  Consequently, there is a well-defined filtration on the orbit space which hence induces a filtration\n\t\\[\n\t{\\mathcal Z} = {\\mathcal Z}_0 \\supset {\\mathcal Z}_1 \\supset \\cdots \\supset {\\mathcal Z}_\\infty = \\cap_{k\\geq 0} {\\mathcal Z}_k\n\t\\]\non the underlying topological space ${\\mathcal Z}$.\n\n\\begin{notation}\n\tIt is common to denote both the ep-groupoid ``$(Z,{\\bar\\m}{Z})$,'' and its object set ``$Z$,'' by the same letter ``$Z$.''\t\n\tWe will refer to the underlying set, the underlying topological space, or the polyfold by the letter ``${\\mathcal Z}$.''\n\tWe will always assume that a topological space ${\\mathcal Z}$ with a polyfold structure is necessarily second countable, paracompact, and Hausdorff.\t\n\tFurthermore, we will write objects as ``$x\\in Z$,'' morphisms as ``$\\phi \\in {\\bar\\m}{Z}$,'' and points as ``$[x]\\in {\\mathcal Z}$'' (due to the identification $\\abs{Z} \\simeq {\\mathcal Z}$). We will write ``$\\phi: x\\to y$'' for a morphism $\\phi \\in {\\bar\\m}{Z}$ with $s(\\phi)=x$ and $t(\\phi)=y$.\n\\end{notation}\n\nThe local topology of a polyfold is related to the local isotropy groups, as demonstrated by the following proposition.\n\n\\begin{proposition}[Natural representation of ${\\bar\\m}{G}(x)$, {\\cite[Thm.~7.1, Prop.~7.6]{HWZbook}}]\n\t\\label{prop:natural-representation}\n\tLet be an ep-groupoid $(Z,{\\bar\\m}{Z})$.  Let $x\\in Z$ with isotropy group ${\\bar\\m}{G}(x)$.  Then for every open neighborhood $V$ of $x$ there exists an open neighborhood $U\\subset V$ of $x$, a group homomorphism $\\Phi : {\\bar\\m}{G}(x)\\rightarrow \\text{Diff}_{\\text{sc}}(U)$, $g\\mapsto  \\Phi (g)$,  and a $\\text{sc}$-smooth map\n\t$\\Gamma: {\\bar\\m}{G}(x)\\times U\\rightarrow {\\bar\\m}{Z}$ such that the following holds.\n\t\\begin{enumerate}\n\t\t\\item $\\Gamma(g,x)=g$.\n\t\t\\item $s(\\Gamma(g,y))=y$ and $t(\\Gamma(g,y))=\\Phi (g)(y)$ for all $y\\in U$ and $g\\in {\\bar\\m}{G}(x)$.\n\t\t\\item If $h: y\\rightarrow z$ is a morphism between points in $U$, then there exists a unique element $g\\in {\\bar\\m}{G}(x)$ satisfying $\\Gamma(g,y)=h$, i.e., \n\t\t\\[\n\t\t\\Gamma: {\\bar\\m}{G}(x)\\times U\\rightarrow \\{\\phi\\in {\\bar\\m}{Z} \\mid   \\text{$s(\\phi)$ and $t(\\phi)\\in U$}\\}\n\t\t\\]\n\t\tis a bijection.\n\t\\end{enumerate}\n\tThe data $(\\Phi,\\Gamma)$ is called the \\textbf{natural representation} of ${\\bar\\m}{G}(x)$.\n\tMoreover, consider the following topological spaces:\n\t\\begin{itemize}\n\t\t\\item ${\\bar\\m}{G}(x) \\backslash U$, equipped with quotient topology defined by the projection $U \\to {\\bar\\m}{G}(x) \\backslash U$,\n\t\t\\item $U \/ \\sim$, where $x \\sim x'$ for $x, x' \\in U$ if there exists a morphism $\\phi \\in {\\bar\\m}{Z}$ with $s(\\phi)=x$ and $t(\\phi)=x$, equipped with the quotient topology defined by the projection $U \\to U \/ \\sim$,\n\t\t\\item $\\abs{U}$, the image of $U$ under the map $Z\\to \\abs{Z}$, equipped with the subspace topology defined by the inclusion $\\abs{U}\\subset \\abs{Z}$.\n\t\\end{itemize}\n\tThen these spaces are all naturally homeomorphic.\n\\end{proposition}\n\n\\subsubsection{Maps between polyfolds}\n\nUsing category-theoretic language, we discuss the definition of map between polyfolds.\n\n\\begin{definition}\n\tA \\textbf{$\\text{sc}^k$ functor} between two polyfold structures\n\t\\[\n\t\\hat{f}:(Z_1,{\\bar\\m}{Z}_1) \\to (Z_2,{\\bar\\m}{Z}_2)\n\t\\]\n\tis a functor on groupoidal categories which moreover is a $\\text{sc}^k$ map when considered on the object and morphism sets.\n\\end{definition}\n\nA $\\text{sc}^k$ functor between two polyfold structures $(Z_1,{\\bar\\m}{Z}_1)$, $(Z_2,{\\bar\\m}{Z}_2)$ with underlying topological spaces ${\\mathcal Z}_1$, ${\\mathcal Z}_2$ induces a continuous map on the orbit spaces $\\abs{\\hat{f}}:\\abs{Z_1} \\to \\abs{Z_2}$, and hence also induces a continuous map $f : {\\mathcal Z}_1 \\to {\\mathcal Z}_2$, as illustrated in the following commutative diagram.\n\t\\[\n\t\\begin{tikzcd}[row sep=small]\n\t\\abs{Z_1} \\arrow[d,phantom,\"\\rotatebox{90}{\\(\\simeq\\)}\"] \\arrow[r,\"\\abs{\\hat{f}}\"] & \\abs{Z_2} \\arrow[d,phantom,\"\\rotatebox{90}{\\(\\simeq\\)}\"] \\\\\n\t{\\mathcal Z}_1 \\arrow[r,\"f\"] & {\\mathcal Z}_2\n\t\\end{tikzcd}\n\t\\]\n\n\\begin{definition}\n\tConsider two topological spaces  ${\\mathcal Z}_1$, ${\\mathcal Z}_2$ with orbifold structures $(Z_1,{\\bar\\m}{Z}_1)$, $(Z_2,{\\bar\\m}{Z}_2)$. We define a \\textbf{$\\text{sc}^k$ map between polyfolds} as a continuous map \n\t\\[\n\tf: {\\mathcal Z}_1 \\to {\\mathcal Z}_2\n\t\\]\n\tbetween the underlying topological spaces of the polyfolds, for which there exists an associated $\\text{sc}^k$ functor\n\t\\[\n\t\\hat{f}: (Z_1,{\\bar\\m}{Z}_1) \\to (Z_2,{\\bar\\m}{Z}_2).\n\t\\]\n\tsuch that $\\abs{\\hat{f}}$ induces $f$.\n\\end{definition}\n\n\\begin{remark}\n\tFrom an abstract point of view a stronger notion of map is needed.   This leads to the definition of \\textit{generalized maps} between orbifold structures, following a category-theoretic localization procedure \\cite[\\S~2.3]{HWZ3}.  Following this, a precise notion of map between two polyfolds is defined using an appropriate equivalence class of a given generalized map between two given polyfold structures \\cite[Def.~16.5]{HWZbook}.\t\n\tWith this in mind, taking an appropriate equivalence class of a given $\\text{sc}^k$-functor between two given polyfold structures is sufficient for giving a well-defined map between two polyfolds.\n\\end{remark}\n\n\\subsubsection{Subgroupoids}\n\nWe state some essential facts about the topology of subgroupoids.\n\n\\begin{definition}\t\n\tLet $(Z,{\\bar\\m}{Z})$ be an ep-groupoid.\n\tWe say that a subset of the object set, $S\\subset Z$, is \\textbf{saturated} if $S = \\pi^{-1} (\\pi(S))$, where $\\pi$ is the quotient map \\eqref{eq:quotient-map}.\n\tWe define a \\textbf{subgroupoid} as the full subcategory $(S,{\\bar\\m}{S})$ associated to a saturated subset of the object set.\n\n\n\\end{definition}\n\nA subgroupoid $(S,{\\bar\\m}{S})$ comes equipped with the subspace topology induced from the ep-groupoid $(Z,{\\bar\\m}{Z})$, in addition to the induced grading.  It does not come with a $\\text{sc}$-smooth structure in general, so the \\'etale condition no longer makes sense.  However, one may observe it inherits the following directly analogous properties.\n\\begin{itemize}\n\t\\item The source and target maps are surjective local homeomorphisms which moreover respect the induced grading. We say that the source and target maps are \\textbf{$\\text{sc}^0$-homeomorphisms} and the subgroupoid $(S,{\\bar\\m}{S})$ is automatically \\textbf{$\\text{sc}^0$-\\'etale}.\n\t\\item For every point $x\\in S$, there exists an open neighborhood $V(x)$ so that the map $t: s^{-1}(\\overline{V(x)}) \\to S$ is a proper mapping. (This can be shown from the definitions, using in addition that if $f:X\\to Y$ is proper, then for any subset $V\\subset Y$ the restriction $f|_{f^{-1}(V)}: f^{-1}(V)\\to V$ is proper.)\n\\end{itemize}\nThus, a subgroupoid is automatically $\\text{sc}^0$-\\'etale in the above sense, as well as proper.\n\n\\begin{remark}\n\t\\label{rmk:local-topology-subgroupoid}\n\tLet $U$ be an open subset of $S$.\n\tWe may consider two topologies on $U$:\n\t\\begin{itemize}\n\t\t\\item $(U, \\tau_S)$, where $\\tau_S$ is the subspace topology induced from the inclusion $\\cup_{i\\in I} M_i \\allowbreak \\hookrightarrow S$,\n\t\t\\item $(U, \\tau_Z)$, where $\\tau_Z$ is the subspace topology induced from the inclusion $\\cup_{i\\in I} M_i \\allowbreak \\hookrightarrow Z$.\n\t\\end{itemize}\n\tThen these two topologies are identical. Moreover, $U\\hookrightarrow S$ is a local homeomorphism.\n\\end{remark}\n\n\\begin{comment}\n\tLet $Y$ be a topological space.  Let $X$ be a subset of $Y$, and equip $X$ with the subspace topology.  Consider an open subset $A\\subset X$, hence $A= X\\cap U$ for some open subset $U$ of $Y$.\n\tWe can equip $A$ with the subspace topology induced from the inclusion $A\\hookrightarrow X$, or the subspace topology induced from the inclusion $A\\hookrightarrow Y$.\n\tThen these two topologies are identical.  Moreover, $A\\hookrightarrow X$ is a local homeomorphism.\n\\end{comment}\n\n\\begin{proposition}\n\t\\label{prop:topology-subgroupoid}\n\tConsider the orbit space of a subgroupoid, $\\abs{S}$.  There are two topologies on this space we may consider:\n\t\\begin{itemize}\n\t\t\\item the subspace topology $\\tau_s$, induced from the inclusion $\\abs{S}\\subset \\abs{Z}$,\n\t\t\\item the quotient topology $\\tau_q$, induced from the projection $S\\to \\abs{S}$.\n\t\\end{itemize}\n\tThese two topologies are identical.\n\\end{proposition}\n\\begin{proof}\nWe show that $\\tau_s = \\tau_q$.\n\t\\begin{itemize}\n\t\t\\item $\\tau_s \\subset \\tau_q$\n\t\\end{itemize}\n\tSuppose $U\\subset \\abs{S}$ and $U\\in \\tau_s$.  Then $U = V\\cap \\abs{S}$ for $V\\subset \\abs{Z}$ open.  By definition, $\\pi^{-1} (V) \\subset Z$ is open.  Moreover, $\\pi^{-1} (U)  = \\pi^{-1} (V) \\cap \\pi^{-1}(S) = \\pi^{-1}(V)\\cap S$.  Hence $\\pi^{-1}(U)$ is open in $S$.  It follows from the definition of the quotient topology that $U\\in \\tau_q$.\n\t\\begin{itemize}\n\t\t\\item $\\tau_q \\subset \\tau_s$\n\t\\end{itemize}\n\tSuppose $U\\subset \\abs{S}$ and $U\\in \\tau_q$.  We will show for every $[x]\\in U$ there exists a subset $B \\subset \\abs{S}$ such that $B \\in \\tau_s$ and $[x] \\in B \\subset U$.  It will then follow that $U\\in\\tau_s$, as desired.\n\t\n\tLet $x\\in \\pi^{-1}(U)$ be a representative of $[x]$.  There exists an open neighborhood $V(x) \\subset Z$ equipped with the natural action by ${\\bar\\m}{G}(x)$ and such that $V(x) \\cap S \\subset \\pi^{-1}(U)$.\n\tObserve that $\\abs{V(x)\\cap S} = \\abs{V(x)} \\cap \\abs{S}$; this follows since $S$ is saturated.\t\n\t\n\tLet $B:= \\abs{V(x)} \\cap \\abs{S} \\subset U$.  Then observe that $\\abs{V(x)} \\subset \\abs{Z}$ is open,\n\tsince the quotient map $\\pi : Z \\to \\abs{Z}$ is an open map (see \\cite[Prop.~7.1]{HWZbook}).\n\tHence $B:=\\abs{V(x)} \\cap \\abs{S}\\subset \\abs{S}$ is open in the subspace topology.  It follows that  $B \\in \\tau_s$ and $[x] \\in B \\subset U$, as desired.\n\\end{proof}\n\nThe following proposition is an analog of Proposition~\\ref{prop:natural-representation} for subgroupoids.\n\n\\begin{proposition}[Induced representation of ${\\bar\\m}{G}(x)$ for a subgroupoid]\n\t\\label{prop:natural-representation-subgroupoid}\n\tLet $(S,{\\bar\\m}{S})$ be a subgroupoid of an ep-groupoid $(Z,{\\bar\\m}{Z})$.  Let $x\\in S$ with isotropy group ${\\bar\\m}{G}(x)$.  Then for every open neighborhood $V$ of $x$ there exists an open neighborhood $U\\subset V$ of $x$, a group homomorphism $\\Phi : {\\bar\\m}{G}(x)\\rightarrow \\text{Homeo}_{\\text{sc}^0}(U)$, $g\\mapsto  \\Phi (g)$,  and a $\\text{sc}^0$-map\n\t$\\Gamma: {\\bar\\m}{G}(x)\\times U\\rightarrow {\\bar\\m}{S}$ such that the following holds.\n\t\\begin{enumerate}\n\t\t\\item $\\Gamma(g,x)=g$,\n\t\t\\item $s(\\Gamma(g,y))=y$ and $t(\\Gamma(g,y))=\\Phi (g)(y)$ for all $y\\in U$ and $g\\in {\\bar\\m}{G}(x)$,\n\t\t\\item if $h: y\\rightarrow z$ is a morphism between points in $U$, then there exists a unique element $g\\in {\\bar\\m}{G}(x)$ satisfying $\\Gamma(g,y)=h$, i.e., \n\t\t\\[\n\t\t\\Gamma: {\\bar\\m}{G}(x)\\times U\\rightarrow \\{\\phi\\in {\\bar\\m}{Z} \\mid   \\text{$s(\\phi)$ and $t(\\phi)\\in U$}\\}\n\t\t\\]\n\t\tis a bijection.\n\t\\end{enumerate}\n\tMoreover, consider the following topological spaces:\n\t\\begin{itemize}\n\t\t\\item ${\\bar\\m}{G}(x) \\backslash U$, equipped with quotient topology via the projection $U \\to {\\bar\\m}{G}(x) \\backslash U$,\n\t\t\\item $U \/ \\sim$, where $x \\sim x'$ for $x, x' \\in U$ if there exists a morphism $\\phi : x \\to x'$, equipped with the quotient topology via the projection $U \\to U \/ \\sim$,\n\t\t\\item $\\abs{U}$, the image of $U$ under the map $S\\to \\abs{S}$, equipped with the subspace topology,\n\t\t\\item $\\abs{U}$, the image of $U$ under the map $Z \\to \\abs{Z}$, equipped with the subspace topology.\n\t\\end{itemize}\n\tThen these spaces are all naturally homeomorphic.\n\\end{proposition}\n\n\\subsubsection{Weighted branched suborbifolds}\n\nView $\\mathbb{Q}^+:= \\mathbb{Q} \\cap [0,\\infty)$ as an ep-groupoid, having only the identities as morphisms. \nConsider a polyfold, consisting of a polyfold structure $(Z,{\\bar\\m}{Z})$ and an underlying topological space ${\\mathcal Z}$.\nConsider a functor $\\hat{\\theta}: (Z,{\\bar\\m}{Z}) \\to \\mathbb{Q}^+$ which induces the function $\\theta:=\\abs{\\hat{\\theta}} :{\\mathcal Z} \\to \\mathbb{Q}^+$.\nObserve that $\\hat{\\theta}$ defines a subgroupoid $(S,{\\bar\\m}{S})\\subset (Z,{\\bar\\m}{Z})$ with object set\n\t\\[\n\tS:= \\operatorname{supp} (\\hat{\\theta}) = \\{x\\in Z\\mid \\hat{\\theta}(x)>0 \\}\n\t\\]\nand with underlying topological space\n\t\\[\n\t{\\mathcal S} := \\operatorname{supp} (\\theta) = \\{[x]\\in {\\mathcal Z} \\mid \\theta([x])>0\\}.\n\t\\]\nMoreover, $(S,{\\bar\\m}{S})$ is a full subcategory of $(Z,{\\bar\\m}{Z})$ whose object set is saturated, i.e., $S= \\pi^{-1} (\\pi(S))$ where $\\pi : Z \\to \\abs{Z}, x\\mapsto [x]$.\n\n\\begin{definition}[{\\cite[Def.~9.1]{HWZbook}}]\n\t\\label{def:weighed-branched-suborbifold}\n\tA \\textbf{weighted branched suborbifold structure} consists of a subgroupoid $(S,{\\bar\\m}{S}) \\subset (Z,{\\bar\\m}{Z})$ defined by a functor $\\hat{\\theta} : (Z,{\\bar\\m}{Z}) \\to \\mathbb{Q}^+$ as above which satisfies the following properties.\n\t\\begin{enumerate}\n\t\t\\item ${\\mathcal S} \\subset {\\mathcal Z}_\\infty$.\n\t\t\\item Given an object $x\\in S$, there exists an open neighborhood $U\\subset Z$ of $x$ and a finite collection $M_i$, $i\\in I$ of finite-dimensional submanifolds of $Z$ (in the sense of \\cite[Def.~4.19]{HWZ2}) such that\n\t\t\\[\n\t\tS \\cap U= \\bigcup_{i \\in I}M_i.\n\t\t\\]\n\t\tWe require that the inclusion maps $\\phi_i: M_i\\hookrightarrow U$ are proper and are {topological embeddings,} and in addition we require that the submanifolds $M_i$ all have the same dimension. \n\t\tThe submanifolds $M_i$ are called \\textbf{local branches} in $U$. \\label{def:local-branches}\n\t\t\\item There exist positive rational numbers $w_i$, $i\\in I$, (called \\textbf{weights}) such that if $y\\in S \\cap U$, then\n\t\t\\[\\hat{\\theta}(y)=\\sum_{\\{i \\in I \\mid   y\\in M_i\\}} w_i.\\]\n\t\\end{enumerate}\n\n\tWe call ${(M_i)}_{i\\in I}$ and ${(w_i)}_{i\\in I}$ a \\textbf{local branching structure}.\n\\end{definition}\n\nBy shrinking the open set $U$ we may assume that the local branches $M_i$ (equipped with the subspace topology induced from $U$) are homeomorphic to open subsets of $\\mathbb{R}^n$.  Hence we may assume that a local branch is given by a subset $M_i\\subset\\mathbb{R}^n$ and an inclusion map $\\phi_i : M_i\\hookrightarrow U$ where $\\phi_i$ is proper and a homeomorphism onto its image.\n\n\n\\begin{definition}\n\t\\label{def:local-orientation}\n\tLet $(S,{\\bar\\m}{S})$ be a weighted branched suborbifold structure. Consider an object $x\\in S$ and a local branching structure $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$ at $x$.\n\tSuppose moreover that each local branch has an \\textit{orientation}, denoted as $(M_i,o_i)$\n\t\n\tWe define a \\textbf{local orientation} at $x$ with respect to the local branching structure $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$ as the following finite formal sum of weighted oriented tangent planes:\n\t\\[\n\t\\sum_{\\{i\\in I \\mid x\\in M_i\\}} w_i \\cdot T_x (M_i,o_i).\n\t\\]\n\\end{definition}\n\n\\begin{definition}\n\t\\label{def:orientation}\n\tLet $(S,{\\bar\\m}{S})$ be a weighted branched suborbifold structure.\n\tWe define an \\textbf{orientation} on $(S,{\\bar\\m}{S})$ as a local orientation at every object $x\\in S$ and local branching structure $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$ at $x$ such that the following holds.\n\t\\begin{enumerate}\n\t\t\\item We require that the local orientation is well-defined and does not depend on choice of local branching structure. Given an object $x\\in S$, suppose we have:\n\t\t\\begin{itemize}\n\t\t\t\\item a local orientation at $x$ with respect to a local branching structure $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$,\n\t\t\t\\item a local orientation at $x$ with respect to a local branching structure $(M'_j)_{j\\in I'}$, $(w'_j)_{j\\in I'}$.\n\t\t\\end{itemize}\n\t\tWe require the finite formal sums of weighted oriented tangent planes to be identical, i.e.,\n\t\t\\[\n\t\t\\sum_{\\{i\\in I \\mid x\\in M_i\\}} w_i \\cdot T_x (M_i,o_i)= \\sum_{\\{j\\in I' \\mid x\\in M'_j\\}} w'_j \\cdot T_x (M'_j,o_j)\n\t\t\\]\n\t\t\\item We require morphism invariance of the local orientations. Given a morphism, $\\phi : x \\to y$ there exists a well-defined tangent map $T\\phi : T_xZ \\to T_yZ$. \n\t\tSuppose we have:\n\t\t\\begin{itemize}\n\t\t\t\\item a local orientation at $x$ with respect to a local branching structure $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$,\n\t\t\t\\item a local orientation at $y$ with respect to a local branching structure $(M'_j)_{j\\in I'}$, $(w'_j)_{j\\in I'}$.\n\t\t\\end{itemize}\t\t\n\t\tThe image of a finite formal sum of weighted oriented tangent planes under this map is again a finite formal sum of weighted oriented tangent planes.\n\t\tWe require invariance of the local orientations under this map, i.e.,\n\t\t\\[\n\t\t\\sum_{\\{j\\in I' \\mid y\\in M'_j\\}} w'_j \\cdot T_y (M'_j,o'_j) = \\sum_{\\{i\\in I \\mid x\\in M_i\\}} w_i \\cdot T\\phi_* (T_x (M_i,o_i)).\n\t\t\\]\n\t\\end{enumerate}\n\\end{definition}\n\nA \\textbf{weighted branched suborbifold structure with boundary} consists of a subgroupoid $(S,{\\bar\\m}{S})\\subset (Z,{\\bar\\m}{Z})$ defined identically to Definition~\\ref{def:weighed-branched-suborbifold} except we allow the possibility that the local branches are manifolds with boundary.\nA \\textbf{local orientation} at an object $x\\in S$ is again defined as in Definition~\\ref{def:local-orientation} as a finite formal sum determined by orientations of the local branches, and again an \\textbf{orientation} is also defined similarly to Definition~\\ref{def:orientation}.\n\n\\subsection{Abstract perturbations in polyfold theory}\n\t\\label{subsec:abstract-perturbations}\n\nAbstract perturbations in polyfold theory are a mixture of two different technologies:\n\\begin{enumerate}\n\t\\item scale calculus generalizations of classical Fredholm theory, involving the development of analogs of Fredholm maps, compact perturbations, and the implicit function theorem for surjective Fredholm operators (originally developed in \\cite{HWZ2});\n\t\\item equivariant transversality through the use of ``multisections;'' due to the presence of nontrivial isotropy, it is generally impossible to obtain transversality through the use of single valued sections, and thus it is necessary to work with multisections (developed in \\cite{cieliebak2003equivariant} and generalized to polyfold theory in \\cite{HWZ3}).\n\\end{enumerate}\n\n\\subsubsection[Strong polyfold bundles and sc+-multisections]{Strong polyfold bundles and $\\text{sc}^+$-multisections}\n\t\\label{subsubsec:polyfold-abstract-perturbations}\n\nIn order to develop a Fredholm theory for polyfolds, it is necessary to formulate the notion of a ``strong polyfold bundle'' over a polyfold.\nLet $P:W\\to Z$ be a strong M-polyfold bundle (see \\cite[Def.~2.26]{HWZbook}).\nRecall that a fiber  $p^{-1} (y) = W_y$ over an object $y \\in O_x$ carries the structure of a $\\text{sc}$-Banach space. Furthermore $W$ is equipped with a double filtration $W_{m,k}$ for $0\\leq k\\leq m+1$, and the filtered spaces\n\t\\begin{gather*}\n\tW[0]:= W_{0,0} \\supset W_{1,1}\\supset \\cdots \\supset W_{i,i} \\supset \\cdots,\t\\\\\n\tW[1]:= W_{0,1} \\supset W_{1,2}\\supset \\cdots \\supset W_{i,i+1}\\supset \\cdots \n\t\\end{gather*}\nare both M-polyfolds in their own rights.\nWith respect to these filtrations, the maps $P[0]: W[0]\\to Z$ and $P[1]:W[1]\\to Z$ are both $\\text{sc}$-smooth.\n\n\\begin{proposition}[{\\cite[Prop.~2.16]{HWZbook}}]\n\t\\label{prop:pullback-bundle}\n\tLet $P: W \\to Z$ be a strong M-polyfold bundle, and let $f: Y \\to Z$ be a $\\text{sc}$-smooth map between M-polyfolds. The pullback $f^* W := \\{(y,w_x) \\in Y \\times W \\mid f(y)=x=P(w_x) \\}$ carries a natural structure of a strong M-polyfold bundle over the M-polyfold $Y$.\n\\end{proposition}\n\nLet $(Z,{\\bar\\m}{Z})$ be a polyfold structure, and consider a strong M-polyfold bundle over the object space, $P:W\\to Z.$\nThe source map $s:{\\bar\\m}{Z}\\to Z$ is a local $\\text{sc}$-diffeomorphism, and hence we may consider the fiber product\n\t\\[\n\t{\\bar\\m}{Z} _s\\times_P W = \\{(\\phi,w)\\in {\\bar\\m}{Z}\\times W\t\\mid\ts(\\phi)=P(w)\t\\}.\n\t\\]\nVia the above proposition, we can also view as ${\\bar\\m}{Z} _s\\times_P W$ as the pullback bundle via $s$ over the morphism space ${\\bar\\m}{Z}$,\n\t\\[\\begin{tikzcd}\n\t{\\bar\\m}{Z} _s\\times_P W \\arrow[r] \\arrow[d] & W \\arrow[d] \\\\\n\t{\\bar\\m}{Z} \\arrow[r, \"s\"] & Z.\n\t\\end{tikzcd}\\]\n\n\\begin{definition}[{\\cite[Def.~8.4]{HWZbook}}]\n\t\\label{def:strong-polyfold-bundle}\n\tA \\textbf{strong polyfold bundle structure} $(W,{\\bar\\m}{W})$ over a polyfold structure $(Z,{\\bar\\m}{Z})$ consists of a strong M-polyfold bundle over the object M-polyfold $P:W\\to Z$ together with a strong bundle map\n\t\\[\n\t\\mu : {\\bar\\m}{Z} _s\\times_P W \\to W\n\t\\]\n\twhich covers the target map $t:{\\bar\\m}{Z} \\to Z$, such that the diagram\n\t\\begin{center}\n\t\t\\begin{tikzcd}\n\t\t{\\bar\\m}{Z} _s\\times_P W \\arrow[r, \"\\mu\"] \\arrow[d] & W \\arrow[d] \\\\\n\t\t{\\bar\\m}{Z} \\arrow[r, \"t\"] & Z\n\t\t\\end{tikzcd}\n\t\\end{center}\n\tcommutes.\n\tFurthermore we require the following:\n\t\\begin{enumerate}\n\t\t\\item $\\mu$ is a surjective local diffeomorphism and linear on fibers,\n\t\t\\item $\\mu(\\operatorname{id}_x,w)= w$ for all $x\\in Z$ and $w\\in W_x$,\n\t\t\\item $\\mu(\\phi \\circ \\gamma ,w)= \\mu (\\phi,\\mu(\\gamma,w))$ for all $\\phi,\\gamma\\in{\\bar\\m}{Z}$ and $w\\in W$ which satisfy\n\t\t\\[\n\t\ts(\\gamma) = P(w),\\qquad t(\\gamma) = s(\\phi) = P(\\mu(\\gamma,w)).\n\t\t\\]\n\t\\end{enumerate}\n\\end{definition}\n\nA strong polyfold bundle structure $(W,{\\bar\\m}{W})$ has polyfold structures in its own right: we may take $W$ as the object set with the grading $W_{i,i}$ or $W_{i,i+1}$, and define the morphism set by ${\\bar\\m}{W}:= {\\bar\\m}{Z} _s\\times_P W$. Moreover, we have source and target maps $s,t: {\\bar\\m}{W} \\to W$ defined as follows:\n\t\\[\n\ts(\\phi,w) := w, \\qquad t(\\phi,w) := \\mu(\\phi,w).\n\t\\]\nWe have a natural smooth projection functor $\\hat{P}: (W,{\\bar\\m}{W}) \\to (Z,{\\bar\\m}{Z})$.\n\n\\begin{definition}\n\tA \\textbf{strong polyfold bundle} consists of a topological space ${\\mathcal W}$ together with a Morita equivalence class of strong polyfold bundle structures $(W,{\\bar\\m}{W})$.\n\\end{definition}\n\nThe double filtration of the fibers is preserved by the structure maps, and hence the orbit space $\\abs{W}$ is equipped with a double filtration\n\t\\[\n\t\\abs{W}_{m,k}, \\quad \\text{for } 0\\leq m \\ \\text{and}\\ 0\\leq k\\leq m+1.\n\t\\]\nWe moreover obtain polyfolds ${\\mathcal W}[0]$ and ${\\mathcal W}[1]$ with the filtrations ${\\mathcal W}[0]_i := {\\mathcal W}_{i,i}$ and ${\\mathcal W}[1]_i := {\\mathcal W}_{i,i+1}$.  Unless specified, ``${\\mathcal W}$'' refers to the first filtration, i.e., ${\\mathcal W}[0]$ and ``$P$'' refers to the projection map with respect to this filtration.\n\n\\begin{definition}[{\\cite[Def.~12.1]{HWZbook}}]\n\tWe define a \\textbf{$\\text{sc}$-smooth Fredholm section} of the strong polyfold bundle $P:{\\mathcal W}\\to{\\mathcal Z}$ as a $\\text{sc}$-smooth map between polyfolds $\\overline{\\partial}: {\\mathcal Z}\\to {\\mathcal W}$ which satisfies $P\\circ \\overline{\\partial} = \\operatorname{id}_{\\mathcal Z}$ (where $\\operatorname{id}_{\\mathcal Z}$ is the identity map on ${\\mathcal Z}$)\n\n\tWe require that $\\overline{\\partial}$ is \\textbf{regularizing}, meaning that if $[x] \\in {\\mathcal Z}_m$ and $\\overline{\\partial} ([x]) \\in {\\mathcal W}_{m,m+1}$ then $[x]\\in {\\mathcal Z}_{m+1}$.\n\tFinally, we require that at every smooth object $x\\in Z$ the germ $(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ is a ``Fredholm germ'' (see \\cite[Def.~3.7]{HWZbook}).\n\\end{definition}\n\n\\begin{definition}\n\t\\label{def:unperturbed-solution-space}\n\tWe define the \\textbf{unperturbed solution space} of $\\overline{\\partial}$ as the set \n\t\t\\[\n\t\t{\\mathcal S}(\\overline{\\partial}) :=\\{ [x]\\in {\\mathcal Z}\\mid \\overline{\\partial}([x]) = 0\\} \\subset {\\mathcal Z},\n\t\t\\]\n\twith topology given by the subspace topology induced from ${\\mathcal Z}$.\n\tThe space ${\\mathcal S}(\\overline{\\partial})$ has an associated subgroupoid structure $(S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}), {\\bar\\m}{S}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}))$ defined as follows:\n\\begin{itemize}\n\t\\item (saturated) object set: $S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}) := \\{\tx\\in Z \\mid \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x)=0\t\\} \\subset Z$,\n\t\\item morphism set: ${\\bar\\m}{S} (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}) := \\{ \\phi \\in {\\bar\\m}{Z}\t\\mid s(\\phi) \\in S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}) \\ (\\text{equivalently } t(\\phi)\\in S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}))\t\\} \\subset\t{\\bar\\m}{Z}$.\n\\end{itemize}\nBoth the object and morphism sets carry the subspace topology induced from the topologies on the object space $Z$ and morphism space ${\\bar\\m}{Z}$.\n\\end{definition}\n\nWe say that the Fredholm section $\\overline{\\partial}$ is \\textbf{proper} if the unperturbed solution space ${\\mathcal S}(\\overline{\\partial})$ is a compact topological space.\n\n\\begin{definition}[{\\cite[Def.~2.24]{HWZbook}}]\n\tA \\textbf{$\\text{sc}^+$-section} is a $\\text{sc}$-smooth map $s: Z \\to W[1]$ which satisfies $P\\circ s = \\operatorname{id}_Z$\n\\end{definition}\n\nThe significance of this definition is captured in the fact that if $(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ is a Fredholm germ and $s$ is a germ of a $\\text{sc}^+$-section around $y$, then $(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}+s,x)$ remains a Fredholm germ.\nThis follows tautologically from the definition of a Fredholm germ (see the comment following \\cite[Def.~2.44]{HWZGW}).\nWe may view the relationship of Fredholm sections and $\\text{sc}^+$-sections in the current theory as the analogs of Fredholm and compact operators in classical functional analysis.\n\nOne can view a ``multisection'' as the rationally weighted characteristic function of an equivariant collection of locally defined single valued sections.\nThis is made precise in the following definition.\n\n\\begin{definition}\n\t\\label{def:sc-multisection}\n\tWe view $\\mathbb{Q}^+:= \\mathbb{Q} \\cap [0,\\infty)$ as an ep-groupoid, having only the identities as morphisms.\n\tA \\textbf{$\\text{sc}^+$-multisection} of a strong polyfold bundle $P:{\\mathcal W}\\to {\\mathcal Z}$ consists of the following:\n\t\t\\begin{itemize}\n\t\t\t\\item a function $\\Lambda:{\\mathcal W} \\to \\mathbb{Q}^+$,\n\t\t\t\\item an associated functor $\\hat{\\Lambda}: W \\to \\mathbb{Q}^+$ where $\\abs{\\hat{\\Lambda}}$ induces $\\Lambda$,\n\t\t\\end{itemize}\n\tsuch that at every $[x]\\in {\\mathcal Z}$ there exists a \\textbf{local section structure} defined as follows.\n\tLet $x\\in Z$ be a representative of $[x]$ and let $U\\subset Z$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of $x$, and consider the restricted strong M-polyfold bundle $P: W|_U \\to U$.\n\tThen there exist finitely many $\\text{sc}^+$-sections $s_1,\\ldots,s_k : U \\to W_U$ (called \\textbf{local sections}) with associated positive rational numbers $\\sigma_1,\\ldots ,\\sigma_k \\in \\mathbb{Q}^+$ (called \\textbf{weights}) which satisfy the following:\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{i=1}^k \\sigma_i =1.$\n\t\t\\item The restriction $\\hat{\\Lambda}|_{W|_{U}}: W_U \\to \\mathbb{Q}^+$ is related to the local sections and weights via the equation \n\t\t\t\\[\n\t\t\t\\hat{\\Lambda}|_{W|_{U}}(w)=\\sum_{i\\in \\{1,\\ldots, k \\mid w=s_i(P(w))\\}} \\sigma_i\n\t\t\t\\]\n\t\twhere the empty sum has by definition the value $0$.\n\t\\end{enumerate}\t\n\\end{definition}\n\nWe define the \\textbf{domain support} of $\\Lambda$ as the subset of ${\\mathcal Z}$ given by\n\t\\[\n\t\\text{dom-supp}(\\Lambda) := \\text{cl}_{\\mathcal Z} (\t\\{\t[x]\\in{\\mathcal Z} \\mid \\exists [w]\\in {\\mathcal W}_{[x]}\\setminus\\{0\\} \\text{ such that } \\Lambda([w])>0\t\\}\t).\n\t\\]\n\\begin{comment}\nThe  \\textbf{domain support} of $\\Lambda$ is the subset of $Z$, defined by \n\\[\n\\text{dom-supp}(\\Lambda) = \\text{cl}_Z (\\{x\\in Z \\mid \\text{there exists $w\\in W_x\\setminus\\{0\\}$ for which  $\\Lambda(w)>0$}\\}).\n\\]\n\nThe \\textbf{support of $\\hat{\\Lambda}$} is the subset of $\\operatorname{supp}(\\hat{\\Lambda})$ of ${\\mathcal W}$ defined by\n\\[\n\\operatorname{supp}(\\hat{\\Lambda}) = \\{\t[w]\\in{\\mathcal W} \\mid \\hat{\\Lambda}(w)>0\t\\}\n\\]\nThe \\textbf{support of $\\Lambda$} is the subset $\\operatorname{supp}(\\Lambda)$ of $W$ defined by\n\\[\n\\operatorname{supp}(\\Lambda)=\\{w\\in W\t\\mid\t\\Lambda(w)>0\\}.\n\\]\n\\end{comment}\n\n\\begin{definition}\n\t\\label{def:perturbed-solution-space}\n\tAssociated to a $\\text{sc}$-smooth Fredholm section $\\overline{\\partial}$ and a $\\text{sc}^+$-multisection $\\Lambda$, \n\twe define the \\textbf{perturbed solution space} as the set\n\t\t\\[\n\t\t{\\mathcal S}(\\overline{\\partial},\\Lambda) :=\\{[x]\\in{\\mathcal Z} \\mid \\Lambda(\\overline{\\partial}([x]))\t>0\t\\}\\subset {\\mathcal Z}\n\t\t\\]\n\twith topology given by the subspace topology induced from ${\\mathcal Z}$. It is equipped with a \\textbf{weight function} $\\Lambda\\circ \\overline{\\partial}:{\\mathcal S}(\\overline{\\partial},\\Lambda) \\to \\mathbb{Q}^+$.\n\tThe space ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ has an associated subgroupoid structure $({\\mathcal S}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}), {\\bar\\m}{{\\mathcal S}}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}))$ with (saturated) object set\n\t\t\\[\n\t\tS(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}) :=\\{x\\in Z \\mid \\hat{\\Lambda} ( \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x))>0\\} \\subset Z\n\t\t\\]\n\tand with morphism set given by \n\t\t\\[\n\t\t{\\bar\\m}{S} (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}, \\hat{\\Lambda}) := \\{ \\phi \\in {\\bar\\m}{Z}\t\\mid s(\\phi) \\in S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda})\t\\} \\subset\t{\\bar\\m}{Z}\n\t\t\\]\n\t(we could equivalently require that $t(\\phi)\\in S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}, \\hat{\\Lambda})$).\n\tIt is equipped with a \\textbf{weight functor} $\\hat{\\Lambda}\\circ \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}:({\\mathcal S}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}), {\\bar\\m}{{\\mathcal S}}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda})) \\to \\mathbb{Q}^+$.\n\\end{definition}\n\nNote that the space ${\\mathcal S}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda})$ or the subgroupoid $({\\mathcal S}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}), {\\bar\\m}{{\\mathcal S}}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},\\hat{\\Lambda}))$ can be respectively encoded entirely via the weight function or the weight functor; such a description is closer to the language used in \\cite{HWZ3} and \\cite{HWZbook}.\n\n\\subsubsection{Transverse perturbations}\n\nAt a local level, it is easy to adapt the functional analytic construction of compact perturbations of Fredholm operators to M-polyfolds; the implicit function theorem for M-polyfolds \\cite[Thm.~3.13]{HWZbook} then guarantees that the zero set of a transversal $\\text{sc}$-Fredholm section has the structure of a finite-dimensional manifold.\nIt is somewhat more involved to adapt these constructions to the global level, as this requires using multisections to obtain equivariance.\n\n\\begin{definition}[{\\cite[Def.~15.2]{HWZbook}}]\n\t\\label{def:transversal-pair}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, $\\overline{\\partial}$ a $\\text{sc}$-smooth Fredholm section, and $\\Lambda$ a $\\text{sc}^+$-multisection.\n\t\n\tConsider a point $[x]\\in {\\mathcal Z}$.  We say $(\\overline{\\partial},\\Lambda)$ is \\textbf{transversal at $[x]$} if, given a local $\\text{sc}^+$-section structure for $\\Lambda$ at a representative $x$, the linearized local expression \n\t\t\\[D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}-s_i)(x):T_x Z \\to W_x\\]\n\tis surjective for all $i\\in I$ with $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x)=s_i(x)$.  We say that $(\\overline{\\partial},\\Lambda)$ is \\textbf{transversal} if it is transversal at every $[x] \\in {\\mathcal S}(\\overline{\\partial},\\Lambda)$.\n\n\t\n\n\t\n\n\\end{definition}\n\nGiven a $\\text{sc}$-Fredholm section it is relatively easy to construct a transversal multisection (see the general position argument of \\cite[Thm.~15.4]{HWZbook}); a key ingredient is \\cite[Lem.~5.3]{HWZbook} which guarantees the existence of locally defined $\\text{sc}^+$-sections which take on a prescribed value at a point.\n\n\\begin{theorem}[{\\cite[Thm.~4.13]{HWZ3}}\\footnote{The original statement of this theorem carries the additional requirement that the perturbed solution set ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ is a compact set.  This requirement is unnecessary, and is not used in the proof.}]\n\t\\label{thm:transversal-pairs-weighted-branched-suborbifolds} \\label{thm:transversality}\n\n\tIf the pair  $(\\overline{\\partial},\\Lambda)$ is transversal, then the perturbed solution set ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ carries in a natural way the structure of a weighted branched suborbifold.\n\\end{theorem}\n\n\\begin{remark}[Relationship between local section structures and local branching structures]\n\t\\label{rmk:relationship-local-section-structures-local-branching-structures}\n\tConsider a weighted branched suborbifold ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ defined by a transversal $\\text{sc}$-smooth Fredholm section $\\overline{\\partial}$ and a $\\text{sc}^+$-multisection $\\Lambda$.\n\tThe relationship between the local section structure for $\\Lambda$ and the local branching structure can be described as follows.\n\tConsider a point $[x]\\in{\\mathcal S}(\\overline{\\partial},\\Lambda)$, and let $U\\subset Z$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of a representative $x$.\t\n\tConsider a local section structure for $\\Lambda$ at $[x]$ consisting of $\\text{sc}^+$-sections $s_i  : U \\to W|_U$ and weights $w_i$ for $i\\in I$.\n\tThe implicit function theorem for M-polyfolds then implies that the sets\n\t\t\\[\n\t\tM_i = (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} -s_i)^{-1}(0)\n\t\t\\]\n\tdefine finite dimensional submanifolds which together with the weights $w_i$ give a local branching structure in $U$.\n\\end{remark}\n\n\\subsubsection{Pairs which control compactness}\n\nGiven a proper $\\text{sc}$-smooth Fredholm section $\\overline{\\partial}$ and a $\\text{sc}^+$-multisection $\\Lambda$, we need some way to control the compactness of the resulting perturbed solution space ${\\mathcal S}(\\overline{\\partial},\\Lambda)$. This can be achieved by requiring that the perturbation $\\Lambda$ is ``small'' in a suitable sense.\n\n\\begin{definition}[{\\cite[Def.~12.2]{HWZbook}}]\n\t\\label{def:auxiliary-norm}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle.\n\tWe define an \\textbf{auxiliary norm} as a $\\text{sc}^0$-map \n\t\\[N:{\\mathcal W}[1] \\to [0,\\infty)\\]\n\twhere we regard $[0,\\infty)$ as a smooth manifold with the trivial ep-groupoid structure (i.e., a polyfold with finite-dimensional local models and trivial isotropy).\n\tIt has an associated $\\text{sc}^0$-functor $\\hat{N}:W[1]\\to [0,\\infty)$ where as usual $\\abs{\\hat{N}}$ induces $N$.\n\tWe require that $\\hat{N}$ satisfies the following conditions.\n\t\\begin{enumerate}\n\t\t\\item The restriction of $\\hat{N}$ to each fiber $W_x[1]$ is a complete norm. (Recall that for each $x\\in Z$, the fiber $W_x[1]$ is a $\\text{sc}$-Banach space.)\n\t\t\\item \\label{property-2-auxiliary-norm} If $\\{h_k\\}$ is a sequence in $W[1]$ such that $\\{\\hat{P}(h_k)\\}$ converges in $Z$ to an object $x$, and if $\\lim_{k\\to \\infty} \\hat{N}(h_k) = 0$, then $\\{h_k\\}$ converges to $0_x \\in W_x[1]$.\n\t\\end{enumerate}\n\\end{definition}\n\n\nGiven a point $[x]\\in{\\mathcal Z}$ we define the \\textbf{pointwise norm} of $\\Lambda$ at $[x]$ with respect to the auxiliary norm $N$ by\n\t\\[\n\tN[\\Lambda] ([x]) := \\max \\{\tN([w])\t\\mid\t[w]\\in {\\mathcal W}[1], \\Lambda ([w])>0, P([w])=[x]\t\\}\n\t\\]\nand moreover define the \\textbf{norm} of $\\Lambda$ with respect to $N$ by\n\t\\[\n\tN[\\Lambda] := \\sup_{[x]\\in{\\mathcal Z}} N[\\Lambda] ([x]).\n\t\\]\n\n\\begin{definition}[{\\cite[Def.~15.4]{HWZbook}}]\n\t\\label{def:pair-which-controls-compactness}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $\\overline{\\partial}$ be a $\\text{sc}$-smooth proper Fredholm section, and let $N :{\\mathcal W}[1]\\to [0,\\infty)$ be an auxiliary norm.\t\n\tConsider an open neighborhood ${\\mathcal U}$ of the unperturbed solution set ${\\mathcal S}(\\overline{\\partial})\\subset {\\mathcal Z}$.\n\tWe say that the pair $(N,{\\mathcal U})$ \\textbf{controls the compactness} of $\\overline{\\partial}$ if the set \n\t\\[\n\tcl_{\\mathcal Z} \\{[x]\\in {\\mathcal U} \\mid \\overline{\\partial} ([x]) \\in {\\mathcal W}[1], N(\\overline{\\partial}([x]))\\leq 1\\} \\subset {\\mathcal Z}\n\t\\]\n\tis compact.\n\\end{definition}\n\n\\begin{remark}\n\t\\label{rmk:shrink-neighborhood}\n\tWe may always shrink the controlling neighborhood ${\\mathcal U}$ of the unperturbed solution set. To be precise, suppose that $(N, {\\mathcal U})$ is a pair which controls compactness, and let ${\\mathcal U}'$ be an open set such that ${\\mathcal S}(\\overline{\\partial})\\subset{\\mathcal U}'\\subset {\\mathcal U}$.  It is immediate from the above definition that the pair $(N, {\\mathcal U}')$ also controls compactness.\n\\end{remark}\n\nGiven a $\\text{sc}$-smooth proper Fredholm section of a strong polyfold bundle, \\cite[Prop.~2.27]{HWZ3} guarantees the existence of auxiliary norms.  The existence of a pair which control compactness then follows from \\cite[Thm.~4.5]{HWZ3} which states that given an auxiliary norm $N$ there always exists an associated neighborhood ${\\mathcal U}$, such that the pair $(N, {\\mathcal U})$ controls compactness.\n\n\\begin{theorem}[{\\cite[Lem.~4.16]{HWZ3}}]\n\t\\label{thm:compactness}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $\\overline{\\partial}$ be a $\\text{sc}$-smooth proper Fredholm section, and let $(N, {\\mathcal U})$ be a pair which controls compactness.\n\t\n\tConsider a $\\text{sc}^+$-multisection $\\Lambda$ and suppose it satisfies the following:\n\t\t\\begin{itemize}\n\t\t\t\\item $N[\\Lambda] \\leq 1$,\n\t\t\t\\item $\\text{dom-supp} (\\Lambda) \\subset {\\mathcal U}$.\n\t\t\\end{itemize}\n\tThen the perturbed solution set ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ is compact.\n\tWe call such a $\\text{sc}^+$-multisection \\textbf{$(N,{\\mathcal U})$-admissible} (compare with \\cite[Def.~15.5]{HWZbook}).\n\\end{theorem}\n\n\\subsubsection{Determinant line bundles and orientations}\n\nWe do not try to give a full account of the polyfold theory on orientations (for that, we refer to \\cite[\\S~6]{HWZbook}).\nHowever, to talk precisely about orientations in our main theorems it is necessary to give a brief summary of the main ideas and definitions.\n\n\\begin{definition}[{\\cite[Defs.~6.3,~6.4]{HWZbook}}]\n\tLet $T: E \\to F$ be a bounded linear Fredholm operator between real Banach spaces. The \\textbf{determinant} of $T$ is the 1-dimensional real vector space\n\t\t\\[\\det T = \\Lambda^{\\max} (\\ker T) \\otimes \\left(\\Lambda^{\\max} (\\operatorname{coker} T)\t\\right)^*.\\]\n\tAn \\textbf{orientation} of $T$ is a choice of orientation of the real line $\\det T$.\n\\end{definition}\n\nLet $P: W\\to Z$ be a strong M-polyfold bundle, and let $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}: Z \\to W$ be a $\\text{sc}$-smooth Fredholm section.\nIn general, there is no intrinsic notion a linearization of the section $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}$ at smooth points $x \\in Z_\\infty$ if $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x) \\neq 0$.\nTo deal with this, one chooses a locally defined $\\text{sc}^+$-section $s$ such that $s(x)=\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x)$; one may then consider the well-defined linearization $D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x) :T_xZ \\to W_x$.\nThe \\textbf{space of linearizations} of $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}$ at $x$ is then defined as the following subset of linear Fredholm operators from $T_x Z \\to W_x:$\n\t\\[\\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x) := \\{\tD(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x) + a \\mid a: T_x Z \\to W_x \\text{ is a }\\text{sc}^+\\text{-operator}\t\\}.\\]\nIt may be observed that $\\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ is a convex subset, and hence is contractible.\n\nTo each linearization we may associate its determinant; in doing so, we may consider the disjoint union\n\t\\[\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x) := \\bigsqcup_{L\\in \\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)} \\{L\\}\\times \\det(L).\\]\nA priori, this set does not have much structure, as although each determinant is a real line, locally the kernel and cokernel of the linearizations may vary in dimension.\nHowever, with some work it is possible to prove the following proposition.\n\n\\begin{proposition}[{\\cite[Prop.~6.11]{HWZbook}}]\n\tThe set $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ has the structure of a topological line bundle over $\\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$.\n\tThe base space $\\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ is contractible and hence $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ has two possible orientations.\n\\end{proposition}\n\nWe may therefore define an \\textbf{orientation of $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}$ at a smooth point $x\\in Z_\\infty$} as a choice of one of the two possible orientations for $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$. We denote such an orientation by $o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$.\n\nAs we vary the smooth points, we need some way to compare the orientations at each point.\nIntuitively, the choice of an orientation at a point should automatically determine an orientation at all nearby points.\nThis intuition is made precise in the theory as a sort of ``local orientation propagation.''\n\n\\begin{theorem}[{\\cite[Thm.~6.1]{HWZbook}}]\n\tConsider a smooth point $x\\in Z_\\infty$. There exists an open neighborhood $U\\subset Z$ such that for any smooth point $y\\in U$ and for any $\\text{sc}$-smooth path $\\phi: [0,1] \\to Z$ with $\\phi(0)=x,$ $\\phi(1)=y$ there exists a well-defined ``local orientation propagation.'' This means that, given an orientation $o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$ of $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ we can associate an orientation $\\phi_* o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$ of $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},y)$, and moreover this association does not depend on the choice of $\\text{sc}$-smooth path.\n\\end{theorem}\n\nWe may therefore define an orientation of a Fredholm section as a fixed choice of orientation at all smooth points which is consistent with the local orientation propagation.\n\n\\begin{definition}[{\\cite[Def.~6.11]{HWZbook}}]\n\t\\label{def:oriented-Fredholm}\n\tLet $P: W\\to Z$ be a strong M-polyfold bundle, and let $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}: Z \\to W$ be a $\\text{sc}$-smooth Fredholm section.\n\tWe define an \\textbf{orientation} of $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}$ as an association for every smooth point $x\\in Z_\\infty$ with an orientation $o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$ of the determinant $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ and which is consistent with the local orientation propagation in the following sense.\n\t\n\tFor any two smooth points $x, y \\in Z_\\infty$ and for any $\\text{sc}$-smooth path $\\phi:[0,1] \\to Z$ with $\\phi(0)=x,$ $\\phi(1)=y$ the orientation $o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},y)}$ is the same as the pushforward orientation $\\phi_* o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$ determined by the local orientation propagation. (Compare with \\cite[Defs.~6.12,~6.13]{HWZbook}.)\n\\end{definition}\n\nWe end with an observation regarding how the above abstract discussion induces orientations on the perturbed solution spaces.\nConsider an oriented $\\text{sc}$-smooth Fredholm section $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}:Z\\to W$ and consider a $\\text{sc}^+$-section locally defined on a neighborhood $U$ of a point $x\\in Z$, $s: U \\to W|_U$.\nSuppose that $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(x) = s(x)$ and suppose that the linearization \n\t\\[D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x) :T_x Z \\to W_x\\]\nis surjective. The implicit function theorem for M-polyfolds implies that $M:= (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)^{-1}(0)$ has the structure of a finite-dimensional manifold.\n\nA choice of orientation $o_{(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)}$ of $\\operatorname{DET}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ determines for any linearization $T\\in \\operatorname{Lin} (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$ a choice of orientation of $\\det T$.\nThen simply observe that $D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x) \\in \\operatorname{Lin}(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$},x)$, and since\n\t\\[\\det ((D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x)) = \\Lambda^{\\max} (\\ker (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x))) = \\Lambda^{\\max} (T_x M)\\]\na choice of orientation for $\\det ((D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s)(x))$ automatically induces an orientation for $M$ at $x$.\n\n\\subsubsection{Regular perturbations and compact cobordism}\n\nIn order to define invariants, a perturbed solution set needs to be both transversally cut out, and compact.\nWe therefore introduce the following definition, given also in \\cite[Cor.~15.1]{HWZbook}.\n\n\\begin{definition}\n\t\\label{def:regular-perturbation}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $\\overline{\\partial}$ be a $\\text{sc}$-smooth proper Fredholm section, and let $(N, {\\mathcal U})$ be a pair which controls compactness.\n\t\n\tSuppose a $\\text{sc}^+$-multisection $\\Lambda$ satisfies both the requirements of Theorem~\\ref{thm:transversality} and Theorem~\\ref{thm:compactness}, i.e.,\n\t\\begin{itemize}\n\t\t\\item $(\\overline{\\partial}, \\Lambda)$ is a transversal pair,\n\t\t\\item $N[\\Lambda] \\leq 1$ and $\\text{dom-supp} (\\Lambda) \\subset {\\mathcal U}$.\n\t\\end{itemize}\n\tWe then say $\\Lambda$ is a \\textbf{regular perturbation} of $\\overline{\\partial}$ with respect to the pair $(N,{\\mathcal U})$.\n\\end{definition}\n\n\\begin{corollary}[{\\cite[Cor.~15.1]{HWZbook}}]\n\t\\label{prop:existence-regular-perturbations}\n\n\t\n\tThere exist regular perturbations $\\Lambda$ of $\\overline{\\partial}$ with respect to the pair $(N,{\\mathcal U})$.\t\n\tTheorems~\\ref{thm:transversality} and \\ref{thm:compactness} immediately imply that the perturbed solution space ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ has the structure of a compact weighted branched suborbifold, with weight function given by $\\Lambda\\circ \\overline{\\partial} : {\\mathcal S}(\\overline{\\partial},\\Lambda) \\to \\mathbb{Q}^+$.\n\\end{corollary}\n\nCompact weighted branched suborbifolds are suitable geometric spaces for defining invariants.\nHowever, it remains to show that such invariants are independent of the choices used to define such a compact weighted branched orbifold, in particular, are independent of:\n\t\\begin{itemize}\n\t\t\\item the choice of regular perturbation,\n\t\t\\item the choice of pair which controls compactness.\n\t\\end{itemize}\n\n\\begin{theorem}[{\\cite[Cor.~15.1]{HWZbook}}]\n\t\\label{thm:cobordism-between-regular-perturbations}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $\\overline{\\partial}$ be a $\\text{sc}$-smooth proper oriented Fredholm section, and let $(N_0, {\\mathcal U}_0)$, $(N_1,{\\mathcal U}_1)$ be two pairs which control compactness.  Suppose that $\\Lambda_0$ is a regular perturbation of $\\overline{\\partial}$ with respect to the pair $(N_0,{\\mathcal U}_0)$, and likewise $\\Lambda_1$ is a regular perturbation of $\\overline{\\partial}$ with respect to the pair $(N_1,{\\mathcal U}_1)$.  Consider the strong polyfold bundle $[0,1]\\times {\\mathcal W} \\to [0,1]\\times {\\mathcal Z}$ and the $\\text{sc}$-smooth proper oriented Fredholm section $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}$ defined by $(t,[z]) \\mapsto (t,\\overline{\\partial}([z]))$.\n\t\n\tThen there exists a pair $(N,{\\mathcal U})$ which controls the compactness of $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}$ and which satisfies the following:\n\t\\begin{enumerate}\n\t\t\\item the auxiliary norm $N:[0,1]\\times {\\mathcal W} \\to \\mathbb{Q}^+$ restricts to $N_0$ on $\\{0\\}\\times {\\mathcal W}$ and restricts to $N_1$ on $\\{1\\}\\times {\\mathcal W}$,\n\t\t\\item the open neighborhood ${\\mathcal U}$ of ${\\mathcal S} (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}})$ satisfies ${\\mathcal U} \\cap (\\{0\\}\\times {\\mathcal Z} )= {\\mathcal U}_0$ and ${\\mathcal U} \\cap (\\{1\\}\\times {\\mathcal Z} )= {\\mathcal U}_1$.\n\t\\end{enumerate}\n\t\n\tIn addition, there exists a regular perturbation $\\tilde{\\Lambda}$ of $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}$ with respect to the pair $(N,{\\mathcal U})$, such that $\\tilde{\\Lambda}|_{\\{0\\}\\times {\\mathcal W}}$ can be identified with $\\Lambda_0$ and likewise $\\tilde{\\Lambda}|_{\\{1\\}\\times {\\mathcal W}}$ can be identified with $\\Lambda_1$.\n\t\n\tIt follows that the perturbed solution set ${\\mathcal S} (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\tilde{\\Lambda})$ has the structure of a compact weighted branched suborbifold, and is a cobordism between perturbed solution sets, in the sense that\n\t\t\\[\n\t\t\\partial {\\mathcal S} (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\tilde{\\Lambda}) = -{\\mathcal S} (\\overline{\\partial},\\Lambda_0) \\sqcup {\\mathcal S} (\\overline{\\partial},\\Lambda_1).\n\t\t\\]\n\\end{theorem}\n\n\\subsection{The branched integral and polyfold invariants}\n\t\\label{subsec:branched-integral-polyfold-invariants}\n\nWe now describe how to define the polyfold invariants through the use of the branched integral. The definition of the branched integral theory on compact oriented weighted branched suborbifolds was originally developed in \\cite{HWZint}.\n\n\\begin{definition}[{\\cite[Def.~4.9]{HWZbook}}]\n\tLet ${\\mathcal Z}$ be a polyfold with an associated polyfold structure $(Z,{\\bar\\m}{Z})$.\n\tThe vector space of $\\text{sc}$-differential $k$-forms $\\Omega^k (Z)$ is the set of $\\text{sc}$-smooth maps \n\t\\[\\omega:\\bigoplus^k_{n=1} TZ\\rightarrow \\mathbb{R}\\]\n\tdefined on the Whitney sum of the tangent of the object space, which are linear in each argument and skew-symmetric.\n\tMoreover, we require that the maps $\\omega$ are morphism invariant in the following sense: for every morphism $\\phi: x\\to y$ in ${\\bar\\m}{Z}_1$ with tangent map $T\\phi:T_xZ\\rightarrow T_yZ$ we require that\n\t\\[\n\t(T\\phi)^*\\omega_y=\\omega_x.\n\t\\]\n\\end{definition}\n\nRecall the definition of ${\\mathcal Z}^i$ as the shifted polyfold with shifted polyfold structure $(Z^i,{\\bar\\m}{Z}^i)$.\nVia the inclusion maps ${\\mathcal Z}^i \\hookrightarrow {\\mathcal Z}$ we may pullback a $\\text{sc}$-differential $k$-form $\\omega$ in $\\Omega^k(Z)$ to $\\Omega^k(Z^i)$, obtaining a directed system\n\\[\n\\Omega^k(Z) \\to \\cdots \\to \\Omega^k(Z^i) \\to \\Omega^k(Z^{i+1}) \\to \\cdots,\n\\]\nwe denote by $\\Omega^k_\\infty (Z)$ the direct limit of this system.\nAs defined in \\cite[p.~149]{HWZbook} there exists an \\textbf{exterior derivative}\n\\[\nd:\\Omega^*(Z^{i+1}) \\to \\Omega^{* +1}(Z^i)\n\\]\nsuch that the composition $d\\circ d = 0$.\nThe exterior derivative commutes with the inclusion maps $Z^i \\hookrightarrow Z^{i+1}$ and hence induces a map\n\\[\nd:\\Omega^*_\\infty(Z) \\to \\Omega^{* +1}_\\infty(Z)\n\\]\nwhich also satisfies $d\\circ d =0$.\n\n\\begin{theorem}[{\\cite[Thm.~9.2]{HWZbook}}]\n\t\\label{def:branched-integral}\n\tLet ${\\mathcal Z}$ be a polyfold with polyfold structure $(Z,{\\bar\\m}{Z})$ which admits $\\text{sc}$-smooth partitions of unity.\n\tGiven a $\\text{sc}$-smooth differential form $\\omega\\in \\Omega^n_\\infty (Z)$ and an $n$-dimensional compact oriented weighted branched suborbifold ${\\mathcal S}\\subset {\\mathcal Z}$.\n\t\n\tThen there exists a well-defined \\textbf{branched integral}, denoted as\n\t\\[\\int_{{\\mathcal S}} \\omega,\\]\n\twhich is partially characterized by the following property. \n\tConsider a point $[x]\\in {\\mathcal S}$ and a representative $x\\in S$ with isotropy group ${\\bar\\m}{G}(x)$. Let $(M_i)_{i\\in I}$, $(w_i)_{i\\in I}$, $(o_i)_{i\\in I}$ be a local branching structure at $x$ contained in a ${\\bar\\m}{G}(x)$-invariant open neighborhood $U\\subset Z$ of $x$.\n\tConsider a $\\text{sc}$-smooth differential form $\\omega\\in \\Omega^n_\\infty (Z)$ and suppose that $\\abs{\\operatorname{supp} \\omega} \\subset \\abs{U}$.\n\tThen\n\t\\[\n\t\\int_{{\\mathcal S}} \\omega = \\frac{1}{\\sharp {\\bar\\m}{G}^\\text{eff}(x)} \\left( \\sum_{i\\in I} w_i \\cdot \\int_{(M_i,o_i)} \\omega\\right)\n\t\\]\n\twhere $\\sharp {\\bar\\m}{G}^\\text{eff}(x)$ is the order of the effective isotropy group and $\\int_{(M_i,o_i)} \\omega$ is the usual integration of the differential $n$-form $\\omega$ on the oriented $n$-dimensional manifold $M_i$.\n\\end{theorem}\n\n\\begin{theorem}[Stokes' theorem, {\\cite[Thm.~9.4]{HWZbook}}]\n\t\\label{thm:stokes}\n\tLet ${\\mathcal Z}$ be a polyfold with polyfold structure $(Z,{\\bar\\m}{Z})$ which admits $\\text{sc}$-smooth partitions of unity.\n\tLet ${\\mathcal S}$ be an $n$-dimensional compact oriented weighted branched suborbifold, and let $\\partial {\\mathcal S}$ be its boundary with induced weights and orientation.  Consider a $\\text{sc}$-differential form $\\omega\\in \\Omega^{n-1}_\\infty (Z)$.\n\tThen \n\t\\[\n\t\\int_{{\\mathcal S}} d\\omega   = \\int_{\\partial {\\mathcal S}} \\omega.\n\t\\]\n\\end{theorem}\n\nThe next theorem follows the same reasoning used to prove \\cite[Thm.~11.8]{HWZbook}.\n\\begin{theorem}[Change of variables]\n\t\\label{thm:change-of-variables}\n\n\n\tLet ${\\mathcal S}_i\\subset {\\mathcal Z}_i$ be $n$-dimensional compact oriented weighted branched suborbifolds with weight functions $\\vartheta_i:{\\mathcal S}_i \\to \\mathbb{Q}^+$ for $i=1,2$.  \n\tLet $(S_i,{\\bar\\m}{S_i})$ be the associated branched suborbifold structures with associated weight functors $\\hat{\\vartheta}_i: (S_i,{\\bar\\m}{S_i}) \\to \\mathbb{Q}^+$ for $i=1,2$.\n\t\n\tLet $g:{\\mathcal Z}_1 \\to {\\mathcal Z}_2$ be a $\\text{sc}$-smooth map between polyfolds, which has a well-defined restriction $g|_{{\\mathcal S}_1}\t: {\\mathcal S}_1 \\to {\\mathcal S}_2$ between the branched suborbifolds. In addition, assume the following:\n\t\\begin{itemize}\n\t\t\\item $g: {\\mathcal S}_1 \\to {\\mathcal S}_2$ is a homeomorphism between the underlying topological spaces,\n\t\t\\item $\\hat{g}: S_1\\to S_2$ is injective and an orientation preserving local homeomorphism,\n\t\t\\item $g$ is weight preserving, i.e., $\\vartheta_2\\circ g=\\vartheta_1$ and $\\hat{\\vartheta}_2 \\circ \\hat{g}=\\hat{\\vartheta}_1$.\n\t\\end{itemize}\n\t\n\tThen given a $\\text{sc}$-smooth differential form $\\omega \\in \\Omega^n_\\infty (Z_2)$,\n\t\\[\n\t\\int_{{\\mathcal S}_2} \\omega = \\int_{{\\mathcal S}_1} g^* \\omega.\n\t\\]\n\\end{theorem}\n\n\\begin{theorem}[Polyfold invariants as branched integrals, {\\cite[Cor.~15.2]{HWZbook}}]\n\tConsider a $\\text{sc}$-smooth map \n\t\\[\n\tf:{\\mathcal Z} \\to {\\mathcal O}\n\t\\]\n\tfrom a polyfold ${\\mathcal Z}$ to an orbifold ${\\mathcal O}$.\n\tWe may define the \\textbf{polyfold invariant} as the homomorphism obtained by pulling back a de Rahm cohomology class from the orbifold and taking the branched integral over a perturbed zero set:\n\t\\[\n\tH^*_{\\dR} (O) \t\\to \\mathbb{R}, \\qquad \\omega \\mapsto \\int_{{\\mathcal S}(p)} f^*\\omega.\n\t\\]\n\tBy Theorem~\\ref{thm:cobordism-between-regular-perturbations} and by Stokes' theorem~\\ref{thm:stokes}, this homomorphism does not depend on the choice of abstract perturbation used to obtain the compact oriented weighted branched suborbifold ${\\mathcal S}(p)$.\n\\end{theorem}\n\n\n\\section{Naturality of polyfold invariants}\n\t\\label{sec:naturality-polyfold-invariants}\n\nIn this section we establish the necessary theory for proving the naturality of polyfold invariants, culminating in Theorem~\\ref{thm:naturality-polyfold-invariants} and in Corollaries~\\ref{cor:naturality-polyfold-gw-invariants} and \\ref{cor:punctures-equal}.\n\n\\subsection{Invariance of domain and branched suborbifolds}\n\t\\label{subsec:invariance-of-domain}\n\nIn the process of considering the naturality of the polyfold invariants, we will encounter a smooth bijection between weighted branched suborbifolds,\n\t\\[\n\tf: {\\mathcal S}_1 \\to {\\mathcal S}_2,\n\t\\]\nwhere $\\dim {\\mathcal S}_1 = \\dim {\\mathcal S}_2$ and ${\\mathcal S}_2$ is a compact topological space.\nWe would like to show that this map is a homeomorphism.\n\nHowever using only knowledge of the topologies of these spaces, it is impossible to show this.\nThe key to resolving this problem is understanding the branched suborbifold structure and how to use this additional structure to prove an invariance of domain result.\nThis result will allow us to assert that the above map is a homeomorphism.\n\nInvariance of domain is a classical theorem of algebraic topology due to Brouwer, and was originally published in 1911.\n\\begin{theorem}[Invariance of domain, {\\cite{brouwer1911beweis}}]\n\t\\label{thm:invariance-of-domain}\n\tLet $U$ be an open subset of $\\mathbb{R}^n$, and let $f: U\\to \\mathbb{R}^n$ be an injective continuous map.  Then $f$ is a homeomorphism between $U$ and $f(U)$.\n\\end{theorem}\n\nThis result can immediately be generalized to manifolds; let $M$ and $N$ be an $n$-dimensional manifolds and let $f: M\\to N$ be an injective continuous map.  Then $f$ is a homeomorphism onto its image.  Moreover, if $f$ is bijective, it is a homeomorphism.  We seek to generalize this result to the branched suborbifolds of our current situation.\n\n\\subsubsection{Local topology of branched submanifolds}\n\n\nAs a starting definition, a \\emph{branched manifold} is a topological space which is locally homeomorphic to a finite union of open subsets of $\\mathbb{R}^n$.\nHowever, such a broad definition of a branched manifold immediately raises the possibility of non-desirable topological properties. Consider the classic example of the \\emph{line with two origins}---although this is a locally Euclidean and second-countable topological space, it is not Hausdorff.\nIn contrast, the branched submanifolds we study are embedded into open subsets of ambient M-polyfolds and have better behaved topologies.\n\n\\begin{lemma}\n\t\\label{lem:topology-of-local-branching-structures}\n\tLet $U$ be a metrizable topological space.\n\tLet $M_i$, $i\\in I$ be a finite collection of finite-dimensional manifolds together with inclusion maps $\\phi_i: M_i \\hookrightarrow U$.\n\tAssume moreover that each $\\phi_i$ is proper and a topological embedding.\n\t\n\tConsider the set defined by the image of the inclusions, $\\cup_{i \\in I} \\phi_i (M_i)$.\n\tThere are two topologies we may consider on this set:\n\t\\begin{itemize}\n\t\t\\item $(\\cup_{i \\in I} \\phi_i(M_i), \\tau_s)$, where $\\tau_s$ is the subspace topology induced from $U$\n\t\t\\item $(\\cup_{i \\in I} \\phi_i(M_i), \\tau_q)$, where $\\tau_q$ is the quotient topology induced by the map $\\sqcup_{i\\in I} \\phi_i : \\sqcup_{i\\in I} M_i \\to \\cup_{i \\in I} \\phi(M_i)$.\n\t\\end{itemize}\n\tThese two topologies are identical.\n\\end{lemma}\n\\begin{proof}\n\tWe show that $\\tau_s = \\tau_q$.\n\t\\begin{itemize}\n\t\t\\item $\\tau_s \\subset \\tau_q$\n\t\\end{itemize}\n\tConsider the following commutative diagram where $q$ is the quotient map, $\\phi_i$ are the continuous inclusion maps $\\phi_i: M_i \\to U$, and $i$ is inclusion map.\n\t\\begin{center}\n\t\t\\begin{tikzcd}\n\t\t\\bigsqcup_{i\\in I} M_i \\arrow[r, \"\\sqcup_{i\\in I}\\phi_i\"] \\arrow[d, \"q\"'] & U \\\\\n\t\t(\\bigcup_{i\\in I} \\phi_i(M_i),\\tau_q) \\arrow[ru, \"i\",hook] \\arrow[r, \"\\operatorname{id}\"] & (\\bigcup_{i \\in I} \\phi_i(M_i), \\tau_s) \\arrow[u, \"i\"', hook]\n\t\t\\end{tikzcd}\n\t\\end{center}\n\tThen by the characteristic property of the quotient topology, $\\sqcup_{i\\in I} \\phi_i$ continuous implies $i:(\\cup_{i \\in I} \\phi_i(M_i), \\tau_q) \\allowbreak \\hookrightarrow \\allowbreak U $ continuous.\n\tBy the definition of the subspace topology, $i:(\\cup_{i \\in I} \\phi_i(M_i), \\tau_q) \\hookrightarrow U $ is continuous.\n\tBy the characteristic property of the subspace topology, $i:(\\cup_{i \\in I} \\phi_i(M_i), \\tau_q) \\hookrightarrow U $ continuous implies\n\t$\\operatorname{id} : (\\cup_{i \\in I} \\phi_i(M_i), \\tau_q) \\to (\\cup_{i \\in I} \\phi_i(M_i), \\tau_s)$\n\tis continuous.\n\t\n\t\\begin{itemize}\n\t\t\\item $\\tau_q \\subset \\tau_s$\n\t\\end{itemize}\n\tBy assumption $U$ is a metrizable space; hence it is also a regular topological space.\n\tThe assumption that each $\\phi_i$ is a topological embedding and is proper implies moreover that the images $\\phi_i(M_i)\\subset U$ are closed in the subspace topology; to see this note that in metric spaces, sequential compactness is equivalent to compactness, and then use properness.\n\t\n\tSuppose $V\\subset \\cup_{i \\in I} \\phi_i(M_i)$ and $V\\in \\tau_q$.  \n\tWe will show for every $x\\in V$ there exists a subset $B \\subset \\cup_{i \\in I} \\phi_i(M_i)$ such that $B \\in \\tau_s$ and $x \\in B \\subset V$.  This implies that $V\\in\\tau_s$, as desired.\n\t\n\tBy the definition of the quotient topology, the set $q^{-1} (V) \\subset \\sqcup_{i\\in I} M_i$ is open and hence $q^{-1} (V) \\cap M_i$ is open in the topology on $M_i$.\tConsider $x$ as a point in $U$ via the set inclusion $\\cup_{i\\in I} \\phi_i(M_i) \\subset U$, since $\\phi_i : M_i\\to U$ is an injection it follows that $q^{-1} (x) = \\{x_{i_1}, \\ldots, x_{i_k} \\}$ where $x_{i_l} \\in M_{i_l}$ for a nonempty subset $\\{i_1,\\ldots , i_k \\} \\subset I$.\n\t\n\tLet $B_\\epsilon (x)\\subset U$ be an $\\epsilon$-ball at $x$.\n\tSince $\\phi_{i_l}$ is a topological embedding it follows that the sets $\\phi^{-1}_{i_l} (B_\\epsilon(x))$ give a neighborhood basis for $M_{i_l}$ at the point $x_{i_l}$.\n\tTherefore, we may take $\\epsilon$ small enough that $\\phi^{-1}_{i_l} (B_\\epsilon(x)) \\subset q^{-1} (V) \\cap M_{i_l}$ for all $i_l \\in \\{i_1,\\ldots , i_k \\}$.\t\n\tSince $U$ is a regular topological space, and since $x$ and $\\phi_j(M_j)$ are disjoint closed subsets of $U$ for $j \\in I \\setminus \\{i_1,\\ldots , i_k \\}$, we can find disjoint open neighborhoods that separate $x$ and $\\phi_j(M_j)$.\n\tThis moreover implies that we may take $\\epsilon$ small enough that $\\phi^{-1}_j (B_\\epsilon (x)) = \\emptyset$ for all $j \\in I \\setminus \\{i_1,\\ldots , i_k \\}$.\n\tFor such an $\\epsilon$, it follows that $\\phi_i^{-1} (B_\\varepsilon(x)) \\subset q^{-1}(V) \\cap M_i$ for all $i\\in I$.\n\t\t\n\tThe desired set is then given by\n\t\t\\[\n\t\tB:= B_\\epsilon (x) \\cap \\bigcup_{i \\in I} \\phi_i (M_i);\n\t\t\\]\n\tit is an open set in the subspace topology on $\\cup_{i \\in I} \\phi_i(M_i)$.\n\tBy construction, $q^{-1}(B) = \\sqcup_{i\\in I} \\phi_i^{-1} (B_\\varepsilon(x)) \\subset \\sqcup_{i\\in I} q^{-1}(V) \\cap M_i = q^{-1} (V)$, therefore $B\\subset V$ as desired.\n\n\t\n\n\t\n\n\\end{proof}\n\nAn open subset of an M-polyfold with the subspace topology is a metrizable topological space, and hence the above lemma applies to the branched suborbifolds of Definition~\\ref{def:weighed-branched-suborbifold}\n\n\\begin{lemma}\n\t\\label{lem:local-homeo-m-polyfold}\n\tLet $S_i$ be $n$-dimensional branched submanifolds of M-polyfolds $Z_i$ for $i=1,2$.\n\tConsider an injective continuous map between these two M-polyfolds,\n\t$\\hat{f}: Z_1 \\hookrightarrow Z_2,$\n\tand suppose that there is a well-defined restriction to the branched submanifolds,\n\t$\\hat{f}|_{S_1}:S_1 \\hookrightarrow S_2.$\n\t\n\tFor every $x\\in S_1$ with $y:= \\hat{f}(x) \\in S_2$, suppose that there exist local branching structures $(M_i)_{i\\in I}$ at $x$ and $(M'_j)_{j\\in I}$ at $y$ which have the same index set $I$. \n\tMoreover, assume that $\\hat{f}$ has a well-defined restriction to the individual local branches for each index $i\\in I$ as follows:\n\t\\[\n\t\\hat{f}|_{M_i} : M_i \\hookrightarrow M'_i.\n\t\\]\n\tThen $\\hat{f}|_{S_1}$ is a local homeomorphism between $S_1$ and $S_2$. Since we have assumed that $\\hat{f}$ is injective, it follows that $\\hat{f}|_{S_1}$ is also a homeomorphism onto its image.\n\\end{lemma}\n\\begin{proof}\n\tLet $x\\in S_1$ which maps to $\\hat{f}(x)\\in S_2$.  By assumption, there exists a local branching structure $(M_i)_{i\\in I}$ in a neighborhood $O_x$ of $x$, and there exists a local branching structure $(M'_j)_{j\\in J}$ in a neighborhood $O_{\\hat{f}(x)}$ of $\\hat{f}(x)$ such that the index sets are the same, $I=J$, and $\\hat{f}$ restricts to a injective continuous map between each branch, i.e.,\n\t\\[\\hat{f}|_{M_i} : M_i \\to M'_i.\\]\n\t\n\tWe may invoke invariance of domain \\ref{thm:invariance-of-domain} to see that the restricted maps $\\hat{f}|_{M_i}$ are homeomorphisms onto their images.\n\tObserve that the open balls $B_\\epsilon(\\hat{f}(x))\\subset O_{\\hat{f}(x)}$ give a neighborhood basis for $M_i'$ at $\\hat{f}(x)$ for all $i\\in I$.  It follows that $\\hat{f}^{-1} (B_\\epsilon (\\hat{f}(x))) \\subset O_x$ give a neighborhood basis for $M_i$ at $x$ for all $i\\in I$.  For $\\epsilon$ small enough, the restricted maps \n\t\\begin{equation}\\label{eq:restriction-to-local-branches}\n\t\\hat{f}|_{M_i \\cap \\hat{f}^{-1} (B_\\epsilon (\\hat{f}(x)))} : M_i \\cap \\hat{f}^{-1} (B_\\epsilon (\\hat{f}(x))) \\to M_i' \\cap B_\\epsilon(\\hat{f}(x))\n\t\\end{equation}\n\tare homeomorphisms for all $i\\in I$.\n\t\n\tDefine a neighborhood of $x$ by $U_x = \\hat{f}^{-1} (B_\\epsilon (\\hat{f}(x)))$; then $N_i := M_i \\cap \\hat{f}^{-1} (B_\\epsilon(\\hat{f}(x)))$ give local branches in $U_x$.  Define a neighborhood of $\\hat{f}(x)$ by $U_{\\hat{f}(x)}:=B_\\epsilon(\\hat{f}(x))$; then $N_i' := M_i' \\cap B_\\epsilon(\\hat{f}(x))$ give local branches in $U_{\\hat{f}(x)}$.  We can now rewrite \\eqref{eq:restriction-to-local-branches} more simply as\n\t\\[\n\t\\hat{f}|_{N_i} : N_i \\to N'_i.\n\t\\]\n\tand note again that the maps $\\hat{f}|_{N_i}$ are homeomorphisms for all $i\\in I$.  Hence the map $\\sqcup_{i\\in I} (\\hat{f}|_{N_i}): \\sqcup_{i\\in I} N_i \\to \\sqcup_{i\\in I} N'_i$ is also a homeomorphism.\n\t\n\tConsider the following commutative diagram of maps.\n\t\\begin{center}\n\t\t\\begin{tikzcd}\n\t\t\\bigsqcup_{i\\in I} N_i \\arrow[d, \"q\"] \\arrow[r, \"\\sqcup (\\hat{f}|_{N_i})\"] & \\bigsqcup_{i\\in I} N'_i \\arrow[d, \"q'\"] \\\\\n\t\t(\\cup_{i\\in I} N_i, \\tau_q) \\arrow[r, \"\\hat{f}|_{\\cup N_i}\"', hook] & (\\cup_{i\\in I} N'_i, \\tau_{q'})\n\t\t\\end{tikzcd}\n\t\\end{center}\n\tWe assert that the map $\\hat{f}|_{\\cup N_i} : (\\cup_{i \\in I} N_i,\\tau_q) \\hookrightarrow (\\cup_{i\\in I} N'_i,\\tau_{q'})$ is a homeomorphism.\n\tIndeed, by assumption $\\hat{f}|_{\\cup N_i}$ is injective.  We can use the fact that $\\sqcup (\\hat{f}|_{N_i})$ is a bijection to see that $\\hat{f}|_{\\cup N_i}$ must also be surjective.\n\tIt is easy to check that $\\hat{f}|_{\\cup N_i}$ is continuous with respect to the quotient topologies $\\tau_q$ and $\\tau_{q'}$.\n\tFurthermore, $\\hat{f}|_{\\cup N_i}$ is an open map.\n\tTo see this, let $U\\subset (\\cup_{i \\in I} N_i,\\tau_q)$ be an open set.  Then $q^{-1} (U) \\subset \\sqcup_{i\\in I} N_i$ is open by the definition of the quotient topology.\n\tSince $\\sqcup (\\hat{f}|_{N_i})$ is a homeomorphism, $(\\sqcup (\\hat{f}|_{N_i})) (q^{-1}(U))$ is open.\n\tCommutativity of the diagram and the fact that both $\\sqcup (\\hat{f}|_{N_i})$ and $\\hat{f}|_{\\cup N_i}$ are bijections implies that $(\\sqcup \\hat{f}|_{N_i}) (q^{-1}(U)) = q'^{-1} (\\hat{f}|_{\\cup N_i} (U))$.\n\tIt therefore follows that $\\hat{f}|_{\\cup N_I} (U)$ is open by the definition of the quotient topology.\n\t\n\tBy Lemma~\\ref{lem:topology-of-local-branching-structures}, the fact that $\\hat{f}|_{\\cup N_i} : (\\cup_{i \\in I} N_i,\\tau_q) \\hookrightarrow (\\cup_{i\\in I} N'_i,\\tau_{q'})$ is a homeomorphism implies that $\\hat{f}|_{\\cup N_i} : (\\cup_{i \\in I} N_i,\\tau_s) \\hookrightarrow (\\cup_{i\\in I} N'_i,\\tau_s)$ is a homeomorphism.\n\tNote that $\\cup_{i \\in I} N_i \\subset S_1$ and $\\cup_{i \\in I} N'_i \\subset S_2$ are both open subsets.\n\tBy Remark~\\ref{rmk:local-topology-subgroupoid}, the inclusion maps $(\\cup_{i \\in I} N_i,\\tau_s)\\hookrightarrow S_1$ and $(\\cup_{i\\in I} N'_i,\\tau_s)\\hookrightarrow S_2$ are both local homeomorphisms.  We now see that the map $\\hat{f} : S_1 \\to S_2$ is a local homeomorphism on an open neighborhood of the point $x\\in S_1$.  Since $x\\in S_1$ was arbitrary, and since $\\hat{f}$ is injective, we can conclude $\\hat{f}$, considered on the object sets, is a local homeomorphism.  It then follows from the \\'etale property that $\\hat{f}$, considered on the morphism sets, is a local homeomorphism.  This proves the claim.\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lem:invariance-of-domain-branched-orbifolds}\n\tLet ${\\mathcal S}_i$ be an $n$-dimensional branched suborbifold of a polyfold ${\\mathcal Z}_i$ for $i=1,2$.\n\tConsider an injective continuous map between these two polyfolds, $f: {\\mathcal Z}_1 \\hookrightarrow {\\mathcal Z}_2$,\n\tand which has an associated functor $\\hat{f}: (Z_1,{\\bar\\m}{Z_1}) \\hookrightarrow (Z_2,{\\bar\\m}{Z_2})$, which is injective and continuous with respect to the object and morphism sets.\n\tIn addition, assume that the functor $\\hat{f}$ is fully faithful.\n\tSuppose that $f$ has a well-defined restriction to the branched suborbifolds $f|_{{\\mathcal S}_1}:{\\mathcal S}_1 \\hookrightarrow {\\mathcal S}_2$; it follows that $\\hat{f}$ restricts to a well-defined functor between the subgroupoids $\\hat{f}|_{S_1} : (S_1,{\\bar\\m}{S}_1) \\to (S_2,{\\bar\\m}{S}_2)$.\n\t\n\tAssume that for every $x\\in S_1$ with $y:= \\hat{f}(x) \\in S_2$, there exist local branching structures $M_i$, $i\\in I$ at $x$ and $M'_j$, $j\\in I$ at $y$ which have the same index set $I$. \n\tMoreover, assume that $\\hat{f}$ has a well-defined restriction to the individual local branches for each index $i\\in I$ as follows:\n\t\\[\n\t\\hat{f}|_{M_i} : M_i \\hookrightarrow M'_i.\n\t\\]\n\t\n\tThen the restriction $f|_{{\\mathcal S}_1}:{\\mathcal S}_1 \\hookrightarrow {\\mathcal S}_2$ is a local homeomorphism. In particular, if $f|_{{\\mathcal S}_1}$ is a bijection, then it is a homeomorphism.\n\\end{lemma}\n\\begin{proof}\t\n\tLet $[x] \\in {\\mathcal S}_1$ and let $f([x]) \\in {\\mathcal S}_2$.  Let $x$ be a representative of $[x]$, hence $\\hat{f}(x)$ is a representative of $f([x])$.  \n\tFrom the proof of Lemma~\\ref{lem:local-homeo-m-polyfold}, we have seen that there exists a local branching structure $(N_i)_{i\\in I}$ at $x$ and a local branching structure $(N'_i)_{i\\in I}$ at $\\hat{f}(x)$ such that $\\hat{f}|_{\\cup N_i} : (\\cup_{i \\in I} N_i,\\tau_s) \\hookrightarrow (\\cup_{i\\in I} N'_i,\\tau_s)$ is a homeomorphism.\n\t\n\tThe proof now follows the same reasoning as Lemma~\\ref{lem:local-homeo-m-polyfold}.\n\tConsider the following commutative diagram of maps.\n\t\\begin{center}\n\t\t\\begin{tikzcd}\n\t\t\\bigcup_{i\\in I} N_i \\arrow[d, \"q\"] \\arrow[r, \"\\hat{f}|_{\\cup N_i}\"] & \\bigcup_{i\\in I} N'_i \\arrow[d, \"q'\"] \\\\\n\t\t(\\abs{\\cup_{i\\in I} N_i},\\tau_q)  \\arrow[r, \"f|_{\\abs{\\cup N_i}}\"', hook] & (\\abs{\\cup_{i\\in I} N'_i}, \\tau_{q'})\n\t\t\\end{tikzcd}\n\t\\end{center}\n\tWe assert that the map $f|_{\\abs{\\cup N_i}}$ is a homeomorphism.\n\tIndeed, by assumption $f|_{\\abs{\\cup N_i}}$ is injective.  We can use the fact that $\\hat{f}|_{\\cup N_i}$ is a bijection to see that $f|_{\\abs{\\cup N_i}}$ must also be surjective.\n\tBy assumption, $f$ is continuous and therefore the restriction $f|_{\\abs{\\cup N_i}}$ is continuous.\n\tFurthermore, $f|_{\\abs{\\cup N_i}}$ is an open map.\n\tTo see this, let $U\\subset \\abs{\\cup_{i\\in I} N_i}$ be an open set.\n\tThen $q^{-1}(U)\\subset \\cup_{i \\in I} N_i$ is open by the definition of the quotient topology.\n\tSince $\\hat{f}|_{\\cup N_i}$ is a homeomorphism,  $(\\hat{f}|_{\\cup N_i})(q^{-1}(U)) \\subset \\cup_{i \\in I} N'_i$ is open.  Commutativity of the diagram and the fact that both $\\hat{f}|_{\\cup N_i}$ and $f|_{\\abs{\\cup N_i}}$ are bijections implies that $(\\hat{f}|_{\\cup N_i}) (q^{-1}(U)) = q'^{-1} (f|_{\\abs{\\cup N_i}}(U))$.\n\tIt therefore follows that $f|_{\\abs{\\cup N_i}}(U)$ is open by the definition of the quotient topology.\n\t\n\tProposition~\\ref{prop:natural-representation-subgroupoid} implies that the inclusion maps $\\abs{\\cup_{i\\in I} N_i} \\hookrightarrow {\\mathcal S}_1$ and $\\abs{\\cup_{i\\in I} N'_i} \\hookrightarrow {\\mathcal S}_2$ are local homeomorphisms.\n\tWe now see that the map $f|_{{\\mathcal S}_1} : {\\mathcal S}_1 \\to {\\mathcal S}_2$ is a local homeomorphism on an open neighborhood of the point $[x]\\in {\\mathcal S}_1$.  Since $[x]\\in {\\mathcal S}_1$ was arbitrary it follows that $f|_{{\\mathcal S}_1}$ is a local homeomorphism.\n\tIt moreover follows that if $f|_{{\\mathcal S}_1}$ is bijective, it is a homeomorphism.  This proves the claim.\n\\end{proof}\n\n\\subsection{Fredholm multisections and abstract perturbations}\n\t\\label{subsec:fredholm-multisections}\n\nIn this subsection we generalize the polyfold abstract perturbation theory from Fredholm sections to Fredholm multisections.\nThis involves minor modifications to the definitions and theorems originally developed in \\cite{HWZ3} and which we recalled in \\S~\\ref{subsec:abstract-perturbations}.\nThis generalization is developed with a specific goal in mind, which is the proof of Theorem~\\ref{thm:naturality}.\n\n\\begin{definition}\n\tLet ${\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle.  We define a \\textbf{$\\text{sc}$-smooth Fredholm multisection} as \n\t\\begin{enumerate}\n\t\t\\item a function $F:{\\mathcal W} \\to \\mathbb{Q}^+$,\n\t\t\\item an associated functor $\\hat{F}: W \\to \\mathbb{Q}^+$ where $\\abs{\\hat{F}}$ induces $F$,\n\t\\end{enumerate}\n\tsuch that at ever $[x]\\in {\\mathcal Z}$ there exists a \\textbf{local Fredholm section structure} defined as follows.\n\tLet $x\\in Z$ be a representative of $[x]$ and let $U\\subset Z$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of $x$, and consider the restricted strong M-polyfold bundle $P: W|_U \\to U$.\n\tThen there exist finitely many $\\text{sc}$-Fredholm sections $f_1,\\ldots,f_k : U \\to W|_U$\n\twith associated positive rational numbers $\\sigma_1,\\ldots ,\\sigma_k \\in \\mathbb{Q}^+$ which satisfy the following:\n\t\\begin{enumerate}\n\t\t\\item $\\sum_{i=1}^k \\sigma_i =1.$\n\t\t\\item The restriction $\\hat{F}|_{W|_U}: W|_U \\to \\mathbb{Q}^+$ is related to the local sections and weights via the equation \n\t\t\t\\[\n\t\t\t\\hat{F}|_{W|_U}(w)=\\sum_{\\{i\\in \\{1,\\ldots, k\\} \\mid w=f_i(p(w))\\}} \\sigma_i\n\t\t\t\\]\n\t\twhere the empty sum has by definition the value $0$.\n\t\\end{enumerate}\n\\end{definition}\n\nWe say that the Fredholm multisection $F$ is \\textbf{proper} if the unperturbed solution set\n\t\\[\n\t{\\mathcal S} (F) := \\{ [z]\\in {\\mathcal Z} \\mid F(0_{[x]})\t>0\t\\} \\subset {\\mathcal Z}\n\t\\]\nis a compact topological space.\n(Notice that the condition $F(0_{[x]})>0$ is equivalent to the condition that $f_i(x) = 0$ for some $i \\in I$ for a given representative $x$ and a local Fredholm section structure $(f_i)_{i\\in I}$, $(\\sigma_i)_{i\\in I}$ at $x$.)\nFurthermore, we can define a weight function on the unperturbed solution set, ${\\mathcal S}(F) \\to \\mathbb{Q}^+$, by $[z] \\mapsto F(0_{[x]})$.\n\n\\begin{example}\n\tFor the applications we have in mind, the $\\text{sc}$-smooth Fredholm multisections are obtained as a pair $(\\overline{\\partial},\\Lambda)$ consisting of:\n\t\\begin{itemize}\n\t\t\\item a $\\text{sc}$-smooth Fredholm section $\\overline{\\partial}:{\\mathcal Z}\\to{\\mathcal W}$,\n\t\t\\item a $\\text{sc}^+$-multisection $\\Lambda :{\\mathcal W} \\to \\mathbb{Q}^+$.\n\t\\end{itemize}\n\tGiven a point $[x]\\in{\\mathcal Z}$, we define a local Fredholm section structure for $(\\overline{\\partial},\\Lambda)$ at $[x]$ as follows.\n\tLet $x\\in Z$ be a representative of $[x]$ and let $U\\subset Z$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of $x$, and consider the restricted strong M-polyfold bundle $P: W|_U \\to U$.\n\tConsider the $\\text{sc}$-smooth Fredholm section $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}: U \\to W|_U$, and let $(s_i)_{i\\in I}$, $(\\sigma_i)_{i\\in I}$ be a local section structure for $\\Lambda$ at $x$.\n\t\n\tThen the local Fredholm section structure is given by $f_i := \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} - s_i$ with associated weight $\\sigma_i$. It follows from \\cite[Thm.~3.2]{HWZbook} that such an $f_i$ is in fact a $\\text{sc}$-smooth Fredholm section.\n\tWe may then define the functor $\\hat{F}$ locally via the equation\n\t\t\\[\n\t\t\\hat{F}|_{W|_U}(w)=\\sum_{i\\in \\{1,\\ldots, k \\mid w=f_i(p(w))\\}} \\sigma_i\n\t\t\\]\n\twhere the empty sum has by definition the value $0$. It is evident this extends to a well-defined functor $\\hat{F}: (W,{\\bar\\m}{W}) \\to \\mathbb{Q}^+$.\n\tFinally, observe the perturbed solution set ${\\mathcal S} (\\overline{\\partial},\\Lambda)$ associated to the pair $(\\overline{\\partial},\\Lambda)$ is the same as the unperturbed solution set ${\\mathcal S} (F)$ associated to the Fredholm multisection $F$, i.e.,\n\t\t\\[\n\t\t\\{ [z]\\in {\\mathcal Z} \\mid \\Lambda(\\overline{\\partial}([x]))\t>0\t\\} = \\{ [z]\\in {\\mathcal Z} \\mid F(0_{[x]})\t>0\t\\}.\n\t\t\\]\n\\end{example}\n\n\\subsubsection{Transverse perturbations of Fredholm multisections}\n\nWe can immediately adapt the main definitions and results of \\S~\\ref{subsec:abstract-perturbations}; there is no difficulty in generalizing the construction of transverse perturbations to Fredholm multisections.\n\n\\begin{definition}\n\tAssociated to a $\\text{sc}$-smooth Fredholm multisection $\\overline{\\partial}$ and a $\\text{sc}^+$-mul\\-ti\\-sec\\-tion $\\Gamma$, \n\twe define the \\textbf{perturbed solution space} as the set\n\t\\[\n\t{\\mathcal S}(F,\\Lambda) :=\\{[z]\\in{\\mathcal Z} \\mid (F\\oplus\\Gamma) (0_{[z]})\t>0\t\\}\\subset {\\mathcal Z}\n\t\\]\n\twith topology given by the subspace topology induced from ${\\mathcal Z}$. It is equipped with the weight function ${\\mathcal S}(F,\\Gamma) \\to \\mathbb{Q}^+,$ $[z]\\mapsto (F\\oplus\\Gamma) (0_{[z]}).$\n\\end{definition}\n\nAlong the same lines as Definition~\\ref{def:transversal-pair}, we can formulate what it means for a Fredholm multisection and a $\\text{sc}^+$-multisection to be transversal.\n\n\\begin{definition}\n\t\\label{def:fredholm-multisection-transversal}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, $F$ a $\\text{sc}$-smooth Fredholm multisection, and $\\Gamma$ a $\\text{sc}^+$-multisection.\n\t\n\tConsider a point $[x]\\in {\\mathcal Z}$.  We say $(F,\\Gamma)$ is \\textbf{transversal at $[x]$} if, given a local Fredholm section structure for $F$ at $[x]$ and given a local $\\text{sc}^+$-section structure for $\\Gamma$ at $[x]$, then the linearized local expression \n\t\t\\[\n\t\tD(f_i-s_j)(x):T_x Z \\to W_x\n\t\t\\]\n\tis surjective for all $i\\in I$, $j\\in J$ with $f_i(x)=s_j(x)$.  We say that $(F,\\Gamma)$ is \\textbf{transversal} if it is transversal at every $[x] \\in {\\mathcal S}(F,\\Gamma)$.\n\\end{definition}\n\nConsider our example of a Fredholm multisection $(\\overline{\\partial},\\Lambda)$ consisting of a Fredholm section and a $\\text{sc}^+$-multisection $\\Lambda$, and let $\\Gamma$ be an additional $\\text{sc}^+$-multisection. Then the sum $\\Lambda \\oplus \\Gamma: {\\mathcal W} \\to \\mathbb{Q}^+$ is a $\\text{sc}^+$-multisection, with local section structure given by $s_i +r_j$ where $(s_i)_{i\\in I}$ is a local section structure for $\\Lambda$ and $(r_j)_{j\\in J}$ is a local section structure for $\\Gamma$.\nWe may now observe that the pair $(\\overline{\\partial}, \\Lambda\\oplus \\Gamma)$ consisting of the Fredholm section $\\overline{\\partial}$ and the $\\text{sc}^+$-multisection $\\Lambda\\oplus \\Gamma$ is transversal in the sense of Definition~\\ref{def:transversal-pair} if an only if the pair $((\\overline{\\partial},\\Lambda), \\Gamma)$ consisting of the Fredholm multisection $(\\overline{\\partial},\\Lambda)$ and the $\\text{sc}^+$-multisection $\\Gamma$ is transversal in the sense of the above Definition~\\ref{def:fredholm-multisection-transversal}.\n\nWe have an analog of Theorem~\\ref{thm:transversal-pairs-weighted-branched-suborbifolds}.\n\n\\begin{proposition}\n\t\\label{prop:fredholm-multisection-transversal-pairs-weighted-branched-suborbifolds}\n\tLet $P:{\\mathcal W}\\rightarrow {\\mathcal Z}$ be a strong polyfold bundle, $F$ a $\\text{sc}$-smooth Fredholm multisection, and $\\Gamma$ a $\\text{sc}^+$-multisection.\n\tIf the pair  $(F,\\Lambda)$ is transversal, then the perturbed solution set ${\\mathcal S}(F,\\Gamma)$ carries in a natural way the structure of a weighted branched suborbifold.\n\\end{proposition}\n\n\\subsubsection{Controlling compactness of Fredholm multisections}\n\nIn contrast to construction of transverse perturbations of Fredholm multisections, where no modification of the underlying definitions or ideas was required, it is somewhat more involved to show how to control the compactness of Fredholm multisections.\nIt is necessary to refer to the earlier work contained in \\cite[\\S~4.2]{HWZ3} in order to obtain complete results in our current situation.\n\n\\begin{definition}\n\tConsider a Fredholm multisection and a point $[z] \\in {\\mathcal Z}$.\n\tLet $(f_i)_{i\\in I}$ be a local section structure for $F$ at a representative $z$.\n\tLet $N:{\\mathcal W}[1]\\to [0,\\infty)$ be an auxiliary norm with associated $\\text{sc}^0$-functor $\\hat{N}:W[1]\\to [0,\\infty)$; as in \\cite[p.~434]{HWZbook}, we may extend $N$ to all of ${\\mathcal W}$ by defining $N([w]):= +\\infty$ for $[w]\\in {\\mathcal W}[0] \\setminus {\\mathcal W}[1]$, and likewise extend $\\hat{N}$ to all of $W$.\n\tWe define the \\textbf{min norm of the Fredholm multisection} $F$ at $[z]$ by the equation\n\t\\[\n\tN_{\\min} (F) [z] := \\min_{i\\in I} \\{\t\\hat{N} (\tf_i(z)\t)\t\\}.\n\t\\]\n\n\\end{definition}\n\n\\begin{definition}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $F$ be a $\\text{sc}$-smooth proper Fredholm multisection, and let $N :{\\mathcal W}\\to [0,\\infty)$ be an extended auxiliary norm.\n\t\n\tConsider an open neighborhood ${\\mathcal U}$ of the unperturbed solution set $\\mathcal{S}(F)\\subset {\\mathcal Z}$.\n\n\tWe say that the pair $(N,{\\mathcal U})$ \\textbf{controls the compactness} of $F$ provided the set \n\t\\[\n\tcl_{\\mathcal Z} \\{[x]\\in {\\mathcal U} \\mid N_{\\min} (F) [x]\\leq 1\\} \\subset {\\mathcal Z}\n\t\\]\n\tis compact.\n\n\\end{definition}\n\n\\begin{proposition}[Analog of {\\cite[Thm.~4.5]{HWZ3}}]\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $F$ be a $\\text{sc}$-smooth proper Fredholm multisection, and let $N :{\\mathcal W}[1]\\to [0,\\infty)$ be an auxiliary norm.  Then there exists an open neighborhood ${\\mathcal U}$ of the unperturbed solution set $\\mathcal{S}(F)$ such that the pair $(N,{\\mathcal U})$ controls the compactness of $F$.\n\\end{proposition}\n\n\\begin{proposition}[Analog of {\\cite[Lem.~4.16]{HWZ3}}]\n\t\\label{prop:fredholm-multisection-compactness}\n\tLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $F$ be a $\\text{sc}$-smooth proper Fredholm multisection, and let $(N,{\\mathcal U})$ be a pair which controls compactness.\n\tIf a $\\text{sc}^+$-multisection $\\Gamma$ satisfies $N[\\Gamma] \\leq 1$ and $\\text{dom-supp} (\\Gamma) \\subset {\\mathcal U}$, then the perturbed solution set $\\mathcal{S}(F,\\Gamma)$ is compact\n\\end{proposition}\n\n\\subsubsection{Regular perturbations and compact cobordism}\n\nLet $P:{\\mathcal W}\\to {\\mathcal Z}$ be a strong polyfold bundle, let $F$ be a $\\text{sc}$-smooth proper Fredholm multisection, and let $(N, {\\mathcal U})$ be a pair which controls compactness.\nWe say a $\\text{sc}^+$-multisection $\\Gamma$ is a \\textbf{regular perturbation} of $F$ with respect to the pair $(N,{\\mathcal U})$ if it satisfies the following:\n\t\\begin{itemize}\n\t\t\\item $(F, \\Gamma)$ is a transversal pair,\n\t\t\\item $N[\\Gamma] \\leq 1$ and $\\text{dom-supp} (\\Gamma) \\subset {\\mathcal U}$.\n\t\\end{itemize}\nAs in \\cite[Cor.~15.1]{HWZbook}, one can prove that there exist regular perturbations $\\Gamma$ of $F$ with respect to the pair $(N,{\\mathcal U})$.\nIt follows from Proposition~\\ref{prop:fredholm-multisection-transversal-pairs-weighted-branched-suborbifolds} and Proposition~\\ref{prop:fredholm-multisection-compactness} that the perturbed solution space ${\\mathcal S}(F,\\Gamma)$ has the structure of a compact weighted branched suborbifold, with weight function given by ${\\mathcal S}(F,\\Gamma) \\to \\mathbb{Q}^+,$ $[z]\\mapsto (F\\oplus \\Gamma)(0_{[z]})$.\n\nFurthermore, as in \\cite[Cor.~15.1]{HWZbook} one can prove the existence of a compact cobordism between perturbed solution sets of regular perturbations.\n\n\\subsubsection{Cobordism from a transversal Fredholm multisection to a regular perturbation}\n\t\\label{subsubsec:cobordism-multisection-regular}\n\nHaving developed the above generalization to Fredholm multisections, we are finally in a position to state the desired specialized result, Proposition~\\ref{prop:cobordism-multisection-regular}.\n\nConsider a strong polyfold bundle $P:{\\mathcal W} \\to {\\mathcal Z}$ and a $\\text{sc}$-smooth proper Fredholm section $\\overline{\\partial}$.\nSuppose that $(N_0,{\\mathcal U}_0)$ is a pair which controls the compactness of $\\overline{\\partial}$.\nConsider a $\\text{sc}^+$-multisection $\\Lambda$ and suppose that $(\\overline{\\partial},\\Lambda)$ is a transversal pair.\n(Note that we do not assume that $\\Lambda$ is admissible to a pair which controls compactness.)\nNow, consider the strong polyfold bundle ${\\mathcal W}\\times[0,1] \\to {\\mathcal Z}\\times[0,1]$, and consider a $\\text{sc}$-smooth Fredholm multisection $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ defined as follows:\n\t\\begin{itemize}\n\t\t\\item $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}$ is the $\\text{sc}$-smooth Fredholm section defined by $([z],s)\\mapsto (\\overline{\\partial}([z]),s)$,\n\t\t\\item $\\tilde{\\Lambda}$ is the $\\text{sc}^+$-multisection defined for $s\\neq 0$ by\n\t\t$([w],s)\\mapsto \\Lambda(1\/s \\cdot [w])$ and for $s=0$ by\n\t\t\t\\[\n\t\t\t([w],0)\t\\mapsto \n\t\t\t\\begin{cases}\n\t\t\t\t1, &\\text{if } [w]=[0], \\\\\n\t\t\t\t0, &\\text{if } [w]\\neq [0],\n\t\t\t\\end{cases}\n\t\t\t\\]\n\t\tand\n\t\twhose local section structure at an object $(x,s)$ is defined by $O_x\\times [0,1] \\to W \\times [0,1]; (x,s) \\mapsto (s\\cdot s_i(x), s)$ (where $(s_i)$ is the original local section structure for $\\Lambda$ at the object $x\\in Z$).\n\t\\end{itemize}\nMoreover, let us assume that the Fredholm multisection $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ is proper, i.e., the solution set ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ is compact.\n\nObserve that the topological boundary of ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ is given by the following set:\n\t\\[\n\t\\partial {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda}) = {\\mathcal S}(\\overline{\\partial}) \\sqcup {\\mathcal S}(\\overline{\\partial},\\Lambda).\n\t\\]\nBy the assumption that $(\\overline{\\partial}, \\Lambda)$ is a transversal pair ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ is a weighted branched orbifold; moreover it is a closed subset of ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\tilde{\\Lambda})$ and is therefore compact.\nWe emphasize that since $\\Lambda$ is not admissible to a pair which controls compactness, it is not a regular perturbation (see Definition~\\ref{def:regular-perturbation}) and hence cannot be used to define polyfold invariants.\nWe can almost consider ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ as a compact cobordism, except $\\overline{\\partial}$ is not assumed to be transverse and hence ${\\mathcal S}(\\overline{\\partial})$ is not assumed to have the structure of a weighted branched suborbifold.\n\nThe following proposition demonstrates how to perturb the solution space ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ in order to obtain a compact cobordism between ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ and a perturbed solution space ${\\mathcal S}(\\overline{\\partial},\\Gamma_0)$ where $\\Gamma_0$ is a \\emph{regular} perturbation.\n\n\\begin{proposition}\n\t\\label{prop:cobordism-multisection-regular}\n    Suppose that $\\Gamma_0$ is a regular perturbation of $\\overline{\\partial}$ with respect to the pair $(N_0,{\\mathcal U}_0)$,\n\tThere exists a pair $(N,{\\mathcal U})$ which controls the compactness of the Fredholm multisection $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ and which satisfies the following:\n\t\\begin{itemize}\n\t\t\\item the auxiliary norm $N: {\\mathcal W}\\times [0,1] \\to \\mathbb{Q}^+$ restricts to $N_0$ on ${\\mathcal W}\\times \\{0\\}$,\n\t\t\\item the open neighborhood ${\\mathcal U}$ of ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ satisfies ${\\mathcal U} \\cap ({\\mathcal Z}\\times \\{0\\}) = {\\mathcal U}_0$.\n\t\\end{itemize}\n\tMoreover, there exists a regular perturbation $\\Gamma$ of $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}},\\tilde{\\Lambda})$ with respect to the pair $(N,{\\mathcal U})$ such that $\\Gamma|_{{\\mathcal W}\\times \\{0\\}}$ can be identified with $\\Gamma_0$ and such that $\\Gamma|_{{\\mathcal W}\\times\\{1\\}} \\equiv 0$.\n\\end{proposition}\n\nThe proof of this proposition follows the same reasoning used to prove Theorem~\\ref{thm:cobordism-between-regular-perturbations}, noting in addition that  we do not need to perturb in a neighborhood of ${\\mathcal Z}\\times\\{1\\}$, as by assumption $(\\overline{\\partial},\\Lambda)$ is a transversal pair.\n\n\\subsection{Intermediary subbundles and naturality of polyfold invariants}\n\t\\label{subsec:intermediary-subbundles-naturality}\n\nConsider a commutative diagram as follows,\n\\begin{equation}\\label{eq:commutative-diagram-naturality}\n\t\\begin{tikzcd}\n\t{\\mathcal W}_1 \\arrow[r, \"\\iota_{\\mathcal W}\"', hook] \\arrow[d, \"\\overline{\\partial}_1\\quad \"'] & {\\mathcal W}_2 \\arrow[d, \"\\quad \\overline{\\partial}_2\"] &  \\\\\n\t{\\mathcal Z}_1 \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[u, bend left] & {\\mathcal Z}_2 \\arrow[u, bend right] & \n\t\\end{tikzcd}\n\\end{equation}\nwhere:\n\\begin{itemize}\n\t\\item ${\\mathcal W}_i \\to {\\mathcal Z}_i$ are strong polyfold bundles for $i=1,2$.\n\t\\item $\\overline{\\partial}_i$ are $\\text{sc}$-smooth proper oriented Fredholm sections of the same index for $i=1,2$.\n\t\\item $\\iota_{\\mathcal Z} :{\\mathcal Z}_1 \\hookrightarrow {\\mathcal Z}_2$ is a $\\text{sc}$-smooth injective map, and the associated functor between polyfold structures $\\hat{\\iota}_{\\mathcal Z} : (Z_1,{\\bar\\m}{Z}_1) \\hookrightarrow (Z_2,{\\bar\\m}{Z}_2)$ is fully faithful and is also an injection on both the object and the morphism sets.\n\t\\item $\\iota_{\\mathcal W}:{\\mathcal W}_1\\hookrightarrow{\\mathcal W}_2$ is a $\\text{sc}$-smooth injective map, and the associated functor between polyfold strong bundle structures $\\hat{\\iota}_{\\mathcal W} :(W_1,{\\bar\\m}{W}_1) \\hookrightarrow (W_2,{\\bar\\m}{W}_2)$ is fully faithful, and is also an injection on both the object and the morphism sets.  Moreover, $\\hat{\\iota}$ is a bundle map (i.e., restricts to a linear map on the fibers).\n\t\\item ${\\mathcal S}(\\overline{\\partial}_2) \\subset \\text{Im} (\\iota_{\\mathcal Z})$.\n\\end{itemize}\n\nIn order to deal with orientations, consider the following.\nConsider a smooth object $x\\in (Z_1)_\\infty$ which maps to $y:= \\hat{\\iota}_{\\mathcal Z} \\in (Z_2)_\\infty$.\nConsider a locally defined $\\text{sc}^+$-section $s:' U \\to W_2$ defined on an open neighborhood $U\\subset Z_2$ of $y$, which satisfies $s'(y) = \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 (y)$.\nAssume that this $\\text{sc}^+$-section has a well-defined restriction $s'|_{U \\cap \\hat{\\iota}_{\\mathcal Z} (Z_1)} : U \\cap \\hat{\\iota}_{\\mathcal Z} (Z_1) \\to \\hat{\\iota}_{\\mathcal W} (W_1)$, which induces a $\\text{sc}^+$-section $s: \\hat{\\iota}_{\\mathcal Z}^{-1}(U) \\to W_1$ which moreover satisfies $s(x) = \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1 (x)$.\nWe therefore have a commutative diagram.\n\t\\[\n\t\\begin{tikzcd}\n\tT_x W_1 \\arrow[r, \"D\\hat{\\iota}_{\\mathcal W}\"'] & T_y W_2 &  \\\\\n\tT_x Z_1 \\arrow[r, \"D\\hat{\\iota}_{\\mathcal Z}\"'] \\arrow[u, \"D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x)\"] & T_y Z_2 \\arrow[u, \"D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y)\"'] & \n\t\\end{tikzcd}\n\t\\]\nConsider the following maps: $D\\hat{\\iota}_{\\mathcal Z}: \\ker (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x))) \\to \\ker (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))$, and $D\\hat{\\iota}_{\\mathcal W}: \\text{Im} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x))) \\to \\text{Im} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))$, which therefore induces a map $\\operatorname{coker} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x))) \\to \\operatorname{coker} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))$.\nThese maps induce a map between the determinant real lines\n\t\\begin{gather*}\n\t\\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x))) = \\Lambda^{\\max} (\\ker (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x)))) \\otimes (\\Lambda^{\\max} (\\operatorname{coker} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x)))))^*,\t\\\\\n\t\\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))) = \\Lambda^{\\max} (\\ker (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))) \\otimes (\\Lambda^{\\max} (\\operatorname{coker} (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y)))\t)^*\t.\n\t\\end{gather*}\n\n\\begin{itemize}\n\t\\item\tAssume that the induced map between the determinants\n\t\t\t\t\\[\\hat{\\iota}_* : \\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s)(x))) \\to \\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s')(y))\\]\n\t\t\tis an isomorphism. Moreover, assume that this isomorphism is orientation preserving, with respect to the chosen orientations of $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1$ at the point $x$ and $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2$ at the point $y$ (see Definition~\\ref{def:oriented-Fredholm}).\n\\end{itemize}\n\nReturning to the main discussion, it follows from commutativity of \\eqref{eq:commutative-diagram-naturality} that $\\iota_{\\mathcal Z}$ restricts to a continuous bijection between the unperturbed solution sets,\n\t\\[\\iota_{\\mathcal Z} |_{{\\mathcal S}(\\overline{\\partial}_1)} : {\\mathcal S}(\\overline{\\partial}_1)\\to {\\mathcal S}(\\overline{\\partial}_2).\\]\nIn fact, this map is a homeomorphism as can be shown via point-set topology, noting that ${\\mathcal S}(\\overline{\\partial}_1)$ is compact and ${\\mathcal S}(\\overline{\\partial}_2)$ is Hausdorff (see \\cite[Rmk.~3.1.15]{MWtopology}).\n\nIn order to compare the polyfold invariants, suppose we also have a commutative diagram\n\t\\begin{equation}\\label{eq:gw-invariant-pair-of-polyfolds}\n\t\\begin{tikzcd}\n\t&  & {\\mathcal O} \\\\\n\t{\\mathcal Z}_1 \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[rru, \"f_1\"] & {\\mathcal Z}_2 \\arrow[ru, \"f_2\"'] & \n\t\\end{tikzcd}\n\t\\end{equation}\nwhere:\n\\begin{itemize}\n\t\\item ${\\mathcal O}$ is a finite-dimensional orbifold.\n\t\\item $f_i$ are $\\text{sc}$-smooth maps for $i=1,2$.\n\\end{itemize}\n\n\\begin{definition}\\label{def:intermediate-subbundle}\n\tWe define an  \\textbf{intermediary subbundle} as a subset ${\\mathcal R} \\subset {\\mathcal W}_2$ which satisfies the following properties.\n\t\\begin{enumerate}\n\t\t\\item Let $(R,{\\bar\\m}{R})$ be the associated subgroupoid of ${\\mathcal R}$. Then for every object $x\\in Z_2$ we require that the fiber $R_x : = R \\cap (W_2)_x$ is a vector subspace of $(W_2)_x$. (Note that we do not require that $R_x$ is complete.)\n\t\t\\item \\label{property-2-intermediary-subbundle} For any point $[x] \\in {\\mathcal Z}_2$, if $\\overline{\\partial}_2 ([x]) \\in {\\mathcal R}$ then $[x] \\in \\iota_{\\mathcal Z}({\\mathcal Z}_1)$. (Equivalently, for any object $x\\in Z_2$, if $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 (x) \\in R$ then $x \\in \\hat{\\iota}_{\\mathcal Z} (Z_1)$.)\n\t\t\\item \\label{property-3-intermediary-subbundle}\n\t\tGiven $[x_0] \\in {\\mathcal S}(\\overline{\\partial}_1) \\simeq {\\mathcal S}(\\overline{\\partial}_2)$, let $V\\subset U\\subset Z_2$ be ${\\bar\\m}{G}(x_0)$-invariant open neighborhoods of a representative $x_0\\in Z_2$ such that $\\overline{V} \\subset U$.\n\t\tWe require that there exist $\\text{sc}^+$-sections\n\t\t\t\\[\n\t\t\ts'_i :U \\to W_2, \\qquad 1\\leq i \\leq k\n\t\t\t\\]\n\t\twhich have well-defined restrictions $s'_i|_{U\\cap \\hat{\\iota}_{\\mathcal Z} (Z_1)} : U\\cap \\hat{\\iota}_{\\mathcal Z} (Z_1) \\to \\hat{\\iota}_{\\mathcal W} (W_1)$. These restrictions induce sections $s_i : \\hat{\\iota}_{\\mathcal Z}^{-1} (U) \\to W_1$ which we require to be $\\text{sc}^+$ with respect to the M-polyfold structures on $Z_1$ and $W_1$.\n\t\tWe require that:\n\t\t\t\\begin{itemize}\n\t\t\t\\item $s'_i (U) \\subset R$,\n\t\t\t\\item $s'_i= 0$ on $U\\setminus V$,\n\t\t\t\\item $\\text{span}\\{s'_1(x_0),\\ldots , s'_k(x_0)\\} \\oplus \\text{Im}(D\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2(x_0)) = (W_2)_{x_0},$\n\t\t\t\\item $\\text{span}\\{s_1(x_0),\\ldots , s_k(x_0)\\} \\oplus \\text{Im}(D\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1(x_0)) = (W_1)_{x_0}.$\n\t\t\t\\end{itemize}\n\t\t\\item In addition, given a pair $(N_2,{\\mathcal U}_2)$ which controls the compactness of $\\overline{\\partial}_2$, we require that these $\\text{sc}^+$-sections satisfy the following:\n\t\t\t\\begin{itemize}\n\t\t\t\\item $\\hat{N}_2[s'_i] \\leq 1,$\n\t\t\t\\item $\\abs {\\operatorname{supp} (s'_i)}\\subset {\\mathcal U}_2$.\n\t\t\t\\end{itemize}\n\t\t\\begin{comment}\n\t\t\\item \\red{Alternative.}\n\t\tGiven $[x] \\in {\\mathcal S}(\\overline{\\partial}_2)$, let $U\\subset Z_2$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of a representative $x$.\n\t\tWe require that there exists a parametrized $\\text{sc}^+$-multisection\n\t\t\t\\[\n\t\t\t\\Lambda : {\\mathcal W}_2\\times B^k_\\varepsilon \\to \\mathbb{Q}^+\n\t\t\t\\]\n\t\tsuch that the composition $\\Lambda \\circ \\iota_{\\mathcal W} : {\\mathcal W}_1 \\times B^k_\\varepsilon \\to \\mathbb{Q}^+$ is also a well-defined $\\text{sc}^+$-multisection.\n\t\t\\end{comment}\n\t\\end{enumerate}\n\\end{definition}\n\nDespite the lengthy properties that a intermediary subbundle must satisfy, in practice such subbundles are easy to construct, as we demonstrate in \\S~\\ref{subsec:independence-sequence} and \\S~\\ref{subsec:independence-punctures}.\n\nWe may now prove Theorem~\\ref{thm:naturality-polyfold-invariants}, which we restate in order to be consistent with our current notation.\n\n\\begin{theorem}\n\t\\label{thm:naturality}\n\tSuppose there exists an intermediary subbundle ${\\mathcal R} \\subset {\\mathcal W}_2$. Then the polyfold invariants for ${\\mathcal Z}_1$ and ${\\mathcal Z}_2$ defined via the branched integral are equal.\n\tThis means that, given a de Rahm cohomology class $\\omega\\in H^*_{\\dR} ({\\mathcal O})$ the branched integrals over the perturbed solution spaces are equal,\n\t\t\\[\n\t\t\\int_{{\\mathcal S} (\\overline{\\partial}_1, p_1)} f_1^* \\omega = \\int_{{\\mathcal S}(\\overline{\\partial}_2,p_2)} f_2^* \\omega,\n\t\t\\]\n\tfor any choices of regular perturbations.\n\\end{theorem}\n\\begin{proof}\nWe prove the theorem in six steps.\n\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 1:} \\emph{We use property \\ref{property-3-intermediary-subbundle} of the intermediary subbundle to construct a transversal $\\text{sc}^+$-multisection with a well-defined transversal restriction.}\n\\end{itemize}\n\nAt the outset, fix pairs $(N_i,{\\mathcal U}_i)$ which control the compactness of $\\overline{\\partial}_i$ for $i=1,2$. Consider a point $[x_0]\\in {\\mathcal S}(\\overline{\\partial}_1)\\simeq {\\mathcal S}(\\overline{\\partial}_2)$ and let $x_0\\in Z_1$ be a representative with isotropy group ${\\bar\\m}{G}(x_0)$. Via the inclusion map $\\hat{\\iota}_{\\mathcal Z}$, we may identify $x_0$ with its image in $Z_2$ and note that we may also identify the isotropy groups.\n\nWe may use property \\ref{property-3-intermediary-subbundle} of the intermediary subbundle to construct an $\\text{sc}^+$-mul\\-ti\\-sec\\-tion functor $\\hat{\\Lambda}'_0:W_2 \\times B_\\varepsilon^k \\to \\mathbb{Q}^+$ \nwith local section structure given by $\\left\\lbrace g * \\left(\\sum_{i=1}^k t_i \\cdot s'_i\\right)\\right\\rbrace_{g\\in {\\bar\\m}{G}(x_0)}$ which satisfies the following.\nThere exists a ${\\bar\\m}{G}(x_0)$-invariant open neighborhood $x_0 \\subset U'_0 \\subset Z_2$ such that at any object $x\\in U'_0$ and for any $g\\in {\\bar\\m}{G}(x_0)$ the linearization of the function \n\t\\begin{align*}\n\tU'_0 \\times B_\\varepsilon^k \t&\\to W_2\\\\\n\t(x, t_1,\\ldots,t_k)\t\t&\\mapsto \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 (x) - g * \\left(\\sum_{i=1}^k t_i \\cdot s'_i(x)\\right)\n\t\\end{align*}\nprojected to the fiber $(W_2)_x$ is surjective.\n\nFurthermore, property \\ref{property-3-intermediary-subbundle} ensures that the functor\n$\\hat{\\Lambda}_0 :=\\hat{\\Lambda}_0' ( \\hat{\\iota}_{\\mathcal W} (\\cdot), \\cdot ) :W_1 \\times B_\\varepsilon^k \\to \\mathbb{Q}^+$ is also a $\\text{sc}^+$-multisection functor, with local section structure $\\left\\lbrace g * \\left(\\sum_{i=1}^k t_i \\cdot s_i\\right)\\right\\rbrace_{g\\in {\\bar\\m}{G}(x_0)}$ where the $\\text{sc}^+$-sections $s_i$ are induced by the well-defined restrictions of the sections $s'_i$. Likewise, there exists a ${\\bar\\m}{G}(x_0)$-invariant open neighborhood $x_0 \\subset U_0 \\subset Z_1$ such that at any object $x\\in U_0$ and for any $g\\in {\\bar\\m}{G}(x_0)$ the linearization of the function $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1(x) - g * \\left(\\sum_{i=1}^k t_i \\cdot s_i(x)\\right)$ projected to the fiber $(W_1)_x$ is surjective.\n\nWe may cover the compact topological space ${\\mathcal S}(\\overline{\\partial}_2)$ by a finite collection of such neighborhoods $\\abs{U'_i}$ of points $[x_i]\\in {\\mathcal S}(\\overline{\\partial}_2)$; we may also cover ${\\mathcal S}(\\overline{\\partial}_1)$ by a finite collection of such neighborhoods $\\abs{U_i}$ of points $[x_i]\\in {\\mathcal S}(\\overline{\\partial}_1)$.\nIt follows that the finite sum of $\\text{sc}^+$-multisections\n\t\\[\n\t\\Lambda_2:= \\bigoplus_i \\Lambda'_i : {\\mathcal W}_2 \\times B_\\varepsilon^N \\to \\mathbb{Q}^+\n\t\\]\nhas the property that: for any point $[x] \\in {\\mathcal Z}_2$ with $\\Lambda_2 \\circ \\overline{\\partial}_2 ([x])>0$, and for any parametrized local section structure $\\{s'_i\\}_{i\\in I}$ at a representative $x$, the linearization of the function $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 (x) - s'_i(x,t)$ projected to the fiber $(W_2)_x$ is surjective.\nLikewise, the finite sum of $\\text{sc}^+$-multisections\n\t\\[\n\t\\Lambda_1 := \\bigoplus_i \\Lambda_i = \\bigoplus_i \\Lambda'_i (\\iota_{\\mathcal W}(\\cdot),\\cdot) : {\\mathcal W}_1 \\times B_\\varepsilon^N \\to \\mathbb{Q}^+\n\t\\]\nhas the property that for any point $[x] \\in {\\mathcal Z}_1$ which satisfies $\\Lambda_1 \\circ \\overline{\\partial}_1 ([x])>0$ and for any parametrized local section structure $\\{s_i\\}_{i\\in I}$ at a representative $x$, the linearization of the function $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1 (x) - s_i(x,t)$ projected to the fiber $(W_1)_x$ is surjective. Observe moreover that the multisection sum commutes with composition and thus $\\Lambda_1(\\cdot,\\cdot) = \\Lambda_2 (\\iota_{\\mathcal W}(\\cdot),\\cdot)$.\n\nFurthermore for $\\varepsilon$ sufficiently small, for any fixed $t_0 \\in B_\\varepsilon^N$ the $\\text{sc}^+$-multisection $\\Lambda_2 (\\cdot, t_0)$ is controlled by the pair $(N_2,{\\mathcal U}_2)$, i.e.,\n\\begin{itemize}\n\t\\item $N_2[\\Lambda_2(\\cdot,t_0)] \\leq 1$,\n\t\\item $\\text{dom-supp}(\\Lambda_2(\\cdot,t_0)) \\subset {\\mathcal U}_2$.\n\\end{itemize}\n\nIn contrast, $\\Lambda_1(\\cdot,t_0)$ will generally not be controlled by the pair $(N_1,{\\mathcal U}_1)$, as in general, \t\n\t\\[\n\t\\text{dom-supp}(\\Lambda_1(\\cdot,t_0)) = \\iota_{\\mathcal Z}^{-1} (\\text{dom-supp}(\\Lambda_2(\\cdot,t_0))) \\nsubseteq {\\mathcal U}_1.\n\t\\]\n\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 2:} \\emph{We show the thickened solution sets satisfy the hypotheses of Lemma~\\ref{lem:invariance-of-domain-branched-orbifolds}, and are therefore homeomorphic.}\n\\end{itemize}\n\nConsider the strong polyfold bundle ${\\mathcal W}_i \\times B_\\varepsilon^N \\to {\\mathcal Z}_i \\times B_\\varepsilon^N$ for $i=1,2$, and let $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_i : {\\mathcal Z}_i \\times B_\\varepsilon^N \\to {\\mathcal W}_i \\times B_\\varepsilon^N$ denote the $\\text{sc}$-smooth proper Fredholm section defined by $([z],t)\\mapsto (\\overline{\\partial}_i([z]),t)$.\nBy construction, $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_i, \\Lambda_i)$ are transversal pairs; hence by Theorem~\\ref{thm:transversal-pairs-weighted-branched-suborbifolds} the thickened solution sets\n\t\\[\n\t{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_i, \\Lambda_i) = \\{ ([z],t) \\in {\\mathcal Z}_i \\times B_\\varepsilon^N \\mid \\Lambda_i (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_i ([z],t)) >0 \\} \\subset {\\mathcal Z}_i\\times B_\\varepsilon^N\n\t\\]\nhave the structure of weighted branched orbifolds.\n\nWe now claim that these thickened solution sets satisfy the hypotheses of Lemma~\\ref{lem:invariance-of-domain-branched-orbifolds}.\nIndeed, commutativity of the diagram \\eqref{eq:commutative-diagram-naturality} together with the equation $\\Lambda_1(\\cdot,\\cdot) = \\Lambda_2 (\\iota_{\\mathcal W}(\\cdot),\\cdot)$ imply that the injective continuous map $\\tilde{\\iota}_{\\mathcal Z}: {\\mathcal Z}_1 \\times B_\\varepsilon^N \\to {\\mathcal Z}_2 \\times B_\\varepsilon^N; ([z],t)\\mapsto (\\iota_{\\mathcal Z}([z]),t)$ has a well-defined restriction to the thickened solution sets,\n\t\\begin{equation}\\label{eq:restriction-to-thickening}\n\t\\tilde{\\iota}_{\\mathcal Z} |_{{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1)} : {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1) \\hookrightarrow {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2, \\Lambda_2).\n\t\\end{equation}\nMoreover, at any $(x,t)\\in S_1(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1,\\hat{\\Lambda}_1)$ which maps to $(y,t)\\in S_2(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2,\\hat{\\Lambda}_2)$, the local section structure $(s_i)$ for $\\hat{\\Lambda}_1$ at $(x,t)$ is induced by the restrictions of the local section structure $(s'_i)$ for $\\hat{\\Lambda}_2$ at $(y,t)$. In particular, we have the following commutative diagram.\n\t\\[\\begin{tikzcd}[column sep = large]\n\tW_1\\times B_\\varepsilon^N \\arrow[r, \"{(\\hat{\\iota}_{\\mathcal W}(\\cdot),\\cdot)}\"', hook] \\arrow[d, \"\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1 - s_i \\quad \"'] & W_2\\times B_\\varepsilon^N \\arrow[d, \"\\quad \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 - s'_i\"] &  \\\\\n\tO_x\\times B_\\varepsilon^N \\arrow[r, \"{(\\hat{\\iota}_{\\mathcal Z}(\\cdot),\\cdot)}\"', hook] \\arrow[u, bend left] & O_y\\times B_\\varepsilon^N \\arrow[u, bend right] & \n\t\\end{tikzcd}\\]\nAs noted in Remark~\\ref{rmk:relationship-local-section-structures-local-branching-structures}, the local section structures and the local branching structures are related via the equations $M_i= (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1 - s_i)^{-1}(0)$, $M'_i = (\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 - s'_i)^{-1}(0)$. Thus it follow from commutativity that we have the required well-defined restriction to the individual local branches.\nAnd now Lemma~\\ref{lem:invariance-of-domain-branched-orbifolds} implies that the map \\eqref{eq:restriction-to-thickening} is a local homeomorphism.\n\nFurthermore, we may observe that by our orientation assumptions the natural induced map $\\tilde{\\iota}_* : \\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_1-s_i)(x))) \\to \\det (D(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2-s'_i)(y))$ is an orientation preserving isomorphism; hence the restriction $\\tilde{\\iota}_{\\mathcal Z} |_{M_i} : M_i \\to M'_i$ is orientation preserving.\n\nWe now show that \\eqref{eq:restriction-to-thickening} is a bijection.\nLet $([y],t)\\in {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2,\\Lambda_2)$, let $(y,t)$ be a representative of $([y],t)$, and consider a local section structure $(s'_i)$ for $\\Lambda_2$ at $(y,t)$.\nIt follows that $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2(y) - s'_i (y,t) = 0$ for some index $i$.\nObserve by construction, $s'_i$ is a finite sum of $\\text{sc}^+$-sections with image contained in the intermediate subbundle $R$, and hence $s'_i(y,t)\\in R$.\nIt follows that $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_2 (y)\\in R$, hence property \\ref{property-2-intermediary-subbundle} of the intermediate subbundle implies that $y\\in \\hat{\\iota}_{\\mathcal Z} (Z_1)$. Therefore, there exists a point $[x]\\in {\\mathcal Z}_1$ such that $\\tilde{\\iota}_{\\mathcal Z} ([x],t) = ([y],t)$. Commutativity of \\eqref{eq:commutative-diagram-naturality} implies that $\\Lambda_1(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1([x]),t) = \\Lambda_2(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2([y],t)) >0$, and therefore $([x],t) \\in {\\mathcal S}_1(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1)$.\nThus, \\eqref{eq:restriction-to-thickening} is a homeomorphism.\n\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 3:} \\emph{For a common regular value $t_0$ the branched integrals of the perturbed solution spaces of $\\overline{\\partial}_1$ and $\\overline{\\partial}_2$ are equal.}\n\\end{itemize}\n\nBy Sard's theorem, we can find a common regular value $t_0 \\in B_\\varepsilon^N$ of the projections ${\\mathcal S} (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1) \\to B_\\varepsilon^N$ and ${\\mathcal S} (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2,\\Lambda_2) \\to B_\\varepsilon^N$.\nFor this common regular value, the perturbed solution sets\n\t\\[\n\t{\\mathcal S}(\\overline{\\partial}_i, \\Lambda_i (\\cdot,t_0)) := \\{\t[z]\\in {\\mathcal Z}_i \\mid \\overline{\\partial}_i(\\Lambda_i([z],t_0))>0\t\\} \\subset {\\mathcal Z}_i\n\t\\]\nhave the structure of weighted branched suborbifolds.\n\nAs we have already noted, $\\Lambda_2(\\cdot,t_0)$ is controlled by the pair $(N_2,{\\mathcal U}_2)$ and hence ${\\mathcal S}(\\overline{\\partial}_2, \\Lambda_2 (\\cdot,t_0))$ is a compact topological space.\nFor such a common regular value, the homeomorphism \\eqref{eq:restriction-to-thickening} has a well-defined restriction to these perturbed solution sets. This restriction is a homeomorphism, and hence ${\\mathcal S}(\\overline{\\partial}_1, \\Lambda_1 (\\cdot,t_0))$ is also a compact topological space (even though in general $\\Lambda_1(\\cdot,t_0)$ will not be controlled by the pair $(N_1,{\\mathcal U}_1)$).\n\nThe restriction $\\tilde{\\iota}_{\\mathcal Z} |_{{\\mathcal S}(\\overline{\\partial}_1, \\Lambda_1 (\\cdot,t_0))}$ satisfies the necessary hypotheses for the change of variables theorem~\\ref{thm:change-of-variables}. \nTherefore for a given $\\text{sc}$-smooth differential form $\\omega\\in \\Omega_\\infty^* ({\\mathcal Z}_2)$ we have\n\t\\begin{equation}\n\t\\label{eq:change-variables}\n\t\\int_{{\\mathcal S}(\\overline{\\partial}_2, \\Lambda_2 (\\cdot,t_0))} \\omega\n\t= \\int_{{\\mathcal S}(\\overline{\\partial}_1, \\Lambda_1 (\\cdot,t_0))} \\tilde{\\iota}_{\\mathcal Z}^* \\omega.\n\t\\end{equation}\nHowever, since in general $\\Lambda_1(\\cdot,t_0)$ is not controlled by a pair, we cannot assume that it is a regular perturbation in the sense of Definition~\\ref{def:regular-perturbation}.\nThis is problematic since Theorem~\\ref{thm:cobordism-between-regular-perturbations} only implies the existence of a compact cobordism between the perturbed solution spaces of two perturbations which are both assumed to be regular perturbations (see Figure~\\ref{fig:cobordism}).\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics{cobordism.eps}\n\t\\caption{Compact cobordism between regular perturbations}\\label{fig:cobordism}\n\\end{figure}\n\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 4:} \\emph{We show that the set \n\t\t\\[\n\t\t{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1 (\\cdot,st_0)) = \\{\t([z],s) \\in {\\mathcal Z}_1\\times [0,1]\t\\mid\t\\Lambda_1(\\overline{\\partial}_1(z),s t_0)>0\t\\}\n\t\t\\]\t\t\n\t\tis compact.}\n\\end{itemize}\n\nLet $\\delta = \\abs{t_0}$.\nThe auxiliary norm $\\tilde{N}_2 : {\\mathcal W}_2[1] \\times \\overline{B_{\\delta}^N} \\to [0,\\infty)$ defined by $\\tilde{N}_2 ([w],t) := N_2([w])$ together with the open set $\\tilde{{\\mathcal U}}_2 := {\\mathcal U}_2 \\times \\overline{B_{\\delta}^N}$ together control the compactness of the extended $\\text{sc}$-smooth Fredholm section $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2$.\nBy construction, $\\Lambda_2$ is controlled by this pair and hence by Theorem~\\ref{thm:compactness} the thickened solution set ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2, \\Lambda_2(\\cdot, t); t\\in \\overline{B_\\delta ^N})$ is a compact topological space.\nTherefore, the closed subset ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2, \\Lambda_2(\\cdot, t) ; t = s \\cdot t_0, s\\in[0,1])$ is also compact.\n\nThe restriction of \\eqref{eq:restriction-to-thickening} yields a homeomorphism  \n\t\\[\n\t{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1(\\cdot, t) ; t = s \\cdot t_0, s\\in[0,1]) \\to {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_2, \\Lambda_2(\\cdot, t) ; t = s \\cdot t_0, s\\in[0,1]).\n\t\\]\nFrom this it is now clear that \n\t\\[\n\t{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1 (\\cdot,st_0)) \\simeq {\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1(\\cdot, t) ; t = s \\cdot t_0, s\\in[0,1])\n\t\\]\nis a compact topological space.\n\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 5:} \\emph{We interpret the pair $(\\overline{\\partial}_1,\\Lambda_1(\\cdot,t_0))$ as a transversal Fredholm multisection, and use Proposition~\\ref{prop:cobordism-multisection-regular} to obtain a compact cobordism to a regular perturbation.}\n\\end{itemize}\n\nWe claim that the hypotheses described in \\S~\\ref{subsubsec:cobordism-multisection-regular} are satisfied.\nIn particular, we must show that the extended Fredholm multisection $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\tilde{\\Lambda}_1 (\\cdot,t_0))$ is proper.\nThis can be seen using step 4; indeed, the solution set ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\tilde{\\Lambda}_1 (\\cdot,t_0))$ described in \\S~\\ref{subsubsec:cobordism-multisection-regular} can be identified with the compact set ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}_1, \\Lambda_1 (\\cdot,st_0))$.\n\nWe may therefore use Proposition~\\ref{prop:cobordism-multisection-regular} to obtain a cobordism from $(\\overline{\\partial}_1,\\Lambda_1(\\cdot,t_0))$ to a regular perturbation $\\Gamma_0: {\\mathcal W}_1 \\to \\mathbb{Q}^+$ of $\\overline{\\partial}_1$.\nGiven a closed $\\text{sc}$-smooth differential form $\\omega\\in \\Omega_\\infty^* ({\\mathcal Z}_1)$, Stokes' theorem~\\ref{thm:stokes} then implies\n\t\\begin{equation}\n\t\\label{eq:step-5-cobordism-stokes}\n\t\\int_{{\\mathcal S}(\\overline{\\partial}_1,\\Gamma_0)}\t\\omega =\\int_{{\\mathcal S}(\\overline{\\partial}_1,\\Lambda_1(\\cdot,t_0))} \\omega.\n\t\\end{equation}\n\t\n\\begin{itemize}[leftmargin=0em]\n\t\\item[]\\textbf{Step 6:} \\emph{We show that the polyfold invariants are equal.}\n\\end{itemize}\n\nLet $\\omega\\in H^*_{\\dR} (O)$ be the de Rahm cohomology class fixed in the statement of the theorem, and used to define the polyfold invariants.\nWe can now compute relate the branched integrals as follows:\n\\begin{align*}\n\t \\int_{{\\mathcal S}(\\overline{\\partial}_2,\\Lambda_2(\\cdot,t_0))} f_2^* \\omega\n\t \t\t& =\t\\int_{{\\mathcal S}(\\overline{\\partial}_1, \\Lambda_1 (\\cdot,t_0))} \\tilde{\\iota}_{\\mathcal Z}^* f_2^*\\omega\t\\\\\n\t \t\t& =\t\\int_{{\\mathcal S}(\\overline{\\partial}_1, \\Lambda_1 (\\cdot,t_0))} f_1^*\\omega\t\\\\\n\t \t\t& =\t\\int_{{\\mathcal S}(\\overline{\\partial}_1,\\Gamma_0)}\tf_1^*\\omega,\n\\end{align*}\nwhere the first equality follows from equation \\eqref{eq:change-variables}, the second equality follows from the commutativity of \\eqref{eq:gw-invariant-pair-of-polyfolds}, and the third equality follows from equation \\eqref{eq:step-5-cobordism-stokes}.\nBy construction, $\\Lambda_2(\\cdot,t_0)$ is a regular perturbation of $\\overline{\\partial}_2$, while $\\Gamma_0$ is a regular perturbation of $\\overline{\\partial}_1$.\nThis proves the theorem.\n\\end{proof}\n\n\\subsection{Gromov--Witten invariants are independent of choice of sequence \\texorpdfstring{$\\delta_i$}{\u03b4i}}\n\t\\label{subsec:independence-sequence}\n\nWe now use Theorem~\\ref{thm:naturality} to show that the Gromov--Witten polyfold invariants are independent of the choice of increasing sequence $(\\delta_i)_{i\\geq 0}\\subset (0,2\\pi)$.\nGiven two sequences $(\\delta_i)\\subset (0,2\\pi)$ and $(\\delta_i')\\subset (0,2\\pi)$ we can always find a third sequence $(\\delta_i'')\\subset (0,2\\pi)$ which satisfies\n\t\\[\n\t\\delta_i \\leq \\delta_i'', \\qquad \\delta_i' \\leq \\delta_i''\n\t\\]\nfor all $i$.  The GW-polyfold associated to the sequence $(\\delta_i'')$ give a refinement of the GW-polyfolds associated to $(\\delta_i)$ and $(\\delta_i')$, in the sense that there are inclusion maps\n\t\\[\n\t{\\mathcal Z}_{A,g,k}^{3,\\delta'_0} \\hookleftarrow {\\mathcal Z}_{A,g,k}^{3,\\delta''_0} \\hookrightarrow {\\mathcal Z}_{A,g,k}^{3,\\delta_0}.\n\t\\]\nIt is therefore sufficient to consider inclusion maps of the form\n${\\mathcal Z}^{3,\\delta_0}_{A,g,k} \\hookrightarrow {\\mathcal Z}^{3,\\delta_0'}_{A,g,k}$\nwith $\\delta_i' \\leq \\delta_i$ for all $i$ and demonstrate that the associated GW-invariants are equal.\n\nTo this end, consider the commutative diagram:\n\\[\\begin{tikzcd}\n{\\mathcal W}^{2,\\delta_0}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal W}\"', hook] \\arrow[d, \"\\overline{\\partial}_J\\quad \"'] & {\\mathcal W}^{2,\\delta_0'}_{A,g,k} \\arrow[d, \"\\quad \\overline{\\partial}_J'\"] &  \\\\\n{\\mathcal Z}^{3,\\delta_0}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[u, bend left] & {\\mathcal Z}^{3,\\delta_0'}_{A,g,k} \\arrow[u, bend right] & \n\\end{tikzcd}\\]\nand observe that it satisfies the same properties as \\eqref{eq:commutative-diagram-naturality}.\nIn addition, consider the commutative diagram:\n\\[\\begin{tikzcd}\n&  & Q^k\\times \\smash{\\overline{\\mathcal{M}}}\\vphantom{\\mathcal{M}}^{\\text{log}}_{g,k} \\\\\n{\\mathcal Z}^{3,\\delta_0}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[rru, \"ev_i \\times \\pi\"] & {\\mathcal Z}^{3,\\delta_0'}_{A,g,k} \\arrow[ru, \"ev_i\\times \\pi\"'] & \n\\end{tikzcd}\\]\nwhich satisfies the same properties as \\eqref{eq:gw-invariant-pair-of-polyfolds}.\n\nNote that if $\\delta_0' < \\delta_0$ the inclusion map $\\iota_{\\mathcal Z}$ is not proper.\nTo see this, exploit the difference in exponential weights to produce a sequence which converges in a local M-polyfold model for ${\\mathcal Z}^{3,\\delta_0}_{A,g,k}$ but diverges in a local M-polyfold model for ${\\mathcal Z}^{3,\\delta_0'}_{A,g,k}$.\nNote also that the pullback strong polyfold bundle is not the same as the standard strong polyfold bundle on ${\\mathcal Z}^{3,\\delta_0}_{A,g,k}$.\n\n\\begin{proposition}\n\t\\label{prop:existence-subbundle-naturality}\n\tThe set\t\n\t\\[\n\t{\\mathcal R} := \\{\t[\\Sigma,j,M,D,u,\\xi] \\in {\\mathcal W}^{2,\\text{deg}_0'}_{A,g,k} \\mid \\operatorname{supp} \\xi \\subset K \\subset \\Sigma\\setminus\\abs{D} \\text{ for some compact } K\t\\}\n\t\\]\n\tis an intermediary subbundle of the strong polyfold bundle ${\\mathcal W}^{2,\\text{deg}_0'}_{A,g,k}$.\n\\end{proposition}\n\\begin{proof}\n\tWe must show that the set ${\\mathcal R}$ satisfies the properties of Definition~\\ref{def:intermediate-subbundle}.\n\tThe first two properties can be easily checked.\n\t\n\tWe show how to construct the $\\text{sc}^+$-sections required by property \\ref{property-3-intermediary-subbundle}.\n\tConsider a stable curve $[\\alpha]=[\\Sigma,j,M,D,u] \\in {\\mathcal S}(\\overline{\\partial}_J') \\subset {\\mathcal Z}^{3,\\text{deg}_0'}_{A,g,k}$ and let let $\\alpha = (\\Sigma,j,M,D,u)$ be a stable map representative.\n\tLet $V_\\alpha \\subset U_\\alpha$ be a ${\\bar\\m}{G}(\\alpha)$-invariant M-polyfold charts centered at $\\alpha$ such that $\\overline{V_\\alpha}\\subset U_\\alpha$.\n\tThis means we have good uniformizing family \n\t\\[\n\t(a,v,\\eta ) \\mapsto (\\Sigma_{a},j(a,v),M_{a},D_{a}, \\oplus_{a} \\exp_u (\\eta)),\\qquad (a,v,\\eta) \\in O_\\alpha.\n\t\\]\n\tLet $K\\to O_\\alpha$ be a local strong bundle model, with $\\text{sc}$-coordinates given by $(a,v,\\eta,\\xi)$ where $\\xi \\in H^{2,\\text{deg}_0'}(\\Sigma,\\Lambda^{0,1} \\otimes_J u^* TQ)$.\n\t\n\tUse Corollary~\\ref{cor:vectors-which-span-cokernel} to choose vectors $v_1,\\ldots, v_k$ which vanish on disk-like regions of the nodal points and such that\n\t\\begin{itemize}\n\t\t\\item $\\text{span}\\{v_1,\\ldots , v_k\\} \\oplus \\text{Im}(D\\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}'(\\alpha)) = H^{2,\\text{deg}_0'}(\\Sigma,\\Lambda^{0,1} \\otimes_J u^* TQ)$,\n\t\t\\item $\\text{span}\\{v_1,\\ldots , v_k\\} \\oplus \\text{Im}(D\\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}(\\alpha)) = H^{2,\\text{deg}_0}(\\Sigma,\\Lambda^{0,1} \\otimes_J u^* TQ)$.\n\t\\end{itemize}\n\t\n\tLet $\\beta: U_\\alpha \\to [0,1]$ be an $\\text{sc}$-smooth cutoff function which satisfies $\\beta\\equiv 1$ near $(0,0,0)$, and $\\beta \\equiv 0$ on $U_\\alpha\\setminus V_\\alpha$.\n\tIn these local $\\text{sc}$-coordinates, the desired $\\text{sc}^+$-sections are defined as follows:\n\t\\[\n\ts_i': U_\\alpha\\to K,\\qquad (a,v,\\eta) \\mapsto (a,v,\\eta, \\beta(a,v,\\eta) \\cdot ( \\rho_{a} (v_i)\t\t)),\n\t\\]\n\twhere $\\rho_{a}$ is the strong bundle projection defined using the hat gluings, see \\cite[p.~117]{HWZbook} and \\cite[pp.~65--67]{HWZGW}\n\t\n\tLet $(N,{\\mathcal U})$ be a pair which controls the compactness of $\\overline{\\partial}_J'$.\n\tBy construction, these $\\text{sc}^+$-sections satisfy $N[s_i'] \\leq C$ for some constant $C\\leq \\infty$, and hence by rescaling the vectors $v_i$ we may assume that $N[s_i'] \\leq 1$. Moreover, by shrinking the support of the cutoff function we may assume that $\\abs{\\operatorname{supp} s_i'} = \\abs{\\operatorname{supp} \\beta } \\subset {\\mathcal U}$.\n\t\n\tBy construction, the $\\text{sc}^+$-section $s_i'$ induces a well-defined restriction $s_i|_{\\hat{\\iota}_{\\mathcal Z}^{-1} (U_\\alpha)}$. Locally this restriction is given by multiplying the $\\text{sc}$-smooth cutoff $\\beta \\circ \\hat{\\iota}_{\\mathcal Z}$ and the locally constant vector $v_i$, hence it is $\\text{sc}^+$.\n\\end{proof}\n\nHaving shown the previous proposition, we may immediately apply Theorem~\\ref{thm:naturality} to see that \nthe polyfold Gromov--Witten invariants do not depend on the choice of an increasing sequence $(\\delta_i)_{i\\geq 0} \\allowbreak \\subset (0,2\\pi)$.\nThis proves Corollary~\\ref{cor:naturality-polyfold-gw-invariants}.\n\n\\subsection[Gromov-Witten invariants are independent of punctures at marked points]{Gromov--Witten invariants are independent of punctures at marked points}\n\\label{subsec:independence-punctures}\n\nWe now recall the regularity estimates that the stable curves of the Gromov--Witten polyfolds as constructed in \\cite{HWZGW} are required to satisfy.\nLet $u: \\Sigma \\to Q$ be a continuous map, and fix a point $z\\in \\Sigma$.\nWe consider a local expression for $u$ as follows. Choose a small disk-like neighborhood $D_z\\subset \\Sigma$ of $z$ such that there exists a biholomorphism $ \\sigma:[0,\\infty)\\times S^1\\rightarrow D_z \\setminus\\{z\\}$.\nLet $\\varphi:U\\rightarrow \\mathbb{R}^{2n}$ be a smooth chart on a neighborhood $U \\subset Q$ of $u(z)$ such that to $\\varphi (u(z))=0$.\nThe local expression \n\\[\n\\tilde{u}: [s_0,\\infty)\\times S^1 \\to \\mathbb{R}^{2n}, \\qquad (s,t) \\mapsto \\varphi\\circ u\\circ \\sigma(s,t)\n\\]\nis defined for $s_0$ large.\nLet $m\\geq 3$ be an integer, and let $\\delta >0$. We say that $u$ is of \\textbf{class $H^{m,\\delta}$ around the point $z\\in \\Sigma$}\nif $e^{\\delta s}  \\tilde{u}$ belongs to the space $L^2([s_0,\\infty)\\times S^1,\\mathbb{R}^{2n})$.\nWe say that $u$ is of \\textbf{class} $H^m_{\\text{loc}}$ \\textbf{around the point $z\\in \\Sigma$} if $u$ belongs to the space $H^m_{\\text{loc}}(D_z)$.\nIf $u$ is of class $H^{m,\\delta}$ at a point $z\\in \\Sigma$ we will refer to that point as a \\textbf{puncture}.\n\nBy definition, any stable map representative $(\\Sigma,j,M,D,u)$ of a stable curve in the Gromov--Witten polyfold ${\\mathcal Z}_{A,g,k}$ is required to be of class $H^{3,\\text{deg}_0}$ at all nodal points.\nThis is required in order to carry out the gluing construction at the nodes of \\cite[\\S~2.4]{HWZGW}.\n\nHowever, in some situations we would like to treat the marked points in the same way as the nodal points.\nNote that allowing a puncture with exponential decay at a specified marked point is a global condition on a Gromov--Witten polyfold.\nHence, we will need to require that the map $u$ is of class $H^{3,\\delta_0}$ at a fixed subset of the marked points (in addition to the nodal points).\n\nGiven a subset $I \\subset \\{1,\\ldots,k\\}$ we can define a GW-polyfold ${\\mathcal Z}^I_{A,g,k}$ where we require that all stable map representatives are of class $H^{3,\\delta_0}$ at the marked points $z_i$ for all $i\\in I$ and of class $H^3_\\text{loc}$ at the remaining marked points.\nGiven another subset $J \\subset\\{1,\\ldots,k\\}$ we can define a GW-polyfold ${\\mathcal Z}^J_{A,g,k}$ in the same manner.\nOn the other hand, we can consider a GW-polyfold ${\\mathcal Y}_{A,g,k}$ where we require that all stable map representatives are of class $H^{3,\\delta_0}$ and also of class $H^3_\\text{loc}$ at all the marked points.\nSuch a GW-polyfold with strict regularity at all marked points gives a refinement of the GW-polyfolds with different choices of punctures $I,J\\subset \\{1,\\ldots,k\\}$ at the marked points, in the sense that there are inclusion maps\n\\[\n{\\mathcal Z}^I_{A,g,k} \\hookleftarrow {\\mathcal Y}_{A,g,k} \\hookrightarrow {\\mathcal Z}^J_{A,g,k}.\n\\]\nIt is sufficient to consider inclusion maps of the form ${\\mathcal Y}_{A,g,k} \\hookrightarrow {\\mathcal Z}^I_{A,g,k}$ and demonstrate that the associated GW-invariants are equal.\n\nTo this end, consider the commutative diagram:\n\\[\\begin{tikzcd}\n{\\mathcal V}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal W}\"', hook] \\arrow[d, \"\\overline{\\partial}_J\\quad \"'] & {\\mathcal W}^I_{A,g,k} \\arrow[d, \"\\quad \\overline{\\partial}_J\"] &  \\\\\n{\\mathcal Y}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[u, bend left] & {\\mathcal Z}^I_{A,g,k} \\arrow[u, bend right] & \n\\end{tikzcd}\\]\nand observe that it satisfies the same properties as \\eqref{eq:commutative-diagram-naturality}.\nIn addition, consider the commutative diagram:\n\\[\\begin{tikzcd}\n&  & Q^k\\times \\smash{\\overline{\\mathcal{M}}}\\vphantom{\\mathcal{M}}^{\\text{log}}_{g,k} \\\\\n{\\mathcal Y}_{A,g,k} \\arrow[r, \"\\iota_{\\mathcal Z}\"', hook] \\arrow[rru, \"ev_i \\times \\pi\"] & {\\mathcal Z}^I_{A,g,k} \\arrow[ru, \"ev_i\\times \\pi\"'] & \n\\end{tikzcd}\\]\nwhich satisfies the same properties as \\eqref{eq:gw-invariant-pair-of-polyfolds}.\n\nThere exist sequences of maps which converge in $H^3_\\text{loc}$ but do not converge in $H^{3,\\delta_0}$.  Consequently, in general the above inclusion map is not proper.  Furthermore, the pullback strong polyfold bundle is not the same as the standard strong polyfold bundle on ${\\mathcal Y}_{A,g,k}$.\n\n\\begin{proposition}\n\tThe set\t\n\t\\[\n\t{\\mathcal R} := \\{\t[\\Sigma,j,M,D,u,\\xi] \\in {\\mathcal W}^I_{A,g,k} \\mid \\operatorname{supp} \\xi \\subset K \\subset \\Sigma\\setminus M \\text{ for some compact } K\t\\}\n\t\\]\n\tis an intermediary subbundle of the strong polyfold bundle ${\\mathcal W}^I_{A,g,k}$.\n\\end{proposition}\n\\begin{proof}\n\tThe proof is identical to the proof of Proposition~\\ref{prop:existence-subbundle-naturality}, except here we use Corollary~\\ref{cor:vectors-which-span-cokernel} to choose vectors $v_i$ which vanish on disk-like regions of the marked points instead of the nodal points.\n\\end{proof}\n\nAgain, combining the previous proposition and Theorem~\\ref{thm:naturality} we see that the polyfold Gromov--Witten invariants do not depend on the choice of puncture at the marked points.\nThis proves Corollary~\\ref{cor:punctures-equal}.\n\n\n\\section{Pulling back abstract perturbations}\n\t\\label{sec:pulling-back-abstract-perturbations}\n\nIn this section we show how to construct a regular perturbation which pulls back to a regular perturbation, culminating in Theorem~\\ref{thm:compatible-pullbacks} and in Corollary~\\ref{cor:pullback-via-permutation}.\n\n\\subsection{Pullbacks of strong polyfold bundles}\n\t\\label{subsec:pullbacks-strong-polyfold-bundles}\n\nLet $P: {\\mathcal W} \\to {\\mathcal Z}$ be a strong polyfold bundle, and let $f: {\\mathcal Y} \\to {\\mathcal Z}$ be a $\\text{sc}$-smooth map between polyfolds.\nConsider the topological pullback\n\t\\[f^* {\\mathcal W} = \\{([x],[w]_{[y]})\t\\mid f([x])=[y]=P([w]_{[y]})\t\\} \\subset {\\mathcal Y} \\times {\\mathcal W}\\]\nequipped with the subspace topology. Since ${\\mathcal Y}$ and ${\\mathcal W}$ are second countable, paracompact, Hausdorff topological spaces, so too is the product ${\\mathcal Y} \\times {\\mathcal W}$ and hence $f^*{\\mathcal W}$ is also a second countable, paracompact, Hausdorff topological spaces.\n\nWe can take the pullback $\\hat{f}^* W$ of the object strong M-polyfold bundle; by Proposition~\\ref{prop:pullback-bundle} this has the structure of a strong M-polyfold bundle over the object space $Y$.\nThe fiber product \n\t\\[{\\bar\\m}{Y} _s \\times_{\\text{proj}_1} \\hat{f}^* W = \\{\t(\\phi,y,w_x) \\mid s(\\phi) = y, \\hat{f}(y)=x\t\\}\\]\nmay be viewed as the strong M-polyfold bundle via the source map $s$ over the morphism space ${\\bar\\m}{Y}$,\n\t\\[\\begin{tikzcd}\n\t{\\bar\\m}{Y} _s \\times_{\\text{proj}_1} \\hat{f}^* W \\arrow[d] \\arrow[r] & \\hat{f}^* W \\arrow[d] &  \\\\\n\t{\\bar\\m}{Y} \\arrow[r, \"s\"'] & Y & \n\t\\end{tikzcd}\\]\n\nWe may define a \\textbf{pullback strong polyfold bundle structure} over the polyfold structure $(Y,{\\bar\\m}{Y})$ as the strong M-polyfold bundle $\\text{proj}_1 : \\hat{f}^*W \\to Y$ together with the bundle map defined as follows:\n\t\\begin{align*}\n\t\\lambda: {\\bar\\m}{Y} _s \\times_{\\text{proj}_1} \\hat{f}^* W & \\to \\hat{f}^*W \\\\\n\t\t\t\t\t\t\t(\\phi,y,w_x)\t& \\mapsto (t(\\phi), \\mu(f(\\phi), w_x))\n\t\\end{align*}\nIt is straightforward to check that this map satisfies the requirements of Definition~\\ref{def:strong-polyfold-bundle}.\n\nGiven a $\\text{sc}$-smooth section $\\overline{\\partial}:{\\mathcal Z}\\to {\\mathcal W}$ there exists a well-defined \\textbf{pullback section} $f^*\\overline{\\partial}:{\\mathcal Y} \\to f^*{\\mathcal W}$.  Between the underlying sets, it is defined by\n\t\\[\n\tf^*\\overline{\\partial} ([x]) = ([x], \\overline{\\partial} \\circ f ([x]) ).\n\t\\]\nIt is automatically regularizing if $\\overline{\\partial}$ is regularizing.\nWe may define the pullback of a $\\text{sc}^+$-multisection as follows.\n\n\\begin{definition}\n\t\\label{def:pullback-multisection}\n\tGiven a $\\text{sc}^+$-multisection $\\Lambda:{\\mathcal W}\\to \\mathbb{Q}^+$ there exists a well-defined \\textbf{pullback $\\text{sc}^+$-multisection} $\\text{proj}_2^*\\Lambda :f^*{\\mathcal W} \\to \\mathbb{Q}^+$.\n\tIt consists of the following:\n\t\\begin{enumerate}\n\t\t\\item the function $\\Lambda \\circ \\text{proj}_2 :f^*{\\mathcal W}\\to\\mathbb{Q}^+$\n\t\t\\item the functor $\\hat{\\Lambda} \\circ \\text{proj}_2 :\\hat{f}^* W\\to \\mathbb{Q}^+$\n\t\t\\item at each $[x]\\in {\\mathcal Z}_1$ there exists a ``pullback local section structure'' for $\\text{proj}_2^*\\Lambda$, defined below.\n\t\\end{enumerate}\n\\end{definition}\n\nGiven $[x]\\in{\\mathcal Z}_1$, the local section structure for $\\text{proj}_2^*\\Lambda$ at $[x]$ is described as follows.\nLet $x\\in Z_1$ be a representative of $[x]$.\nLet $y:=\\hat{f}(x)\\in Z_2$, and let $U_y\\subset Z_2$ be a ${\\bar\\m}{G}(y)$-invariant open neighborhood of $y$, and let \n$s_1,\\ldots,s_k : U_y \\to W$ be a local section structure for $\\hat{\\Lambda}$ at $y$ with associated weights $\\sigma_1,\\ldots ,\\sigma_k$.\nLet $U_x\\subset Z_1$ be a ${\\bar\\m}{G}(x)$-invariant open neighborhood of $x$ such that $\\hat{f}(U_x)\\subset U_y$, and consider the restricted strong M-polyfold bundle $(\\hat{f}^*W)|_{U_x} \\to U_x$.\nThen the pullback of the local sections $\\hat{f}^*s_1,\\ldots,\\hat{f}^*s_k :U_x \\to \\hat{f}^* W$ with the associated weights $\\sigma_1,\\ldots ,\\sigma_k$ gives the local section structure for $\\text{proj}_2^*\\Lambda$ at $[x]$.\n\nIndeed, it tautologically follows from the original assumption that $s_1,\\ldots,s_k$, $\\sigma_1,\\ldots,\\sigma_k$ is a local section structure for $\\Lambda$ at $[y]$ that\n\\begin{enumerate}\n\t\\item $\\sum_{i=1}^k \\sigma_i =1$ \n\t\\item the local expression $\\text{proj}_2^* \\hat{\\Lambda}:\\hat{f}^*W|_{U_x}\\to \\mathbb{Q}^+$ is related to the local sections and weights via the equation \n\t\\[\n\t\\text{proj}_2^* \\hat{\\Lambda} (x',w_{y'}) = \\sum_{\\{i\\in \\{1,\\ldots, k \\} \\mid (x',w_{y'}) = \\hat{f}^*s_i (\\text{proj}_1 (x',w_{y'} ) )\\}} \\sigma_i\n\t\\]\n\tfor all $(x',w_{y'}) \\in (\\hat{f}^* W)|_{U_x}$ (which necessarily satisfy $\\hat{f}(x')= y' = P(w_{y'})$).\n\\end{enumerate}\n\n\\subsection{The topological pullback condition and pullbacks of pairs which control compactness}\n\t\\label{subsec:topological-pullback-condition-controlling-compactness}\n\nSuppose at the outset that we have a $\\text{sc}$-smooth map $f: {\\mathcal Y} \\to {\\mathcal Z}$ and a strong polyfold bundle $P:{\\mathcal W} \\to {\\mathcal Z}$ with a $\\text{sc}$-smooth proper Fredholm section $\\overline{\\partial}$.\nConsider the pullback of this bundle and of this section via the map $f$ yielding the following commutative diagram:\n\t\\[\\begin{tikzcd}\n\tf^* {\\mathcal W} \\arrow[d, \"f^* \\overline{\\partial}\\quad\"'] \\arrow[r, \"\\text{proj}_2\"'] & {\\mathcal W} \\arrow[d, \"\\quad \\overline{\\partial}\"] &  \\\\\n\t{\\mathcal Y} \\arrow[r, \"f\"'] \\arrow[u, bend left] & {\\mathcal Z}. \\arrow[u, bend right] & \n\t\\end{tikzcd}\\]\nAssume that the $\\text{sc}$-smooth section $f^*\\overline{\\partial}$ is a proper Fredholm section; such an assumption is not automatic from the above setup, however it is natural in the context of polyfold maps one might encounter.\\footnote{An alternative to outright assuming that $f^*\\overline{\\partial}$ is a Fredholm section would be to formulate a precise notion of a ``Fredholm map'' for a map between polyfolds, and then require that $f$ is such a map. This would also be natural in the context of polyfold maps one might encounter.}\n\nGiven a pair $(N,{\\mathcal U})$ which controls the compactness of $\\overline{\\partial}$ we show in this subsection how to obtain a pullback of this pair, which will control the compactness of $f^*\\overline{\\partial}$.\n\n\\begin{proposition}\n\tLet $N :{\\mathcal W}[1] \\to [0,\\infty)$ be an auxiliary norm. The pullback of $N$, given by\n\t\\[\n\t\\text{proj}_2^* N : f^* {\\mathcal W}[1] \\to [0,\\infty)\n\t\\]\n\tdefines an auxiliary norm on the pullback strong polyfold bundle $f^* {\\mathcal W}\\to {\\mathcal Y}$.\n\\end{proposition}\n\\begin{proof}\n\tThis is immediate from the definitions.  In particular, property \\ref{property-2-auxiliary-norm} of Definition~\\ref{def:auxiliary-norm} can be checked as follows.  Let $(x_k, w_k)$ be a sequence in $\\hat{f}^* W[1]$, such that $x_k$ converges to $x$ in $Y$, and suppose $\\text{proj}_2^* \\hat{N} (x_k, w_k) \\to 0$.  Then $w_k$ is a sequence in $W[1]$ such that $\\hat{f}(x_k)$ converges to $\\hat{f}(x)$ in $Z$, and $\\hat{N} (w_k) = \\text{proj}_2^* \\hat{N} (x_k, w_k) \\to 0$, and hence $w_k\\to 0_{\\hat{f}(x)}$.  Thus $(x_k,w_k)\\to (x,0_{\\hat{f}(x)})$, as required.\n\\end{proof}\n\n\\begin{definition}\\label{topological-pullback-condition}\n\tWe say that $f$ satisfies the \\textbf{topological pullback condition} if for all $[y] \\in {\\mathcal S}(\\overline{\\partial})\\subset {\\mathcal Z}$ and for any open neighborhood ${\\mathcal V} \\subset {\\mathcal Y}$ of the fiber $f^{-1} ([y])$ there exists an open neighborhood ${\\mathcal U}_{[y]}\\subset {\\mathcal Z}$ of $[y]$ such that $f^{-1} ({\\mathcal U}_{[y]}) \\subset {\\mathcal V}$.\n\tNote that if $f^{-1} ([y])=\\emptyset$, this implies that there exists an open neighborhood ${\\mathcal U}_{[y]}$ of $[y]$ such that $f^{-1} ({\\mathcal U}_{[y]})=\\emptyset$.\n\\end{definition}\n\n\\begin{proposition}\n\t\\label{prop:simultaneous-compactness}\n\tSuppose that the map $f:{\\mathcal Y} \\to {\\mathcal Z}$ satisfies the topological pullback condition.  \n\tThen there exists a pair $(N,{\\mathcal U})$ which controls the compactness of $\\overline{\\partial}$ such that the pair $(\\text{proj}_2^* N, f^{-1} ({\\mathcal U}) )$ controls the compactness of $f^* \\overline{\\partial}$.\n\t\n\tFurthermore, if a $\\text{sc}^+$-multisection $\\Lambda: {\\mathcal W}\\to \\mathbb{Q}^+$ satisfies $N [\\Lambda ] \\leq 1$ and $\\text{dom-supp} (\\Lambda) \\subset {\\mathcal U}$, then its pullback $\\text{proj}_2^* \\Lambda :f^*{\\mathcal W} \\to \\mathbb{Q}^+$ satisfies $\\text{proj}_2^* N [\\text{proj}_2^* \\Lambda ] \\leq 1$ and $\\text{dom-supp} (\\text{proj}_2^* \\Lambda) \\allowbreak \\subset \\allowbreak f^{-1} ({\\mathcal U})$.\n\\end{proposition}\n\\begin{proof}\n\tLet $(N, {\\mathcal V})$ be a pair which controls the compactness of $\\overline{\\partial}$.\n\tBy the previous proposition we know that the pullback $\\text{proj}_2^* N :f^* {\\mathcal W}[1]\\to [0,\\infty)$ is an auxiliary norm.\n\tWe may then apply \\cite[Prop.~4.5]{HWZ3} to assert the existence of a neighborhood ${\\mathcal U}'\\subset {\\mathcal Y}$ of ${\\mathcal S}(f^*\\overline{\\partial})$ such that the pair $(\\text{proj}_2^* N, {\\mathcal U}')$ controls the compactness of $f^*\\overline{\\partial}$.\n\t\n\tAt every $[y] \\in {\\mathcal S}(\\overline{\\partial})$, observe that $f^{-1} ([y]) \\subset {\\mathcal S}(f^*\\overline{\\partial}) \\subset {\\mathcal U}'$.  We can use the topological pullback condition to choose a neighborhood ${\\mathcal U}_{[y]}$ of $[y]$ such that $f^{-1} ({\\mathcal U}_{[y]}) \\subset {\\mathcal U}'\\subset {\\mathcal Y}$ and moreover such that ${\\mathcal U}_{[y]} \\subset {\\mathcal V}\\subset {\\mathcal Z}$. \n\tWe define an open neighborhood by ${\\mathcal U} := \\cup_i \\ {\\mathcal U}_{[y]_i}$ for every $[y]_i\\in {\\mathcal S}(\\overline{\\partial})$.\n\t\n\tThen $(N,{\\mathcal U})$ is the desired pair.\n\tIndeed, ${\\mathcal U}$ is an open neighborhood of the unperturbed solution set ${\\mathcal S}(\\overline{\\partial})$.\n\tAnd ${\\mathcal U}\\subset {\\mathcal V}$ since for every $[y]_i$ we have ${\\mathcal U}_{[y]_i}\\subset {\\mathcal V}$.  Hence we have ${\\mathcal S}(\\overline{\\partial}) \\subset {\\mathcal U}\\subset {\\mathcal V}$ therefore it follows from Remark~\\ref{rmk:shrink-neighborhood} that $(N,{\\mathcal U})$ controls the compactness of $\\overline{\\partial}$.\n\t\n\tObserve that ${\\mathcal S}(f^*\\overline{\\partial}) = f^{-1} ({\\mathcal S}(\\overline{\\partial})) \\subset f^{-1}({\\mathcal U})$.  By the construction of ${\\mathcal U}$ we have $f^{-1} ({\\mathcal U}) \\subset {\\mathcal U}'$.  Hence we have ${\\mathcal S}(f^*\\overline{\\partial}) \\subset f^{-1}({\\mathcal U}) \\subset {\\mathcal U}'$ therefore it follows from Remark~\\ref{rmk:shrink-neighborhood} that $(\\text{proj}_2^* N, f^{-1} ({\\mathcal U}))$ controls the compactness of $f^* \\overline{\\partial}$.\n\t\n\tFinally, the claim regarding the pullback of a $\\text{sc}^+$-multisection is immediate from the construction.\n\\end{proof}\n\n\\subsection{Construction of regular perturbations which pullback to regular perturbations}\n\t\\label{subsec:construction-regular-perturbation-which-pullback}\n\nWith the same assumptions and setup as in the previous subsection (i.e., our map satisfies the topological pullback condition), we show in this subsection how to construct regular perturbations which will pullback to regular perturbations\n\n\\begin{theorem}\n\t\\label{thm:compatible-pullbacks}\n\tWe can construct a regular perturbation $\\Lambda: {\\mathcal W} \\to \\mathbb{Q}^+$ which pulls back to a regular perturbation $\\text{proj}_2^* \\Lambda$. This means that the perturbations satisfy the following conditions.\n\t\\begin{enumerate}\n\t\t\\item $(\\overline{\\partial}, \\Lambda)$ and $(f^* \\overline{\\partial}, \\text{proj}_2^* \\Lambda)$ are both transversal pairs.\n\t\t\\item There exists a pair $(N,{\\mathcal U})$ which controls the compactness of $\\overline{\\partial}$ such that the pair $(\\text{proj}_2^* N, f^{-1} ({\\mathcal U}) )$ controls the compactness of $f^* \\overline{\\partial}$. Then:\n\t\t\\begin{itemize}\n\t\t\t\\item $N [\\Lambda ] \\leq 1$ and $\\text{dom-supp} (\\Lambda) \\subset {\\mathcal U}$\n\t\t\t\\item $\\text{proj}_2^* N [\\text{proj}_2^* \\Lambda ] \\leq 1$ and $\\text{dom-supp} (\\text{proj}_2^* \\Lambda) \\subset f^{-1} ({\\mathcal U})$.\n\t\t\\end{itemize}\n\t\\end{enumerate}\n\n\\end{theorem}\n\\begin{proof}\n\nWe now give an explicit construction of a $\\text{sc}^+$-multisection $\\Lambda : {\\mathcal W} \\to \\mathbb{Q}^+$ such that $(\\overline{\\partial}, \\Lambda)$ and $(f^* \\overline{\\partial}, \\text{proj}_2^* \\Lambda)$ are both transversal pairs.  Our approach is based on the general position argument of \\cite[Thm.~15.4]{HWZbook}.\n\n\\noindent\\emph{Local construction.}\nWe construct a $\\text{sc}^+$-multisection $\\Lambda_0 : {\\mathcal W} \\to \\mathbb{Q}^+$ which will be transversal at a point $[y_0] \\in {\\mathcal S}(\\overline{\\partial}) \\subset {\\mathcal Z}$.\nLet $U(y_0) \\subset Z$ be a ${\\bar\\m}{G}(y_0)$-invariant open neighborhood of $y_0$ and moreover let $V(y_0)\\subset U(y_0)$ be a ${\\bar\\m}{G}(y_0)$-invariant open neighborhood of $y_0$ such that $\\overline{V(y_0)} \\subset U(y_0)$.\n\nChoose smooth vectors $v_1,\\ldots, v_k \\in W_{y_0}$ such that\n\t\\[\n\t\\text{span} \\{v_1,\\ldots,v_k \\} \\oplus D\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_{y_0} (T_{y_0} Z) = W_{y_0}.\n\t\\]\n\nFor each smooth vector $v_1,\\ldots, v_k$ we can use \\cite[Lem.~5.3]{HWZbook} to define $\\text{sc}^+$-sections $s_i : U(y_0) \\to W$ such that\n\\begin{itemize}\n\t\\item $s_i= 0$ on $U(y_0)\\setminus V(y_0)$,\n\t\\item $s_i(y_0) = v_i$.\n\\end{itemize}\nFurthermore, to ensure that the resulting multisection will be controlled by the pair $(N,{\\mathcal U})$ we require that\n\\begin{itemize}\n\t\\item $N[s_i] \\leq 1,$\n\t\\item $\\text{supp}(s_i)\\subset {\\mathcal U}$.\n\\end{itemize}\n\nWe may use these locally constructed $\\text{sc}^+$-sections to define a $\\text{sc}^+$-multisection functor \n\t\\[\\hat{\\Lambda}'_0:W \\times B_\\varepsilon^k \\to \\mathbb{Q}^+\\]\nwith local section structure given by \n$\\left\\lbrace g * \\left(\\sum_{i=1}^k t_i \\cdot s_i\\right)\\right\\rbrace_{g\\in {\\bar\\m}{G}(y_0)}$\nwhich satisfies the following.\nThere exists a ${\\bar\\m}{G}(y_0)$-invariant open neighborhood $y_0 \\subset U'_0 \\subset Z$ such that at any object $y\\in U'_0$ and for any $g\\in {\\bar\\m}{G}(y_0)$ the linearization of the function \n\t\\begin{align*}\n\tU'_0 \\times B_\\varepsilon^k \t&\\to W\\\\\n\t(y, t_1,\\ldots,t_k)\t\t&\\mapsto \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}(y) - g * \\left(\\sum_{i=1}^k t_i \\cdot s_i(y)\\right)\n\t\\end{align*}\nprojected to the fiber $W_y$ is surjective.\n\nWe now construct a $\\text{sc}^+$-multisection whose pullback $\\text{proj}_2^* \\Lambda_0 : f^*{\\mathcal W} \\to \\mathbb{Q}^+$ will be transversal at a point $[x_0] \\in {\\mathcal S}(f^*\\overline{\\partial}) \\subset {\\mathcal Y}$.\nConsider a point $[x_0]\\in {\\mathcal S}(f^*\\overline{\\partial}) \\subset {\\mathcal Y}$ which maps to $[y_0]:=f([x_0])\\in {\\mathcal S}(\\overline{\\partial})\\subset {\\mathcal Z}$.\nLet $x_0\\in S(\\hat{f}^*\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$})$ be a representative of $[x_0]$, and let $U(x_0) \\subset Y$ be a ${\\bar\\m}{G}(x_0)$-invariant open neighborhood of $x_0$.\nThen $y_0:= \\hat{f}(x_0) \\in S(\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$})$ is a representative of $[y_0]$.\nLet $U(y_0) \\subset Z$ be a ${\\bar\\m}{G}(y_0)$-invariant open neighborhood of $y_0$ and moreover let $V(y_0)\\subset U(y_0)$ be a ${\\bar\\m}{G}(y_0)$-invariant open neighborhood of $y_0$ such that $\\overline{V(y_0)} \\subset U(y_0)$.\nBy shrinking the open set $U(x_0)$, we may assume that the local expression $\\hat{f}: U(x_0)\\to U(y_0)$ is well-defined.\n\nThe fibers $(\\hat{f}^*W)_{x_0}$ and $W_{y_0}$ may be identified via $\\text{proj}_2$.\nChoose smooth vectors $v_1,\\ldots, v_k \\in W_{y_0}$ such that\n\t\\[\\text{span} \\{\\text{proj}_2^{-1} (v_1),\\ldots,\\text{proj}_2^{-1} (v_k) \\} \\oplus D\\hat{f}^*\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$}_{x_0} (T_{x_0} Y) = (\\hat{f}^*W)_{x_0}.\\]\n\nFor each smooth vector $v_1,\\ldots, v_k$ we may use \\cite[Lem.~5.3]{HWZbook} to define $\\text{sc}^+$-sections $s_i : U(y_0) \\to W$ such that\n\\begin{itemize}\n\t\\item $s_i\\equiv 0$ on $U(y_0)\\setminus V(y_0)$,\n\t\\item $s_i(y_0) = v_i$.\n\\end{itemize}\nFurthermore, to ensure that the resulting multisection will be controlled by the pair $(N,{\\mathcal U})$ we require that\n\\begin{itemize}\n\t\\item $N[s_i] \\leq 1,$\n\t\\item $\\text{supp}(s_i)\\subset {\\mathcal U}$.\n\\end{itemize}\n\nWe may use these locally defined $\\text{sc}^+$-sections to define a $\\text{sc}^+$-multisection functor $\\hat{\\Lambda}_0:\tW \\times B_\\varepsilon^k \\to \\mathbb{Q}^+$ \nwith local section structure given as follows. \n\t\\[\\left\\lbrace g * \\left(\\sum_{i=1}^k t_i \\cdot s_i\\right)\\right\\rbrace_{g\\in {\\bar\\m}{G}(y_0)}\\]\nBy construction, the pullback $\\text{sc}^+$-multisection functor\n$\\text{proj}_2^* \\hat{\\Lambda}_0:\t\\hat{f}^* W \\times B_\\varepsilon^k \\to \\mathbb{Q}^+$ \nhas local section structure given as follows. \n\t\\[\\left\\lbrace \\hat{f}^* \\left(g * \\left(\\sum_{i=1}^k t_i \\cdot s_i\\right)\\right) \\right\\rbrace_{g\\in {\\bar\\m}{G}(y_0)}\\]\nIt may be observed that there exists a ${\\bar\\m}{G}(x_0)$-invariant open neighborhood $x_0 \\subset U_0\\subset Y$ such that at any object $x\\in U_0$ and for any $g\\in {\\bar\\m}{G}(y_0)$ the linearization of the function \n\t\\begin{align*}\n\tU_0 \\times B_\\varepsilon^k \t&\\to \\hat{f}^*W\\\\\n\t(x, t_1,\\ldots,t_k)\t&\\mapsto \\hat{f}^* \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} (x) - \\hat{f}^* \\left( g * \\left(\\sum_{i=1}^k t_i \\cdot s_i\\right)\\right) (x) \\\\\n\t\t\t\t\t\t&\\phantom{\\mapsto} = \\left(x, \\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} (\\hat{f}(x)) - g * \\left(\\sum_{i=1}^k t_i \\cdot s_i(\\hat{f}(x)) \\right) \\right)\n\t\\end{align*}\nprojected to the fiber $(\\hat{f}^* W)_x$ is surjective.\n\n\\noindent\\emph{Global construction.}\nWe may cover the compact topological space ${\\mathcal S}(\\overline{\\partial})$ by a finite collection of such neighborhoods $\\abs{U'_i}$ of points $[y_i]\\in {\\mathcal S}(\\overline{\\partial})$; we may also cover ${\\mathcal S}(f^*\\overline{\\partial})$ by a finite collection of such neighborhoods $\\abs{U_i}$ of points $[x_i]\\in {\\mathcal S}(f^*\\overline{\\partial})$.\nIt follows that the finite sum of $\\text{sc}^+$-multisections\n\t\\[\n\t\\Lambda:= \\bigoplus_i \\Lambda_i : {\\mathcal W} \\times B_\\varepsilon^N \\to \\mathbb{Q}^+\n\t\\]\nhas the property that: for any point $[y] \\in {\\mathcal Z}$ with $\\Lambda \\circ \\overline{\\partial} ([y])>0$, and for any parametrized local section structure $\\{s_i\\}_{i\\in I}$ at a representative $y$, the linearization of the function $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} (y) - s_i(y,t)$ projected to the fiber $W_y$ is surjective.\n\nLikewise, the pullback $\\text{sc}^+$-multisection $\\text{proj}_2^* \\Lambda : f^*{\\mathcal W} \\times B_\\varepsilon^N \\to \\mathbb{Q}^+$\nhas the property that for any point $[x] \\in {\\mathcal Y}$ which satisfies $\\text{proj}_2^* \\Lambda \\circ f^*\\overline{\\partial} ([x])>0$ and for any parametrized local section structure $\\{s_i\\}_{i\\in I}$ at a representative $x$, the linearization of the function $\\stackon[3pt]{$\\overline{\\partial}$}{$\\hat{}$} (x) - s_i(x,t)$ projected to the fiber $(\\hat{f}^*W)_x$ is surjective. \n\nFurthermore for $\\varepsilon$ sufficiently small, for any fixed $t_0 \\in B_\\varepsilon^N$ the $\\text{sc}^+$-multisection $\\Lambda (\\cdot, t_0)$ is controlled by the pair $(N,{\\mathcal U})$, i.e.,\n\\begin{itemize}\n\t\\item $N[\\Lambda(\\cdot,t_0)] \\leq 1$,\n\t\\item $\\text{dom-supp}(\\Lambda(\\cdot,t_0)) \\subset {\\mathcal U}$.\n\\end{itemize}\nIt follows from Proposition~\\ref{prop:simultaneous-compactness} that the pullback $\\text{sc}^+$-multisection $\\text{proj}_2^* \\Lambda (\\cdot, t_0)$ is controlled by the pullback $(\\text{proj}_2^*N,f^{-1}({\\mathcal U}))$.\n\n\\noindent\\emph{Common regular value.}\nConsider the strong polyfold bundle ${\\mathcal W} \\times B_\\varepsilon^N \\to {\\mathcal Z} \\times B_\\varepsilon^N$ and the pullback strong polyfold bundle $f^*{\\mathcal W} \\times B_\\varepsilon^N \\to {\\mathcal Y} \\times B_\\varepsilon^N$.\nLet $\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}} : {\\mathcal Z} \\times B_\\varepsilon^N \\to {\\mathcal W} \\times B_\\varepsilon^N$ denote the $\\text{sc}$-smooth proper Fredholm section defined by $([y],t)\\mapsto (\\overline{\\partial}([y]),t)$, and let $\\tilde{f}^*\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}} :{\\mathcal Y} \\times B_\\varepsilon^N \\to f^*{\\mathcal W}\\times B_\\varepsilon^N$ denote the $\\text{sc}$-smooth proper Fredholm section defined by $([x],t) \\mapsto ([x],\\overline{\\partial}(f([x])),t)$.\nBy construction, $(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\Lambda)$ and $(\\tilde{f}^*\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\text{proj}_2^* \\Lambda)$ are transversal pairs; hence by Theorem~\\ref{thm:transversal-pairs-weighted-branched-suborbifolds} the thickened solution sets\n\t\\begin{align*}\n\t{\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\Lambda) &= \\{ ([y],t) \\in {\\mathcal Z} \\times B_\\varepsilon^N \\mid \\Lambda (\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}} ([y],t)) >0 \\} \\subset {\\mathcal Z} \\times B_\\varepsilon^N,\t\\\\\n\t{\\mathcal S}(\\tilde{f}^*\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\text{proj}_2^* \\Lambda) &= \\{ ([x],t) \\in {\\mathcal Y} \\times B_\\varepsilon^N \\mid \\text{proj}_2^*\\Lambda (\\tilde{f}^*\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}} ([x],t)) >0 \\} \\subset {\\mathcal Y} \\times B_\\varepsilon^N\n\t\\end{align*}\nhave the structure of weighted branched orbifolds.\n\nBy Sard's theorem, we can find a common regular value $t_0 \\in B_\\varepsilon^N$ of the projections ${\\mathcal S}(\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\Lambda) \\to B_\\varepsilon^N$ and ${\\mathcal S}(\\tilde{f}^*\\scalerel*{\\tilde{\\overline{\\partial}}}{\\hat{M}}, \\text{proj}_2^* \\Lambda)\\to B_\\varepsilon^N$.\nThen the $\\text{sc}^+$-multisection $\\Lambda(\\cdot,t_0) : {\\mathcal W} \\to \\mathbb{Q}^+$ and its pullback $\\text{proj}_2^*\\Lambda(\\cdot,t_0) :f^*{\\mathcal W} \\to \\mathbb{Q}^+$ are the desired regular perturbations.\n\\end{proof}\n\nThe significance of this theorem is the following.\nBoth perturbed solution sets ${\\mathcal S}(f^*\\overline{\\partial}, \\text{proj}_2^* \\Lambda )$ and ${\\mathcal S}(\\overline{\\partial},\\Lambda)$ have the structure of compact oriented weighted branched suborbifolds.  \nMoreover, the restriction of $f$ gives a well-defined continuous function between these perturbed solution spaces, i.e.,\n\t\\[\n\tf|_{{\\mathcal S}(f^*\\overline{\\partial}, \\text{proj}_2^* \\Lambda )}: {\\mathcal S}(f^*\\overline{\\partial}, \\text{proj}_2^* \\Lambda ) \\to {\\mathcal S}(\\overline{\\partial},\\Lambda).\n\t\\]\nFurthermore, $f$ is weight preserving in the sense that the weight functions are related via pullback by the following equation $ ( \\Lambda \\circ \\overline{\\partial}) \\circ f= \\text{proj}_2^* \\Lambda \\circ f^* \\overline{\\partial}$.\n\n\\subsection{The permutation maps between perturbed Gromov--Witten moduli spaces}\n\t\\label{subsec:permutation-map}\n{As we have explained in the introduction, naively there does not exist a permutation map for an arbitrary choice of abstract perturbation.}\n\nFix a permutation $\\sigma \\in S_k,\\  \\sigma: \\{1,\\ldots, k\\}\\to \\{1,\\ldots,k\\}$.\nAssociated to this permutation we can define a $\\text{sc}$-smooth permutation map between the Gromov--Witten polyfold\n\\[\n\\sigma: {\\mathcal Z}_{A,g,k} \\to {\\mathcal Z}_{A,g,k}, \\qquad [\\Sigma,j,M,D,u]  \\mapsto [\\Sigma,j,M^\\sigma,D,u]\n\\]\nwhere $M = \\{z_1,\\ldots,z_k\\}$ and where $M^\\sigma := \\{z'_1,\\ldots,z'_k\\}$, $z'_i:= z_{\\sigma(i)}$.\n\nConsider the pullback via $\\sigma$ of the strong bundle ${\\mathcal W}_{A,g,k} \\to {\\mathcal Z}_{A,g,k}$ and the Cauchy--Riemann section $\\overline{\\partial}_J$, as illustrated in the below commutative diagram.\n\t\\[\n\t\\begin{tikzcd}\n\t\\sigma^* {\\mathcal W}_{A,g,k} \\arrow[d, \"\\sigma^* \\overline{\\partial}_J \\quad\"'] \\arrow[r, \"\\text{proj}_2\"'] & {\\mathcal W}_{A,g,k} \\arrow[d, \"\\quad \\overline{\\partial}_J\"] &  \\\\\n\t{\\mathcal Z}_{A,g,k} \\arrow[r, \"\\sigma\"'] \\arrow[u, bend left] & {\\mathcal Z}_{A,g,k} \\arrow[u, bend right] & \n\t\\end{tikzcd}\n\t\\]\nThe map $\\sigma$ is a homeomorphism when considered on the underlying topological spaces, and hence satisfies the topological pullback condition.\nBy applying Theorem~\\ref{thm:compatible-pullbacks} we immediately obtain Corollary~\\ref{cor:pullback-via-permutation}.\n\nIt follows that the permutation map restricts to a well-defined map between the perturbed Gromov--Witten moduli spaces, \n\\[\n\\sigma|_{{\\mathcal S}_{A,g,k} (\\overline{\\partial}_J, \\text{proj}_2^* \\Lambda)} : {\\mathcal S}_{A,g,k} (\\overline{\\partial}_J, \\text{proj}_2^* \\Lambda) \\to {\\mathcal S}_{A,g,k} (\\overline{\\partial}_J,\\Lambda).\n\\]\nConsidered on the underlying topological spaces, this map is a homeomorphism.  Considered on the branched ep-subgroupoid structures, the associated functor\n\\[\n\\hat{\\sigma}|_{S_{A,g,k} (\\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}, \\text{proj}_2^* \\hat{\\Lambda})}: S_{A,g,k} (\\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}, \\text{proj}_2^* \\hat{\\Lambda}) \\to S_{A,g,k} (\\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J},\\hat{\\Lambda})\n\\]\nis a local diffeomorphism, and moreover is injective.\nThe restricted permutation map $\\sigma$ and its associated functor $\\hat{\\sigma}$ are both weight preserving, i.e.,\n$(\\Lambda\\circ \\overline{\\partial}_J) \\circ \\sigma = \\text{proj}_2^*\\Lambda \\circ \\overline{\\partial}_J$ and $(\\hat{\\Lambda}\\circ \\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}) \\circ \\hat{\\sigma} = \\text{proj}_2^* \\hat{\\Lambda} \\circ \\scalerel*{\\hat{\\overline{\\partial}}_J}{\\hat{M}_J}$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Document History}\n\\begin{itemize}\n    \\item[\\textbf{v1.0}] Initial release\n    \\item[\\textbf{v1.1}] Added discussion of conflicting report to the introduction\\\\ Added discussion of cost pitfalls to Section~\\ref{sec:results:requirements:motivation}\n    \\item[\\textbf{v1.2}] Added new data for the application type (Section~\\ref{subsec:application_type})\n\\end{itemize}\n\n\n\\newpage\n\\pagenumbering{roman}\n\\setcounter{tocdepth}{4}\n\\begin{spacing}{1.3}\n\\tableofcontents\n\\end{spacing}\n\n\n\\newpage\n\\thispagestyle{plain}\n\\section*{Executive Summary}\nThe serverless computing paradigm promises many desirable properties for cloud applications---low-cost, fine-grained deployment, and management-free operation.\nConsequently, the paradigm has underwent rapid growth: there currently exist tens of serverless platforms and all global cloud providers host serverless operations.\nTo help tune existing platforms, guide the design of new serverless approaches, and overall contribute to understanding this paradigm, in this work we present a long-term, comprehensive effort to identify, collect, and characterize serverless use cases.\nWe survey 89 use cases, sourced from white and grey literature, and from consultations with experts in areas such as scientific computing.\nWe study each use case using 24 characteristics, including general aspects, but also workload, application, and requirements. When the use cases employ workflows, we further analyze their characteristics.\nOverall, we hope our study will be useful for both academia and industry, and encourage the community to further share and communicate their use cases.\n\\\\\n\n\\noindent\\textbf{Keywords}\\\\\n\n\\noindent serverless use cases, cloud computing, serverless computing, serverless applications, workflows, requirements analysis, empirical research\\\\\n\n\\noindent\\textbf{Disclaimer}\\\\\n\n\\noindent SPEC, the SPEC logo and the names SPEC CPU2017, SERT, SPEC Cloud IaaS 2018 are trademarks of the Standard Performance Evaluation Corporation (SPEC). \nSPEC Research and SPEC RG Cloud are service marks of SPEC. Additional product and service names mentioned herein may be the trademarks of their respective owners. \nCopyright \u00a9 1988-2020 Standard Performance Evaluation Corporation (SPEC). \nAll rights reserved.\n\n\n\\mainmatter\n\\setlength{\\parindent}{0cm}\n\n\\input{sections\/Introduction.tex}\n\n\\input{sections\/Methodology.tex}\n\n\\input{sections\/Results.tex}\n\n\\input{sections\/Threats.tex}\n\n\\input{sections\/Conclusion.tex}\n\n\n\\cleardoublepage\n\\pagenumbering{gobble}\n\\pagestyle{empty}\n\\renewcommand\\bibname{References}\n\n\\bibliographystyle{IEEEtranSA}\n\n\\section{Conclusion and Future Work}\n\\label{sec:conclusion}\\label{sec:futurework}\n\nThe emergence of serverless computing has already led to a diverse design space, with tens of serverless platforms and the participation of all major cloud providers.\nWe identify in this work the need for a systematic, comprehensive study of serverless use cases, which could help the development of serverless techniques and solutions in the fields of software engineering, distributed systems, and performance engineering. \\\\\n\nWe have proposed a systematic process to identify, collect, and characterize serverless use cases.\nTo identify use cases, the process considers open-source software projects, peer-reviewed literature, self-published material, and domain knowledge.\nTo collect the use cases, the process proposes a structured repository, from which reviewers take and characterize each use case alongside 24 features of interest.\nEach use case is covered by the following types of features:\n(a) \\textit{general} characteristics, such as platform, application type and domain, whether the use case was observed in production, and whether the use case provides open-source software;\n(b) \\textit{workload} characteristics, such as the execution pattern, burstiness, types of triggers, and data volume;\n(c) \\textit{application} characteristics, such as programming language(s) used to develop it, the resource bounds, whether the application depends on external services, etc.;\n(d) the \\textit{requirements} posed by the use case, such as locality and latency, or the performance-cost trade-off; and\n(e) \\textit{workflow} characteristics, including structure, size, and internal parallelism. \\\\\n\nUsing this process, we have collected and characterized a total of 89~ serverless use cases from four different sources. Our systematic and comprehensive study reveals that:\n\\begin{enumerate}\n\n    \\item We find a dominating portion of serverless use cases already being in production with AWS as the most popular platform and web services being the most common application domain. \n\n    \\item Serverless workloads tend to exhibit on-demand execution patterns exemplified by 81\\% bursty workloads, which makes their load hard to predict.\n    \n    \\item Most cloud functions (67\\%) are short-running, with running times in the order of milliseconds or seconds, thus requiring serverless frameworks that impose small overheads when running functions.\n    \n    \\item Cost savings (both in terms of infrastructure and operation costs) are a bigger driver for the adoption of cost than the offered performance and scalability gains.\n    \n    \\item \n    We observe an increasing trend toward ever-larger, ever more complex workflows, indicating the need to move toward (cloud-native) workflow engines.\n\\end{enumerate}\n\n\nLast, but not least, we see this study as a step toward a community-wide policy of sharing and discussing about use cases. Persisting beyond our effort, such use cases could stimulate a new wave of serverless designs, facilitate meaningful tuning and benchmarking, and overall prove useful for both academia and industry.\nWe therefore extend an open invitation to prospective new collaborators in the SPEC-RG Cloud group.\n\n\n\n\\section{Introduction}\\label{sec:intro}\\label{sec:introduction}\n\nServerless computing is an emerging technology with increasing impact on our modern society, and increasing adoption by both academia and industry~\\cite{FutureScape, Markets, 10.5555\/3027041.3027047}.\nThe key promise of serverless computing is to make computing services more accessible, fine-grained, and affordable~\\cite{DBLP:journals\/internet\/EykTTVUI18,DBLP:journals\/corr\/abs-1902-03383} by managing operational concerns~\\cite{10.1145\/3368454}.\nMajor cloud providers, such as Amazon, Microsoft, Google, and IBM already offer capable serverless platforms.\nHowever, serverless computing, and its common Function-as-a-Service~(FaaS) realization, still raises many important challenges that may reduce adoption. These challenges have been recognized and discussed in fields such as software engineering, distributed systems, performance engineering~\\cite{DBLP:conf\/middleware\/EykIST17,DBLP:conf\/wosp\/EykIAGE18,DBLP:conf\/cidr\/HellersteinFGSS19}.\nThis work focuses on a first step to alleviate these challenges:\n{\\it understanding serverless applications through a variety of use cases}.\n\\newline\n\nServerless computing enables developers to focus on implementing business logic, leaving the operational concerns to cloud providers. In turn, the providers turn to automation, which they achieve through \ncapable serverless platforms, such as AWS Lambda, Azure Functions, or Google Cloud Functions, and IBM Cloud Functions (based on Apache OpenWhisk).\nServerless platforms already support fine-grained function deployment, detailed resource allocation, and to some extent also autoscaling~\\cite{10.1145\/3368454}.\nHowever, more sophisticated operational features have started to emerge such as:\n(a) complex function composition and even full {\\it workflows}, \n(b) eventing and provider-managed messaging, \n(c) low-latency scheduling, \n(d) file storage and database setup, \n(e) streaming and locality-aware deployment, and \n(f) versioning and logging solutions. \nThese features facilitate the serverless {\\it application lifecyle}, and help further decreases the time-to-market for serverless applications~\\cite{leitner2019MixedMethod}.\n\\newline\n\nResearchers and industry practitioners have an urgent need for serverless use cases.\nThe variety of already existing platforms and support from the major cloud providers indicate the presence of many serverless applications.\nHowever, relatively little is known about their characteristics or behavior.\nFor an emerging technology such as serverless computing, researchers, \nengineers, and \nplatform providers \ncould use descriptions of {\\it use cases}---{\\it which applications?}, {\\it where and how was this technology already successfully applied?}, and {\\it what are the characteristics of these use cases?}---to guide their drive for discovery and improvement {\\it in the right direction}.\n\\textit{Researchers} can study different use cases related to the same application to extract meaningful patterns and trigger new designs. They can also identify representative use cases, which can later be used for the evaluation of novel approaches and in empirical studies. \n\\textit{Engineers} require descriptions in which areas serverless computing was already successfully applied, which helps to decide whether to adopt serverless computing for other projects. Additionally, existing solutions can serve as blueprints for similar use cases. \\textit{Platform providers} require knowledge of how their products are used, to optimize them and gaps in adoption can point out deficits in their current offerings.\n\\newline\n\nThere are only a few, and sometimes conflicting, reports addressing important questions such as why developers build serverless applications, when serverless applications are well suited, or how serverless applications are implemented in practice. For example, there are reports of significant cost savings by switching to serverless applications~\\cite{adzic2017serverless, Levinson2020}, but also articles suggesting higher cost in some scenarios compared to traditional hosting~\\cite{eivy2017wary}. There are also reports of successfully serverless applications for data-intensive applications~\\cite{witte2019serverless, 10.1145\/3307339.3343462}, despite other articles claiming that serverless is not well suited for data-intensive applications~\\cite{DBLP:conf\/cidr\/HellersteinFGSS19}. Some people suggest that containers are superior to serverless for latency-critical tasks~\\cite{Thorn}, but there are also reports of people successfully applying serverless for latency-critical user-facing traffic~\\cite{Droplr}. Having concrete information on these topics would be valuable for managers to guide decisions on whether a serverless application can be a suitable solution for a specific use case.\n\\newline\n\nHowever much needed, serverless use cases have not been studied systematically so far.\nFor serverless computing, existing research has focused on serverless platforms and their performance properties~\\cite{yussupov:19}.\nSeveral studies currently exist about the features, architecture, and performance properties of these platforms~\\cite{DBLP:journals\/internet\/EykIGEBVTSHA19,10.1007\/978-3-319-99819-0_11, figiela2018performance, lee2018evaluation, lloyd2018serverless, wang2018peeking}.\nShahrad et al.~\\cite{shahrad2020serverless} characterize the aggregated performance properties of the entire production FaaS workload from Microsoft Azure Functions, but do not provide details on individual use cases.\nA recent mixed-method empirical study investigates how developers use serverless computing, focusing on the issues (pain points) they experienced~\\cite{leitner2019MixedMethod}.\nAnother multivocal literature review discusses common patterns in the architecture of serverless applications~\\cite{taibi2020serverless}.\nTo the best of our knowledge, the only existing collection of serverless use cases is an article by Castro et al.~\\cite{10.1145\/3368454}, which introduces ten use cases collected from non-peer-reviewed (\\textit{grey}) literature.\n\\newline\n\nIn this technical report, we collect a total of 89~ serverless use cases from four different sources. 32~ use cases are from open-source projects, 23~ from white literature, 28~ from grey literature, and 6~ from the area of scientific computing. Each use case is reviewed by a pair of reviewers in regard to 24 characteristics, such as execution pattern, workflow coordination, use of external services, and motivation for adopting serverless.\nThe full dataset containing all use cases and their characteristics is publicly available as a persistent Zenodo repository~\\cite{dataset}. \n\\newline\n\n\nIn the next section, we discuss our process for use-case collection and characterization. In Section~\\ref{sec:results}, we describe the 24~ characteristics we reviewed for each use case and the results of this review. Section~\\ref{sec:validity} discusses threats to validity and mitigation strategies.\nFinally, Section~\\ref{sec:conclusion} concludes this technical report, and discusses promising future research directions based on the finding of this study. \n\n\\section{Study Design}\n\\label{sec:method}\n\nThis section summarizes our overall study process, describes the data sources to identify primary studies, the selection strategy with inclusion and exclusion criteria, the characteristics review protocol, and the discussion and consolidation phase covering inter-reviewer agreement.\n\n\\subsection{Process Overview}\n\\label{sec:process}\n\n\\Cref{fig:overview} summarizes the use case analysis process.\nFirstly, we compiled an extensive list of potentially relevant use cases from four different data sources (see \\Cref{sec:process:source}), namely open source projects, white literature, grey literature, and scientific computing.\nSecondly, we applied our selection criteria (see \\Cref{subsec:selection}) to classify and filter only relevant use cases in the context of this study.\nThis resulted in 83 use cases from publicly available sources and 6~ scientific use cases from internal sources, where we had access to domain experts.\nThirdly, we defined a list of interesting characteristics including potential values and perform reviews to extract the actual values from available documentation  (see \\Cref{subsec:data_extraction}).\nFor all public sources, 2 randomly assigned researchers out of a pool of 7 available authors conducted two redundant reviews for each use case.\nEach scientific use case was reviewed by a single domain expert.\nSubsequently, we calculated the inter-reviewer agreements for all redundant reviews and resolved any conflicting values during discussion and consolidation (see \\Cref{subsec:data_synthesis}).\nThis resulted in a total of 89~ analyzed use cases.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=\\linewidth]{figures\/Schema.pdf}\n    \\caption{Process overview for use case analysis.}\n    \\label{fig:overview}\n\\end{figure}\n\n\\subsection{Data Sources}\n\\label{sec:process:identification}\\label{sec:process:source}\\label{sec:process:sourcing}\n\nReports on use cases for serverless applications appear in many different forms ranging from peer-reviewed academic papers, open-source projects, blog posts, podcasts, talks, provider-reported success stories to direct exchange with application developers. Therefore, we collect use cases from a variety of different sources. We also aim to not have a dominant source that contributes the lions share of the use cases and therefore introduces a strong selection bias. Based on this, we do not aim for an exhaustive collection of use cases, but collect use cases from the following different sources with the goal of obtaining a large varied sample:\n\n\\begin{itemize}\n    \\item \\textbf{Open-source projects:} Many serverless open-source projects are currently available on GitHub.\n    As a starting point for the open-source projects, we used an existing data set from \\cite{pavlov:19}. This data set was scraped from GitHub using GHTorrent~\\cite{Gousi13}, an offline mirror of the  GitHub public event time line. It excludes unrelated or insignificant projects, based on a keyword search\\footnote{Keywords: \\textit{aws, aws lambda, amazon lambda, lambda functions, azure, openwhisk, serverless, google cloud functions, microsoft azure, azure functions, ibm blue mix, bluemix, oracle fn, oracle cloud fn, kubeless, ibm cloud functions, fn project}} and also excludes any projects that started prior to the launch of AWS Lambda (the first major serverless platform). From this data set, we removed small and inactive projects based on the number of files, commits, contributors, and watchers. As this still left us with many projects that only mention one of the keywords (e.g., ''In the future, we are looking to use AWS lambda for the image resizing.''), we manually filtered the resulting data set to include only projects that are deployed as serverless applications. This resulted in a total of 32~ use cases from open-source projects.\n    \n    \\item \\textbf{White literature:} There is also a growing interest in serverless applications from academia, which results in a number of scientific publications, i.e., journal papers, conference papers and workshop papers describing serverless use cases. For white literature, we based our search on an existing community-curated dataset on literature for serverless computing consisting of over 180 articles from 2016 to 2019~\\cite{sldataset}. First, we filtered the articles based on title and abstract. In a second iteration, we filtered out any articles that implement only a single function for evaluation purposes or do not include sufficient detail to enable a review. As the authors were familiar with a few additional publications describing serverless applications, we contributed these articles to the community-curated dataset and included them in this study. This resulted in a total of 23~ use cases from white literature.\n    \n    \\item \\textbf{Grey literature:} In  software engineering, the discourse in not limited to scientific articles but extends to grey literature, such as blog posts, forum discussions and podcasts~\\cite{10.1145\/2915970.2916008, 10.1145\/2786805.2803200}. For serverless computing, there are a number of blog posts by companies or individuals, talks at industry conferences and provider-reported success stories, as the development of serverless computing was initially mostly industry driven. We filtered the case studies reported by the major serverless providers (AWS\\footnote{\\url{https:\/\/aws.amazon.com\/solutions\/case-studies\/}}, Azure\\footnote{\\url{https:\/\/azure.microsoft.com\/en-in\/case-studies\/}}, Google\\footnote{\\url{https:\/\/cloud.google.com\/customers}} and IBM\\footnote{\\url{https:\/\/www.ibm.com\/case-studies\/}}) and selected those that used mostly serverless solutions. We also included the ten use cases reported in a recent article on the rise of serverless computing~\\cite{10.1145\/3368454}, which to the best of our knowledge is the largest collection of grey literature on serverless use cases. We further extended this collection with grey literature articles describing serverless use cases that the authors were already familiar with. This process resulted in a total of 28~ use cases from grey literature.\n    \n    \\item \\textbf{Scientific computing:} There is also an increasing interest in serverless solutions from the scientific computing community (e.g., by NASA~\\cite{nasa}). However, most of these use cases are still at an early stage and therefore there is little public data available for them. One of the authors of this paper is currently employed at the German Aerospace Center~(DLR), which allowed us to collect information about several projects at DLR that are either currently moving to serverless solutions or are planning to do so. Additionally, a use case from the German Electron Syncrotron~(DESY) could be included. This resulted in a total of 6~ use cases from the area of scientific computing.\n\\end{itemize}\nSome use cases are contained in multiple sources, e.g., a use case might have a GitHub repository that matches our keywords and is also used in the evaluation of an academic paper. For these use cases, we assign them only to a single source using the following ranking: open-source projects > grey literature > white literature. For the scientific use cases, there are no overlaps with the other use case sources.\n\n\\subsection{Use Case Selection}\n\\label{subsec:selection}\n\nWe defined the following inclusion (I) and exclusion (E) criteria for our study:\n\\begin{itemize}\n    \\item[I1] Concrete serverless use cases, as we are interested in real-world example applications.\n    \\item[I2] Use cases described in sufficient detail to conduct a meaningful review (i.e., excluding vague high-level case studies mainly focusing on a specific serverless platform or solution, but lacking technical detail).\n    \\item[E1] Serverless platforms (e.g., Apache OpenWhisk) and frameworks (e.g., Serverless Framework\\footnote{\\url{https:\/\/www.serverless.com\/}}), as these are not concrete workloads.\n    \\item[E2] Boilerplate code and simple technology demonstrations as often found in official serverless provider documentation, as these do not constitute full-fledged use cases.\n    \\item[E3] Academic papers on the same use case. For example, there are a number of academic papers that discuss serverless neural networks serving~\\cite{ishakian2018serving, bhattacharjee2019barista, tu2018pay}. In this case we only include a single representative paper.\n\\end{itemize}\n\n\\subsection{Characteristics Review}\n\\label{subsec:data_extraction}\nWe first determined and formalized the set of investigated characteristics. In an initial round, all authors individually suggested characteristics they consider interesting. In a next round, we merged similar characteristics and kept all characteristics that at least two authors considered relevant. This process resulted in 24 characteristics, which can be divided into five groups: general characteristics, workload characteristics, application characteristics, requirement characteristics, and workflow characteristics. \\emph{General characteristics} aim to quantify the structure of our data set and include characteristics such as ``Is the use case open-source?'', ``Is the use case currently deployed in production?'', and ``What domain is this use case from?''. \\emph{Workload characteristics} aim to describe the traffic pattern and request properties of the use case, e.g., ``Is the workload bursty?'', ``What is the data volume per request?'', and ``Is the application workload triggered by HTTP requests, cloud events, or regularly scheduled?''. \\emph{Application characteristics} describe the structure and properties of the serverless application itself and focuses on characteristics such as ``How many functions does the application consist of?'', ``What programming languages are used?'', and ``Which managed cloud services does the application use?''. \\emph{Requirement characteristics} describe the requirements from the stakeholders, such as ``Is latency relevant?'', ``Does the application have to run in a specific region, replicated in multiple regions or even on edge devices?'', and ``What is the reason for the adoption of serverless computing?''. Finally, \\emph{workflow characteristics} describe the properties of workflows within the serverless applications, e.g., ``Is the use case a workflow?'', ``How many functions does the workflow consist of?'', and ``How is the workflow execution coordinated?''. \\\\\n\nBased on a group discussion, we defined an exhaustive set of potential values for each characteristics. For example, for the characteristic ``How are executions triggered?'' we defined the potential values ``HTTP request'', ``Cloud event', ``Scheduled'' and ``Manual''. Additionally, for every characteristic we introduced the values ``Unknown'' and ``Not applicable''. ``Unknown'' indicates that the documentation of the use case does not contain enough information to determine this characteristic. ``Not applicable'' is used when a characteristic does not make sense for a use cases, for example all workflow characteristics are only applicable to use cases that contain a workflow. For some characteristics, we were not able to define a set of potential values prior to reviewing the use cases. For these characteristics, we used text fragments during the review. Using thematic coding~\\cite{coffey1996making, guest2011applied}, we extracted codes and treated those as the values for these characteristics. For example, for the characteristic ``What is the reason for the adoption of serverless computing?'' thematic coding resulted in the codes ``NoOps'', ``Scalability'', ``Performance'', ``Maintainability'' and ``Simplify Development''. This process enabled us to extract quantifiable results from the textual descriptions. \\\\\n\nWe randomly assigned each use case to 2 reviewers out of a pool of 7 available reviewers from the authors.\nWe manually adjusted a few reviewer assignments to minimize the number of coinciding reviewer pairs (i.e., avoid that many use cases are reviewed by the same two reviewers).\nSubsequently, each reviewer individually assigned values to all characteristics of its assigned use cases.\n\n\\subsection{Discussion and Consolidation}\n\\label{subsec:data_synthesis}\nAfter completing the initial round of reviews, we calculate the fleiss kappa to quantify the level of agreement between the reviewers~\\cite{gwet2014handbook}. Due to the nature of the fleiss kappa, we excluded all characteristic assignments, where at least one reviewer assigned multiple values for a characteristic (e.g., if a use case execution is triggered both via HTTP requests and cloud events), the characteristics using thematic coding as well as the numeric characteristic ``How many functions does the application consist of?''. As characteristics have a different number of possible values, we calculated an individual fleiss kappa value for each characteristic and then the weighted average across these individual kappa fleiss values. This results in a kappa fleiss value of 0.48, which can be interpreted as ``moderate agreement''~\\cite{landis1977measurement}.\n\nIn the following discussion and consolidation phase, the reviewers compared their notes and tried to reach a consensus for the characteristics with conflicting assignments. In a few cases, the two reviewers had different interpretations of a characteristics. These conflicts were discussed among all authors to ensure that characteristic interpretations were consistent. However for most conflicts, the consolidation was a quick process as the most frequent type of conflict was that one reviewer found additional documentation that the other reviewer did not find.\nFollowing this process, we were able to resolve all conflicts, resulting in a collection of 89~ use cases described by 24 characteristics. \\\\\n\nFor the scientific use cases, a different approach was necessary as many of them were not publicly available yet. Therefore, these use cases are reviewed by a single domain expert, which is either involved in the development of the use case or in direct contact with the development. For each of the scientific use cases there is also a textual description (see Appendix Section~\\ref{appendix}).\n\\subsection{Application Characteristics}\n\\label{sec:results:application}\nThis section characterizes the how the applications use cloud functions.\nIn the following, we analyze the applications regarding: the number of distinct functions in them, the function run times, the resource bounds of the functions, the programming languages used to implement the functions, the upgrade frequency of the cloud functions, and their interactions with external cloud services.\n\n\\subsubsection{Number of Distinct Functions}\n\\para{Description}\nThe business logic of serverless applications is contained within serverless functions and connects to a variety of managed cloud services. Similarly to microservices, the appropriate granularity of serverless functions is a controversial topic. Opinions range from wrapping each programming function as a serverless function, each API endpoint as a serverless function to full microservices as a serverless function. In this characteristic we investigate the number of distinct functions within the use cases. As this characteristics targets the development perspective, we count a function that is executed multiple times within an application as a single function. For some use cases, the only information available is along the lines of ''more than X functions\", which we count as X for this analysis.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/number_of_functions.pdf}\n    \\caption{Histogram of the number of functions per use case, single outlier at 170 not shown.}\n    \\label{fig:number_of_functions}\n\\end{figure}\n\n\\para{Results} About a third (32\\%) of the analyzed use cases consist of only a single function, as shown in\n\\autoref{fig:number_of_functions}.\nFurther, about one-fifth (21\\%) consist of two functions, a tenth (12\\%) of three functions, another tenth (11\\%) of four functions, and 5\\% of five functions. \nLarger sizes are very rare and without causing a mode in the empirical distribution: there exists in our analysis only one use case for each of the sizes 6, 7, 13, 15, 16, and, notably, 170; there exist only two use-cases for each of the sizes 8, 9, and 10. The use case with more than 170 functions is the back-end for the mobile app of a now defunct start-up. \nOverall, 82\\% of all use cases consist of five functions or less. \nFurthermore, 93\\% of the use cases that consists of ten functions or less. \\\\\n\n\\para{Discussion}\nOur results determine that serverless application use a low number of serverless functions, with 82\\% of all use cases consisting of five or less functions and 93\\% of the use cases consisting of less than ten functions. There are two potential reasons for this. First, the serverless application models reduces the amount of code developers have to write, as it allows them to focus on business logic while all other concerns are taken care of by the cloud provider and managed cloud services. Secondly, this seems to indicate that developers are currently choosing a rather large granularity for the size of serverless functions. However, determining the optimal granularity for serverless functions is still an open research challenge.\n\n\\subsubsection{Function Runtime}\n\\para{Description}\nThe run time of the cloud functions may have important impact on optimization choices of the serverless frameworks running these functions. We classified the run time of the functions in the use cases as: \\emph{short} (order of milliseconds or seconds) and \\emph{long} (order of minutes).\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/function_runtime.pdf}\n    \\caption{Function runtime distribution among the surveyed use cases.}\n    \\label{fig:function_runtime}\n\\end{figure}\n\n\\para{Results}\nThe majority of the functions in our survey are short (ms, s), 67\\%; only 22\\% of them have a run time in the order of minutes (see Figure~\\ref{fig:function_runtime}). We could not assess this characteristic in 10\\% of the use cases we studied. \nAll but one of the long-running functions is triggered on demand (as opposed to scheduled), with half of them falling into the scientific computing domain.\nThe long-running functions that did not fall within the scientific computing domain, are mostly operations or side-tasks, not business critical.\nFinally, these long functions are not high-volume on demand APIs (only one was classified as such).\\\\\n\n\\para{Discussion}\nThe overhead associated with running a function is larger, in proportion, for the case of functions with short run times.\nThis supports the large number of efforts concentrated in reducing this overhead. \nA limitation of our results is that, as the platforms impose a run time limit in the order of minutes, there may be a bias towards short running functions that would not exist if there were no time limits to the function run times. \\\\\n\n\\subsubsection{Resource Bounds}\n\\para{Description}\nWe wanted to know if the functions' run time were limited by \\emph{I\/O}, \\emph{CPU}, both (\\emph{hybrid}), the \\emph{network}, or an \\emph{external service} (e.g., cloud database).\nInformation on the workload mix can be useful for studies regarding the scheduling of functions and the routing of execution requests. \\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/resource_bounds.pdf}\n    \\caption{Distribution of the resource bounds among the surveyed use cases.}\n    \\label{fig:resource_bounds}\n\\end{figure}\n\n\\para{Results}\nMost use cases did not explicitly state this information (\\emph{unknown} is 72\\%, see Figure~\\ref{fig:resource_bounds}). For the use cases that did report this information, the I\/O-bound functions and CPU-bound functions are equally represented in our survey (9\\% I\/O, 8\\% CPU, 6\\% hybrid, 4\\% external service and 1\\% network). \\\\\n\n\\para{Discussion}\nThe percentage of use cases reporting this information is too small for us to derive any statistically significant analysis of the results. \\\\\n\n\\subsubsection{Programming Languages}\n\\para{Description}\nThis characteristic refers to the main programming languages used to write code for FaaS functions in a given application.\nFaaS providers typically offer a set of officially supported runtimes (e.g., Node.js for JavaScript).\nThese execution environments of FaaS functions determine the operating system and pre-installed software libraries.\nSome providers support further languages through custom runtimes, often in the form of Docker images following a documented interface.\nNotice that the programming language might differ from the technical function runtime as so called \\emph{shims} can be used to invoke a target language through a wrapper runtime (e.g., invoking C++ through Node.js via system calls). \\\\\n\n\\para{Results}\nJavaScript (32\\%) is the most common programming language for FaaS functions tied with Python (32\\%).\nLess common languages include Java (9\\%), C and C++ (8\\%), C\\# (6\\%), Go (3\\%), and Ruby (1\\%).\nWe were unable to determine the language for 25\\% of the use cases due to lacking technical descriptions. \\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/languages.pdf}\n    \\caption{Programming language distribution among the surveyed use cases. Some use cases use multiple programming languages.}\n    \\label{fig:languages}\n\\end{figure}\n\n\\para{Discussion}\nThe ranking of languages in our results follows a general trend also observed in other surveys. \nA study on FaaS industrial practices (N=161) \\cite{leitner2019MixedMethod} and initial results of the latest Serverless Community survey (N=109)\\footnote{Question 25 in \\url{https:\/\/www.nuweba.com\/blog\/serverless-community-survey-2020-results}} indicate that JavaScript is 20\\% more popular than Python on used languages in FaaS applications.\nHowever, they largely follow the same ranking but suggest higher popularity of Java over C\\#.\nOur results are plausible and confirm that JavaScript and Python are the most widely supported programming languages in FaaS.\nFurther, the remaining languages (i.e., Java, C, etc.) all belong to the world's most popular languages according to the TIOBE index\\footnote{\\url{https:\/\/www.tiobe.com\/tiobe-index\/}}, although they are often not the primary choice for the new FaaS paradigm.\n\n\n\\subsubsection{Function Upgrade Frequency} \n\\para{Description}\nHow frequently the code of the functions is updated has implications to software engineering and to the mechanisms used by the framework to upgrade the code in the functions that are already deployed.\nWe used two classification levels for this property: \\emph{rarely} and \\emph{often}; \\emph{unknown} indicates that this information cannot be obtained from the use case. \\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/upgrade.pdf}\n    \\caption{Function upgrade frequency distribution among the surveyed use cases.}\n    \\label{fig:upgrade}\n\\end{figure}\n\n\\para{Results}\nMost use cases did not explicitly state this information (unknown is 70\\%, see Figure~\\ref{fig:upgrade}). For the use cases that did report this information, the functions are updated rarely (26\\% rarely, 3\\% often).\\\\\n\n\\para{Discussion}\nThe percentage of use cases reporting this information is too small for us to derive any statistically significant analysis of the results. \\\\\n\n\\subsubsection{Use of External Services}\n\\label{subsec:external_services}\n\\para{Description}\nFaaS functions are often integrated into an ecosystem of serverless external services.\nPersistency services include cloud storage for blob data (e.g., Amazon S3 for images) and cloud database for structured data storage and querying (e.g., Amazon DynomoDB or Google Cloud SQL).\nA cloud API gateway exposes HTTP endpoints and can trigger FaaS functions upon incoming HTTP requests.\nMessaging services include cloud pub\/sub for durable asynchronous messaging (e.g., Amazon SNS), cloud queue for reliable FIFO-ordered messaging (e.g., Amazon SQS), and cloud streaming for real-time data ingestion and processing (e.g., Amazon Kinesis).\nCloud logging and monitoring refers to applications that explicitly process log data because we implicitly assume some essential logging infrastructure for FaaS functions (e.g., Amazon CloudWatch for AWS Lambda).\nCloud ML covers machine learning services, such as Amazon Rekognition for image or video analysis.\nNotice that we abstracted from vendor-specific services to cross-platform terminology (e.g., AWS S3 becomes Cloud Storage). \\\\\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/external_services.pdf}\n    \\caption{Distribution of used external services among the surveyed use cases. Many use cases use multiple external services.}\n    \\label{fig:external_services}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:external_services} shows that cloud storage (61\\%) and cloud database (47\\%) are the most popular external services, followed by the cloud API gateway (18\\%) and messaging services (10-17\\%).\nFor 12\\% of the use cases, we could not identify any external service integration. \\\\\n\n\\para{Discussion}\nGiven the ephemeral nature of FaaS functions, it is unsurprising that persistency services are the most popular external services, which is consistent with other survey results~\\cite{leitner2019MixedMethod}.\nHowever, the API gateway receives surprisingly little attention, especially when compared to the 46\\% of the use cases using HTTP triggers (see \\Cref{subsec:trigger_types}).\nWe suspect the use of an API gateway is often implicitly assumed, thus not explicitly mentioned, and therefore not comprehensively captured here.\nOverall, we conclude that currently used external services almost exclusively focus on technical aspects (e.g., storage or messaging) and more specialized services (e.g., cloud ML) are very uncommon among our surveyed use cases.\n\n\\subsection{General Characteristics}\n\\label{sec:results:general}\n\nIn this section, we analyse general characteristics of serverless use cases: the supported platform(s) and application types. Furthermore, we check if a serverless use case is in production yet and its availability as open source. Last, we report on the distribution across application domains for the analysed use cases.\n\n\\subsubsection{Platform}\n\\para{Description}\nIn November 2014, Amazon released the first commercial Function-as-a-Service platform with AWS Lambda and started the serverless trend. Two years later in 2016, Microsoft Azure, Google Cloud, and IBM Cloud released their own Function-as-a-Service platforms. There are also a number of open-source Function-as-a-Service platforms, such as Knative, OpenWhisk and OpenLambda. Selecting a deployment platform is a major decision for serverless applications, as there is a strong vendor lock-in that makes changing the deployment platform at a later point in time difficult. In this study, we grouped the deployment platforms into \\emph{AWS}, \\emph{Azure}, \\emph{IBM Cloud}, \\emph{Google Cloud}, and \\emph{Private Cloud}. \\\\\n\n\\begin{figure}[h]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/platform.pdf}\n    \\caption{Distribution of deployment platform among the surveyed use cases. Some use cases support multiple deployment platforms.}\n    \\label{fig:platform}\n\\end{figure}\n\n\\para{Results}\nAmong the use cases we surveyed, AWS is the clear choice-leader, with 80\\% of the use cases choosing AWS as their deployment platform. The other cloud vendors are far behind, with Azure at 10\\%, IBM at 7\\% and Google Cloud with 3\\%. 8\\% of the use cases use a private cloud, with the majority of them being scientific use cases. A total of five use cases can be deployed across multiple cloud platforms.\\\\\n\n\\para{Discussion}\nThat AWS is by far the most popular choice for serverless deployment among the cases we surveyed can probably be attributed to AWS having a two year head-start with offering this technology as a commercial service.\nA consequence of the earlier head-start is that \nthere was more time to develop and report about serverless applications using AWS serverless technology. Additionally, AWS has the largest market share when it comes to general cloud computing~\\cite{gartner}, which gives it a larger existing user base that can move applications to serverless.\\\\\n\nThe very low adoption of private clouds outside of the scientific workflows is in strong contrast to the large number of open-source Function-as-a-Service frameworks that have been developed. A large appeal of the serverless application model is the reduction of operational concerns, so we hypothesize that the increase in operational concerns that comes with maintaining a fleet of servers and an open-source Function-as-a-Service frameworks is deterring the adoption of these frameworks. Additionally, most serverless applications make use of many managed services (storage, databases, messaging, logging, streaming, etc.) which are not available directly in a private cloud environment.\\\\\n\nIt is interesting that five of the use cases we studied can be deployed across multiple cloud platforms. This goes in contrast to the commonly reported vendor lock-in of serverless computing~\\cite{adzic2017serverless, eivy2017wary}. However, upon closer inspection, three of these use cases are computation frameworks that provide additional layers of abstraction on top of commercial Function-as-a-Service and another one is a monitoring framework that utilizes serverless technologies. Therefore, we conclude that while there are some frameworks that can operate across multiple cloud platforms, most serverless applications can only be deployed on the cloud platform they were initially developed for. This provides further evidence that vendor lock-in exists for serverless applications.\n\n\\subsubsection{Application Type} \n\\label{subsec:application_type}\n\\para{Description}\nWe categorized each use case according to their type of serverless application. \nThe motivation behind this was to explore for what kind of tasks a serverless approach is typically employed---whether there are certain application types that are dominant or application types that are notably missing in current use cases.\nTo evaluate this usage aspect, we added the \\textit{Application Type} metric in which we provided options of typical application types to group the use cases in:\n\n\\begin{enumerate}\n    \\item \\emph{Operations \\& Monitoring:} Consists of the use cases that deploy serverless application to assist in operating software systems. Examples of such applications include automation of the test or deployment pipelines, failure mitigation or remediation, or controlling the state of running systems. This label superseeds all other labels.\n    \\item \\emph{Stream\/async Processing:} Groups the serverless applications that perform an asynchronous task, which also includes any processing of events from an event bus or stream.\n    \\item \\emph{Batch Task:} This is a special case of the \\texttt{stream\/async processing} category, which encompasses tasks that are executed in large batches. This label superseeds the \\texttt{stream\/async processing} label.\n    \\item \\emph{API:} Contains use cases that employ serverless to implement an API, such as a REST API or a GraphQL API. The exact nature of this API is not relevant here, rather that it is called synchronously, so the caller is waiting for a response.\n    \\item \\emph{Unknown:} Denotes the use cases that did not provide enough information about what type of serverless applications were employed.\n\\end{enumerate}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/application_type.pdf}\n    \\caption{Application type distribution among the surveyed use cases.}\n    \\label{fig:application_type}\n\\end{figure}\n\\para{Results}\nFigure~\\ref{fig:application_type} provides an overview of the collected results for the \\textit{application type} metric. We find that serverless is commonly used to implement \\texttt{APIs} (28\\%), \\texttt{stream\/async processing} (27\\%), \\texttt{batch tasks} (23\\%), and \\texttt{operations\/monitoring tasks} (20\\%). We were able to determine this characteristic for all but 2\\% of the survey applications.\n\n\\para{Discussion}\nServerless is commonly recommended for operations tasks, which also shows in our results as 20\\% of the use cases implement operations and monitoring applications. However, we also find large shares of APIs, asynchronous processing, and batch tasks. This shows that serverless is not seen as a niche technology that fits a special use case, but rather as a broadly applicable solution.\\\\\n\n\\subsubsection{In Production}\n\\label{subsec:production}\n\\para{Description}\nAnother characteristic that we analyzed is whether or not a specific use case  was actually deployed in production environments.\nOur motivation behind this is to evaluate how reliable the given characteristics actually are, or how representative they are for the real applications running in practice.\nTherefore, this metric can be seen as a kind of validation of our results.\n\nFor this characteristic, we work with three possible values: \n\\begin{itemize}\n    \\item \\emph{Yes}: There is clear evidence or statements claiming that the specific use case is already deployed in production.\n    \\item \\emph{No}: There is strong evidence that the specific use case is not deployed in production.\n    \\item \\emph{Unknown}: There can be no information found to support either ``Yes\" or ``No\".\n\\end{itemize}\nHowever, in our analysis an ``Unknown\" has almost certainly to be seen as a ``No\", as if we can not find evidence supporting that a use case was not applied in production, we have to assume that it is not.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/production.pdf}\n    \\caption{Percentage of the survey use cases that are deployed in production.}\n    \\label{fig:production}\n\\end{figure}\n\n\\para{Results}\nThe results of this characteristic are shown in Figure~\\ref{fig:production}.\nWe observe that more that half (55\\%) of analyzed use cases are actually designed for and deployed in production.\nRoughly a quarter (29\\%) is not used in production, and for the remaining 16\\% of use cases no clear answer could be found. \nHowever, as discussed above, we should assume that all ``Unknown'' use cases belong to the ``No'' field, leaving us with a total of 45\\% that do not have strong claims supporting their application in production.\\\\\n\n\\para{Discussion}\nAs more than half of the use cases included in this study are actually used in production, this can be seen as a strong indicator that the results of our analysis are representative and that the developed best practices have been applied in practice.\nFurthermore, many of the approaches not actively used in production actually originate from white literature. \nAs most of the white literature papers just present prototypical studies and evaluation use cases, their share of non-production use-cases is significantly higher. \nHowever, this in turn increases the respective share of in-production use cases for grey literature and GitHub Projects.\n\n\\subsubsection{Open Source}\n\n\\para{Description}\nOpen source indicates whether the source code of the FaaS function or application is publicly available.\nOpen source software is a valuable contribution for education, reuse, and testing.\nThis is exemplified through FaaS providers sharing their own vision for high-level reference architectures\\footnote{\\url{https:\/\/aws.amazon.com\/lambda\/resources\/reference-architectures\/}} and fostering cataloging of example applications\\footnote{\\url{https:\/\/aws.amazon.com\/serverless\/serverlessrepo\/}}. \\emph{Yes} indicates that a use case is open-source and \\emph{No} indicates that no source code is available.\nNotice that we barely check the availability of open source artifacts and cannot make any claims about completeness, maintenance levels, or use of appropriate licenses for this characteristic.\\\\\n\n\\para{Results} Figure~\\ref{fig:open_source} shows that 53\\% of the use cases are open source and 47\\% are closed source. \\\\\n\n\\para{Discussion}\nOpen source software is typically hosted on GitHub and similarly common for use cases deployed in production.\nInterestingly, open source software is comparably widespread among the 49 use cases deployed in production with 49\\%.\nWe expected a clearer tendency of use cases deployed in production to remain closed source, which is probably due to our selection strategy favoring open source use cases.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/open_source.pdf}\n    \\caption{Percentage of the survey use cases that are open-source.}\n    \\label{fig:open_source}\n\\end{figure}\n\n\\subsubsection{Domain} \n\\para{Description}\nFor each use case, we classified their application domain based as one of the following: \\emph{IoT}, \\emph{entertainment}, \\emph{scientific computing}, \\emph{WebServices}, \\emph{public authority}, \\emph{university}, \\emph{FinTech}, \\emph{cross-domain}, or \\emph{other}.\nCross-domain are those use cases that are generic and could be useful across more than one domain; for example, a generic image identification service which could be used in IoT, scientific computing, WebServices, public authority, and university domains.\n\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/domain.pdf}\n    \\caption{Distribution of the domain of the surveyed use cases.}\n    \\label{fig:domain}\n\\end{figure}\n\n\\para{Results}\nUnsurprisingly, WebServices is the most common application domain in our survey (33\\%, see Figure~\\ref{fig:domain}).\nThis is followed by cross-domain use cases (24\\%), which we mostly found via our GitHub search.\nThe other groups with high representation in our review were scientific computing (16\\%) and IoT (10\\%).\nThe significant presence of scientific computing cases is a result of our conscientious efforts in including scientific computing use cases in our survey.\n\\\\\n\n\\para{Discussion}\nWe find that there is a wide variety of application domains represented in our survey.\nThis is a strength of our study, as others can use our insights to make decisions about the design and implementation of serverless frameworks that are applicable to a broad variety of domains.\n\\subsection{Requirements Characteristics}\n\\label{sec:results:requirements}\nIn the following section, we analyze the different requirements and expectations that users have towards the serverless platforms when moving towards those.\nWe discuss the main motivation drivers, as well as the trade-off between cost and performance, and requirements towards latency and locality of the invocations.\n\n\\subsubsection{Motivation}\n\\label{sec:results:requirements:motivation}\n\\para{Description}\nThis characteristic aims at capturing the motivation of the respective engineers and therefore quantifies why they decided to host their application in a serverless environment.\nFor this, we developed six main motivation fields and grouped each use case into one or multiple of those fields, depending on the motivation the authors gave in the description.\nIf no conclusive motivation could be found, we put \\emph{Unknown}.\nThe main motivations we found are:\n\\begin{itemize}\n    \\item \\emph{Cost}: Running the application in a serverless platform significantly reduces operation cost in comparison to traditional cloud hosting.\n    \\item \\emph{NoOps}: Deploying a serverless application has the advantage of saving operation effort.\n    \\item \\emph{Scalability}: The increased scalability of serverless platforms is advantageous for the application.\n    \\item \\emph{Performance}: The performance of the application, i.e., throughputs and response times, is better when running on a serverless platform.\n    \\item \\emph{Simplify Development}: The development cycle as well as the release structure is easier using serverless applications.\n    \\item \\emph{Maintainability}: Deploying an application in a serverless cloud saves maintenance effort.\n    \\item \\emph{Scalability}: The increased scalability of serverless platforms is advantageous for the application.\n\\end{itemize}\n\n\\para{Results}\nThe results of our study can be observed in Figure~\\ref{fig:motivation}. \nThe biggest drivers for the adoption of serverless in our use cases are cost (33\\%), the reduced operation effort (24\\%), and the offered scalability (24\\%).\nTwo further significant motivation behind the adoption seems to be the performance benefits (13\\%) and the simplified development (9\\%). \nHowever, the maintainability (2\\%) only plays a minor role. \nFor 30\\% of the use cases, no specific motivation could be determined.\\\\\n\n\n\\para{Discussion} As the time savings by employing the NoOps paradigm of the serverless platforms can be converted to personnel costs, we observe that saving effort and costs seems to be a bigger contributor to the adoption of serverless than the offered performance and scalability improvements (although they are closely behind on second place).\n\nIt is important to note here, that there are many common pitfalls which can make serverless functions cost-inefficient. First, right now most providers bill by rounding up the execution time to the nearest 100ms. While this is negligible for most functions, this can be quite inefficient for very short-running functions. For example, if a functions runs for 10ms, it is billed for 100ms, which increases the billed duration tenfold. Secondly, most providers offer different function memory sizes and scale the other allocated resources such as CPU, I\/O capacity and network bandwidth accordingly. A recent survey reports that about 50\\% of serverless functions use the minimum size of 128MB~\\cite{datadog}, which is reported to be inefficient for most serverless functions. Thirdly, at a very large scale, the raw infrastructure costs are significantly larger than for a traditional VM-based solution~\\cite{eivy2017wary}. However, one might argue that the total cost of ownership could still be lower for the serverless solution due to the reduced operational overhead. In general, the the economic benefits of serverless computing heavily depend on the execution behavior and volumes of the application workload~\\cite{eivy2017wary}.\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/motivation.pdf}\n    \\caption{Distribution of the motivation behind adopting serverless among the surveyed use cases. Some use cases have multiple motivations.}\n    \\label{fig:motivation}\n\\end{figure}\n\n\\subsubsection{Cost\/Performance Trade-off}\n\\label{sec:results:requirements:tradeoff}\n\\para{Description}\nThe cost\/performance trade-off describes whether a use case tends to focus rather on cost optimization (i.e., \\emph{cost-focused}) or rather on performance optimization (i.e., \\emph{performance-focused}).\nThe trade-off is \\emph{undefined} if cost and performance are equally important and \\emph{unknown} if we could find no evidence towards any previously mentioned value in the provided use case description.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/tradeoff.pdf}\n    \\caption{Distribution of the cost\/performance trade-off among the surveyed use cases.}\n    \\label{fig:tradeoff}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:tradeoff} shows that cost is generally more important than performance for 41\\% of the use cases.\nCost-focused use cases are also twice as common compared to performance-focused use case (23\\%).\nFor 15\\% of the use cases, cost and performance are equally important.\nFinally, the trade-off remains unknown for 22\\% of the use cases.\\\\\n\n\\para{Discussion}\nThe clear focus on cost optimization is plausible given that cost is a strong motivation for adopting serverless (see \\Cref{sec:results:requirements:motivation}).\nServerless solutions, such as FaaS, were also associated with lower perceived total cost in another study~\\cite{leitner2019MixedMethod}.\n\n\\subsubsection{Is Latency Relevant?}\n\\para{Description}\nA diversity of use cases comes with a broad spectrum of expectations or even requirements on latency as a central performance metric. So we posed ourselves the question about the relevance of latency across the analysed serverless use cases. For this characteristic, we distinguish between four levels  plus \\emph{Unknown}. The levels are:\n\\begin{itemize}\n    \\item \\emph{Not important}: For these use cases we found evidence that latency does not play a central role. Delays and variations in latency are acceptable without disturbing the mode of operation.  \n    \\item \\emph{For complete use case}: Latency plays a relevant role for the whole use case on a level of mostly unspecified expectations on latency and its variations over time, e.g., expected to exhibit a latency for convenient human user interaction.\n    \\item \\emph{For parts of the use case}: The use case includes parts where latency is irrelevant and other parts where latency is of concern following the understanding of the level above as mostly unspecified expectation.  \n    \\item \\emph{Real-time}: We select the level \\emph{real-time} if evidence was found that there are soft latency requirements specified. Replies that take longer than a given upper time limit are becoming useless and are not further processed. This interpretation of \\emph{real-time} is not implying safety critical states when latency requirements are violated. \n\\end{itemize}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/latency_relevant.pdf}\n    \\caption{Distribution of latency importance among the surveyed use cases.}\n    \\label{fig:latency_relevant}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:latency_relevant} shows that in more than one third (36\\%) of the use cases, latency does not play a role. On the other side, for 58\\% of the analysed use cases, latency is of relevance (joining the respective levels). In 27\\% this is only for parts of the use cases. \nThe portion of use cases with real-time requirements on latency is small with only 2\\%.\nNo clear assignment of was reasonable for 3\\% of use cases. \\\\\n\n\\para{Discussion}\nServerless computing is especially convenient for triggered or scheduled background tasks that need to run from time to time without any latency requirements. But issues around function cold-starts and limited function life time have not shown to be a showstopper for serverless uses cases that expect a certain degree of stable latency, e.g. for a smooth interaction with human users. Also, over time, we would expect more examples for serverless use cases that come with stricter soft-real-time requirements as the platforms continue to mature. We doubt that serverless computing will accommodate use cases in production with hard real-time requirements and safety critical implications in case of violations. \n\n\n\\subsubsection{Locality Requirements}\n\n\\para{Description}\nMigration of any application from dedicated servers in a possibly self-owned data center to a compute infrastructure managed by a cloud provider reduces the control over the locality where the code runs and data is persisted. From the early days of cloud computing on this remains still a possible issue or even show-stopper. \nWe analyse the serverless use cases if requirements on locality are imposed. The reasons for locality requirements can differ broadly from regulatory to performance related ones.\nHere, we distinguish between four levels of locality requirements plus the case ``unknown'': \n\n\\begin{itemize}\n    \\item \\emph{None}: There is evidence that for the given use case, no locality requirements are imposed.\n    \\item \\emph{Multi-region}: The serverless application is or should be deployed in multiple regions, e.g. for improved latency or in tailored variations for specific geographic regions.\n    \\item \\emph{Specific-region}: The serverless applications is required to run in a specific region.\n    \\item \\emph{Edge}: The application or parts of it should run closer to a user or IoT device in an edge infrastructure\n\\end{itemize}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/locality_requirement.pdf}\n    \\caption{Locality requirement distribution among the surveyed use cases.}\n    \\label{fig:locality_requirement}\n\\end{figure}\n\n\\para{Results}\nWhile we have unclear or unspecified locality requirements for 34\\% of the use cases, Figure~\\ref{fig:locality_requirement} shows that the biggest portion (44\\%) comes with no locality requirements. For 21\\% of the use cases, we found locality requirements. Out of those, 8\\% are deployed across regions, while 10\\% are run in specific regions. The remaining 3\\% are tailored serverless solutions for edge computing. \\\\\n\n\n\\para{Discussion}\nAs serverless technologies and applications are maturing, we expect to see more business-critical elements of serverless applications in daily operation. The portion of serverless use-cases that comes with region-specific requirements will grow respectively. Furthermore, the use of serverless technologies for Edge computing can be seen as a trend of growing importance. Thus, we think it is likely to see a growing importance of the locality requirement ``Edge''. At the current time, for the dominating part of use cases, locality requirements are apparently not specified or not given yet. \n\\subsection{Workflow Characteristics}\n\\label{sec:results:workflow}\\label{sec:results:structural}\n\nMany serverless use cases cannot use a single serverless function to meet their functional and non-functional requirements. \nInstead, such use cases require the execution of multiple functions, \nexpressed and orchestrated as \\textit{serverless workflows}. In this section, we investigate the characteristics of serverless workflows. However, not all use cases include workflows. Thus, we first investigate in  Section~\\ref{sec:results:wf:is} which use cases are based on serverless workflows, and from then on we only report results for the use cases that do (the \\textit{workflow use cases}).\n\\newline\n\n\\subsubsection{Is it a Workflow?}\\label{sec:results:wf:is}\n\\para{Description}\nWe evaluate here the prevalence of serverless workflows among the surveyed use cases. A use case is categorized as a workflow (bar \\textit{Yes} in Figure~\\ref{fig:is_workflow}) if for a part or all of its functionality multiple serverless functions are needed. If not, the use case is not based on a workflow (\\textit{No}). Use cases where this could not be determined are assigned \\textit{Unknown}. \\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/is_workflow.pdf}\n    \\caption{Percentage of use cases including workflows, among the surveyed use cases.}\n    \\label{fig:is_workflow}\n\\end{figure}\n\n\\para{Results}\nAs depicted in Figure~\\ref{fig:is_workflow}, we observe that nearly a third (31\\%) of the use cases include serverless workflows. The other use cases (69\\%) are simple enough that one or a couple of independent serverless functions can fully provide the desired functionality. No use case was labeled Unknown.\\\\\n\n\\para{Discussion}\nThe relative prevalence of workflows in use cases is important, as it hints that serverless use cases are getting more and more complex. The evolution of use cases in fields such as grid computing and more recently cloud computing is indicative that, once workflows become acceptable practice, they become increasingly more prevalent~\\cite{hey2009fourth,isom2012your,deelman2018future}. Interesting too is the lack of use cases categorized as Unknown, which indicates that the presence or absence of workflows for any relevant use case is one of the clearest questions to answer. \n\n\\subsubsection{Workflow Coordination}\\label{sec:results:wf:coord}\n\n\n\\para{Description}\nThe use of workflows currently does not constrain the method used for orchestration. There are various approaches to ensure that tasks---or serverless functions---are executed in a coordinated way, e.g., functions can use events to trigger the start of a new task or for other purposes, a task can act as a coordinator for a specific workflow structure, or a workflow engine can orchestrate arbitrary workflows. To evaluate which of these approaches is prevalent, we surveyed their use across workflow use cases. More formally, for this part we categorized orchestration techniques as follows: \n\n\\begin{enumerate}\n    \\item \\emph{Event} groups all use cases that rely on event-driven orchestration. In this approach, workflows are constructed by configuring functions to be triggered to execute on the arrival of the completion (or failure) events of other functions. Typically, this method requires functions to explicitly listen for and publish their results or errors to a message queue---though in some cases platforms this functionality is built-in and no explicit interaction with an external message queue is needed.  \n    \n    \\item \\emph{Local coordinator} groups all the applications which rely on programmed, user-side logic to take care of the orchestration. An application running on the user's machine, such as a GUI or the client-side JavaScript running on a web page, invokes the functions in the appropriate order, and ensures that each function is executed with the correct configuration and input data.\n    \n    \\item \\emph{Workflow engine} contains the use cases that delegate the coordination to a dedicated workflow management system. This workflow engine has functionality to ensure the correct orchestration, along with higher-order concerns, such as (data) provenance, monitorability, and task scheduling optimizations. The workflows typically need to be specified in a consistent format using a set of workflow primitives that are supported by the workflow engine. Compared to the local coordinator, a workflow engine can also be seen as an external, persistent coordinator.\n    \n    \\item \\emph{Unknown} captures the use cases where we could not determine the coordination approach, for example because of lacking or lack of documentation.\n\\end{enumerate}\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/coordination.pdf}\n    \\caption{Distribution of workflow coordination approach among the surveyed use cases.}\n    \\label{fig:coordination}\n\\end{figure}\n\n\\para{Results}\nAs depicted in Figure~\\ref{fig:coordination}, half of the serverless workflows rely on events for the coordination (50\\%). Slightly less prevalent, about one-third (32\\%) of the use cases rely on a dedicated workflow engine to ensure correct coordination. Only a few (2\\%) defer the coordination to a local coordinator. For 16\\% of the workflow-based use cases the approach could not be determined.\\\\\n\n\\para{Discussion}\nOur results indicate that event-driven workflows are currently most prevalent. Inspecting the individual use cases, we find that this is in part caused by implicit workflows; use cases that do not explicitly construct workflows, but instead configure the function triggers in such a way that these form simple pipelines. While this approach can address simple workflows and especially chains of a few tasks,\nexperience from the fields of grid and cloud computing indicates using this approach will not scale to future workflows. We further find that, as the workflows grow in complexity, workflow engines are more often used to coordinate the workflows. In contrast to cloud-side coordination techniques, the use of local coordinators is unpopular because it distributes to the user more complex logic and, in part, because it is difficult to maintain. Furthermore, such an approach could be less reliable in operation -- cloud-side coordination techniques and workflow engines are carefully engineered for fault-tolerance, which significantly exceeds the typical development effort of local coordinators.\n\n\n\\subsubsection{Workflow Structure}\\label{sec:results:wf:structure}\n\n\\para{Description}\nThe complexity of a workflow is mostly determined by its structure. A {\\it bag of tasks} is a simple workflow (in mathematical terms: a degenerate workflow), which consists of a set of tasks that can be executed in any arbitrary order. \nAnother common workflow structure is the {\\it sequential workflow}, where all tasks need to be executed sequentially. \nWe further define {\\it complex workflows} as workflows that include significantly more complex structure than the previous types, including (multi-stage) gather and scatter operations, workflows with conditional execution of (some) tasks, workflows with loops, and fully dynamic workflows.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/workflow_structure.pdf}\n    \\caption{Distribution of the workflow structure among the surveyed use cases.}\n    \\label{fig:workflow_structure}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:workflow_structure} depicts the results.\nSequential workflows are the most popular workflow structure for serverless applications, with 40\\% of the workflow use cases including them. \nIncluding sequential workflows and bags of tasks (14\\%), over half (54\\%) of the workflow use cases only include non-complex workflows. \nAbout one-quarter (26\\%) of the serverless applications are complex workflows. \nLast, for over one-fifth (21\\%) of the workflow use cases we were not able to determine the workflow structure.\\\\\n\n\\para{Discussion}\nFor serverless applications, simple workflow structures (bag of tasks and sequential workflows) are more than twice as common as more complex workflow structures. We hypothesize that this is because serverless applications are currently mostly used for comparatively simple tasks and rarely for complex data analysis.\nAnother possible explanation is the lack of workflow engines (Section~\\ref{sec:results:wf:coord}); it is difficult to orchestrate arbitrarily complex workflows without such an engine. \\\\\n\n\nThe lack of bags of tasks can probably attributed to the fact that serverless applications come with built-in scalability when the functions can be conveniently executed in parallel. For example, resizing a collection of images can be conveniently implemented as a bag of many tasks, where each task invokes the same image-resizing function. In this case, the serverless platform has the capability to execute this case, without further need for orchestration from the user.\n\n\\subsubsection{Workflow Size} \\label{sec:results:wf:size}\n\n\\para{Description} Next, we study workflow size, expressed as the number of tasks in the workflow. \nWe aggregate all use cases into three groups:\n\\begin{enumerate}\n    \\item \\textit{Small workflows}, containing 2--10 functions,\n    \\item \\textit{Medium-size workflows}, invoking 10--1000 functions, and\n    \\item \\textit{Large workflows}, comprised of more than 1000 function invocations. \n\\end{enumerate}\n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/workflow_size.pdf}\n    \\caption{Workflow size distribution among the surveyed use cases.}\n    \\label{fig:workflow_size}\n\\end{figure}\n\n\\para{Results} Figure~\\ref{fig:workflow_size} depicts the results of our analysis. \nThe majority (59\\%) of analyzed workflows are small workflows.\nAround one fifth of use cases (19\\%) are medium-size, and only few (3\\%) qualify as large workflows.\nNearly one-fifth of the workflows (19\\%) could not be assigned to one of the groups. \\\\\n\n\\para{Discussion} Our results suggest that a majority of workflow executions are small; because these workflows are only composed of ten or less individual function executions, they are also relatively short-lived. This is consistent with the characteristics of early workflows in engineering and in scientific prototypes~\\cite{conf\/cgiw\/OstermannIPFE08}. Only about one-fifth of the workflows are medium or large-sized. Similarly to the previous section, we hypothesize that orchestrating workflows of this size is dependent on the presence of an automated facility, such as a workflow engine. (The other hypothesis introduced in Section, that serverless workflows are currently used for relatively simpler tasks, does not limit the size of the workflow -- in the earlier example, the image-resizing workflow can run 10,000s or even 100,000s of functions~\\cite{bharathi2008characterization, deelman2018future}.)\n\n\n\\subsubsection{Workflow Internal Parallelism}\n\n\\para{Description}\nFor those use cases in which a workflow of serverless functions was present, we further analyzed whether they present \\emph{internal parallelism}---at least an instance of multiple functions running in parallel---or not. \\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/parallelism.pdf}\n    \\caption{Percentage of workflows with internal parallelism among the surveyed use cases.}\n    \\label{fig:parallelism}\n\\end{figure}\n\n\\para{Results}\nFrom Figure~\\ref{fig:parallelism}, we observe that most workflows (52\\%) exhibit internal parallelism; about one-third (31\\%) of the workflows are simpler, exhibiting no internal parallelism.\nWe could not obtain this information (\\emph{unknown}) for 16\\% of the workflows. \\\\\n\n\\para{Discussion}\nThe high prevalence of workflows with at least some level of internal parallelism calls for workflow managers that are native to---or well integrated with---the serverless framework, to facilitate workflow composition and management yet deliver parallelism with low overhead.\n\\\\\n\\subsection{Workload Characteristics}\n\\label{sec:results:workload}\n\nThis section characterizes the nature of workloads imposed on serverless applications through human users or technical invokers.\nIn the following, we discuss execution patterns, burstiness, trigger types, and common data volumes of our reviewed use cases.\n\n\\subsubsection{Execution Pattern}\n\n\\para{Description}\nFunctions or workflows of functions can be triggered \\emph{on-demand} as a direct result of a user interacting with the application, or they can be \\emph{scheduled} to be run at specific times.\nFor the on-demand workflows, we further classify them as regular \\emph{on-demand} or \\emph{high-volume on-demand}.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/execution_pattern.pdf}\n    \\caption{Execution pattern distribution among the surveyed use cases. Some use cases are executed both on a schedule and on-demand.}\n    \\label{fig:execution_pattern}\n\\end{figure}\n\n\\para{Results}\nMost (workflows of) functions are triggered on-demand, with scheduled triggers being used in only 17\\% of the use cases we analyzed (see Figure~\\ref{fig:execution_pattern}).\nOut of the on-demand execution patterns, close to half are high-volume, business-critical invocations. \\\\\n\n\n\\para{Discussion}\nThe high prevalence of high-volume on-demand triggers calls for special study in minimizing function start-up times, and auto-scaling mechanisms.\nIn addition, half of the workflows that are scheduled fall into the application type category \\emph{operations} (see \\cref{subsec:application_type}), highlighting how the serverless model has been adopted---in many cases---to automate operations, software management, and DevOps pipelines.\n\\\\\n\n\\subsubsection{Burstiness}\n\\para{Description}\nThe workload of a function can be either bursty or non-bursty.\nA bursty workload follows a workload pattern that includes certain sudden and unexpected load spikes, or alternatively a significant amount of sustained noise and variation in intensity.\nWe classify a use case as bursty (i.e., \\emph{yes} for burstiness) if its workload typically includes or can include burst patterns in some situations and as non-bursty (i.e, \\emph{no}) if the workload is almost guaranteed to never receive burst patterns (e.g., if all executions are scheduled and known in advance).\nIf the burstiness of a use case is \\emph{unknown}, then the use case was either under-specified, or can be both bursty or non-bursty depending on the specific area of application.\nNote that in any scenario that involves a set of human users, we consider the workload pattern to be bursty, as user behavior can almost never the scheduled or reliably controlled, leading to a possibly bursty behavior.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/bursty.pdf}\n    \\caption{Percentage of the survey use cases that have a bursty workload.}\n    \\label{fig:bursty}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:bursty} depicts that a large majority (81\\%) of the analyzed workload patterns are classified as bursty.\nAdditionally, a 16\\% share have a clear non-bursty workload pattern, while a small minority of 3\\% could not be attributed to be either bursty or non-bursty.\\\\\n\n\\para{Discussion}\nAs one of the strengths of serverless computing is its seamless and almost infinite scalability, together with the general ease of operations it comes as no surprise that most of the use cases indeed experience bursty workload patterns.\nEarly adopters that want to test out the new emerging paradigm are more likely to choose a use case that is optimized for the serverless offering. \nAt the same time, engineers that have been struggling with bursty workloads and face regular performance issues are also more likely to migrate or adopt their application towards the a serverless clouds than applications that run smoothly.\n\nAn interesting comparison here would be to compare the share of bursty workloads executed on serverless platforms versus the share of bursty workloads of conventional applications.\nHowever, we can still conclude that a large majority of use cases designed for or applied to serverless platforms are experiencing bursty workloads and hence make use of the seamless elasticity that these services offer.\n\n\n\\subsubsection{Trigger Types}\n\\label{subsec:trigger_types}\n\n\\para{Description}\nTrigger types refer to alternative ways of invoking a FaaS function and are closely related to external services (see \\Cref{subsec:external_services}).\nAn \\emph{HTTP request} can trigger a FaaS function, which then processes the request and generates an HTTP response.\nThis HTTP routing is often implemented through API gateways.\nA \\emph{cloud event} describes a state change happening in a connected cloud service, such as a file upload to cloud storage or a modified value in a cloud database.\nSuch cloud events can be configured to trigger new function executions.\nA \\emph{scheduled} trigger invokes a FaaS function at a defined and potentially recurring time.\nThe category of \\emph{manually} triggered functions refers to human-initiated executions typically executed on-demand.\nNotice that some use cases combine multiple trigger types and thus the sum of proportions exceeds 100\\%. \\\\\n\n\\para{Results}\nFigure~\\ref{fig:trigger} reveals that the most common trigger types are HTTP request (46\\%) and cloud event (39\\%).\nFar less common are scheduled (12\\%) and manual (9\\%) execution triggers.\nWe were unable to derive the trigger type for 3\\% of the use cases from their insufficient descriptions.\\\\\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/trigger.pdf}\n    \\caption{Trigger type distribution among the surveyed use cases. Some use cases have multiple trigger types. }\n    \\label{fig:trigger}\n\\end{figure}\n\n\\para{Discussion}\nWe compare our results to the trigger types reported for the production workload of Microsoft Azure functions~\\cite{shahrad2020serverless}.\nScheduled triggers and HTTP triggers are both 16-20\\% more common among Azure FaaS applications in comparison to our analyzed use cases focusing on AWS (85\\%).\nThe results for the remaining categories very closely (<=2\\%) match (after mapping some cumulative categories).\nWe conclude that the order of categories is in line with current production workloads reported for Azure but note that some values might be higher in practice.\nSuch an underestimation is plausible given that we derive our results from potentially incomplete sources.\nHowever, the explicit grouping of functions into applications in the Azure FaaS implementation possibly leads to different function groupings compared to our AWS-focused use cases.\n\n\\subsubsection{Data Volume}\n\n\\para{Description}\nThe data volume defines what load will be on network and storage devices. \nThe motivation here is to analyze whether there are any clusterings of data usages or certain patterns that are generally avoided.\nWe categorized the different use cases into five different categories: Volumes of \\emph{less than 1 MB} per execution, \\emph{less than 10 MB}, \\emph{less than 100 MB}, \\emph{less than 1 GB}, and \\emph{more than 1 GB}.\nAdditionally, there is also the \\emph{unknown} category, if data volume could not be assessed.\nNote that the data volume refers to executions of the entire workflow.\nFurthermore, as exact numbers were seldom found in the sources, this categorization is often based on the estimate of our reviewers.\\\\  \n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{figures\/results\/data_volume.pdf}\n    \\caption{Data volume distribution among the surveyed use cases.}\n    \\label{fig:data_volume}\n\\end{figure}\n\n\\para{Results}\nFigure~\\ref{fig:data_volume} depicts the distribution of use cases among the different classes. \nAlmost half of the use cases (44\\%) fall in the smallest category of data volumes of less than 1 MB. \nThe second categorization transmitting more than 1 MB of data, but less than 10 MB, also make the second largest group with a fraction of 13\\%.\nEven less use cases (3\\%) consider a data volume between 10 and 100 MB. \nHowever, the following group between 100 MB and 1 GB increases in popularity, and finally the share of use cases transmitting 1 GB or more to the serverless platform increases to be the second-largest group (13\\%).\nAdditionally, 18\\% of use-cases could not have a specific data volume assigned and therefore do not count into any of the enumerated groups.\\\\\n\n\\para{Discussion}\nGenerally, the different data volumes are relatively distributed and do not cast a clear picture.\nThere is definitely a use case for any data volume characteristic. Therefore platforms should not strive to optimize themselves towards any specific limitation here.\nThat said, most of the use cases transmit less than 1 MB of data per workflow execution. \nNote that this group also includes all use cases that might not send any data at all. \nTherefore, the large majority of serverless use cases that we surveyed does not work with big amounts of data. \nHowever, there is also the exact opposite group of use cases working with vast vast amounts of data of 1 GB and more per workflow execution.\n\\section{Analysis Results: On the Characteristics of Serverless Use Cases}\n\\label{sec:results}\n\nWe describe in this section the results of our characterization and analysis of serverless use cases. Overall, we cover a diverse set of  characteristics, identifying the values commonly used in practice and further analyzing their impact on serverless practice.\n\n\\subsection{Main Findings}\n\\label{sec:results:main}\n\n\nOur main findings are:\n\\begin{enumerate}\n\n    \\item \\emph{General Characteristics:} We find AWS as the currently dominating for platform for serverless applications (80\\%). The dominating application domain is web services (33\\%), with 40\\% of the analysed workloads being business-critical and at least 55\\% of them in production already.\n    \n    \\item \\emph{Application Characteristics:}\n    82\\% of all use cases consist of applications that use five or less different functions. Most (67\\%) of these functions are short-running, with running times in the order of milliseconds or seconds. JavaScript and Python are the most used programming languages for cloud functions (each used by 32\\% of the cases we studied).\n    These applications depend on a wide variety of cloud services, with the three most used ones being cloud storage (used by 61\\% of the applications) and cloud database (47\\%); cloud API gateway (18\\%) and cloud pub-sub (17\\%) are also widely used.\n    \n    \\item \\emph{Requirements Characteristics:} The reduced operation cost of serverless platforms (33\\%), the reduced operation effort (24\\%), the scalability (24\\%), and performance gains (13\\%) are the main drivers of serverless adoption. \n    In comparison, cost savings seems to be a stronger motivator than the performance benefits. \n    At the same time, 58\\% of use cases have latency requirements, 2\\% even have real-time demands, while only 36\\% are latency insensitive.\n    Locality requirements are only relevant for 21\\% of the total use cases.\n    \n    \\item \\emph{Workload Characteristics:}\n    81\\% of the analyzed use cases exhibit bursty workloads.\n    This highlights the overall trend of serverless workloads to feature unpredictable on-demand workloads, typically triggered through lightweight (<1MB) HTTP requests.\n\n    \\item \\emph{Workflow Characteristics:}\n    Although the presence of workflows is already sizable (31\\% of the use cases), most workflows are of simple structure, small, and short-lived. \n    This is likely to change, as demand follows natural trends and orchestration methods move toward (cloud-native) workflow engines.\n    \n\\end{enumerate}\n\n\\input{sections\/results\/General.tex}\n\n\\input{sections\/results\/Workload.tex}\n\n\\input{sections\/results\/Application.tex}\n\n\\input{sections\/results\/Requirements.tex}\n\n\\input{sections\/results\/Workflow.tex}\n\\section{Threats to Validity}\n\\label{sec:validity}\\label{sec:threats}\nWe discuss potential threats to validity and mitigation strategies for internal validity, construct validity, and external validity.\n\n\\subsection{Internal Validity}\nManual data extraction can lead to inaccurate or incomplete data.\nTo mitigate this threat, we established and discussed a review protocol prior to reviewing, continuously discussed upcoming questions during the review process, and performed redundant reviews through multiple reviewers.\nIn our review protocol, we established an exhaustive list of potential values for each characteristic and configured automated validation, which immediately highlighted deviations from these values.\nFor characteristics with thematic coding, we continuously refined their values in regular meetings during the review process.\nTo address potential individual bias, we performed two independent reviews for each use case, quantified the inter-rater agreement after an initial review round through Fleiss' Kappa, and resolved each disagreement in an extended discussion and consolidation phase.\n\n\\subsection{Construct Validity}\nTo align the goal of this study (i.e., comprehensive understanding of existing serverless use cases) with the data extraction, we compiled a list of 24 characteristics covering 5 different aspect groups.\nWe conducted and discussed this selection process together in an international working group with authors from 5 different institutions but other researchers might consider different characteristics as relevant.\n\n\\subsection{External Validity}\nOur study was designed to cover use cases from open source projects, white literature, and grey literature but we cannot claim generalizability to all serverless use cases.\nFor open source projects, we filtered non-trivial projects from the most popular open source repository (i.e., GitHub) but might have missed projects published in other repositories.\nHowever, we are unaware of such other repositories and also did not discover any among our other use cases from white and grey literature.\nOur white literature collection is based on a curated dataset on serverless literature and complemented with articles known to the authors but we might have missed more recent articles uncovered in the dataset and unknown to all authors.\nGrey literature use cases mostly focus on provider-reported case studies, an existing collection of grey literature use cases, and sources known to the authors.\nWe only partially cover corporate use cases as many of them remain unpublished and others provide insufficient details to conduct a meaningful review, which is similar to FaaS platforms~\\cite{DBLP:journals\/internet\/EykIGEBVTSHA19}.\nOur scientific computing use cases are limited to the aerospace domain originating from a national aerospace institution.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nHumans' unique abilities such as adaptive behavior in dynamic environments, and social interaction and moral judgment capabilities, make them essential elements of many control loops. On the other hand, compared to humans, automation provides higher computational performance and multi-tasking capabilities without any fatigue, stress, or boredom \\cite{Noth16,Kor15}. Although they have their own individual strengths, humans and automation also demonstrate several weaknesses. Humans may have anxiety, fear and may become unconscious during an operation. Furthermore, in the tasks that require increased attention and focus, humans tend to provide high gain control inputs that can cause undesired oscillations. One example of this phenomenon, for example, is the occurrence of pilot induced oscillations (PIO), where undesired and sustained oscillations are observed due to an abnormal coupling between the aircraft and the pilot \\cite{YilKol10, Yil11a,AcoYil14,TohYil18}. Similarly, automation may fail due to an uncertainty, fault or cyber-attack \\cite{Li14}. Thus, it is more preferable to design systems where humans and automation work in harmony, complementing each other, resulting in a structure that benefits from the advantages of both.\n\nTo achieve a reliable human-automation harmony, a mathematically rigorous human operator model is paramount. A human operator model helps develop safe control systems, and provide a better prediction of human actions and limitations \\cite{Hul13,Yuc18,Emre19, Zha19}. \nQuasi-linear model \\cite{Mcr57} is one of the first human operator models, which consists of a describing function and a remnant signal accounting for nonlinear behavior. An overview of this model is provided in \\cite{Mcr74}. In some applications, where the linear behavior may be dominant, the nonlinear part of this model can be ignored, and the resulting lead-lag-type compensator is used in closed loop stability analysis \\cite{Neal71}. The crossover model, proposed in \\cite{Mcr63}, is another important human operator model in the aerospace domain. It is motivated from the empirical observations that human pilots adapt their responses in such a way that the overall system dynamics resembles that of a well designed feedback system \\cite{Bee08}. A generalized crossover model which mimics human behavior when controlling a fractional order plant is proposed in \\cite{Mar17}. In \\cite{War16}, crossover model is employed to provide information about the human intent for the controller. In \\cite{Gil09}, the dynamics of the operator is represented as a spring-damper-mass system. \n\n\nControl theoretical operator models drawing from the optimal and adaptive control theories are also proposed by several authors. \nOptimal human models are based on the idea that a well trained human operator behaves in an optimal manner \\cite{Wier69, Kle70, Na12, Lon13, Hu19}. On the other hand, adaptive models, such as the ones proposed in \\cite{Hess09, Hess15} and \\cite{TohYil19Human}, aim to replicate the adaptation capability of humans in uncertain and dynamics environments. In \\cite{Hess09} and \\cite{Hess15}, adaptation rules are proposed based on expert knowledge. The adaptive model proposed in \\cite{Hess09} is applied to change the parameters of the pilot model based on force feedback from a smart inceptor \\cite{Xu19}. \nA survey on various pilot models can be found in \\cite{Lon14} and \\cite{Xu17}.\n\n\n\n\n\n Several approaches are also developed for human model parameter identification. In \\cite{Zaal11}, a two-step method using wavelets and a windowed maximum likelihood estimation is exploited for the estimation of time-varying pilot model parameters. In \\cite{Duar17}, a linear parameter varying model identification framework is incorporated to estimate time-varying human state space representation matrices. Subsystem identification is used in \\cite{Zha15} to model human control strategies. In \\cite{Van15}, a human operator model for preview tracking tasks is derived from measurement data.\n\n\nIn this paper, we build upon the earlier successful pilot models and propose an adaptive human pilot model that modifies its behavior based on plant uncertainties. This model distinguishes itself from earlier adaptive models by having mathematically derived laws to achieve a cross-over-model-like behavior, instead of employing expert knowledge. This allows a rigorous stability proof, using the Lyapunov-Krasovskii stability criteria, of the overall closed loop system. To validate the model, a setup, including a joystick and a monitor, is used. The participant data collected through this experimental setup is subjected to visual and statistical analyses to evaluate the accuracy of the proposed model. Initial research results of this study were presented in \\cite{TohYil19Human}, where the details of the mathematical proof and human experimental validation studies were missing.\n\n\nThis paper is organized as follows. In Section \\ref{sec:prob}, the problem statement is given. Obtaining reference model parameters, which determine the properties of the cross-over model, is discussed in Section \\ref{sec:reference}. Section \\ref{sec:human} presents the human control strategy together with a stability analysis. Experimental set-up, results, and a statistical analysis are provided in Section \\ref{sec:experiment}. Finally, a summary is given in Section \\ref{sec:conclusion}.\n\n\n\\section{PROBLEM STATEMENT}\\label{sec:prob}\n\nAccording to McRuer's crossover model \\cite{Mcr67}, human pilots in the control loop behave in a way that results in an open loop transfer function\n\\begin{equation}\\label{eq:e1x}\nY_{OL}(s)=Y_{h}(s)Y_{p}(s)=\\frac{\\omega_ce^{-\\tau s}}{s},\n\\end{equation}\nnear the crossover frequency, $ \\omega_c $, where $ Y_{h} $ is the transfer function of the human pilot and $ Y_{p} $ is the transfer function of the plant. $ \\tau $ is the effective time delay, including transport delays and high frequency neuromuscular lags.\n\nConsider the following plant dynamics\n\\begin{equation}\\label{eq:e2x}\n\\dot{x}_p(t)=A_px_p(t)+B_p u_p(t),\n\\end{equation}\nwhere $ x_p\\in \\mathbb{R}^{n_p} $ is the plant state vector, $ u_p\\in \\mathbb{R} $ is the input vector, $ A_p\\in \\mathbb{R}^{n_p\\times n_p} $ is an unknown state matrix and $ B_p\\in \\mathbb{R}^{n_p} $ is an unknown input matrix.\n\nThe human \\textit{neuromuscular model} \\cite{Mag71, Van04} is represented in state space form as\n\\begin{equation}\\label{eq:e3x}\n\\begin{aligned}\n\\dot{x}_h(t)&=A_hx_h(t)+B_hu(t-\\tau) \\\\\ny_h(t)&=C_hx_h(t)+D_hu(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhere $ x_h\\in \\mathbb{R}^{n_h} $ is the neuromuscular state vector, $ A_h\\in \\mathbb{R}^{n_h\\times n_h} $ is the state matrix, $ B_h\\in \\mathbb{R}^{n_h} $ is the input matrix, $ C_h\\in \\mathbb{R}^{1\\times n_h} $ is the output matrix and $ D_h\\in \\mathbb{R} $ is the control output matrix. $ u\\in \\mathbb{R} $ is the neuromuscular input vector, which represents the control decisions taken by the human and fed to the neuromuscular system, $ y_h\\in \\mathbb{R} $ is the output vector, and $ \\tau\\in \\mathbb{R}^+ $ is a known, constant delay. The neuromuscular model parameters are assumed to be known and the output of the model, $ y_h $, is used as the plant input $ u_p $ in (\\ref{eq:e2x}), that is $ y_h=u_p $ (see Fig. \\ref{fig:f1-2}). \n\\begin{figure}\n\t\\vspace{0.3cm}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{Blockdiagxx.jpg}    %\n\t\t\\caption{The block diagram of the human adaptive behavior and decision making in a closed loop system.} \n\t\t\\label{fig:f1-2}\n\t\\end{center}\n\\end{figure}\n\n\nBy aggregating the human pilot and plant states, we obtain the combined open loop human neuromuscular and plant dynamics as\n\\begin{equation}\\label{eq:e4x}\n\\begin{aligned}\n\\underbrace{\\begin{bmatrix}\n\t\\dot{x}_h(t) \\\\ \\dot{x}_p(t)\n\t\\end{bmatrix}}_{\\dot{x}_{hp}(t)}&=\n\\underbrace{\\begin{bmatrix}\n\tA_h & 0_{n_h\\times n_p} \\\\ B_p C_h & A_p\n\t\\end{bmatrix}}_{A_{hp}}\\underbrace{\\begin{bmatrix}\n\tx_h(t) \\\\ x_p(t)\n\t\\end{bmatrix}}_{x_{hp(t)}}\\\\ &+\n\\underbrace{\\begin{bmatrix}\n\tB_h \\\\ B_pD_h\n\t\\end{bmatrix}}_{B_{hp}}u(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhich can be written in the following compact form\n\\begin{equation}\\label{eq:e5x}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_{hp}x_{hp}(t)+B_{hp}u(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhere $ x_{hp}=[x_h^T\\ x_p^T]^T\\in \\mathbb{R}^{(n_p+n_h)} $, $ A_{hp}\\in \\mathbb{R}^{(n_p+n_h)\\times (n_p+n_h)} $, $ B_{hp}\\in \\mathbb{R}^{(n_p+n_h)} $. \n\\begin{assumption}\n\tThe pair $ (A_{hp}, B_{hp}) $ is controllable.\n\\end{assumption}\n\nThe goal is to obtain the input $ u(t) $ in (\\ref{eq:e3x}), which is the human pilot control decision variable, such that the closed loop system consisting of the adaptive human pilot model and the plant follow the output of a unity feedback reference model with an open loop crossover model transfer function. The closed loop transfer function of the reference model is therefore calculated as\n\\begin{equation}\\label{eq:e21xqx}\n\\begin{aligned}\nG_{cl}(s)=\\frac{\\frac{\\omega_c}{s}e^{-\\tau s}}{1+\\frac{\\omega_c}{s}e^{-\\tau s}}=\\frac{\\omega_c e^{-\\tau s}}{s+\\omega_c e^{-\\tau s}}.\n\\end{aligned} \n\\end{equation}\nAn approximation of (\\ref{eq:e21xqx}) can be given as  \n\\begin{equation}\\label{eq:e21xqqx}\n\\begin{aligned}\n\\hat{G}_{cl}(s)=\\frac{b_ms^m+b_{m-1}s^{m-1}+...+b_0}{s^n+a_{n-1}s^{n-1}+...+a_0}e^{-\\tau s},\n\\end{aligned} \n\\end{equation}\nwhere $ n=n_h+n_p $ and $ m\\leq n $ are positive real constants, and $ a_{i} $ and $ b_j $ for $ i=0, ..., n-1 $ and $ j=0, ..., m-1 $, are real constants to be estimated. \nThe reference model then can be obtained as the state space representation of (\\ref{eq:e21xqqx}) as\n\\begin{equation}\\label{eq:e7x}\n\\begin{aligned}\n\\dot{x}_{m}(t)&=A_{m}x_{m}(t)+B_{m}r(t-\\tau),\\\\\n\\end{aligned} \n\\end{equation}\nwhere $ x_m\\in \\mathbb{R}^{(n_h+n_p)} $ is the reference model state vector, $ A_m\\in \\mathbb{R}^{(n_h+n_p)\\times (n_h+n_p)} $ is the state matrix, $ B_m\\in \\mathbb{R}^{(n_h+n_p)\\times m_h} $ is the input matrix, and $ r\\in \\mathbb{R}^{m_h} $ is the reference input.\n\n\\section{REFERENCE MODEL PARAMETERS}\\label{sec:reference}\nThe crossover transfer function (\\ref{eq:e1x}) contains the crossover frequency, $ \\omega_c $, which is not known a priori. Experimental data, showing the reference input ($ r(t) $) frequency bandwidth, $ \\omega_i $, versus crossover frequency $ \\omega_c $, is provided in \\cite{Bee08} and \\cite{Mcr67}, for plant transfer functions $ K $, $ K\/s $ and $ K\/s^2 $. We fit polynomials to these experimental results to obtain the crossover frequency of the open loop transfer function given a reference input frequency bandwidth. These polynomials are given in Table I. It is noted that when the reference input has multiple frequency components, the highest frequency is used to calculate the crossover frequency.\n\n\n\\begin{remark}\n\tIn this work, we use the polynomial relationships provided in Table I for zero, first and second order plant dynamics with nonzero poles and zeros. Further experimental work can be conducted to obtain a more precise relationship between the crossover and reference input frequencies, but this is currently out of the scope of this work.\n\\end{remark}\n  \n\n\n\n\\begin{table}\n\t\\vspace{0.3cm}\n\t\\caption{}\n\t\\centering\n\t\\begin{tabular}{ | c | c | }\n\t\t\\hline \n\t\tPlant transfer & Crossover frequency of the \\\\\n\t\tfunction & open loop transfer function (rad\/s)\\\\\n\t\t\\hline\n\t\t$ K $ & $ \\omega_c=0.067\\omega_i^2+0.099\\omega_i+4.8 $ \\\\ \\hline\n\t\t$ K\/s $ & $ \\omega_c=0.14\\omega_i+4.3 $ \\\\ \\hline\n\t\t$ K\/s^2 $ & $ \\omega_c=-0.0031\\omega_i^4-0.072\\omega_i^3+0.29\\omega_i^2$ \\\\ & $-0.13\\omega_i+3 $ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\n\n\\section{HUMAN PILOT CONTROL DECISION COMMAND}\\label{sec:human}\n\n\nThe adaptive human pilot control decision command, $ u(t) $, is determined as\n\\begin{equation}\\label{eq:e8x}\nu(t)=K_rK_xx_{hp}(t+\\tau)+K_rr(t)\n\\end{equation}\nwhere $ K_x\\in \\mathbb{R}^{1\\times (n_h+n_p)} $, and $ K_r\\in \\mathbb{R}^{m_h\\times m_h} $. Using (\\ref{eq:e8x}) and (\\ref{eq:e5x}), the closed loop dynamics can be obtained as\n\\begin{equation}\\label{eq:e9x}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=(A_{hp}+B_{hp}K_rK_x)x_{hp}(t)+B_{hp}K_rr(t-\\tau).\n\\end{aligned} \n\\end{equation} \n\nEquation (\\ref{eq:e8x}) describes a non-causal decision command which requires future values of the states. This problem can be eliminated by solving the differential equation (\\ref{eq:e5x}) as a $ \\tau $-seconds ahead predictor as\n\\begin{equation}\\label{eq:e9xx}\n\\begin{aligned}\nx_{hp}(t+\\tau)&=e^{A_{hp}\\tau}x_{hp}(t)+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta)d\\eta. \n\\end{aligned} \n\\end{equation}\n\n\\begin{assumption}\nThere exist ideal parameters $ K_r^* $ and $ K_x^* $ satisfying the following matching conditions\n\\begin{equation}\\label{eq:e10x}\n\\begin{aligned}\n&A_{hp}+B_{hp}K_r^*K_x^*=A_m\\\\\n&B_{hp}K_r^*=B_m.\n\\end{aligned} \n\\end{equation}\n\\end{assumption} \n\n\n\nBy substituting (\\ref{eq:e9xx}) into (\\ref{eq:e8x}), the human pilot control decision input can be written as \n\\begin{equation}\\label{eq:e11xxxy}\n\\begin{aligned}\nu(t)&=K_rK_xe^{A_{hp}\\tau}x_{hp}(t)\\\\\n&+K_rK_x\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta)d\\eta+K_rr(t).\n\\end{aligned} \n\\end{equation}\nBy defining $ \\theta_x(t) $ and $ \\lambda(t,\\eta) $ as\n\\begin{equation}\\label{eq:e11xxxq}\n\\begin{aligned}\n\\theta_x(t)&=K_r(t)K_x(t)e^{A_{hp}\\tau},\\\\\n\\lambda(t,\\eta)&=K_r(t)K_x(t)e^{-A_{hp}\\eta}B_{hp},\n\\end{aligned} \n\\end{equation}\n(\\ref{eq:e11xxxy}) can be rewritten as (see fig. \\ref{fig:f1-2})\n\\begin{equation}\\label{eq:e11xxx}\n\\begin{aligned}\nu(t)=\\theta_x(t)x_{hp}(t)+\\int_{-\\tau}^{0}\\lambda(t,\\eta)u(t+\\eta)d\\eta+K_r(t)r(t).\n\\end{aligned} \n\\end{equation}\nThe ideal values of $ \\theta_x $ and $ \\lambda $ can be obtained as\n\\begin{equation}\\label{eq:e11xxxxx}\n\\begin{aligned}\n\\theta_x^*&=K_r^*K_x^*e^{A_{hp}\\tau}\\\\ \n\\lambda^*(\\eta)&=K_r^*K_x^*e^{-A_{hp}\\eta}B_{hp}.\n\\end{aligned} \n\\end{equation}\nSince $ A_{hp} $ and $ B_{hp} $ are unknown, $ \\theta_x $ and $ \\lambda $ need to be estimated. The closed loop dynamics can be obtained using (\\ref{eq:e5x}) and (\\ref{eq:e11xxx}) as\n\\begin{equation}\\label{eq:e11xxxx}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_{hp}x_{hp}(t)+B_{hp}\\theta_x(t-\\tau)x_{hp}(t-\\tau)\\\\ &+\\int_{-\\tau}^{0}B_{hp}\\lambda(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta\\\\ &+B_{hp}K_rr(t-\\tau),\n\\end{aligned} \n\\end{equation}\n\nDefining the deviations of the adaptive parameters from their ideal values as $ \\tilde{\\theta}_x=\\theta_x-\\theta_x^* $ and $ \\tilde{\\lambda}=\\lambda-\\lambda^* $, and adding and subtracting $ A_mx_{hp}(t) $ to (\\ref{eq:e11xxxx}), and using (\\ref{eq:e10x}), we obtain that\n\\begin{equation}\\label{eq:e12xxz}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_mx_{hp}(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)\\Big(e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\ &+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\ &+B_{hp}K_r(t-\\tau)r(t-\\tau).\\\\\n\\end{aligned} \n\\end{equation}\n Using (\\ref{eq:e9xx}), (\\ref{eq:e12xxz}) is rewritten as\n\\begin{equation}\\label{eq:e12xx}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_mx_{hp}(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)r(t-\\tau).\n\\end{aligned} \n\\end{equation}\nDefining the tracking error as $ e(t)=x_{hp}-x_{m} $, and subtracting (\\ref{eq:e7x}) from (\\ref{eq:e12xx}), and using (\\ref{eq:e10x}), and following a similar procedure given in \\cite{NarAnn12}, it is obtained that\n\\begin{equation}\\label{eq:e12xxxz}\n\\begin{aligned}\n\\dot{e}(t)&=\\dot{x}_{hp}-\\dot{x}_{m}\\\\\n&=A_me(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\ &+B_{hp}(K_r(t-\\tau)-K_r^*)r(t-\\tau)\\\\\n&=A_me(t)+\\big( -B_{hp}K_r^*K_x^*\\\\\n&+B_{hp}(K_r^*-K_r^*+K_r(t-\\tau))K_x(t-\\tau)\\big)x_{hp}(t) \n\\\\&+B_{hp}(K_r(t-\\tau)-K_r^*)r(t-\\tau)\\\\\n&=A_me(t)+B_{m}(K_x(t-\\tau)-K_x^*)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}K_r(t-\\tau)-1)K_x(t-\\tau)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}K_r(t-\\tau)-1)r(t-\\tau)\\\\\n&=A_me(t)+B_{m}(\\tilde{K}_x(t-\\tau)x_{hp}(t)\\\\\n&\\hspace{-0.1cm}+B_{m}({K_r^*}^{-1}-K_r^{-1}(t-\\tau))K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}-K_r^{-1}(t-\\tau))K_r(t-\\tau)r(t-\\tau).\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e9xx}) and defining $ \\Phi={K_r^*}^{-1}-K_r^{-1} $, we can rewrite (\\ref{eq:e12xxxz}) as\n\\begin{equation}\\label{eq:e12xxxzz}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+{B_mK_r^*}^{-1}(K_r^*K_x(t-\\tau)-K_r^*K_x^*)\\\\\n&\\hspace{-0.1cm}\\times\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_m\\Phi(t-\\tau)\\Big( K_r(t-\\tau)K_x(t-\\tau)\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+K_r(t-\\tau)r(t-\\tau) \\Big).\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e11xxxxx}) and (\\ref{eq:e12xxxzz}), we obtain that\n\\begin{equation}\\label{eq:e12xxxzzz}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_mK_x(t-\\tau)\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&-B_m{K_r^*}^{-1}\\Big( \\theta_x^*x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}\\lambda^*(\\eta)u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\\\\\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e11xxxq}), (\\ref{eq:e12xxxzzz}) can be rewritten as\n\\begin{equation}\\label{eq:e12xxxq}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_m\\Big( \\big( K_r^{-1}(t-\\tau)\\theta_x(t-\\tau)-{K_r^*}^{-1}\\theta_x^*\\big)\\\\\n&\\times x_{hp}(t-\\tau)+\\int_{-\\tau}^{0}\\big( K_r^{-1}(t-\\tau)\\lambda(t-\\tau,\\eta)\\\\\n&-{K_r^*}^{-1}\\lambda^*(\\eta) \\big) u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\n\\end{aligned} \n\\end{equation}\nDefining $ \\theta_1=K_r^{-1}\\theta_x $ and $ \\lambda_1=K_r^{-1}\\lambda $, and using their deviations from their ideal values, $ \\tilde{\\theta}_1=\\theta_1-\\theta_1^* $ and $ \\tilde{\\lambda}_1=\\lambda_1-\\lambda_1^* $, where \n$ \\theta_1^*={K_r^*}^{-1}\\theta_x^* $ and $ \\lambda_1^*={K_r^*}^{-1}\\lambda^* $, (\\ref{eq:e12xxxq}) can be rewritten as\n\\begin{equation}\\label{eq:e12xxx}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_{m}\\tilde{\\theta}_1(t-\\tau)x_{hp}(t-\\tau)\\\\ &+B_m\\int_{-\\tau}^{0}\\tilde{\\lambda}_1(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta \\\\ \n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\n\\end{aligned} \n\\end{equation}\nThe following lemma will be necessary to prove the main theorem of this article.\n\n\\begin{lemma}\\label{lem1}\nSuppose that the continuous function $ u(t) $ is given as\n\\begin{equation}\\label{eq:e21xx}\n\\begin{aligned}\nu(t)=f(t)+\\int_{-\\tau}^{0}\\lambda(t,\\eta)u(t+\\eta)d\\eta,\n\\end{aligned} \n\\end{equation}\nwhere $ u,f: [t_0-\\tau,\\infty]\\rightarrow R $, and $ \\lambda:[t_0, \\infty)\\times [-\\tau, 0]\\rightarrow R $. Then\n\\begin{equation}\\label{eq:e22xx}\n\\begin{aligned}\n|u(t)|\\leq 2(\\bar{f}+c_0c_1)e^{c_0^2(t-t')},\\ \\forall t_j'\\geq t_i',\n\\end{aligned} \n\\end{equation}\nif constants $ t_i',\\bar{f}, c_0, c_1\\in R^+ $ exist such that $ |f(t)|\\leq \\bar{f} $, \n\\begin{equation}\\label{eq:e23xx}\n\\begin{aligned}\n\\int_{-\\tau}^{0}\\lambda^2(t,\\eta)d\\eta\\leq c_0^2 \\ \\ for\\  t\\in [t_i',t_j'),\n\\end{aligned} \n\\end{equation}\nand\n\\begin{equation}\\label{eq:e24xx}\n\\begin{aligned}\n\\int_{-\\tau}^{0}u^2(t+\\eta)d\\eta\\leq c_1^2 \\ \\ \\forall\nt\\leq t_i'.\n\\end{aligned} \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe proof of Lemma \\ref{lem1} can be found in \\cite{YilAnn10}.\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{thm1}\nGiven the initial conditions $ \\tilde{\\theta}_1(\\xi)$, $ \\tilde{\\lambda}_1(\\xi,\\eta) $, $ \\Phi(\\xi) $ and $ x_{hp}(\\xi) $ for $ \\xi\\in [-\\tau, 0] $, and $ u(\\zeta) $ for $ \\zeta\\in[-2\\tau,0] $, there exists a $ \\tau^* $ such that for all $ \\tau\\in[0,\\tau^*] $, the controller (\\ref{eq:e11xxx}) with the adaptive laws\n\\begin{equation}\\label{eq:14thm}\n\\dot{{\\theta}}_1^T(t)=-x_{hp}(t-\\tau)e(t)^TPB_{m},\n\\end{equation} \n\\begin{equation}\\label{eq:15thm}\n\\dot{{\\Phi}}^T(t)=-u(t-\\tau)e(t)^TPB_{m},\\\\\n\\end{equation} \n\\begin{equation}\\label{eq:16thm}\n\\dot{\\lambda}_1^T(t,\\eta)=-u(t+\\eta-\\tau)e(t)^TPB_{m},\n\\end{equation} \nwhere $ P $ is the symmetric positive definite matrix satisfying\nthe Lyapunov equation $ A_m^TP +P A_m = -Q $ for a symmetric\npositive definite matrix $ Q $, which can be employed to obtain controller parameters using $ \\dot{K}_r=\\text{Proj}(K_r\\dot{\\Phi}K_r) $, $\\theta_x(t)=K_r(t)\\theta_1(t) $ and $ \\lambda(t)=K_r(t)\\lambda_1(t) $, make the pilot neuromuscular and plant aggregate system (\\ref{eq:e5x}) follow the crossover reference model (\\ref{eq:e7x}) asymptotically, i.e, $ lim_{t\\to \\infty} x_{hp}(t)=x_m(t) $, while keeping all the signals bounded.\n\\end{theorem}\n\\begin{proof}\nConsider a Lyapunov-Krasovskii functional \\cite{YilAnn10}\n\\begin{equation}\\label{eq:e12x}\n\\begin{aligned}\nV(t)&=e^TPe+\\text{tr}({\\Phi}^T(t){\\Phi}(t))+\\text{tr}(\\tilde{\\theta}_1^T(t)\\tilde{\\theta}_1(t))\\\\ &+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(\\xi)\\dot{\\tilde{\\theta}}_1(\\xi))d\\xi dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\text{tr}(\\dot{{\\Phi}}^T(\\xi)\\dot{{\\Phi}}(\\xi))d\\xi dv\\\\\n&+\\int_{-\\tau}^{0}\\text{tr}(\\tilde{\\lambda}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ &+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(\\xi,\\eta)\\dot{\\tilde{\\lambda}}_1(\\xi,\\eta))d\\eta d\\xi dv.\n\\end{aligned} \n\\end{equation}\nThe derivative of $ V(t) $ can be calculated as\n\\begin{equation}\\label{eq:e14x}\n\\begin{aligned}\n\\dot{V}(t)&=\\dot{e}^T(t)^TPe(t)+e^T(t)P\\dot{e}(t)+2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t))\\\\ &+2\\text{tr}(\\dot{{\\Phi}}^T(t){\\Phi}(t))+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ \n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nSubstituting (\\ref{eq:e12xxx}) into (\\ref{eq:e14x}) and using the Lyapunov equation $ A_m^TP +P A_m = -Q $, it is obtained that\n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)+2e^T(t)PB_m\\tilde{\\theta}_1(t-\\tau)x_{hp}(t-\\tau)\\\\\n&+2e^T(t)PB_m\\int_{-\\tau}^{0}\\tilde{\\lambda}_1(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta\\\\\n&+2e^T(t)PB_m\\Phi(t-\\tau)u(t-\\tau)\\\\\n&+2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t))+2\\text{tr}(\\dot{{\\Phi}}^T(t){\\Phi}(t)) \\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ \n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\notag\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:e14xx}\n\\begin{aligned}\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nUsing $ g(t)-g(t-\\tau)=\\int_{-\\tau}^{0}\\dot{g}(t+v)dv $, (\\ref{eq:e14xx}) can be rewritten as\n\\begin{equation}\\label{eq:e14xxxx}\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)\\\\\n&+2\\text{tr}\\Big(x_{hp}(t-\\tau)e^T(t)PB_m\\tilde{\\theta}_1(t)+\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t)\\Big)\\\\\n&+2\\text{tr}\\Big(u(t-\\tau)e^T(t)PB_m\\Phi(t)+\\dot{{\\Phi}}^T(t){\\Phi}(t)\\Big)\\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}\\Big(u(t+\\eta-\\tau)e^T(t)PB_m\\tilde{\\lambda}_1(t,\\eta)\\\\ &+\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta)\\Big)d\\eta\\\\\n&-2e^T(t)PB_m(\\int_{-\\tau}^{0}\\dot{\\tilde{\\theta}}_1(t+v)dv)x_{hp}(t-\\tau)\\\\\n&-2e^T(t)PB_m(\\int_{-\\tau}^{0}\\dot{{\\Phi}}(t+v)dv)u(t-\\tau)\\\\\n&-2e^T(t)PB_m\\Big(\\int_{-\\tau}^{0}(\\int_{-\\tau}^{0}\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)dv)\\\\\n&\\times u(t+\\eta-\\tau)d\\eta\\Big)\\\\\n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nBy substituting (\\ref{eq:14thm})-(\\ref{eq:16thm}) into (\\ref{eq:e14xxxx}), it is obtained that\n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)\\\\\n&-2\\int_{-\\tau}^{0}\\text{tr}(x_{hp}(t-\\tau)e(t)^TPB_{m}\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ &-2\\int_{-\\tau}^{0}\\text{tr}(u(t-\\tau)e(t)^TPB_{m}\\dot{{\\Phi}}(t+v))dv\\\\\n&-2\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(u(t+\\eta-\\tau)e(t)^TPB_{m}\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))dvd\\eta\\\\\n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv\\notag \n\\end{aligned} \n\\end{equation}\n\\begin{equation}\\label{eq:e17x}\n\\begin{aligned}\n&=-{e}^T(t)Qe(t)+\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t+v)\\\\\n&+\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t) -\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v)\\Big)dv\\\\\n&+\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t+v)+\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t)\\\\\n&-\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v)\\Big)dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)\\\\\n&+\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta)-\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)\\Big)d\\eta dv.\n\\end{aligned} \n\\end{equation}\nUsing the trace property $ \\text{tr}(A+B)=\\text{tr}(A)+\\text{tr}(B) $, and the algebraic inequality $ a^2\\geq 2ab-b^2 $ for two scalars $ a $ and $ b $, it can be shown that $ \\text{tr}(2A^TB+A^TA-B^TB)\\leq 2\\text{tr}(A^TA) $. Using these inequalities, (\\ref{eq:e17x}) can be rewritten as\n\\begin{equation}\\label{eq:e17xx}\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))dv\\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nBy substituting (\\ref{eq:14thm})-(\\ref{eq:16thm}) into (\\ref{eq:e17xx}), and using the trace operator property $ \\text{tr}(AB)=\\text{tr}(BA) $ for two square matrices $ A $ and $ B $, (\\ref{eq:e17xx}) can be rewritten as\n\\begin{equation}\\label{eq:e18x}\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)\\\\\n&+2\\tau \\text{tr}\\big(e(t)x_{hp}^T(t-\\tau)x_{hp}(t-\\tau)e(t)^TPB_{m}B_{m}^TP\\big)\\\\ \n&+2\\tau \\text{tr}\\big(e(t)u^T(t-\\tau)u(t-\\tau)e(t)^TPB_{m}B_{m}^TP\\big)\\\\\n&+2\\tau\\int_{-\\tau}^{0} \\text{tr}\\big(e(t)u^T(t+\\eta-\\tau)u(t+\\eta-\\tau)e(t)^T\\\\\n&\\times PB_{m}B_{m}^TP\\big)d\\eta.\n\\end{aligned} \n\\end{equation}\nUsing $ \\text{tr}(AB)\\leq \\text{tr}(A)\\text{tr}(B) $ for two positive semidefinite matrices $ A $ and $ B $, and $ \\text{tr}(X^TX)=||X||_F^2 $ for a matrix $ X $, an upper bound for (\\ref{eq:e18x}) can be derived as \n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)\\\\\n&+2\\tau \\text{tr}(e(t)x_{hp}^T(t-\\tau)x_{hp}(t-\\tau)e(t)^T)\\text{tr}(PB_{m}B_{m}^TP)\\\\ \n&+2\\tau \\text{tr}\\big(e(t)u^T(t-\\tau)u(t-\\tau)e(t)^T\\big)\\text{tr}\\big(PB_{m}B_{m}^TP\\big)\\\\\n&+2\\tau\\int_{-\\tau}^{0} \\text{tr}\\big(e(t) u^T(t-\\tau+\\eta)u(t-\\tau+\\eta)e(t)^T\\big)\\\\\n&\\times \\text{tr}\\big(PB_{m}B_{m}^TP\\big)d\\eta\\\\\n&\\leq -\\lambda_{min}(Q)||e(t)||^2\\\\\n&+2\\tau||x_{hp}(t-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2\\\\\n&+2\\tau ||u(t-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2\\\\\n&+2\\tau\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2d\\eta \\notag\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:e19x}\n\\begin{aligned}\n&\\leq -\\lambda_{min}(Q)||e(t)||^2\\\\\n&+2\\tau||x_{hp}(t-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2\\\\\n&+2\\tau ||u(t-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2\\\\\n&+2\\tau\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2d\\eta\\\\\n&=||B_{m}^TP||_F^2||e(t)||^2\\Big( -\\frac{\\lambda_{min}(Q)}{||B_{m}^TP||_F^2}\\\\\n&+2\\tau\\big( ||x_{hp}(t-\\tau)||^2+||u(t-\\tau)||^2\\\\\n&+\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2d\\eta \\big) \\Big).\n\\end{aligned} \n\\end{equation}\nDefining $ q\\equiv \\frac{\\lambda_{min}(Q)}{||B_{m}^TP||_F^2} $, the inequality\n\\begin{equation}\\label{eq:e20x}\n\\begin{aligned}\nq&-2\\tau\\big( ||x_{hp}(t-\\tau)||^2+||u(t-\\tau)||^2+\\\\\n&+\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2d\\eta \\big)> 0.\n\\end{aligned} \n\\end{equation}\nneeds to be satisfied for the non-positiveness of $ \\dot{V} $.\nAssuming that $ x_{hp} $ and $ u $ are bounded in the interval $ [t_0-2\\tau,t_0) $, the rest of the proof is divided into the following four steps:\n\n\\noindent\n\\textbf{Step 1} In this step, the negative semi-definiteness of the Lyapunov-Krasovskii functional's (\\ref{eq:e12x}) time derivative in the interval $ [t_0-\\tau,t_0) $ is shown which leads to the boundedness of the the signals in this interval. In addition, an upper bound for $ u $ in the interval $ [t_0-2\\tau,t_0) $ is given.\n\nSuppose that\n\\begin{equation}\n\\begin{aligned}\n\\sup_{\\xi\\in[t_0-\\tau, t_0)}&||x_{hp}(\\xi)||^2\\leq \\gamma_1\\\\\n\\sup_{\\xi\\in[t_0-2\\tau, t_0)}&||u(\\xi)||^2\\leq \\gamma_2\n\\end{aligned} \n\\end{equation} \nfor some positive $ \\gamma_1, \\gamma_2$, and a $ \\tau_1 $ is given such that\n\\begin{equation}\n\\begin{aligned}\n2\\tau_1(\\gamma_1+\\gamma_2+\\tau_1\\gamma_2)<q.\n\\end{aligned} \n\\end{equation} \nThen the following inequality is satisfied:\n\\begin{equation}\n\\begin{aligned}\nq&-2\\tau\\big( ||x_{hp}(\\xi-\\tau)||^2+||u(\\xi-\\tau)||^2+\\\\\n&+\\int_{-\\tau}^{0}||u(\\xi+\\eta-\\tau)||^2d\\eta \\big)> 0,\\\\ &\\forall \\xi\\in[t_0,t_0+\\tau), \\forall\\tau\\in[0, \\tau_1].\n\\end{aligned} \n\\end{equation}\nIt follows that $ V(t) $, defined in (\\ref{eq:e12x}), is non-increasing for $ t\\in[t_0, t_0+\\tau) $. Thus, we have\n\\begin{equation}\n\\begin{aligned}\n\\lambda_{min}(P)||e(\\xi)||^2\\leq e(\\xi)^TPe(\\xi)\\leq V(t_0),\n\\end{aligned} \n\\end{equation}\nwhich leads to\n\\begin{equation}\n\\begin{aligned}\n||x_{hp}(\\xi)||-||x_m(\\xi)||\\leq ||e(\\xi)||\\leq \\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}.\n\\end{aligned} \n\\end{equation}\nThen, we have\n\\begin{equation}\n\\begin{aligned}\n||x_{hp}(\\xi)||\\leq\\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}+||x_{m}(\\xi)||,\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\nWe also have the inequality\n\\begin{align}\\label{eq:x47}\n&||{\\Phi}(\\xi)||^2\\leq V(t_0)\\implies ||{K_r^*}^{-1}-K_r^{-1}(\\xi)||^2\\leq V(t_0)\\notag\\\\\n&\\implies ||K_r^{-1}(\\xi)||\\leq \\sqrt{V(t_0)}+||{K_r^*}^{-1}||.\n\\end{align}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\nIt is noted that the boundedness of $ \\Phi={K_r^*}^{-1}-K_r^{-1} $ does not guarantee the boundedness of $ \\tilde{K}_r $. In order to guarantee the boundedness of $ \\tilde{K}_r $ independent of the boundedness of $ \\Phi $, the projection algorithm \\cite{Eug13} is employed as \n\\begin{equation}\\label{eq:e19xzzz}\n\\begin{aligned}\n\\dot{K}_r=\\text{Proj}\\big(K_r,-K_rB_m^TPe(t)u^T(t-\\tau)K_r\\big),\n\\end{aligned} \n\\end{equation}\nwith an upper bound $ K_{max} $, that is $ ||{K}_r||\\leq K_{max} $. \nThus, a lower bound for $ ||K_r^{-1}(\\xi)|| $ can be calculated using the following algebraic manipulations: \n \\begin{align}\\label{eq:47x}\n &K_r(\\xi)K_r^{-1}(\\xi)=I\\Rightarrow ||K_r(\\xi)K_r^{-1}(\\xi)||=1\\notag \\\\\n &\\Rightarrow 1\\leq ||K_r(\\xi)||||K_r^{-1}(\\xi)||\\leq K_{max}||K_r^{-1}(\\xi)||\\notag \\\\\n & \\Rightarrow \\frac{1}{K_{max}}\\leq ||K_r^{-1}(\\xi)||.\n \\end{align}\nDefining $ k_1=\\sqrt{V(t_0)}+||{K_r^*}^{-1}|| $, and using (\\ref{eq:x47}), it is obtained that\n\\begin{align}\\label{eq:47}\n\\frac{1}{K_{max}}\\leq ||K_r^{-1}(\\xi)||\\leq k_1,\\ \\ \\ \\xi\\in[t_0,t_0+\\tau).\n\\end{align}\nTherefore, $ K_r $ is always bounded and $ K_r^{-1}(\\xi) $ is bounded for $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\n\nFurthermore, using the definitions of $ \\theta_x, \\theta_1, \\lambda,\\lambda_1 $ given in Theorem \\ref{thm1}, and the non-increasing Lyapunov functional (\\ref{eq:e12x}), it can be concluded that\n\\begin{align}\\label{eq:48}\n||\\tilde{\\theta}_1(\\xi)||_F^2\\leq V(t_0)&\\implies ||\\tilde{K}_r^{-1}(\\xi)\\tilde{\\theta}_x(\\xi)||_F^2\\leq V(t_0),\n\\end{align}\n\\begin{align}\\label{eq:49}\n\\int_{-\\tau}^{0}||\\tilde{\\lambda}_1(\\xi,\\eta)||_F^2 &d\\eta \\leq V(t_0)\\\\\n&\\implies \\int_{-\\tau}^{0}||K_r^{-1}(\\xi)\\tilde{\\lambda}(\\xi,\\eta)||_F^2d\\eta\\leq V(t_0),\\notag\n\\end{align} \nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $. Using (\\ref{eq:48}) and (\\ref{eq:49}), it can be obtained that\n\\begin{equation}\n\\begin{aligned}\n||\\tilde{\\theta}_x(\\xi)||_F^2&\\leq K_{max}^2V(t_0),\\\\\n\\int_{-\\tau}^{0}||\\tilde{\\lambda}(\\xi,\\eta)||_F^2d\\eta&\\leq K_{max}^2V(t_0).\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\n\nTo simplify the notation, we define\n\\begin{equation}\n\\begin{aligned}\nI_0\\equiv \\text{max}&\\Big( \\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}+\\sup_{[t_0,t_0+\\tau)}||x_{m}(\\xi)||\\\\ &, K_{max}\\sqrt{V(t_0)}, K_{max}^2V(t_0) \\Big),\n\\end{aligned} \n\\end{equation}\nwhere $ R_{max} $ is the upper bound of the reference input $ r(t) $.\n\nAn upper bound on the control signal $ u(t) $ for $ t\\in[t_0, t_0+\\tau) $ can be derived by using Lemma \\ref{lem1} and (\\ref{eq:e11xxx}). In particular, setting $ t_i'=t_0 $, $ t_j'=t_0+\\tau $, $ c_0^2=V(t_0) $, we obtain that\n\\begin{equation}\\label{eq:e50}\n\\begin{aligned}\n|u(\\xi)|\\leq 2\\Big( \\bar{f}+\\big( \\int_{-\\tau}^{0}u^2(t_0+\\eta)d\\eta \\big)^{1\/2}I_0 \\Big)e^{I_0\\tau},\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $, where $ \\bar{f} $, which is the upper bound of $ \\theta_x(t)x_{hp}(t)+K_r(t)r(t) $, depends only on $ I_0 $. Defining $ g(\\gamma_2, I_0, \\tau)\\equiv 2(\\bar{f}+\\gamma_2I_0\\sqrt{\\tau})e^{I_0\\tau} $, (\\ref{eq:e50}) can be rewritten as\n\\begin{equation}\n\\begin{aligned}\n|u(\\xi)|\\leq g(\\gamma_2, I_0, \\tau),\\ \\forall \\xi\\in[t_0,t_0+\\tau).\n\\end{aligned} \n\\end{equation}\n\n\nThe rest of the proof is similar to the one given in \\cite{YilAnn10}. Below, a summary of the next steps are given.\n\n\\noindent\n\\textbf{Step 2} A delay range $ [0, \\tau_2] $ is found that satisfies the condition (\\ref{eq:e20x}) over the interval $ [t_0, t_0+2\\tau) $ as\n\\begin{align}\\label{eq:step2}\n\t2\\tau_2\\left( I_0^2+\\left( max\\left( \\gamma_2,g\\left( \\gamma_2,I_0,\\tau_2 \\right)  \\right)  \\right)^2 (1+\\tau_2) \\right) < q,\n\\end{align} \nwhich leads to $ ||x_{hp}(\\xi)||<I_0 $, $ \\forall\\xi\\in[t_0, t_0+2\\tau) $, $ \\forall \\tau\\in [0, \\bar{\\tau}_2] $, $ \\bar{\\tau}_2=min\\{\\tau_1, \\tau_2\\} $.\n\n\\noindent\n\\textbf{Step 3} It is shown in this step that the bound on $ u $ over the interval $ [t_0, t_0+\\tau) $ depends only on $ A_{hp} $, $ B_{hp} $, $ T $ and $ \\tau $, where $ T $ is a value between $ t_0 $ and $ \\tau $. Denoting this upper bound as $ U(I_0) $, we have $ |u(t)|\\leq U(I_0) $, $ t\\in[t_0,t_0+\\tau) $.\n\n\\noindent\n\\textbf{Step 4} Using the calculated upper bound for $ u $ in the previous step, a delay range $ [0, \\tau_2] $ is found that satisfies the condition \n\\begin{align}\\label{eq:step4}\n2\\tau_3\\left( I_0^2+\\left( max\\left( U(I_0),g\\left( U(I_0),I_0,\\tau_3 \\right)  \\right)  \\right)^2 (1+\\tau_3) \\right) < q.\n\\end{align} \nFor $ \\tau^*=min(\\bar{\\tau_2}, \\tau_3) $, $ ||x_{hp}(\\xi)||\\leq I_0 $ and $ |u(\\xi)|\\leq U(I_0) $ for all $ \\xi \\in [t_0, t_0+\\tau] $, $ \\forall \\tau\\in [0, \\tau^*] $.\n\n\n\nThe above four steps show that $ x_{hp}(\\xi) $ and $ u(\\xi) $ are bounded for $ \\forall\\xi\\in[t_0, t_0+k\\tau] $, for $ k=1 $ and $ \\tau  \\in (0, \\tau^*]$. By assuming that $ x_{hp} $ and $ u $ are bounded for a given $ k $, the rest of the proof consists of showing that the boundedness of these variables hold for $ k+1 $. Using this assumption and repeating steps 1-4, which leads to satisfying (\\ref{eq:step4}), we conclude that the Lyapunov function is non-increasing and $ ||x_{hp}(\\xi)|| \\leq I_0 $, and $ |u(\\xi)|\\leq g(U(I_0),I_0,\\tau) $ for $ \\xi\\in [t_0, t_0+(k+1)\\tau] $, $ \\tau \\leq \\tau^* \\leq \\tau_3 $.\nThis completes the boundedness proof using induction.\nThen, using Barbalat's Lemma, it can be shown that the error between the human-in-the-loop system output $ x_{hp} $ and the reference model output $ x_m $ converges to zero.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\section{Experimental Results}\\label{sec:experiment}\n\\subsection{Experimental environment}\n\nIn order to test the proposed adaptive human model against data, an experimental setup consisting of a Logitech Extreme 3D Pro joystick and a Toshiba Portege-Z30-B laptop with Intel Core i7 CPU is used (see Fig. \\ref{fig:setup}). \n\n\n\n\nThe tracking task is performed by an operator monitoring the compensatory display, which provides information about the error between the target to be followed, and follower, which is the output of the plant (see Fig. \\ref{fig:attitudeind}). The operator provides the input $ u_p $ (see Fig. \\ref{fig:f1-2}) through the joystick, which is fed to the plant using MATLAB SIMULINK (R2018b). In return, the response of the plant is calculated and shown on the laptop screen in real-time.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{setup.jpg}    %\n\t\t\\caption{Experimental setup.} \n\t\t\\label{fig:setup}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{attitudeindx.jpg}    %\n\t\t\\caption{Compensatory display.} \n\t\t\\label{fig:attitudeind}\n\t\\end{center}\n\\end{figure}\n\nThe reference signal $ r(t) $ is generated as a sum of the sinusoids with frequencies of 0.1, 0.3, 0.5, 0.7, 1, 1.3 and 1.5 rad\/s with the same amplitude of 0.2 and without phase shift.\n\n\nThree classes of plant models, having zero, first and second order transfer functions are used in the experiments. In this section, we first give a detailed analysis of the first order plant case and then provide a summary of the results of the other cases in tables. The nominal\nfirst order plant used in the experiments is $ Y_p(s)=\\frac{4}{s+1} $, which is similar to the one used in \\cite{Hess09}. The uncertainty is introduced to the plant model by modifying the gain and the pole location by $ 50\\% $ to obtain $ Y_p(s)=\\frac{6}{s+0.5} $. \n\n\n\n\n\nTo form the reference model (\\ref{eq:e7x}), two parameters, namely the crossover frequency and the time-delay, need to be determined. The highest frequency component of the reference signal is $ \\omega_i=1.5 $ rad\/s. Employing Table I for the first order plant $ Y_p $, the crossover frequency is calculated as $ \\omega_c=4.5 $ rad\/s. The delay is determined by using the mean value of the operators' delay, which is $ \\tau=0.3 $ s. Therefore, the closed loop transfer function of the reference model is calculated as\n\t\\begin{equation}\\label{eq:e21xq}\n\t\\begin{aligned}\n\tG_{cl}(s)=\\frac{\\frac{4.5}{s}e^{-0.3s}}{1+\\frac{4.5}{s}e^{-0.3 s}}=\\frac{4.5 e^{-0.3 s}}{s+4.5 e^{-0.3 s}}.\n\t\\end{aligned} \n\t\\end{equation}\n\t\n\tSimilar to (\\ref{eq:e21xqqx}), an approximate transfer function is obtained as\n\t\\begin{equation}\\label{eq:e21xqq}\n\t\\begin{aligned}\n\t\\hat{G}_{cl}(s)=\\frac{3.881s+24.24}{s^2+0.6834s+24.72}e^{-0.3s}.\n\t\\end{aligned} \n\t\\end{equation}\n\tFigure \\ref{fig:bode1} shows a comparison between (\\ref{eq:e21xq}) and (\\ref{eq:e21xqq}), and demonstrates that the approximation works well for almost all frequencies.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{bode1-eps-converted-to.pdf}    %\n\t\t\\caption{Bode plot of the reference model and its approximation.} \n\t\t\\label{fig:bode1}\n\t\\end{center}\n\\end{figure}\n\n\n\n\nThe neuromuscular dynamics is taken as $ Y_h(s)=\\frac{s+3}{s+2}e^{-0.3s} $, where the time delay $ \\tau=0.3 $ is the effective time delay, including human decision making delay and neuromuscular lags.\n\n\\begin{remark}\\label{rem3}\n\tIn this paper, we assume that the neuromuscular dynamics are given. The procedure for finding the neuromuscular model can be found in \\cite{Mag71, Van04}.  \n\\end{remark}\n\n\\subsection{The behavior of the adaptive model}\n\n\n The error between the plant output and the reference model is illustrated in Figure \\ref{fig:error}. The effect of uncertainty injection can be seen at $ t=70 $ s. Figures \\ref{fig:theta}, \\ref{fig:lambda}, and \\ref{fig:Kr} illustrates the adaptive human model parameters. To understand the amount of agreement between these results and the human experimental trials, visual and statistical analyses are provided in the following sections.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{error-eps-converted-to.pdf}    %\n\t\t\\caption{Time evolution of the error between the output of the plant controlled by the adaptive model, and the reference model output.} \n\t\t\\label{fig:error}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{theta-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameters $ \\theta_{x1} $ and $ \\theta_{x2} $.} \n\t\t\\label{fig:theta}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{lambda-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameters $ \\lambda_i,\\ i=1, 2, 3 $ and $ 4 $.} \n\t\t\\label{fig:lambda}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{Kr-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameter $ K_r $.} \n\t\t\\label{fig:Kr}\n\t\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Participants and experimental procedure\n\nEleven participants (6 women and 5 men) from the graduate and undergraduate student pools of Bilkent University participated the experiment. All of the participants read and signed the ``informed consent to participate\" document. This study is approved by Bilkent University Ethics Committee for research with human participants. Before the experiments, to familiarize the participants with the experimental setup, and its environment, consisting of the display and the joystick, each participant was asked to follow a given reference via joystick inputs for the duration of $ 200 $ seconds. To prevent learning during these warm-up runs, the reference input, uncertainty injection times and the uncertainty types were chosen differently from the ones used in the real experimental runs. Specifically, the reference signal for the warm-up runs consisted of the sum of the sinusoids with frequencies of $ 0.1, 0.5, 1 $ and $ 1.5 $ rad\/s with the same amplitude of $ 0.2 $ and without phase shift. The plant dynamics at the beginning of the warm-up run was $ \\frac{2}{s^2+3s+2} $. At $ t = 45 $ s, the dynamics changed to $ \\frac{5}{s+2} $ in a step like manner (suddenly). It changed to $ \\frac{3}{s+1} $ at around $ t = 90 $ s using a sigmoid function (gradually), and again changed to a zero order dynamics at $ 150 $ s (suddenly). \n\n\n\n\n\n  \n\n\n\n\n\\subsection{A visual analysis of the adaptive model}\n\nLet $ f_{p1}(t), f_{p2}(t), ..., f_{pk}(t) $ be the plant outputs when participants $ p1, p2, ..., pk $ are in the loop, respectively. For each $ f_{pi}(t) $, $ t=T_1, T_2, ..., T_N $, where $ T_j, j=1, 2, ..., N $, represents a sampling instant. At each sampling instant $ T_j $, the minimum, the maximum and the mean values of the plant outputs when participants are in the loop can be obtained as\n\\begin{align}\nf_{p_{min}}(T_j)&=\\min_{i=1, 2, ..., k}f_{pi}(T_j),\\ \\ j=1, ..., N,\\label{eq:estat4} \\\\\nf_{p_{max}}(T_j)&=\\max_{i=1, 2, ..., k}f_{pi}(T_j),\\ \\ j=1, ..., N, \\label{eq:estat5} \\\\\nf_{p_{mean}}(T_j)&=\\frac{\\sum_{i=1}^{k}f_{pi}(T_j)}{k}, \\ \\ j=1, ..., N,\n\\end{align} \n where $ k=11 $ is the number of participants. Figure \\ref{fig:minmaxadapt} shows the evolutions of  $ f_{p_{min}} $ and $ f_{p_{max}} $, together with $ f_{ad}(t)\\in \\mathbb{R}^N $, which is the plant output when adaptive human model is in the loop, where $ t= T_1, T_2, ..., T_N $. It is seen that the plant output when adaptive human model is in the loop almost always stays between the maximum and the minimum values of the plant output when participants are in the loop. Furthermore, Figure \\ref{fig:meanadapt} demonstrates that $ f_{p_{mean}} $ and $ f_{ad} $ evolve reasonably close to each other.\n  \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{minmaxadapt1-eps-converted-to.pdf}    %\n\t\t\\caption{Plant output, $ x_{hp} $, when adaptive human model is in the loop vs. minimum and maximum values of plant output when participants are in the loop.} \n\t\t\\label{fig:minmaxadapt}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{meanadapt1-eps-converted-to.pdf}    %\n\t\t\\caption{Plant output, $ x_{hp} $, when adaptive human model is in the loop vs. mean value of plant output when participants are in the loop.} \n\t\t\\label{fig:meanadapt}\n\t\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Statistical analysis of the adaptive model using confidence intervals}\n\nThe difference between the plant output when the $ i^{\\text{th}} $ participant is in the loop and when the adaptive human model is in the loop is defined as\n\\begin{equation}\\label{eq:estat1}\n\\begin{aligned}\nd_i\\equiv f_{ad}-f_{pi}, \\ \\ i=1, ..., k,\n\\end{aligned} \n\\end{equation}\nwhere $ d_i=[d_i(T_1), ..., d_i(T_N)]^T\\in \\mathbb{R}^N, i=1, ..., k $, is called the $ i^{\\text{th}} $ difference. The mean and the standard deviation of the $ i $th difference is obtained as\n\\begin{align}\n\\bar{d}_i&= \\frac{\\sum_{j=1}^{N}d_i(T_j)}{N}, \\ \\ i=1, ..., k,\\label{eq:estat2} \\\\\ns_i&=\\sqrt{\\frac{\\sum_{j=1}^{N}(d_i(T_j)-\\bar{d}_i)^2}{N-1}}, \\ \\ i=1, ..., k.\\label{eq:estat3}\n\\end{align}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{normalscores-eps-converted-to.pdf}    %\n\t\t\\caption{Normal-scores plot} \n\t\t\\label{fig:normalscores}\n\t\\end{center}\n\\end{figure}\n\nThe normal-scores plot for $ \\bar{d}_i $ is given in Figure \\ref{fig:normalscores}. The figure does not show any significant deviation from the normal distribution. This shows us that the data do not suggest that the \\textit{population} of mean-errors, $ \\bar{d}_i $, deviates significantly from normal distribution. The \\textit{sample} mean and the \\textit{sample} standard deviation of $ \\bar{d}_i $'s can be obtained as\n\\begin{align}\n\\bar{d}&=\\frac{\\sum_{i=1}^{k}\\bar{d}_i}{k}, \\label{eq:estat6}\\\\\ns&=\\sqrt{\\frac{\\sum_{i=1}^{k}(\\bar{d}_i-\\bar{d})^2}{k-1}}. \\label{eq:estat7}\n\\end{align}\nLet $ \\mu_0 $ be the mean value of the \\textit{population} of mean-errors, given as\n\\begin{align}\\label{eq:estat8}\n\\mu_0\\equiv \\frac{\\sum_{i=1}^{K}\\bar{d}_i}{K},\n\\end{align}\nwhere $ K $ is the \\textit{population} size. Since normal-scores plot, given in Figure \\ref{fig:normalscores}, didn't provide any counter evidence, assuming that the distribution of the set of data $ \\{\\bar{d}_1, ..., \\bar{d}_K\\} $ is normal with mean $ \\mu_0 $, $ \\mu_0 $ satisfies the following probability \\cite{Joh19}\n\\begin{align}\\label{eq:estat9}\nP\\left[\\bar{d}-t_{\\alpha\/2}\\frac{s}{\\sqrt{k}}<\\mu_0<\\bar{d}+t_{\\alpha\/2}\\frac{s}{\\sqrt{k}} \\right]=1-\\alpha,\n\\end{align}\nwhere $ \\bar{d} $ and $ s $ are obtained from (\\ref{eq:estat6}-\\ref{eq:estat7}), $ k $ is the number of participants, $ \\alpha $ is the significance level, and $ t_{\\alpha\/2} $ is the upper $ \\alpha\/2 $ point of the $ t $ distribution with degree of freedom $ k-1 $, which can be obtained from the $ t $-distribution table. Since the number of participants, $ k=11 $, is less than $ 30 $, it is appropriate to use the $ t $-distribution.\nUsing $ \\alpha=0.05 $, obtaining $ t_{\\alpha\/2} $ from the $ t $-distribution table as $ 2.228 $, and calculating $ \\bar{d} $ as $ -0.0068 $ and $ s $ as $ 0.0379 $, it can be concluded using (\\ref{eq:estat9}) that we are $ 95\\% $ confident that $ \\mu_0 $ is in the interval $ (-0.0323, 0.0187) $. This shows that the mean $ \\mu_0 $ of the \\textit{population}'s mean deviation from the adaptive human model is reasonably close to zero. \n\n\n\nSimilarly, the variance, $ \\sigma_0^2 $, of the \\textit{population}'s mean deviation from the adaptive human model satisfies the following probability \n\\cite{Joh19}\n\\begin{align}\\label{eq:estat11}\nP\\left[\\frac{(k-1)s^2}{\\chi_{\\alpha\/2}^2}<\\sigma_0^2<\\frac{(k-1)s^2}{\\chi_{1-\\alpha\/2}^2} \\right]=1-\\alpha,\n\\end{align}\nwhere $ \\chi_{\\alpha\/2}^2 $ is the upper $ {\\alpha\/2} $ point of the $ \\chi^2 $ distribution with degree of freedom $ k-1 $ and can be obtained from the $ \\chi^2 $ distribution table. Calculating $ s $ from (\\ref{eq:estat7}), using $ \\alpha = 0.05 $, and obtaining $ \\chi_{\\alpha\/2}^2 $ and $ \\chi_{1-\\alpha\/2}^2 $ from the $ \\chi^2 $ table with $ 10 $ degrees of freedom, it can be concluded using (\\ref{eq:estat11}) that we are $ 95\\% $ confident that $ \\sigma_0 $ is in the interval $ (0.0265,\\ 0.0663) $. This shows that the standard deviation $ \\sigma_0 $ of the \\textit{population}'s mean deviation from the adaptive human model is reasonably small. \n\n\n\n\n\\subsection{Statistical analysis of the adaptive model using hypothesis testing}\n\n\n\n\n\nIn this analysis, we test whether the hypothesis ``the mean value of the \\textit{population} mean-errors, or the mean deviations from the adaptive model,\" is zero. In other words, our null hypothesis, $ H_0 $, is given as\n\\begin{align}\nH_0: \\mu_0=0,\n\\end{align}\nwhere $ \\mu_0 $ is defined in (\\ref{eq:estat8}). The alternative hypothesis, $ H_1 $, is given as $ H_1: \\mu\\neq 0 $. Similar to the confidence interval analysis, assuming that $ \\mu_0 $ is the mean of a normally distributed set of data $ \\{\\bar{d}_1, ..., \\bar{d}_K\\} $ where $ K $ is the \\textit{population} size, the hypothesis $ H_0 $ is rejected if,\n\\begin{align}\\label{eq:estat10}\n\\left| \\frac{(\\bar{d}-\\mu_0)\\sqrt{k}}{s}\\right| \\geq t_{\\alpha\/2},\n\\end{align}\nwhere $ \\bar{d} $ and $ s $ are obtained from (\\ref{eq:estat6}-\\ref{eq:estat7}), $ k $ is the number of participants and $ t_{\\alpha\/2} $ is the upper $ \\alpha\/2 $ point of the $ t $ distribution with degree of freedom $ k-1 $ \\cite{Joh19}. Using the significance level $ \\alpha=0.05 $ and degree of freedom $ k-1=10 $, obtaining $ t_{0.025}=2.228 $ from the $ t $-distribution table, calculating $ \\bar{d}=-0.0068 $ and $ s=0.038 $ using (\\ref{eq:estat6}) and (\\ref{eq:estat7}), respectively, and substituting $ \\mu_0=0 $ and $ k=11 $, the left hand side of (\\ref{eq:estat10}) can be calculated as $ 0.5935 $, which is less than $ t_{\\alpha\/2} $. Therefore, we cannot reject $ H_0 $. We retain $ H_0 $ and conclude that $ H_1 $ fails to be proved.\n \nSince we are retaining the null hypothesis, we want to minimize the probability $ \\beta $ of incorrectly retaining the null hypothesis. This means that we want our test's power, $ 1-\\beta $, to be large, such as $ 0.95 $. What is the minimum required deviation of the \\textit{population} mean from $ 0 $, represented as $ \\mu_1 $, that would make our test to incorrectly retain the null hypothesis with $ 0.05 $ probability, i.e. $ \\beta=0.05 $? To calculate this, we first write the rejection region, $ R $, using (\\ref{eq:estat10}) as\n\\begin{align}\nR:\\ \\left| \\frac{(\\bar{d}-\\mu_0)\\sqrt{k}}{s}\\right| \\geq t_{\\alpha\/2}\\ \n \\implies R:\\ |\\bar{d}|\\geq 0.0255.\n\\end{align}\nDefining $ T=\\frac{(\\bar{d}-\\mu_1)\\sqrt{k}}{s} $, for $ \\beta=0.05 $, we need\n\\begin{align}\\label{eq:estat12}\n&P\\left[ \\frac{(-0.0255-\\mu_1)\\sqrt{k}}{s}<T<\\frac{(0.0255-\\mu_1)\\sqrt{k}}{s} \\right]  \\notag \\\\\n&=\\frac{\\beta}{2}=0.025.\n\\end{align}\nUsing the $ t $-table, it can be found that the minimum $ |\\mu_1| $ that satisfies (\\ref{eq:estat12}) is $ 0.051 $. This means that our test can detect an $ 0.051 $ deviation from the mean value of the mean-error between the adaptive human model and the participant data when the probability of the test to incorrectly conclude that the model and the data are compatible ($ \\mu_0=0 $) is only $ 5\\% $. \n\n\\begin{table}\\label{table2}\n\n\t\\caption{Sudden uncertainty}\n\t\\centering\n\t\\begin{tabular}{ | c | c | c | c | }\n\t\t\\hline \n\t\t& 0 order & 1st order & 2nd order \\\\\n\t\t\\hline \n\t\tTF before $ 70 $ s & $ 4 $ & $ \\frac{4}{s+1} $ & $ \\frac{4}{(s+1)(s+5)} $ \\\\\n\t\t\\hline\n\t\tTF after $ 70 $ s & $ 6 $ & $ \\frac{6}{s+0.5} $ & $ \\frac{6}{(s+0.5)(s+2.5)} $ \\\\\n\t\t\\hline\n\t\t$ \\bar{d} $ & $ 0.0085 $ & $ -0.0068 $ & $ 0.0011 $ \\\\\n\t\t\\hline \n\t\t$ s $ & $ 0.0339 $ & $ 0.0379 $ & $ 0.0252 $ \\\\\n\t\t\\hline\n\t\tMean conf. int. & $ (-0.014, 0.03) $ & $ (-0.032, 0.02) $ & $ (-0.016, 0.018) $ \\\\\n\t\t\\hline\n\t\tSt.d. conf. int. & $ (0.024, 0.06) $ &  $ (0.026, 0.066) $ & $ (0.018, 0.044) $ \\\\\n\t\t\\hline\n\t\tHypothesis test & $ H_0 $ is retained & $ H_0 $ is retained  & $ H_0 $ is retained \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\\begin{table}\\label{table3}\n\n\t\\caption{Gradual uncertainty}\n\t\\centering\n\t\\begin{tabular}{ | c | c | c | c | }\n\t\t\\hline \n\t\t& 0 order & 1st order & 2nd order \\\\\n\t\t\\hline \n\t\tTF before $ 70 $ s & $ 4 $ & $ \\frac{4}{s+1} $ & $ \\frac{4}{(s+1)(s+5)} $ \\\\\n\t\t\\hline\n\t\tTF after $ 70 $ s & $ 6 $ & $ \\frac{6}{s+0.5} $ & $ \\frac{6}{(s+0.5)(s+2.5)} $ \\\\\n\t\t\\hline\n\t\t$ \\bar{d} $ & $ 0.0154 $ & $ 0.0026 $ & $ 0.004 $ \\\\\n\t\t\\hline \n\t\t$ s $ & $ 0.038 $ & $ 0.034 $ & $ 0.03 $ \\\\\n\t\t\\hline\n\t\tMean conf. int. & $ (-0.01, 0.04) $ & $ (-0.02, 0.025) $ & $ (-0.016, 0.023) $ \\\\\n\t\t\\hline\n\t\tSt.d. conf. int. & $ (0.026, 0.067) $ & $ (0.0235, 0.06) $ & $ (0.02, 0.05) $ \\\\\n\t\t\\hline\n\t\tHypothesis test & $ H_0 $ is retained & $ H_0 $ is retained  & $ H_0 $ is retained \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\n\n\nAnalyses of the experimental results where a first order plant dynamics is used with a sudden uncertainty injection is provided above. All of the results, including the ones for the other cases, where plants with different orders and sudden\/gradual uncertainty injections, are summarized in Tables II and III.\nThe data collected from the participants can be reached at \\href{http:\/\/www.syslab.bilkent.edu.tr\/research}{http:\/\/www.syslab.bilkent.edu.tr\/research}.\n\n\n\n\n\\section{SUMMARY}\\label{sec:conclusion}\n\nIn this paper, an adaptive human pilot model based on model reference adaptive control principles is proposed. This model mimics the pilot decision making process by making sure that the overall closed loop system follows the crossover model in the presence of plant uncertainties. The stability of the system is shown using the Lyapunov-Krasovskii stability criteria. Furthermore, experiments with human operators are conducted to validate the model. Detailed visual and statistical analyses of the experimental results show that the adaptive model creates similar system responses as the human operators. \n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n\n\\section{INTRODUCTION}\n\nHumans' unique abilities such as adaptive behavior in dynamic environments, and social interaction and moral judgment capabilities, make them essential elements of many control loops. On the other hand, compared to humans, automation provides higher computational performance and multi-tasking capabilities without any fatigue, stress, or boredom \\cite{Noth16,Kor15}. Although they have their own individual strengths, humans and automation also demonstrate several weaknesses. Humans may have anxiety, fear and may become unconscious during an operation. Furthermore, in the tasks that require increased attention and focus, humans tend to provide high gain control inputs that can cause undesired oscillations. One example of this phenomenon, for example, is the occurrence of pilot induced oscillations (PIO), where undesired and sustained oscillations are observed due to an abnormal coupling between the aircraft and the pilot \\cite{YilKol10, Yil11a,AcoYil14,TohYil18}. Similarly, automation may fail due to an uncertainty, fault or cyber-attack \\cite{Li14}. Thus, it is more preferable to design systems where humans and automation work in harmony, complementing each other, resulting in a structure that benefits from the advantages of both.\n\nTo achieve a reliable human-automation harmony, a mathematically rigorous human operator model is paramount. A human operator model helps develop safe control systems, and provide a better prediction of human actions and limitations \\cite{Hul13,Yuc18,Emre19, Zha19}. \nQuasi-linear model \\cite{Mcr57} is one of the first human operator models, which consists of a describing function and a remnant signal accounting for nonlinear behavior. An overview of this model is provided in \\cite{Mcr74}. In some applications, where the linear behavior may be dominant, the nonlinear part of this model can be ignored, and the resulting lead-lag-type compensator is used in closed loop stability analysis \\cite{Neal71}. The crossover model, proposed in \\cite{Mcr63}, is another important human operator model in the aerospace domain. It is motivated from the empirical observations that human pilots adapt their responses in such a way that the overall system dynamics resembles that of a well designed feedback system \\cite{Bee08}. A generalized crossover model which mimics human behavior when controlling a fractional order plant is proposed in \\cite{Mar17}. In \\cite{War16}, crossover model is employed to provide information about the human intent for the controller. In \\cite{Gil09}, the dynamics of the operator is represented as a spring-damper-mass system. \n\n\nControl theoretical operator models drawing from the optimal and adaptive control theories are also proposed by several authors. \nOptimal human models are based on the idea that a well trained human operator behaves in an optimal manner \\cite{Wier69, Kle70, Na12, Lon13, Hu19}. On the other hand, adaptive models, such as the ones proposed in \\cite{Hess09, Hess15} and \\cite{TohYil19Human}, aim to replicate the adaptation capability of humans in uncertain and dynamics environments. In \\cite{Hess09} and \\cite{Hess15}, adaptation rules are proposed based on expert knowledge. The adaptive model proposed in \\cite{Hess09} is applied to change the parameters of the pilot model based on force feedback from a smart inceptor \\cite{Xu19}. \nA survey on various pilot models can be found in \\cite{Lon14} and \\cite{Xu17}.\n\n\n\n\n\n Several approaches are also developed for human model parameter identification. In \\cite{Zaal11}, a two-step method using wavelets and a windowed maximum likelihood estimation is exploited for the estimation of time-varying pilot model parameters. In \\cite{Duar17}, a linear parameter varying model identification framework is incorporated to estimate time-varying human state space representation matrices. Subsystem identification is used in \\cite{Zha15} to model human control strategies. In \\cite{Van15}, a human operator model for preview tracking tasks is derived from measurement data.\n\n\nIn this paper, we build upon the earlier successful pilot models and propose an adaptive human pilot model that modifies its behavior based on plant uncertainties. This model distinguishes itself from earlier adaptive models by having mathematically derived laws to achieve a cross-over-model-like behavior, instead of employing expert knowledge. This allows a rigorous stability proof, using the Lyapunov-Krasovskii stability criteria, of the overall closed loop system. To validate the model, a setup, including a joystick and a monitor, is used. The participant data collected through this experimental setup is subjected to visual and statistical analyses to evaluate the accuracy of the proposed model. Initial research results of this study were presented in \\cite{TohYil19Human}, where the details of the mathematical proof and human experimental validation studies were missing.\n\n\nThis paper is organized as follows. In Section \\ref{sec:prob}, the problem statement is given. Obtaining reference model parameters, which determine the properties of the cross-over model, is discussed in Section \\ref{sec:reference}. Section \\ref{sec:human} presents the human control strategy together with a stability analysis. Experimental set-up, results, and a statistical analysis are provided in Section \\ref{sec:experiment}. Finally, a summary is given in Section \\ref{sec:conclusion}.\n\n\n\\section{PROBLEM STATEMENT}\\label{sec:prob}\n\nAccording to McRuer's crossover model \\cite{Mcr67}, human pilots in the control loop behave in a way that results in an open loop transfer function\n\\begin{equation}\\label{eq:e1x}\nY_{OL}(s)=Y_{h}(s)Y_{p}(s)=\\frac{\\omega_ce^{-\\tau s}}{s},\n\\end{equation}\nnear the crossover frequency, $ \\omega_c $, where $ Y_{h} $ is the transfer function of the human pilot and $ Y_{p} $ is the transfer function of the plant. $ \\tau $ is the effective time delay, including transport delays and high frequency neuromuscular lags.\n\nConsider the following plant dynamics\n\\begin{equation}\\label{eq:e2x}\n\\dot{x}_p(t)=A_px_p(t)+B_p u_p(t),\n\\end{equation}\nwhere $ x_p\\in \\mathbb{R}^{n_p} $ is the plant state vector, $ u_p\\in \\mathbb{R} $ is the input vector, $ A_p\\in \\mathbb{R}^{n_p\\times n_p} $ is an unknown state matrix and $ B_p\\in \\mathbb{R}^{n_p} $ is an unknown input matrix.\n\nThe human \\textit{neuromuscular model} \\cite{Mag71, Van04} is represented in state space form as\n\\begin{equation}\\label{eq:e3x}\n\\begin{aligned}\n\\dot{x}_h(t)&=A_hx_h(t)+B_hu(t-\\tau) \\\\\ny_h(t)&=C_hx_h(t)+D_hu(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhere $ x_h\\in \\mathbb{R}^{n_h} $ is the neuromuscular state vector, $ A_h\\in \\mathbb{R}^{n_h\\times n_h} $ is the state matrix, $ B_h\\in \\mathbb{R}^{n_h} $ is the input matrix, $ C_h\\in \\mathbb{R}^{1\\times n_h} $ is the output matrix and $ D_h\\in \\mathbb{R} $ is the control output matrix. $ u\\in \\mathbb{R} $ is the neuromuscular input vector, which represents the control decisions taken by the human and fed to the neuromuscular system, $ y_h\\in \\mathbb{R} $ is the output vector, and $ \\tau\\in \\mathbb{R}^+ $ is a known, constant delay. The neuromuscular model parameters are assumed to be known and the output of the model, $ y_h $, is used as the plant input $ u_p $ in (\\ref{eq:e2x}), that is $ y_h=u_p $ (see Fig. \\ref{fig:f1-2}). \n\\begin{figure}\n\t\\vspace{0.3cm}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{Blockdiagxx.jpg}    %\n\t\t\\caption{The block diagram of the human adaptive behavior and decision making in a closed loop system.} \n\t\t\\label{fig:f1-2}\n\t\\end{center}\n\\end{figure}\n\n\nBy aggregating the human pilot and plant states, we obtain the combined open loop human neuromuscular and plant dynamics as\n\\begin{equation}\\label{eq:e4x}\n\\begin{aligned}\n\\underbrace{\\begin{bmatrix}\n\t\\dot{x}_h(t) \\\\ \\dot{x}_p(t)\n\t\\end{bmatrix}}_{\\dot{x}_{hp}(t)}&=\n\\underbrace{\\begin{bmatrix}\n\tA_h & 0_{n_h\\times n_p} \\\\ B_p C_h & A_p\n\t\\end{bmatrix}}_{A_{hp}}\\underbrace{\\begin{bmatrix}\n\tx_h(t) \\\\ x_p(t)\n\t\\end{bmatrix}}_{x_{hp(t)}}\\\\ &+\n\\underbrace{\\begin{bmatrix}\n\tB_h \\\\ B_pD_h\n\t\\end{bmatrix}}_{B_{hp}}u(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhich can be written in the following compact form\n\\begin{equation}\\label{eq:e5x}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_{hp}x_{hp}(t)+B_{hp}u(t-\\tau),\n\\end{aligned} \n\\end{equation}\nwhere $ x_{hp}=[x_h^T\\ x_p^T]^T\\in \\mathbb{R}^{(n_p+n_h)} $, $ A_{hp}\\in \\mathbb{R}^{(n_p+n_h)\\times (n_p+n_h)} $, $ B_{hp}\\in \\mathbb{R}^{(n_p+n_h)} $. \n\\begin{assumption}\n\tThe pair $ (A_{hp}, B_{hp}) $ is controllable.\n\\end{assumption}\n\nThe goal is to obtain the input $ u(t) $ in (\\ref{eq:e3x}), which is the human pilot control decision variable, such that the closed loop system consisting of the adaptive human pilot model and the plant follow the output of a unity feedback reference model with an open loop crossover model transfer function. The closed loop transfer function of the reference model is therefore calculated as\n\\begin{equation}\\label{eq:e21xqx}\n\\begin{aligned}\nG_{cl}(s)=\\frac{\\frac{\\omega_c}{s}e^{-\\tau s}}{1+\\frac{\\omega_c}{s}e^{-\\tau s}}=\\frac{\\omega_c e^{-\\tau s}}{s+\\omega_c e^{-\\tau s}}.\n\\end{aligned} \n\\end{equation}\nAn approximation of (\\ref{eq:e21xqx}) can be given as  \n\\begin{equation}\\label{eq:e21xqqx}\n\\begin{aligned}\n\\hat{G}_{cl}(s)=\\frac{b_ms^m+b_{m-1}s^{m-1}+...+b_0}{s^n+a_{n-1}s^{n-1}+...+a_0}e^{-\\tau s},\n\\end{aligned} \n\\end{equation}\nwhere $ n=n_h+n_p $ and $ m\\leq n $ are positive real constants, and $ a_{i} $ and $ b_j $ for $ i=0, ..., n-1 $ and $ j=0, ..., m-1 $, are real constants to be estimated. \nThe reference model then can be obtained as the state space representation of (\\ref{eq:e21xqqx}) as\n\\begin{equation}\\label{eq:e7x}\n\\begin{aligned}\n\\dot{x}_{m}(t)&=A_{m}x_{m}(t)+B_{m}r(t-\\tau),\\\\\n\\end{aligned} \n\\end{equation}\nwhere $ x_m\\in \\mathbb{R}^{(n_h+n_p)} $ is the reference model state vector, $ A_m\\in \\mathbb{R}^{(n_h+n_p)\\times (n_h+n_p)} $ is the state matrix, $ B_m\\in \\mathbb{R}^{(n_h+n_p)\\times m_h} $ is the input matrix, and $ r\\in \\mathbb{R}^{m_h} $ is the reference input.\n\n\\section{REFERENCE MODEL PARAMETERS}\\label{sec:reference}\nThe crossover transfer function (\\ref{eq:e1x}) contains the crossover frequency, $ \\omega_c $, which is not known a priori. Experimental data, showing the reference input ($ r(t) $) frequency bandwidth, $ \\omega_i $, versus crossover frequency $ \\omega_c $, is provided in \\cite{Bee08} and \\cite{Mcr67}, for plant transfer functions $ K $, $ K\/s $ and $ K\/s^2 $. We fit polynomials to these experimental results to obtain the crossover frequency of the open loop transfer function given a reference input frequency bandwidth. These polynomials are given in Table I. It is noted that when the reference input has multiple frequency components, the highest frequency is used to calculate the crossover frequency.\n\n\n\\begin{remark}\n\tIn this work, we use the polynomial relationships provided in Table I for zero, first and second order plant dynamics with nonzero poles and zeros. Further experimental work can be conducted to obtain a more precise relationship between the crossover and reference input frequencies, but this is currently out of the scope of this work.\n\\end{remark}\n  \n\n\n\n\\begin{table}\n\t\\vspace{0.3cm}\n\t\\caption{}\n\t\\centering\n\t\\begin{tabular}{ | c | c | }\n\t\t\\hline \n\t\tPlant transfer & Crossover frequency of the \\\\\n\t\tfunction & open loop transfer function (rad\/s)\\\\\n\t\t\\hline\n\t\t$ K $ & $ \\omega_c=0.067\\omega_i^2+0.099\\omega_i+4.8 $ \\\\ \\hline\n\t\t$ K\/s $ & $ \\omega_c=0.14\\omega_i+4.3 $ \\\\ \\hline\n\t\t$ K\/s^2 $ & $ \\omega_c=-0.0031\\omega_i^4-0.072\\omega_i^3+0.29\\omega_i^2$ \\\\ & $-0.13\\omega_i+3 $ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\n\n\\section{HUMAN PILOT CONTROL DECISION COMMAND}\\label{sec:human}\n\n\nThe adaptive human pilot control decision command, $ u(t) $, is determined as\n\\begin{equation}\\label{eq:e8x}\nu(t)=K_rK_xx_{hp}(t+\\tau)+K_rr(t)\n\\end{equation}\nwhere $ K_x\\in \\mathbb{R}^{1\\times (n_h+n_p)} $, and $ K_r\\in \\mathbb{R}^{m_h\\times m_h} $. Using (\\ref{eq:e8x}) and (\\ref{eq:e5x}), the closed loop dynamics can be obtained as\n\\begin{equation}\\label{eq:e9x}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=(A_{hp}+B_{hp}K_rK_x)x_{hp}(t)+B_{hp}K_rr(t-\\tau).\n\\end{aligned} \n\\end{equation} \n\nEquation (\\ref{eq:e8x}) describes a non-causal decision command which requires future values of the states. This problem can be eliminated by solving the differential equation (\\ref{eq:e5x}) as a $ \\tau $-seconds ahead predictor as\n\\begin{equation}\\label{eq:e9xx}\n\\begin{aligned}\nx_{hp}(t+\\tau)&=e^{A_{hp}\\tau}x_{hp}(t)+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta)d\\eta. \n\\end{aligned} \n\\end{equation}\n\n\\begin{assumption}\nThere exist ideal parameters $ K_r^* $ and $ K_x^* $ satisfying the following matching conditions\n\\begin{equation}\\label{eq:e10x}\n\\begin{aligned}\n&A_{hp}+B_{hp}K_r^*K_x^*=A_m\\\\\n&B_{hp}K_r^*=B_m.\n\\end{aligned} \n\\end{equation}\n\\end{assumption} \n\n\n\nBy substituting (\\ref{eq:e9xx}) into (\\ref{eq:e8x}), the human pilot control decision input can be written as \n\\begin{equation}\\label{eq:e11xxxy}\n\\begin{aligned}\nu(t)&=K_rK_xe^{A_{hp}\\tau}x_{hp}(t)\\\\\n&+K_rK_x\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta)d\\eta+K_rr(t).\n\\end{aligned} \n\\end{equation}\nBy defining $ \\theta_x(t) $ and $ \\lambda(t,\\eta) $ as\n\\begin{equation}\\label{eq:e11xxxq}\n\\begin{aligned}\n\\theta_x(t)&=K_r(t)K_x(t)e^{A_{hp}\\tau},\\\\\n\\lambda(t,\\eta)&=K_r(t)K_x(t)e^{-A_{hp}\\eta}B_{hp},\n\\end{aligned} \n\\end{equation}\n(\\ref{eq:e11xxxy}) can be rewritten as (see fig. \\ref{fig:f1-2})\n\\begin{equation}\\label{eq:e11xxx}\n\\begin{aligned}\nu(t)=\\theta_x(t)x_{hp}(t)+\\int_{-\\tau}^{0}\\lambda(t,\\eta)u(t+\\eta)d\\eta+K_r(t)r(t).\n\\end{aligned} \n\\end{equation}\nThe ideal values of $ \\theta_x $ and $ \\lambda $ can be obtained as\n\\begin{equation}\\label{eq:e11xxxxx}\n\\begin{aligned}\n\\theta_x^*&=K_r^*K_x^*e^{A_{hp}\\tau}\\\\ \n\\lambda^*(\\eta)&=K_r^*K_x^*e^{-A_{hp}\\eta}B_{hp}.\n\\end{aligned} \n\\end{equation}\nSince $ A_{hp} $ and $ B_{hp} $ are unknown, $ \\theta_x $ and $ \\lambda $ need to be estimated. The closed loop dynamics can be obtained using (\\ref{eq:e5x}) and (\\ref{eq:e11xxx}) as\n\\begin{equation}\\label{eq:e11xxxx}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_{hp}x_{hp}(t)+B_{hp}\\theta_x(t-\\tau)x_{hp}(t-\\tau)\\\\ &+\\int_{-\\tau}^{0}B_{hp}\\lambda(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta\\\\ &+B_{hp}K_rr(t-\\tau),\n\\end{aligned} \n\\end{equation}\n\nDefining the deviations of the adaptive parameters from their ideal values as $ \\tilde{\\theta}_x=\\theta_x-\\theta_x^* $ and $ \\tilde{\\lambda}=\\lambda-\\lambda^* $, and adding and subtracting $ A_mx_{hp}(t) $ to (\\ref{eq:e11xxxx}), and using (\\ref{eq:e10x}), we obtain that\n\\begin{equation}\\label{eq:e12xxz}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_mx_{hp}(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)\\Big(e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\ &+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\ &+B_{hp}K_r(t-\\tau)r(t-\\tau).\\\\\n\\end{aligned} \n\\end{equation}\n Using (\\ref{eq:e9xx}), (\\ref{eq:e12xxz}) is rewritten as\n\\begin{equation}\\label{eq:e12xx}\n\\begin{aligned}\n\\dot{x}_{hp}(t)&=A_mx_{hp}(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)r(t-\\tau).\n\\end{aligned} \n\\end{equation}\nDefining the tracking error as $ e(t)=x_{hp}-x_{m} $, and subtracting (\\ref{eq:e7x}) from (\\ref{eq:e12xx}), and using (\\ref{eq:e10x}), and following a similar procedure given in \\cite{NarAnn12}, it is obtained that\n\\begin{equation}\\label{eq:e12xxxz}\n\\begin{aligned}\n\\dot{e}(t)&=\\dot{x}_{hp}-\\dot{x}_{m}\\\\\n&=A_me(t)-B_{hp}K_r^*K_x^*x_{hp}(t)\\\\ &+B_{hp}K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\ &+B_{hp}(K_r(t-\\tau)-K_r^*)r(t-\\tau)\\\\\n&=A_me(t)+\\big( -B_{hp}K_r^*K_x^*\\\\\n&+B_{hp}(K_r^*-K_r^*+K_r(t-\\tau))K_x(t-\\tau)\\big)x_{hp}(t) \n\\\\&+B_{hp}(K_r(t-\\tau)-K_r^*)r(t-\\tau)\\\\\n&=A_me(t)+B_{m}(K_x(t-\\tau)-K_x^*)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}K_r(t-\\tau)-1)K_x(t-\\tau)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}K_r(t-\\tau)-1)r(t-\\tau)\\\\\n&=A_me(t)+B_{m}(\\tilde{K}_x(t-\\tau)x_{hp}(t)\\\\\n&\\hspace{-0.1cm}+B_{m}({K_r^*}^{-1}-K_r^{-1}(t-\\tau))K_r(t-\\tau)K_x(t-\\tau)x_{hp}(t)\\\\\n&+B_{m}({K_r^*}^{-1}-K_r^{-1}(t-\\tau))K_r(t-\\tau)r(t-\\tau).\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e9xx}) and defining $ \\Phi={K_r^*}^{-1}-K_r^{-1} $, we can rewrite (\\ref{eq:e12xxxz}) as\n\\begin{equation}\\label{eq:e12xxxzz}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+{B_mK_r^*}^{-1}(K_r^*K_x(t-\\tau)-K_r^*K_x^*)\\\\\n&\\hspace{-0.1cm}\\times\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_m\\Phi(t-\\tau)\\Big( K_r(t-\\tau)K_x(t-\\tau)\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+K_r(t-\\tau)r(t-\\tau) \\Big).\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e11xxxxx}) and (\\ref{eq:e12xxxzz}), we obtain that\n\\begin{equation}\\label{eq:e12xxxzzz}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_mK_x(t-\\tau)\\Big( e^{A_{hp}\\tau}x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}e^{-A_{hp}\\eta}B_{hp}u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&-B_m{K_r^*}^{-1}\\Big( \\theta_x^*x_{hp}(t-\\tau)\\\\\n&+\\int_{-\\tau}^{0}\\lambda^*(\\eta)u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\\\\\n\\end{aligned} \n\\end{equation}\nUsing (\\ref{eq:e11xxxq}), (\\ref{eq:e12xxxzzz}) can be rewritten as\n\\begin{equation}\\label{eq:e12xxxq}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_m\\Big( \\big( K_r^{-1}(t-\\tau)\\theta_x(t-\\tau)-{K_r^*}^{-1}\\theta_x^*\\big)\\\\\n&\\times x_{hp}(t-\\tau)+\\int_{-\\tau}^{0}\\big( K_r^{-1}(t-\\tau)\\lambda(t-\\tau,\\eta)\\\\\n&-{K_r^*}^{-1}\\lambda^*(\\eta) \\big) u(t+\\eta-\\tau)d\\eta \\Big)\\\\\n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\n\\end{aligned} \n\\end{equation}\nDefining $ \\theta_1=K_r^{-1}\\theta_x $ and $ \\lambda_1=K_r^{-1}\\lambda $, and using their deviations from their ideal values, $ \\tilde{\\theta}_1=\\theta_1-\\theta_1^* $ and $ \\tilde{\\lambda}_1=\\lambda_1-\\lambda_1^* $, where \n$ \\theta_1^*={K_r^*}^{-1}\\theta_x^* $ and $ \\lambda_1^*={K_r^*}^{-1}\\lambda^* $, (\\ref{eq:e12xxxq}) can be rewritten as\n\\begin{equation}\\label{eq:e12xxx}\n\\begin{aligned}\n\\dot{e}(t)&=A_me(t)+B_{m}\\tilde{\\theta}_1(t-\\tau)x_{hp}(t-\\tau)\\\\ &+B_m\\int_{-\\tau}^{0}\\tilde{\\lambda}_1(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta \\\\ \n&+B_{m}\\Phi(t-\\tau)u(t-\\tau).\n\\end{aligned} \n\\end{equation}\nThe following lemma will be necessary to prove the main theorem of this article.\n\n\\begin{lemma}\\label{lem1}\nSuppose that the continuous function $ u(t) $ is given as\n\\begin{equation}\\label{eq:e21xx}\n\\begin{aligned}\nu(t)=f(t)+\\int_{-\\tau}^{0}\\lambda(t,\\eta)u(t+\\eta)d\\eta,\n\\end{aligned} \n\\end{equation}\nwhere $ u,f: [t_0-\\tau,\\infty]\\rightarrow R $, and $ \\lambda:[t_0, \\infty)\\times [-\\tau, 0]\\rightarrow R $. Then\n\\begin{equation}\\label{eq:e22xx}\n\\begin{aligned}\n|u(t)|\\leq 2(\\bar{f}+c_0c_1)e^{c_0^2(t-t')},\\ \\forall t_j'\\geq t_i',\n\\end{aligned} \n\\end{equation}\nif constants $ t_i',\\bar{f}, c_0, c_1\\in R^+ $ exist such that $ |f(t)|\\leq \\bar{f} $, \n\\begin{equation}\\label{eq:e23xx}\n\\begin{aligned}\n\\int_{-\\tau}^{0}\\lambda^2(t,\\eta)d\\eta\\leq c_0^2 \\ \\ for\\  t\\in [t_i',t_j'),\n\\end{aligned} \n\\end{equation}\nand\n\\begin{equation}\\label{eq:e24xx}\n\\begin{aligned}\n\\int_{-\\tau}^{0}u^2(t+\\eta)d\\eta\\leq c_1^2 \\ \\ \\forall\nt\\leq t_i'.\n\\end{aligned} \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe proof of Lemma \\ref{lem1} can be found in \\cite{YilAnn10}.\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{thm1}\nGiven the initial conditions $ \\tilde{\\theta}_1(\\xi)$, $ \\tilde{\\lambda}_1(\\xi,\\eta) $, $ \\Phi(\\xi) $ and $ x_{hp}(\\xi) $ for $ \\xi\\in [-\\tau, 0] $, and $ u(\\zeta) $ for $ \\zeta\\in[-2\\tau,0] $, there exists a $ \\tau^* $ such that for all $ \\tau\\in[0,\\tau^*] $, the controller (\\ref{eq:e11xxx}) with the adaptive laws\n\\begin{equation}\\label{eq:14thm}\n\\dot{{\\theta}}_1^T(t)=-x_{hp}(t-\\tau)e(t)^TPB_{m},\n\\end{equation} \n\\begin{equation}\\label{eq:15thm}\n\\dot{{\\Phi}}^T(t)=-u(t-\\tau)e(t)^TPB_{m},\\\\\n\\end{equation} \n\\begin{equation}\\label{eq:16thm}\n\\dot{\\lambda}_1^T(t,\\eta)=-u(t+\\eta-\\tau)e(t)^TPB_{m},\n\\end{equation} \nwhere $ P $ is the symmetric positive definite matrix satisfying\nthe Lyapunov equation $ A_m^TP +P A_m = -Q $ for a symmetric\npositive definite matrix $ Q $, which can be employed to obtain controller parameters using $ \\dot{K}_r=\\text{Proj}(K_r\\dot{\\Phi}K_r) $, $\\theta_x(t)=K_r(t)\\theta_1(t) $ and $ \\lambda(t)=K_r(t)\\lambda_1(t) $, make the pilot neuromuscular and plant aggregate system (\\ref{eq:e5x}) follow the crossover reference model (\\ref{eq:e7x}) asymptotically, i.e, $ lim_{t\\to \\infty} x_{hp}(t)=x_m(t) $, while keeping all the signals bounded.\n\\end{theorem}\n\\begin{proof}\nConsider a Lyapunov-Krasovskii functional \\cite{YilAnn10}\n\\begin{equation}\\label{eq:e12x}\n\\begin{aligned}\nV(t)&=e^TPe+\\text{tr}({\\Phi}^T(t){\\Phi}(t))+\\text{tr}(\\tilde{\\theta}_1^T(t)\\tilde{\\theta}_1(t))\\\\ &+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(\\xi)\\dot{\\tilde{\\theta}}_1(\\xi))d\\xi dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\text{tr}(\\dot{{\\Phi}}^T(\\xi)\\dot{{\\Phi}}(\\xi))d\\xi dv\\\\\n&+\\int_{-\\tau}^{0}\\text{tr}(\\tilde{\\lambda}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ &+\\int_{-\\tau}^{0}\\int_{t+v}^{t}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(\\xi,\\eta)\\dot{\\tilde{\\lambda}}_1(\\xi,\\eta))d\\eta d\\xi dv.\n\\end{aligned} \n\\end{equation}\nThe derivative of $ V(t) $ can be calculated as\n\\begin{equation}\\label{eq:e14x}\n\\begin{aligned}\n\\dot{V}(t)&=\\dot{e}^T(t)^TPe(t)+e^T(t)P\\dot{e}(t)+2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t))\\\\ &+2\\text{tr}(\\dot{{\\Phi}}^T(t){\\Phi}(t))+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ \n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nSubstituting (\\ref{eq:e12xxx}) into (\\ref{eq:e14x}) and using the Lyapunov equation $ A_m^TP +P A_m = -Q $, it is obtained that\n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)+2e^T(t)PB_m\\tilde{\\theta}_1(t-\\tau)x_{hp}(t-\\tau)\\\\\n&+2e^T(t)PB_m\\int_{-\\tau}^{0}\\tilde{\\lambda}_1(t-\\tau,\\eta)u(t+\\eta-\\tau)d\\eta\\\\\n&+2e^T(t)PB_m\\Phi(t-\\tau)u(t-\\tau)\\\\\n&+2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t))+2\\text{tr}(\\dot{{\\Phi}}^T(t){\\Phi}(t)) \\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta))d\\eta\\\\ \n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\notag\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:e14xx}\n\\begin{aligned}\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nUsing $ g(t)-g(t-\\tau)=\\int_{-\\tau}^{0}\\dot{g}(t+v)dv $, (\\ref{eq:e14xx}) can be rewritten as\n\\begin{equation}\\label{eq:e14xxxx}\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)\\\\\n&+2\\text{tr}\\Big(x_{hp}(t-\\tau)e^T(t)PB_m\\tilde{\\theta}_1(t)+\\dot{\\tilde{\\theta}}_1^T(t)\\tilde{\\theta}_1(t)\\Big)\\\\\n&+2\\text{tr}\\Big(u(t-\\tau)e^T(t)PB_m\\Phi(t)+\\dot{{\\Phi}}^T(t){\\Phi}(t)\\Big)\\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}\\Big(u(t+\\eta-\\tau)e^T(t)PB_m\\tilde{\\lambda}_1(t,\\eta)\\\\ &+\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\tilde{\\lambda}_1(t,\\eta)\\Big)d\\eta\\\\\n&-2e^T(t)PB_m(\\int_{-\\tau}^{0}\\dot{\\tilde{\\theta}}_1(t+v)dv)x_{hp}(t-\\tau)\\\\\n&-2e^T(t)PB_m(\\int_{-\\tau}^{0}\\dot{{\\Phi}}(t+v)dv)u(t-\\tau)\\\\\n&-2e^T(t)PB_m\\Big(\\int_{-\\tau}^{0}(\\int_{-\\tau}^{0}\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)dv)\\\\\n&\\times u(t+\\eta-\\tau)d\\eta\\Big)\\\\\n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nBy substituting (\\ref{eq:14thm})-(\\ref{eq:16thm}) into (\\ref{eq:e14xxxx}), it is obtained that\n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&=-{e}^T(t)Qe(t)\\\\\n&-2\\int_{-\\tau}^{0}\\text{tr}(x_{hp}(t-\\tau)e(t)^TPB_{m}\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ &-2\\int_{-\\tau}^{0}\\text{tr}(u(t-\\tau)e(t)^TPB_{m}\\dot{{\\Phi}}(t+v))dv\\\\\n&-2\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(u(t+\\eta-\\tau)e(t)^TPB_{m}\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))dvd\\eta\\\\\n&+\\tau \\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v))dv\\\\ \n&+\\tau\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))-\\int_{-\\tau}^{0}\\text{tr}(\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v))dv\\\\\n&+\\tau\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta\\\\\n&-\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta))d\\eta dv\\notag \n\\end{aligned} \n\\end{equation}\n\\begin{equation}\\label{eq:e17x}\n\\begin{aligned}\n&=-{e}^T(t)Qe(t)+\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t+v)\\\\\n&+\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t) -\\dot{\\tilde{\\theta}}_1^T(t+v)\\dot{\\tilde{\\theta}}_1(t+v)\\Big)dv\\\\\n&+\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t+v)+\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t)\\\\\n&-\\dot{{\\Phi}}^T(t+v)\\dot{{\\Phi}}(t+v)\\Big)dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}\\text{tr}\\Big(2\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)\\\\\n&+\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta)-\\dot{\\tilde{\\lambda}}_1^T(t+v,\\eta)\\dot{\\tilde{\\lambda}}_1(t+v,\\eta)\\Big)d\\eta dv.\n\\end{aligned} \n\\end{equation}\nUsing the trace property $ \\text{tr}(A+B)=\\text{tr}(A)+\\text{tr}(B) $, and the algebraic inequality $ a^2\\geq 2ab-b^2 $ for two scalars $ a $ and $ b $, it can be shown that $ \\text{tr}(2A^TB+A^TA-B^TB)\\leq 2\\text{tr}(A^TA) $. Using these inequalities, (\\ref{eq:e17x}) can be rewritten as\n\\begin{equation}\\label{eq:e17xx}\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\theta}}_1^T(t)\\dot{\\tilde{\\theta}}_1(t))dv\\\\\n&+\\int_{-\\tau}^{0}2\\text{tr}(\\dot{{\\Phi}}^T(t)\\dot{{\\Phi}}(t))dv\\\\\n&+\\int_{-\\tau}^{0}\\int_{-\\tau}^{0}2\\text{tr}(\\dot{\\tilde{\\lambda}}_1^T(t,\\eta)\\dot{\\tilde{\\lambda}}_1(t,\\eta))d\\eta dv.\n\\end{aligned} \n\\end{equation}\nBy substituting (\\ref{eq:14thm})-(\\ref{eq:16thm}) into (\\ref{eq:e17xx}), and using the trace operator property $ \\text{tr}(AB)=\\text{tr}(BA) $ for two square matrices $ A $ and $ B $, (\\ref{eq:e17xx}) can be rewritten as\n\\begin{equation}\\label{eq:e18x}\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)\\\\\n&+2\\tau \\text{tr}\\big(e(t)x_{hp}^T(t-\\tau)x_{hp}(t-\\tau)e(t)^TPB_{m}B_{m}^TP\\big)\\\\ \n&+2\\tau \\text{tr}\\big(e(t)u^T(t-\\tau)u(t-\\tau)e(t)^TPB_{m}B_{m}^TP\\big)\\\\\n&+2\\tau\\int_{-\\tau}^{0} \\text{tr}\\big(e(t)u^T(t+\\eta-\\tau)u(t+\\eta-\\tau)e(t)^T\\\\\n&\\times PB_{m}B_{m}^TP\\big)d\\eta.\n\\end{aligned} \n\\end{equation}\nUsing $ \\text{tr}(AB)\\leq \\text{tr}(A)\\text{tr}(B) $ for two positive semidefinite matrices $ A $ and $ B $, and $ \\text{tr}(X^TX)=||X||_F^2 $ for a matrix $ X $, an upper bound for (\\ref{eq:e18x}) can be derived as \n\\begin{equation\n\\begin{aligned}\n\\dot{V}(t)&\\leq-{e}^T(t)Qe(t)\\\\\n&+2\\tau \\text{tr}(e(t)x_{hp}^T(t-\\tau)x_{hp}(t-\\tau)e(t)^T)\\text{tr}(PB_{m}B_{m}^TP)\\\\ \n&+2\\tau \\text{tr}\\big(e(t)u^T(t-\\tau)u(t-\\tau)e(t)^T\\big)\\text{tr}\\big(PB_{m}B_{m}^TP\\big)\\\\\n&+2\\tau\\int_{-\\tau}^{0} \\text{tr}\\big(e(t) u^T(t-\\tau+\\eta)u(t-\\tau+\\eta)e(t)^T\\big)\\\\\n&\\times \\text{tr}\\big(PB_{m}B_{m}^TP\\big)d\\eta\\\\\n&\\leq -\\lambda_{min}(Q)||e(t)||^2\\\\\n&+2\\tau||x_{hp}(t-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2\\\\\n&+2\\tau ||u(t-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2\\\\\n&+2\\tau\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)e(t)^T||_F^2||B_{m}^TP||_F^2d\\eta \\notag\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\\label{eq:e19x}\n\\begin{aligned}\n&\\leq -\\lambda_{min}(Q)||e(t)||^2\\\\\n&+2\\tau||x_{hp}(t-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2\\\\\n&+2\\tau ||u(t-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2\\\\\n&+2\\tau\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2||e(t)||^2||B_{m}^TP||_F^2d\\eta\\\\\n&=||B_{m}^TP||_F^2||e(t)||^2\\Big( -\\frac{\\lambda_{min}(Q)}{||B_{m}^TP||_F^2}\\\\\n&+2\\tau\\big( ||x_{hp}(t-\\tau)||^2+||u(t-\\tau)||^2\\\\\n&+\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2d\\eta \\big) \\Big).\n\\end{aligned} \n\\end{equation}\nDefining $ q\\equiv \\frac{\\lambda_{min}(Q)}{||B_{m}^TP||_F^2} $, the inequality\n\\begin{equation}\\label{eq:e20x}\n\\begin{aligned}\nq&-2\\tau\\big( ||x_{hp}(t-\\tau)||^2+||u(t-\\tau)||^2+\\\\\n&+\\int_{-\\tau}^{0}||u(t+\\eta-\\tau)||^2d\\eta \\big)> 0.\n\\end{aligned} \n\\end{equation}\nneeds to be satisfied for the non-positiveness of $ \\dot{V} $.\nAssuming that $ x_{hp} $ and $ u $ are bounded in the interval $ [t_0-2\\tau,t_0) $, the rest of the proof is divided into the following four steps:\n\n\\noindent\n\\textbf{Step 1} In this step, the negative semi-definiteness of the Lyapunov-Krasovskii functional's (\\ref{eq:e12x}) time derivative in the interval $ [t_0-\\tau,t_0) $ is shown which leads to the boundedness of the the signals in this interval. In addition, an upper bound for $ u $ in the interval $ [t_0-2\\tau,t_0) $ is given.\n\nSuppose that\n\\begin{equation}\n\\begin{aligned}\n\\sup_{\\xi\\in[t_0-\\tau, t_0)}&||x_{hp}(\\xi)||^2\\leq \\gamma_1\\\\\n\\sup_{\\xi\\in[t_0-2\\tau, t_0)}&||u(\\xi)||^2\\leq \\gamma_2\n\\end{aligned} \n\\end{equation} \nfor some positive $ \\gamma_1, \\gamma_2$, and a $ \\tau_1 $ is given such that\n\\begin{equation}\n\\begin{aligned}\n2\\tau_1(\\gamma_1+\\gamma_2+\\tau_1\\gamma_2)<q.\n\\end{aligned} \n\\end{equation} \nThen the following inequality is satisfied:\n\\begin{equation}\n\\begin{aligned}\nq&-2\\tau\\big( ||x_{hp}(\\xi-\\tau)||^2+||u(\\xi-\\tau)||^2+\\\\\n&+\\int_{-\\tau}^{0}||u(\\xi+\\eta-\\tau)||^2d\\eta \\big)> 0,\\\\ &\\forall \\xi\\in[t_0,t_0+\\tau), \\forall\\tau\\in[0, \\tau_1].\n\\end{aligned} \n\\end{equation}\nIt follows that $ V(t) $, defined in (\\ref{eq:e12x}), is non-increasing for $ t\\in[t_0, t_0+\\tau) $. Thus, we have\n\\begin{equation}\n\\begin{aligned}\n\\lambda_{min}(P)||e(\\xi)||^2\\leq e(\\xi)^TPe(\\xi)\\leq V(t_0),\n\\end{aligned} \n\\end{equation}\nwhich leads to\n\\begin{equation}\n\\begin{aligned}\n||x_{hp}(\\xi)||-||x_m(\\xi)||\\leq ||e(\\xi)||\\leq \\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}.\n\\end{aligned} \n\\end{equation}\nThen, we have\n\\begin{equation}\n\\begin{aligned}\n||x_{hp}(\\xi)||\\leq\\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}+||x_{m}(\\xi)||,\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\nWe also have the inequality\n\\begin{align}\\label{eq:x47}\n&||{\\Phi}(\\xi)||^2\\leq V(t_0)\\implies ||{K_r^*}^{-1}-K_r^{-1}(\\xi)||^2\\leq V(t_0)\\notag\\\\\n&\\implies ||K_r^{-1}(\\xi)||\\leq \\sqrt{V(t_0)}+||{K_r^*}^{-1}||.\n\\end{align}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\nIt is noted that the boundedness of $ \\Phi={K_r^*}^{-1}-K_r^{-1} $ does not guarantee the boundedness of $ \\tilde{K}_r $. In order to guarantee the boundedness of $ \\tilde{K}_r $ independent of the boundedness of $ \\Phi $, the projection algorithm \\cite{Eug13} is employed as \n\\begin{equation}\\label{eq:e19xzzz}\n\\begin{aligned}\n\\dot{K}_r=\\text{Proj}\\big(K_r,-K_rB_m^TPe(t)u^T(t-\\tau)K_r\\big),\n\\end{aligned} \n\\end{equation}\nwith an upper bound $ K_{max} $, that is $ ||{K}_r||\\leq K_{max} $. \nThus, a lower bound for $ ||K_r^{-1}(\\xi)|| $ can be calculated using the following algebraic manipulations: \n \\begin{align}\\label{eq:47x}\n &K_r(\\xi)K_r^{-1}(\\xi)=I\\Rightarrow ||K_r(\\xi)K_r^{-1}(\\xi)||=1\\notag \\\\\n &\\Rightarrow 1\\leq ||K_r(\\xi)||||K_r^{-1}(\\xi)||\\leq K_{max}||K_r^{-1}(\\xi)||\\notag \\\\\n & \\Rightarrow \\frac{1}{K_{max}}\\leq ||K_r^{-1}(\\xi)||.\n \\end{align}\nDefining $ k_1=\\sqrt{V(t_0)}+||{K_r^*}^{-1}|| $, and using (\\ref{eq:x47}), it is obtained that\n\\begin{align}\\label{eq:47}\n\\frac{1}{K_{max}}\\leq ||K_r^{-1}(\\xi)||\\leq k_1,\\ \\ \\ \\xi\\in[t_0,t_0+\\tau).\n\\end{align}\nTherefore, $ K_r $ is always bounded and $ K_r^{-1}(\\xi) $ is bounded for $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\n\nFurthermore, using the definitions of $ \\theta_x, \\theta_1, \\lambda,\\lambda_1 $ given in Theorem \\ref{thm1}, and the non-increasing Lyapunov functional (\\ref{eq:e12x}), it can be concluded that\n\\begin{align}\\label{eq:48}\n||\\tilde{\\theta}_1(\\xi)||_F^2\\leq V(t_0)&\\implies ||\\tilde{K}_r^{-1}(\\xi)\\tilde{\\theta}_x(\\xi)||_F^2\\leq V(t_0),\n\\end{align}\n\\begin{align}\\label{eq:49}\n\\int_{-\\tau}^{0}||\\tilde{\\lambda}_1(\\xi,\\eta)||_F^2 &d\\eta \\leq V(t_0)\\\\\n&\\implies \\int_{-\\tau}^{0}||K_r^{-1}(\\xi)\\tilde{\\lambda}(\\xi,\\eta)||_F^2d\\eta\\leq V(t_0),\\notag\n\\end{align} \nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $. Using (\\ref{eq:48}) and (\\ref{eq:49}), it can be obtained that\n\\begin{equation}\n\\begin{aligned}\n||\\tilde{\\theta}_x(\\xi)||_F^2&\\leq K_{max}^2V(t_0),\\\\\n\\int_{-\\tau}^{0}||\\tilde{\\lambda}(\\xi,\\eta)||_F^2d\\eta&\\leq K_{max}^2V(t_0).\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $.\n\nTo simplify the notation, we define\n\\begin{equation}\n\\begin{aligned}\nI_0\\equiv \\text{max}&\\Big( \\sqrt{\\frac{V(t_0)}{\\lambda_{min}(P)}}+\\sup_{[t_0,t_0+\\tau)}||x_{m}(\\xi)||\\\\ &, K_{max}\\sqrt{V(t_0)}, K_{max}^2V(t_0) \\Big),\n\\end{aligned} \n\\end{equation}\nwhere $ R_{max} $ is the upper bound of the reference input $ r(t) $.\n\nAn upper bound on the control signal $ u(t) $ for $ t\\in[t_0, t_0+\\tau) $ can be derived by using Lemma \\ref{lem1} and (\\ref{eq:e11xxx}). In particular, setting $ t_i'=t_0 $, $ t_j'=t_0+\\tau $, $ c_0^2=V(t_0) $, we obtain that\n\\begin{equation}\\label{eq:e50}\n\\begin{aligned}\n|u(\\xi)|\\leq 2\\Big( \\bar{f}+\\big( \\int_{-\\tau}^{0}u^2(t_0+\\eta)d\\eta \\big)^{1\/2}I_0 \\Big)e^{I_0\\tau},\n\\end{aligned} \n\\end{equation}\nfor $ \\forall \\xi\\in[t_0,t_0+\\tau) $, where $ \\bar{f} $, which is the upper bound of $ \\theta_x(t)x_{hp}(t)+K_r(t)r(t) $, depends only on $ I_0 $. Defining $ g(\\gamma_2, I_0, \\tau)\\equiv 2(\\bar{f}+\\gamma_2I_0\\sqrt{\\tau})e^{I_0\\tau} $, (\\ref{eq:e50}) can be rewritten as\n\\begin{equation}\n\\begin{aligned}\n|u(\\xi)|\\leq g(\\gamma_2, I_0, \\tau),\\ \\forall \\xi\\in[t_0,t_0+\\tau).\n\\end{aligned} \n\\end{equation}\n\n\nThe rest of the proof is similar to the one given in \\cite{YilAnn10}. Below, a summary of the next steps are given.\n\n\\noindent\n\\textbf{Step 2} A delay range $ [0, \\tau_2] $ is found that satisfies the condition (\\ref{eq:e20x}) over the interval $ [t_0, t_0+2\\tau) $ as\n\\begin{align}\\label{eq:step2}\n\t2\\tau_2\\left( I_0^2+\\left( max\\left( \\gamma_2,g\\left( \\gamma_2,I_0,\\tau_2 \\right)  \\right)  \\right)^2 (1+\\tau_2) \\right) < q,\n\\end{align} \nwhich leads to $ ||x_{hp}(\\xi)||<I_0 $, $ \\forall\\xi\\in[t_0, t_0+2\\tau) $, $ \\forall \\tau\\in [0, \\bar{\\tau}_2] $, $ \\bar{\\tau}_2=min\\{\\tau_1, \\tau_2\\} $.\n\n\\noindent\n\\textbf{Step 3} It is shown in this step that the bound on $ u $ over the interval $ [t_0, t_0+\\tau) $ depends only on $ A_{hp} $, $ B_{hp} $, $ T $ and $ \\tau $, where $ T $ is a value between $ t_0 $ and $ \\tau $. Denoting this upper bound as $ U(I_0) $, we have $ |u(t)|\\leq U(I_0) $, $ t\\in[t_0,t_0+\\tau) $.\n\n\\noindent\n\\textbf{Step 4} Using the calculated upper bound for $ u $ in the previous step, a delay range $ [0, \\tau_2] $ is found that satisfies the condition \n\\begin{align}\\label{eq:step4}\n2\\tau_3\\left( I_0^2+\\left( max\\left( U(I_0),g\\left( U(I_0),I_0,\\tau_3 \\right)  \\right)  \\right)^2 (1+\\tau_3) \\right) < q.\n\\end{align} \nFor $ \\tau^*=min(\\bar{\\tau_2}, \\tau_3) $, $ ||x_{hp}(\\xi)||\\leq I_0 $ and $ |u(\\xi)|\\leq U(I_0) $ for all $ \\xi \\in [t_0, t_0+\\tau] $, $ \\forall \\tau\\in [0, \\tau^*] $.\n\n\n\nThe above four steps show that $ x_{hp}(\\xi) $ and $ u(\\xi) $ are bounded for $ \\forall\\xi\\in[t_0, t_0+k\\tau] $, for $ k=1 $ and $ \\tau  \\in (0, \\tau^*]$. By assuming that $ x_{hp} $ and $ u $ are bounded for a given $ k $, the rest of the proof consists of showing that the boundedness of these variables hold for $ k+1 $. Using this assumption and repeating steps 1-4, which leads to satisfying (\\ref{eq:step4}), we conclude that the Lyapunov function is non-increasing and $ ||x_{hp}(\\xi)|| \\leq I_0 $, and $ |u(\\xi)|\\leq g(U(I_0),I_0,\\tau) $ for $ \\xi\\in [t_0, t_0+(k+1)\\tau] $, $ \\tau \\leq \\tau^* \\leq \\tau_3 $.\nThis completes the boundedness proof using induction.\nThen, using Barbalat's Lemma, it can be shown that the error between the human-in-the-loop system output $ x_{hp} $ and the reference model output $ x_m $ converges to zero.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\\section{Experimental Results}\\label{sec:experiment}\n\\subsection{Experimental environment}\n\nIn order to test the proposed adaptive human model against data, an experimental setup consisting of a Logitech Extreme 3D Pro joystick and a Toshiba Portege-Z30-B laptop with Intel Core i7 CPU is used (see Fig. \\ref{fig:setup}). \n\n\n\n\nThe tracking task is performed by an operator monitoring the compensatory display, which provides information about the error between the target to be followed, and follower, which is the output of the plant (see Fig. \\ref{fig:attitudeind}). The operator provides the input $ u_p $ (see Fig. \\ref{fig:f1-2}) through the joystick, which is fed to the plant using MATLAB SIMULINK (R2018b). In return, the response of the plant is calculated and shown on the laptop screen in real-time.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{setup.jpg}    %\n\t\t\\caption{Experimental setup.} \n\t\t\\label{fig:setup}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{attitudeindx.jpg}    %\n\t\t\\caption{Compensatory display.} \n\t\t\\label{fig:attitudeind}\n\t\\end{center}\n\\end{figure}\n\nThe reference signal $ r(t) $ is generated as a sum of the sinusoids with frequencies of 0.1, 0.3, 0.5, 0.7, 1, 1.3 and 1.5 rad\/s with the same amplitude of 0.2 and without phase shift.\n\n\nThree classes of plant models, having zero, first and second order transfer functions are used in the experiments. In this section, we first give a detailed analysis of the first order plant case and then provide a summary of the results of the other cases in tables. The nominal\nfirst order plant used in the experiments is $ Y_p(s)=\\frac{4}{s+1} $, which is similar to the one used in \\cite{Hess09}. The uncertainty is introduced to the plant model by modifying the gain and the pole location by $ 50\\% $ to obtain $ Y_p(s)=\\frac{6}{s+0.5} $. \n\n\n\n\n\nTo form the reference model (\\ref{eq:e7x}), two parameters, namely the crossover frequency and the time-delay, need to be determined. The highest frequency component of the reference signal is $ \\omega_i=1.5 $ rad\/s. Employing Table I for the first order plant $ Y_p $, the crossover frequency is calculated as $ \\omega_c=4.5 $ rad\/s. The delay is determined by using the mean value of the operators' delay, which is $ \\tau=0.3 $ s. Therefore, the closed loop transfer function of the reference model is calculated as\n\t\\begin{equation}\\label{eq:e21xq}\n\t\\begin{aligned}\n\tG_{cl}(s)=\\frac{\\frac{4.5}{s}e^{-0.3s}}{1+\\frac{4.5}{s}e^{-0.3 s}}=\\frac{4.5 e^{-0.3 s}}{s+4.5 e^{-0.3 s}}.\n\t\\end{aligned} \n\t\\end{equation}\n\t\n\tSimilar to (\\ref{eq:e21xqqx}), an approximate transfer function is obtained as\n\t\\begin{equation}\\label{eq:e21xqq}\n\t\\begin{aligned}\n\t\\hat{G}_{cl}(s)=\\frac{3.881s+24.24}{s^2+0.6834s+24.72}e^{-0.3s}.\n\t\\end{aligned} \n\t\\end{equation}\n\tFigure \\ref{fig:bode1} shows a comparison between (\\ref{eq:e21xq}) and (\\ref{eq:e21xqq}), and demonstrates that the approximation works well for almost all frequencies.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{bode1-eps-converted-to.pdf}    %\n\t\t\\caption{Bode plot of the reference model and its approximation.} \n\t\t\\label{fig:bode1}\n\t\\end{center}\n\\end{figure}\n\n\n\n\nThe neuromuscular dynamics is taken as $ Y_h(s)=\\frac{s+3}{s+2}e^{-0.3s} $, where the time delay $ \\tau=0.3 $ is the effective time delay, including human decision making delay and neuromuscular lags.\n\n\\begin{remark}\\label{rem3}\n\tIn this paper, we assume that the neuromuscular dynamics are given. The procedure for finding the neuromuscular model can be found in \\cite{Mag71, Van04}.  \n\\end{remark}\n\n\\subsection{The behavior of the adaptive model}\n\n\n The error between the plant output and the reference model is illustrated in Figure \\ref{fig:error}. The effect of uncertainty injection can be seen at $ t=70 $ s. Figures \\ref{fig:theta}, \\ref{fig:lambda}, and \\ref{fig:Kr} illustrates the adaptive human model parameters. To understand the amount of agreement between these results and the human experimental trials, visual and statistical analyses are provided in the following sections.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{error-eps-converted-to.pdf}    %\n\t\t\\caption{Time evolution of the error between the output of the plant controlled by the adaptive model, and the reference model output.} \n\t\t\\label{fig:error}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{theta-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameters $ \\theta_{x1} $ and $ \\theta_{x2} $.} \n\t\t\\label{fig:theta}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{lambda-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameters $ \\lambda_i,\\ i=1, 2, 3 $ and $ 4 $.} \n\t\t\\label{fig:lambda}\n\t\\end{center}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=8.5cm]{Kr-eps-converted-to.pdf}    %\n\t\t\\caption{Evolution of human adaptive parameter $ K_r $.} \n\t\t\\label{fig:Kr}\n\t\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Participants and experimental procedure\n\nEleven participants (6 women and 5 men) from the graduate and undergraduate student pools of Bilkent University participated the experiment. All of the participants read and signed the ``informed consent to participate\" document. This study is approved by Bilkent University Ethics Committee for research with human participants. Before the experiments, to familiarize the participants with the experimental setup, and its environment, consisting of the display and the joystick, each participant was asked to follow a given reference via joystick inputs for the duration of $ 200 $ seconds. To prevent learning during these warm-up runs, the reference input, uncertainty injection times and the uncertainty types were chosen differently from the ones used in the real experimental runs. Specifically, the reference signal for the warm-up runs consisted of the sum of the sinusoids with frequencies of $ 0.1, 0.5, 1 $ and $ 1.5 $ rad\/s with the same amplitude of $ 0.2 $ and without phase shift. The plant dynamics at the beginning of the warm-up run was $ \\frac{2}{s^2+3s+2} $. At $ t = 45 $ s, the dynamics changed to $ \\frac{5}{s+2} $ in a step like manner (suddenly). It changed to $ \\frac{3}{s+1} $ at around $ t = 90 $ s using a sigmoid function (gradually), and again changed to a zero order dynamics at $ 150 $ s (suddenly). \n\n\n\n\n\n  \n\n\n\n\n\\subsection{A visual analysis of the adaptive model}\n\nLet $ f_{p1}(t), f_{p2}(t), ..., f_{pk}(t) $ be the plant outputs when participants $ p1, p2, ..., pk $ are in the loop, respectively. For each $ f_{pi}(t) $, $ t=T_1, T_2, ..., T_N $, where $ T_j, j=1, 2, ..., N $, represents a sampling instant. At each sampling instant $ T_j $, the minimum, the maximum and the mean values of the plant outputs when participants are in the loop can be obtained as\n\\begin{align}\nf_{p_{min}}(T_j)&=\\min_{i=1, 2, ..., k}f_{pi}(T_j),\\ \\ j=1, ..., N,\\label{eq:estat4} \\\\\nf_{p_{max}}(T_j)&=\\max_{i=1, 2, ..., k}f_{pi}(T_j),\\ \\ j=1, ..., N, \\label{eq:estat5} \\\\\nf_{p_{mean}}(T_j)&=\\frac{\\sum_{i=1}^{k}f_{pi}(T_j)}{k}, \\ \\ j=1, ..., N,\n\\end{align} \n where $ k=11 $ is the number of participants. Figure \\ref{fig:minmaxadapt} shows the evolutions of  $ f_{p_{min}} $ and $ f_{p_{max}} $, together with $ f_{ad}(t)\\in \\mathbb{R}^N $, which is the plant output when adaptive human model is in the loop, where $ t= T_1, T_2, ..., T_N $. It is seen that the plant output when adaptive human model is in the loop almost always stays between the maximum and the minimum values of the plant output when participants are in the loop. Furthermore, Figure \\ref{fig:meanadapt} demonstrates that $ f_{p_{mean}} $ and $ f_{ad} $ evolve reasonably close to each other.\n  \n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{minmaxadapt1-eps-converted-to.pdf}    %\n\t\t\\caption{Plant output, $ x_{hp} $, when adaptive human model is in the loop vs. minimum and maximum values of plant output when participants are in the loop.} \n\t\t\\label{fig:minmaxadapt}\n\t\\end{center}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{meanadapt1-eps-converted-to.pdf}    %\n\t\t\\caption{Plant output, $ x_{hp} $, when adaptive human model is in the loop vs. mean value of plant output when participants are in the loop.} \n\t\t\\label{fig:meanadapt}\n\t\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Statistical analysis of the adaptive model using confidence intervals}\n\nThe difference between the plant output when the $ i^{\\text{th}} $ participant is in the loop and when the adaptive human model is in the loop is defined as\n\\begin{equation}\\label{eq:estat1}\n\\begin{aligned}\nd_i\\equiv f_{ad}-f_{pi}, \\ \\ i=1, ..., k,\n\\end{aligned} \n\\end{equation}\nwhere $ d_i=[d_i(T_1), ..., d_i(T_N)]^T\\in \\mathbb{R}^N, i=1, ..., k $, is called the $ i^{\\text{th}} $ difference. The mean and the standard deviation of the $ i $th difference is obtained as\n\\begin{align}\n\\bar{d}_i&= \\frac{\\sum_{j=1}^{N}d_i(T_j)}{N}, \\ \\ i=1, ..., k,\\label{eq:estat2} \\\\\ns_i&=\\sqrt{\\frac{\\sum_{j=1}^{N}(d_i(T_j)-\\bar{d}_i)^2}{N-1}}, \\ \\ i=1, ..., k.\\label{eq:estat3}\n\\end{align}\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\includegraphics[width=9.0cm]{normalscores-eps-converted-to.pdf}    %\n\t\t\\caption{Normal-scores plot} \n\t\t\\label{fig:normalscores}\n\t\\end{center}\n\\end{figure}\n\nThe normal-scores plot for $ \\bar{d}_i $ is given in Figure \\ref{fig:normalscores}. The figure does not show any significant deviation from the normal distribution. This shows us that the data do not suggest that the \\textit{population} of mean-errors, $ \\bar{d}_i $, deviates significantly from normal distribution. The \\textit{sample} mean and the \\textit{sample} standard deviation of $ \\bar{d}_i $'s can be obtained as\n\\begin{align}\n\\bar{d}&=\\frac{\\sum_{i=1}^{k}\\bar{d}_i}{k}, \\label{eq:estat6}\\\\\ns&=\\sqrt{\\frac{\\sum_{i=1}^{k}(\\bar{d}_i-\\bar{d})^2}{k-1}}. \\label{eq:estat7}\n\\end{align}\nLet $ \\mu_0 $ be the mean value of the \\textit{population} of mean-errors, given as\n\\begin{align}\\label{eq:estat8}\n\\mu_0\\equiv \\frac{\\sum_{i=1}^{K}\\bar{d}_i}{K},\n\\end{align}\nwhere $ K $ is the \\textit{population} size. Since normal-scores plot, given in Figure \\ref{fig:normalscores}, didn't provide any counter evidence, assuming that the distribution of the set of data $ \\{\\bar{d}_1, ..., \\bar{d}_K\\} $ is normal with mean $ \\mu_0 $, $ \\mu_0 $ satisfies the following probability \\cite{Joh19}\n\\begin{align}\\label{eq:estat9}\nP\\left[\\bar{d}-t_{\\alpha\/2}\\frac{s}{\\sqrt{k}}<\\mu_0<\\bar{d}+t_{\\alpha\/2}\\frac{s}{\\sqrt{k}} \\right]=1-\\alpha,\n\\end{align}\nwhere $ \\bar{d} $ and $ s $ are obtained from (\\ref{eq:estat6}-\\ref{eq:estat7}), $ k $ is the number of participants, $ \\alpha $ is the significance level, and $ t_{\\alpha\/2} $ is the upper $ \\alpha\/2 $ point of the $ t $ distribution with degree of freedom $ k-1 $, which can be obtained from the $ t $-distribution table. Since the number of participants, $ k=11 $, is less than $ 30 $, it is appropriate to use the $ t $-distribution.\nUsing $ \\alpha=0.05 $, obtaining $ t_{\\alpha\/2} $ from the $ t $-distribution table as $ 2.228 $, and calculating $ \\bar{d} $ as $ -0.0068 $ and $ s $ as $ 0.0379 $, it can be concluded using (\\ref{eq:estat9}) that we are $ 95\\% $ confident that $ \\mu_0 $ is in the interval $ (-0.0323, 0.0187) $. This shows that the mean $ \\mu_0 $ of the \\textit{population}'s mean deviation from the adaptive human model is reasonably close to zero. \n\n\n\nSimilarly, the variance, $ \\sigma_0^2 $, of the \\textit{population}'s mean deviation from the adaptive human model satisfies the following probability \n\\cite{Joh19}\n\\begin{align}\\label{eq:estat11}\nP\\left[\\frac{(k-1)s^2}{\\chi_{\\alpha\/2}^2}<\\sigma_0^2<\\frac{(k-1)s^2}{\\chi_{1-\\alpha\/2}^2} \\right]=1-\\alpha,\n\\end{align}\nwhere $ \\chi_{\\alpha\/2}^2 $ is the upper $ {\\alpha\/2} $ point of the $ \\chi^2 $ distribution with degree of freedom $ k-1 $ and can be obtained from the $ \\chi^2 $ distribution table. Calculating $ s $ from (\\ref{eq:estat7}), using $ \\alpha = 0.05 $, and obtaining $ \\chi_{\\alpha\/2}^2 $ and $ \\chi_{1-\\alpha\/2}^2 $ from the $ \\chi^2 $ table with $ 10 $ degrees of freedom, it can be concluded using (\\ref{eq:estat11}) that we are $ 95\\% $ confident that $ \\sigma_0 $ is in the interval $ (0.0265,\\ 0.0663) $. This shows that the standard deviation $ \\sigma_0 $ of the \\textit{population}'s mean deviation from the adaptive human model is reasonably small. \n\n\n\n\n\\subsection{Statistical analysis of the adaptive model using hypothesis testing}\n\n\n\n\n\nIn this analysis, we test whether the hypothesis ``the mean value of the \\textit{population} mean-errors, or the mean deviations from the adaptive model,\" is zero. In other words, our null hypothesis, $ H_0 $, is given as\n\\begin{align}\nH_0: \\mu_0=0,\n\\end{align}\nwhere $ \\mu_0 $ is defined in (\\ref{eq:estat8}). The alternative hypothesis, $ H_1 $, is given as $ H_1: \\mu\\neq 0 $. Similar to the confidence interval analysis, assuming that $ \\mu_0 $ is the mean of a normally distributed set of data $ \\{\\bar{d}_1, ..., \\bar{d}_K\\} $ where $ K $ is the \\textit{population} size, the hypothesis $ H_0 $ is rejected if,\n\\begin{align}\\label{eq:estat10}\n\\left| \\frac{(\\bar{d}-\\mu_0)\\sqrt{k}}{s}\\right| \\geq t_{\\alpha\/2},\n\\end{align}\nwhere $ \\bar{d} $ and $ s $ are obtained from (\\ref{eq:estat6}-\\ref{eq:estat7}), $ k $ is the number of participants and $ t_{\\alpha\/2} $ is the upper $ \\alpha\/2 $ point of the $ t $ distribution with degree of freedom $ k-1 $ \\cite{Joh19}. Using the significance level $ \\alpha=0.05 $ and degree of freedom $ k-1=10 $, obtaining $ t_{0.025}=2.228 $ from the $ t $-distribution table, calculating $ \\bar{d}=-0.0068 $ and $ s=0.038 $ using (\\ref{eq:estat6}) and (\\ref{eq:estat7}), respectively, and substituting $ \\mu_0=0 $ and $ k=11 $, the left hand side of (\\ref{eq:estat10}) can be calculated as $ 0.5935 $, which is less than $ t_{\\alpha\/2} $. Therefore, we cannot reject $ H_0 $. We retain $ H_0 $ and conclude that $ H_1 $ fails to be proved.\n \nSince we are retaining the null hypothesis, we want to minimize the probability $ \\beta $ of incorrectly retaining the null hypothesis. This means that we want our test's power, $ 1-\\beta $, to be large, such as $ 0.95 $. What is the minimum required deviation of the \\textit{population} mean from $ 0 $, represented as $ \\mu_1 $, that would make our test to incorrectly retain the null hypothesis with $ 0.05 $ probability, i.e. $ \\beta=0.05 $? To calculate this, we first write the rejection region, $ R $, using (\\ref{eq:estat10}) as\n\\begin{align}\nR:\\ \\left| \\frac{(\\bar{d}-\\mu_0)\\sqrt{k}}{s}\\right| \\geq t_{\\alpha\/2}\\ \n \\implies R:\\ |\\bar{d}|\\geq 0.0255.\n\\end{align}\nDefining $ T=\\frac{(\\bar{d}-\\mu_1)\\sqrt{k}}{s} $, for $ \\beta=0.05 $, we need\n\\begin{align}\\label{eq:estat12}\n&P\\left[ \\frac{(-0.0255-\\mu_1)\\sqrt{k}}{s}<T<\\frac{(0.0255-\\mu_1)\\sqrt{k}}{s} \\right]  \\notag \\\\\n&=\\frac{\\beta}{2}=0.025.\n\\end{align}\nUsing the $ t $-table, it can be found that the minimum $ |\\mu_1| $ that satisfies (\\ref{eq:estat12}) is $ 0.051 $. This means that our test can detect an $ 0.051 $ deviation from the mean value of the mean-error between the adaptive human model and the participant data when the probability of the test to incorrectly conclude that the model and the data are compatible ($ \\mu_0=0 $) is only $ 5\\% $. \n\n\\begin{table}\\label{table2}\n\n\t\\caption{Sudden uncertainty}\n\t\\centering\n\t\\begin{tabular}{ | c | c | c | c | }\n\t\t\\hline \n\t\t& 0 order & 1st order & 2nd order \\\\\n\t\t\\hline \n\t\tTF before $ 70 $ s & $ 4 $ & $ \\frac{4}{s+1} $ & $ \\frac{4}{(s+1)(s+5)} $ \\\\\n\t\t\\hline\n\t\tTF after $ 70 $ s & $ 6 $ & $ \\frac{6}{s+0.5} $ & $ \\frac{6}{(s+0.5)(s+2.5)} $ \\\\\n\t\t\\hline\n\t\t$ \\bar{d} $ & $ 0.0085 $ & $ -0.0068 $ & $ 0.0011 $ \\\\\n\t\t\\hline \n\t\t$ s $ & $ 0.0339 $ & $ 0.0379 $ & $ 0.0252 $ \\\\\n\t\t\\hline\n\t\tMean conf. int. & $ (-0.014, 0.03) $ & $ (-0.032, 0.02) $ & $ (-0.016, 0.018) $ \\\\\n\t\t\\hline\n\t\tSt.d. conf. int. & $ (0.024, 0.06) $ &  $ (0.026, 0.066) $ & $ (0.018, 0.044) $ \\\\\n\t\t\\hline\n\t\tHypothesis test & $ H_0 $ is retained & $ H_0 $ is retained  & $ H_0 $ is retained \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\\begin{table}\\label{table3}\n\n\t\\caption{Gradual uncertainty}\n\t\\centering\n\t\\begin{tabular}{ | c | c | c | c | }\n\t\t\\hline \n\t\t& 0 order & 1st order & 2nd order \\\\\n\t\t\\hline \n\t\tTF before $ 70 $ s & $ 4 $ & $ \\frac{4}{s+1} $ & $ \\frac{4}{(s+1)(s+5)} $ \\\\\n\t\t\\hline\n\t\tTF after $ 70 $ s & $ 6 $ & $ \\frac{6}{s+0.5} $ & $ \\frac{6}{(s+0.5)(s+2.5)} $ \\\\\n\t\t\\hline\n\t\t$ \\bar{d} $ & $ 0.0154 $ & $ 0.0026 $ & $ 0.004 $ \\\\\n\t\t\\hline \n\t\t$ s $ & $ 0.038 $ & $ 0.034 $ & $ 0.03 $ \\\\\n\t\t\\hline\n\t\tMean conf. int. & $ (-0.01, 0.04) $ & $ (-0.02, 0.025) $ & $ (-0.016, 0.023) $ \\\\\n\t\t\\hline\n\t\tSt.d. conf. int. & $ (0.026, 0.067) $ & $ (0.0235, 0.06) $ & $ (0.02, 0.05) $ \\\\\n\t\t\\hline\n\t\tHypothesis test & $ H_0 $ is retained & $ H_0 $ is retained  & $ H_0 $ is retained \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table} \n\n\n\nAnalyses of the experimental results where a first order plant dynamics is used with a sudden uncertainty injection is provided above. All of the results, including the ones for the other cases, where plants with different orders and sudden\/gradual uncertainty injections, are summarized in Tables II and III.\nThe data collected from the participants can be reached at \\href{http:\/\/www.syslab.bilkent.edu.tr\/research}{http:\/\/www.syslab.bilkent.edu.tr\/research}.\n\n\n\n\n\\section{SUMMARY}\\label{sec:conclusion}\n\nIn this paper, an adaptive human pilot model based on model reference adaptive control principles is proposed. This model mimics the pilot decision making process by making sure that the overall closed loop system follows the crossover model in the presence of plant uncertainties. The stability of the system is shown using the Lyapunov-Krasovskii stability criteria. Furthermore, experiments with human operators are conducted to validate the model. Detailed visual and statistical analyses of the experimental results show that the adaptive model creates similar system responses as the human operators. \n\n                                 \n                                 \n                                 \n                                 \n                                 \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe study the Marshall-Peierls sign rule (MPSR) for the frustrated\nspin-${1\\over2}$ Heisenberg antiferromagnet\n\\begin{equation}\n\\label{H12}\n{\\rm \\hat H} =J_1 \\sum_{\\langle {\\bf NN} \\rangle} {\\bf S_i} {\\bf S}_{\\bf\nj} + J_2 \\sum_{\\langle {\\bf NNN} \\rangle} {\\bf S_i} {\\bf S}_{\\bf j},\n\\end{equation}\nwhere $\\langle {\\bf NN}\\rangle$ and $\\langle {\\bf NNN}\\rangle$ denote\nnearest-neighbor and next-nearest-neighbor bonds on a linear chain or a\nsquare lattice.\n\nAccording to Marshall's early work in 1955 \\cite{Marshall:1955} we know\nthe relative phases of the Ising basis states building the ground state\nwave function of a bipartite spin-${1\\over2}$ Heisenberg antiferromagnet\n(MPSR, for more details see below). Later on Lieb and Mattis generalized\nthe theorem to arbitrary site spins and bipartite lattices without\ntranslational symmetries \\cite{Lieb:1961,Lieb:1962}. The knowledge of\nthe sign is of great importance for the construction of variational wave\nfunctions (see e.g. \\cite{Richter:zpb93}) and for quantum Monte Carlo\nprocedures which may have the so-called minus-sign problem if the MPSR\nis violated (see e.g.\\cite{Raedt}). In particular, the possible\nviolation of the MPSR in frustrated systems is a serious problem for\nvariational and quantum  Monte Carlo procedures.\n\nFor the considered model (\\ref{H12}) the MPSR can be proved for $J_2 \\le\n0$. In the frustrated model ($J_2 > 0$) the MPSR can be destroyed\n\\cite{Richter:zpb93,Kitatani:92,Richter:epl94}. However, in a recent\nwork \\cite{Richter:zpb93,Richter:epl94} we have presented general\narguments that the MPSR may survive for relatively large $J_2$. These\narguments are based on exact diagonalization results for small clusters\n(number of sites $N \\le 24$), as well as on the spin wave approximation.\nFollowing these arguments Zeng and Parkinson \\cite{Zeng:prb95} use the\nMPSR as a new way of investigating the spatial periodicity in the ground\nstate of frustrated spin chains. Furthermore, they studied the breakdown\nof the MPSR as a function of the chain length and the frustrating $J_2$.\nBy finite size extrapolation they estimated a finite critical value for\n$J_2$ for an infinite chain of about $0.03J_1$ where the MPSR is\nviolated in the ground state.\n\nBecause of the exponential growth of the number of basis states the\ndirect numerical calculation of the singlet ground state is limited to\nsmall clusters and the conclusions obtained from small systems seem not\nto be quite reliable.\n\nIn this paper we exploit the observation that the MPSR holds not only\nfor the singlet ground state but also for every lowest eigenstate in any\nsubspace with higher quantum number $M \\le {N\\over2}$ of the z-component\nof the total spin. In these subspaces the number of basis states is\nreduced and it is possible to diagonalize much larger systems. With this\ndata an approximation to the thermodynamic limit is more reliable. Below\nwe present data up to $N\\le144$ and we can conclude that the MPSR\nsurvives indeed a finite frustration $J_2$ at least for states with\nhigher quantum numbers $M$.\n\n\\section{Marshall Peierls sign rule}\nIn the unfrustrated limit $J_2=0$ the lowest eigenstate of the\nHamiltonian (\\ref{H12}) in each subspace of fixed eigenvalue $M$ of the\nspin operator $S_{total}^z$ reads\n\\begin{equation}\n\\label{GZ:Phi}\n\\Psi_{M} = \\sum_{m}{c_{m}^{(M)}|m \\rangle} \\hspace{0.2cm},\n\\hspace{0.5cm} c_{m}^{(M)}>0 \\hspace{0.2cm}.\n\\end{equation}\nHere the Ising states $|m\\rangle$ are defined by\n\\begin{equation}\n\\label{m}\n|m\\rangle \\equiv (-1)^{S_A-M_A}|m_1\\rangle \\otimes |m_2 \\rangle \\otimes\n\\cdots \\otimes |m_N\\rangle \\hspace{0.2cm},\n\\end{equation}\nwhere $|m_i\\rangle,\\hspace{0.2cm}i=1,\\cdots,N$, are the eigenstates of\nthe site spin operator $S_{i}^{z}$ ($ -s_{i} \\leq m_{i} \\leq s_i$),\n$S_A= \\sum_{i\\in A}s_i$, $M_{A(B)}=\\sum_{i\\in A(B)}m_i$, $M=M_A+M_B$.\nThe lattice consists of two equivalent sublattices $A$ and $B$.\n$s_i\\equiv s$, $i=1,\\cdots,N$, are the site spins. The summations in\nEq.(\\ref{GZ:Phi}) are restricted by the condition $\\sum_{i=1}^N m_i=M$.\nThe wave function (\\ref{GZ:Phi}) is not only an eigenstate of the\nunfrustrated Hamiltonian ($J_2=0$) and $S_{total}^z$ but simultaneously\nof the square of the total spin ${\\bf S}_{total}^2$ with quantum number\n$S=\\mid \\!M \\! \\mid$. Because $c_m^{(M)}>0$ is valid for each $m$ from\nthe basis set (\\ref{m}) it is impossible to build up other orthonormal\nstates without using negative amplitudes $c_m^{(M)}$. Hence the ground\nstate wave function $\\Psi_M$ is nondegenerated.\n\nFor the lowest energy eigenvalues $E(S)$ belonging to the subspace $M$\nwe have the Lieb-Mattis level-ordering\n\\begin{equation}\n\\label{e}\nE(S)<E(S+1) \\hspace{0.2cm}, \\hspace{0.2cm} S\\geq 0 \\hspace{0.5cm},\n\\end{equation}\ni.e. the ground state is a singlet. Note that this level ordering might\nbe violated by frustration. However, a lot of numerical calculations\nshow the same level ordering also for strongly frustrated systems\n\\cite{Bernu92,Richter95}.\n\n\n\\section{Analytic Solutions}\nNow we include frustration ($J_2>0$). In the subspace with maximum\n$M=N\/2$ the MPSR is never violated in any dimension. Here the only\npossible state is the fully polarized ferromagnetic state which does not\nchange with increasing $J_2$.\n\nIn the next subspace $M=(N\/2)-1$ analytic solutions are found for linear\nchains and square lattices. In this subspace we deal with the so-called\none-magnon state, where the wave function can be expressed as a\nBloch-wave with a given $\\vec k$.\n\n{\\it Linear Chain} - In this case the solution can be found by comparing\nthe energies as a function of the wave vector $\\vec k$.\n\\begin{equation}\nE(k) = J_1 ( {N\\over 4} -1 + \\cos(k) ) + J_2 ( {N\\over 4} -1 +  \\cos(2k) )\n\\end{equation}\nwith $\\vec k = {2 \\pi\\over N} i , i=0,\\pm1,\\pm2,\\ldots,+{N\\over2} $.\nThe comparison of $E(\\pi)$ and $E({2 \\pi\\over N}({N\\over2}-1))$ yields the\nequation for the critical $J_2$\n\\begin{equation}\nJ_2^c = J_1 {1 + \\cos \\left[ \\pi (1- {2 \\over N}) \\right] \\over\n                1 - \\cos \\left[ 2\\pi (1- {2 \\over N}) \\right] } .\n\\end{equation}\nIn the limit $N \\rightarrow \\infty$ one obtains $J_2^c={1\\over4}J_1$.\n\n{\\it Square lattice} -\nFor small $J_2$ in the considered subspace  the lowest energy is\nobtained for  $\\vec k=(\\pi,\\pi)$ and reads $E_1=J_1(N-8)+J_2N$. The\ncorresponding eigenfunction fulfills the MPSR. For larger $J_2$ we have\nto distinguish between two cases: (a) If the number of spins in the\nsublattice $N\/2$ is even we find a transition at $J_2=(J_1\/2)$ to a\ntwofold degenerated ground state with $\\vec k =(\\pi,0)$ or $\\vec k\n=(0,\\pi)$ with an energy $E_2=J_1(N-4)+J_2(N-8)$. This state violates\nthe MPSR, i.e. we have $J_2^c={1\\over2}J_1$. Notice, that  the\neigenfunctions with $\\vec k =(\\pi,0)$ or $\\vec k =(0,\\pi)$ fulfill the\nso-called product-MPSR \\cite{Richter:zpb93}. (b) If the number of spins\nin the sublattice is odd (e.g. $N=26$), the situation is more\ncomplicated. The energy levels cross each other for $J_2^{c}$ slightly\ngreater than ${1\\over2}J_1$.\n\n\\section{Numerical Results}\nIn subspaces with lower quantum numbers $M < (N\/2)-1$ we cannot find\nsimple expressions for the eigenvalues and eigenfunctions. Hence, we\npresent exact diagonalization data for $M\\le(N\/2)-2$. Using a modified\nLanczos procedure we calculate in every subspace $M$ the state with the\nlowest energy $E_0(M)$. Since the number of Ising basis states increases\nexponentially as ${N \\choose N-M}$, one can calculate $E_0(M)$ for {\\bf\nall} $M={N\\over2},\\ldots,0$ only for relatively small systems (in our\ncase $N \\le 26$). However, in subspaces with larger $M$ we are able to\npresent data for $N$ up to $144$. In all cases we use periodic boundary\nconditions. Analyzing the eigenfunction with respect to the MPSR we can\ndetermine numerically the critical $J_2^{c}$ where the MPSR is violated.\n\n{\\it Linear Chain} - In Fig.\\ref{fig1} we show $J_2^{c}$ as a function\nof $1\/N$. For $M(N) = (N\/2)-1$ the analytic result is drawn. For the\nnext $M(N) = (N\/2)-2$ the data show a similar behavior with the same\ncritical value of $J_2={1\\over4}$ for $N\\rightarrow \\infty$. By\nconsidering the numerical data an analytic solution can be predicted\n\\begin{equation}\nJ_2^c = J_1{1 + \\cos \\left[ \\pi (1- {2 \\over (N-1)}) \\right] \\over\n                1 - \\cos \\left[ 2\\pi (1- {2 \\over (N-1)}) \\right] } .\n\\end{equation}\nThe following subspaces $M(N)=(N\/2)-p$, $p>2$ show a different behavior\nwith different critical values for $J_2$ if $N\\rightarrow \\infty$. These\ncritical values decrease for increasing $p$ but evidently a finite\nregion with a non-violated MPSR is preserved.\n\nIn Fig.\\ref{fig2} the critical  $J^c_2$ is shown as a function of a\nrenormalized \\linebreak $M_r = M(N)\/(N\/2) $ for small systems\n($N=8,\\ldots,26$) over the full range of $M_r$. $M_r=1$ is the ground\nstate subspace for a ferromagnet and $M_r=0$ for an antiferromagnet. The\nfinite size extrapolation for the ground state with $M_r=0$ yields a\nsmall but finite critical value $J^c_2 \\approx 0.03J_1$ which\ncorresponds to the value estimated by Zeng and Parkinson in\n\\cite{Zeng:prb95}. The monotonic decrease of $J_2^{c}$ with decreasing\n$M_r$ indicates a finite region of a validity of the MPSR for all $M_r$.\n\n{\\it Square lattice} - In Fig.\\ref{fig3} we show $J_2^{c}$ as a function\nof $1\/N$. Here the $N$ dependence is less regular and a finite size\nextrapolation is much more difficult. This is mainly connected with the\nshape of the periodic boundaries. For some of the finite lattices the\nboundaries are parallel to the x- and y-axis (e.g. for $N=$4x4,\n6x6,...,12x12) whereas for other lattices the boundaries are inclined\n(e.g. $N=18,20,32$, see e.g \\cite{Oitmaa79}). The impression of an\noscillating behavior of $J_2^c$ versus $1\/N$ stems just from the\nalternation of parallel and inclined finite lattices. Nevertheless, it\nis evident that the critical $J^c_2$ goes to a finite value for\n$N\\rightarrow \\infty$. An extrapolation to the thermodynamic limit for\nthe antiferromagnetic ground state, i.e. subspace $M=0$, is almost\nimpossible for the square lattice. However, if we assume for $M=0$ that\nthe $J_2^c$ is almost independent of $1\/N$ for $N>16$ (as it is\nsuggested by Fig.\\ref{fig3} and by spin wave theory) we could estimate\nfrom our data for $N=10,16,18,20$ a critical value of about $0.2 \\ldots\n0.3$.\n\nFig.\\ref{fig4} supports this estimation. Here the critical $J_2^c$ is\nshown for different small lattices $N \\le 34$ as a function of $M_r$. It\nis seen that for $M_r \\le 0.6$ the critical $J_2^c$ does not strongly\ndepend on $M_r$ (in contrast to the linear chain, where we have a\nmonotonic decrease) and gives for all the lattices a value of about\n$0.3J_1$ for the antiferromagnetic ground state ($M_r=0$).\n\n\\section{Conclusion}\nFor linear chains and square lattices we have shown, that in subspaces\nwith large quantum number $M$ of the spin operator $S_{total}^z$, the\nMarshall-Peierls sign rule is preserved up to a fairly large frustration\nparameter $J_2^{c}$.\n\nMoreover, for linear chains the finite size extrapolation is quite\nreliable even for the singlet ground state and yields for {\\bf all} $M$\na finite parameter region for $J_2$ where the MPSR is valid.\n\nFor square lattices we observe in general higher critical values $J_2^c$\ncompared to linear chains. From this observation and from the\nextrapolation for subspaces with higher $M$ we argue that for square\nlattices the MPSR is stable against a finite frustration in all\nsubspaces, too.\n\n\n\\section{Acknowledgments}\nThis work has been supported by the Deutsche Forschungsgemeinschaft\n(Project No. Ri 615\/1-2) and the Bulgarian Science Foundation (Grant\nF412\/94).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sIntroduction}\n\nA complex $b$-manifold is a smooth manifold with boundary together with a complex $b$-structure. The latter is a smooth involutive subbundle ${}^b{\\!}T^{0,1}\\mathcal M$ of the complexification $\\mathbb C{}^b{\\!}T\\mathcal M$ of Melrose's $b$-tangent bundle \\cite{RBM1,RBM2} with the property that\n\\begin{equation*}\n\\mathbb C{}^b{\\!}T\\mathcal M = {}^b{\\!}T^{0,1}\\mathcal M + \\overline{{}^b{\\!}T^{0,1}\\mathcal M}\n\\end{equation*}\nas a direct sum. Manifolds with complex $b$-structures generalize the situation that arises as a result of spherical and certain anisotropic (not complex) blowups of complex manifolds at a discrete set of points or along a complex submanifold, cf. \\cite[Section 2]{Me3}, \\cite{Me5}, as well as (real) blow-ups of complex analytic  varieties with only point singularities.\n\nThe interior of $\\mathcal M$ is a complex manifold. Its $\\overline \\partial$-complex determines a $b$-elliptic complex, the ${}^b\\!\\overline \\partial$-complex, on sections of the exterior powers of the dual of ${}^b{\\!}T^{0,1}\\mathcal M$, see Section~\\ref{sPreliminaries}. The indicial families $\\overline\\D(\\sigma)$ of the ${}^b\\!\\overline \\partial$-operators at a connected component $\\mathcal N$ of $\\partial \\mathcal M$ give, for each $\\sigma$, an elliptic complex, see Section~\\ref{sIndicialComplex}. Their cohomology at the various values of $\\sigma$ determine the asymptotics at $\\mathcal N$ of tempered representatives of cohomology classes of the ${}^b\\!\\overline \\partial$-complex, in particular of tempered holomorphic functions. \n\nEach boundary component $\\mathcal N$ of $\\mathcal M$ inherits from ${}^b{\\!}T^{0,1}\\mathcal M$ the following objects in the $C^\\infty$ category:\n\\begin{enumerate}\n\\item an involutive vector subbundle $\\overline\\Vee\\subset \\mathbb C T\\mathcal N$ such that $\\mathcal V+\\overline\\Vee=\\mathbb C T\\mathcal N$;\n\\item a real nowhere vanishing vector field $\\mathcal T$ such that $\\mathcal V\\cap\\overline\\Vee=\\Span_\\mathbb C\\mathcal T$;\n\\item a class $\\pmb \\beta$ of sections of $\\smash[t]{\\overline\\Vee}^*$,\n\\end{enumerate}\nwhere the elements of $\\pmb \\beta$ have additional properties, described in (4) below. The vector bundle $\\overline\\Vee$, being involutive, determines a complex of first order differential operators $\\overline\\Dee$ on sections of the exterior powers of $\\smash[t]{\\overline\\Vee}^*$, elliptic because of the second property in (1) above. To that list add\n\\begin{enumerate}\n\\item [(4)] If $\\beta\\in \\pmb \\beta$ then $\\overline\\Dee\\beta=0$ and $\\Im\\langle \\beta,\\mathcal T\\rangle=-1$, and if $\\beta$, $\\beta'\\in\\pmb\\beta$, then $\\beta'-\\beta=\\overline\\Dee u$ with $u$ real-valued.\n\\end{enumerate}\nThese properties, together with the existence of a Hermitian metric on $\\overline\\Vee$ invariant under $\\mathcal T$ make $\\mathcal N$ behave in many ways as the circle bundle of a holomorphic line bundle over a compact complex manifold. These analogies are investigated in  \\cite{Me6,Me7,Me8,Me9}. The last of these papers contains a detailed account of circle bundles from the perspective of these boundary structures. The paper \\cite{Me4}, a predecessor of the present one, contains some facts studied here in more detail.\n\n\n\\medskip\nThe paper is organized as follows. Section~\\ref{sPreliminaries} deals with the definition of complex $b$-structure and Section~\\ref{sHolomorphicVectorBundles} with holomorphic vector bundles over complex $b$-manifolds (the latter term just means that the $b$-tangent bundle takes on a primary role over that of the usual tangent bundle). The associated Dolbeault complexes are defined in these sections accordingly.\n\nSection~\\ref{sBoundaryStructure} is a careful account of the structure inherited by the boundary.\n\nIn Section~\\ref{sLocalInvariants} we show that complex $b$-structures have no formal local invariants at boundary points. The issue here is that we do not have a Newlander-Nirenberg theorem that is valid in a neighborhoods of a point of the boundary, so no explicit local model for $b$-manifolds.\n\nSection~\\ref{sIndicialComplex} is devoted to general aspects of $b$-elliptic first order complexes $A$. We introduce here the set $\\spec_{b,\\mathcal N}^q(A)$, the boundary spectrum of the complex in degree $q$ at the component $\\mathcal N$ of $\\mathcal M$, and prove basic properties of the boundary spectrum (assuming that the boundary component $\\mathcal N$ is compact), including some aspects concerning Mellin transforms of $A$-closed forms. Some of these ideas are illustrated using the $b$-de Rham complex.\n\nSection~\\ref{sUnderlyingCRcomplexes} is a systematic study of the $\\overline \\partial_b$-complex of CR structures on $\\mathcal N$ associated with elements of the class $\\pmb \\beta$. Each $\\beta\\in \\pmb\\beta$ defines a CR structure, $\\overline \\K_\\beta=\\ker \\beta$. Assuming that $\\overline\\Vee$ admits a $\\mathcal T$-invariant Hermitian metric, we show that there is $\\beta\\in \\pmb \\beta$ such that the CR structure $\\overline \\K_\\beta$ is $\\mathcal T$-invariant.\n\nIn Section~\\ref{sSpectrum} we assume that $\\overline\\Vee$ is $\\mathcal T$-invariant and show that for $\\mathcal T$-invariant CR structures, a theorem proved in \\cite{Me9} gives that the cohomology spaces of the associated $\\overline \\partial_b$-complex, viewed as the kernel of the Kohn Laplacian at the various degrees, split into eigenspaces of $-i\\mathcal {L}_\\mathcal T$. The eigenvalues of the latter operator are related to the indicial spectrum of the ${}^b\\!\\overline \\partial$-complex.\n\nIn Section~\\ref{sIndicialCohomology} we prove a precise theorem on the indicial cohomology and spectrum for the ${}^b\\!\\overline \\partial$-complex under the assumption that $\\overline\\Vee$ admits a $\\mathcal T$-invariant Hermitian metric.\n\nFinally, we have included a very short appendix listing a number of basic definitions in connection with $b$-operators. \n\n\n\\section{Complex $b$-structures}\\label{sPreliminaries}\n\nLet $\\mathcal M$ be a smooth manifold with smooth boundary. An almost CR $b$-structure on $\\mathcal M$ is a subbundle $\\overline\\W$ of the complexification, $\\mathbb C{}^b{\\!}T\\mathcal M\\to\\mathcal M$ of the $b$-tangent bundle of $\\mathcal M$ (Melrose \\cite{RBM1,RBM2}) such that\n\\begin{equation}\\label{bCRstructure}\n\\mathcal W\\cap \\overline\\W=0\n\\end{equation}\nwith $\\mathcal W=\\overline\\W$. If in addition \n\\begin{equation}\\label{bElliptic}\n\\mathcal W + \\overline\\W=\\mathbb C{}^b{\\!}T\\mathcal M\n\\end{equation}\nthen we say that $\\overline\\W$ is an almost complex $b$-structure and write ${}^b{\\!}T^{0,1}\\mathcal M$ instead of $\\overline\\W$ and ${}^b{\\!}T^{1,0}\\mathcal M$ for its conjugate. As is customary, the adverb ``almost'' is dropped if $\\mathcal W$ is involutive. Note that since $C^\\infty(\\mathcal M;{}^b{\\!}T\\mathcal M)$ is a Lie algebra, it makes sense to speak of involutive subbundles of ${}^b{\\!}T\\mathcal M$ (or its complexification). \n\n\\begin{definition}\nA complex $b$-manifold is a manifold together with a complex $b$-structure.\n\\end{definition}\n\nBy the Newlander-Nirenberg Theorem~\\cite{NeNi57}, the interior of complex $b$-manifold is a complex manifold. However, its boundary is not a CR manifold; rather, as we shall see, it naturally carries a family of CR structures parametrized by the defining functions of $\\partial\\mathcal M$ in $\\mathcal M$ which are positive in $\\open \\mathcal M$.\n\n\\medskip\nThat $C^\\infty(\\mathcal M;{}^b{\\!}T\\mathcal M)$ is a Lie algebra is an immediate consequence of the definition of the $b$-tangent bundle, which indeed can be characterized as being a vector bundle ${}^b{\\!}T\\mathcal M\\to\\mathcal M$ together with a vector bundle homomorphism \n\\begin{equation*}\n\\mathrm{ev}:{}^b{\\!}T\\mathcal M\\to T\\mathcal M\n\\end{equation*}\ncovering the identity such that the induced map\n\\begin{equation*}\n\\mathrm{ev}_*:C^\\infty(\\mathcal M;{}^b{\\!}T\\mathcal M)\\to C^\\infty(\\mathcal M;T\\mathcal M)\n\\end{equation*}\nis a $C^\\infty(\\mathcal M;\\mathbb R)$-module isomorphism onto the submodule $C^\\infty_{\\tan}(\\mathcal M;T\\mathcal M)$ of smooth vector fields on $\\mathcal M$ which are tangential to the boundary of $\\mathcal M$. Since $C^\\infty_{\\tan}(\\mathcal M,T\\mathcal M)$ is closed under Lie brackets, there is an induced Lie bracket on $C^\\infty(\\mathcal M;{}^b{\\!}T\\mathcal M)$ The homomorphism $\\mathrm{ev}$ is an isomorphism over the interior of $\\mathcal M$, and its restriction to the boundary,\n\\begin{equation}\\label{evbM}\n\\mathrm{ev}_{\\partial\\mathcal M}:{}^b{\\!}T_{\\partial\\mathcal M}\\mathcal M\\to T\\partial\\mathcal M\n\\end{equation}\nis surjective. Its kernel, a fortiori a rank-one bundle, is spanned by a canonical section denoted $\\mathfrak r\\partial_\\mathfrak r$. Here and elsewhere, $\\mathfrak r$ refers to any smooth defining function for $\\partial \\mathcal M$ in $\\mathcal M$, by convention positive in the interior of $\\mathcal M$.\n\nAssociated with a complex $b$-structure on $\\mathcal M$ there is a Dolbeault complex. Let ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M$ denote the $q$-th exterior power of the dual of ${}^b{\\!}T^{0,1}M$. Then the operator\n\\begin{equation*}\n\\cdots\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M)\\xrightarrow{{}^b\\!\\overline \\partial} C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}\\mathcal M) \\to \\cdots\n\\end{equation*}\nis define by \n\\begin{multline}\\label{CartanFormula}\n(q+1)\\,{}^b\\!\\overline \\partial\\phi(V_0,\\dotsc,V_q)=\\sum_{j=0}^q V_j\\phi(V_0,\\dotsc,\\hat V_j,\\dotsc,V_q)\\\\\n +\\sum_{j<k}(-1)^{j+k}\\phi([V_j,V_k],V_0,\\dotsc,\\hat V_j,\\dotsc,\\hat V_k,\\dotsc,V_q)\n\\end{multline}\nas with the standard de Rham differential (see Helgason~\\cite[p.~21]{He}) whenever $\\phi$ is a smooth section of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal M$ and $V_0,\\dotsc,V_q\\in C^\\infty(\\mathcal M;{}^b{\\!}T^{0,1}\\mathcal M)$. In this formula $V_j$ acts on functions via the vector field $\\mathrm{ev}_*V_j$. The involutivity of ${}^b{\\!}T^{0,1}\\mathcal M$ is used in the the terms involving brackets, of course. The same proof that $d\\circ d=0$ works here to give that ${}^b\\!\\overline \\partial^2=0$. The formula\n\\begin{equation}\n{}^b\\!\\overline \\partial(f\\phi)=f\\,{}^b\\!\\overline \\partial\\phi+{}^b\\!\\overline \\partial f\\wedge \\phi\\text{ for } \\phi\\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal M) \\text{ and } f\\in C^\\infty(\\mathcal M),\n\\end{equation}\nimplies that ${}^b\\!\\overline \\partial$ is a  first order operator.\n\nSince we do not have at our disposal holomorphic frames (near the boundary) for the bundles of forms of type $(p,q)$ for $p>0$, we define ${}^b\\!\\overline \\partial$ on forms of type $(p,q)$ with $p>0$ with the aid of the $b$-de Rham complex, exactly as in Foland and Kohn~\\cite{FK} for standard complex structures and de Rham complex. The $b$-de Rham complex, we recall from Melrose~\\cite{RBM2}, is the complex associated with the dual, $\\mathbb C{}^b{\\!}T^*\\mathcal M$, of $\\mathbb C{}^b{\\!}T\\mathcal M$,\n\\begin{equation*}\n\\cdots\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^r\\mathcal M)\\xrightarrow{{}^b\\!d} C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{r+1}\\mathcal M) \\to \\cdots\n\\end{equation*}\nwhere ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal M$ denotes the $r$-th exterior power of $\\mathbb C{}^b{\\!}T^*\\mathcal M$. The operators ${}^b\\!d$ are defined by the same formula as \\eqref{CartanFormula}, now however with the $V_j\\in C^\\infty(\\mathcal M;\\mathbb C{}^b{\\!}T\\mathcal M)$. On functions $f$ we have \n\\begin{equation*}\n{}^b\\!d f=\\mathrm{ev}^*df.\n\\end{equation*}\nMore generally,\n\\begin{equation*}\n\\mathrm{ev}^* \\circ d = {}^b\\!d\\circ \\mathrm{ev}^*\n\\end{equation*}\nin any degree. Also,\n\\begin{equation}\\label{FirstOrder}\n{}^b\\!d(f\\phi)=f\\,{}^b\\!d\\phi+{}^b\\!d f\\wedge \\phi\\text{ for } \\phi\\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^r\\mathcal M) \\text{ and } f\\in C^\\infty(\\mathcal M).\n\\end{equation}\nIt is convenient to note here that for $f\\in C^\\infty(\\mathcal M)$,\n\\begin{equation}\\label{VanishingOnBdy}\n\\text{${}^b\\!d f$ vanishes on $\\partial \\mathcal M$ if $f$ does.}\n\\end{equation}\n\nNow, with the obvious definition,\n\\begin{equation}\\label{SpittingOfDeRham}\n{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^r\\mathcal M=\\bigoplus_{p+q=r}{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M.\n\\end{equation}\nUsing the special cases \n\\begin{gather*}\n {}^b\\!d:C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1})\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{1,1})+C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,2}),\\\\\n {}^b\\!d:C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{1,0})\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{2,0})+C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{1,1}),\n\\end{gather*}\nconsequences of the involutivity of ${}^b{\\!}T^{0,1}\\mathcal M$ and its conjugate, one gets \n\\begin{equation*}\n{}^b\\!d\\phi\\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p+1,q}\\mathcal M)\\oplus C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q+1}\\mathcal M)\\quad \\text{if } \\phi\\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M)\n\\end{equation*}\nfor general $(p,q)$. Let $\\pi_{p,q}:{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k\\mathcal M\\to{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M$ be the projection according to the decomposition \\eqref{SpittingOfDeRham}, and define\n\\begin{equation*}\n{}^b\\!\\dee=\\pi_{p+1,q}{}^b\\!d,\\quad {}^b\\!\\overline \\partial=\\pi_{q,p+1}{}^b\\!d,\n\\end{equation*}\nso ${}^b\\!d={}^b\\!\\dee+{}^b\\!\\overline \\partial$. The operators ${}^b\\!\\overline \\partial$ are identical to the $\\overline \\partial$-operators over the interior of $\\mathcal M$ and with the previously defined ${}^b\\!\\overline \\partial$ operators on $(0,q)$-forms, and give a complex\n\\begin{equation}\\label{bdeebarComplex}\n\\cdots\n\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M) \\xrightarrow{{}^b\\!\\overline \\partial}\nC^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q+1}\\mathcal M)\n\\to\\cdots\n\\end{equation}\nfor each $p$.  On functions $f:\\mathcal M\\to\\mathbb C$,\n\\begin{equation}\\label{bdeebarOnFunctions}\n{}^b\\!\\overline \\partial f = \\pi_{0,1}\\, {}^b\\!d f.\n\\end{equation}\nThe formula\n\\begin{equation}\\label{bdeebarFirstOrderPrime}\\tag{\\ref{FirstOrder}$'$}\n{}^b\\!\\overline \\partial f\\phi={}^b\\!\\overline \\partial f\\wedge \\phi+f{}^b\\!\\overline \\partial\\phi ,\\quad f\\in C^\\infty(\\mathcal M),\\ \\phi\\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M),\n\\end{equation}\na consequence of \\eqref{FirstOrder}, implies that ${}^b\\!\\overline \\partial$ is a first order operator. As a consequence of \\eqref{VanishingOnBdy}, \n\\begin{equation}\\label{bdeebarVanishingOnBdy}\\tag{\\ref{VanishingOnBdy}$'$}\n\\text{${}^b\\!\\overline \\partial f$ vanishes on $\\partial \\mathcal M$ if $f$ does.}\n\\end{equation}\n\n\n\n\\medskip\nThe operators of the $b$-de Rham complex are first order operators because of \\eqref{FirstOrder}, and \\eqref{VanishingOnBdy} implies that these are $b$-operators, see \\eqref{TotallyChar}. Likewise, \\eqref{bdeebarFirstOrderPrime} and \\eqref{bdeebarVanishingOnBdy} imply that in any bidegree, the operator $\\phi\\mapsto \\mathfrak r^{-1}\\,{}^b\\!\\overline \\partial \\,\\mathfrak r\\phi$ has coefficients smooth up to the boundary, so\n\\begin{equation}\\label{bdeebarOnpq}\n{}^b\\!\\overline \\partial\\in \\Diff^1_b(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q+1}\\mathcal M),\n\\end{equation}\nsee \\eqref{TotallyChar}. We also get from these formulas that the $b$-symbol of ${}^b\\!\\overline \\partial$ is\n\\begin{equation}\\label{bsymbdeebar}\n\\bsym({}^b\\!\\overline \\partial)(\\xi)(\\phi)=i \\pi_{0,1}(\\xi)\\wedge \\phi, \\quad x\\in\\mathcal M,\\ \\xi\\in {}^b{\\!}T^*_x\\mathcal M,\\ \\phi\\in {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}_x\\mathcal M,\n\\end{equation}\nsee \\eqref{TheBSymbol}. Since $\\pi_{0,1}$ is injective on the real $b$-cotangent bundle (this follows from \\eqref{bElliptic}), the complex \\eqref{bdeebarComplex} is $b$-elliptic.\n\n\n\\section{Holomorphic vector bundles}\\label{sHolomorphicVectorBundles}\nThe notion of holomorphic vector bundle in the $b$-category is a translation of the standard one using connections. Let $\\rho:F\\to\\mathcal M$ be a complex vector bundle. Recall from \\cite{RBM2} that a $b$-connection on $F$ is a linear operator\n\\begin{equation*}\n{}^b\\!\\nabla:C^\\infty(\\mathcal M;F)\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^1\\mathcal M \\otimes F)\n\\end{equation*}\nsuch that\n\\begin{equation}\\label{Connection}\n{}^b\\!\\nabla f\\phi=f\\, {}^b\\!\\nabla\\phi+ {}^b\\!d f \\otimes \\phi\n\\end{equation}\nfor each $\\phi \\in C^\\infty(\\mathcal M;F)$ and $f\\in C^\\infty(\\mathcal M)$. This property automatically makes ${}^b\\!\\nabla$ a $b$-operator.\n\nA standard connection $\\nabla:C^\\infty(\\mathcal M;F)\\to C^\\infty(\\mathcal M;\\raise2ex\\hbox{$\\mathchar\"0356$}^1\\mathcal M \\otimes F)$ determines a $b$-connection by composition with\n\\begin{equation*}\n\\mathrm{ev}^*\\otimesI:\\raise2ex\\hbox{$\\mathchar\"0356$}^1\\mathcal M \\otimes F\\to {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^1\\mathcal M \\otimes F,\n\\end{equation*}\nbut $b$-connections are more general than standard connections. Indeed, the difference between the latter and the former can be any smooth section of the bundle $\\Hom(F,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^1\\mathcal M\\otimes F)$. A $b$-connection ${}^b\\!\\nabla$ on $F$ arises from a standard connection if and only if ${}^b\\!\\nabla_{\\mathfrak r\\partial_\\mathfrak r}=0$ along $\\partial\\mathcal M$.\n\nAs in the standard situation, the $b$-connection ${}^b\\!\\nabla$ determines operators\n\\begin{equation}\\label{ExtConnection}\n{}^b\\!\\nabla:C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k\\mathcal M \\otimes F)\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{k+1}\\mathcal M \\otimes F)\n\\end{equation}\nby way of the usual formula translated to the $b$ setting:\n\\begin{equation}\\label{ExtConnectionBis}\n{}^b\\!\\nabla(\\alpha\\otimes \\phi) = (-1)^k \\alpha \\wedge {}^b\\!\\nabla \\phi + {}^b\\!d\\alpha \\wedge \\phi,\\quad \\phi \\in C^\\infty(\\mathcal M;F),\\ \\alpha\\in {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k\\mathcal M.\n\\end{equation}\nSince\n\\begin{equation*}\n{}^b\\!\\nabla \\mathfrak r\\alpha\\otimes \\phi=\\mathfrak r\\, {}^b\\!\\nabla(\\alpha\\otimes \\phi) + {}^b\\!d \\mathfrak r \\wedge \\alpha\\otimes \\phi\n\\end{equation*}\nis smooth and vanishes on $\\partial\\mathcal M$, also\n\\begin{equation*}\n{}^b\\!\\nabla\\in \\Diff^1_b(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k\\mathcal M\\otimes F,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{k+1}\\mathcal M\\otimes F).\n\\end{equation*}\nThe principal $b$-symbol of \\eqref{ExtConnection}, easily computed using \\eqref{ExtConnectionBis} and\n\\begin{equation*}\n\\bsym({}^b\\!\\nabla)({}^b\\!d f)(\\phi) = \\lim_{\\tau\\to\\infty} \\frac{e^{-i\\tau f}}{\\tau} {}^b\\!\\nabla e^{i\\tau f}\\phi\n\\end{equation*}\nfor $f\\in C^\\infty(\\mathcal M;\\mathbb R)$ and $\\phi \\in C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k\\mathcal M \\otimes F)$, is\n\\begin{equation*}\n\\bsym({}^b\\!\\nabla)(\\xi)(\\phi) = i\\xi\\wedge \\phi,\\quad \\xi\\in {}^b{\\!}T_x^*\\mathcal M,\\ \\phi \\in {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k_x\\mathcal M \\otimes F_x,\\ x\\in \\mathcal M.\n\\end{equation*}\n\nAs expected, the connection is called holomorphic if the component in ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,2}\\mathcal M\\otimes F$ of the curvature operator\n\\begin{equation*}\n\\Omega={}^b\\!\\nabla^2:C^\\infty(\\mathcal M;F)\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^2\\mathcal M\\otimes F),\n\\end{equation*}\nvanishes. Such a connection gives $F$ the structure of a complex $b$-manifold. Its complex $b$-structure can be described locally as in the standard situation, as follows. Fix a frame $\\eta_\\mu$ for $F$ and let the $\\omega^\\nu_\\mu$ be the local sections of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$ such that\n\\begin{equation*}\n{}^b\\!\\overline \\partial \\eta_\\mu = \\sum_\\nu \\omega^\\nu_\\mu\\otimes \\eta_\\nu.\n\\end{equation*}\nDenote by $\\zeta^\\mu$ the fiber coordinates determined by the frame $\\eta_\\mu$. Let $V_1,\\dotsc, V_{n+1}$ be a frame of ${}^b{\\!}T^{0,1}\\mathcal M$ over $U$, denote by $\\tilde V_j$ the sections of $\\mathbb C{}^b{\\!}T F$ over $\\rho^{-1}(U)$ which project on the $V_j$ and satisfy $\\tilde V_j\\zeta^\\mu=\\tilde V_j\\overline \\zeta^\\mu=0$ for all $\\mu$, and by $\\partial_{\\zeta^\\mu}$ the vertical vector fields such that $\\partial_{\\zeta^\\mu}\\zeta^\\nu=\\delta^\\nu_\\mu$ and $\\partial_{\\zeta^\\mu}\\overline \\zeta^\\nu=0$. Then the sections\n\\begin{equation}\\label{LocalbT01E}\n\\tilde V_j-\\sum_{\\mu,\\nu}\\zeta^\\mu \\langle \\omega^\\nu_\\mu, V_j\\rangle \\partial_{\\zeta^\\nu},\\ j=1,\\dotsc,n+1,\\quad \\partial_{\\overline \\zeta^\\nu},\\ \\nu=1,\\dotsc,k\n\\end{equation}\nof $\\mathbb C{}^b{\\!}T F$ over $\\rho^{-1}(U)$ form a frame of ${}^b{\\!}T^{0,1}F$. As in the standard situation, the involutivity of this subbundle of $\\mathbb C{}^b{\\!}T F$ is equivalent to the condition on the vanishing of the $(0,2)$ component of the curvature of ${}^b\\!\\nabla$. A vector bundle $F\\to\\mathcal M$ together with the complex $b$-structure determined by a choice of holomorphic $b$-connection (if one exists at all) is a holomorphic vector bundle.\n\n\nThe $\\overline \\partial$ operator of a holomorphic vector bundle is\n\\begin{equation*}\n{}^b\\!\\overline \\partial = (\\pi_{0,q+1}\\otimes I)\\circ {}^b\\!\\nabla : C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M\\otimes F)\\to C^\\infty(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}\\mathcal M \\otimes F).\n\\end{equation*}\nAs is the case for standard complex structures, the condition on the curvature of ${}^b\\!\\nabla$ implies that these operators form a complex, $b$-elliptic since\n\\begin{equation*}\n\\bsym({}^b\\!\\overline \\partial)(\\xi)(\\phi) = i\\pi_{0,1}(\\xi)\\wedge \\phi,\\quad \\xi\\in {}^b{\\!}T_x^*\\mathcal M,\\ \\phi \\in {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^k_x\\mathcal M \\otimes F_x,\\ x\\in \\mathcal M\n\\end{equation*}\nand $\\pi_{0,1}(\\xi)=0$ for $\\xi\\in {}^b{\\!}T^*\\mathcal M$ if and only if $\\xi=0$.\n\nAlso as usual, a $b$-connection ${}^b\\!\\nabla$ on a Hermitian vector bundle $F\\to\\mathcal M$ with Hermitian form $h$ is Hermitian if\n\\begin{equation*}\n{}^b\\!d h(\\phi,\\psi)= h({}^b\\!\\nabla \\phi,\\psi)+h(\\phi,{}^b\\!\\nabla\\psi)\n\\end{equation*}\nfor every pair of smooth sections $\\phi$, $\\psi$ of $F$. In view of the definition of ${}^b\\!d$ this means that for every $v\\in \\mathbb C{}^b{\\!}T\\mathcal M$ and sections as above,\n\\begin{equation*}\n\\mathrm{ev}(v) h(\\phi,\\psi)= h({}^b\\!\\nabla_{\\!v} \\phi,\\psi)+h(\\phi,{}^b\\!\\nabla_{\\!\\overline v}\\psi)\n\\end{equation*}\nOn a complex $b$-manifold $\\mathcal M$, if an arbitrary connection ${}^b\\!\\nabla'$ and the Hermitian form $h$ are given for a vector bundle $F$, holomorphic or not, then there is a unique \\emph{Hermitian} $b$-connection ${}^b\\!\\nabla$ such that $\\pi_{0,1}{}^b\\!\\nabla = \\pi_{0,1}{}^b\\!\\nabla'$. Namely, let $\\eta_\\mu$ be a local orthonormal frame of $F$, let\n\\begin{equation*}\n(\\pi_{0,1}\\otimesI)\\circ {}^b\\!\\nabla'\\eta_\\mu = \\sum_\\nu\\omega^\\nu_\\mu \\otimes \\eta_\\nu,\n\\end{equation*}\nand let ${}^b\\!\\nabla$ be the connection defined in the domain of the frame by\n\\begin{equation}\\label{HermitianConnectionForms}\n{}^b\\!\\nabla\\eta_\\mu =(\\omega^\\nu_\\mu-\\overline \\omega^\\mu_\\nu)\\otimes \\eta_\\nu.\n\\end{equation}\nIf the matrix of functions $Q=[q^\\mu_\\lambda]$ is unitary and $\\tilde \\eta_\\lambda=\\sum_\\mu q^\\mu_\\lambda \\eta_\\mu$, then \n\\begin{equation*}\n(\\pi_{0,1}\\otimesI)\\circ {}^b\\!\\nabla'\\tilde \\eta_\\lambda = \\sum_\\nu\\tilde \\omega^\\sigma_\\lambda \\otimes \\tilde \\eta_\\sigma\n\\end{equation*}\nwith\n\\begin{equation*}\n\\tilde \\omega^\\sigma_\\lambda = \\sum_\\mu \\overline q^\\mu_\\sigma\\,{}^b\\!\\overline \\partial q^\\mu_\\lambda + \\sum_{\\mu,\\nu} \\overline q^\\mu_\\sigma q^\\nu_\\lambda\\omega^\\mu_\\nu,\n\\end{equation*}\nusing \\eqref{Connection}, that $Q^{-1}=[\\overline q^\\mu_\\lambda]$, and that $\\pi_{0,1}{}^b\\!d f={}^b\\!\\overline \\partial f$. Thus\n\\begin{align*}\n\\tilde \\omega^\\sigma_\\lambda-\\overline {\\tilde \\omega}^\\lambda_\\sigma\n&= \\sum_\\mu (\\overline q^\\mu_\\sigma\\,{}^b\\!\\overline \\partial q^\\mu_\\lambda -  q^\\mu_\\lambda\\,{}^b\\!\\dee \\overline q^\\mu_\\sigma)  + \\sum_{\\mu,\\nu} (\\overline q^\\mu_\\sigma q^\\nu_\\lambda\\omega^\\mu_\\nu -  q^\\mu_\\lambda \\overline q^\\nu_\\sigma\\overline \\omega^\\mu_\\nu)\\\\\n&=\\sum_\\mu ({}^b\\!\\overline \\partial q^\\mu_\\lambda + {}^b\\!\\dee q^\\mu_\\lambda)\\overline q^\\mu_\\sigma  + \\sum_{\\mu,\\nu} q^\\nu_\\lambda ( \\omega^\\mu_\\nu - \\overline \\omega^\\nu_\\mu)\\overline q^\\mu_\\sigma\\\\\n&=\\sum_\\mu {}^b\\!d q^\\mu_\\lambda + \\overline q^\\mu_\\sigma  + \\sum_{\\mu,\\nu} q^\\nu_\\lambda ( \\omega^\\mu_\\nu - \\overline \\omega^\\nu_\\mu)\\overline q^\\mu_\\sigma\n\\end{align*}\nusing that $\\overline{{}^b\\!\\overline \\partial f}={}^b\\!\\dee \\overline f$ and that $\\sum_\\mu  q^\\mu_\\lambda\\,{}^b\\!\\dee \\overline q^\\mu_\\sigma = - \\sum_\\mu {}^b\\!\\dee q^\\mu_\\lambda\\, \\overline q^\\mu_\\sigma$ because $\\sum_\\mu q^\\mu_\\lambda \\overline q^\\mu_\\sigma$ is constant, and that ${}^b\\!\\overline \\partial q^\\mu_\\lambda + {}^b\\!\\dee q^\\mu_\\lambda={}^b\\!d q^\\mu_\\lambda$. Thus there is a globally defined Hermitian connection locally given by \\eqref{HermitianConnectionForms}. We leave to the reader to verify that this connection is Hermitian. Clearly ${}^b\\!\\nabla$ is the unique Hermitian connection such that $\\pi_{0,1}{}^b\\!\\nabla = \\pi_{0,1}{}^b\\!\\nabla'$. When ${}^b\\!\\nabla'$ is a holomorphic connection, ${}^b\\!\\nabla$ is the unique Hermitian holomorphic connection.\n\n\\begin{lemma}\nThe vector bundles ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,0}\\mathcal M$ are holomorphic.\n\\end{lemma}\n\nWe prove this by exhibiting a holomorphic $b$-connection. Fix an auxiliary Hermitian metric on ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,0}\\mathcal M$ and pick an orthonormal frame $(\\eta_\\mu)$ of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,0}\\mathcal M$ over some open set $U$. Let $\\omega^\\nu_\\mu$ be the unique sections of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$ such that\n\\begin{equation*}\n{}^b\\!\\overline \\partial\\eta_\\mu =\\sum_\\nu \\omega^\\nu_\\mu\\wedge\\eta_\\nu,\n\\end{equation*}\nand let ${}^b\\!\\nabla$ be the $b$-connection defined on $U$ by the formula \\eqref{HermitianConnectionForms}. As in the previous paragraph, this gives a globally defined $b$-connection. That it is holomorphic follows from\n\\begin{equation*}\n{}^b\\!\\overline \\partial \\omega^\\nu_\\mu + \\sum_\\lambda \\omega^\\nu_\\lambda\\wedge \\omega^\\lambda_\\mu = 0,\n\\end{equation*}\na consequence of ${}^b\\!\\overline \\partial^2=0$. Evidently, with the identifications ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M\\otimes {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,0}\\mathcal M={}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{p,q}\\mathcal M$, $\\pi_{p,q+1}{}^b\\!\\nabla$ is the ${}^b\\!\\overline \\partial$ operator in \\eqref{bdeebarOnpq}.\n\n\n\\section{The boundary a complex $b$-manifold}\\label{sBoundaryStructure}\n\nSuppose that $\\mathcal M$ is a complex $b$-manifold and $\\mathcal N$ is a component of its boundary. We shall assume $\\mathcal N$ compact, although for the most part this is not necessary.\n\nThe homomorphism\n\\begin{equation*}\n\\mathrm{ev}:\\mathbb C{}^b{\\!}T\\mathcal M\\to \\mathbb C T\\mathcal M\n\\end{equation*}\nis an isomorphism over the interior of $\\mathcal M$, and its restriction to $\\mathcal N$ maps onto $\\mathbb C T\\mathcal N$ with kernel spanned by $\\mathfrak r\\partial_\\mathfrak r$. Write\n\\begin{equation*}\n\\mathrm{ev}_\\mathcal N:\\mathbb C{}^b{\\!}T_{\\mathcal N}\\mathcal M\\to \\mathbb C T\\mathcal N\n\\end{equation*}\nfor this restriction and \n\\begin{equation}\\label{bdyIso}\n\\Phi:{}^b{\\!}T^{0,1}_{\\mathcal N}\\mathcal M\\to\\overline\\Vee\n\\end{equation}\nfor of the restriction of $\\mathrm{ev}_\\mathcal N$ to ${}^b{\\!}T^{0,1}_{\\mathcal N}\\mathcal M$. From \\eqref{bCRstructure} and the fact that the kernel of $\\mathrm{ev}_\\mathcal N$ is spanned by the real section $\\mathfrak r\\partial_{\\mathfrak r}$ one obtains that $\\Phi$ is injective, so its image,\n\\begin{equation*}\n\\overline\\Vee=\\Phi({}^b{\\!}T^{0,1}_{\\mathcal N}\\mathcal M)\n\\end{equation*}\nis a subbundle of $\\mathbb C T\\mathcal N$.\n\nSince ${}^b{\\!}T^{0,1}\\mathcal M$ is involutive, so is $\\overline\\Vee$, see \\cite[Proposition 3.12]{Me3}. From \\eqref{bElliptic} and the fact that $\\mathrm{ev}_\\mathcal N$ maps onto $\\mathbb C T\\mathcal N$, one obtains that\n\\begin{equation}\\label{VbarIsElliptic}\n\\mathcal V + \\overline\\Vee=\\mathbb C T\\mathcal N, \n\\end{equation}\nsee \\cite[Lemma 3.13]{Me3}. Thus\n\n\\begin{lemma}\n$\\overline\\Vee$ is an elliptic structure.\n\\end{lemma}\n\nThis just means what we just said: $\\overline\\Vee$ is involutive and \\eqref{VbarIsElliptic} holds, see Treves~\\cite{Tr81,Tr92}; the sum need not be direct. All elliptic structures are locally of the same kind, depending only on the dimension of $\\mathcal V\\cap \\overline\\Vee$. This is a result of Nirenberg~\\cite{Ni57} (see also H\\\"ormander~\\cite{Ho65}) extending the Newlander-Nirenberg theorem. In the case at hand, $\\overline\\Vee\\cap \\mathcal V$ has rank $1$ because of the relation\n\\begin{equation*}\n\\rk_\\mathbb C(\\mathcal V\\cap \\overline\\Vee)=2\\rk_\\mathbb C\\overline\\Vee-\\dim\\mathcal N\n\\end{equation*}\nwhich holds whenever \\eqref{VbarIsElliptic} holds. \n\\begin{equation}\\label{HypoanalyticChart}\n\\display{300pt}{Every $p_0\\in \\mathcal N$ hs a neighborhood in which there coordinates $x^1,\\dotsc,x^{2n},t$ such that with $z^j=x^j+\\mathfrak m x^{j+n}$, the vector fields \n\\begin{equation*}\n\\hspace*{-60pt}\\frac{\\partial}{\\partial \\overline z^1},\\dotsc,\\frac{\\partial}{\\partial \\overline z^n},\\ \\frac{\\partial }{\\partial t}\n\\end{equation*}\nspan $\\overline\\Vee$ near $p_0$. The function $(z^1,\\dotsc,z^n,t)$ is called a hypoanalytic chart (Baouendi, Chang, and Treves~\\cite{BCT83}, Treves~\\cite{Tr92}).}\n\\end{equation}\n\nThe intersection $\\overline\\Vee\\cap \\mathcal V$ is, in the case we are discussing, spanned by a canonical globally defined real vector field.\nNamely, let $\\mathfrak r\\partial_\\mathfrak r$ be the canonical section of ${}^b{\\!}T\\mathcal M$ along $\\mathcal N$. There is a unique section $J\\mathfrak r\\partial_\\mathfrak r$ of ${}^b{\\!}T\\mathcal M$ along $\\mathcal N$ such that $\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r \\partial_\\mathfrak r$ is a section of ${}^b{\\!}T^{0,1}\\mathcal M$ along $\\mathcal N$. Then\n\\begin{equation*}\n\\mathcal T=\\mathrm{ev}_\\mathcal N(J\\mathfrak r\\partial_\\mathfrak r)\n\\end{equation*}\nis a nonvanishing real vector field in $\\mathcal V\\cap \\overline\\Vee$, (see \\cite[Lemma 2.1]{Me4}). Using the isomorphism \\eqref{bdyIso} we have\n\\begin{equation*}\n\\mathcal T=\\Phi(J(\\mathfrak r\\partial_\\mathfrak r)-i\\mathfrak r\\partial_\\mathfrak r).\n\\end{equation*}\n\nBecause $\\overline\\Vee$ is involutive, there is yet another complex, this time associated with the exterior powers of the dual of $\\overline\\Vee$:\n\\begin{equation}\\label{bdyComplex}\n\\cdots\n\\to C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q \\smash[t]{\\overline\\Vee}^*)\\xrightarrow{\\overline\\Dee}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\smash[t]{\\overline\\Vee}^*)\n\\to\\cdots,\n\\end{equation}\nwhere $\\overline\\Dee$ is defined by the formula \\eqref{CartanFormula} where now the $V_j$ are sections of $\\overline\\Vee$. The complex \\eqref{bdyComplex} is elliptic because of \\eqref{VbarIsElliptic}. For a function $f$ we have $\\overline\\Dee f=\\iota^*df$, where $\\iota^*:\\mathbb C T^*\\mathcal N\\to \\smash[t]{\\overline\\Vee}^*$ is the dual of the inclusion homomorphism $\\iota:\\overline\\Vee\\to\\mathbb C T\\mathcal N$.\n\nFor later use we show:\n\\begin{lemma}\\label{ConstantSolutions}\nSuppose that $\\mathcal N$ is compact and connected. If $\\zeta:\\mathcal N\\to\\mathbb C$ solves $\\overline\\Dee\\zeta=0$, then $\\zeta$ is constant.\n\\end{lemma}\n\n\\begin{proof}\nLet $p_0$ be an extremal point of $|\\zeta|$. Fix a hypoanalytic chart $(z,t)$ for $\\overline\\Vee$ centered at $p_0$. Since $\\overline\\Dee \\zeta=0$, $\\zeta(z,t)$ is independent of $t$ and $\\partial_{\\overline z^\\nu}\\zeta=0$. So there is a holomorphic function $Z$ defined in a neighborhood of $0$ in $\\mathbb C^n$ such that $\\zeta=Z\\circ z$. Then $|Z|$ has a maximum at $0$, so $Z$ is constant near $0$. Therefore $\\zeta$ is constant, say $\\zeta(p)=c$, near $p_0$. Let $C=\\set{p:\\zeta(p)=c}$, a closed set. Let $p_1\\in C$. Since $p_1$ is also an extremal point of $\\zeta$, the above argument gives that $\\zeta$ is constant near $p_1$, therefore equal to $c$. Thus $C$ is open, and consequently $\\zeta$ is constant on $\\mathcal N$.\n\\end{proof}\n\n\n\nSince the operators ${}^b\\!\\overline \\partial:C^\\infty(\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M)\\to C^\\infty(\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}\\mathcal M)$ are totally characteristic, they induce operators\n\\begin{equation*}\n{}^b\\!\\overline \\partial_b:C^\\infty(\\mathcal N,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}_\\mathcal N\\mathcal M)\\to C^\\infty(\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}_\\mathcal N\\mathcal M),\n\\end{equation*}\nsee \\eqref{Pb}; these boundary operators define a complex because of \\eqref{PQb}. By way of the dual\n\\begin{equation}\\label{DualbdyIso}\n\\Phi^*:\\smash[t]{\\overline\\Vee}^*\\to {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}_\\mathcal N\\mathcal M\n\\end{equation}\nof the isomorphism \\eqref{bdyIso} the operators ${}^b\\!\\overline \\partial_b$ become identical to the operators of the $\\overline\\Dee$-complex \\eqref{bdyComplex}: The diagram\n\\begin{equation*}\n\\begin{CD}\n\\cdots &@>>> C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q \\smash[t]{\\overline\\Vee}^*)&@>\\overline\\Dee>>C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\smash[t]{\\overline\\Vee}^*)&@>>>&\\cdots\\\\\n&& &@V{\\Phi^*}VV&@VV{\\Phi^*}V&\\\\\n\\cdots &@>>> C^\\infty(\\mathcal N,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}_\\mathcal N\\mathcal M)&@>{}^b\\!\\overline \\partial_b>>C^\\infty(\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}_\\mathcal N\\mathcal M)&@>>>&\\cdots\n\\end{CD}\n\\end{equation*}\nis commutative and the vertical arrows are isomorphisms. This can be proved by writing the ${}^b\\!\\overline \\partial$ operators using Cartan's formula \\eqref{CartanFormula} for ${}^b\\!\\overline \\partial$ and $\\overline\\Dee$ and comparing the resulting expressions.\n\nLet $\\mathfrak r:\\mathcal M\\to\\mathbb R$ be a smooth defining function for $\\partial \\mathcal M$, $\\mathfrak r>0$ in the interior of $\\mathcal M$. Then ${}^b\\!\\overline \\partial \\mathfrak r$ is smooth and vanishes on $\\partial \\mathcal M$, so $\\frac{{}^b\\!\\overline \\partial \\mathfrak r}{\\mathfrak r}$ is also a smooth ${}^b\\!\\overline \\partial$-closed section of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$. Thus we get a $\\overline\\Dee$-closed element\n\\begin{equation}\\label{DefinitionBeta}\n\\beta_\\mathfrak r=[\\Phi^*]^{-1}\\frac{{}^b\\!\\overline \\partial \\mathfrak r}{\\mathfrak r} \\in C^\\infty(\\partial \\mathcal M;\\smash[t]{\\overline\\Vee}^*).\n\\end{equation}\nBy definition,\n\\begin{equation*}\n\\langle \\beta_\\mathfrak r,\\mathcal T\\rangle=\\langle \\frac{{}^b\\!\\overline \\partial \\mathfrak r}{\\mathfrak r},J(\\mathfrak r\\partial_\\mathfrak r)-i\\mathfrak r\\partial_\\mathfrak r\\rangle.\n\\end{equation*}\nExtend the section $\\mathfrak r\\partial_\\mathfrak r$ to a section of ${}^b{\\!}T\\mathcal M$ over a neighborhood $U$ of $\\mathcal N$ in $\\mathcal M$ with the property that $\\mathfrak r\\partial_\\mathfrak r\\rr=\\mathfrak r$. In $U$ we have\n\\begin{equation*}\n\\langle {}^b\\!\\overline \\partial \\mathfrak r,J(\\mathfrak r\\partial_\\mathfrak r)-i\\mathfrak r\\partial_\\mathfrak r\\rangle = (J(\\mathfrak r\\partial_\\mathfrak r)-i\\mathfrak r\\partial_\\mathfrak r)\\mathfrak r = J(\\mathfrak r\\partial_\\mathfrak r)\\mathfrak r -i\\mathfrak r.\n\\end{equation*}\nThe function $J(\\mathfrak r\\partial_\\mathfrak r)\\mathfrak r$ is smooth, real-valued, and vanishes along the boundary. So $\\mathfrak r^{-1}J(\\mathfrak r\\partial_\\mathfrak r)\\mathfrak r$ is smooth, real-valued. Thus\n\\begin{equation*}\n\\langle\\beta_\\mathfrak r,\\mathcal T\\rangle = a_\\mathfrak r-i\n\\end{equation*}\non $\\mathcal N$ for some smooth function $a_\\mathfrak r:\\mathcal N\\to\\mathbb R$, see \\cite[Lemma 2.5]{Me4}.\n\nIf $\\mathfrak r'$ is another defining function for $\\partial \\mathcal M$, then $\\mathfrak r'=\\mathfrak r e^u$ for some smooth function $u:\\mathcal M\\to\\mathbb R$. Then\n\\begin{equation*}\n{}^b\\!\\overline \\partial \\mathfrak r'=e^u\\,{}^b\\!\\overline \\partial \\mathfrak r+ e^u\\mathfrak r\\,{}^b\\!\\overline \\partial u\n\\end{equation*}\nand it follows that\n\\begin{equation*}\n\\beta_{\\mathfrak r'}=\\beta_\\mathfrak r+ \\overline\\Dee u.\n\\end{equation*}\nIn particular,\n\\begin{equation*}\na_{\\mathfrak r'}=a_\\mathfrak r+ \\mathcal T u.\n\\end{equation*}\nLet $\\mathfrak a_t$ denote the one-parameter group of diffeomorphisms generated by $\\mathcal T$.\n\n\\begin{proposition}\\label{Averages}\nThe functions $a^{\\sup}_\\mathrm{av}$, $a^{\\inf}_\\mathrm{av}:\\mathcal N\\to \\mathbb R$ defined by\n\\begin{equation*}\na^{\\sup}_\\mathrm{av}(p) =\\limsup_{t\\to\\infty}\\frac{1}{2t}\\int_{-t}^t a_{\\mathfrak r}(\\mathfrak a_s(p))\\,ds,\\quad a^{\\inf}_\\mathrm{av}(p) =\\liminf_{t\\to\\infty}\\frac{1}{2t}\\int_{-t}^t a_{\\mathfrak r}(\\mathfrak a_s(p))\\,ds\n\\end{equation*}\nare invariants of the complex $b$-structure, that is, they are independent of the defining function $\\mathfrak r$. The equality $a^{\\sup}_\\mathrm{av} = a^{\\inf}_\\mathrm{av}$ holds for some $\\mathfrak r$ if and only if it holds for all $\\mathfrak r$. \n\\end{proposition}\n\nIndeed,\n\\begin{equation*}\n\\lim_{t\\to\\infty}\\Big( \\frac{1}{2t}\\int_{-t}^t a_{\\mathfrak r'}(\\mathfrak a_s(p))\\,ds - \\frac{1}{2t}\\int_{-t}^t a_{\\mathfrak r}(\\mathfrak a_s(p))\\,ds\\Big) = \\lim_{t\\to\\infty}\\frac{1}{2t}\\int_{-t}^t \\frac{d}{ds}u(\\mathfrak a_s(p))\\,ds = 0\n\\end{equation*}\nbecause $u$ is bounded (since $\\mathcal N$ is compact).\n\nThe functions $a^{\\sup}_\\mathrm{av}$, $a^{\\inf}_\\mathrm{av}$ are constant on orbits of $\\mathcal T$, but they may not be smooth. \n\n\n\n\\begin{example}\\label{AnisotropicSphere}\nLet $\\mathcal N$ be the unit sphere in $\\mathbb C^{n+1}$ centered at the origin. Write $(z^1,\\dotsc,z^{n+1})$ for the standard coordinates in $\\mathbb C^{n+1}$. Fix $\\tau_1,\\dotsc,\\tau_{n+1}\\in \\mathbb R\\backslash 0$, all of the same sign, and let\n\\begin{equation*}\n\\mathcal T= i\\sum_{j=1}^{n+1} \\tau_j(z^j\\partial_{z^j}-\\overline z^j\\partial_{\\overline z^j}).\n\\end{equation*}\nThis vector field is real and tangent to $\\mathcal N$. Let $\\overline \\K$ be the standard CR structure of $\\mathcal N$ as a submanifold of $\\mathbb C^{n+1}$ (the part of $T^{0,1}\\mathbb C^{n+1}$ tangential to $\\mathcal N$). The condition that the $\\tau_j$ are different from $0$ and have the same sign ensures that $\\mathcal T$ is never in $\\mathcal K\\oplus \\overline \\K$. Indeed, the latter subbundle of $\\mathbb C T\\mathcal N$ is the annihilator of the pullback to $\\mathcal N$ of $i \\overline \\partial \\sum_{\\ell=1}^{n+1}|z^\\ell|^2$. The pairing of this form with $\\mathcal T$ is\n\\begin{equation*}\n\\langle i \\sum_{\\ell=1}^{n+1} z^\\ell d\\overline z^\\ell,i\\sum_{j=1}^{n+1} \\tau_j(z^j\\partial_{z^j}-\\overline z^j\\partial_{\\overline z^j})\\rangle = \\sum_{j=1}^{n+1} \\tau_j|z^j|^2,\n\\end{equation*}\na function that vanishes nowhere if and only if all $\\tau_j$ are different from zero and have the same sign. Thus $\\overline\\Vee=\\overline \\K\\oplus \\Span_\\mathbb C\\mathcal T$ is a subbundle of $\\mathbb C T\\mathcal N$ of rank $n+1$ with the property that $\\mathcal V+\\overline\\Vee=\\mathbb C T\\mathcal N$. To show that $\\overline\\Vee$ is involutive we first note that $\\overline \\K$ is the annihilator of the pullback to $\\mathcal N$ of the span of the differentials $dz^1,\\dotsc,dz^{n+1}$. Let $\\mathcal {L}_\\mathcal T$ denote the Lie derivative with respect to $\\mathcal T$. Then $\\mathcal {L}_\\mathcal T dz^j =i \\tau_j dz^j$, so if $L$ is a CR vector field, then so is $[L,\\mathcal T]$. Since in addition $\\overline \\K$ and $\\Span_\\mathbb C \\mathcal T$ are themselves involutive, $\\overline\\Vee$ is involutive. Thus $\\overline\\Vee$ is an elliptic structure with $\\mathcal V\\cap \\overline\\Vee=\\Span_\\mathbb C\\mathcal T$. Let $\\beta$ be the section of $\\smash[t]{\\overline\\Vee}^*$ which vanishes on $\\overline \\K$ and satisfies $\\langle \\beta,\\mathcal T\\rangle=-i$. Let $\\overline\\Dee$ denote the operators of the associated differential complex. Then $\\overline\\Dee\\beta=0$, since $\\beta$ vanishes on commutators of sections of $\\overline \\K$ (since $\\overline \\K$ is involutive) and on commutators of $\\mathcal T$ with sections of $\\overline \\K$ (since such commutators are in $\\overline \\K$).\n\nIf the $\\tau_j$ are positive (negative), this example may be viewed as the boundary of a blowup (compactification) of $\\mathbb C^{n+1}$, see \\cite{Me5}.\n\\end{example}\n\n\\medskip\nLet now $\\rho:F\\to\\mathcal M$ be a holomorphic vector bundle. Its ${}^b\\!\\overline \\partial$-complex also determines a complex along $\\mathcal N$,\n\\begin{equation}\\label{defDeebarBundle}\n\\cdots \\to C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N)\\xrightarrow{\\overline\\Dee} C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N)\\to\\cdots,\n\\end{equation}\nwhere $\\overline\\Dee$ is defined using the boundary operators ${}^b\\!\\overline \\partial_b$ and the isomorphism \\eqref{DualbdyIso}:\n\\begin{equation}\\label{defDeebarE}\n\\overline\\Dee (\\phi\\otimes \\eta) = (\\Phi^*)^{-1}{}^b\\!\\overline \\partial_b [\\Phi^* (\\phi\\otimes \\eta)]\n\\end{equation}\nwhere $\\Phi^*$ means $\\Phi^*\\otimesI$. These operators can be expressed locally in terms of the operators of the complex \\eqref{bdyComplex}. Fix a smooth frame $\\eta_\\mu$, $\\mu=1,\\dotsc,k$, of $F$ in a neighborhood $U\\subset \\mathcal M$ of $p_0\\in \\mathcal N$, and suppose\n\\begin{equation*}\n{}^b\\!\\overline \\partial \\eta_\\mu = \\sum_\\nu\\omega^\\nu_\\mu\\otimes \\eta_\\nu.\n\\end{equation*}\nThe $\\omega^\\nu_\\mu$ are local sections of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$, and if $\\sum_\\mu \\phi^\\mu\\otimes \\eta_\\mu$ is a section of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M\\otimes F$ over $U$, then\n\\begin{equation*}\n{}^b\\!\\overline \\partial\\sum\\phi^\\mu\\otimes\\eta_\\mu = \\sum_\\nu ({}^b\\!\\overline \\partial \\phi^\\nu +\\sum_\\mu \\omega^\\nu_\\mu\\wedge \\phi^\\mu)\\otimes \\eta_\\nu.\n\\end{equation*}\nTherefore, using the identification \\eqref{DualbdyIso}, the boundary operator ${}^b\\!\\overline \\partial_b$ is the operator given locally by\n\\begin{equation}\\label{DeebarE}\n\\overline\\Dee\\sum\\phi^\\mu\\otimes\\eta_\\mu = \\sum_\\nu (\\overline\\Dee \\phi^\\nu +\\sum_\\mu \\omega^\\nu_\\mu\\wedge \\phi^\\mu)\\otimes \\eta_\\nu\n\\end{equation}\nwhere now the $\\phi^\\mu$ are sections of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$, the $\\omega^\\nu_\\mu$ are the sections of $\\smash[t]{\\overline\\Vee}^*$ corresponding to the original $\\omega^\\nu_\\mu$ via $\\Phi^*$, and $\\overline\\Dee$ on the right hand side of the formula is the operator associated with $\\overline\\Vee$.\n\nThe structure bundle ${}^b{\\!}T^{0,1} F$ is locally given as the span of the sections \\eqref{LocalbT01E}. Applying the evaluation homomorphism $\\mathbb C{}^b{\\!}T_{\\partial F} F\\to \\mathbb C T\\partial F$ (over $\\mathcal N$) to these sections gives vector fields on $F_\\mathcal N$ forming a frame for the elliptic structure $\\overline\\Vee_F$ inherited by $F_\\mathcal N$. Writing $V_j^0=\\mathrm{ev} V_j$, this frame is just\n\\begin{equation}\\label{LocalVbarE}\n\\tilde V_j^0-\\sum_{\\mu,\\nu}\\zeta^\\mu \\langle \\omega^\\nu_\\mu, V_j^0\\rangle \\partial_{\\zeta^\\nu},\\ j=1,\\dotsc,n+1,\\quad \\partial_{\\overline \\zeta^\\nu},\\ \\nu=1,\\dotsc,k,\n\\end{equation}\nwhere now the $\\omega^\\nu_\\mu$ are the forms associated to the $\\overline\\Dee$ operator of $F_\\mathcal N$. Alternatively, one may take the $\\overline\\Dee$ operators of $F_\\mathcal N$ and use the formula \\eqref{DeebarE} to define a subbundle of $\\mathbb C TF$ locally as the span of the vector fields \\eqref{LocalVbarE}, a fortiori an elliptic structure on $F_\\mathcal N$, involutive because\n\\begin{equation*}\n\\overline\\Dee\\omega^\\nu + \\sum_\\lambda \\omega^\\nu_\\lambda\\wedge \\omega^\\lambda_\\mu = 0.\n\\end{equation*}\n\nTo obtain a formula for the canonical real vector field $\\mathcal T_F$ in $\\overline\\Vee_F$, let $J_F$ be the almost complex $b$-structure of ${}^b{\\!}T F$ and consider again the sections \\eqref{LocalbT01E}; they are defined in an open set $\\rho^{-1}(U)$, $U$ a neighborhood in $\\mathcal M$ of a point of $\\mathcal N$. Since the elements $\\partial_{\\overline\\zeta^\\nu}$ are sections of ${}^b{\\!}T^{0,1}F$,\n\\begin{equation}\\label{JEVert}\nJ_F\\Re\\partial_{\\overline\\zeta^\\nu}=\\Im\\partial_{\\overline\\zeta^\\nu}.\n\\end{equation}\nPick a defining function $\\mathfrak r$ for $\\mathcal N$. Then $\\tilde\\mathfrak r=\\rho^*\\mathfrak r$ is a defining function for $F_\\mathcal N$. We may take $V_{n+1}=\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r$ along $U\\cap \\mathcal N$. Then $\\tilde V_{n+1}=\\tilde \\mathfrak r\\partial_{\\tilde \\mathfrak r}+i \\widetilde {J\\mathfrak r \\partial_\\mathfrak r}$ along $\\rho^{-1}(U)\\cap F_\\mathcal N$ and so\n\\begin{multline*}\nJ_F\\Re\\big(\\tilde \\mathfrak r\\partial_{\\tilde \\mathfrak r} + i \\widetilde{J\\mathfrak r\\partial_\\mathfrak r}-\\sum_{\\mu,\\nu} \\zeta^\\mu\\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle\\partial_{\\zeta^\\nu}\\big) =\\\\ \\Im\\big(\\tilde \\mathfrak r\\partial_{\\tilde \\mathfrak r} + i \\widetilde{J\\mathfrak r\\partial_\\mathfrak r}-\\sum_{\\mu,\\nu} \\zeta^\\mu\\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle\\partial_{\\zeta^\\nu}\\big)\n\\end{multline*}\nalong $\\rho^{-1}(U)\\cap F_\\mathcal N$. Using \\eqref{JEVert} this gives\n\\begin{equation*}\nJ_F\\tilde \\mathfrak r\\partial_{\\tilde \\mathfrak r} =\n\\widetilde{J\\mathfrak r\\partial_\\mathfrak r} - 2\\Im\\sum _{\\mu,\\nu} \\zeta^\\mu\\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle\\partial_{\\zeta^\\nu}.\n\\end{equation*}\nApplying the evaluation homomorphism gives\n\\begin{equation}\\label{TE}\n\\mathcal T_F =\n\\tilde\\mathcal T - 2\\Im\\sum _{\\mu,\\nu} \\zeta^\\mu\\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle\\partial_{\\zeta^\\nu}\n\\end{equation}\nwhere $\\tilde \\mathcal T$ is the real vector field on $\\rho^{-1}(U\\cap \\mathcal N)=\\rho^{-1}(U)\\cap F_\\mathcal N$ which projects on $\\mathcal T$ and satisfies $\\tilde \\mathcal T\\zeta^\\mu=0$ for all $\\mu$.\n\nLet $h$ be a Hermitian metric on $F$, and suppose that the frame $\\eta_\\mu$ is orthonormal. Applying $\\mathcal T_E$ as given in \\eqref{TE} to the function $|\\zeta|^2=\\sum|\\zeta^\\mu|^2$ we get that $\\mathcal T_F$ is tangent to the unit sphere bundle of $F$ if and only if\n\\begin{equation*}\n\\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle - \\overline{\\langle\\omega^\\mu_\\nu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle} = 0\n\\end{equation*}\nfor all $\\mu, \\nu$. Equivalently, in terms of the isomorphism \\eqref{DualbdyIso},\n\\begin{equation}\\label{ExactMetricCondition}\n\\langle(\\Phi^*)^{-1}\\omega^\\nu_\\mu,\\mathcal T \\rangle + \\overline{\\langle(\\Phi^*)^{-1}\\omega^\\mu_\\nu,\\mathcal T\\rangle} = 0\\quad \\text{ for all }\\mu,\\nu.\n\\end{equation}\n\\begin{definition}\\label{ExactMetric}\nThe Hermitian metric $h$ will be called exact if \\eqref{ExactMetricCondition} holds.\n\\end{definition}\n\nThe terminology in Definition \\ref{ExactMetric} is taken from the notion of exact Riemannian $b$-metric of Melrose~\\cite[pg. 31]{RBM2}. For such metrics, the Levi-Civita $b$-connection has the property that ${}^b\\!\\nabla_{\\mathfrak r\\partial_\\mathfrak r}=0$ [op. cit., pg. 58]. We proceed to show that the Hermitian holomorphic connection of an exact Hermitian metric on $F$ also has this property. Namely, suppose that $h$ is an exact Hermitian metric, and let $\\eta_\\mu$ be an orthonormal frame of $F$. Then for the Hermitian holomorphic connection we have\n\\begin{equation*}\n\\langle\\omega^\\nu_\\mu-\\overline \\omega^\\mu_\\nu,\\mathfrak r\\partial_\\mathfrak r\\rangle = \\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r\\rangle - \\overline{\\langle \\omega^\\mu_\\nu,\\mathfrak r\\partial_\\mathfrak r\\rangle}\n=\\frac{1}{2}\\big( \\langle\\omega^\\nu_\\mu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle - \\overline{\\langle \\omega^\\mu_\\nu,\\mathfrak r\\partial_\\mathfrak r+i J\\mathfrak r\\partial_\\mathfrak r\\rangle} \\big)\n\\end{equation*}\nusing that the $\\omega^\\nu_\\mu$ are of type $(0,1)$. Thus  ${}^b\\!\\nabla_{\\mathfrak r\\partial_\\mathfrak r}=0$.\n\n\\section{Local invariants}\\label{sLocalInvariants}\n\nComplex structures have no local invariants: every point of a complex $n$-manifold has a neighborhood biholomorphic to a ball in $\\mathbb C^n$ It is natural to ask the same question about complex $b$-structures, namely,\n\\begin{equation*}\n\\display{300pt}{is there a local model depending only on dimension for every complex $b$-stucture? }\n\\end{equation*}\nIn lieu of a Newlander-Nirenberg theorem, we show that complex $b$-structures have no local formal invariants at the boundary. More precisely:\n\n\\begin{proposition}\\label{NoLocalFormalInvariants}\nEvery $p_0\\in \\mathcal N$ has a neighborhood $V$ in $\\mathcal M$ on which there are smooth coordinates $x^j$, $j=1,\\dotsc,2n+2$ centered at $p_0$ with $x^{n+1}$ vanishing on $V\\cap\\mathcal N$ such that with\n\\begin{equation}\\label{NoLocalFormalInvariants1}\n\\overline L^0_j=\\frac{1}{2}(\\partial_{x^j}+i\\partial_{x^{j+n+1}}),\\ j\\leq n,\\quad\n\\overline L^0_{n+1} = \\frac{1}{2}(x^{n+1}\\partial_{x^{n+1}}+i\\partial_{x^{2n+2}})\n\\end{equation}\nthere are smooth functions $\\gamma^j_k$ vanishing to infinite order on $V\\cap\\mathcal N$ such that\n\\begin{equation*}\n\\overline L_j=\\overline L_j^0+\\sum_{k=1}^{n+1}\\gamma^k_j L_k^0\n\\end{equation*}\nis a frame for ${}^b{\\!}T^{0,1}\\mathcal M$ over $V$.\n\\end{proposition}\n\nThe proof will require some preparation. Let $\\mathfrak r:\\mathcal M\\to\\mathbb R$ be a defining function for $\\partial\\mathcal M$. Let $p_0\\in \\mathcal N$, pick a hypoanalytic chart $(z,t)$ (cf. \\eqref{HypoanalyticChart}) centered at $p_0$ with $\\mathcal T t=1$. Let $U\\subset \\mathcal N$ be a neighborhood of $p_0$ contained in the domain of the chart, mapped by it to $B\\times(-\\delta,\\delta)\\subset \\mathbb C^n\\times\\mathbb R$, where $B$ is a ball with center $0$ and $\\delta$ is some small positive number. For reference purposes we state\n\n\\begin{lemma}\\label{LocalExactBStructure} On such $U$, the problem\n\\begin{equation*}\n\\overline\\Dee \\phi = \\psi, \\quad \\psi\\in C^\\infty(U;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\smash[t]{\\overline\\Vee}^*|_U)\\text{ and }\\overline\\Dee \\psi=0\n\\end{equation*}\nhas a solution in $C^\\infty(U;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*|_U)$.\n\\end{lemma}\n\nExtend the functions $z^j$ and $t$ to a neighborhood of $p_0$ in $\\mathcal M$. Shrinking $U$ if necessary, we may assume that in some neighborhood $V$ of $p_0$ in $\\mathcal M$ with $V\\cap\\partial\\mathcal M=U$, $(z,t,\\mathfrak r)$ maps $V$ diffeomorphically onto $B\\times(-\\delta,\\delta)\\times [\\neutral{]}0,\\varepsilon\\neutral{(})$ for some $\\delta$, $\\varepsilon>0$. Since the form $\\beta_\\mathfrak r$ defined in \\eqref{DefinitionBeta} is $\\overline\\Dee$-closed, there is $\\alpha\\in C^\\infty(U)$ such that\n\\begin{equation*}\n-i\\overline\\Dee \\alpha=\\beta_\\mathfrak r.\n\\end{equation*}\nExtend $\\alpha$ to $V$ as a smooth function. The section\n\\begin{equation}\\label{StartNoInv}\n{}^b\\!\\overline \\partial(\\log \\mathfrak r+i\\alpha)=\\frac{{}^b\\!\\overline \\partial \\mathfrak r}{\\mathfrak r}+i{}^b\\!\\overline \\partial\\alpha\n\\end{equation}\nof ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$ over $V$ vanishes on $U$, since $\\beta_\\mathfrak r+i \\overline\\Dee \\alpha=0$. So there is a smooth section $\\phi$ of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$ over $V$ such that\n\\begin{equation*}\n{}^b\\!\\overline \\partial(\\log \\mathfrak r +i\\alpha)=\\mathfrak r e^{i\\alpha}\\phi.\n\\end{equation*}\nSuppose $\\zeta:U\\to\\mathbb C$ is a solution of $\\overline\\Dee \\zeta=0$ on $U$, and extend it to $V$. Then ${}^b\\!\\overline \\partial\\zeta$ vanishes on $U$, so again we have\n\\begin{equation*}\n{}^b\\!\\overline \\partial\\zeta=\\mathfrak r e^{i\\alpha}\\psi.\n\\end{equation*}\nfor some smooth section $\\psi$ of ${}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,1}\\mathcal M$ over $V$. The following lemma will be applied for $f_0$ equal to $\\log\\mathfrak r+i\\alpha$ or each of the functions $z^j$.\n\n\\begin{lemma}\nLet $f_0$ be smooth in $V\\backslash U$ and suppose that ${}^b\\!\\overline \\partial f_0=\\mathfrak r e^{i\\alpha}\\psi_1$ with $\\psi_1$ smooth on $V$. Then there is $f:V\\to\\mathbb C$ smooth vanishing at $U$ such that ${}^b\\!\\overline \\partial(f_0+f)$ vanishes to infinite order on $U$.\n\\end{lemma}\n\n\\begin{proof} Suppose that $f_1,\\dotsc,f_{N-1}$ are defined on $V$ and that\n\\begin{equation}\\label{StepN}\n{}^b\\!\\overline \\partial \\sum_{k=0}^{N-1}(\\mathfrak r e^{i\\alpha})^k f_k = (\\mathfrak r e^{i\\alpha})^N\\psi_N\n\\end{equation}\nholds with $\\psi_N$ smooth in $V$; by the hypothesis, \\eqref{StepN} holds when $N=1$. Using \\eqref{StartNoInv} we get that ${}^b\\!\\overline \\partial(\\mathfrak r e^{i\\alpha})=(\\mathfrak r e^{i\\alpha})^2\\phi$, therefore\n\\begin{equation*}\n0={}^b\\!\\overline \\partial\\big((\\mathfrak r e^{i\\alpha})^N\\psi_N) = (\\mathfrak r e^{i\\alpha})^N[{}^b\\!\\overline \\partial\\psi_N + N\\mathfrak r e^{i\\alpha}\\phi\\wedge \\psi_N],\n\\end{equation*}\nwhich implies that ${}^b\\!\\overline \\partial\\psi_N=0$ on $U$.  With arbitrary $f_N$ we have\n\\begin{equation*}\n{}^b\\!\\overline \\partial\\sum_{k=0}^{N}(\\mathfrak r e^{i\\alpha})^k f_k = (\\mathfrak r e^{i\\alpha})^N(\\psi_N  + {}^b\\!\\overline \\partial f_N + N \\mathfrak r e^{i\\alpha}f_N \\phi).\n\\end{equation*}\nSince $\\overline\\Dee\\psi_N=0$ and $H_{\\overline\\Dee}^1(U)=0$ by Lemma~\\ref{LocalExactBStructure}, there is a smooth function $f_N$ defined in $U$ such that $\\overline\\Dee f_N=-\\psi_N$ in $U$. So there is $\\chi_N$ such that $\\psi_N + {}^b\\!\\overline \\partial f_N = \\mathfrak r e^{i\\alpha}\\chi_N$. With such $f_N$, \\eqref{StepN} holds with $N+1$ in place of $N$ and some $\\psi_{N+1}$. Thus there is a sequence $\\{f_j\\}_{j=1}^\\infty$ such that \\eqref{StepN} holds for each $N$. Borel's lemma then gives $f$ smooth with\n\\begin{equation*}\nf\\sim \\sum_{k=1}^\\infty (\\mathfrak r e^{i\\alpha})^k f_k\\quad\\text{ on }U\n\\end{equation*}\nsuch that $\\overline\\Dee(f_0+f)$ vanishes to infinite order on $U$.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition~\\ref{NoLocalFormalInvariants}]\nApply the lemma with $f_0=\\log\\mathfrak r+i\\alpha$ to get a function $f$ such that ${}^b\\!\\overline \\partial (f_0+f)$ vanishes to infinite order at $U$. Let\n\\begin{equation*}\nx^{n+1}=\\mathfrak r e^{-\\Im\\alpha+\\Re f},\\quad x^{2n+2}=\\Re\\alpha+\\Im f.\n\\end{equation*}\nThese functions are smooth up to $U$.\n\nApplying the lemma to each of the functions $f_0=z^j$, $j=1,\\dotsc,n$ gives smooth functions $\\zeta^j$ such that $\\zeta^j=z^j$ on $U$ and ${}^b\\!\\overline \\partial \\zeta^j=0$ to infinite order at $U$. Define\n\\begin{equation*}\nx^j=\\Re \\zeta^j,\\quad x^{j+n+1}=\\Im\\zeta^j,\\quad j=1,\\dotsc,n.\n\\end{equation*}\nThe functions $x^j$, $j=1\\dots,2n+2$ are independent, and the forms\n\\begin{equation*}\n\\eta^j={}^b\\!d\\zeta^j,j=1\\dots,n,\\quad \\eta^{n+1}=\\frac{1}{x^{n+1}e^{i x^{2n+2}}}{}^b\\!d[x^{n+1}e^{i x^{2n+2}}]\n\\end{equation*}\ntogether with their conjugates form a frame for $\\mathbb C{}^b{\\!}T\\mathcal M$ near $p_0$. Let $\\eta^j_{1,0}$ and $\\eta^j_{0,1}$ be the $(1,0)$ and $(0,1)$ components of $\\eta^j$ according to the complex $b$-structure of $\\mathcal M$. Then\n\\begin{equation*}\n\\eta^j_{0,1}=\\sum_k p^j_k \\eta^k +q^j_k \\overline {\\eta}^k.\n\\end{equation*}\nSince $\\eta^j_{0,1}={}^b\\!\\overline \\partial\\zeta^j$ vanishes to infinite order at $U$,\nthe coefficients $p^j_k$ and $q^j_k$ vanish to infinite order at $U$. Replacing this formula for $\\eta^j_{0,1}$ in $\\eta^j=\\eta^j_{1,0}+\\eta^j_{0,1}$\nget\n\\begin{equation*}\n\\sum_k(\\delta^j_k-p^j_k )\\eta^k -\\sum_k q^j_k \\overline \\eta^k= \\eta^j_{1,0}.\n\\end{equation*}\nThe matrix $I-[p^j_k]$ is invertible with inverse of the form $I+[P^j_k]$ with $P^j_k$ vanishing to infinite order at $U$. So\n\\begin{equation}\\label{TheForms}\n\\eta^j -\\sum_k \\gamma^j_k \\overline \\eta^k=\\sum_k(\\delta^j_k+P^j_k) \\eta^k_{1,0}\n\\end{equation}\nwith suitable $\\gamma^j_k$ vanishing to infinite order on $U$. Define the vector fields $\\overline L_j^0$ as in \\eqref{NoLocalFormalInvariants1}. The vector fields\n\\begin{equation*}\n\\overline L_j=\\overline L_j^0+\\sum_k \\gamma^k_j L_k^0, \\quad j=1,\\dotsc,n+1\n\\end{equation*}\nare independent and since $\\langle \\overline L_j^0,\\eta^k\\rangle=0$ and $\\langle L_j^0,\\eta^k\\rangle=\\delta^k_j$, they annihilate each of the forms on the left hand side of \\eqref{TheForms}. So they annihilate the forms $\\eta^k_{1,0}$, which proves that the $\\overline L_j$ form a frame of ${}^b{\\!}T^{0,1}\\mathcal M$.\n\\end{proof}\n\n\\section{Indicial complexes}\\label{sIndicialComplex}\n\nThroughout this section we assume that $\\mathcal N$ is a connected component of the boundary of a compact manifold $\\mathcal M$. Let\n\\begin{equation}\\label{GenericComplex}\n\\cdots\\to C^\\infty(\\mathcal M;E^q)\\xrightarrow{A_q} C^\\infty(\\mathcal M;E^{q+1})\\to\\cdots\n\\end{equation}\nbe a $b$-elliptic complex of operators $A_q\\in \\Diff^1_b(\\mathcal M;E^q,E^{q+1})$; the $E^q$, $q=0,\\dotsc,r$, are vector bundles over $\\mathcal M$.\n\nNote that since $A_q$ is a first order operator,\n\\begin{equation}\\label{FirstOrderChar}\nA_q(f\\phi)=f A_q\\phi-i\\, \\bsym(A_q)({}^b\\!d f)(\\phi).\n\\end{equation}\nThis formula follows from the analogous formula for the standard principal symbol and the definition of principal $b$-symbol. It follows from \\eqref{FirstOrderChar} and \\eqref{VanishingOnBdy} that $A_q$ defines an operator\n\\begin{equation*}\nA_{b,q}:\\Diff^1(\\mathcal N;E^q_{\\mathcal N},E^{q+1}_{\\mathcal N}).\n\\end{equation*}\nFix a smooth defining function $\\mathfrak r:\\mathcal M\\to\\mathbb R$ for $\\partial \\mathcal M$, $\\mathfrak r>0$ in the interior of $\\mathcal M$, let \n\\begin{equation*}\n\\mathcal A_q(\\sigma): \\Diff^1_b(\\mathcal N;E^q_{\\mathcal N},E^{q+1}_{\\mathcal N}),\\quad \\sigma\\in \\mathbb C\n\\end{equation*}\ndenote the indicial family of $A_q$ with respect to $\\mathfrak r$, see \\eqref{IndicialFamily}. Using \\eqref{FirstOrderChar} and defining\n\\begin{equation*}\n\\Lambda_{\\mathfrak r,q}=\\bsym(A_q)(\\frac{{}^b\\!d \\mathfrak r}{\\mathfrak r}),\n\\end{equation*}\nthe indicial family of $A_q$ with respect to $\\mathfrak r$ is\n\\begin{equation}\\label{DefGenericIndOp}\n\\mathcal A_q(\\sigma) = A_{b,q}+\\sigma\\Lambda_{\\mathfrak r,q}:C^\\infty(\\mathcal N;E^q_\\mathcal N)\\to C^\\infty(\\mathcal N;E^{q+1}_\\mathcal N).\n\\end{equation}\nBecause of \\eqref{PQb}, these operators form an elliptic complex\n\\begin{equation}\\label{IndicialComplex}\n\\cdots\n\\to C^\\infty(\\mathcal N;E^q_\\mathcal N) \\xrightarrow{\\mathcal A_q(\\sigma)}\nC^\\infty(\\mathcal N;E^{q+1}_\\mathcal N)\n\\to\\cdots\n\\end{equation}\nfor each $\\sigma$ and each connected component $\\mathcal N$ of $\\partial \\mathcal M$. The operators depend on $\\mathfrak r$, but the cohomology groups at a given $\\sigma$ for different defining functions $\\mathfrak r$ are isomorphic. Indeed, if $\\mathfrak r'$ is another defining function for $\\partial \\mathcal M$, then $\\mathfrak r'=e^u\\mathfrak r$ for some smooth real-valued function $u$, and a simple calculation gives\n\\begin{equation*}\n(A_{b,q}+\\sigma \\Lambda_{\\mathfrak r,q})(e^{i\\sigma u}\\phi)=e^{i\\sigma u}(A_{b,q}+\\sigma \\Lambda_{\\mathfrak r',q})\\phi.\n\\end{equation*}\nIn analogy with the definition of boundary spectrum of an elliptic operator $A\\in \\Diff^m_b(\\mathcal M;E,F)$, we have\n\n\\begin{definition}\nLet $\\mathcal N$ be a connected component of $\\partial \\mathcal M$. The family of complexes \\eqref{IndicialComplex}, $\\sigma\\in \\mathbb C$, is the indicial complex of \\eqref{GenericComplex} at $\\mathcal N$. For each $\\sigma\\in \\mathbb C$ let $H^q_{\\mathcal A(\\sigma)}(\\mathcal N)$ denote the $q$-th cohomology group of \\eqref{IndicialComplex} on $\\mathcal N$. The $q$-th boundary spectrum of the complex \\eqref{GenericComplex} at $\\mathcal N$ is the set\n\\begin{equation*}\n\\spec_{b,\\mathcal N}^q(A)=\\set{\\sigma\\in \\mathbb C: H^q_{\\mathcal A(\\sigma)}(\\mathcal N)\\ne 0}.\n\\end{equation*}\nThe $q$-th boundary spectrum of $A$ is $\\spec_b^q(A) = \\bigcup_{\\mathcal N}\\spec_{b,\\mathcal N}^q(A)$.\n\\end{definition}\n\nThe spaces $H^q_{\\mathcal A(\\sigma)}(\\mathcal N)$ are finite-dimensional because \\eqref{IndicialComplex} is an elliptic complex and $\\mathcal N$ is compact. It is convenient to isolate the behavior of the indicial complex according to the components of the boundary, since the sets $\\spec_{b,\\mathcal N}^q(A)$ can vary drastically from component to component.\n\n\\medskip\nSuppose that $\\mathcal M$ is a complex $b$-manifold. Recall that since\n\\begin{equation*}\n{}^b\\!\\overline \\partial \\in \\Diff^1_b(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}\\mathcal M),\n\\end{equation*}\nthere are induced boundary operators\n\\begin{equation*}\n{}^b\\!\\overline \\partial_b\\in \\Diff^1(\\mathcal N;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}_\\mathcal N\\mathcal M,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}_\\mathcal N\\mathcal M)\n\\end{equation*}\nwhich via the isomorphism \\eqref{bdyIso} become the operators of the $\\overline\\Dee$-complex \\eqref{bdyComplex}. Combining \\eqref{bdeebarOnFunctions} and \\eqref{bsymbdeebar} we get\n\\begin{equation*}\n\\bsym({}^b\\!\\overline \\partial)(\\frac{{}^b\\!d \\mathfrak r}{\\mathfrak r})(\\phi) = i \\frac{{}^b\\!\\overline \\partial\\mathfrak r}{\\mathfrak r}\\wedge\\phi\n\\end{equation*}\nand using \\eqref{DefinitionBeta} we may identify $\\widehat{{}^b\\!\\overline \\partial_b}(\\sigma)$, given by \\eqref{DefGenericIndOp}, with the operator\n\\begin{equation}\\label{DefIndOp}\n\\overline\\D(\\sigma)\\phi = \\overline\\Dee \\phi + i \\sigma\\beta_\\mathfrak r\\wedge \\phi.\n\\end{equation}\nIf $E\\to\\mathcal M$ is a holomorphic vector bundle, then the indicial family of\n\\begin{equation*}\n{}^b\\!\\overline \\partial\\in \\Diff^1_b(\\mathcal M;{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q}\\mathcal M\\otimes E,{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}^{0,q+1}\\mathcal M\\otimes E)\n\\end{equation*}\nis again given by \\eqref{DefIndOp}, but using the operator $\\overline\\Dee$ of the complex \\eqref{defDeebarBundle}.\n\n\\medskip\nReturning to the general complex \\eqref{GenericComplex}, fix a smooth positive $b$-density $\\mathfrak m$ on $\\mathcal M$ and a Hermitian metric on each $E^q$. Let $\\mathcal A_q^\\star(\\sigma)$ be the indicial operator of the formal adjoint, $A^\\star_q$, of $A_q$. The Laplacian $\\square_q$ of the complex \\eqref{GenericComplex} in degree $q$ belongs to $\\Diff^2_b(\\mathcal M;E^q\\mathcal M)$, is $b$-elliptic, and its indicial operator is\n\\begin{equation*}\n\\widehat \\square_q(\\sigma)=\n\\mathcal A_q^\\star(\\sigma) \\mathcal A_q(\\sigma)+\\mathcal A_{q-1}(\\sigma) \\mathcal A_{q-1}^\\star(\\sigma).\n\\end{equation*}\nThe $b$-spectrum of $\\square_q$ at $\\mathcal N$, see Melrose~\\cite{RBM2}, is the set\n\\begin{equation*}\n\\spec_{b,\\mathcal N}(\\square_q) = \\set{\\sigma\\in \\mathbb C:\\widehat\\square_q(\\sigma):C^\\infty(\\mathcal N;E^q_\\mathcal N)\\to C^\\infty(\\mathcal N;E^q_\\mathcal N)\\text{ is not invertible}}.\n\\end{equation*}\nNote that unless $\\sigma$ is real, $\\widehat \\square_q(\\sigma)$ is not the Laplacian of the complex \\eqref{IndicialComplex}.\n\n\\begin{proposition}\\label{DiscretenesOfBSpectrum}\nFor each $q$, $\\spec_{b,\\mathcal N}^q(A)\\subset \\spec_{b,\\mathcal N}(\\square_q)$. \\end{proposition}\n\nNote that the set $\\spec_{b,\\mathcal N}(\\square_q)$ depends on the choice of Hermitian metrics and $b$-density used to construct the Laplacian, but that the subset $\\spec_{b,\\mathcal N}^q(A)$ is independent of such choices. For a general $b$-elliptic complex \\eqref{GenericComplex} it may occur that $\\spec_{b,\\mathcal N}^q(A)\\ne \\spec_{b,\\mathcal N}(\\square_q)$. In Example \\ref{IndComplexbd} we show that $\\spec_{b,\\mathcal N}^q({}^b\\!d)\\subset \\set{0}$. As is well known, $\\spec_{b,\\mathcal N}(\\Delta_q)$ is an infinite set if $\\dim\\mathcal M>1$. At the end of this section we will give an example where $\\spec_{b,\\mathcal N}^0({}^b\\!\\overline \\partial)$ is an infinite set. A full discussion of $\\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial)$ for any $q$ and other aspects of the indicial complex of complex $b$-structures is given in Section~\\ref{sIndicialCohomology}.\n\n\\begin{proof}[Proof of Proposition~\\ref{DiscretenesOfBSpectrum}]\nSince $\\square_q$ is $b$-elliptic, the set $\\spec_{b,\\mathcal N}(\\square_q)$ is closed and discrete. Let $H^2(\\mathcal N;E^q_\\mathcal N)$ be the $L^2$-based Sobolev space of order $2$. For $\\sigma\\notin \\spec_{b,\\mathcal N}(\\square_q)$ let\n\\begin{equation*}\n\\mathcal G_q(\\sigma):L^2(\\mathcal N;E^q_\\mathcal N)\\to H^2(\\mathcal N;E^q_\\mathcal N)\n\\end{equation*}\nbe the inverse of $\\widehat \\square_q(\\sigma)$. The map $\\sigma\\mapsto \\mathcal G_q(\\sigma)$ is meromorphic with poles in $\\spec_b(\\square_q)$. Since\n\\begin{equation*}\n\\mathcal A_q^\\star(\\sigma)= [\\mathcal A_q(\\overline \\sigma)]^\\star\n\\end{equation*}\nthe operators $\\widehat \\square_q(\\sigma)$ are the Laplacians of the complex \\eqref{IndicialComplex} when $\\sigma$ is real. Thus for $\\sigma\\in \\mathbb R\\backslash (\\spec_{b,\\mathcal N}(\\square_q)\\cup\\spec_{b,\\mathcal N}(\\square_{q+1}))$ we have\n\\begin{equation*}\n\\mathcal A_{q}(\\sigma) \\mathcal G_{q}(\\sigma)=\\mathcal G_{q+1}(\\sigma)\\mathcal A_q(\\sigma),\\quad\n\\mathcal A_q(\\sigma)^\\star \\mathcal G_{q+1}(\\sigma) = \\mathcal G_q(\\sigma)\\mathcal A_q^\\star(\\sigma)\n\\end{equation*}\nby standard Hodge theory. Since all operators depend holomorphically on $\\sigma$, the same equalities hold for $\\sigma\\in \\mathfrak R=\\mathbb C\\backslash(\\spec_{b,\\mathcal N}(\\square_q)\\cup \\spec_{b,\\mathcal N}(\\square_{q+1}))$. It follows that\n\\begin{equation*}\n\\mathcal A_q^\\star(\\sigma)\\mathcal A_q(\\sigma) \\mathcal G_q(\\sigma) =\n\\mathcal G_q(\\sigma) \\mathcal A_q^\\star(\\sigma)\\mathcal A_q(\\sigma)\n\\end{equation*}\nin $\\mathfrak R$. By analytic continuation the equality holds on all of $\\mathbb C\\backslash \\spec_{b,\\mathcal N}(\\square_q)$. Thus if $\\sigma_0\\notin \\spec_{b,\\mathcal N}(\\square_q)$ and $\\phi$ is a $\\mathcal A_q(\\sigma_0)$-closed section, $\\mathcal A_q(\\sigma_0)\\phi=0$, then the formula\n\\begin{equation*}\n\\phi=[\\mathcal A_q^\\star(\\sigma_0) \\mathcal A_q(\\sigma_0)+\\mathcal A_{q-1}(\\sigma_0) \\mathcal A_{q-1}^\\star(\\sigma_0)]\\mathcal G_q(\\sigma_0) \\phi\n\\end{equation*}\nleads to\n\\begin{equation*}\n\\phi=\\mathcal A_{q-1}(\\sigma_0)[\\mathcal A_{q-1}^\\star(\\sigma_0) \\mathcal G_{q}(\\sigma_0)\\phi].\n\\end{equation*}\nTherefore $\\sigma_0\\notin \\spec_{b,\\mathcal N}^q(A)$.\n\\end{proof}\n\nSince $\\square_q$ is $b$-elliptic, the set $\\spec _{b,\\mathcal N}(\\square_q)$ is discrete and intersects each horizontal strip $a\\leq\\Im\\sigma\\leq b$ in a finite set (Melrose~\\cite{RBM2}). Consequently:\n\n\\begin{corollary}\nThe sets $\\spec_{b,\\mathcal N}^q(A)$, $q=0,1\\dotsc$, are closed, discrete, and intersect each horizontal strip $a\\leq\\Im\\sigma\\leq b$ in a finite set.\n\\end{corollary}\n\nWe note in passing that the Euler characteristic of the complex \\eqref{IndicialComplex} vanishes for each $\\sigma$. Indeed, let $\\sigma_0\\in \\mathbb C$. The Euler characteristic of the $\\mathcal A(\\sigma_0)$-complex is the index of\n\\begin{equation*}\n\\mathcal A(\\sigma_0) + \\mathcal A(\\sigma_0)^\\star:\\bigoplus_{q\\text{ even}}C^\\infty(\\mathcal N;E^q)\\to \\bigoplus_{q\\text{ odd}}C^\\infty(\\mathcal N;E^q).\n\\end{equation*}\nThe operator $\\mathcal A_q(\\sigma)$ is equal to $A_{b,q} + \\sigma \\Lambda_{\\mathfrak r,q}$, see \\eqref{DefGenericIndOp}. Thus $\\mathcal A_q(\\sigma)^\\star = A_{b,q}^\\star+\\overline \\sigma \\Lambda_{\\mathfrak r,q}^\\star$, and it follows that for any $\\sigma$,\n\\begin{equation*}\n\\mathcal A(\\sigma) + \\mathcal A(\\sigma)^\\star=\n\\mathcal A(\\sigma_0) + \\mathcal A(\\sigma_0)^\\star\n+(\\sigma-\\sigma_0)\\Lambda_\\mathfrak r+(\\overline \\sigma-\\overline \\sigma_0)\\Lambda_\\mathfrak r^\\star\n\\end{equation*}\nis a compact perturbation of $\\mathcal A(\\sigma_0) + \\mathcal A(\\sigma_0)^\\star$. Therefore, since the index is invariant under compact perturbations, the index of $\\mathcal A(\\sigma) + \\mathcal A(\\sigma)^\\star$ is independent of $\\sigma$. Then it vanishes, since it vanishes when $\\sigma\\notin \\bigcup_q \\spec_{b,\\mathcal N}^q(A)$.\n\n\\medskip\nLet $\\mathfrak {Mero}^q(\\mathcal N)$ be the sheaf of germs of $C^\\infty(\\mathcal N;E^q)$-valued meromorphic functions on $\\mathbb C$ and let $\\mathfrak {Hol}^q(\\mathcal N)$ be the subsheaf of germs of holomorphic functions. Let $\\mathfrak S^q(\\mathcal N)=\\mathfrak {Mero}^q(\\mathcal N)\/\\mathfrak {Hol}^q(\\mathcal N)$. The holomorphic family $\\sigma\\mapsto \\mathcal A_q(\\sigma)$ gives a sheaf homomorphism $\\mathcal A_q:\\mathfrak {Mero}^q(\\mathcal N)\\to\\mathfrak {Mero}^{q+1}(\\mathcal N)$ such that $\\mathcal A_q(\\mathfrak {Hol}^q(\\mathcal N))\\subset \\mathfrak {Hol}^{q+1}(\\mathcal N)$ and $\\mathcal A_{q+1}\\circ \\mathcal A_{q}=0$, so we have a complex\n\\begin{equation}\\label{BdyMeroCohomologySheaf}\n\\cdots\\to \\mathfrak S^q(\\mathcal N)\\xrightarrow{\\mathcal A_q}\\mathfrak S^{q+1}(\\mathcal N)\\to\\cdots.\n\\end{equation}\n\nThe cohomology sheafs $\\mathfrak H^q_A(\\mathcal N)$ of this complex contain more refined information about the cohomology of the complex $A$.\n\n\\begin{proposition}\nThe sheaf $\\mathfrak H^q_\\mathcal A(\\mathcal N)$ is supported on $\\spec_{b,\\mathcal N}^q(A)$.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\sigma_0\\in \\mathbb C$ be such that $H^q_{\\mathcal A(\\sigma_0)}(\\mathcal N)=0$ and let\n\\begin{equation}\\label{SingPart}\n\\phi(\\sigma)=\\sum_{k=1}^{\\mu}\\frac{\\phi_k}{(\\sigma-\\sigma_0)^{k}},\n\\end{equation}\n$\\mu>0$, $\\phi_k\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*)$, represent the $\\mathcal A$-closed element $[\\phi]$ of the stalk of $\\mathfrak S^q(\\mathcal N)$ over $\\sigma_0$. The condition that $\\mathcal A_q[\\phi]=0$ means that $\\mathcal A_q(\\sigma)\\phi(\\sigma)$ is holomorphic, that is,\n\\begin{equation*}\n\\frac{\\mathcal A_q(\\sigma_0)\\phi_\\mu}{(\\sigma-\\sigma_0)^\\mu} +\\sum_{k=1}^{\\mu-1}\\frac{\\mathcal A_q(\\sigma_0)\\phi_k+\\Lambda_{\\mathfrak r,q}\\phi_{k+1}}{(\\sigma-\\sigma_0)^{k}}=0.\n\\end{equation*}\nIn particular $\\mathcal A_q(\\sigma_0)\\phi_\\mu=0$. Since $H^q_{\\mathcal A(\\sigma_0)}(\\mathcal N)=0$, there is $\\psi_\\mu\\in C^\\infty(\\mathcal N;E^{q-1})$ such that $\\mathcal A_{q-1}(\\sigma_0)\\psi_\\mu=\\phi_\\mu$. This shows that if $\\mu=1$, then $[\\phi]$ is exact, and that if $\\mu>1$, then letting $\\phi'(\\sigma)=\\phi(\\sigma)-\\mathcal A_{q-1}(\\sigma)\\psi_\\mu\/(\\sigma-\\sigma_0)^\\mu$, that $\\phi$ is cohomologous to an element $[\\phi']$ represented by a sum as in \\eqref{SingPart} with $\\mu-1$ instead of $\\mu$. By induction, $[\\phi]$ is exact.\n\\end{proof}\n\n\\begin{definition}\\label{CohomologySheafs}\nThe cohomology sheafs $\\mathfrak H^q_{A}(\\mathcal N)$ of the complex \\eqref{BdyMeroCohomologySheaf} will be referred to as the indicial cohomology sheafs of the complex $A$. If $[\\phi]\\in \\mathfrak h^q_A(\\mathcal N)$ is a nonzero element of the stalk over $\\sigma_0$, the smallest $\\mu$ such that there is a meromorphic function \\eqref{SingPart} representing $[\\phi]$ will be called the order of the pole of $[\\phi]$.\n\\end{definition}\n\nThe relevancy of this notion of pole lies in that it predicts, for any given cohomology class of the complex $A$, the existence of a representative with the most regular leading term (the smallest power of log that must appear in the expansion at the boundary). We will see later (Proposition~\\ref{bdeebarCohomologySheaf}) that for the $b$-Dolbeault complex, under a certain geometric assumption, the order of the pole of $[\\phi]\\in \\mathfrak H^q_{{}^b\\!\\overline \\partial}(\\mathcal N)\\backslash 0$ is $1$.\n\n\\begin{example}\\label{IndComplexbd}\nFor the $b$-de Rham complex one has $\\spec_{b,\\mathcal N}^q({}^b\\!d)\\subset \\set{0}$ and\n\\begin{equation*}\nH^q_{\\mathcal D(0)}(\\mathcal N)=H^q_{\\mathrm {dR}}(\\mathcal N)\\oplus H^{q-1}_{\\mathrm {dR}}(\\mathcal N)\n\\end{equation*}\nfor each component $\\mathcal N$ of $\\partial\\mathcal M$, and that every element of the stalk of $\\mathfrak H^q_{{}^b\\!d}(\\mathcal N)$ over $0$ has a representative with a simple pole. By way of the residue we get an isomorphism from the stalk over $0$ onto $H^q_{\\mathrm {dR}}(\\mathcal N)$.\n\nSince the map \\eqref{evbM} is surjective with kernel spanned by $\\mathfrak r\\partial_\\mathfrak r$, the dual map\n\\begin{equation}\\label{evbMDual}\n\\mathrm{ev}_{\\mathcal N}^*: T^*{\\mathcal N}\\to {}^b{\\!}T_{\\mathcal N}^*\\mathcal M\n\\end{equation}\nis injective with image the annihilator, $\\mathcal H$, of $\\mathfrak r\\partial\\mathfrak r$. Let $\\mathbf i_{\\mathfrak r\\partial_\\mathfrak r}:{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^q\\mathcal M\\to {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^{q-1}\\mathcal M$ denote interior multiplication by $\\mathfrak r\\partial_\\mathfrak r$ Then $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal H=\\ker(\\mathbf i_{\\mathfrak r\\partial_\\mathfrak r}:{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^q\\mathcal M\\to {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^{q-1}\\mathcal M)$. The isomorphism \\eqref{evbMDual} gives isomorphisms\n\\begin{equation*}\n\\mathrm{ev}_{\\mathcal N}^*: \\raise2ex\\hbox{$\\mathchar\"0356$}^q{\\mathcal N}\\to \\mathcal H^q\n\\end{equation*}\nfor each $q$. Fix a defining function $\\mathfrak r$ for $\\mathcal N$ and let $\\Pi:{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^q\\mathcal M \\to {}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^q\\mathcal M$ be the projection on $\\mathcal H^q$ according to the decomposition\n\\begin{equation*}\n{}^b{\\!}\\raise2ex\\hbox{$\\mathchar\"0356$}_{\\mathcal N}^q\\mathcal M=\\mathcal H^q\\oplus \\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\mathcal H^{q-1},\n\\end{equation*}\nthat is,\n\\begin{equation*}\n\\Pi\\phi = \\phi-\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge \\mathbf i_{\\mathfrak r\\partial_\\mathfrak r}\\phi.\n\\end{equation*}\nIf $\\phi^0\\in C^\\infty(\\mathcal N,\\mathcal H^q)$ and $\\phi^1\\in C^\\infty(\\mathcal N,\\mathcal H^{q-1})$, then\n\\begin{equation*}\n{}^b\\!d(\\phi^0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi^1)=\\Pi\\,{}^b\\!d\\phi^0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge(-\\Pi\\,{}^b\\!d\\phi^1).\n\\end{equation*}\nSince\n\\begin{equation*}\n\\mathfrak r^{-i\\sigma}{}^b\\!d\\mathfrak r^{i\\sigma}\\phi = {}^b\\!d \\phi +i \\sigma\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi,\n\\end{equation*}\nthe indicial operator $\\mathcal D(\\sigma)$ of ${}^b\\!d$ is\n\\begin{equation*}\n\\mathcal D(\\sigma)(\\phi_0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi^1) = \\Pi\\,{}^b\\!d\\phi^0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge(i \\sigma\\phi^0-\\Pi\\, {}^b\\!d\\phi^1).\n\\end{equation*}\nIf $\\mathcal D(\\sigma)(\\phi_0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi^1)=0$, then of course $\\Pi{}^b\\!d\\phi^0=0$ and $i \\sigma\\phi^0=\\Pi{}^b\\!d\\phi^1$, and it follows that if $\\sigma\\ne 0$, then\n\\begin{equation*}\n(\\phi_0+\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi^1) = \\mathcal D(\\sigma)\\frac{1}{i\\sigma}\\phi^1.\n\\end{equation*}\nThus all cohomology groups of the complex $\\mathcal D(\\sigma)$ vanish if $\\sigma\\ne 0$, i.e., $\\spec_{b,\\mathcal N}^q({}^b\\!d)\\subset \\set{0}$.\n\nIt is not hard to verify that\n\\begin{equation*}\n\\Pi{}^b\\!d\\,\\mathrm{ev}_{\\mathcal N}^*=\\mathrm{ev}_{\\mathcal N}^*d.\n\\end{equation*}\nSince\n\\begin{equation*}\n\\mathfrak r^{-i\\sigma}{}^b\\!d\\mathfrak r^{i\\sigma}\\phi = {}^b\\!d \\phi +i \\sigma\\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\phi,\n\\end{equation*}\nthe indicial operator of ${}^b\\!d$ at $\\sigma=0$ can be viewed as the operator\n\\begin{equation*}\n\\begin{bmatrix}\nd & 0\\\\\n0 & -d\n\\end{bmatrix}:\n\\begin{matrix}\n\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal N\\\\\n\\oplus\\\\\n\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\mathcal N\n\\end{matrix} \\to\n\\begin{matrix}\n\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\mathcal N\\\\\n\\oplus\\\\\n\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\mathcal N\n\\end{matrix}.\n\\end{equation*}\nFrom this we get the cohomology groups of $\\mathcal D(0)$ in terms of the de Rham cohomology of $\\mathcal N$:\n\\begin{equation*}\nH^q_{\\mathcal D(0)}(\\mathcal N)=H^q_{\\mathrm {dR}}(\\mathcal N)\\oplus H^{q-1}_{\\mathrm {dR}}(\\mathcal N).\n\\end{equation*}\nThus the groups $H^q_{\\mathcal D(0)}(\\mathcal N)$ do not vanish for $q=0$, $1$, $\\dim\\mathcal M-1$, $\\dim\\mathcal M$ but may vanish for other values of $q$.\n\nWe now show that every element of the stalk of $\\mathfrak H^q_{{}^b\\!d}(\\mathcal N)$ over $0$ has a representative with a simple pole at $0$. Suppose that\n\\begin{equation}\\label{Representative}\n\\phi(\\sigma)=\\sum_{k=1}^\\mu \\frac{1}{\\sigma^k}\\left(\\phi^0_k+\\frac{{}^b\\!d \\mathfrak r}{\\mathfrak r}\\wedge \\phi^1_k\\right)\n\\end{equation}\nis such that $\\mathcal D(\\sigma)\\phi(\\sigma)$ is holomorphic. Then\n\\begin{equation*}\n\\sum_{k=1}^\\mu \\frac{1}{\\sigma^k}\\left(d\\phi^0_k - \\frac{{}^b\\!d \\mathfrak r}{\\mathfrak r}\\wedge d\\phi^1_k\\right) + \\frac{{}^b\\!d \\mathfrak r}{\\mathfrak r}\\wedge\\left(\\sum_{k=1}^{\\mu-1} \\frac{i}{\\sigma^k}\\phi^0_{k+1}\\right) = 0,\n\\end{equation*}\nhence $d\\phi^0_1=0$, $d\\phi^1_\\mu=0$ and $\\phi^0_k=-i d\\phi^1_{k-1}$, $k=2,\\dotsc,\\mu$. Let\n\\begin{equation*}\n\\psi(\\sigma)=-i \\sum_{k=2}^{\\mu+1} \\frac{1}{\\sigma^k} \\phi^1_{k-1}.\n\\end{equation*}\nThen\n\\begin{align*}\n\\mathcal D(\\sigma)\\psi(\\sigma)\n&= -i \\sum_{k=2}^{\\mu+1} \\frac{1}{\\sigma^k} d\\phi^1_{k-1} + \\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\sum_{k=2}^{\\mu+1}\\frac{1}{\\sigma^{k-1}}\\phi^1_{k-1}\\\\\n&= \\sum_{k=2}^{\\mu} \\frac{1}{\\sigma^k} \\phi^0_k + \\frac{{}^b\\!d\\mathfrak r}{\\mathfrak r}\\wedge\\sum_{k=1}^{\\mu}\\frac{1}{\\sigma^k}\\phi^1_k\n\\end{align*}\nso\n\\begin{equation*}\n\\phi(\\sigma)-\\mathcal D(\\sigma)\\psi(\\sigma)=\\frac{1}{\\sigma}\\phi^0_1.\n\\end{equation*}\nThe map that sends the class of the $\\mathcal D(\\sigma)$-closed element \\eqref{Representative} to the class of $\\phi^0_1$ in $H^q_{\\mathrm {dR}}(\\mathcal N)$ is an isomorphism.\n\\end{example}\n\n\\begin{example}\nAs we just saw, the boundary spectrum of the ${}^b\\!d$ complex in degree $0$ is just $\\set{0}$. In contrast, $\\spec_{b,\\mathcal N}^0({}^b\\!\\overline \\partial)$ may be an infinite set. We illustrate this in the context of Example \\ref{AnisotropicSphere}. The functions\n\\begin{equation*}\nz^\\alpha=(z^1)^{\\alpha_1}\\dotsm (z^{n+1})^{\\alpha_{n+1}},\n\\end{equation*}\nwhere the $\\alpha_j$ are nonnegative integers, are CR functions that satisfy\n\\begin{equation*}\n\\mathcal T z^\\alpha=i (\\sum \\tau_j\\alpha_j) z^\\alpha.\n\\end{equation*}\nThis implies that\n\\begin{equation*}\n\\overline\\Dee z^\\alpha + i (-i \\sum \\tau_j\\alpha_j) \\beta z^\\alpha=0\n\\end{equation*}\nwith $\\beta$ as in Example \\ref{AnisotropicSphere}, so the numbers $\\sigma_\\alpha=(-i \\sum \\tau_j\\alpha_j)$ belong to $\\spec_{b,\\mathcal N}^0({}^b\\!\\overline \\partial)$. \n\nFor the sake of completeness we also show that if $\\sigma\\in \\spec_{b,\\mathcal N}^0({}^b\\!\\overline \\partial)$, then $\\sigma=\\sigma_\\alpha$ for some $\\alpha$ as above. To see this, suppose that $\\zeta:S^{2n+1}\\to\\mathbb C$ is not identically zero and satisfies\n\\begin{equation*}\n\\overline\\Dee \\zeta+i \\sigma\\zeta\\beta=0\n\\end{equation*}\nfor some $\\sigma\\ne 0$. Then $\\zeta$ is smooth, because the principal symbol of $\\overline\\Dee$ on functions is injective. Since $\\langle\\beta,\\mathcal T\\rangle=-i$,\n\\begin{equation*}\nT\\zeta+ \\sigma\\zeta=0.\n\\end{equation*}\nThus $\\zeta(\\mathfrak a_t(p))=e^{-\\sigma t}\\zeta(p)$ for any $p$. Since $|\\zeta(\\mathfrak a_t(p))|$ is bounded as a function of $t$ and $\\zeta$ is not identically $0$, $\\sigma$ must be purely imaginary. Since $\\zeta$ is a CR function, it extends uniquely to a holomorphic function $\\tilde\\zeta$ on $B=\\set{z\\in \\mathbb C^{n+1}:\\|z\\|<1}$, necessarily smooth up to the boundary. Let $\\zeta_t=\\zeta\\circ \\mathfrak a_t$. This is also a smooth CR function, so it has a unique holomorphic extension $\\tilde \\zeta_t$ to $B$. The integral curve through $z_0=(z^1_0,\\dotsc,z^{n+1}_0)$ of the vector field $\\mathcal T$ is\n\\begin{equation*}\nt\\mapsto \\mathfrak a_t(z_0)=(e^{i \\tau_1 t} z^1_0,\\dotsc,e^{i \\tau_{n+1} t} z^{n+1}_0)\n\\end{equation*}\nExtending the definition of $\\mathfrak a_t$ to allow arbitrary $z\\in \\mathbb C^{n+1}$ as argument we then have that $\\tilde \\zeta_t=\\tilde \\zeta\\circ \\mathfrak a_t$. Then\n\\begin{equation*}\n\\partial_t\\tilde \\zeta_t+ \\sigma\\tilde \\zeta_t=0\n\\end{equation*}\ngives\n\\begin{equation*}\n\\tilde\\zeta(z)=\\sum_{\\set{\\alpha:\\pmb \\tau\\cdot\\alpha=i \\sigma}} c_\\alpha z^\\alpha\n\\end{equation*}\nfor $|z|<1$, where $\\pmb\\tau=(\\tau_1,\\dotsc,\\tau_{n+1})$. Thus $\\sigma=-i\\sum\\tau_j\\alpha_j$ as claimed. Note that $\\Im \\sigma$ is negative (positive) if the $\\tau_j$ are positive (negative) and $\\alpha\\ne0$.\n\\end{example}\n\n\\section{Underlying CR complexes}\\label{sUnderlyingCRcomplexes}\n\nAgain let $\\mathfrak a:\\mathbb R\\times\\mathcal N\\to\\mathcal N$ be the flow of $\\mathcal T$. Let $\\mathcal {L}_\\mathcal T$ denote the Lie derivative with respect to $\\mathcal T$ on de Rham $q$-forms or vector fields and let $\\mathbf i_\\mathcal T$ denote interior multiplication by $\\mathcal T$ of de Rham $q$-forms or of elements of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$.\n\nThe proofs of the following two lemmas are elementary.\n\n\\begin{lemma}\nIf $\\alpha$ is a smooth section of the annihilator of $\\overline\\Vee$ in $\\mathbb C T^*\\mathcal N$, then $(\\mathcal {L}_{\\mathcal T}\\alpha)|_{\\overline\\Vee}=0$. Consequently, for each $p\\in \\mathcal N$ and $t\\in \\mathbb R$, $d\\mathfrak a_t:\\mathbb C T_p\\mathcal N\\to \\mathbb C T_{\\mathfrak a_t(p)}\\mathcal N$ maps $\\overline\\Vee_p$ onto $\\overline\\Vee_{\\mathfrak a_t(p)}$.\n\\end{lemma}\n\nIt follows that there is a well defined smooth bundle homomorphism $\\mathfrak a_t^*: \\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^* \\to \\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$ covering $\\mathfrak a_{-t}$. In particular, one can define the Lie derivative $\\mathcal {L}_{\\mathcal T}\\phi$ with respect to $\\mathcal T$ of an element in $\\phi \\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*)$. The usual formula holds:\n\n\\begin{lemma}\nIf $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*)$, then $\\mathcal {L}_{\\mathcal T}\\phi=\\mathbf i_{\\mathcal T}\\overline\\Dee\\phi+\\overline\\Dee\\mathbf i_{\\mathcal T}\\phi$. Consequently, for each $t$ and $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*)$, $\\overline\\Dee \\mathfrak a_t^*\\phi = \\mathfrak a_t^*\\overline\\Dee\\phi$.\n\\end{lemma}\n\n\\medskip\nFor any defining function $\\mathfrak r$ of $\\mathcal N$ in $\\mathcal M$, $\\overline \\K_{\\mathfrak r}=\\ker\\beta_\\mathfrak r$ is a CR structure of CR codimension $1$: indeed, $\\mathcal K_\\mathfrak r\\cap \\overline \\K_\\mathfrak r\\subset \\Span_\\mathbb C\\mathcal T$ but since $\\langle\\beta_\\mathfrak r,\\mathcal T\\rangle$ vanishes nowhere, we must have $\\overline \\K\\cap \\mathcal K=0$. Since $\\mathcal K\\oplus \\overline \\K\\oplus\\Span_\\mathbb C\\mathcal T=\\mathbb C T\\mathcal N$, the CR codimension is $1$. Finally, if $V,W\\in C^\\infty(\\mathcal N;\\overline \\K_{\\mathfrak r})$, then \n\\begin{equation*}\n\\langle\\beta_\\mathfrak r,[V,W]\\rangle=V\\langle\\beta_\\mathfrak r,W\\rangle-W\\langle\\beta_\\mathfrak r,V\\rangle-2\\overline\\Dee\\beta(V,W),\n\\end{equation*}\nSince the right hand side vanishes, $[V,W]$ is again a section of $\\overline \\K_\\mathfrak r$.\n\nSince $\\overline\\Vee=\\overline \\K_\\mathfrak r\\oplus \\Span_\\mathbb C \\mathcal T$, the dual of $\\overline \\K_\\mathfrak r$ is canonically isomorphic to the kernel of $\\mathbf i_\\mathcal T:\\smash[t]{\\overline\\Vee}^*\\to\\mathbb C$. We will write $\\overline \\K^*$ for this kernel. More generally, $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*$ and the kernel, $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$, of $\\mathbf i_\\mathcal T:\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\to\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\smash[t]{\\overline\\Vee}^*$ are canonically isomorphic. The vector bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$ are independent of the defining function $\\mathfrak r$. We regard the $\\overline \\partial_b$-operators of the CR structure as operators\n\\begin{equation*}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*)\\to C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K^*).\n\\end{equation*}\nThey do depend on $\\mathfrak r$ but we will not indicate this in the notation. \n\nTo get a formula for $\\overline \\partial_b$, let\n\\begin{equation*}\n\\tilde \\beta_\\mathfrak r=\\frac{i }{i -a_\\mathfrak r}\\beta_\\mathfrak r\n\\end{equation*}\n(so that $\\langle i \\tilde \\beta_\\mathfrak r,\\mathcal T\\rangle=1$). The projection $\\Pi_\\mathfrak r:\\raise2ex\\hbox{$\\mathchar\"0356$}^q \\smash[t]{\\overline\\Vee}^*\\to\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$ on $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$ according to the decomposition\n\\begin{equation}\\label{DecompositionOfVeebar}\n\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*=\\raise2ex\\hbox{$\\mathchar\"0356$}^q \\overline \\K^* \\oplus i \\tilde \\beta_\\mathfrak r\\wedge \\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K^*\n\\end{equation}\nis\n\\begin{equation}\\label{DefinitionOfPi}\n\\Pi_\\mathfrak r\\phi=\\phi-i\\tilde \\beta_\\mathfrak r\\wedge \\mathbf i_\\mathcal T\\phi.\n\\end{equation}\n\n\\begin{lemma}\nWith the identification of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*$ with $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$ described above, the $\\overline \\partial_b$-operators of the CR structure $\\overline \\K_\\mathfrak r$ are given by\n\\begin{equation}\\label{DefinitionOfdeebarb}\n\\overline \\partial_b\\phi=\\Pi_\\mathfrak r\\overline\\Dee\\phi\\quad \\text{ if }\\phi\\in C^\\infty(\\mathcal N,\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*),\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof} Suppose that $(z,t)$ is a hypoanalytic chart for $\\overline\\Vee$ on some open set $U$, with $\\mathcal T t=1$. So $\\partial_{\\overline z^\\mu}$, $\\mu=1\\dotsc,n$, $\\mathcal T=\\partial_t$ is a frame for $\\overline\\Vee$ over $U$ with dual frame $\\overline\\Dee\\overline z^\\mu$, $\\overline\\Dee t$. If\n\\begin{equation*}\n\\beta_\\mathfrak r=\\sum_{\\mu=1}^n\\beta_\\mu\\overline\\Dee \\overline z^\\mu+\\beta_0\\overline\\Dee t.\n\\end{equation*}\nthen\n\\begin{equation*}\n\\overline L_\\mu=\\partial_{\\overline z^\\mu}-\\frac{\\beta_\\mu}{\\beta_0}\\partial_t,\\quad\\mu=1,\\dotsc,n\n\\end{equation*}\nis a frame for $\\overline \\K_\\mathfrak r$ over $U$. Let $\\overline \\eta^\\mu$ denote the dual frame (for $\\overline \\K_\\mathfrak r^*$). Since the $\\overline L_\\mu$ commute, $\\overline \\partial_b\\overline \\eta^\\mu=0$, so if $\\phi=\\sum'_{|I|=q}\\phi_I\\,\\overline \\eta^I$, then (with the notation as in eg. Folland and Kohn~\\cite{FK})\n\\begin{equation*}\n\\overline \\partial_b \\phi=\\sideset{}{'}\\sum_{|J|=q+1}\\,\\sideset{}{'}\\sum_{|I|=q}\\sum_\\mu \\sign {\\mu I}{J}\\overline L_\\mu\\phi_I\\,\\overline \\eta^J.\n\\end{equation*}\nOn the other hand, the frame of $\\smash[t]{\\overline\\Vee}^*$ dual to the frame $\\overline L_\\mu$, $\\mu=1,\\dotsc,n$, $\\mathcal T$ of $\\overline\\Vee$ is $\\overline\\Dee \\overline z^\\mu$, $i\\tilde \\beta_\\mathfrak r$, and the identification of $\\overline \\K_\\mathfrak r^*$ with $\\overline \\K^*$ maps the $\\eta^\\mu$ to the $\\overline\\Dee \\overline z^\\mu$. So, as a section of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$,\n\\begin{equation*}\n\\phi=\\sideset{}{'}\\sum_{|I|=q} \\phi_I\\, \\overline\\Dee\\overline z^I\n\\end{equation*}\nand\n\\begin{equation*}\n\\overline\\Dee\\phi=\\sideset{}{'}\\sum_{|J|=q+1}\\,\\sideset{}{'}\\sum_{|I|=q} \\sign{\\mu I}{J} \\overline L_\\mu\\phi_I \\,\\overline\\Dee\\overline z^J+\ni\\tilde \\beta_\\mathfrak r\\wedge\\sideset{}{'}\\sum_{|I|=q} \\mathcal T\\phi_I\\,\\overline\\Dee\\overline z^I.\n\\end{equation*}\nThus $\\Pi_\\mathfrak r\\overline\\Dee\\phi$ is the section of $\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K^*$ associated with $\\overline \\partial_b\\phi$ by the identifying map.\n\\end{proof}\n\nUsing \\eqref{DefinitionOfPi} in \\eqref{DefinitionOfdeebarb} and the fact that $\\mathbf i_\\mathcal T\\overline\\Dee\\phi=\\mathcal {L}_\\mathcal T\\phi$ if $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*)$ we get\n\\begin{equation}\\label{FormulaForDeeOnK*}\n\\overline \\partial_b\\phi=\\overline\\Dee\\phi-i \\tilde \\beta_\\mathfrak r\\wedge\\mathcal {L}_\\mathcal T\\phi \\quad \\text{ if }\\phi\\in C^\\infty(\\mathcal N,\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*).\n\\end{equation}\n\nThe $\\overline\\Dee$ operators can be expressed in terms of the $\\overline \\partial_b$ operators. Suppose $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*)$. Then $\\phi=\\phi^0+i \\tilde \\beta_\\mathfrak r\\wedge \\phi^1$ with unique $\\phi^0 \\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q \\overline \\K^*)$ and $\\phi^1 \\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1} \\overline \\K^*)$, and\n\\begin{equation*}\n\\overline\\Dee\\phi^0=\\overline \\partial_b\\phi^0+i\\tilde \\beta_\\mathfrak r\\wedge\\mathcal {L}_\\mathcal T\\phi^0,\n\\end{equation*}\nsee \\eqref{FormulaForDeeOnK*}. Using\n\\begin{equation*}\n\\overline\\Dee\\tilde \\beta_\\mathfrak r=\\frac{\\overline\\Dee a_\\mathfrak r}{i - a_\\mathfrak r}\\wedge\\tilde \\beta_\\mathfrak r\n\\end{equation*}\nand \\eqref{FormulaForDeeOnK*} again we get\n\\begin{equation*}\n\\overline\\Dee(i \\tilde \\beta_\\mathfrak r\\wedge \\phi^1)=\ni\\tilde \\beta_\\mathfrak r\\wedge \\big(-\\frac{\\overline\\Dee a_\\mathfrak r}{i - a_\\mathfrak r}\\wedge\\phi^1 - \\overline\\Dee\\phi^1 \\big)=\ni\\tilde \\beta_\\mathfrak r\\wedge \\big(-\\frac{\\overline \\partial_b a_\\mathfrak r}{i - a_\\mathfrak r}\\wedge\\phi^1 - \\overline \\partial_b\\phi^1 \\big).\n\\end{equation*}\nThis gives\n\\begin{equation}\\label{DeeAsMatrix}\n\\overline\\Dee=\n\\begin{bmatrix}\n\\overline \\partial_b & 0\\\\\n\\mathcal {L}_\\mathcal T & -\\overline \\partial_b -\\dfrac{\\overline \\partial_b a_\\mathfrak r}{i -a_\\mathfrak r}\n\\end{bmatrix}\n:\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*)\\\\ \\oplus \\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K^*)\n\\end{matrix}\n\\to\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K^*)\\\\ \\oplus \\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*)\n\\end{matrix}.\n\\end{equation}\n\nSince $\\mathcal T$ itself is $\\mathcal T$-invariant, $\\mathbf i_\\mathcal T\\mathfrak a_t^*=a_t^*\\mathbf i_\\mathcal T$: the subbundle $\\overline \\K^*$ of $\\smash[t]{\\overline\\Vee}^*$ is invariant under $\\mathfrak a_t^*$ for each $t$. This need not be true of $\\overline \\K_\\mathfrak r$, i.e., the statement that for all $t$, $d\\mathfrak a_t(\\overline \\K_\\mathfrak r)\\subset \\overline \\K_\\mathfrak r$, equivalently,\n\\begin{equation*}\nL\\in C^\\infty(\\mathcal M;\\overline \\K_\\mathfrak r)\\implies [\\mathcal T,L]\\in C^\\infty(\\mathcal M;\\overline \\K_\\mathfrak r),\n\\end{equation*}\nmay fail to hold. Since $\\overline\\Dee\\beta_\\mathfrak r=0$, the formula\n\\begin{equation*}\n0=\\mathcal T\\langle \\beta_\\mathfrak r,L\\rangle - L\\langle \\beta_\\mathfrak r,\\mathcal T\\rangle - \\langle \\beta_\\mathfrak r,[\\mathcal T,L]\\rangle\n\\end{equation*}\nwith $L\\in C^\\infty(\\mathcal N;\\overline \\K_\\mathfrak r)$ gives that $\\overline \\K_\\mathfrak r$ is invariant under $d\\mathfrak a_t$ if and only if $La_\\mathfrak r=0$ for each CR vector field, that is, if and only if $a_\\mathfrak r$ is a CR function. This proves the equivalence between the first and last statements in the following lemma. The third statement is the most useful.\n\n\\begin{lemma}\\label{Invariances}\nLet $\\mathfrak r$ be a defining function for $\\mathcal N$ in $\\mathcal M$ and let $\\overline \\partial_b$ denote the operators of the associated CR complex. The following are equivalent:\n\\begin{enumerate}\n\\item The function $a_\\mathfrak r$ is CR;\n\\item $\\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r=0$;\n\\item $\\mathcal {L}_\\mathcal T\\overline \\partial_b-\\overline \\partial_b\\mathcal {L}_\\mathcal T=0$;\n\\item $\\overline \\K_\\mathfrak r$ is $\\mathcal T$-invariant.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFrom $\\beta_\\mathfrak r=(a_\\mathfrak r-i)i\\tilde \\beta_\\mathfrak r$ and $\\mathcal {L}_\\mathcal T\\beta_\\mathfrak r=\\overline\\Dee a_\\mathfrak r$ we obtain\n\\begin{equation*}\n\\overline\\Dee a_\\mathfrak r=(\\mathcal {L}_\\mathcal T a_\\mathfrak r)i \\tilde \\beta_\\mathfrak r+(a_\\mathfrak r-i)i \\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r,\n\\end{equation*}\nso\n\\begin{equation*}\n\\overline \\partial_b a_\\mathfrak r=\\overline\\Dee a_\\mathfrak r-(\\mathcal {L}_\\mathcal T a_\\mathfrak r) i \\tilde \\beta_\\mathfrak r =(a_\\mathfrak r-i)i \\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r.\n\\end{equation*}\nThus $a_\\mathfrak r$ is CR if and only if $\\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r=0$.\n\nUsing $\\mathcal {L}_\\mathcal T\\overline\\Dee=\\overline\\Dee\\mathcal {L}_\\mathcal T$ and the definition of $\\overline \\partial_b$ we get\n\\begin{equation*}\n\\mathcal {L}_\\mathcal T \\overline \\partial_b\\phi\n=\\mathcal {L}_\\mathcal T(\\overline\\Dee\\phi-i\\tilde \\beta_\\mathfrak r\\wedge \\mathcal {L}_\\mathcal T\\phi)\n=\\overline \\partial_b\\mathcal {L}_\\mathcal T\\phi-i(\\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r)\\wedge \\mathcal {L}_\\mathcal T\\phi\n\\end{equation*}\nfor $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*)$. Thus $\\mathcal {L}_\\mathcal T\\overline \\partial_b-\\overline \\partial_b\\mathcal {L}_\\mathcal T=0$ if and only if $\\mathcal {L}_\\mathcal T\\tilde \\beta_\\mathfrak r=0$.\n\\end{proof}\n\n\n\\begin{lemma}\\label{WithInvariantMetric}\nSuppose that $\\overline\\Vee$ admits a $\\mathcal T$-invariant metric. Then there is a defining function $\\mathfrak r$ for $\\mathcal N$ in $\\mathcal M$ such that $a_\\mathfrak r$ is constant. If $\\mathfrak r$ and $\\mathfrak r'$ are defining functions such that $a_\\mathfrak r$ and $a_{\\mathfrak r'}$ are constant, then $a_\\mathfrak r = a_{\\mathfrak r'}$. This constant will be denoted $\\mathfrak a_\\mathrm{av}$.\n\\end{lemma}\n\n\\begin{proof}\nLet $h$ be a metric as stated. Let $\\mathcal H^{0,1}$ be the subbundle of $\\overline\\Vee$ orthogonal to $\\mathcal T$. This is $\\mathcal T$-invariant, and since the metric is $\\mathcal T$-invariant, $\\mathcal H^{0,1}$ has a $\\mathcal T$-invariant metric. This metric gives canonically a metric on $\\mathcal H^{1,0}=\\overline {\\mathcal H^{0,1}}$. Using the decomposition $\\mathbb C T\\mathcal N=\\mathcal H^{1,0}\\oplus\\mathcal H^{0,1}\\oplus \\Span_\\mathbb C \\mathcal T$ we get a $\\mathcal T$-invariant metric on $\\mathbb C T\\mathcal N$ for which the decomposition is orthogonal. This metric is induced by a Riemannian metric $g$. Let $\\mathfrak m_0$ be the corresponding Riemannian density, which is $\\mathcal T$-invariant because $g$ is. Since $\\overline\\Dee$, $h$, and $\\mathfrak m_0$ are $\\mathcal T$-invariant, so are the formal adjoint $\\overline\\Dee^\\star$ of $\\overline\\Dee$ and the Laplacians of the $\\overline\\Dee$-complex, and if $G$ denotes the Green's operators for these Laplacians, then $G$ is also $\\mathcal T$-invariant, as is the orthogonal projection $\\Pi$ on the space of $\\overline\\Dee$-harmonic forms. Arbitrarily pick a defining function $\\mathfrak r$ for $\\mathcal N$ in $\\mathcal M$. Then\n\\begin{equation*}\na_\\mathfrak r-G\\overline\\Dee^\\star\\overline\\Dee a_\\mathfrak r=\\Pi a_\\mathfrak r\n\\end{equation*}\nwhere $\\Pi a_\\mathfrak r$ is a constant function by Lemma~\\ref{ConstantSolutions}.\nSince $\\beta_\\mathfrak r$ is $\\overline\\Dee$-closed, $\\overline\\Dee a_\\mathfrak r=\\mathcal {L}_\\mathcal T\\beta_\\mathfrak r$. Thus $G\\overline\\Dee^\\star\\overline\\Dee a_\\mathfrak r= \\mathcal T G\\overline\\Dee^\\star \\beta_\\mathfrak r$, and since $a_\\mathfrak r$ is real valued and $\\mathcal T$ is a real vector field,\n\\begin{equation*}\na_\\mathfrak r- \\mathcal T \\Re G\\overline\\Dee^\\star\\beta_\\mathfrak r=\\Re\\Pi a_\\mathfrak r.\n\\end{equation*}\nExtend the function $u=\\Re G\\overline\\Dee^\\star\\beta_\\mathfrak r$ to $\\mathcal M$ as a smooth real-valued function. Then $\\mathfrak r'=e^{-u}\\mathfrak r$ has the required property.\n\nSuppose that $\\mathfrak r$, $\\mathfrak r'$ are defining functions for $\\mathcal N$ in $\\mathcal M$ such that $a_\\mathfrak r$ and $a_{\\mathfrak r'}$ are constant. Then these functions are equal by Proposition \\ref{Averages}. \n\\end{proof}\n\nNote that if for some $\\mathfrak r$, the subbundle $\\overline \\K_\\mathfrak r$ is $\\mathcal T$-invariant and admits a $\\mathcal T$ invariant Hermitian metric, then there is a $\\mathcal T$-invariant metric on $\\overline\\Vee$.\n\n\\medskip\nSuppose now that $\\rho:F\\to\\mathcal M$ is a holomorphic vector bundle over $\\mathcal M$. Using the operators\n\\begin{equation*}\n\\overline\\Dee:C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N)\\to C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N),\n\\end{equation*}\nsee \\eqref{defDeebarE}, define operators\n\\begin{equation}\\label{deebarbE}\n\\cdots\\to C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)\\xrightarrow{\\overline \\partial_b} C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K^*\\otimes F_\\mathcal N)\\to\\cdots\n\\end{equation}\nby\n\\begin{equation*}\n\\overline \\partial_b \\phi =\\Pi_\\mathfrak r\\overline\\Dee\\phi,\\quad\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)\n\\end{equation*}\nwhere $\\Pi_\\mathfrak r$ means $\\Pi_\\mathfrak r\\otimes I$ with $\\Pi_\\mathfrak r$ defined by \\eqref{DefinitionOfPi}. The operators \\eqref{deebarbE} form a complex. Define also\n\\begin{equation*}\n\\mathcal {L}_\\mathcal T = \\mathbf i_\\mathcal T\\overline\\Dee + \\overline\\Dee\\mathbf i_\\mathcal T\n\\end{equation*}\nwhere $\\mathbf i_\\mathcal T$ stands for $\\mathbf i_\\mathcal T\\otimes I$. Then\n\\begin{equation*}\n\\mathbf i_\\mathcal T \\mathcal {L}_\\mathcal T = \\mathcal {L}_\\mathcal T\\mathbf i_\\mathcal T,\\quad \\mathcal {L}_\\mathcal T\\overline\\Dee=\\overline\\Dee\\mathcal {L}_\\mathcal T.\n\\end{equation*}\nThe first of these identities implies that the image of $C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)$ by $\\mathcal {L}_\\mathcal T$ is contained in $C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)$.\nWith these definitions, $\\overline\\Dee$ as an operator\n\\begin{equation*}\n\\overline\\Dee :\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)\\\\ \\oplus \\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K^*\\otimes F_\\mathcal N)\n\\end{matrix}\n\\to\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K^*\\otimes F_\\mathcal N)\\\\ \\oplus \\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F_\\mathcal N)\n\\end{matrix}.\n\\end{equation*}\nis given by the matrix in \\eqref{DeeAsMatrix} with the new meanings for $\\overline \\partial_b$ and $\\mathcal {L}_\\mathcal T$.\n\nAssume that there is a $\\mathcal T$-invariant Riemannian metric on $\\mathcal N$,  that $\\mathfrak r$ has be chosen so that $a_\\mathfrak r$ is constant, that $\\overline \\K_\\mathfrak r$ is orthogonal to $\\mathcal T$, and that $\\mathcal T$ has unit length. Then the term involving $\\overline \\partial_b a_\\mathfrak r$ in the matrix \\eqref{DeeAsMatrix} is absent, and since $\\mathbb D^2=0$,\n\\begin{equation*}\n\\mathcal {L}_\\mathcal T\\overline \\partial_b=\\overline \\partial_b\\mathcal {L}_\\mathcal T.\n\\end{equation*}\nWrite $h_{\\smash[t]{\\overline\\Vee}^*}$ for the metric induced on the bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$ or $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$.\n\nIf $\\eta_\\mu$, $\\mu=1,\\dotsc,k$ is a local frame of $F_\\mathcal N$ over an open set $U\\subset \\mathcal N$ and $\\phi$ is a local section of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N$ over $U$, then for some smooth sections $\\phi^\\mu$ of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$ and $\\omega^\\nu_\\mu$ of $\\smash[t]{\\overline\\Vee}^*$ over $U$,\n\\begin{equation*}\n\\phi= \\sum_\\mu \\phi^\\mu \\otimes \\eta_\\mu,\\quad \\overline\\Dee \\sum_\\mu \\phi^\\mu \\otimes \\eta_\\mu = \\sum_\\nu (\\overline\\Dee\\phi^\\nu + \\sum_\\mu \\omega^\\nu_\\mu \\wedge \\phi^\\mu)\\otimes \\eta_\\nu.\n\\end{equation*}\nThis gives\n\\begin{equation*}\n\\overline \\partial_b \\sum_\\mu \\phi^\\mu \\otimes \\eta_\\mu = \\sum_\\nu (\\overline \\partial_b\\phi^\\nu + \\sum_\\mu \\Pi_\\mathfrak r\\omega^\\nu_\\mu \\wedge \\phi^\\mu)\\otimes \\eta_\\nu\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathcal {L}_\\mathcal T \\sum_\\mu \\phi^\\mu \\otimes \\eta_\\mu = \\sum_\\nu (\\mathcal {L}_\\mathcal T\\phi^\\nu + \\sum_\\mu \\langle\\omega^\\nu_\\mu,\\mathcal T\\rangle \\phi^\\mu)\\otimes \\eta_\\nu.\n\\end{equation*}\n\nSuppose now that $h_F$ is a Hermitian metric on $F$. With this metric and the metric $h_{\\smash[t]{\\overline\\Vee}^*}$ we get Hermitian metrics $h$ on each of the bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N$. If $\\eta_\\mu$ is an orthonormal frame of $F_\\mathcal N$ and $\\phi=\\sum\\phi^\\mu\\otimes\\eta_\\mu$, $\\psi=\\sum\\psi^\\mu\\otimes\\eta_\\mu$ are sections of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N$, then\n\\begin{equation*}\nh(\\phi,\\psi) = \\sum_\\nu h_{\\smash[t]{\\overline\\Vee}^*}(\\phi^\\mu,\\psi^\\mu).\n\\end{equation*}\nTherefore\n\\begin{align*}\nh(\\mathcal {L}_\\mathcal T\\phi,\\psi)&+h(\\phi,\\mathcal {L}_\\mathcal T\\psi)\\\\\n&= \\sum_\\nu h_{\\smash[t]{\\overline\\Vee}^*}(\\mathcal {L}_\\mathcal T\\phi^\\nu + \\sum_\\mu \\langle\\omega^\\nu_\\mu,\\mathcal T\\rangle \\phi^\\mu,\\psi^\\nu)\n+ \\sum_\\mu h_{\\smash[t]{\\overline\\Vee}^*}(\\phi^\\mu, \\mathcal {L}_\\mathcal T\\psi^\\mu + \\langle\\omega^\\mu_\\nu,\\mathcal T\\rangle \\psi^\\nu)\\\\\n&= \\sum_\\nu \\mathcal T h_{\\smash[t]{\\overline\\Vee}^*}(\\phi^\\nu,\\psi^\\nu) + \\sum_{\\mu,\\nu}  ( \\langle\\omega^\\nu_\\mu,\\mathcal T\\rangle + \\overline{\\langle\\omega^\\mu_\\nu,\\mathcal T\\rangle}) h_{\\smash[t]{\\overline\\Vee}^*}(\\phi^\\mu,\\psi^\\nu)\\\\\n&= \\mathcal T h(\\phi,\\psi) + \\sum_{\\mu,\\nu}  ( \\langle\\omega^\\nu_\\mu,\\mathcal T\\rangle + \\overline{\\langle\\omega^\\mu_\\nu,\\mathcal T\\rangle}) h_{\\smash[t]{\\overline\\Vee}^*}(\\phi^\\mu,\\psi^\\nu).\n\\end{align*}\nThus $\\mathcal T h(\\phi,\\psi) = h(\\mathcal {L}_\\mathcal T\\phi,\\psi) +h(\\phi,\\mathcal {L}_\\mathcal T\\psi)$ if and only if\n\\begin{equation}\\label{TangencyOfTE}\n\\langle\\omega^\\nu_\\mu,\\mathcal T\\rangle + \\overline{\\langle\\omega^\\mu_\\nu,\\mathcal T\\rangle}=0\\text{ for all }\\mu,\\nu.\n\\end{equation}\nThis condition is \\eqref{ExactMetricCondition}; just note that by the definition of $\\overline\\Dee$, the forms $(\\Phi^*)^{-1}\\omega^\\nu_\\mu$ in \\eqref{ExactMetricCondition} are the forms that we are denoting $\\omega^\\nu_\\mu$ here. Thus \\eqref{TangencyOfTE} holds if and only if $h_F$ is an exact Hermitian metric, see Definition \\eqref{ExactMetric}.\n\nConsequently,\n\n\\begin{lemma} The statement\n\\begin{equation}\\label{LieTbis}\n\\mathcal T h(\\phi,\\psi) = h(\\mathcal {L}_\\mathcal T\\phi,\\psi)+h(\\phi,\\mathcal {L}_\\mathcal T\\psi)\\quad \\forall \\phi,\\psi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N)\n\\end{equation}\nholds if and only the Hermitian metric $h_F$ is exact.\n\\end{lemma}\n\n\n\n\\section{Spectrum}\\label{sSpectrum}\n\nSuppose that $\\overline\\Vee$ admits an invariant Hermitian metric. Let $\\mathfrak r$ be a defining function for $\\mathcal N$ in $\\mathcal M$ such that $a_\\mathfrak r$ is constant. By  Lemma \\eqref{Invariances} $\\overline \\K_\\mathfrak r$ is $\\mathcal T$-invariant, so the restriction of the metric to this subbundle gives a $\\mathcal T$-invariant metric; we use the induced metric on the bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$ in the following. As in the proof of Lemma~\\ref{WithInvariantMetric}, there is a $\\mathcal T$-invariant density $\\mathfrak m_0$ on $\\mathcal N$. \n\nLet $\\rho:F\\to\\mathcal M$ be a Hermitian holomorphic vector bundle, assume that the Hermitian metric of $F$ is exact, so with the induced metric $h$ on the vector bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F_\\mathcal N$, \\eqref{LieTbis} holds. We will write $F$ in place of $F_\\mathcal N$. \n\nLet $\\overline \\partial_b^\\star$ be the formal adjoint of the ${}^b\\!\\overline \\partial$ operator \\eqref{deebarbE} with respect to the inner on the bundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F$ and the density $\\mathfrak m_0$, and let $\\square_{b,q} = \\overline \\partial_b\\deebarb^\\star + \\overline \\partial_b^\\star\\overline \\partial_b$ be the formal $\\overline \\partial_b$-Laplacian. Since $-i\\mathcal {L}_\\mathcal T$ is formally selfadjoint and commutes with $\\overline \\partial_b$, $\\mathcal {L}_\\mathcal T$ commutes with $\\square_{b,q}$. Let\n\\begin{equation*}\n\\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)=\\ker\\square_{b,q}=\\set{\\phi\\in L^2(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F):\\square_{b,q}\\phi=0}\n\\end{equation*}\nand let\n\\begin{equation*}\n\\Dom_q(\\mathcal {L}_\\mathcal T)=\\set{\\phi\\in \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)\\text{ and }\\mathcal {L}_\\mathcal T\\phi\\in \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)}.\n\\end{equation*}\nThe spaces $\\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)$ may be of infinite dimension, but in any case they are closed subspaces of $L^2(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F)$, so they may be regarded as Hilbert spaces on their own right. If $\\phi\\in \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)$, the condition $\\mathcal {L}_\\mathcal T\\phi\\in \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)$ is equivalent to the condition\n\\begin{equation*}\n\\mathcal {L}_\\mathcal T\\phi\\in L^2(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*\\otimes F).\n\\end{equation*}\nSo we have a closed operator\n\\begin{equation}\\label{LieOnB-Harmonic}\n-i \\mathcal {L}_\\mathcal T:\\Dom_q(\\mathcal {L}_\\mathcal T)\\subset \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)\\to \\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F).\n\\end{equation}\nThe fact that $\\square_{b,q}-\\mathcal {L}_\\mathcal T^2$ is elliptic, symmetric, and commutes with $\\mathcal {L}_\\mathcal T$ implies that \\eqref{LieOnB-Harmonic} is a selfadjoint Fredholm operator with discrete spectrum (see \\cite[Theorem 2.5]{Me9}).\n\n\\begin{definition}\\label{BCohomologyWithCoeffs}\nLet $\\spec^q_0(-i \\mathcal {L}_\\mathcal T)$ be the spectrum of the operator \\eqref{LieOnB-Harmonic}, and let $\\mathscr H^q_{\\overline \\partial_b,\\tau}(\\mathcal N;F)$ be the eigenspace of $-i\\mathcal {L}_\\mathcal T$ in $\\mathscr H^q_{\\overline \\partial_b}(\\mathcal N;F)$ corresponding to the eigenvalue $\\tau$.\n\\end{definition}\n\nLet $\\pmb\\tau$ denote the principal symbol of $-i\\mathcal T$. Then the principal symbol of $\\mathcal {L}_\\mathcal T$ acting on sections of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K^*$ is $\\pmb\\tauI$. Because $\\square_{b,q}-\\mathcal {L}_\\mathcal T^2$ is elliptic, $\\Char(\\square_{b,q})$, the characteristic variety of $\\square_{b,q}$, lies in $\\pmb\\tau\\ne 0$. Let\n\\begin{equation*}\n\\Char^\\pm(\\square_{b,q})=\\set{\\nu\\in\\Char (\\square_{b,q}):\\pmb \\tau(\\nu)\\gtrless 0}.\n\\end{equation*}\nBy \\cite[Theorem 4.1]{Me9}, if $\\square_{b,q}$ is microlocally hypoelliptic on $\\Char^\\pm(\\square_{b,q})$, then \n\\begin{equation*}\n\\set{\\tau\\in \\spec^q_0(-i \\mathcal {L}_\\mathcal T):\\tau\\gtrless0}\n\\end{equation*}\nis finite. We should perhaps point out that $\\Char(\\square_{b,q})$ is equal to the characteristic variety, $\\Char(\\overline \\K_r)$, of the CR structure. \n\nAs a special case consider the situation where $F$ is the trivial line bundle. Let $\\theta_\\mathfrak r$ be the real $1$-form on $\\mathcal N$ which vanishes on $\\overline \\K_\\mathfrak r$ and satisfies $\\langle\\theta_\\mathfrak r,\\mathcal T\\rangle =1$; thus $\\theta_\\mathfrak r$ is smooth, spans $\\Char(\\overline \\K_\\mathfrak r)$, and has values in $\\Char^+(\\overline \\K_\\mathfrak r)$. The Levi form of the structure is\n\\begin{equation*}\n\\Levi_{\\theta_\\mathfrak r}(v,w)=-i d\\theta_\\mathfrak r(v,\\overline w), \\quad v,\\ w\\in \\mathcal K_{\\mathfrak r,p},\\ p\\in \\mathcal N.\n\\end{equation*}\nSuppose that $\\Levi_{\\theta_\\mathfrak r}$ is nondegenerate, with $k$ positive and $n-k$ negative eigenvalues. It is well known that then $\\square_{b,q}$ is microlocally hypoelliptic at $\\nu\\in\\Char\\mathcal K_\\mathfrak r$ for all $q$ except if $q=k$ and $\\pmb \\tau(\\nu)<0$ or if $q=n-k$ and $\\pmb \\tau(\\nu)>0$.\n\nThen the already mentioned Theorem~4.1 of \\cite{Me9} gives:\n\n\\begin{theorem}[{\\cite[Theorem 6.1]{Me9}}]\\label{WeakVanishing}\nSuppose that $\\overline\\Vee$ admits a Hermitian metric and that for some defining function $\\mathfrak r$ such that $\\mathfrak a_\\mathfrak r$ is constant, $\\Levi_{\\theta_\\mathfrak r}$ is nondegenerate with $k$ positive and $n-k$ negative eigenvalues. Then\n\\begin{enumerate}\n\\item $\\spec_0^q(-i \\mathcal {L}_\\mathcal T)$ is finite if $q\\ne k,\\ n-k$;\n\\item $\\spec_0^k(-i\\mathcal {L}_\\mathcal T)$ contains only finitely many positive elements, and \\item $\\spec_0^{n-k}(-i\\mathcal {L}_\\mathcal T)$ contains only finitely many negative elements.\n\\end{enumerate}\n\\end{theorem}\n\n\\section{Indicial cohomology}\\label{sIndicialCohomology}\n\nSuppose that there is a $\\mathcal T$-invariant Hermitian metric $\\tilde h$ on $\\overline\\Vee$. By Lemma~\\ref{WithInvariantMetric} there is a defining function $\\mathfrak r$ such that $\\langle\\beta_\\mathfrak r,\\mathcal T\\rangle$ is constant, equal to $a_\\mathrm{av}-i$. Therefore $\\overline \\K_\\mathfrak r$ is $\\mathcal T$-invariant. Let $h$ be the metric on $\\overline\\Vee$ which coincides with $\\tilde h$ on $\\overline \\K_\\mathfrak r$, makes the decomposition $\\overline\\Vee=\\overline \\K_\\mathfrak r\\oplus \\Span_\\mathbb C\\mathcal T$ orthogonal, and for which $\\mathcal T$ has unit length. The metric $h$ is $\\mathcal T$-invariant. We fix $\\mathfrak r$ and such a metric, and let $\\mathfrak m_0$ be the Riemannian measure associated with $h$. The decomposition \\eqref{DecompositionOfVeebar} of $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*$ is an orthogonal decomposition.\n\nRecall that $\\overline\\D(\\sigma)\\phi=\\overline\\Dee \\phi+i\\sigma\\beta_\\mathfrak r\\wedge \\phi$. Since $a_\\mathfrak r=a_\\mathrm{av}$ is constant (in particular CR),\n\\begin{equation*}\n\\overline\\D(\\sigma)(\\phi^0 + i \\tilde \\beta_\\mathfrak r\\wedge \\phi^1) =\n\\overline \\partial_b\\phi_0+i\\tilde \\beta_\\mathfrak r\\wedge \\big[\\big(\\mathcal {L}_\\mathcal T+(1+i a_\\mathrm{av})\\sigma\\big)\\phi^0 -\\overline \\partial_b\\phi^1\\big]\n\\end{equation*}\nif $\\phi^0\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*)$ and $\\phi^1\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K_\\mathfrak r^*)$. So $\\overline\\D(\\sigma)$ can be regarded as the operator\n\\begin{equation}\\label{CalDasMarix}\n\\overline\\D(\\sigma)=\\begin{bmatrix}\n\\overline \\partial_b & 0\\\\\n\\mathcal {L}_\\mathcal T+(1+i a_\\mathrm{av})\\sigma & -\\overline \\partial_b\n\\end{bmatrix}:\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*) \\\\ \\oplus\\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K_\\mathfrak r^*)\n\\end{matrix}\n\\to\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K_\\mathfrak r^*) \\\\ \\oplus\\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q}\\overline \\K_\\mathfrak r^*).\n\\end{matrix}\n\\end{equation}\nSince the subbundles $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r$ and $\\tilde \\beta \\wedge \\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1} \\overline \\K_\\mathfrak r$ are orthogonal with respect to the metric induced by $h$ on $\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline\\Vee$, the formal adjoint of $\\overline\\D(\\sigma)$ with respect to this metric and the density $\\mathfrak m_0$ is\n\\begin{equation*}\n\\overline\\D(\\sigma)^\\star=\n\\begin{bmatrix}\n\\overline \\partial_b^\\star & -\\mathcal {L}_\\mathcal T+(1-i a_\\mathrm{av})\\overline \\sigma \\\\\n0& -\\overline \\partial_b^\\star\n\\end{bmatrix}:\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q+1}\\overline \\K_\\mathfrak r^*) \\\\ \\oplus\\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q}\\overline \\K_\\mathfrak r^*)\n\\end{matrix}\n\\to\n\\begin{matrix}\nC^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*) \\\\ \\oplus\\\\ C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K_\\mathfrak r^*)\n\\end{matrix}\n\\end{equation*}\nwhere $\\overline \\partial_b^\\star$ is the formal adjoint of $\\overline \\partial_b$. So the Laplacian, $\\square_{\\overline\\D(\\sigma),q}$, of the $\\overline\\D(\\sigma)$-complex is the diagonal operator with diagonal entries $P_q(\\sigma)$, $P_{q-1}(\\sigma)$ where\n\\begin{equation*}\nP_q(\\sigma)=\\square_{b,q}+(\\mathcal {L}_\\mathcal T+(1+i a_\\mathrm{av})\\sigma)(-\\mathcal {L}_\\mathcal T+(1-i a_\\mathrm{av})\\overline \\sigma)\n\\end{equation*}\nacting on $C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*)$ and $P_{q-1}(\\sigma)$ is the ``same'' operator, acting on sections of $\\raise2ex\\hbox{$\\mathchar\"0356$}^{q-1}\\overline \\K_\\mathfrak r^*$; recall that $\\mathcal {L}_\\mathcal T$ commutes with $\\overline \\partial_b$ and since $\\mathcal {L}_\\mathcal T^\\star=-\\mathcal {L}_\\mathcal T$, also with $\\overline \\partial_b^\\star$, and that $a_\\mathrm{av}$ is constant. Note that $P_q(\\sigma)$ is an elliptic operator.\n\nSuppose that $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*)$ is a nonzero element of $\\ker P_q(\\sigma)$; the complex number $\\sigma$ is fixed. Since $P_q(\\sigma)$ is elliptic, $\\ker P_q(\\sigma)$ is a finite dimensional space, invariant under $-i \\mathcal {L}_\\mathcal T$ since the latter operator commutes with $P_q(\\sigma)$. As an operator on $\\ker P_q(\\sigma)$, $-i \\mathcal {L}_\\mathcal T$ is selfadjoint, so there is a decomposition of $\\ker P_q(\\sigma)$ into eigenspaces of $-i\\mathcal {L}_\\mathcal T$. Thus\n\\begin{equation*}\n\\phi=\\sum_{j=1}^N \\phi_j, \\quad -i \\mathcal {L}_\\mathcal T\\phi_j=\\tau_j\\phi_j\n\\end{equation*}\nwhere the $\\tau_j$ are distinct real numbers and $\\phi_j\\in \\ker P_q(\\sigma)$, $\\phi_j\\ne 0$. In particular,\n\\begin{equation*}\n\\square_{b,q}\\phi_j + (\\mathcal {L}_\\mathcal T+(1+i a_\\mathrm{av})\\sigma)(-\\mathcal {L}_\\mathcal T+(1-i a_\\mathrm{av})\\overline \\sigma)\\phi_j = 0,\n\\end{equation*}\nfor each $j$, that is,\n\\begin{equation*}\n\\square_{b,q}\\phi_j+ |i\\tau_j+(1+i a_\\mathrm{av})\\sigma|^2\\phi_j = 0.\n\\end{equation*}\nSince $\\square_{b,q}$ is a nonnegative operator and $\\phi_j\\ne 0$, $i\\tau_j+(1+i a_\\mathrm{av})\\sigma=0$ and $\\phi_j\\in \\ker\\square_{b,q}$. Since $\\sigma$ is fixed, all $\\tau_j$ are equal, which means that $N=1$. Conversely, if $\\phi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\overline \\K_\\mathfrak r^*)$ belongs to $\\ker \\square_{b,q}$ and $-i\\mathcal {L}_\\mathcal T\\phi=\\tau \\phi$, then $P_q(\\sigma)\\phi=0$ with $\\sigma$ such that $\\tau=(i-a_\\mathrm{av})\\sigma$.\n\nLet $\\mathscr H^q_{\\overline\\D(\\sigma)}(\\mathcal N)$ be the kernel of $\\square_{\\overline\\D(\\sigma),q}$.\n\n\\begin{theorem}\\label{TheCohomology}\nSuppose that $\\overline\\Vee$ admits a $\\mathcal T$-invariant metric and let $\\mathfrak r$ be a defining function for $\\mathcal N$ in $\\mathcal M$ such that $\\langle \\beta_\\mathfrak r,\\mathcal T\\rangle=a_\\mathrm{av}-i$ is constant. Then\n\\begin{equation*}\n\\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial) = (i-a_\\mathrm{av})^{-1}\\spec_0^q(-i \\mathcal {L}_\\mathcal T)\\cup (i-a_\\mathrm{av})^{-1}\\spec_0^{q-1}(-i \\mathcal {L}_\\mathcal T),\n\\end{equation*}\nand if $\\sigma\\in \\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial)$, then, with the notation in Definition \\ref{BCohomologyWithCoeffs}\n\\begin{equation*}\n\\mathscr H^q_{\\overline\\D(\\sigma)}(\\mathcal N)=\\mathscr H^q_{\\overline \\partial_b,\\tau(\\sigma)}(\\mathcal N)\\oplus \\mathscr H^{q-1}_{\\overline \\partial_b,\\tau(\\sigma)}(\\mathcal N)\n\\end{equation*}\nwith $\\tau(\\sigma)=(i-a_\\mathrm{av})\\sigma$.\n\\end{theorem}\n\nIf the CR structure $\\overline \\mathcal K_\\mathfrak r$ is nondegenerate, Proposition~\\ref{WeakVanishing} gives more specific information on $\\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial)$. In particular,\n\\begin{proposition}\nWith the hypotheses of Theorem~\\ref{TheCohomology}, suppose that $\\Levi_{\\theta_\\mathfrak r}$ is nondegenerate with $k$ positive and $n-k$ negative eigenvalues. If $k>0$, then $\\spec_{b,\\mathcal N}^0 \\subset \\set{\\sigma\\in \\mathbb C:\\Im\\sigma\\leq 0}$, and if $n-k>0$, then $\\spec_{b,\\mathcal N}^0({}^b\\!\\overline \\partial) \\subset \\set{\\sigma\\in \\mathbb C:\\Im\\sigma\\geq 0}$.\n\\end{proposition}\n\n\\begin{remark}\nThe $b$-spectrum of the Laplacian of the ${}^b\\!\\overline \\partial$-complex in any degree can be described explicitly in terms of the joint spectra $\\spec(-i\\mathcal {L}_\\mathcal T,\\square_{b,q})$. We briefly indicate how. With the metric $h$ and defining function $\\mathfrak r$ as in the first paragraph of this section, suppose that $h$ is extended to a metric on ${}^b{\\!}T^{0,1}\\mathcal M$. This gives a Riemannian $b$-metric on $\\mathcal M$ that in turn gives a $b$-density $\\mathfrak m$ on $\\mathcal M$. With these we get formal adjoints ${}^b\\!\\overline \\partial^\\star$ whose indicial families $\\overline\\D^\\star(\\sigma)$ are related to those of ${}^b\\!\\overline \\partial$ by\n\\begin{equation*}\n\\overline\\D^\\star(\\sigma) = \\widehat{{}^b\\!\\overline \\partial^\\star}(\\sigma)= [\\widehat{{}^b\\!\\overline \\partial}(\\overline \\sigma)]^\\star = \\overline\\D(\\overline \\sigma)^\\star.\n\\end{equation*}\nBy \\eqref{CalDasMarix},\n\\begin{equation*}\n\\overline\\D^\\star(\\sigma)=\n\\begin{bmatrix}\n\\overline \\partial_b^\\star & -\\mathcal {L}_\\mathcal T+(1-i a_\\mathrm{av}) \\sigma \\\\\n0& -\\overline \\partial_b^\\star\n\\end{bmatrix}.\n\\end{equation*}\nUsing this one obtains that the indicial family of the Laplacian $\\square_q$ of the ${}^b\\!\\overline \\partial$-complex in degree $q$ is a diagonal operator with diagonal entries $P'_q(\\sigma)$, $P'_{q-1}(\\sigma)$ with\n\\begin{equation*}\nP'_q(\\sigma)=\\square_{b,q}+(\\mathcal {L}_\\mathcal T+(1+i a_\\mathrm{av})\\sigma)(-\\mathcal {L}_\\mathcal T+(1-i a_\\mathrm{av})\\sigma)\n\\end{equation*}\nand the analogous operator in degree $q-1$. The set $\\spec_b(\\square_q)$ is the set of values of $\\sigma$ for which either $P'_q(\\sigma)$ or $P'_{q-1}(\\sigma)$ is not injective. These points can written in terms of the points $\\spec(-i\\mathcal {L}_\\mathcal T,\\square_{b})$ as asserted. In particular one gets\n\\begin{equation*}\n\\spec_b(\\square_q)\\subset \\set{\\sigma: |\\Re \\sigma|\\leq |a_\\mathrm{av}||\\Im\\sigma|}\n\\end{equation*}\nwith $\\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial)$ being a subset of the boundary of the set on the right.\n\\end{remark}\n\n\nWe now discuss the indicial cohomology sheaf of ${}^b\\!\\overline \\partial$, see Definition \\ref{CohomologySheafs}. We will show:\n\n\\begin{proposition}\\label{bdeebarCohomologySheaf}\nLet $\\sigma_0\\in \\spec_{b,\\mathcal N}^q({}^b\\!\\overline \\partial)$. Every element of the stalk of $\\mathfrak H^q_{{}^b\\!\\overline \\partial}(\\mathcal N)$ at $\\sigma_0$ has a representative of the form\n\\begin{equation*}\n\\frac{1}{\\sigma-\\sigma_0}\n\\begin{bmatrix}\n\\phi^0 \\\\ 0\n\\end{bmatrix}\n\\end{equation*}\nwhere $\\phi^0\\in \\mathscr H^q_{\\overline \\partial_b,\\tau_0}(\\mathcal N)$, $\\tau_0= (i-a_\\mathrm{av})\\sigma_0$.\n\\end{proposition}\n\n\\begin{proof}\nLet\n\\begin{equation}\\label{RepresentativeA}\n\\phi(\\sigma)=\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\phi^0_k \\\\ \\phi^1_k\n\\end{bmatrix}\n\\end{equation}\nrepresent an element in the stalk at $\\sigma_0$ of the sheaf of germs of $C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^q\\smash[t]{\\overline\\Vee}^*\\otimes F)$-valued meromorphic functions on $\\mathbb C$ modulo the subsheaf of holomorphic elements. Letting $\\alpha=1+i a_\\mathrm{av}$ we have\n\\begin{equation*}\n\\overline\\D(\\sigma)\\phi(\\sigma) =\n\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\overline \\partial_b \\phi^0_k \\\\\n\\big(\\mathcal {L}_\\mathcal T + \\alpha\\sigma_0\\big) \\phi^0_k-\\overline \\partial_b \\phi^1_k\n\\end{bmatrix}\n+ \\sum_{k=0}^{\\mu-1} \\frac{\\alpha}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n0 \\\\\n\\phi^0_{k+1}\n\\end{bmatrix},\n\\end{equation*}\nso the condition that $\\overline\\D(\\sigma)\\phi(\\sigma)$ is holomorphic is equivalent to\n\\begin{equation}\\label{TopEqs}\n\\overline \\partial_b\\phi^0_k=0,\\ k=1,\\dotsc,\\mu\n\\end{equation}\nand\n\\begin{equation}\\label{BottomEqs}\n\\begin{gathered}\n(\\mathcal {L}_\\mathcal T + \\alpha\\sigma_0) \\phi^0_\\mu-\\overline \\partial_b \\phi^1_\\mu = 0,\\\\ (\\mathcal {L}_\\mathcal T + \\alpha\\sigma_0) \\phi^0_k-\\overline \\partial_b \\phi^1_k + \\alpha\\phi^0_{k+1}=0,\\ k=1,\\dotsc,\\mu-1.\n\\end{gathered}\n\\end{equation}\n\nLet $P_{q'}=\\square_{b,q'}-\\mathcal {L}_\\mathcal T^2$ in any degree $q'$. For any $(\\tau,\\lambda)\\in \\mathbb R^2$ and $q'$ let\n\\begin{equation*}\n\\mathcal E^{q'}_{\\tau,\\lambda}=\\set{\\psi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q'}\\smash[t]{\\overline\\Vee}^*\\otimes F):P_{q'}\\psi=\\lambda\\psi,\\ -i \\mathcal {L}_\\mathcal T \\psi=\\tau\\psi}.\n\\end{equation*}\nThis space is zero if $(\\tau,\\lambda)$ is not in the joint spectrum $\\Sigma^{q'} = \\spec^{q'}(-i\\mathcal {L}_\\mathcal T,P_{q'})$. Each $\\phi^i_k$ decomposes as a sum of elements in the spaces $\\mathcal E^{q-i}_{\\tau,\\lambda}$, $(\\tau,\\lambda)\\in \\Sigma^{q-i}$. Suppose that already $\\phi^i_k\\in \\mathcal E^{q-i}_{\\tau,\\lambda}$:\n\\begin{equation*}\nP_{q-i}\\phi^i_k=\\lambda \\phi^i_k,\\quad -i\\mathcal {L}_\\mathcal T \\phi^i_k=\\tau\\phi^i_k,\\quad i=0,1,\\ k=1,\\dotsc,\\mu.\n\\end{equation*}\nThen \\eqref{BottomEqs} becomes\n\\begin{equation}\\label{BottomEqsEigen}\n\\begin{gathered}\n(i\\tau + \\alpha\\sigma_0) \\phi^0_\\mu-\\overline \\partial_b \\phi^1_\\mu = 0,\\\\ (i\\tau + \\alpha\\sigma_0) \\phi^0_k-\\overline \\partial_b \\phi^1_k + \\alpha\\phi^0_{k+1}=0,\\ k=1,\\dotsc,\\mu-1.\n\\end{gathered}\n\\end{equation}\nIf $\\tau\\ne \\tau_0$, then $i\\tau + \\alpha\\sigma_0\\ne 0$, and we get $\\phi^0_k=\\overline \\partial_b \\psi^0_k$ for all $k$ with\n\\begin{equation*}\n\\psi^0_k = \\sum_{j=0}^{\\mu-k} \\frac{(-\\alpha)^j}{(i\\tau + \\alpha\\sigma_0)^{j+1}}\\phi^1_{k+j}.\n\\end{equation*}\nTrivially\n\\begin{equation*}\n\\big(\\mathcal {L}_\\mathcal T + \\alpha\\sigma_0\\big) \\psi^0_\\mu = \\phi^1_\\mu\n\\end{equation*}\nand also\n\\begin{equation*}\n\\big(\\mathcal {L}_\\mathcal T + \\alpha\\sigma_0\\big) \\psi^0_k + \\alpha\\psi^0_{k+1} = \\phi^1_k,\\quad k=1,\\dotsc,\\mu-1,\n\\end{equation*}\nso\n\\begin{equation*}\n\\phi(\\sigma) - \\overline\\D(\\sigma)\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix} \\psi^0_k\\\\\n0\n\\end{bmatrix}=0\n\\end{equation*}\nmodulo an entire element.\n\nSuppose now that the $\\phi^i_k$ are arbitrary and satisfy \\eqref{TopEqs}-\\eqref{BottomEqs}. The sum\n\\begin{equation}\\label{FourierSeries}\n\\phi^i_k=\\sum_{(\\tau,\\lambda)\\in \\Sigma^{q-i}} \\phi^i_{k,\\tau,\\lambda},\\quad \\phi^i_{k,\\tau,\\lambda}\\in \\mathcal E^{q-i}_{\\tau,\\lambda}\n\\end{equation}\nconverges in $C^\\infty$, indeed for each $N$ there is $C_{i,k,N}$ such that\n\\begin{equation}\\label{RapidDecay}\n\\sup_{p\\in \\mathcal N}\\|\\phi^i_{k,\\tau,\\lambda}(p)\\|\\leq C_{i,k,N}(1+\\lambda)^{-N}\\quad\\text{ for all }\\tau,\\lambda.\n\\end{equation}\nSince $\\overline\\D(\\sigma)$ preserves the spaces $\\mathcal E^q_{\\tau,\\lambda}\\oplus \\mathcal E^{q-1}_{\\tau,\\lambda}$, the relations \\eqref{BottomEqsEigen} hold for the $\\phi^i_{k,\\tau,\\lambda}$ for each $(\\tau,\\lambda)$. Therefore,\nwith\n\\begin{equation}\\label{PsiFourierSeries}\n\\psi^0_k = \\sum_{\\substack{(\\tau,\\lambda)\\in\\Sigma^{q-1}\\\\\\tau\\ne \\tau_0}}\\sum_{j=0}^{\\mu-k} \\frac{(-\\alpha)^j}{(i\\tau + \\alpha\\sigma_0)^{j+1}}\\phi^1_{k+j,\\tau,\\lambda}\n\\end{equation}\nwe have formally that\n\\begin{equation*}\n\\phi(\\sigma)-\\overline\\D(\\sigma) \\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix} \\psi_k\\\\\n0\n\\end{bmatrix} = \\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix} \\tilde \\phi^0_k\\\\\n\\tilde \\phi^1_k\n\\end{bmatrix}\n\\end{equation*}\nwith\n\\begin{equation}\\label{AtTau0}\n\\tilde \\phi^i_k=\\sum_{\\substack{(\\tau,\\lambda)\\in \\Sigma^{q-1}\\\\\\tau=\\tau_0}} \\phi^i_{k,\\tau,\\lambda},\\quad \\phi^i_{k,\\tau,\\lambda}\\in \\mathcal E^{q-i}_{\\tau,\\lambda}.\n\\end{equation}\nHowever, the convergence in $C^\\infty$ of the series \\eqref{PsiFourierSeries} is questionable since there may be a sequence $\\set{(\\tau_\\ell,\\lambda_\\ell)}_{\\ell=1}^\\infty \\subset \\spec(-i\\mathcal {L}_\\mathcal T,P_{q-1})$ of distinct points such that $\\tau_\\ell\\to\\tau_0$ as $\\ell\\to\\infty$, so that the denominators $i \\tau_\\ell+\\alpha\\sigma_0$ in the formula for $\\psi^0_k$ tend to zero so fast that for some nonnegative $N$, $\\lambda_\\ell^{-N}\/(i \\tau_\\ell+\\alpha\\sigma_0)$ is unbounded. To resolve this difficulty we will first show that $\\phi(\\sigma)$ is $\\overline\\D(\\sigma)$-cohomologous (modulo holomorphic terms) to an element of the same form as $\\phi(\\sigma)$ for which in the series \\eqref{FourierSeries} the terms $\\phi^i_{k,\\tau,\\lambda}$ vanish if $\\lambda-\\tau^2>\\varepsilon$; the number $\\varepsilon>0$ is chosen so that\n\\begin{equation}\\label{ChoiceOfEps}\n(\\tau_0,\\lambda)\\in \\Sigma^q\\cup\\Sigma^{q-1} \\implies \\lambda=\\tau_0^2\\text{ or }\\lambda\\geq\\tau_0^2+\\varepsilon.\n\\end{equation}\nRecall that $\\spec^{q'}(-i\\mathcal {L}_\\mathcal T,P_{q'})\\subset \\set{(\\tau,\\lambda):\\lambda\\geq \\tau^2}$.\n\nFor any $V\\subset \\bigcup_{q'}\\Sigma^{q'}$ let\n\\begin{equation*}\n\\Pi^{q'}_V:L^2(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q'}\\smash[t]{\\overline\\Vee}^*\\otimes F)\\to L^2(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q'}\\smash[t]{\\overline\\Vee}^*\\otimes F)\n\\end{equation*}\nbe the orthogonal projection on $\\bigoplus_{(\\tau,\\lambda)\\in V} \\mathcal E^{q'}_{\\tau,\\lambda}$. If $\\psi\\in C^\\infty(\\mathcal N;\\raise2ex\\hbox{$\\mathchar\"0356$}^{q'}\\smash[t]{\\overline\\Vee}^*\\otimes F)$, then the series\n\\begin{equation*}\n\\Pi^{q'}_V \\psi=\\sum_{(\\tau,\\lambda)\\in V}\\psi_{\\tau,\\lambda}, \\quad\\psi_{\\tau,\\lambda}\\in \\mathcal E^{q'}_{\\tau,\\lambda}\n\\end{equation*}\nconverges in $C^\\infty$. It follows that $\\square_{b,q'}$ and $\\mathcal {L}_\\mathcal T$ commute with $\\Pi^{q'}_V$ and that $\\overline \\partial_b\\Pi^{q'}_V=\\Pi^{q'+1}_V\\overline \\partial_b$. Since the $\\Pi^{q'}_V$ are selfadjoint, also $\\overline \\partial_b^\\star\\Pi^{q'+1}_V = \\Pi^{q'}_V\\overline \\partial_b^\\star$.\n\nLet\n\\begin{equation*}\nU=\\set{(\\tau,\\lambda)\\in \\Sigma^q\\cup \\Sigma^{q-1}:\\lambda <\\tau^2+\\varepsilon},\\quad U^c=\\Sigma^q\\cup \\Sigma^{q-1}\\backslash U.\n\\end{equation*}\nThen, for any sequence $\\set{(\\tau_\\ell,\\lambda_\\ell)}\\subset U$ of distinct points we have $|\\tau_\\ell|\\to\\infty$ as $\\ell\\to\\infty$. Define\n\\begin{equation*}\nG^{q'}_{U^c}\\psi =\\sum_{(\\tau,\\lambda)\\in U^c}\\frac{1}{\\lambda-\\tau^2}\\psi_{\\tau,\\lambda}\n\\end{equation*}\nIn this definition the denominators $\\lambda-\\tau^2$ are bounded from below by $\\varepsilon$, so $G^{q'}_{U^c}$ is a bounded operator in $L^2$ and maps smooth sections to smooth sections because the components of such sections satisfy estimates as in \\eqref{RapidDecay}. The operators are analogous to Green operators: we have\n\\begin{equation}\\label{PseudoGreen1}\n\\square_{b,q'}G^{q'}_{U^c} = G^{q'}_{U^c} \\square_{b,q'}=I -\\Pi^{q'}_U\n\\end{equation}\nso if $\\overline \\partial_b\\psi=0$, then\n\\begin{equation}\\label{PseudoGreen2}\n\\square_{b,q'}G^{q'}_{U^c}\\psi = \\overline \\partial_b\\deebarb^\\star G^{q'}_{U^c}\\psi\n\\end{equation}\nsince $\\overline \\partial_b G^{q'}_{U^c}= G^{q'+1}_{U^c}\\overline \\partial_b$.\n\nWrite $\\phi(\\sigma)$ in \\eqref{RepresentativeA} as\n\\begin{equation*}\n\\phi(\\sigma) =\\Pi_{U^c}\\phi(\\sigma)+ \\Pi_{U}\\phi(\\sigma)\n\\end{equation*}\nwhere\n\\begin{equation*}\n\\Pi_{U^c}\\phi(\\sigma) =\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\Pi^q_{U^c}\\phi^0_k \\\\ \\Pi^{q-1}_{U^c}\\phi^1_k\n\\end{bmatrix},\\quad\n\\Pi_U\\phi(\\sigma)=\n\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\Pi^q_U\\phi^0_k \\\\ \\Pi^{q-1}_U\\phi^1_k\n\\end{bmatrix}.\n\\end{equation*}\nSince $\\overline\\D(\\sigma)\\phi(\\sigma)$ is holomorphic, so are $\\overline\\D(\\sigma)\\Pi_{U^c}\\phi(\\sigma)$ and $\\overline\\D(\\sigma)\\Pi_U\\phi(\\sigma)$.\n\nWe show that $\\Pi_{U^c}\\phi(\\sigma)$ is exact modulo holomorphic functions. Using \\eqref{TopEqs}, \\eqref{PseudoGreen1}, and \\eqref{PseudoGreen2}, $\\Pi^q_{U^c}\\phi^0_k = \\overline \\partial_b^\\star \\overline \\partial_b \\Pi^q_{U^c}\\phi^0_k$. Then\n\\begin{equation*}\n\\Pi_{U^c}\\phi(\\sigma)-\\overline\\D(\\sigma)\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\overline \\partial_b^\\star G^q_{U^c}\\Pi^q_U\\phi^0_k \\\\ 0\n\\end{bmatrix}\n=\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n0 \\\\ \\hat\\phi^1_k\n\\end{bmatrix}\n\\end{equation*}\nmodulo a holomorphic term for some $\\hat\\phi^1_k$ with $\\Pi^{q-1}_{U^c}\\hat\\phi^1_k=\\hat\\phi^1_k$. The element on the right is $\\overline\\D(\\sigma)$-closed modulo a holomorphic function, so its components satisfy \\eqref{TopEqs}, \\eqref{BottomEqs}, which give that the $\\tilde\\phi^1_k$ are $\\overline \\partial_b$-closed. Using again \\eqref{PseudoGreen1} and \\eqref{PseudoGreen2} we see that $\\Pi_{U^c}\\phi(\\sigma)$ represent an exact element.\n\nWe may thus assume that $\\Pi^q_{U^c}\\phi(\\sigma)=0$. If this is the case, then the series \\eqref{PsiFourierSeries} converges in $C^\\infty$, so $\\phi(\\sigma)$ is cohomologous to the element\n\\begin{equation*}\n\\tilde\\phi(\\sigma)=\\sum_{k=1}^\\mu \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\tilde \\phi^0_k \\\\ \\tilde \\phi^1_k\n\\end{bmatrix}\n\\end{equation*}\nwhere the $\\tilde \\phi^i_k$ are given by \\eqref{AtTau0} and satisfy $\\Pi^{q-i}_{U^c}\\tilde \\phi^i_k=0$. By \\eqref{ChoiceOfEps}, $\\tilde \\phi^i_k\\in \\mathcal E^{q-i}_{\\tau_0,\\tau_0^2}$. In particular, $\\square_{b,q-i}\\phi^i_k=0$.\n\nAssuming now that already $\\phi^i_k\\in \\mathcal E^{q-i}_{\\tau_0,\\tau_0^2}$, the formulas \\eqref{BottomEqsEigen} give (since $\\tau=\\tau_0$ and $i\\tau_0+\\alpha\\sigma_0=0$)\n\\begin{equation*}\n\\overline \\partial_b\\phi^1_\\mu=0,\\quad \\phi^0_k = \\overline \\partial_b \\frac{1}{\\alpha}\\phi^1_{k-1},\\ k=2,\\dotsc,\\mu.\n\\end{equation*}\nThen\n\\begin{equation*}\n\\phi(\\sigma)-\\frac {1}{\\alpha}\\overline\\D(\\sigma)\\sum_{k=2}^{\\mu+1} \\frac{1}{(\\sigma-\\sigma_0)^k}\n\\begin{bmatrix}\n\\phi^1_{k-1} \\\\ 0\n\\end{bmatrix}\n=\\frac{1}{\\sigma-\\sigma_0}\n\\begin{bmatrix}\n\\phi^0_1 \\\\ 0\n\\end{bmatrix}\n\\end{equation*}\nwith $\\square_{b,q}\\phi^0_1=0$.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}