diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzaiuk" "b/data_all_eng_slimpj/shuffled/split2/finalzzaiuk" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzaiuk" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{intro}\n\nLet $\\fg$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $\\fh$ and the Weyl group $W$. Let $P_+$ be the set of dominant integral weights. For $\\lambda \\in P_+$, let $L(\\lambda)$ be the irreducible, integrable, highest weight representation of $\\fg$ with highest weight $\\lambda$. For a positive integer $s$, define the {\\em saturated tensor semigroup} as\n\\begin{align*}\n\\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\,\nN>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}.\n\\end{align*}\nThe aim of this paper is to begin a systematic study of $\\Gamma_s$ in the infinite dimensional symmetrizable Kac-Moody case. \nIn this paper, we produce a set of necessary inequalities satisfied by $\\Gamma_s$, which we describe now. Let $X =G^{\\min}\/B$ be the standard full KM-flag variety\nassociated to $\\fg$, where $G^{\\min}$ is the `minimal' Kac-Moody group with Lie algebra $\\fg$ and $B$ is the standard Borel subgroup of \n$G^{\\min}$. For $w\\in W$, let $X_w = \\overline{BwB\/B}\\subset X$ be the corresponding Schubert variety. Let $\\{\\eps^w\\}_{w\\in W} \\subset H^*(X,\\bz)$ be the \n(Schubert) basis dual (with respect to the standard pairing) to the basis of the singular homology of $X$ given by the fundamental classes of $X_w$. \nThe following result is our first main theorem valid for any symmetrizable $\\fg$ (cf. Theorem \\ref{thm1}).\n\\begin{thm}\nLet\n $(\\lambda_1,\\dots,\\lambda_s,\\mu) \\in \\Gamma_s$. Then, \nfor any $u_1, \\dots , u_s, v\\in W$ such that $n^v_{u_1, \\dots, u_s}\\neq 0$, where\n$$\\eps^{u_1} \\dots \\eps^{u_s} = \\sum_w n^w_{u_1, \\dots, u_s}\\,\\eps^w,$$\nwe have \n\\[\\left(\\sum^s_{j=1} \\lam_j (u_jx_i)\\right) - \\mu (vx_i) \\geq 0, \\,\\,\\,\\text{for any}\\,\\, x_i,\\]\nwhere \n $x_i\\in \\fh$ is dual to the simple roots of $\\fg$.\n\\end{thm}\nThe proof of the theorem relies on the Kac-Moody analogue of the Borel-Weil theorem and the Geometric Invariant Theory (specifically the Hilbert-Mumford index). We conjecture that the above inequalities are sufficient as well to describe $\\Gamma_s$. In fact, we conjecture a much sharper result,\nwhere much fewer inequalities suffice to describe the semigroup $\\Gamma_s$. To explain our conjecture, we need some more notation. \n\nLet $P\\supset B$ be a (standard) parabolic subgroup and let $X_P:=G^{\\min}\/P$ be the corresponding partial flag variety. Let $W_P$ be the Weyl group of $P$ (which is, by definition, the Weyl group of the Levi $L$ of $P$) and let $W^P$ be the set of minimal length coset representatives \nof cosets in $W\/W_P$. \nThe projection map $X\\to X_P$ induces an injective homomorphism $H^*(X_P, \\bz) \\to H^*(X, \\bz)$ and $H^*(X_P, \\bz) $ has the Schubert basis\n$\\{\\eps^w_P\\}_{w\\in W^P}$ such that $\\eps^w_P$ goes to $\\eps^w$ for any $w\\in W^P$. As defined by Belkale-Kumar [BK, $\\S$6] in the finite dimensional case (and extended here in Section 7 for any symmetrizable Kac-Moody case), there is a new deformed product $\\odot_0$ in\n$H^*(X_P, \\bz)$, which is commutative and associative. Now, we are ready to state our conjecture (see Conjecture \\ref{conj1}). \n\\begin{conjecture} \nLet $\\fg$ be any indecomposable symmetrizable Kac-Moody Lie algebra\nand let $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in P_+^{s+1}$. Assume further that none of $\\lambda_j$ is $W$-invariant and $\\mu-\\sum_{j=1}^s \\lambda_j\\in Q$, where $Q$ is the root lattice of $G$. \nThen, the following are equivalent:\n\n(a) $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$.\n\n(b) For every standard maximal parabolic subgroup $P$ in $G^{\\min}$ and every choice of\n$s+1$-tuples $(w_1, \\dots, w_s, v)\\in (W^P)^{s+1}$ such that $\\epsilon_P^v$ occurs with coefficient $1$ in \nthe deformed product\n$$\\epsilon_P^{w_1}\\odot_0\\, \\cdots \\,\\odot_0 \\epsilon_P^{w_s}\n\\in \\bigl(H^*(X_P,\\Bbb{Z}), \\odot_0\\bigr),$$\n the following inequality holds:\n \\[\n\\bigl(\\sum_{j=1}^s \\lambda_j(w_jx_{P})\\bigr)-\\mu(vx_P)\\geq 0, \\tag{$I^P_{(w_1,\\dots, w_s,v)}$}\n\\]\nwhere $\\alpha_{i_P}$ is the (unique) simple root not in the Levi of $P$\nand $x_P:=x_{i_P}$.\n\\end{conjecture}\n\nThis conjecture is motivated from its validity in the finite case due to Belkale-Kumar [BK, Theorem 22]. (For a survey of these results in the finite case, see [K$_5$].) So far, the only evidence of its validity in the infinite dimensional case is shown for $s=2$ and $\\fg$ of types $A_1^{(1)}$ and $A_2^{(2)}$ (cf. Theorems \\ref{thm7.5} and \\ref{thm8.6}). In these cases, we explicitly determine $\\Gamma_2$ and thereby show the validity of the conjecture. \n\nA positive integer $d_o$ is called a {\\em saturation factor} for $\\fg$ if for any $\\Lambda$, $\\Lambda'$, $\\Lambda'' \\in P_+$\nsuch that $\\Lambda-\\Lambda'-\\Lambda''\\in Q$ and \n $L(N\\Lambda)$ is a submodule of $L(N\\Lambda')\\otimes L(N\\Lambda'')$, for some $N\\in \\bz_{>0}$, then \n $L(d_o\\Lambda)$ is a submodule of $L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$.\n\nWe prove the following result on saturation factors (cf. Corollaries \\ref{cor6.4} and \\ref{cor8.7}).\n\\begin{thm} For $A_1^{(1)}$, any integer $d_o>1$ is a saturation factor. For $A_2^{(2)}$, $4$ is a saturation factor.\n\\end{thm}\n\nThe proof in these affine rank-2 cases makes use of basic representation theory of the Virasoro algebra (in particular, Lemma \\ref{virasoro}). \nLet $\\delta$ be the smallest positive imaginary root of $\\fg$. To determine the saturated tensor semigroup, we show that it is enough to know the components of $L(\\lambda_1)\\otimes L(\\lambda_2)$ which are $\\delta$-maximal, i.e., the components $L(\\mu)\\subset L(\\lambda_1)\\otimes L(\\lambda_2)$ such that $L(\\mu+n\\delta) \\nsubseteq L(\\lambda_1) \\otimes L(\\lambda_2)$ for any $n>0$. Let $m^\\mu_{\\lambda_1,\\lambda_2}$ be the multiplicity of $L(\\mu)$ in $L(\\lambda_1) \\otimes L(\\lambda_2)$. If $L(\\mu)$ is a $\\delta$-maximal component of $L(\\lambda_1)\\otimes L(\\lambda_2)$, then $\\sum_{n\\in\\bz_{\\leq 0}} L(\\mu+n\\delta)^{\\oplus m^{\\mu+n\\delta}_{\\lambda_1,\\lambda_2}}$ is a unitarizable coset module for the Virasoro algebra arising from the Sugawara construction for the diagonal embedding $\\fg \\hookrightarrow \\fg \\oplus\\fg$. Proposition \\ref{maximum} for $A_1^{(1)}$ (and the analogous Proposition \\ref{maxS} for $A_2^{(2)}$) determining the maximal $\\delta$-components plays a crucial role in the proofs.\n\n\\vskip2ex\n\\noindent\n{\\bf Acknowledgements.} We thank Evgeny Feigin and Victor Kac for some helpful correspondences. Both the authors were partially supported by the NSF grant number DMS-1201310.\n\\section{Notation}\\label{sec2}\n\nWe take the base field to be the field of complex numbers $\\bc$. By a variety, we\n mean an algebraic variety over $\\bc$, which is reduced but not necessarily irreducible.\n\nLet $G$ be any symmetrizable Kac-Moody group over $\\bc$ completed\nalong\n the negative roots (as opposed to completed along the positive roots as in [K$_3$,\n Chapter 6]) and $G^{\\min}\\subset G$ be the `minimal' Kac-Moody group as in\n [K$_3$, \\S7.4]. Let $B$ be the standard (positive) Borel\nsubgroup, $B^{-}$ the standard negative Borel subgroup, $H=B\\cap B^{-}$ the\nstandard maximal torus and $W$ the Weyl group (cf.\n[K$_3$, Chapter 6]). Let $U$ (resp. $U^-$) be the unipotent radical $[B,B]$ (resp. $[B^-,B^-]$) of\n$B$ (resp. $B^-$). Let\n \\[\n\\bar{X} = G\/B\n \\]\nbe the `thick' flag variety which contains the standard KM-flag\nvariety\n \\[ X = G^{\\min}\/B. \\]\nIf $G$ is not of finite type, $\\bar{X}$ is an infinite\ndimensional non quasi-compact scheme (cf. [Ka, \\S4]) and $X$ is an\nind-projective variety (cf. [K$_3$, \\S7.1]). The group $G^{\\min}$ acts on $\\bar{X}$ and $X$. \n\nMore generally, for any \nstandard parabolic subgroup $P\\supset B$, define the partial flag variety\n \\[ X_P = G^{\\min}\/P, \\]\nand \n\\[\\bar{X}_P=G\/P.\\]\n\nRecall that if $W_P$ is the Weyl group of $P$ (which is, by definition, the Weyl\nGroup $W_L$ of its Levi subgroup $L$), then in each coset of $W\/W_P$ we have a unique member $w$ of minimal length.\n Let $W^P$ be the set of the minimal length representatives\nin the cosets of $W\/W_P$.\n\nFor any $w\\in W^P$, define the Schubert cell:\n\\[\nC_w^P:= BwP\/P \\subset G\/P\n \\]\nendowed with the reduced subscheme structure.\nThen, it is a locally closed subvariety of the ind-variety $G\/P$ isomorphic with the affine\nspace $\\Bbb A^{\\ell(w)}, \\ell(w)$ being the length of $w$ (cf. [K$_3$, $\\S$7.1]). Its closure is denoted by $X^P_w$, \nwhich is an irreducible (projective) subvariety\nof $G\/P$ of dimension $\\ell(w)$. We denote the point $wP\\in C_w^P$ by $\\dot{w}$.\nWe abbreviate $C^B_w, X_w^B$ by $C_w, X_w$ respectively.\n\nSimilarly, define the opposite Schubert cell\n$$\nC^{w}_P:={B^{-}wP\/P}\\subset \\bar{X}_P,\n$$\nand the opposite Schubert variety\n$$\nX^{w}_P:=\\overline{C^w}\\subset \\bar{X}_P,\n$$\nboth endowed with the reduced subscheme structures.\nThen, $X^{w}_P$ is a finite codimensional irreducible subscheme\nof $\\bar{X}_P$ (cf. [K$_3$, Section 7.1] and [Ka, \\S4]). As above, we abbreviate $C_B^w, X^w_B$ by $C^w, X^w$ respectively.\n\nFor any integral\nweight $\\lambda$ (i.e., any character $e^{\\lambda}$ of $H$), we have\na $G^{\\min}$-equivariant line bundle $\\mathcal{L}_B(\\lambda)$ on $X$\nassociated to the character $e^{-\\lambda}$ of $H$. Similarly, we have \na $G$-equivariant line bundle $\\mathcal{L}_{B^-}(\\lambda)$ on $X^-:=G\/B^-$\nassociated to the character $e^{\\lambda}$ of $H$. \n\n By the Bruhat decomposition\n$$X_P=\\sqcup_{w\\in W^P}\\,C_w^P,$$\nthe singular homology $H_*(X_P, \\bz)$ of $X_P$ with integral coefficients\nhas a basis $\\{\\mu(X_w^P)\\}_{w\\in W^P}$, where $\\mu(X_w^P)\\in H_{2\\ell(w)}(X_P, \\bz)$ denotes the \nfundamental class of $X_w^P$. Let $\\{\\epsilon^w_P\\}_{w\\in W^P}$ be the dual \nbasis of the singular cohomology $H^*(X_P, \\bz)$ under the standard pairing of cohomology with homology, i.e., \n$$\\epsilon^u_P(\\mu(X_v^P))=\\delta_{u,v},\\,\\,\\,\\text{for any} \\,\\,u,v\\in W^P.$$\nThus, $\\epsilon^w_P\\in H^{2\\ell(w)}(X_P, \\bz)$. If $P=B$, we abbreviate \n$\\epsilon^u_P$ by $\\epsilon^u$. \n\nLet $\\Delta=\\{\\alpha_1,\\ldots,\\alpha_{r}\\}\\subset \\mathfrak{h}^{*}$ be the\nset of simple roots,\n$\\{\\alpha_1^{\\vee},\\ldots,\\alpha^{\\vee}_{r}\\}\\subset \\mathfrak{h}$\nthe set of simple coroots and $\\{s_1,\\ldots, s_{r}\\}\\subset W$ the\ncorresponding simple reflections, where $\\mathfrak{h}:=\\Lie H$. Let\n$\\rho\\in X(H)$ be any weight satisfying\n$$\n\\rho(\\alpha^{\\vee}_{i})=1,\\quad\\text{for all}\\quad 1\\leq i\\leq r,\n$$\nwhere $X(H)$ is the character group of $H$ (identified as a subgroup of $\\fh^*$ \nvia the derivative). \nWhen $G$ is a finite dimensional semisimple group, $\\rho$ is unique,\nbut for a general Kac-Moody group $G$, it may not be unique.\n\nChoose elements $x_i\\in \\fh$ such that \n\\beqn \\label{eq1} \\alpha_j(x_i)=\\delta_{i,j}, \\,\\,\\,\\text{for any}\\,\\, 1\\leq i,j\\leq r.\n\\eeqn\nObserve that $x_i$ may not be unique.\n\nDefine the set of {\\it dominant integral weights} \n$$P_+:=\\{\\lambda\\in X(H): \\lambda (\\alpha_i^\\vee)\\in \\bz_+ \\,\\forall \\,\n1\\leq i\\leq r\\},$$\n and the set of {\\it dominant integral regular weights} \n$$P_{++}:=\\{\\lambda\\in X(H): \\lambda (\\alpha_i^\\vee)\\in \\bz_{\\geq 1} \\,\\forall \\,\n1\\leq i\\leq r\\},$$\nwhere $\\bz_+$ is the set of non-negative integers. The integrable highest \nweight (irreducible) modules of $G^{\\min}$ are parameterized by $P_+$. For $\\lambda\n\\in P_+$, let $L(\\lambda)$ be the corresponding integrable highest weight (irreducible) $G$-module \nwith highest weight $\\lambda$. \n\n\\section{Necessary Inequalities for the Saturated Tensor Semigroup}\\label{sec3}\nFix a positive integer $s$ and define the {\\em saturated tensor semigroup} $\\Gamma_s=\\Gamma_s(G)$:\n\\beqn\n\\Gamma_s:= \\{(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in P_+^{s+1}: \\exists\\,\nN>1 \\,\\,\\text{with}\\,\\, L(N\\mu)\\subset L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s)\\}.\n\\eeqn\nIt is indeed a semigroup by the anlogue of the Borel-Weil theorem for the Kac-Moody case (see the identity \\eqref{ne3.3}\nin the proof of Theorem \\ref{thm1}).\nWe give a certain set of inequalities satisfied by $\\Gamma_s$. But, we first recall some basic results about the Hilbert-Mumford index.\n\n\\begin{definition}\\label{git} Let $S$ be any (not necessarily reductive) algebraic group\nacting on a (not necessarily projective) variety $\\exx$ and let $\\elal$ be\nan $S$-equivariant line bundle on $\\exx$. Let $O(S)$ be the set of all one parameter\nsubgroups (for short OPS) in $S$.\n Take any $x\\in \\exx$ and\n $\\delta \\in O(S)$ such that the limit\n $\\lim_{t\\to 0}\\delta(t)x$\nexists in $\\exx$ (i.e., the morphism ${\\delta}_x:\\Bbb{G}_m\\to \\exx$ given by\n$t\\mapsto \\delta(t)x$ extends to a morphism $\\tilde{\\delta}_x : \\Bbb{A}^1\\to \\exx$).\nThen, following Mumford, define a number $\\mu^{\\elal}(x,\\delta)$ as follows:\nLet $x_o\\in \\exx$ be the point $\\tilde{\\delta}_x(0)$. Since $x_o$ is $\\Bbb{G}_m$-invariant\nvia $\\delta$, the fiber of $\\elal$ over $x_o$ is a\n$\\Bbb{G}_m$-module; in particular, it is given by a character of $\\Bbb{G}_m$. This integer is defined as $\\mu^{\\elal}(x,\\delta)$.\n\\end{definition}\n\nWe record the following standard properties of $\\mu^{\\elal}(x,\\delta)$ (cf.\n [MFK, Chap. 2, $\\S$1]):\n\\begin{proposition}\\label{propn14} For any $x\\in \\exx$ and $\\delta \\in O(S)$ such that $\\lim_{t\\to 0}\\delta(t)x$\nexists in $\\exx$, we have the following (for any $S$-equivariant line bundles\n$\\elal, \\elal_1, \\elal_2$):\n\\begin{enumerate}\n\\item[(a)]\n$\\mu^{\\elal_1\\otimes\\elal_2}(x,\\delta)=\\mu^{\\elal_1}(x,\\delta)+\\mu^{\\elal_2}(x,\\delta).$\n\\item[(b)] If there exists $\\sigma\\in H^0(\\exx,\\elal)^S$ such that $\\sigma(x) \\neq 0$, then $\\mu^{\\elal}(x,\\delta)\\geq 0.$\n\\item[(c)] If $\\mu^{\\elal}(x,\\delta)=0$, then any element of $H^0(\\exx,\\elal)^S$\nwhich does not vanish at $x$ does not vanish at $\\lim_{t\\to 0}\\delta(t)x$ as well.\n\\item[(d)] For any $S$-variety $\\exx'$ together with an $S$-equivariant morphism $f:\\exx'\\to \\exx$ and any $x'\\in \\exx'$ such that $\\lim_{t\\to 0}\\delta(t)x'$\nexists in $\\exx'$, we have\n$\\mu^{f^*\\elal}(x',\\delta)=\\mu^{\\elal}(f(x'),\\delta).$\n\\item[(e)] (Hilbert-Mumford criterion) Assume that $\\exx$ is projective, $S$ is\n connected and reductive\nand $\\elal$ is ample. Then, $x\\in\\exx$ is semistable (with respect to $\\elal$) if\nand only if $\\mu^{\\elal}(x,\\delta)\\geq 0$, for all $\\delta\\in O(S)$.\n\nIn particular, if $x\\in \\exx$ is semistable and $\\delta$-fixed, then\n$\\mu^{\\elal}(x,\\delta)= 0$.\n\\end{enumerate}\n\\end{proposition}\n\nThe following theorem is one of our main results giving a collection of necessary inequalities defining the semigroup\n$\\Gamma_s$.\n \\begin{theorem} \\label{thm1} Let $G$ be any symmetrizable Kac-Moody group and let $(\\lam_1, \\cdots ,\\lam_s, \\mu )\\in \\Gamma_s$.\n Then, for any $u_1, \\dots ,u_s, v\\in W$ such that $n^v_{u_1, \\dots, u_s}\\neq 0$, where\n$$\\eps^{u_1} \\cdots \\eps^{u_s}\n= \\sum_w n^w_{u_1, \\dots, u_s}\\,\\eps^w\\in H^*(X, \\bz),$$ we have\n \\[\n\\bigl(\\sum^s_{j=1} \\lam_j (u_jx_i)\\bigr) - \\mu (vx_i) \\geq 0, \\quad\\text{ for any }x_i,\n \\]\nwhere $x_i$ is defined by the equation \\eqref{eq1}.\n \\end{theorem}\n\n \\begin{proof}\nLet\n \\[\nZ := \\bigl\\{ (\\bar{g}_1, \\dots ,\\bar{g}_s)\\in {(X^-)}^{s}: g_1X^{u_1}\n\\cap \\cdots \\cap g_sX^{u_s}\\cap X_v \\neq \\emptyset\\bigr\\} ,\n \\]\nwhere $X^-:=G\/B^-$ and $\\bar{g}_j= g_jB^-$. \n Then, $Z$ contains a nonempty open set by Proposition \\ref{prop5}.\n(In fact, by Proposition \\ref{prop5}, $Z = (X^-)^{s}$, but we do not need this stronger result.)\n\nTake a nonzero $\\sig\\in H^0 \\bigl( (X^-)^{s}\\times X, \\cl^N\n\\bigr)^{G^{\\min}}$, where\n$$\\cl := \\cl_{B^-}(\\lam_1)\\boxtimes \\cdots\\boxtimes \\cl_{B^-}(\\lam_s)\\boxtimes\n\\cl_B (\\mu ).$$\nSuch a nonzero $\\sigma$ exists, for some $N>0$, since by [K$_3$, Corollary 8.3.12(a) and Lemma 8.3.9], \n\\begin{align} \\label{ne3.3}\nH^0 \\bigl( (X^-)^{s}\\times X, \\cl^N\n\\bigr)^{G^{\\min}}&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee\\otimes L(N\\mu), \\bc \\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), [L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee]^*\\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), [L(N\\lambda_1)^\\vee\\otimes \\dots \\otimes L(N\\lambda_s)^\\vee]^\\vee \\bigr)\\notag\\\\\n&\\simeq \\Hom_{G^{\\min}}\\bigl( L(N\\mu), L(N\\lambda_1)\\otimes \\dots \\otimes L(N\\lambda_s) \\bigr)\\notag\\\\\n&\\neq 0,\n\\end{align}\nsince $(\\lambda_1, \\dots, \\lambda_s,\\mu)\\in \\Gamma_s$, where, for a $G^{\\min}$-module $M$, $M^\\vee$ denotes the direct sum of the $H$-weight spaces of the full dual module $M^*$. \n\nPick $(\\bar{g}_1, \\dots ,\\bar{g}_s)\\in Z$ such that $\\sig (\\bar{g}_1,\n\\dots ,\\bar{g}_s, \\bar{1})\\neq 0$, where \n$\\bar{1} =1\\cdot B$. Since $(\\bar{g}_1, \\dots ,\\bar{g}_s)\\in Z$, there exists $u'_1 \\geq u_1, \\cdots , u'_s \\geq u_s$ and $v' \\leq v$\nsuch that $g_1C^{u'_1} \\cap \\cdots \\cap g_s C^{u'_s}\n \\cap C_{v'}$ is nonempty. Now, pick $g\\in G^{\\min}$ such that\n \\beqn \\label{e101}\ngB \\in g_1C^{u'_1} \\cap \\cdots \\cap g_s C^{u'_s}\n \\cap C_{v'}.\n \\eeqn\nBy Proposition \\ref{propn14}, for any $\\delta \\in O(G^{\\min})$, $\\mu^{\\cl}(\\bar{x}, \\del\n(t))\\geq 0$, where $\\bar{x} = (\\bar{g}_1, \\dots ,\\bar{g}_s, \\bar{1})$\n(since $\\sig (\\bar{x})\\neq 0$). By the following Lemma \\ref{lem2},\napplied to the OPS $\\delta (t)=gt^{x_i}g^{-1}$, we get\n \\beqn \\label{eqn02}\n\\bigl(\\sum^s_{j=1} \\lam_j (u'_jx_i)\\bigr) - \\mu (v' x_i) \\geq 0. \n \\eeqn\nBut, by [K$_3$, Lemma 8.3.3],\n \\[ (u'_j)^{-1}\\lam_j \\leq u_j^{-1} (\\lam_j). \\]\nThus,\n \\[ \\lam_j(u'_j x_i) \\leq \\lam_j(u_jx_i). \\]\nSimilarly,\n \\[ \\mu (v'x_i) \\geq \\mu (v x_i). \\]\nThus, from \\eqref{eqn02}, we get\n \\[ \\bigl(\\sum_{j=1}^s \\lam_j(u_jx_i) \\bigr)- \\mu (vx_i) \\geq 0. \\]\n\n This proves the theorem. \n \\end{proof}\n\n \\begin{lemma} \\label{lem2} Let $g\\in G^{\\min}$ be as in the equation \\eqref{e101}.\nConsider the one parameter subgroup\n $\n\\del (t) = gt^{x_i} g^{-1}\\in O(G^{\\min}).\n $ Then,\n\n(a) $\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t)) = \\lam_j(u_j' x_i)$.\n\n(b) $\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = -\\mu (v'x_i)$.\n \\end{lemma}\n\n \\begin{proof} (a) \n $\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t))= \\mu^{\\cl_{B^-}\n (\\lam_j)}(g^{-1}g_jB^-, t^{x_i})$.\\\\\nBy assumption, $g^{-1}_jg\\in U^- u'_jB$. Write\n \\[\ng_j^{-1}g = b_j^-u'_jp_j, \\quad\\text{ for some } \\,b_j^-\\in U^-,\\, p_j \\in B.\n \\]\nThus,\n \\[ 1 = g^{-1}g_jb_j^- u_j' p_j. \\]\nLet\n \\[ b_j(t) = b_j^- u'_j t^{-x_i} (u'_j)^{-1} (b_j^-)^{-1} \\in B^- . \\]\nThen,\n \\beqn\\label{eqn01}\nt^{x_i} g^{-1}g_jb_j(t) = t^{x_i}p_j^{-1} t^{-x_i} (u'_j)^{-1} (b_j^-)^{-1}.\n \\eeqn\nConsider the $G_m$-invariant section (via $t^{x_i}$) of $\\cl_{B^-}(\\lam_j):$\n \\begin{align*}\n\\hat{\\sig}(t) &= \\bigl( t^{x_i}\\, g^{-1}g_j, 1\\bigr) \\mod B^-\\\\\n&= \\bigl( t^{x_i}\\, g^{-1}g_jb_j(t), \\lam_j (b_j(t)^{-1})\\bigr) \\mod B^- .\n \\end{align*}\nClearly, $\\lt_{t\\to 0} \\,t^{x_i}\\, g^{-1}g_jb_j(t)$ exists in $G$ by \\eqref{eqn01}.\n\nNow,\n \\begin{align*}\n\\lam_j \\bigl( b_j(t)^{-1}\\bigr) &= \\lam_j\\bigl( b_j^- u'_j t^{x_i} (u'_j)^{-1}\n(b_j^-)^{-1}\\bigr)\\\\\n &= \\lam_j \\bigl( t^{u_j' x_i}\\bigr) .\n \\end{align*}\nThis gives\n \\[\n\\mu^{\\cl_{B^-}(\\lam_j)} (g_jB^-, \\del (t)) = \\lam_j (u'_j(x_i)) .\n \\]\n\nThis proves the (a) part of the lemma.\n\\vskip1ex\n\n (b) $\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = \\mu^{\\cl_B(\\mu )}\n (g^{-1}B, t^{x_i})$. By assumption,\n \\[\n g\\in B v'\\cdot B.\n \\]\nWrite\n \\[\ng = bv'p, \\quad\\text{for }b\\in U, p\\in B.\n \\]\nThus,\n \\[ 1 = g^{-1} b v'p. \\]\nLet\n \\[ b(t) = bv't^{-x_i}(v')^{-1}b^{-1} \\in B. \\]\nNow,\n \\[\nt^{x_i}g^{-1} b(t) = t^{x_i}p^{-1} t^{-x_i}(v' )^{-1}b^{-1}.\n \\]\nThus,\n \\[\n\\lt_{t\\to 0} \\,t^{x_i}g^{-1}b(t) \\text{ exists in } G^{\\min}.\n \\]\n\nConsider the $G_m$-invariant section (via $t^{x_i}$)\n \\begin{align*}\n\\hat{\\sig}(t) &= (t^{x_i}g^{-1}, 1) \\mod B\\\\\n&= \\bigl( t^{x_i}g^{-1}b(t), \\mu (b(t))\\bigr) \\mod B.\n \\end{align*}\nNow,\n \\begin{align*}\n\\mu (b(t)) &= \\mu (bv' t^{-x_i} (v')^{-1}b^{-1})\\\\\n&= \\mu (t^{-v' x_i}).\n \\end{align*}\nThis gives\n \\[\n\\mu^{\\cl_B(\\mu )}(1\\cdot B, \\del (t)) = -\\mu (v' (x_i)).\n \\]\n\n This proves the (b)-part and hence the lemma is proved.\n \\end{proof}\n\n \n \\begin{definition} \\label{n2.1}\n For a quasi-compact scheme $Y$, an $\\co_{Y}$-module $\\cs$ is called {\\it coherent}\n if it is finitely presented as an $\\co_{Y}$-module and any $\\co_{Y}$-submodule of finite\n type admits a\nfinite presentation.\n\n An $\\co_{\\bar{X}}$-module $\\cs$ is called {\\it coherent} if\n $\\cs_{|V^S}$ is a\ncoherent $\\co_{V^S}$-module for any finite ideal $S\\subset W$ (where a subset $S\\subset W$\nis called an {\\it ideal} if\n for $x\\in S$ and $y\\leq x\\Rightarrow y\\in S$), where $V^S$ is the quasi-compact open subset\n of $\\bar{X}$ defined by\n $$V^S = \\bigcup_{w\\in S} wU^- B\/B.$$\n Let $K^0(\\bar{X})$ denote the Grothendieck group of\n coherent $\\co_{\\bar{X}}$-modules $\\cs$.\n\n Similarly,\ndefine $K_0(X) := \\lim_{n\\to\\infty} K_0(X_n)$, where $\\{\nX_n\\}_{n\\geq 1}$ is the filtration of $X$ giving the ind-projective\nvariety structure (i.e., $X_n = \\bigcup_{\\ell (w)\\leq n} C_w$) and\n$K_0(X_n)$ is the Grothendieck group of coherent\nsheaves on the projective variety $X_n$.\n\nWe also define\n \\[\nK^{\\optop}(X) := \\Invlt_{n\\to\\infty} K^{\\optop}(X_n),\n \\]\nwhere $K^{\\optop}(X_n)$ is the topological $K$-group of the\n projective variety $X_n$.\n\nLet $*:K^{\\optop}(X_n)\\to K^{\\optop}(X_n)$ be the involution induced from\nthe operation which takes a vector bundle to its dual. This,\n of course, induces the involution $*$ on $K^{\\optop}(X)$.\n\nFor any $w\\in W$,\n \\[ [\\co_{X_w}] \\in K_0(X). \\]\n \\end{definition}\n\n \\begin{lemma} $\\bigl\\{ [\\co_{X_w}]\\bigr\\}_{w\\in W}$ forms a basis of $K_0(X)$ as a $\\bz$-module.\n \\end{lemma}\n\n \\begin{proof} By [CG, \\S 5.2.14 and Theorem 5.4.17], the result follows.\n \\end{proof}\n\n For $u\\in W$, by [KS, \\S 2], $\\co_{X^u}$ is a coherent $\\co_{\\bar{X}}$-module.\n In particular, $\\co_{\\bar{X}}$ is a coherent $\\co_{\\bar{X}}$-module.\n\n\n\nDefine a pairing\n$$\n\\langle \\, ,\\, \\rangle : K^0(\\bar{X}) \\otimes K_0(X) \\to \\bz,\\,\\,\n\\langle[\\cs], [\\cf]\\rangle = \\sum_i (-1)^i \\chi \\bigl (X_n, \\tor_\ni^{\\co_{\\bar{X}}}\n (\\cs,\\cf ) \\bigr),$$\nif $\\cs$ is a coherent sheaf on $\\bar{X}$ and $\\cf$\nis a coherent sheaf on ${X}$ supported in $X_n$ (for\nsome $n$), where $\\chi$ denotes the \nEuler-Poincar\\'{e} characteristic.\nThen, as in [K$_4$, Lemma 3.4], \n the above pairing is well defined.\n \nBy [KS, Proof of Proposition 3.4], for any $u\\in W$,\n \\beqn\\label{eq1.0}\n\\ext^k_{\\co_{\\bar{X}}} (\\co_{X^u}, \\co_{\\bar{X}}) =0 \\quad\\forall k\\neq \\ell (u).\n \\eeqn\nDefine the sheaf\n \\[\n\\om_{X^u} := \\ext^{\\ell (u)}_{\\co_{\\bar{X}}}\n\\bigl(\\co_{X^u}, \\co_{\\bar{X}} \\bigr)\\otimes\\cl (-2\\rho ),\n \\]\n which, by the analogy with the Cohen-Macaulay (for short CM) schemes of finite type, will be called\n the {\\it dualizing sheaf} of $X^u$.\n\n\nNow, set the sheaf on $\\bar{X}$\n \\begin{align*}\n\\xi^u &:= \\cl (\\rho )\\om_{X^u} \\\\\n&= \\cl (-\\rho ) \\ext^{\\ell (u)}_{\\co_{\\bar{X}}}\n(\\co_{X^u}, \\co_{\\bar{X}} ).\n \\end{align*}\nThen, as proved in [K$_4$, Proposition 3.5], for any $u,w\\in W$,\n\\beqn\\label{e106}\n\\langle[\\xi^u], [\\co_{X_w}]\\rangle = \\delta_{u,w}.\n\\eeqn\nWith these preliminaries, we are ready to prove the following result.\n \\begin{proposition} \\label{prop5} With the notation as in the proof of Theorem \\ref{thm1},\n$Z = (X^-)^{s}$, if $\\eps^v $ occurs in $\\eps^{u_1}\n\\cdots\\eps^{u_s}$ with\n nonzero coefficient.\n \\end{proposition}\n\n \\begin{proof} We give the proof in the case $s=2$. The proof for general\n $s$ is similar.\n\nFor $u,v\\in W$, express\n \\[\n\\eps^u\\eps^v = \\sum_{ \\substack{w\\\\ \\ell (w)=\\ell (u)+\\ell (v)} } n^w_{u,v} \\eps^w.\n \\]\nExpress the product in topological $K$-theory $K^{\\optop}(X)$ of $X=G^{\\min}\/B$:\n \\[\n\\psi^u_o\\psi^v_o = \\sum_{\\ell (w)\\geq\\ell (u)+\\ell (v)} m^w_{u,v} \\psi^w_o,\n \\]\nwhere $\\psi^w := *\\tau^{w^{-1}}$ ($\\tau^w$ being the Kostant-Kumar `basis'\nof $K^{\\optop}_H(X)$ as in [KK, Remark 3.14]) and $\\{\\psi^{w}_o\\}_{w \\in W}$ is the corresponding `basis' of \n$K^{\\optop}(X)\\simeq \\bz\\otimes_{R(H)}\\,K^{\\optop}_H(X),$ cf. [KK, Proposition 3.25]). \n\nThen, by [KK, Proposition 2.30],\n \\beqn\\label{e102}\nn_{u,v}^w = m_{u,v}^w, \\quad\\text{if }\\ell (w) = \\ell (u)+\\ell (v).\n \\eeqn\nLet $\\Delta: X \\to X\\times X$ be the diagonal map. Then, by [K$_4$, Proposition 4.1] and the identity \\eqref{e106}, for any $u,v,w \\in W$, $g_1,g_2\\in G^{\\min}$,\n \\begin{align*}\nm^w_{u,v} &= \\langle [\\xi^u\\boxtimes\\xi^v], [\\Del_*\\co_{X_w}]\\rangle \\\\\n&= \\langle [\\xi^u\\boxtimes\\xi^v], [(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_* \\co_{X_w})]\\rangle,\n \\end{align*}\nsince $[(g_1^{-1}, g_2^{-1})\\cdot\\Del_*\\co_{X_w}]= [\\Del_*\\co_{X_w}]$ as elements\nof $K_0(X\\times X)$. Thus,\n \\begin{align}\\label{e103}\n m^w_{u,v} &= \\langle [\\xi^u\\boxtimes\\xi^v], [(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_*\\co_{X_w})]\\rangle\n \\\\ &:= \\sum_i (-1)^i \\chi (\\bar{X}\\times \\bar{X}, \\tor_i^{\\co_{\\bar{X}\\times \\bar{X}}} \\Bigl(\\xi^u\\boxtimes\\xi^v,(g_1^{-1}, g_2^{-1}) \\cdot (\\Del_* \\co_{X_w})\\Bigr) .\\notag\n \\end{align}\n\nNow, by definition, the support of $\\xi^u$ is contained in $X^u$ and hence the\nsupport of the sheaf\n \\[\n\\cs_i := \\tor_i^{\\co_{\\bar{X}\\times \\bar{X}}} \\bigl( \\xi^u\\boxtimes\\xi^v, (g_1^{-1}, g_2^{-1})\\cdot \\Del_* \\co_{X_w}\\bigr)\n \\]\nis contained in\n \\beqn\\label{e104}\nX^u\\times X^v \\cap \\bigl((g_1^{-1}, g_2^{-1})\\cdot \\Del (X_w)\\bigr),\n \\eeqn\nwhich is empty if\n \\beqn\\label{e105}\n(g_1X^u) \\cap (g_2X^v) \\cap X_w = \\emptyset .\n \\eeqn\nThus, if the equation \\eqref{e105} is true, then the Tor sheaf $\\cs_i =0$ $\\forall i\\geq 0$. Thus, if \nthe equation \\eqref{e105} is satisfied, \n \\[ m_{u,v}^w =0. \\]\nNow, assume that $\\ell (w) = \\ell (u)+\\ell (v)$. Then, by the equation \\eqref{e102},\n \\[ n^w_{u,v} =0, \\quad\\text{ if the equation \\eqref{e105} is satisfied}. \\]\nBut, since by assumption, $n^w_{u,v} \\neq 0$, we see that\n \\[\n(g_1X^u) \\cap (g_2X^v )\\cap X_w \\neq \\emptyset , \\;\\text{ for any } g_1,g_2\\in G^{\\min}.\n \\]\nBut since $G^{\\min}\/(G^{\\min} \\cap B^-) \\simto X^-$, we get the proposition.\n \\end{proof}\n\n\n\\section{Tensor Product Decomposition for Affine Kac-Moody Lie Algebras}\n\n\\subsection{The Virasoro Algebra}\n\nWe recall the definition of the Virasoro algebra and its basic representation theory, which we need. \nThe {\\it Virasoro algebra} $\\mathrm{Vir}$ has a basis $\\{C,\\, L_{n}\\;:\\; n\\in\\mathbb{Z}\\}$\nover $\\bc$ \nand the Lie bracket is given by \n\\[[L_{m},L_{n}]=(m-n)L_{m+n}+\\frac{1}{12}(m^{3}-m)\\delta_{m,-n}C\\,\\,\n\\text{and}\\, [\\mathrm{Vir,C]=0}.\\] \n\nLet $\\Vir_0:= \\mathbb{C}L_{0}\\oplus\\mathbb{C}C$. Then, a Vir module\n$V$ is said to be a {\\it highest weight representation} if there exists a $\\Vir_0$-eigenvector $v_o\\in V$\nsuch that $L_{n}v_o=0$ for $n\\in\\mathbb{Z}_{>0}$ and $U(\\bigoplus_{n<0}\\mathbb{C}L_{n})v_o=V$.\nSuch a $V$ is said to have {\\it highest weight} $\\lambda\\in \\Vir_0^{*}$\nif $Xv_o=\\lambda(X)v_o$, for all $X\\in \\Vir_0$. (It is easy to see that such a $v_o$ is unique up to a scalar multiple and hence\n$\\lambda$ is unique.)\nThe irreducible\nhighest weight representations of Vir are in 1-1 correspondence with elements\nof $\\Vir_0^{*}$ given by the highest weight. \nDenote the basis of $\\Vir_0^*$ dual to the basis $\\{L_{0},C\\}$ of $\\Vir_0$ as $\\{h,z\\}$. \nFor any $\\mu\\in \\Vir_0^*$, denote the $\\mu$-th weight space \nof $V$ by $V_\\mu$, i.e.,\n\\[V_\\mu:=\\{v\\in V: X\\cdot v=\\mu(X)v\\,\\, \\forall X\\in \\Vir_0\\}.\\]\n\nDefine a Vir module $V$ to be {\\it unitarizable} if there exists a positive\ndefinite Hermitian form $(\\cdot\\,,\\,\\cdot)$ on $V$ so that $(L_{n}v\\,,\\, w)=(v\\,,\\, L_{-n}w)$\nfor all $n\\in\\mathbb{Z}$ and $(Cv\\,,\\, w)=(v\\,,\\, Cw)$. It is easy\nto see that if $M$ is a $\\Vir$-submodule of $V$, then $M^{\\perp}$\nis also a submodule. Hence, any unitarizable representation of Vir\nis completely reducible. Note that for a unitarizable highest weight Vir-representation\n$V$ with highest weight $\\lambda$, if $v_o$ is a highest weight vector,\nthen \n\\beqn \\label{e4.2}\n0\\leq(L_{-n}v_o\\,,\\, L_{-n}v_o)=(L_{n}L_{-n}v_o\\,,\\, v_o)=(2n\\lambda(L_{0})+\\frac{1}{12}(n^{3}-n)\\lambda(C))(v_o\\,,\\, v_o)\n\\eeqn\n for all $n>0$. Therefore, both $\\lambda(L_{0})$ and $\\lambda(C)$\nmust be nonnegative real numbers. \n\n\n\\begin{lem}\\label{virasoro}\nLet $V$ be a unitarizable, highest weight (irreducible) representation of $Vir$\nwith highest weight $\\lambda$. \n\n(a) If $\\lambda(L_0)\\neq 0$, then $V_{\\lambda+nh}\\neq 0$, for any $n\\in \\bz_+$. \n\n(b) If \n$\\lambda(L_0)= 0$ and $\\lambda (C)\\neq 0$, then $V_{\\lambda+nh}\\neq 0$, for any $n\\in \\bz_{>1}$ and \n$V_{\\lambda+h} =0$.\n\n(c) If \n$\\lambda(L_0)= \\lambda (C) = 0$, then $V$ is one dimensional. \n\\end{lem}\n\\begin{proof}\nIf $\\lambda(L_0)\\neq 0$, then by the equation \\eqref{e4.2} (since both of $\\lambda (L_0)$ and $\\lambda(C)\\in \\br_+$), \n$L_{-n}v_o\\neq 0$, for any $n\\in \\bz_+$. \n\nIf $\\lambda(L_0)=0$ and $\\lambda(C)\\neq 0$, then again by the equation \\eqref{e4.2}, \n$L_{-n}v_o\\neq 0$, for any $n\\in \\bz_{>1}$. Also, $L_{-1}v_o=0$. \n\n If $\\lambda(L_0)=\\lambda(C)= 0$, then (by the equation \\eqref{e4.2} again), \n$L_{-n}v_o= 0$, for any $n\\in \\bz_{\\geq 1}$. This shows that $V$ is one dimensional.\n\\end{proof}\n\\subsection{Tensor product decomposition: A general method}\n\nLet $\\mathfrak{g}$ be the untwisted affine Kac-Moody Lie algebra associated to a finite dimensional simple \nLie algebra $\\overset{\\circ}{\\mathfrak{g}}$, i.e.,\n\\[\\fg=\\bigl(\\overset{\\circ}{\\mathfrak{g}}\\otimes \\bc[t,t^{-1}]\\bigr) \\oplus \\bc c\\oplus \\bc d.\\]\nLet $\\overset{\\circ}{\\mathfrak{h}}$ be a Cartan subalgebra of $\\overset{\\circ}{\\mathfrak{g}}$. Then, \n\\[\\fh:=\\overset{\\circ}{\\mathfrak{h}}\\otimes 1\\oplus\\bc c\\oplus \\bc d\\]\nis the standard Cartan subalgebra of $\\fg$.\n Let $\\delta\\in \\fh^*$ be the smallest positive imaginary root of \n$\\fg$ (so that the positive imaginary roots of $\\fg$ are precisely $\\{n\\delta, n\\in \\bz_{\\geq 1}\\}$). Then,\n$\\delta$ is given by $\\delta_{|\\overset{\\circ}{\\mathfrak{h}}\\oplus \\bc c}\\equiv 0$ and $\\delta(d)=1$. \nFor any $\\Lambda \\in P_+$, let $P(\\Lambda)$ \nbe the set of weights of $L(\\Lambda)$ and let $P^o(\\Lambda)$ be the set of $\\delta$-maximal weights of $L(\\Lambda)$, i.e.,\n\\[\nP^o(\\Lambda)=\\left\\{ \\lambda\\in\\mathfrak{h}^{*}:\\lambda \\in P(\\Lambda) \\,\\,\\text{but}\\,\\, \\lambda+n\\delta \\notin P(\\Lambda)\\,\\,\n\\text{for any}\\,\\,n>0\\right\\}.\n\\]\nFor any $\\lambda \\in X(H)$, define the $\\delta$-{\\it character of $L(\\Lambda)$ through} $\\lambda$ by\n\\[c_{\\Lambda,\\lambda}=\\sum_{n\\in\\mathbb{Z}}\\dim L(\\Lambda)_{\\lambda+n\\delta}\\,e^{n\\delta}.\\]\nSince $\\delta$ is $W$-invariant,\n\\beqn \\label{e4.1}\nc_{\\Lambda,\\lambda}=c_{\\Lambda, w\\lambda}, \\,\\,\\text{for any}\\, w\\in W.\n\\eeqn\nMoreover, $P^o(\\Lambda)$ is $W$-stable.\n It\nis obvious that \n\\beqn \\label{e13}\nch\\, L(\\Lambda)=\\sum_{\\lambda\\in P^o(\\Lambda)}c_{\\Lambda,\\lambda}e^{\\lambda}.\n\\eeqn\nBy [K$_3$, Exercise 13.1.E.8], for any $\\lambda\\in P(\\Lambda')$ and $\\Lambda''\\in P_+$, $\\Lambda''+\\lambda+\\rho$ belongs to the Tits cone. Hence,\nthere exists $v\\in W$ such that $v^{-1}(\\Lambda''+\\lambda+\\rho)\\in P_+$. Moreover, if $\\Lambda''+\\lambda+\\rho$ has nontrivial $W$-isotropy, then its isotropy group must contain a reflection (cf. [K$_3$, Proposition 1.4.2(a)]). Thus, for such a $\\lambda\\in P(\\Lambda')$, i.e., if $\\Lambda''+\\lambda+\\rho$ has nontrivial $W$-isotropy,\n\\beqn\\label{e14} \\sum_{w\\in W}\\,\\varepsilon(w) e^{w(\\Lambda''+\\lambda+ \\rho)}=0.\\eeqn\nDefine \n$$ \\bar{P}_+:=\\{\\Lambda \\in P_+: \\Lambda(d)=0\\}.$$\nFor any $m\\in \\bz_+$, let \n\\[P_+^{(m)}:=\\{\\Lambda\\in P_+: \\Lambda (c)=m\\},\\]\nand let \n\\[\\bar{P}_+^{(m)}:=\\bar{P}_+\\cap P_+^{(m)}.\\]\n Then, \n${\\bar{P}}_+^{(m)}$ provides a set of representatives in $P_+^{(m)}$ mod $(P_+\\cap \\bc\\delta)$. \n\nFor any $\\Lambda, \\Lambda',\\Lambda''\\in P_+$, define\n\\begin{align*} T_{\\Lambda}^{\\Lambda',\\Lambda''}=\\{&\\lambda \\in P^o(\\Lambda'): \\exists v_{\\Lambda,\\Lambda'', \\lambda}\\in W\\,\\,\\text{and}\\, \nS_{\\Lambda,\\Lambda'', \\lambda}\\in \\bz \\,\\,\\text{with}\\\\ \n&\\, \\lambda+\\Lambda''+\\rho=v_{\\Lambda,\\Lambda'', \\lambda}(\\Lambda+\\rho)+ \nS_{\\Lambda,\\Lambda'', \\lambda}\\delta\\}.\n\\end{align*}\nObserve that since $\\Lambda+\\rho +n\\delta \\in P_{++}$ for any $n\\in \\bz$, such a $v_{\\Lambda,\\Lambda'', \\lambda}$ and $S_{\\Lambda,\\Lambda'', \\lambda}$ are unique by [K$_3$, Proposition 1.4.2 (a), (b)]\n(if they exist). Also, observe that \n\\beqn \\label{eq1001} T_{\\Lambda}^{\\Lambda',\\Lambda''}=\\emptyset,\\,\\,\\text{unless}\\, \\Lambda (c)=\\Lambda'(c)+\\Lambda''(c)\\,\\,\\,\\text{and}\\,\\, \n\\Lambda'+\\Lambda''-\\Lambda\\in Q,\\eeqn\nwhere $Q$ is the root lattice of $\\fg$.\n\\begin{prop} \\label{tensor} For any $\\Lambda'$ and $\\Lambda''\\in P_+$, \n\\[ch\\,\\bigl( L(\\Lambda')\\otimes L(\\Lambda'')\\bigr) = \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}ch\\, L(\\Lambda)\\bigl(\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\bigr),\\]\nwhere $m:=\\Lambda'(c)+\\Lambda''(c)$. \n\nMoreover, $\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}$ is the character of a unitary representation (though, in general, not irreducible) of the Virasoro algebra \n$\\mathrm{Vir}$ with central charge \n$$\\dim \\overset{\\circ}{\\mathfrak{g}}\\cdot\\bigl(\\frac{m'}{m'+g}+\\frac{m''}{m''+g}-\\frac{m}{m+g}\\bigr),$$\nwhere $m':=\\Lambda'(c), m'':=\\Lambda''(c)$ and $g$ is the dual Coxeter number of $\\overset{\\circ}{\\mathfrak{g}}$.\n\\end{prop}\n\\begin{proof}\nBy the Weyl-Kac character formula (cf. [K$_3$, Theorem 2.2.1]) and the identity \\eqref{e13}, for any $\\Lambda', \\Lambda''\\in P_+$, \n\\begin{align*}\n\\left(\\sum_{w\\in W}\\varepsilon(w)e^{w\\rho}\\right)&\\cdot ch\\, L(\\Lambda')\\cdot ch\\, L(\\Lambda'')\\\\\n&=\\left(\\sum_{\\lambda\\in P^o(\\Lambda')}c_{\\Lambda',\\lambda}e^{\\lambda}\\right)\\cdot\\left(\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda''+\\rho)}\\right)\\\\\n&=\\sum_{\\lambda\\in P^o(\\Lambda')}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda''+\\lambda+\\rho)},\n\\,\\, \\text{by}\\, \\eqref{e4.1}\\\\\n&= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)e^{w(v_{\\Lambda,\\Lambda'',\\lambda}(\\Lambda+\\rho))+S_{\\Lambda,\\Lambda'',\\lambda}\\delta},\\,\\,\\text{by}\\, \\eqref{e14}\\\\\n &= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}c_{\\Lambda',\\lambda}\\sum_{w\\in W}\\varepsilon(w)\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})e^{w(\\Lambda+\\rho)}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\\\\n &= \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}\\sum_{w\\in W}\\varepsilon(w)e^{w(\\Lambda+\\rho)}\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}.\n\\end{align*}\nThus,\n\\[ch\\, \\bigl(L(\\Lambda')\\otimes L(\\Lambda'')\\bigr) = \\sum_{\\Lambda\\in \\bar{P}_{+}^{(m)}}ch\\, L(\\Lambda)\\bigl(\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}\\bigr).\n\\]\nTo prove the second part of the proposition, use [KR, Proposition 10.3].\nThis proves the proposition.\n\\end{proof}\n\\begin{remark} For an affine Kac-Moody Lie algebra $\\fg$, if we consider the tensor product decomposition of $L(\\Lambda')\\otimes L(\\Lambda'')$ with respect to the derived subalgebra $\\fg'$ (i.e., without the $d$-action), then the components $L(\\Lambda)$ are precisely of the form \n$\\Lambda\\in \\Lambda'+\\Lambda''+\\overset{\\circ}{Q}$, where $\\overset{\\circ}{Q}$ is the root lattice of $\\overset{\\circ}{\\mathfrak{g}}$\n(cf. [KW]). Thus, the determination of the eigen semigroup and the saturated eigen semigroup is fairly easy for $\\fg'$.\n\\end{remark}\n\nLet $\\theta=\\sum_{i=1}^\\ell h_i\\alpha_i$ be the highest root of $\\overset{\\circ}{\\mathfrak{g}}$ (with respect to a choice of the positive roots), written as a linear combination of the simple roots $\\{\\alpha_1, \\dots, \\alpha_\\ell\\}$ of $\\overset{\\circ}{\\mathfrak{g}}$. Let\n\\[ S:= \\{\\sum_{i=0}^\\ell \\, n_i\\alpha_i: n_i\\geq 0 \\,\\,\\,\\text{for any}\\, i\\,\\,\\,\\text{and}\\, 0\\leq n_i< h_i \\,\\,\\text{for some}\\,\\, 0\\leq i\\leq \\ell \\},\\]\nwhere $h_0:=1$.\n\\begin{prop} \\label{prop4.1} Let $\\fg$ be an untwisted affine Kac-Moody Lie algebra as above. Then, \nfor any $\\Lambda \\in P_+$ with $\\Lambda (c)>0$, \n\\[P^o(\\Lambda)_+= S(\\Lambda)\\cap P_+,\\]\nwhere $P^o(\\Lambda)_+:=P^o(\\Lambda)\\cap P_+$ and\n$S(\\Lambda)=\\{\\Lambda-\\beta: \\beta \\in S\\}.$\n\\end{prop}\n\\begin{proof}\nTake $\\lambda \\in S(\\Lambda)$. Then, for any $n\\geq 1$, \n\\[\n\\Lambda-(\\lambda+n\\delta)=\\bigl(\\sum_{i=0}^\\ell\\, n_{i}\\alpha_{i}\\bigr)-n\\delta=(n_{0}-n)\\alpha_{0}+ \\sum_{i=1}^\\ell\\,(n_{i}-nh_{i})\\alpha_{i},\n\\]\n since $\\alpha_{0}:=\\delta-\\theta$. Now, the coefficient of some $\\alpha_{i}$\nin the above sum is negative, for any positive $n$, since $\\lambda \\in S(\\Lambda)$. Thus, \n $\\lambda+n\\delta$\ncould not be a weight of $L(\\Lambda)$ for any positive $n$. Therefore, if $\\lambda \\in P(\\Lambda)\\cap S(\\Lambda)$, then it is $\\delta$-maximal.\n\nBy [Kac, Proposition 12.5(a)], if $\\Lambda(c)\\neq 0$, then $ S(\\Lambda) \\cap P_+\\subset P(\\Lambda).$ Therefore,\n $ S(\\Lambda) \\cap P_+\\subset P^o(\\Lambda)_+.$\n\nConversely, take $\\lambda \\in P^o(\\Lambda)_+$. Then, $\\lambda \\in P(\\Lambda) \\cap P_+$ and \n$\\lambda+\\delta \\notin P(\\Lambda)$. Express $\\lambda=\\Lambda - n_0\\alpha_0-\\sum_{i=1}^\\ell\\, n_i\\alpha_i$, for some \n$n_i\\in \\bz_+$. Then, \n\\[\\lambda+\\delta=\\Lambda - (n_0-1)\\alpha_0-\\sum_{i=1}^\\ell\\, (n_i-h_i)\\alpha_i.\\]\nAgain applying [Kac, Proposition 12.5(a)], $\\lambda+\\delta\\notin P(\\Lambda)$ if and only if $\\lambda+\\delta\\not\\leq \\Lambda $,\ni.e., for some $0\\leq i\\leq \\ell$, $n_i< h_i$. Thus, $\\lambda \\in S(\\Lambda)$. This proves the proposition.\n\\end{proof} \n\n\n\n\\section{$A_{1}^{(1)}$ Case}\n\nIn this section, we consider $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}=\\left(\\bigoplus_{n\\in\\mathbb{Z}}\\mathbb{C}t^{n}\\otimes\n\\mathfrak{sl}_{2}\\right)\\oplus\\mathbb{C}c\\oplus\\bc d$.\nIn this case $\\mathfrak{h}^{*}=\\mathbb{C}\\alpha\\oplus\\mathbb{C}\\delta\\oplus\\mathbb{C}\\Lambda_{0}$, where \n$\\alpha$ is the simple root of $\\mathfrak{sl}_{2}$ and \n${\\Lambda_0}_{|\\overset{\\circ}{\\mathfrak{h}}\\oplus \\bc d}\\equiv 0$ and $\\Lambda_0(c)=1$. Then, $\\Lambda_0$ is a zeroeth fundamental weight. \nThe simple roots of $\\widehat{\\mathfrak{sl}_{2}}$ are \n $\\alpha_0:=\\delta-\\alpha$ and $\\alpha_1:=\\alpha$. The simple coroots are \n $\\alpha_0^\\vee :=c-\\alpha^\\vee$ and $\\alpha_1^\\vee :=\\alpha^\\vee$. It is easy to see that an element of $\\fh^*$ of the form $m\\Lambda_0+\\frac{j}{2}\\alpha$ belongs to $P_+$ if and only if $m,j\\in \\bz_+$ and $m\\geq j$. \n\n\nSpecializing Proposition \\ref{prop4.1} to the case of $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$, we get the following.\n\n\\begin{corollary} \\label{cor5.1} For $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$\nand $\\Lambda=m\\Lambda_{0}+\\frac{j}{2}\\alpha\\in P_+$, \n\\beqn \\label{e5.2}\nP^o(\\Lambda)_+=\\left\\{ \\Lambda-k\\alpha,\\,\\Lambda-l(\\delta -\\alpha)\\;:\\; k,l\\in \\bz_+ \\,\\,\\text{and}\\,\\, k\\leq\\frac{j}{2},\\, l\\leq\\frac{m-j}{2}\\right\\}. \n\\eeqn\n\\end{corollary}\n\\begin{proof} \nThe corollary follows from Proposition \\ref{prop4.1} since $m_1\\Lambda_0+\\frac{m_2}{2}\\alpha +\n m_3 \\delta $ belongs to $P_+$ if and only if $m_1,m_2\\in \\bz_+$ and $m_1\\geq m_2$.\n\\end{proof}\n\nLet $\\pi$ be the projection $\\mathfrak{h}^{*}=\\mathbb{C}\\Lambda_{0}\\oplus \\bc \\alpha\\oplus\\mathbb{C}\\delta\n\\to \\mathbb{C}\\Lambda_{0}\\oplus \\bc \\alpha$. \n\\begin{lemma} \\label{lemma5.1} Let $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$. For $\\Lambda=m\\Lambda_{0}+\\frac{j}{2}\\alpha \\in P_+$ (i.e., \n$m,j\\in \\bz_+$ and $m\\geq j$) such that $m>0$,\n\\beqn\\label{e5.1} \\pi(P^o(\\Lambda))=\\{\\Lambda+k\\alpha: k\\in \\bz\\}.\n\\eeqn\nMoreover, for any $k\\in \\bz$, let $n_k$ be the unique integer such that $\\Lambda+k\\alpha+n_k\\delta\\in P^o(\\Lambda)$. Then, writing\n$k=qm+r, 0\\leq r0$. \nThen,\n\\begin{align*}\n\\pi\\left(T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}\\right)=\\{ \\Lambda'+k\\alpha\\;:\\; k\\in\\mathbb{Z},\\, & k\\equiv\\frac{1}{2}\\left(j-j'-j''\\right) \\\\\n&\\text{or}\\,\\, k\\equiv -\\frac{1}{2}\\left(j+j'+j''\\right)-1\\,\\text{mod}\\,M\\} ,\n\\end{align*}\nwhere $M:=m+2$. In particular, by the equation \\eqref{eq1001}, $T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}$ is nonempty if and\nonly if $\\frac{j-j'-j''}{2}\\in \\bz$. \n\nMoreover, for \n$\\lambda=\\Lambda'+k\\alpha+n_k\\delta \\in T_{\\Lambda}^{\\Lambda',\\Lambda''}$,\n\\[\nv_{\\Lambda,\\Lambda'',\\,\\lambda}=\\begin{cases}\nT_{\\frac{k-\\frac{1}{2}\\left(j-j'-j''\\right)}{M}\\alpha^\\vee}, & \\text{if}\\; k\\equiv\\frac{1}{2}\\left(j-j'-j''\\right)\\,\\mod\\, M\\\\\ns_{1}T_{-\\frac{k+\\frac{1}{2}\\left(j+j'+j''\\right)+1}{M}\\alpha^\\vee },& \\text{if}\\; k\\equiv-\\frac{1}{2}\\left(j+j'+j''\\right)-1\\,\\text{mod}\\, M,\n\\end{cases}\n\\]\nwhere $T_{n\\alpha^\\vee}$ is defined by the equation \\eqref{e5.3}. Further,\n\\[\n S_{\\Lambda,\\Lambda'',\\lambda}=n_k+\\frac{\\left(k-\\frac{1}{2}\\left(j-j'-j''\\right)\\right)\\left(k+\\frac{1}{2}\\left(j+j'+j''\\right)+1\\right)}{M}.\n\\]\n\\end{lemma}\n\\begin{proof}\nFollows from the fact that $W=\\stackrel{\\circ}{W}\\rtimes\\mathbb{Z}\\alpha^\\vee$\nand that $\\rho=2\\Lambda_{0}+\\frac{1}{2}\\alpha$.\\end{proof}\nWe have the following very crucial result.\n\\begin{prop}\\label{maximum}\nFix $\\Lambda, \\Lambda'$ and $\\Lambda''$ as in Lemma \\ref{lemma5.1'} and asume that $\\frac{j-j'-j''}{2}\\in \\bz$ and both of $m',m''>0$. Then, the maximum of \n$\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}\\,\\,\\text{and}\\, \\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\} $\n is achieved precisely when $\\pi(\\lambda)=\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha$. \n\\end{prop}\n\\begin{proof} By Lemma \\ref{lemma5.1'} and the explicit description of the length function of $T_{n\\alpha^\\vee}$ (cf. [K$_3$, Exercise 13.1.E.3]), \n$$\\pi\\{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\,\\Lambda''}:\\varepsilon (v_{\\Lambda,\\Lambda'',\\lambda})=1\\}=\\{\\Lambda'+k_l\\alpha:l\\in \\bz\\},$$\nwhere $M:=m+2$ and $k_l:=\\frac{j-j'-j''}{2}+lM$. \nTake $\\lambda=\\Lambda'+k_l\\alpha\\in \\pi(T_{\\Lambda}^{\\Lambda',\\,\\Lambda''})$ for $l\\in \\bz$.\n Write $k_l=q_lm'+r_l$ for $q_l\\in \\bz$ and $0\\leq r_l< m'$. Then, by Lemmas \\ref{lemma5.1}, \\ref{lemma5.2'} and \\ref{lemma5.1'}, for $\\lambda=\\Lambda'+k_l\\alpha$ (setting $J:=\\frac{j-j'-j''}{2}$), \n\\begin{align*}\nS_{\\Lambda,\\Lambda'',\\lambda} & = n_{r_l}-\\frac{(J+j'+lM+r_l)(J+lM-r_l)}{m'}+l(lM+1+j)\\\\\n&=l^2M(1-\\frac{M}{m'})+l(1+j-\\frac{M(j-j'')}{m'})-\\frac{(j-j'')^2-j'^2}{4m'}+\\frac{r_l^2}{m'}\n+\\frac{r_lj'}{m'}+n_{r_l}\\\\\n&= l^2M(1-\\frac{M}{m'})+l(1+j-\\frac{M}{m'}(j-j''))-\\frac{(j-j'')^2-j'^2}{4m'}+p(k_l),\n\\end{align*}\nwhere \n\\[\np(k_l):= \\frac{r_l^2}{m'}+\\frac{r_l}{m'}j' +n_{k_l}.\n\\]\nLet $P=P_{m',j'}:\\mathbb{R}\\rightarrow\\mathbb{R}$ be the following function:\n\\begin{eqnarray*}\n P(s) & := & \\begin{cases}\n\\frac{(s-\\frac{m'}{2}k)^{2}}{m'}-\\frac{(j')^2}{4m'}, & \\text{if}\\;\\left|s-\\frac{m'}{2}k\\right|\\leq\\frac{j'}{2}\\;\\text{for some }k\\in2\\mathbb{Z}\\\\\n\\frac{(s-\\frac{m'}{2}k)^{2}}{m'}-\\frac{(m'-j')^2}{4m'}, & \\text{if}\\;\\left|s-\\frac{m'}{2}k\\right|\\leq\\frac{m'-j'}{2}\\;\\text{for some }k\\in2\\mathbb{Z}+1.\n\\end{cases}\n\\end{eqnarray*}\nLet $k_s\\in \\bz$ be such a $k$. (Of course, $k_s$ depends upon $m'$ and $j'$.)\n\\begin{claim}\n$P(s) = p (s-\\frac{j'}{2})$ for $s \\in \\frac{j'}{2}+\\mathbb{Z}$.\n\\end{claim}\n\\begin{proof}\nClearly, both of $P$ and $p$ are periodic with period $m'$. So, it is enough to show that $P(s) = p (s-\\frac{j'}{2})$, for $s-\\frac{j'}{2}$\nequal to any of the integral points of the interval $[-j', m'-j']$. By Lemma \\ref{lemma5.2'} and the identity \\eqref{ne18}, for\nany integer $-j'\\leq r\\leq 0$,\n$$p (r)= \\frac{1}{m'}r(r+j'),$$\nand for any integer $0\\leq r\\leq m'-j'$, \n$$p (r)= \\frac{r(r+j')}{m'}-r.$$\nFrom this, the claim follows immediately. \n\\end{proof}\n Fix $m'>0$. Let \n\\begin{align*}\nI :=\\{(t,j',m'',j'',j)\\in\\mathbb{R}^5 \\,:\\, &0\\leq j' \\leq m',\\, 1 \\leq m'',\\\\\n &0\\leq j'' \\leq m'',\\, 0\\leq j\\leq m'+m''\\}.\n\\end{align*}\nDefine $F: I \\rightarrow \\mathbb{R}$ by\n\\begin{eqnarray*}\nF: (t,j',m'',j'',j) &\\mapsto& t^{2}M(1-\\frac{M}{m'})+t\\bigl(j(1-\\frac{M}{m'})+1+\\frac{M}{m'}j''\\bigr)\\\\\n&&+\\frac{(j')^2-(j-j'')^{2}}{4m'} +P(\\frac{1}{2}\\left(j-j''\\right)+tM).\n\\end{eqnarray*}\nThus, $F$ is a continuous, piecewise smooth function with failure of differentiability along the set $$\\{(t,j',m'',j'',j)\\in I\\,:\\, \\frac{1}{2}(j\\pm j'-j'')+tM\\in m'\\mathbb{Z} \\}.$$\n\\begin{claim}\\label{claim1}\nLet $\\Delta (t)=\\Delta(t, j',m'',j'', j):=F(t+1,j',m'',j'',j)-F(t,j',m'',j'',j)$. Then, on $I$,\n\\begin{enumerate}\n\\item $\\Delta$ is a nonincreasing function of $t$\n\\item $\\Delta$ is increasing with respect to $j''$\n\\item $\\Delta$ is nonincreasing in $j$\n\\item\\label{delclaimb} $\\Delta(0)$ is decreasing in $m''$\n\\item\\label{delclaima} $\\Delta(-1)$ is nondecreasing in $m''$.\\end{enumerate}\n\\end{claim}\n\n\\begin{proof}\nWe compute and give bounds for the partial derivatives of $\\Delta$, where they exist. \n\\begin{eqnarray*}\n\\Delta(t) & =&\\,2tM(1-\\frac{M}{m'})+\\bigl((j+M)(1-\\frac{M}{m'})+1+\\frac{M}{m'}j''\\bigr)\\\\\n&&+P(tM+M+\\frac{1}{2}(j-j''))-P(tM+\\frac{1}{2}(j-j'')).\n\\end{eqnarray*}\nHence, \n\\begin{eqnarray*}\n\\partial_t\\Delta(t) & = & 2M(1-\\frac{M}{m'})+M\\bigl(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\bigr)\\\\\n & = & 2M(1-\\frac{M}{m'})+2\\frac{M}{m'}(M-\\frac{m'}{2}k_{1}+\\frac{m'}{2}k_{0})\\\\\n & = & 2M(1-\\frac{k_{1}-k_{0}}{2}),\n\\end{eqnarray*}\nwhere $k_1:=k_{(t+1)M+\\frac{1}{2}(j-j'')}$ and $k_0:=k_{tM+\\frac{1}{2}(j-j'')}$.\nSince $2\\leq k_{1}-k_{0}$, we see that $\\partial_t\\Delta\\leq0$, wherever $\\partial_t \\Delta$ exists. \n Since $\\Delta$ is continuous everywhere and differentiable \non all but a discrete set, $\\Delta$ is nonincreasing in $t$.\n\\[\n\\partial_{j''}\\Delta(t)=\\frac{M}{m'}-\\frac{1}{2}\\left(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\right).\n\\]\nNow, $|P'|\\leq1$, so $\\frac{M}{m'}+1\\geq\\partial_{j''}\\Delta\\geq\\frac{M}{m'}-1=\\frac{m''+2}{m'}>0$.\n\nFor (3):\n\\begin{eqnarray*}\n\\partial_{j}\\Delta(t) & = & 1-\\frac{M}{m'}+\\frac{1}{2}\\bigl(P'(tM+M+\\frac{1}{2}(j-j''))-P'(tM+\\frac{1}{2}(j-j''))\\bigr)\\\\\n & = & 1-\\frac{M}{m'}+\\frac{1}{m'}\\left(M-\\frac{m'}{2}k_{1}+\\frac{m'}{2}k_{0}\\right)\\\\\n & = & 1-\\frac{k_{1}-k_{0}}{2}\\leq 0.\n\\end{eqnarray*}\n (\\ref{delclaimb}) and (\\ref{delclaima}) follow from the following calculation:\n\\begin{align*}\n\\partial_{m''}\\Delta=&2t(1-2\\frac{M}{m'})+(1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j))\\\\\n&+(t+1)P'(tM+M+\\frac{1}{2}(j-j''))-tP'(tM+\\frac{1}{2}(j-j'')).\n\\end{align*}\nHence, \n\\begin{eqnarray*}\n\\partial_{m''}\\Delta(0) & = & 1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)+P'(M+\\frac{1}{2}(j-j''))\\\\\n & \\leq & 1-2\\frac{M}{m'}+\\frac{m''}{m'}+1\\\\\n & = & \\frac{-m''-4}{m'}<0,\n\\end{eqnarray*}\nand \n\\begin{eqnarray*}\n\\partial_{m''}\\Delta(-1) & = & -2(1-2\\frac{M}{m'})+(1-2\\frac{M}{m'}+\\frac{1}{m'}(j''-j))+P'(-M+\\frac{1}{2}(j-j''))\\\\\n & = & -1+2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)+P'(-M+\\frac{1}{2}(j-j''))\\\\\n & = & -1+2\\frac{M}{m'}+\\frac{1}{m'}(j''-j)-2\\frac{M}{m'}+\\frac{1}{m'}(j-j'')-k_{0}\\\\\n & = & -1-k_{0}.\n\\end{eqnarray*}\nNote that $k_{0}\\leq-1$ since $-\\frac{(j-j'')}{2}-M<-\\frac{m'}{2}$.\nThus, $\\partial_{m''}\\Delta(-1)\\geq0$.\\end{proof}\n\\begin{claim}\n{\\it The maximum of $F=F(-, j',m'',j'',j):\\,\\mathbb{Z}\\rightarrow\\mathbb{R}$ occurs at $0$.}\n\\end{claim}\n\\begin{proof}\nWe show that $\\Delta(-1)>0 > \\Delta(0)$. Since $\\Delta$ is\nnonincreasing in $t$, it would follow that $F(0)>F(t)$ for all $t\\in\\mathbb{Z}_{\\neq0}$.\n\nLet us begin with $\\Delta(-1)$. By the previous claim \\ref{claim1}, $\\Delta(-1)$\nis as small as possible when $m''=1$, $j''=0$, and $j=m'+1$. So,\nlet us compute with these values:\n\n\\begin{eqnarray*}\n\\Delta(-1) & \\geq & \\frac{6}{m'}+1+P(\\frac{1}{2}m'+\\frac{1}{2})-P(-2-\\frac{1}{2}m'-\\frac{1}{2})\\\\\n & = & \\frac{6}{m'}+1+\\frac{(\\frac{1}{2}m'+\\frac{1}{2}-\\frac{1}{2}m'k_{1})^{2}}{m'}-\\frac{(2+\\frac{1}{2}m'+\\frac{1}{2}+\n\\frac{1}{2}m'k_{0})^{2}}{m'}\\\\\n&+&\\begin{cases}\n\\frac{m'}{4}-\\frac{j'}{2} & \\text{ if }k_{0}\\text{ odd},\\, k_{1}\\text{ even}\\\\\n0 & \\text{ if }k_{1}-k_{0}\\text{ even}\\\\\n\\frac{j'}{2}-\\frac{m'}{4} & \\text{ if }k_{1}\\text{ odd},\\, k_{0}\\text{ even}.\n\\end{cases}\n\\end{eqnarray*}\nNote that for $m'\\geq 5$, the possible values of $(k_{1},k_{0})$ are\n$(1,-1)$; $(1,-2)$; or $(2,-2)$. So, the result, that $\\Delta (-1)>0$, is established\nby considering such pairs directly and by cases for smaller $m'$.\n\nFor $\\Delta(0)$, we take $m''=1$, $j''=1$, and $j=0$.\n\\begin{eqnarray*}\n\\Delta(0) & = & \\bigl(\\frac{-3(3+m')}{m'}+1+\\frac{3+m'}{m'}\\bigr)+P(\\frac{1}{2}+2+m')-P(-\\frac{1}{2})\\\\\n & =&1 - \\frac{2(3+m')}{m'}+P(\\frac{1}{2}+2+m')-P(-\\frac{1}{2})\\\\\n & = & 1-\\frac{2(3+m')}{m'}+\\frac{(\\frac{1}{2}+2+m'-\\frac{1}{2}m'k_{1})^{2}}{m'}-\\frac{(\\frac{1}{2}+\\frac{1}{2}m'k_{0})^{2}}{m'}\n\\\\ &+&\\begin{cases}\n\\frac{m'}{4}-\\frac{j'}{2} & \\text{ if }k_{0}\\text{ odd},\\, k_{1}\\text{ even}\\\\\n0 & \\text{ if }k_{1}-k_{0}\\text{ even}\\\\\n\\frac{j'}{2}-\\frac{m'}{4} & \\text{ if }k_{1}\\text{ odd},\\, k_{0}\\text{ even}.\n\\end{cases}\n\\end{eqnarray*}\nFor $m'\\geq 5$, the possible values of $(k_{1}, k_{0})$ are\n $(3,-1)$; $(3,0)$; or $(2,0)$. So, again the result, that $\\Delta (0)<0$, is established\nby considering such pairs directly and by cases for smaller $m'$.\n\\end{proof}\n This completes the proof of the proposition.\n\\end{proof}\n\\begin{rem}\\label{newremark}\nWe have shown that $F(l,j',m'',j'',j) = S_{\\Lambda,\\Lambda'',\\lambda}$ for integral values of $l$. If $l$ is not an integer, then $\\lambda_l := \\Lambda' + (lM+J)\\alpha$ may not be in $\\pi(T_\\Lambda^{\\Lambda',\\Lambda''})$, in which case $ S_{\\Lambda,\\Lambda'',\\lambda_l}$ is not defined. On the other hand, if $\\lambda_l \\in \\pi(T_\\Lambda^{\\Lambda',\\Lambda''})$, we note that the equality $F(l,j',m'',j'',j) = S_{\\Lambda,\\Lambda'',\\lambda_l}$ holds, as can be seen by letting $k_l = lM -\\frac{1}{2} (j+j'+j'')-1$ in the above proof.\n\\end{rem}\n\nNow, let us apply the same analysis to the case that $\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=-1$.\nBy Lemma \\ref{lemma5.1'}, this corresponds to $k_l=-\\frac{1}{2}\\left(j+j'+j''\\right)-1+lM$.\n For \n$\\lambda=\\Lambda'+k_l\\alpha$, let us denote the function $S_{\\Lambda,\\Lambda'',\\lambda}$ by $G_\\bz(l)=G_\\bz(l, j',m'',j'',j)$. Thus,\n$G_\\bz:\\bz \\to \\bz$. \n\\begin{lemma} \\label{lemma5.9} Define the function $G=G(-, j',m'',j'',j):\\br \\to \\br$ by \n\\[G(t, j',m'',j'',j)=F(t-\\frac{j+1}{M}, j',m'',j'',j).\\]\nThen, $G_{|\\bz}=G_\\bz.$\n\nHence, $S_{\\Lambda,\\Lambda'',\\lambda}$\nhas a maximum when $l=0$ or $l=1$.\n\\end{lemma}\n\\begin{proof}\nBy the proof of Proposition \\ref{maximum} and Remark \\ref{newremark}, $S_{\\Lambda,\\Lambda'',\\lambda+(j+1)\\alpha} = F(l)$, for $\\lambda=\\Lambda'+k_l\\alpha$. Since $\\lambda = \\Lambda' +(-\\frac{1}{2}\\left(j+j'+j''\\right)-1+lM)\\alpha$, by Proposition \\ref{maximum}, $S_{\\Lambda,\\Lambda'',\\lambda} = F(l-\\frac{j+1}{M})$.\n This proves the lemma.\n\\end{proof}\n\nFrom Lmma \\ref{lemma5.9} and the definition of $F$, it is easy to see that\n\\begin{equation}\\label{eq22}\nG(1-t,m'-j',m'',m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j)=G(t, j',m'', j'',j),\n\\end{equation}\nfor any $t\\in \\br$.\nHence, if the maximum of $G_\\bz$ occurs at\n1, it is equal to \n\\begin{equation}\\label{eq23}\nG(0, m'-j',m'',m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j).\n\\end{equation}\nWe also record the following identity, which is easy to prove from the definition of $F$.\n\\begin{equation}\\label{neq23}\nF (0,m'-j',m'', m''-j'',m'+m''-j)+\\frac{1}{2}(j'+j''-j)=F(0,j',m'', j'',j).\n\\end{equation}\n\nAs a corollary of Proposition \\ref{maximum} and Lemma \\ref{lemma5.9}, we get the following `Non-Cancellation Lemma'.\n\\begin{corollary}\\label{corcancel} Let $\\Lambda, \\Lambda',\\Lambda''$ be as in Proposition \\ref{maximum} and let\n\\begin{align*}\n\\mu^{\\Lambda',\\Lambda''}_{\\Lambda}&:=\\max\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}\\,\\,\\,\\text{and}\\,\\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\},\\\\\n\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}&:=\\max\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda} :\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}\\,\\,\\,\\text{and}\\,\\,\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=\n-1\\right\\}. \n\\end{align*}\nAssume that ${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$. Then,\n\\[{\\mu}^{\\Lambda'',\\Lambda'}_{\\Lambda}\\neq\\bar{\\mu}^{\\Lambda'',\\Lambda'}_{\\Lambda}.\\]\n\\end{corollary}\n\\begin{proof}\nWe proceed in two cases:\n\n{\\em Case I.} Suppose the maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ \noccurs when $\\pi(\\lambda)=\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha$ (cf. Lemma \\ref{lemma5.9}).\nThis means that the $\\delta$-maximal weights of $L(\\Lambda')$ through\n$\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha$ and through\n$\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha$ have the same $\\delta$\ncoordinate (cf. Proposition \\ref{maximum}). By (next) Lemma \\ref{lemma5.11}, \nwe know that this occurs if and only if \none of the following two conditions are satisfied:\n\n(1) $\\left|\\frac{1}{2}\\left(j-j''\\right)\\right|\\leq\\frac{j'}{2}$\nand $\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}$, or \n\n(2) $\\frac{1}{2}\\left(j+j''\\right)+1=\\frac{1}{2}\\left(j-j''\\right)$.\n\nThe latter is clearly impossible, while the former condition is fulfilled\nprecisely when $\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}$.\n\n So, for the equality ${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ in this case,\nthe neccesary and sufficient condition is: \n\\beqn \\label{e5.10.3} \\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2}.\n\\eeqn\n\n{\\em Case II.} Suppose the maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ \noccurs when $\\pi(\\lambda )= \\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1-M)\\alpha$.\nThen, by the identities \\eqref{eq23} and \\eqref{neq23}, we get \n\\beqn\\label{eq24} G (0,m'-j',m'', m''-j'',m'+m''-j)=F (0,m'-j',m'', m''-j'',m'+m''-j).\n\\eeqn\nSo, from the case I, we get in this case II, \n${\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}=\\bar{\\mu}^{\\Lambda',\\Lambda''}_{\\Lambda}$ if and only if\n\\beqn\\label{eq25}\n\\frac{1}{2}\\bigl((m'+m''-j)+(m''-j'')\\bigr)+1\\leq\\frac{1}{2}(m'-j').\n\\eeqn\nSo, if either of the inequalities \\eqref{e5.10.3} or \\eqref{eq25} is satisfied, then none of them can be satisfied \nfor the triple $(\\Lambda, \\Lambda',\\Lambda'')$ replaced by $(\\Lambda, \\Lambda'',\\Lambda')$. This proves the corollary.\n\\end{proof}\n\\begin{lem}\\label{lemma5.11}\nSuppose $\\Lambda'-(\\frac{1}{2}\\left(j+j'+j''\\right)+1)\\alpha+n_1 \\delta$ and $\\Lambda'+\\frac{1}{2}\\left(j-j'-j''\\right)\\alpha+ n_2 \\delta$ are $\\delta$-maximal weights of $L(\\Lambda')$. Then $n_1 = n_2$ if and only if \n\\[\n\\left|\\frac{1}{2}\\left(j-j''\\right)\\right|\\leq\\frac{j'}{2} \\quad \\text{and}\\,\\,\\,\\,\n\\frac{1}{2}\\left(j+j''\\right)+1\\leq\\frac{j'}{2},\n\\]\nor $\\frac{1}{2}\\left(j+j''\\right)+1=\\frac{1}{2}\\left(j-j''\\right)$.\n\\end{lem}\n\\begin{proof}\nFix an integer $n$ and consider the set $P_n = \\{ \\nu \\in P(\\Lambda') \\, : \\, \\Lambda' - \\nu = k\\alpha + n \\delta,\\, k \\in \\bz \\}$. We give a description of $P_n \\cap P^o(\\Lambda')$. Clearly, $P_n = \\{\\lambda, \\lambda -\\alpha, \\dots, \\lambda - \\langle\\lambda,\\alpha^\\vee\\rangle\\alpha \\}$ for some $\\lambda=\\lambda_n$ and that this $\\lambda$ is uniquely determined by $n$ (cf. [K$_3$, Exercise 2.3.E.2]). Suppose that some $\\mu \\in P_n$ is not $\\delta$-maximal, then none of $\\{\\mu, \\dots, \\mu - \\langle\\mu, \\alpha^\\vee\\rangle\\alpha \\}$ are $\\delta$-maximal, since\nif $\\mu+k\\delta\\in P(\\Lambda')$, then the whole string $\\{\\mu +k\\delta, \\dots, \\mu +k\\delta- \\langle\\mu, \\alpha^\\vee\\rangle\\alpha \\}\n\\subset P(\\Lambda')$. In particular, if $\\lambda - \\alpha$ is $\\delta$-maximal, then so is $\\lambda$. Hence, $\\mathfrak{g}_{\\delta-\\alpha}L(\\Lambda')_\\lambda = 0$ and $\\mathfrak{g}_{\\alpha}L(\\Lambda')_\\lambda = 0$. Therefore, $\\lambda$ is the highest weight $\\Lambda'$. Thus, $P_n \\cap P^o(\\Lambda')$ is either empty, or $\\lambda=\\Lambda'$ (in the case that $n = 0$), or the set $\\{\\lambda,s_1\\lambda\\}$. From this and Corollary \\ref{cor5.1} the lemma follows easily.\n\\end{proof}\n\n\\section{Saturation factor for the $A_{1}^{(1)}$ Case}\n\n We assume that $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$ in this section.\n\\begin{defn}\nLet $\\Lambda'\\in P_{+}^{(m')}, \\Lambda''\\in P_{+}^{(m'')}$ and $\\Lambda\\in P_{+}^{(m'+m'')}$. Then, we call $L(\\Lambda+n\\delta)$\nthe \\textit{$\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nthrough $\\Lambda$} if $L(\\Lambda+n\\delta)$ is a submodule of $L(\\Lambda')\\otimes L(\\Lambda'')$\nbut $L(\\Lambda+m\\delta)$ is not a component for any $m>n$.\\end{defn}\n\\begin{thm} \\label{thm6.2} Let $\\Lambda', \\Lambda'', \\Lambda$ be as in Proposition \\ref{maximum} . Then, \n$L(\\Lambda+n\\delta)$ is a $\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nif $n=\\min(n_{1},n_{2})$, where $n_{1}$ is such that $\\Lambda-\\Lambda''+n_{1}\\delta\\in P^o(\\Lambda')$\nand $n_{2}$ is such that $\\Lambda-\\Lambda'+n_{2}\\delta\\in P^o(\\Lambda'')$.\\end{thm}\n\\begin{proof} This follows immediately by combining Propositions \\ref{tensor}, \\ref{maximum} and Lemma \\ref{lemma5.1'}.\n\\end{proof}\n\\begin{lem} Fix a positive integer $N$. \nLet $\\Lambda\\in \\bar{P}_+$ and let $\\lambda\\in\\Lambda+Q$, where $Q$ is the root lattice $\\mathbb{Z}\\alpha\\oplus\\mathbb{Z}\\delta$ of\n$\\widehat{\\mathfrak{sl}_{2}}$. \nThen, $N\\lambda\\in P^o(N\\Lambda)$ if\nand only if $\\lambda\\in P^o(\\Lambda)$. \n\\end{lem}\n\\begin{proof}\nThe validity of the lemma is clear for $\\lambda\\in P^o(\\Lambda)_+$ from Corollary \\ref{cor5.1}. But since\n$ P^o(\\Lambda)=W\\cdot ( P^o(\\Lambda)_+)$,\n and the action of $W$ on $\\fh^*$ is linear, the lemma follows for any $\\lambda \\in P^o(\\Lambda)$.\\end{proof}\n\\begin{cor}\\label{cor6.4}\nLet $d_o\\in\\mathbb{Z}_{>1}$. Let $\\Lambda$, $\\Lambda'$, $\\Lambda'' \\in P_+$\nbe such that $\\Lambda-\\Lambda'-\\Lambda''\\in Q$ and \n $L(N\\Lambda)$ is a submodule of $L(N\\Lambda')\\otimes L(N\\Lambda'')$, for some $N\\in \\bz_{>0}$. \nThen, $L(d_o\\Lambda)$ is a submodule of $L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$.\n\nSuch a $d_o$ is called a {\\em saturation factor}. \n\\end{cor}\n\\begin{proof} If $\\Lambda'(c)=0$ or $\\Lambda''(c)=0$, then \n$$L(N\\Lambda')\\otimes L(N\\Lambda'')\\simeq L(N(\\Lambda'+\\Lambda'')),$$\nfor any $N\\geq 1$. Thus, the corollary is clearly true in this case. So, let us assume that both of \n$\\Lambda'(c)>0$ and $\\Lambda''(c)>0$. \nLet $L(N\\Lambda+n\\delta)$ be the $\\delta$-maximal component of $L(N\\Lambda')\\otimes L(N\\Lambda'')$\nthrough $L(N\\Lambda)$, for some $n\\geq 0$. For any $\\Psi\\in P_+$, let $\\bar{\\Psi}\\in \\bar{P}_+$ be the projection $\\pi(\\Psi)$ defined just before Lemma \\ref{lemma5.1}. Applying Theorem \\ref{thm6.2} to $\\bar{\\Lambda}', \\bar{\\Lambda}'', \\bar{\\Lambda}$, and observing that \n\\beqn \\label{eq6.4.0} L(\\bar{\\Psi}+k\\delta)\\simeq L(\\bar{\\Psi})\\otimes L(k\\delta)\\eeqn\nand $L(k\\delta)$ is one dimensional, we get that there is a $\\delta$-maximal component $L(\\Lambda+\\tilde{n}\\delta)$ of \n$L(\\Lambda')\\otimes L(\\Lambda'')$ through $L(\\Lambda)$, for some (unique) $\\tilde{n}\\in \\bz$. \n\nAgain applying Theorem \\ref{thm6.2} to $N\\bar{\\Lambda}', N\\bar{\\Lambda}'', N\\bar{\\Lambda}$, and observing \n(using Corollary \\ref{cor5.1}) that \n\\beqn \\label{eq6.4.1} P^o(N\\bar{\\Psi})\\supset NP^o(\\bar{\\Psi}),\\eeqn\nwe get that $L(N\\Lambda+N\\tilde{n}\\delta)$ is the $\\delta$-maximal component of \n$L(N\\Lambda')\\otimes L(N\\Lambda'')$ through $L(N\\Lambda)$. Thus, $n=N\\tilde{n}$. In particular,\n\\beqn\\label{e6.3} \\tilde{n}\\geq 0.\\eeqn\n Let \n\\beqn\\label{e6.4}\n\\sum_{\\lambda\\in T_{\\bar{\\Lambda}}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\bar{\\Lambda},\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\bar{\\Lambda},\\Lambda'',\\lambda}\\delta}=\\sum_{k\\in\\mathbb{Z}_+}c_{k}e^{(\\Lambda(d)+\\tilde{n}-k)\\delta},\n\\eeqn\nfor some $c_k\\in \\bz_+$ with $c_0$ nonzero. \nBy Proposition \\ref{tensor}, this is the character of a unitarizable \nVirasoro representation with each irreducible component having the same nonzero central charge. Thus, by Lemma \n\\ref{virasoro}, for any $k>1$, we get \n$c_{k}\\neq0$. \n\nBy the above argument, $L(d_o\\Lambda+d_o\\tilde{n}\\delta)$ is the $\\delta$-maximal component of \n$L(d_o\\Lambda')\\otimes L(d_o\\Lambda'')$ through $L(d_o\\Lambda)$. If $\\tilde{n}=0$, we get that \n $$L(d_o\\Lambda) \\subset L(d_o\\Lambda')\\otimes L(d_o\\Lambda'').$$\nIf $\\tilde{n}>0$, then $d_o\\tilde{n}$ being $>1$, by the analogue of \\eqref{e6.4} for \n$d_o\\Lambda', d_o\\Lambda''$ and $d_o\\Lambda$,\n$L(d_o\\Lambda) \\subset L(d_o\\Lambda')\\otimes L(d_o\\Lambda'').$ (Here we have used that \n$L_0=-d+p$ on any $\\fg$-isotypical component of $L(\\Lambda')\\otimes L(\\Lambda'')$ with highest weight in $\\Lambda+\\bz \\delta$, for \na number $p$ depending only upon $\\bar{\\Lambda}, \\Lambda'$ and $\\Lambda''$, cf. [KR, Identity 10.25 on page 116].)\nThis proves the corollary.\n\\end{proof}\n\n\\begin{rem}\nWe note that $L(2\\Lambda_{0}-\\delta)$ is not a component of $L(\\Lambda_{0})\\otimes L(\\Lambda_{0})$\n(cf. [Kac, Exercise 12.16]).\nBut, of course, $L(2\\Lambda_{0})$ is a $\\delta$-maximal component. By the identity \\eqref{e6.4}, we know that $L(2d_o\\Lambda_{0}-d_o\\delta)$\nmust be a component of $L(d_o\\Lambda_{0})\\otimes L(d_o\\Lambda_{0})$, for any $d_o>1$. \nSo $d_o$ can not be taken to be $1$ in Corollary \\ref{cor6.4}.\n\\end{rem}\n\n\\section{A Conjecture}\n\nIn this section, $G$ is any symmetrizable Kac-Moody group. \nWe recall the following definition of the deformed product due to Belkale-Kumar [BK]. (Even though they gave the definition in the finite case, the same definition works in the symmetrizable Kac-Moody case, though with only one parameter.)\n\\begin{definition}\nLet $P$ be any standard parabolic subgroup of $G$. Recall from Section 2 that $\\{\\epsilon^w_P\\}_{w\\in W^P}$ is a\nbasis of the singular cohomology $H^*(X_P, \\bz)$. \nWrite the standard cup product in $H^*(X_P, \\Bbb Z)$ in this basis as follows:\n\\begin{equation}\\label{constants}\n\\epsilon_P^u\\cdot \\epsilon_P^v=\\sum_{w\\in W^P} n^w_{u,v}\\epsilon_P^w,\\,\\,\\,\\text{for some (unique)}\\, n^w_{u,v}\\in \\bz.\n\\end{equation}\nIntroduce the indeterminate ${\\tau}$ and define a deformed cup product $\\odot$\nas follows:\n\\begin{equation}\\label{10n}\n\\epsilon_P^u \\odot \\epsilon_P^v=\n\\sum_{w\\in W^P} \n{\\tau}^{(u^{-1}\\rho+v^{-1}\\rho -w^{-1}\\rho -\\rho)(x_P)}\nn^w_{u,v} \\epsilon_P^w,\n\\end{equation}\nwhere $x_P:=\\sum_{\\alpha_i\\in\n\\Delta\\setminus\\Delta(P)}\\,x_i$, $\\Delta(P)$ is the set of simple roots of the Levi $L$ of $P$ and, as in Section 2, $\\Delta$ is \nthe set of simple roots of $G$. \n\nThe following lemma is a generalization of the corresponding result in the finite case (cf. [BK, Proposition 17]). \n\\begin{proposition} (a) The product $\\odot$ is associative and clearly commutative. \n\n(b) \nWhenever $n^w_{u,v}$ is nonzero,\nthe exponent of $\\tau$ in the above is a nonnegative integer. \n\\end{proposition}\n\\begin{proof} The proof of the associativity of $\\odot$ is identical to the proof given in [BK, Proof of Proposition 17 (b)]. \n\n(b) The proof of this part follows the proof of [BK, Theorem 43]. Consider the\ndecreasing filtration $\\ca =\\{\\ca_m\\}_{m\\geq 0}$ of\n$H^*(X_P, \\bc )$ defined as follows:\n \\[\n\\ca_m := \\bigoplus_{w\\in W^P:(\\rho-w^{-1}\\rho) (x_P)\\geq m} \\bc \\epsilon_P^w.\n \\]\nA priori $\\{\\ca_m\\}_{m\\geq 0}$ may not be a multiplicative filtration. \n\nWe next introduce another filtration $\\{\\bar\\cf_m\\}_{m\\geq 0}$ of\n$H^*(X_P,\\bc )$ in terms of the Lie algebra cohomology. Recall\nthat $H^*(X_P,\\bc )$ can be identified canonically with the Lie\nalgebra cohomology $H^*(\\fg ,\\fl )$, where $\\fl$ is the Lie algebra of the Levi subgroup\n$L$ of $P$ (cf. [K$_2$, Theorem 1.6]). The underlying cochain\ncomplex $C\\u. =C\\u. (\\fg ,\\fl )$ for $H^*(\\fg ,\\fl )$ can be\nwritten as\n \\[\nC\\u. := [\\wed\\u. (\\fg \/\\fl )^*]^{\\fl} = \\Hom_{\\fl} \\bigl( \\wed\\u.\n(\\fu_P)\\otimes\\wed\\u. (\\fu_P^-), \\bc \\bigr),\n \\]\nwhere $\\fu_P$ (resp. $\\fu_P^-$) is the nil-radical of the Lie algebra of $P$ (resp. the opposite parabolic subgroup $P^-$). \nDefine a decreasing multiplicative filtration $\\cf =\\{\\cf_m\\}_{m\\geq 0}$ of the\ncochain\ncomplex $C\\u.$ by subcomplexes:\n \\[\n\\cf_m := \\Hom_{\\fl}\\Biggl( \\frac{\\wed\\u. (\\fu_P)\\otimes\\wed\\u.\n(\\fu^-_P)}{\\bigoplus_{s+t\\leq m-1}\n\\wed\\u._{(s)}(\\fu_P)\\otimes\\wed\\u._{(t)}(\\fu^-_P)}, \\bc\\Biggr) ,\n \\]\nwhere $\\wed\\u._{(s)}(\\fu_P)$ (resp. $\\wed\\u._{(s)}(\\fu^-_P)$) denotes the\nsubspace of $\\wed\\u. (\\fu_P)$ (resp. $\\wed\\u. (\\fu^-_P)$) spanned by the\n$\\fh$-weight vectors of weight $\\beta$ with {\\em $P$-relative height}\n$$\\text{ht}_P (\\beta ) := \\mid\\beta (x_P)\\mid=s.$$\n\nNow, define the filtration $\\bar\\cf =\\{ \\bar\\cf_m\\}_{m\\geq 0}$ of\n$H^*(\\fg ,\\fl )\\simeq H^*(X_P, \\bc)$ by\n \\[\n\\bar\\cf_m := \\text{Image of } H^*(\\cf_m) \\to H^*(C\\u. ).\n \\]\nThe filtration $\\cf$ of $C\\u.$ gives rise to the cohomology\nspectral sequence $\\{ E_r\\}_{r\\geq 1}$ converging to $H^*(C\\u.\n)=H^*(X_P,\\bc )$. By [K$_3$, Proof of Proposition 3.2.11], for any $m\\geq\n0$,\n \\[\nE^m_1 = \\bigoplus_{s+t=m} [H\\u._{(s)}(\\fu_P)\\otimes\nH\\u._{(t)}(\\fu^-_P)]^{\\fl},\n \\]\nwhere $H\\u._{(s)}(\\fu_P)$ denotes the cohomology of the subcomplex\n$(\\wed\\u._{(s)}(\\fu_P))^*$ of the standard cochain complex\n$\\wed\\u. (\\fu_P)^*$ associated to the Lie algebra $\\fu_P$ and\nsimilarly for $H\\u._{(t)}(\\fu^-_P)$. Moreover, by loc. cit., the\nspectral sequence degenerates at the $E_1$ term, i.e.,\n \\begin{equation} \\label{eqn10.1}\nE_1^m = E^m_{\\infty}.\n \\end{equation}\nFurther, by the definition of $P$-relative height,\n \\[\n[H\\u._{(s)}(\\fu_P) \\otimes H\\u._{(t)}(\\fu^-_P)]^{\\fl} \\neq 0 \\Rightarrow\ns=t.\n \\]\nThus,\n \\begin{align*}\nE^m_1 &= 0, \\qquad\\quad\\text{unless $m$ is even and}\\\\\nE_1^{2m} &= [H\\u._{(m)}(\\fu_P) \\otimes H\\u._{(m)}(\\fu^-_P)]^{\\fl} .\n \\end{align*}\nIn particular, from (\\ref{eqn10.1}) and the general properties of spectral sequences\n(cf. [K$_3$, Theorem E.9]), we have a canonical algebra isomorphism:\n \\begin{equation} \\label{eqn10.2}\n\\gr (\\bar\\cf ) \\simeq \\bigoplus_{m\\geq 0} \\bigl[ H\\u._{(m)}(\\fu_P) \\otimes\nH\\u._{(m)}(\\fu^-_P)\\bigr]^{\\fl} ,\n \\end{equation}\nwhere $\\bigl[ H\\u._{(m)}(\\fu_P) \\otimes H\\u._{(m)}(\\fu^-_P)\\bigr]^{\\fl}$\nsits inside $\\gr(\\bar\\cf )$ precisely as the homogeneous part of degree\n$2m$; homogeneous parts of $\\gr(\\bar\\cf )$ of odd degree being zero.\n\nFinally, we claim that, for any $m\\geq 0$,\n \\begin{equation} \\label{eqn10.3}\n\\ca_m = \\bar\\cf_{2m} :\n \\end{equation}\n\nFollowing Kumar [K$_1$], take the d-$\\partial$ harmonic representative\n $\\hat{s}^w$ in $C\\u.$ for the cohomology class $\\epsilon_P^w$. An\nexplicit expression is given in [K$_1$, Proposition 3.17]. From this explicit expression, we easily see that\n \\begin{equation} \\label{eqn10.4}\n\\ca_m \\subset \\bar\\cf_{2m}, \\,\\,\\,\\text{for all}\\,\\, m\\geq 0.\n \\end{equation}\nMoreover, from the definition of $\\ca$, for any $m\\geq 0$,\n \\[\n\\dim \\frac{\\ca_m}{\\ca_{m+1}} = \\# \\bigl\\{ w\\in W^P : (\\rho\n-w^{-1}\\rho) (x_P)=m\\bigr\\} .\n \\]\nAlso, by the isomorphism (\\ref{eqn10.2}) and [K$_3$, Theorem 3.2.7],\n $$\n\\dim \\frac{\\bar\\cf_{2m}}{\\bar\\cf_{2m+1}} = \\# \\bigl\\{ w\\in W^P: (\\rho-w^{-1}\\rho)\n(x_P) = m\\bigr\\}. $$\n\nThus,\n \\beqn \\label{eq77}\n\\dim \\frac{\\ca_m}{\\ca_{m+1}} = \\dim \\frac{\\bar\\cf_{2m}}{\\bar\\cf_{2m+1}} .\n \\eeqn\nOf course, \n\\beqn\\label{eq78} \\ca_0 = \\bar\\cf_0.\n\\eeqn\nThus, combining the equations \\eqref{eqn10.4}, \\eqref{eq77} and \\eqref{eq78}, we get \\eqref{eqn10.3}. \nIt is easy to see that the filtration $\\{\\bar{\\cf}_{2m}\\}_{m\\geq 0}$ is multiplicative and hence so is\n$\\{\\ca_m \\}_{m\\geq 0}$. This proves the (b) part of the proposition.\n \\end{proof}\n\nThe cohomology of $X_P$\n obtained by setting ${\\tau}=0$ in\n$(H^*(X_P, \\Bbb Z)\\otimes\\Bbb{Z}[{\\tau}],\\odot)$ is denoted by\n$(H^*(X_P, \\Bbb Z),\\odot_0)$. Thus, as a $\\Bbb Z$-module, it is the same as the singular cohomology\n $H^*(X_P, \\Bbb Z)$ and under the product $\\odot_0$ it is associative (and commutative).\n\\end{definition}\n\nThe following conjecture is a generalization of the corresponding result in the finite case due to Belkale-Kumar [BK, Theorem 22].\n\\begin{conjecture} \\label{conj1}\nLet $G$ be any indecomposable symmetrizable Kac-Moody group (i.e., its generalized Cartan matrix is indecomposable, cf. [K$_3$, $\\S$ 1.1]) \nand let $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in P_+^{s+1}$. Assume further that none of $\\lambda_j$ is $W$-invariant and $\\mu-\\sum_{j=1}^s \\lambda_j\\in Q$, where $Q$ is the root lattice of $G$. \nThen, the following are equivalent:\n\n(a) $(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$.\n\n(b) For every standard maximal parabolic subgroup $P$ in $G$ and every choice of\n$s+1$-tuples $(w_1, \\dots, w_s, v)\\in (W^P)^{s+1}$ such that $\\epsilon_P^v$ occurs with coefficient $1$ in \nthe deformed product\n$$\\epsilon_P^{w_1}\\odot_0\\, \\cdots \\,\\odot_0 \\epsilon_P^{w_s}\n\\in \\bigl(H^*(X_P,\\Bbb{Z}), \\odot_0\\bigr),$$\n the following inequality holds:\n \\[\\label{eqn29}\n\\bigl(\\sum_{j=1}^s \\lambda_j(w_jx_{P})\\bigr)-\\mu(vx_P)\\geq 0, \\tag{$I^P_{(w_1,\\dots, w_s,v)}$}\n\\]\nwhere $\\alpha_{i_P}$ is the (unique) simple root in $\\Delta\\setminus \\Delta (P)$\nand $x_P:=x_{i_P}$.\n\\end{conjecture}\n\\begin{remark} (a) By Theorem \\ref{thm1}, the above inequalities $I^P_{(w_1,\\dots, w_s,v)}$ are indeed satisfied for any \n$(\\lambda_1, \\dots, \\lambda_s, \\mu)\\in \\Gamma_s$. \n\n(b) If $G$ is an affine Kac-Moody group, then the condition \nthat $\\lambda\\in P_+$ is $W$-invariant is equivalent to the condition that $\\lambda(c)=0$.\n\\end{remark}\n\n\\begin{theorem} \\label{thm7.5} Let $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$. \nLet $\\lambda,\\mu,\\nu \\in P_+$ be such that $\\lambda+\\mu-\\nu\\in Q$ and both of $\\lambda(c)$ and $\\mu(c)$ are nonzero. Then, the following are equivalent:\n\n(a) $(\\lambda,\\mu,\\nu) \\in \\Gamma_2$.\n\n(b) The following set of inequalities is satisfied for all $w\\in W$ and $i=0,1$.\n\\begin{align*}\n\\lambda(x_i)+\\mu(wx_i)-\\nu(wx_i) &\\geq 0, \\,\\,\\,\\text{and}\\\\\n\\lambda(wx_i)+\\mu(x_i)-\\nu(wx_i) &\\geq 0.\n\\end{align*}\nIn particular, Conjecture \\ref{conj1} is true for $\\mathfrak{g}=\\widehat{\\mathfrak{sl}_{2}}$ and $s=2$.\n\\end{theorem}\n\\begin{proof}\nBy Lemma \\ref{lemma5.1}, there exist (unique) $n_1, n_2\\in \\bz$ such that \n\\[\n\\nu - \\mu + n_1 \\delta \\in P^o(\\lambda), \\,\\,\\,\\,\\text{and}\\,\\,\\,\n\\nu - \\lambda +n_2 \\delta \\in P^o(\\mu).\n\\]\nLet $n:=$ min $(n_1,n_2)$. \nBy our description of the $\\delta$-maximal components as in Theorem \\ref{thm6.2} applied to $\\bar{\\lambda},\\bar{\\mu}, \\bar{\\nu}$\nand using the identity \\eqref{eq6.4.0}, we see that \n$L(\\nu+n\\delta)$ is a $\\delta$-maximal component of $L(\\lambda)\\otimes L(\\mu)$. Thus, by the equation \\eqref{eq6.4.1}, for any $N\\geq 1$, \n$L(N\\nu+Nn\\delta)$ is a $\\delta$-maximal component of $L(N\\lambda)\\otimes L(N\\mu)$. In particular, by Proposition \\ref{tensor} and Lemma \n\\ref{virasoro}, \n\\beqn\\label{eq7.4.1} L(N\\nu) \\subset L(N\\lambda)\\otimes L(N\\mu) \\,\\,\\,\\text{ for some}\\,\\, N>1\\,\\,\\,\\text{ if and only if}\\,\\, n\\geq 0.\n\\eeqn\nBy [Kac, Proposition 12.5 (a)], if a weight $\\gamma+k\\delta\\in P(\\lambda)$ (for some $k\\in \\bz_+$), then $\\gamma\\in P(\\lambda)$. Thus, \n\\beqn\\label{eq7.4.2} n\\geq 0 \\,\\,\\,\\text{ if and only if}\\,\\, \\nu\\in \\bigl(P(\\lambda)+\\mu\\bigr)\\cap \\bigl(P(\\mu)+\\lambda\\bigr).\n\\eeqn\nWe next show that \n\\beqn\\label{eq7.4.3} P(\\lambda)=(\\lambda + Q)\\cap C_\\lambda,\n\\eeqn\nwhere $C_\\lambda:=\\{\\gamma\\in \\fh^*: \\lambda(x_i)-\\gamma(wx_i)\\geq 0 \\,\\,\\,\\text{for all}\\,\\, w\\in W \\,\\,\\,\\text{and all} \\,\\, x_i\\}$.\nClearly, $$\n P(\\lambda)\\subset(\\lambda + Q)\\cap C_\\lambda.$$\nSince $\\lambda+Q$ and $C_\\lambda$ are $W$-stable, and $\\lambda+Q$ is contained in the Tits cone (by [K$_3$, Exercise 13.1.E.8(a)]),\n$(\\lambda+Q)\\cap C_\\lambda= W\\cdot \\bigl((\\lambda+Q)\\cap C_\\lambda\\cap P_+\\bigr)$.\n\nConversely,\ntake $\\gamma\\in (\\lambda + Q)\\cap C_\\lambda \\cap P_+$. Then, \n$(\\lambda-\\gamma) (x_i)\\geq 0$ and $(\\lambda-\\gamma) (c)= 0$ and hence $\\lambda-\\gamma \\in \\oplus_i\\,\\bz_+\\alpha_i$, i.e., \n$\\lambda \\geq \\gamma$. Thus, by [Kac, Proposition 12.5(a)], $\\gamma \\in P(\\lambda)$. This proves \\eqref{eq7.4.3}. \nNow, combining \\eqref{eq7.4.1}, \\eqref{eq7.4.2} and \\eqref{eq7.4.3}, we get \n$ L(N\\nu) \\subset L(N\\lambda)\\otimes L(N\\mu)$ for some $N>1$ if and only if for all \n $w\\in W$ and $i=0,1$,\n\\[\n\\lambda(x_i)-(\\nu-\\mu)(wx_i)\\geq 0, \\,\\,\\,\\text{and}\\,\\,\\,\n\\mu(x_i)-(\\nu - \\lambda)(wx_i) \\geq 0.\\]\nThis proves the equivalence of (a) and (b) in the theorem. \n\nTo prove the `In particular' statement of the theorem, let $P_0$ (resp. $P_1$) be the maximal parabolic subgroup of $G=\\widehat{\\SL_{2}}$\nwith $\\Delta(P_0)=\\{\\alpha_1\\}$ (resp. $\\Delta(P_1)=\\{\\alpha_0\\}$). For any $n\\geq 0$, let \n$$w_n:=\\dots s_0s_1s_0 \\,\\,(\\text{$n$-factors})\\,\\,\\,\\text{and}\\,\\,\\,v_n:=\\dots s_1s_0s_1 \\,\\,(\\text{$n$-factors}).$$\nThen, by [K$_3$, Exercise 11.3.E.4], $H^*(G\/P_0)$ has a $\\bz$-basis $\\{\\epsilon^{n}_{P_0}\\}_{n\\geq 0}$, where \n$\\epsilon^{n}_{P_0}:=\\epsilon^{w_n}_{P_0}$. Moreover,\n$$\\epsilon^{n}_{P_0}\\cdot \\epsilon^{m}_{P_0}=\\binom {n+m}{n}\\epsilon^{n+m}_{P_0}.$$\nIn particular, $\\epsilon^{n+m}_{P_0}$ appears with coefficient one in $\\epsilon^{n}_{P_0}\\cdot \\epsilon^{m}_{P_0}$ if and only if \nat least one of $n$ or $m$ is $0$. \n\nBy using the diagram automorphism of $\\widehat{\\SL_{2}}$, one similarly gets that $H^*(G\/P_1)$ has a $\\bz$-basis $\\{\\epsilon^{n}_{P_1}\\}_{n\\geq 0}$, where \n$\\epsilon^{n}_{P_1}:=\\epsilon^{v_n}_{P_1}$, with the product given by\n$$\\epsilon^{n}_{P_1}\\cdot \\epsilon^{m}_{P_1}=\\binom {n+m}{n}\\epsilon^{n+m}_{P_1}.$$\nMoreover, from the definition of the deformed product $\\odot_0$, clearly \n$$\\epsilon^{0}_{P_0}\\odot_0 \\epsilon^{m}_{P_0}= \\epsilon^{0}_{P_0}\\cdot \\epsilon^{m}_{P_0},$$\nand similarly for $P_1$. From this the `In particular' statement of the theorem follows.\n\\end{proof}\n\\begin{remark} (1) It is easy to see that if $\\lambda=m\\delta$ for some $m\\in \\bz$, then the equivalence of (a) and (b) in the above theorem breaks down.\n\n(2) Though we have proved Conjecture \\ref{conj1} for $\\widehat{\\SL_{2}}$ only for $s=2$, it is quite likely that a similar proof will prove it for \nany $s$ (for $\\widehat{\\SL_{2}}$).\n\\end{remark}\n\n\\section{The $A_2^{(2)}$ case}\nBy a method similar to that of $A_1^{(1)}$, we handle the $A_2^{(2)}$ case, with minor modifications where necessary. \nWrite $\\mathfrak{h}=\\mathbb{C}c\\oplus\\mathbb{C}\\alpha^{\\vee}\\oplus\\mathbb{C}d$\nand $\\mathfrak{h}^{*}=\\mathbb{C}\\omega_{0}\\oplus\\mathbb{C}\\alpha\\oplus\\mathbb{C}\\delta$,\nwhere $\\alpha(\\alpha^{\\vee})=2$, $\\delta(d)=1$, $\\omega_{0}(c)=1$,\nand all other values are $0$. Then $(\\mathfrak{h},\\{\\alpha_0:=\\delta-2\\alpha, \\alpha_1:=\\alpha\\}, \\{\\alpha_0^\\vee:=c-\\frac{1}{2}\\alpha^{\\vee},\\alpha_1^\\vee:=\\alpha^{\\vee}\\})$ is a realization of the GCM \n\\[ \\left( \\begin{array}{cc}\n2 & -1\\\\\n-4 & 2 \\end{array} \\right)\\] \nof $A_2^{(2)}$.\nThe fundamental weights are $\\omega_{0}$ and $\\omega_{1}=\\frac{1}{2}\\omega_{0}+\\frac{1}{2}\\alpha$. This easily allows one to compute the dominant $\\delta$-maximal weights. Analogous to Corollary \\ref{cor5.1}, we have the following:\n\\begin{lemma}\\label{domdelmaxweights} Let $\\lambda$ be a dominant integral weight. Then, the dominant $\\delta$-maximal\nweights of $L(\\lambda)$ are the dominant weights of the form \n\\[\nP_+\\cap \\left\\{ \\lambda-j\\alpha,\\,\\lambda+k(2\\alpha-\\delta),\\,\\lambda+\\alpha-\\delta+l(2\\alpha-\\delta)\\,:\\, j,k,l\\in\\mathbb{Z}_{\\geq0}\\right\\} .\n\\]\nMoreover, $P^o(\\lambda)$ is the $W$-orbit of the above.\n\\end{lemma}\n\nAgain, to determine the saturated tensor cone, it is enough to describe the $\\delta$-maximal components. Thus, to determine the $\\delta$-maximal components, by virtue of proposition \\ref{tensor}, we must find the highest $\\delta$-degree term in $\\sum_{\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''}}\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})c_{\\Lambda',\\lambda}e^{S_{\\Lambda,\\Lambda'',\\lambda}\\delta}$. This computation is done in a somewhat similar manner as in the $A_1^{(1)}$ case, but there are some important modifications.\nFirst, we need to use two different piecewise smooth functions to describe the $\\delta$-maximal weights of $L(\\lambda)$. An upper function $A^+$ interpolates the $\\delta$-maximal weights which are in the $W$-orbit of the dominant weights of the form\n\\[\n\\left\\{ \\lambda-j\\alpha,\\,\\lambda+k(2\\alpha-\\delta)\\,:\\, j,k\\in\\mathbb{Z}_{\\geq0}\\right\\}, \n\\]\nwhile another function $A^-$ interpolates the $\\delta$-maximal weights in the $W$-orbit of the dominant weights of the form\n\\[\n\\left\\{ \\lambda-j\\alpha,\\,\\lambda+\\alpha-\\delta+l(2\\alpha-\\delta)\\,:\\, j,l\\in\\mathbb{Z}_{\\geq0}\\right\\}. \n\\]\n\nNow, all of the arguments made in the $\\widehat{\\mathfrak{sl}_2}$ case must be made for two extensions of $S_{\\Lambda,\\Lambda'',\\lambda}$ to non-integral values, using $A^+$ and $A^-$ respectively. Let $\\Lambda :=m_{0}\\omega_{0}+m_{1}\\omega_{1}$, $\\Lambda' :=m'_{0}\\omega_{0}+m'_{1}\\omega_{1}$, and $\\Lambda'' :=m''_{0}\\omega_{0}+m''_{1}\\omega_{1}$. The following result is an analogue of \nProposition \\ref{maximum} and Lemma \\ref{lemma5.9} for the $A_2^{(2)}$ case.\n\\begin{prop}\\label{maxS} Let $\\Lambda, \\Lambda', \\Lambda''$ be as above. Assume that both of $\\Lambda'(c)$ and $\\Lambda''(c)>0$ and $\\Lambda-\\Lambda'-\\Lambda''\\in Q$, where \n$Q=\\bz \\alpha+\\bz \\delta$ is the root lattice of $A_2^{(2)}$ . \nThen, the maximum $\\mu^{\\Lambda',\\Lambda''}_\\Lambda$ of the set\n\\[ \n\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''},\\;\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=1\\right\\}\n\\] occurs when $\\lambda\\equiv\\Lambda'+\\frac{1}{2}\\left(m_{1}-m_{1}'-m_{1}''\\right)\\alpha \\mod\\mathbb{C}\\delta$.\nThe maximum $\\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$ of the set \n\\[\n\\left\\{ S_{\\Lambda,\\Lambda'',\\lambda}:\\;\\lambda\\in T_{\\Lambda}^{\\Lambda',\\Lambda''},\\;\\varepsilon(v_{\\Lambda,\\Lambda'',\\lambda})=-1\\right\\}\n\\]\noccurs when $\\lambda\\equiv\\Lambda'-\\bigl(\\frac{1}{2}(m_{1}'+m_{1}''+m_{1})+1\\bigr)\\alpha \\mod \\mathbb{C}\\delta $\nor when $\\lambda\\equiv\\Lambda'-\\bigl(\\frac{1}{2}(m_{1}'+m_{1}''+m_{1})-2(\\Lambda'(c)+\\Lambda''(c)+1)\\bigr)\\alpha \\mod \\mathbb{C}\\delta $. \n\\end{prop}\n\n\\begin{corollary}\\label{a22cancellation}\nLet $\\Lambda, \\Lambda', \\Lambda''$ be as in Proposition \\ref{maxS}. Assume further that \n$\\Lambda'(c) \\geq 2$, $\\Lambda''(c) \\geq 2$, $m'_1,m''_1 \\neq 1$. Then, if ${\\mu}^{\\Lambda',\\Lambda''}_\\Lambda = \\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$, we have \n\\[ {\\mu}^{\\Lambda'',\\Lambda'}_\\Lambda \\neq \\bar{\\mu}^{\\Lambda'',\\Lambda'}_\\Lambda.\\]\n\\end{corollary}\n\nThe proof of Corollary \\ref{a22cancellation} requires a description of the situations in which \n ${\\mu}^{\\Lambda',\\Lambda''}_\\Lambda = \\bar{\\mu}^{\\Lambda',\\Lambda''}_\\Lambda$. We reduce these situations to certain cases, and show that in most of these cases, if the roles of $\\Lambda'$ and $\\Lambda''$ are interchanged, then (as in the $\\widehat{\\mathfrak{sl}_2}$ case) the equality does not occur. In the remaining cases, we show that $\\Lambda'(c) < 2$, $\\Lambda''(c) < 2$, $m'_1 =1$, or $m''_1 = 1$.\n\n\\begin{thm} Let $\\Lambda,\\Lambda',\\Lambda''$ be as in Proposition \\ref{maxS}. Then, \n$L(\\Lambda+n\\delta)$ is a $\\delta$-maximal component of $L(\\Lambda')\\otimes L(\\Lambda'')$\nif $n=\\min(n_{1},n_{2})$, where $n_{1}$ is such that $\\Lambda-\\Lambda''+n_{1}\\delta\\in P^o(\\Lambda')$ and\n$n_{2}$ is such that $\\Lambda-\\Lambda'+n_{2}\\delta\\in P^o(\\Lambda'')$.\n\\end{thm}\n\\begin{lem} Fix a positive integer $N$. \nLet $\\Lambda\\in \\bar{P}_+$ and let $\\lambda\\in\\Lambda+Q$. \nThen, $N\\lambda\\in P^o(N\\Lambda)$ if\nand only if $\\lambda\\in P^o(\\Lambda)$. \n\\end{lem}\nCombining the above results, we get a description of $\\Gamma_2$, which is identical to that of $\\widehat{\\mathfrak{sl}_2}$ \n(cf. Theorem \\ref{thm7.5}).\n\\begin{theorem} \\label{thm8.6} Let $\\mathfrak{g}=A_2^{(2)}$. \nLet $\\lambda,\\mu,\\nu \\in P_+$ be such that $\\lambda+\\mu-\\nu\\in Q$ and both of $\\lambda(c)$ and $\\mu(c)$ are nonzero. Then, the following are equivalent:\n\n(a) $(\\lambda,\\mu,\\nu) \\in \\Gamma_2$.\n\n(b) The following set of inequalities is satisfied for all $w\\in W$ and $i=0,1$.\n\\begin{align*}\n\\lambda(x_i)+\\mu(wx_i)-\\nu(wx_i) &\\geq 0, \\,\\,\\,\\text{and}\\\\\n\\lambda(wx_i)+\\mu(x_i)-\\nu(wx_i) &\\geq 0.\n\\end{align*}\nIn particular, Conjecture \\ref{conj1} is true for this case as well for $s=2$. \n\\end{theorem}\n The `In particular' statement of the above theorem follows by using the description of the cup product in the cohomology of the full flag variety of $A_2^{(2)}$ given by Kitchloo [Ki]. \n\nIt is clear that if the level of $L(\\Lambda')$ or $L(\\Lambda'')$ is zero, then the tensor product has a single component. Thus, it is already saturated. Assume now that the levels of both of $L(\\Lambda')$ and $L(\\Lambda'')$ are $>0$. Then, since there are representations of level $\\frac{1}{2}$, the conditions of Corollary \\ref{a22cancellation} are satisfied for any $N\\Lambda$, $N\\Lambda'$, $N\\Lambda''$ with $\\Lambda-\\Lambda'-\\Lambda''\\in Q$, provided $N \\geq 4$. Hence:\n\\begin{cor} \\label{cor8.7} For $A_2^{(2)}$, $4$ is a saturation factor.\n\\end{cor}\n\\begin{remark} \nWhen the Kac-Moody Lie algebra $\\fg$ is infinite dimensional, then the saturated tensor semigroup $\\Gamma_s$ is {\\em not} finitely generated,\nfor any $s\\geq 2$. Thus, it is not clear a priori that there exists a saturation factor for such a $\\fg$.\n\\end{remark}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Summary}\n\nSymplectic field theory (SFT), introduced by H. Hofer, A. Givental and Y. Eliashberg in 2000 ([EGH]), is a very large project and can be viewed as a topological quantum field theory approach to Gromov-Witten theory. Besides providing a unified view on established pseudoholomorphic curve theories like symplectic Floer homology, contact homology and Gromov-Witten theory, it leads to numerous new applications and opens new routes yet to be explored. \\\\\n \nWhile symplectic field theory leads to algebraic invariants with very rich algebraic structures, which are currently studied by a large group of researchers, for all the geometric applications found so far it was sufficient to work with simpler invariants like cylindrical contact homology. Although cylindrical contact homology is not always defined, it is much easier to compute, not only since it involves just moduli spaces of holomorphic cylinders but also due to the simpler algebraic formalism. While the rich algebraic formalism of the higher invariants of symplectic field theory seems to be too complicated for concrete geometric applications, it was pointed out by Eliashberg in his ICM 2006 plenary talk ([E]) that the integrable systems of rational Gromov-Witten theory very naturally appear in rational symplectic field theory by using the link between the rational symplectic field theory of circle bundles in the Morse-Bott version and the rational Gromov-Witten potential of the underlying symplectic manifold. Indeed, after introducing gravitational descendants as in Gromov-Witten theory, it is precisely the rich algebraic formalism of SFT with its Weyl and Poisson structures that provides a natural link between symplectic field theory and (quantum) integrable systems. In particular, in the case where the contact manifold is a circle bundle over a closed symplectic manifold, the rich algebraic formalism of symplectic field theory seems to provide the right framework to understand the deep relation between Gromov-Witten theory and integrable systems, at least in the genus zero case. \\\\\n\nWhile in the Morse-Bott case in [E] it follows from the corresponding statements for the Gromov-Witten descendant potential that the sequences of commuting operators and Poisson-commuting functions are independent of auxiliary choices like almost complex structure and abstract perturbations, for the case of general contact manifolds it is well-known that the SFT Hamiltonian however in general explicitly depend on choices like contact form, cylindrical almost complex structure and coherent abstract perturbations and hence is not an invariant for the contact manifold itself. But before we can come down to the question of invariance, we first need to give a rigorous definition of gravitational descendants in the context of symplectic field theory. \\\\\n\nWhile in Gromov-Witten theory the gravitational descendants were defined by integrating powers of the first Chern class of the tautological line bundle over the moduli space, which by Poincare duality corresponds to counting common zeroes of sections in this bundle, in symplectic field theory, more generally every holomorphic curves theory where curves with punctures and\/or boundary are considered, we are faced with the problem that the moduli spaces generically have codimension-one boundary, so that the count of zeroes of sections in general depends on the chosen sections in the boundary. It follows that the integration of the first Chern class of the tautological line bundle over a single moduli space has to be replaced by a construction involving all moduli space at once. Note that this is similar to the choice of coherent abstract perturbations for the moduli spaces in symplectic field theory in order to achieve transversality for the Cauchy-Riemann operator. Keeping the interpretation of descendants as common zero sets of sections in powers of the tautological line bundles (which will turn out to be particularly useful when one studies the topological meaning of descendants by localizing on special divisors, see [FR]), we define in this paper the notion of {\\it coherent collections of sections} in the tautological line bundles over all moduli spaces, which just formalizes how the sections chosen for the lower-dimensional moduli spaces should affect the section chosen for a moduli spaces on its boundary. To be more precise, since the sections should be invariant under obvious symmetries like reordering of the punctures and the marked points, we actually need to work with multi-sections in order to meet both the symmetry and the transversality assumption. We will then define {\\it descendants of moduli spaces} $\\overline{\\operatorname{\\mathcal{M}}}^j\\subset\\overline{\\operatorname{\\mathcal{M}}}$, which we obtain inductively as zero sets of these coherent collections of sections $(s_j)$ in the tautological line bundles over the descendant moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}\\subset\\overline{\\operatorname{\\mathcal{M}}}$, and define {\\it descendant Hamiltonians} $\\operatorname{\\mathbf{H}}^1_{i,j}$ by integrating chosen closed differential forms $\\theta_i$ over $\\overline{\\operatorname{\\mathcal{M}}}^j$. For these we prove the following theorem. \\\\ \n\\\\\n{\\bf Theorem:} {\\it Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0) \\end{equation*} \n{\\it in the full SFT homology algebra with differential $D^0=[\\operatorname{\\mathbf{H}}^0,\\cdot]: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, which commute with respect to the commutator bracket on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$,} \n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}] = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\nIn contrast to the Morse-Bott case considered in [E] it follows that, when the differential in symplectic field theory counting holomorphic curves without additional marked points is no longer zero, the sequences of generating functions no longer commute with respect to the bracket, but only commute {\\it after passing to homology.} On the other hand, in the same way as the rational symplectic field theory of a contact manifold is defined by counting only curves with genus zero, we immediately obtain a rational version of the above statement by expanding $\\operatorname{\\mathbf{H}}^0$ and the $\\operatorname{\\mathbf{H}}^1_{i,j}$ in powers of the formal variable $\\hbar$ for the genus. \\\\\n\\\\\n{\\bf Corollary:} {\\it Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^0,d^0), \\end{equation*} \n{\\it in the rational SFT homology algebra with differential $d^0=\\{\\operatorname{\\mathbf{h}}^0,\\cdot\\}: \\operatorname{\\mathfrak{P}}^0\\to\\operatorname{\\mathfrak{P}}^0$, which commute with respect to the Poisson bracket on $H_*(\\operatorname{\\mathfrak{P}}^0,d^0)$,} \n\\begin{equation*} \\{\\operatorname{\\mathbf{h}}^1_{i,j},\\operatorname{\\mathbf{h}}^1_{k,\\ell}\\} = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\nAs we already outlined above, in contrast to the circle bundle case we have to expect that the sequence of descendant Hamiltonians depends on the auxiliary choices like contact form, cylindrical almost complex structure and coherent abstract polyfold perturbations. Here we prove the following natural invariance statements. \\\\\n\\\\\n{\\bf Theorem:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$ , abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting systems of commuting operators $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ which maps $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ to $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$.}\\\\ \n\\\\\nNote that this theorem is an extension of the theorem in [EGH] stating that for different choices of auxiliary data the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic. As above we clearly also get a rational version of the invariance statement: \\\\\n\\\\\n{\\bf Corollary:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$, abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Poisson algebras $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ which maps $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ to $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$.} \\\\\n\nAs concrete example beyond the case of circle bundles discussed in [E] we consider the symplectic field theory of a closed geodesic. For this recall that in [F2] the author introduces the symplectic field theory of a closed Reeb orbit $\\gamma$, which is defined by counting only those holomorphic curves which are branched covers of the orbit cylinder $\\operatorname{\\mathbb{R}}\\times\\gamma$ in $\\operatorname{\\mathbb{R}}\\times V$. In [F2] we prove that these orbit curves do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the subset of orbit curves over a fixed orbit is closed under taking boundaries and gluing, which follows from the fact that they are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. It follows that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. In particular, in the same way as we define sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ and $\\operatorname{\\mathbf{h}}^1_{i,j}$ by counting general curves in the symplectization of a contact manifold, we can define sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{\\gamma,i,j}$ and $\\operatorname{\\mathbf{h}}^1_{\\gamma,i,j}$ by just counting branched covers of the orbit cylinder over $\\gamma$ with signs (and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for $\\gamma$. We further prove that for branched covers of orbit cylinders over any closed Reeb orbit the gravitational descendants indeed have a geometric interpretation in terms of branching conditions, which generalizes the work of [OP] used in [E] for the circle. \\\\\n\nSince all the considered holomorphic curves factor through the embedding of the closed Reeb orbit into the contact manifold, it follows that it only makes sense to consider differential forms of degree zero or one. While it follows from the result $\\operatorname{\\mathbf{h}}^0_{\\gamma}=0$ in [F2] that the sequences $\\operatorname{\\mathbf{h}}^1_{\\gamma,i,j}$ indeed commute with respect to the Poisson bracket (before passing to homology), the same proof as in [F2] shows that every descendant Hamiltonian in the sequence vanishes if the differential form is of degree zero. For differential forms of degree one the strategy of the proof however no longer applies and it is indeed shown in [E] that for $\\gamma=V=S^1$ and $\\theta=dt$ we get nontrivial contributions from branched covers. In this paper we want to determine the corresponding Poisson-commuting sequence in the special case where the contact manifold is the unit cotangent bundle $S^*Q$ of a ($m$-dimensional) Riemannian manifold $Q$, so that every closed Reeb orbit $\\gamma$ on $V=S^*Q$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$. For this we denote by $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ be the graded Weyl subalgebra of the Weyl algebra $\\operatorname{\\mathfrak{W}}^0$, which is generated only by those $p$- and $q$-variables $p_n=p_{\\gamma^n}$, $q_n=q_{\\gamma^n}$ corresponding to Reeb orbits which are multiple covers of the fixed orbit $\\gamma$ {\\it and which are good in the sense of [BM]}. In the same way we further introduce the Poisson subalgebra $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ of $\\operatorname{\\mathfrak{P}}^0$. Setting $q_{-n}=p_n$ we prove the following \\\\\n\\\\\n{\\bf Theorem:} {\\it Assume that the contact manifold is the unit cotangent bundle $V=S^*Q$ of a Riemannian manifold $Q$, so that the closed Reeb orbit $\\gamma$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$, and that the string of differential forms just consists of a single one-form which integrates to one around the orbit. Then the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to the system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\;\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\nNote that in the case of the circle $\\bar{\\gamma}=Q=S^1$ the degree condition is automatically fulfilled and we just get back the sequence of descendant Hamiltonians for the circle in [E], which agrees with the sequence of Poisson-commuting integrals of the dispersionless KdV integrable hierarchy. Forgetting about the appearing sign issues, it follows that the sequence $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is obtained from the sequence for the circle by removing all summands with the wrong, that is, not maximal degree, so that the system is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. \\\\\n\n\\noindent{\\bf Remark:} Note that the signs in the formula for $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ are determined by the linearized Reeb flow around $\\gamma$ and a choice of \norientations for all multiples of $\\gamma$. For this recall from [BM] that in order to orient moduli spaces in symplectic field theory one additionally needs to choose orientations for all occuring Reeb orbits, while the resulting invariants are independent of these auxiliary choices. While the precise formula for the functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ depends on these choices, the resulting systems of Poisson-commuting functions for different choices are indeed isomorphic, since changing the orientations for some orbits $\\gamma^k$ leads to an automorphism of the underlying Poisson algebra. Apart from the fact that the commutativity condition $\\{\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j},\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},k}\\}=0$ clearly leads to relations between the different $\\epsilon(\\vec{n})$, observe that a choice of orientation for $\\gamma$ does {\\it not} lead to a canonical choice of orientations for its multiples $\\gamma^k$. While we expect that it is in general very hard to write down a set of signs $\\epsilon(\\vec{n})$ explicitly, for all the geometric applications we have in mind and the educational purposes as a test model beyond the Gromov-Witten case we are rather interested in proving vanishing results as the one below than giving precise formulas. \\\\\n\nWhile in the case of the circle we obtain a complete set of integrals, it is important that our theorem allows us to prove the following vanishing result, which simply follows from the fact that for hyperbolic Reeb orbits the Conley-Zehnder index is multiplicative. \\\\\n\\\\\n{\\bf Corollary:} {\\it Assume that the closed geodesic $\\bar{\\gamma}$ represents a hyperbolic Reeb orbit in the unit cotangent bundle of a surface $Q$. Then $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}=0$ and hence $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}=0$ for all $j>0$.} \\\\\n\nNote that this result is actually true for $\\dim Q > 1$. While for $\\dim Q > 2$ the result directly follows from index reasons, in the case when $\\dim Q=2$ a simple computation shows that all moduli spaces with $2j+1$ punctures possibly contribute to the descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$. Since in this case the Fredholm index is $2j-1$ and hence for $j>0$ strictly smaller than the dimension of the underlying nonregular moduli space of branched covers, which is $4j-2$, transversality cannot be satisfied but the cokernels of the linearized operators fit together to give an obstruction bundle of rank $2j-1$. \\\\\n\nApart from using the geometric interpretation of gravitational descendants for branched covers of orbit cylinders over a closed Reeb orbit in terms of branching conditions mentioned above, the second main ingredient for the proof is the idea in [CL] to compute the symplectic field theory of $V=S^*Q$ from the string topology of the underlying Riemannian manifold $Q$ by studying holomorphic curves in the cotangent bundle $T^*Q$. More precisely, we compute the symplectic field theory of a closed Reeb orbit $\\gamma$ in $S^*Q$ including differential forms and gravitational descendants by studying branched covers of the trivial half-cylinder connecting the closed Reeb orbit in the unit cotangent bundle with the underlying closed geodesic in the cotangent bundle $T^*Q$ with special branching data, where the latter uses the geometric interpretation of gravitational descendants. In order to give a complete proof we also prove the neccessary transversality theorems using finite-dimensional obstruction bundles over the underlying nonregular moduli spaces. While on the SFT side one has very complicated obstruction bundles over nonregular moduli spaces of arbitary large dimension, on the string side all relevant nonregular moduli spaces already turn out to be discrete, so that the obstruction bundles disappear if the Fredholm index is right. It follows that the system of Poisson-commuting function for a closed geodesic is completely determined by the KdV hierarchy and the Morse indices of the closed geodesic and its iterates. \\\\\n\nThis paper is organized as follows. \\\\\n\nSection one is concerned with the definition and the basic results about gravitational descendants in symplectic field theory. After we recalled the basic definitions of symplectic field theory in subsection 1.1, we define gravitational descendants in subsection 1.2 using the coherent collections of sections and prove that the resulting sequences of descendant Hamiltonians commute after passing to homology. In subsection 1.3 we prove the desired invariance statement and discuss the important case of circle bundles in the Morse-Bott setup outlined in [E] in 1.4. \\\\\n\nAfter we treated the general case in section one, section two is concerned with a concrete example beyond the case of circle bundles, the symplectic field theory of a closed geodesic, which naturally generalizes the case of the circle in [E]. After we have recalled the definition of symplectic field theory for a closed Reeb orbit including the results from [F2] in subsection 2.1, we show in subsection 2.2 that for branched covers of orbit cylinders the gravitational descendants have a geometric interpretation in terms of branching conditions. After outlining that there exists a version of the isomorphism in [CL] involving the symplectic field theory of a closed Reeb orbit in the unit cotangent bundle, we study the moduli space of branched covers of the corresponding trivial half-cylinder in the cotangent bundle in subsection 2.3. Since we meet the same transversality problems as in [F2], we study the neccessary obstruction bundle setup including Banach manifolds and Banach space bundles in subsection 2.3. In subsection 2.4 we finally prove the above theorem by studying branched covers of the trivial half-cylinder with special branching behavior. \\\\ \n\\\\\n{\\bf Acknowledgements:} This research was supported by the German Research Foundation (DFG). \nThe author thanks K. Cieliebak, Y. Eliashberg, K. Fukaya, M. Hutchings and P. Rossi for useful discussions.\n \n\\section{Symplectic field theory with gravitational descendants}\n\n\\subsection{Symplectic field theory}\nSymplectic field theory (SFT) is a very large project, initiated by Eliashberg,\nGivental and Hofer in their paper [EGH], designed to describe in a unified way \nthe theory of pseudoholomorphic curves in symplectic and contact topology. \nBesides providing a unified view on well-known theories like symplectic Floer \nhomology and Gromov-Witten theory, it shows how to assign algebraic invariants \nto closed contact manifolds $(V,\\xi=\\{\\lambda=0\\})$: \\\\\n \nRecall that a contact one-form $\\lambda$ defines a vector field $R$ on $V$ by \n$R\\in\\ker d\\lambda$ and $\\lambda(R)=1$, which \nis called the Reeb vector field. We assume that \nthe contact form is Morse in the sense that all closed orbits of the \nReeb vector field are nondegenerate in the sense of [BEHWZ]; in particular, the set \nof closed Reeb orbits is discrete. The invariants are defined by counting \n$\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$ which are asymptotically cylindrical over \nchosen collections of Reeb orbits $\\Gamma^{\\pm}=\\{\\gamma^{\\pm}_1,...,\n\\gamma^{\\pm}_{n^{\\pm}}\\}$ as the $\\operatorname{\\mathbb{R}}$-factor tends to $\\pm\\infty$, see [BEHWZ]. \nThe almost complex structure $\\underline{J}$ on the cylindrical \nmanifold $\\operatorname{\\mathbb{R}}\\times V$ is required to be cylindrical in the sense that it is \n$\\operatorname{\\mathbb{R}}$-independent, links the two natural vector fields on $\\operatorname{\\mathbb{R}}\\times V$, namely the \nReeb vector field $R$ and the $\\operatorname{\\mathbb{R}}$-direction $\\partial_s$, by $\\underline{J}\\partial_s=R$, and turns \nthe distribution $\\xi$ on $V$ into a complex subbundle of $TV$, \n$\\xi=TV\\cap \\underline{J} TV$. We denote by $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ the corresponding compactified\nmoduli space of genus $g$ curves with $r$ additional marked points ([BEHWZ],[EGH]). \nPossibly after choosing abstract perturbations using polyfolds (see [HWZ]), obstruction \nbundles ([F2]) or domain-dependent structures ([F1]) following the ideas in [CM] we get that \n$\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ is a branched-labelled orbifold with boundaries and corners of dimension \nequal to the Fredholm index of the Cauchy-Riemann operator for $\\underline{J}$. \n{\\it Note that in the same way as we will not discuss transversality for the general case but just refer to the upcoming papers on polyfolds by Hofer and his co-workers, in what follows we will for simplicity assume that every moduli space is indeed a manifold with boundaries and corners, since we expect that all the upcoming constructions can be generalized in an appropriate way.} \\\\ \n \nLet us now briefly introduce the algebraic formalism of SFT as described in [EGH]: \\\\\n \nRecall that a multiply-covered Reeb orbit $\\gamma^k$ is called bad if \n$\\operatorname{CZ}(\\gamma^k)\\neq\\operatorname{CZ}(\\gamma)\\mod 2$, where $\\operatorname{CZ}(\\gamma)$ denotes the \nConley-Zehnder index of $\\gamma$. Calling a Reeb orbit $\\gamma$ {\\it good} if it is not bad we assign to every good Reeb orbit $\\gamma$ two formal \ngraded variables $p_{\\gamma},q_{\\gamma}$ with grading \n\\begin{equation*} \n|p_{\\gamma}|=m-3-\\operatorname{CZ}(\\gamma),|q_{\\gamma}|=m-3+\\operatorname{CZ}(\\gamma) \n\\end{equation*} \nwhen $\\dim V = 2m-1$. In order to include higher-dimensional moduli spaces we further assume that a string \nof closed (homogeneous) differential forms $\\Theta=(\\theta_1,...,\\theta_N)$ on $V$ is chosen and assign to \nevery $\\theta_i\\in\\Omega^*(V)$ a formal variables $t_i$ \nwith grading\n\\begin{equation*} |t_i|=2 -\\deg\\theta_i. \\end{equation*} \nFinally, let $\\hbar$ be another formal variable of degree $|\\hbar|=2(m-3)$. \\\\\n\nLet $\\operatorname{\\mathfrak{W}}$ be the graded Weyl algebra over $\\operatorname{\\mathbb{C}}$ of power series in the variables \n$\\hbar,p_{\\gamma}$ and $t_i$ with coefficients which are polynomials in the \nvariables $q_{\\gamma}$, which is equipped with the associative product $\\star$ in \nwhich all variables super-commute according to their grading except for the \nvariables $p_{\\gamma}$, $q_{\\gamma}$ corresponding to the same Reeb orbit $\\gamma$, \n\\begin{equation*} [p_{\\gamma},q_{\\gamma}] = \n p_{\\gamma}\\star q_{\\gamma} -(-1)^{|p_{\\gamma}||q_{\\gamma}|} \n q_{\\gamma}\\star p_{\\gamma} = \\kappa_{\\gamma}\\hbar.\n\\end{equation*}\n($\\kappa_{\\gamma}$ denotes the multiplicity of $\\gamma$.) Following [EGH] we further introduce \nthe Poisson algebra $\\operatorname{\\mathfrak{P}}$ of formal power series in the variables $p_{\\gamma}$ and $t_i$ with \ncoefficients which are polynomials in the variables $q_{\\gamma}$ with Poisson bracket given by \n\\begin{equation*} \n \\{f,g\\} = \\sum_{\\gamma}\\kappa_{\\gamma}\\Bigl(\\frac{\\partial f}{\\partial p_{\\gamma}}\\frac{\\partial g}{\\partial q_{\\gamma}} -\n (-1)^{|f||g|}\\frac{\\partial g}{\\partial p_{\\gamma}}\\frac{\\partial f}{\\partial q_{\\gamma}}\\Bigr). \n\\end{equation*}\n\nAs in Gromov-Witten theory we want to organize all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$\ninto a generating function $\\operatorname{\\mathbf{H}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}$, called {\\it Hamiltonian}. In order to include also higher-dimensional \nmoduli spaces, in [EGH] the authors follow the approach in Gromov-Witten theory to integrate the chosen differential forms \n$\\theta_1,...,\\theta_N$ over the moduli spaces after pulling them back under the evaluation map from target manifold $V$. The \nHamiltonian $\\operatorname{\\mathbf{H}}$ is then defined by\n\\begin{equation*}\n \\operatorname{\\mathbf{H}} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\n \\operatorname{ev}_1^*\\theta_{i_1}\\wedge...\\wedge\\operatorname{ev}_r^*\\theta_{i_r}\\; \\hbar^{g-1}t^Ip^{\\Gamma^+}q^{\\Gamma^-}\n\\end{equation*}\nwith $t^I=t_{i_1}...t_{i_r}$, $p^{\\Gamma^+}=p_{\\gamma^+_1}...p_{\\gamma^+_{n^+}}$ and $q^{\\Gamma^-}=q_{\\gamma^-_1}...q_{\\gamma^-_{n^-}}$. \nExpanding \n\\begin{equation*} \\operatorname{\\mathbf{H}}=\\hbar^{-1}\\sum_g \\operatorname{\\mathbf{H}}_g \\hbar^g \\end{equation*} \nwe further get a rational Hamiltonian $\\operatorname{\\mathbf{h}}=\\operatorname{\\mathbf{H}}_0\\in\\operatorname{\\mathfrak{P}}$, which counts only curves with genus zero. \\\\\n\nWhile the Hamiltonian $\\operatorname{\\mathbf{H}}$ explicitly depends on the chosen contact form, the cylindrical almost complex structure, the differential forms and abstract polyfold perturbations making all moduli spaces regular, it is outlined in [EGH] how to construct algebraic invariants, which just depend on the contact structure and the cohomology classes of the differential forms. \\\\\n \n\\subsection{Gravitational descendants}\nFor the relation to integrable systems it is outlined in [E] that, as in Gromov-Witten theory, symplectic field theory must be enriched by considering so-called {\\it gravitational descendants} of the {\\it primary} Hamiltonian $\\operatorname{\\mathbf{H}}$. \\\\\n\nBefore we give a rigorous definition of gravitational descendants in SFT, we recall the definition from \nGromov-Witten theory. Denote by $\\overline{\\operatorname{\\mathcal{M}}}_r=\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(X,J)$ the compactified moduli space of closed $J$-holomorphic curves in the closed symplectic manifold $X$ of genus $g$ with $r$ marked points (and fixed homology class). Following [MDSa] we introduce over $\\overline{\\operatorname{\\mathcal{M}}}_r$ so-called {\\it tautological line bundles} $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\overline{\\operatorname{\\mathcal{M}}}_r$ in the noncompactified moduli space is given by the cotangent line to the underlying, possibly unstable closed nodal Riemann surface $S$ at the $i$.th marked point,\n\\begin{equation*} (\\operatorname{\\mathcal{L}}_i)_{(u,z_1,...,z_r)} = T^*_{z_i}S,\\;\\;i=1,...,r. \\end{equation*}\nTo be more formal, observe that there exists a canonical map $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ by forgetting the $(r+1)$.st marked point and stabilizing the map, where the fibre over the curve $(u,z_1,...,z_r)$ agrees with the curve itself. Then the tautological line bundle $\\operatorname{\\mathcal{L}}_i$ can be defined as the pull-back of the vertical cotangent line bundle of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ under the canonical section $\\sigma_i:\\overline{\\operatorname{\\mathcal{M}}}_r\\to\\overline{\\operatorname{\\mathcal{M}}}_{r+1}$ mapping to the $i$.th marked point in the fibre. Note that while the vertical cotangent line bundle is rather a sheaf than a true bundle since it becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are indeed true line bundles as the marked points are different from the nodes and hence these sections avoid the singular loci. \\\\\n\nDenoting by $c_1(\\operatorname{\\mathcal{L}}_i)$ the first Chern class of the complex line bundle $\\operatorname{\\mathcal{L}}_i$, one then considers for the descendant potential of Gromov-Witten theory integrals of the form\n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_r} \\operatorname{ev}_1^*\\theta_{i_1}\\wedge c_1(\\operatorname{\\mathcal{L}}_1)^{j_1}\\wedge ... \\wedge \\operatorname{ev}_r^*\\theta_{i_r}\\wedge c_1(\\operatorname{\\mathcal{L}}_r)^{j_r}, \\end{equation*}\nwhere $(i_k,j_k)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, which can again be organized into a generating function. \\\\\n\nLike pulling-back cohomology classes from the target manifold, the introduction of the tautological line bundles hence \nhas the effect that the generating function also sees the higher-dimensional moduli spaces. On the other hand, \nin contrast to the former, the latter refers to partially fixing the complex structure on the underlying punctured \nRiemann surface. \\\\\n\nBefore we can turn to the definition of gravitational descendants in SFT, it will turn out to be useful to give an alternative definition, where the integration of the powers of the first Chern classes is replaced by considering zero sets of sections. Restricting for notational simplicity to the case with one marked point, we can define by induction over $j\\in\\operatorname{\\mathbb{N}}$ a nested sequence of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j+1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}})^j = \\frac{1}{j!} \\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i. \\end{equation*}\n\nFor $j=1$ observe that, since the first Chern class of a line bundle agrees with its Euler class, the homology class obtained by integrating $c_1(\\operatorname{\\mathcal{L}})$ over the compactified moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ can be represented by the zero set of a generic section $s_1$ in $\\operatorname{\\mathcal{L}}$. Note that here we use that $\\overline{\\operatorname{\\mathcal{M}}}_1$ represents a pseudo-cycle and hence has no codimension-one boundary strata. In other words, we find that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}}) \\;=\\; \\int_{\\overline{\\operatorname{\\mathcal{M}}}^1_1} \\operatorname{ev}^*\\theta_i, \\end{equation*}\nwhere $\\overline{\\operatorname{\\mathcal{M}}}^1_1=s_1^{-1}(0)$. \\\\\n\nNow consider the restriction of the tautological line bundle $\\operatorname{\\mathcal{L}}$ to $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. Instead of describing the integration of powers of the first Chern class in terms of common zero sets of sections in the same line bundle $\\operatorname{\\mathcal{L}}$, it turns out to be more geometric (see 2.2) to choose a section $s_j$ not in $\\operatorname{\\mathcal{L}}$ but in its $j$-fold (complex) tensor product $\\operatorname{\\mathcal{L}}^{\\otimes j}$ and define \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0) \\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1. \\end{equation*} \nSince $c_1(\\operatorname{\\mathcal{L}}^{\\otimes j}) = j\\cdot c_1(\\operatorname{\\mathcal{L}})$ it follows that \n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i = j\\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1} \\operatorname{ev}^*\\theta_i\\wedge c_1(\\operatorname{\\mathcal{L}}) \\end{equation*}\nso that by induction\n\\begin{equation*} \\int_{\\overline{\\operatorname{\\mathcal{M}}}_1} \\operatorname{ev}^*\\theta_i \\wedge c_1(\\operatorname{\\mathcal{L}})^j = \\frac{1}{j!} \\cdot \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_1} \\operatorname{ev}^*\\theta_i \\end{equation*}\nas desired. \\\\\n\nWhile the result of the integration is well-known to be independent of the choice of the almost complex structure and the abstract polyfold perturbations, it also follows that the result is independent of the precise choice of the sequence of sections $s_1,...,s_j$. Like for the almost complex structure and the perturbations this results from the fact that the moduli spaces studied in Gromov-Witten theory have no codimension-one boundary. \\\\\n\nOn the other hand, it is well-known that the moduli spaces in SFT typically have codimension-one boundary, so that now the result of the integration will not only depend on the chosen contact form, cylindrical almost complex structure and abstract polyfold perturbations, but also additionally explicitly depend on the chosen sequences of sections $s_1,...,s_j$. While the Hamiltonian is hence known to depend on all extra choices, it is well-known from Floer theory that we can expect to find algebraic invariants independent of these choices. \\\\\n\nWhile the problem of dependency on contact form, cylindrical almost complex structure and abstract polyfold perturbations is sketched in [EGH], we will now show how to include gravitational descendants into their algebraic constructions. For this we will define {\\it descendants of moduli spaces}, which we obtain as zero sets of {\\it coherent collections of sections} in the tautological line bundles over all moduli spaces. \\\\\n\nFrom now on let $\\overline{\\operatorname{\\mathcal{M}}}_r$ denote the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ studied in SFT for chosen collections \nof Reeb orbits $\\Gamma^+,\\Gamma^-$. In complete analogy to Gromov-Witten theory we can introduce $r$ tautological line \nbundles $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\overline{\\operatorname{\\mathcal{M}}}_r$ is again given \nby the cotangent line to the underlying, possibly unstable nodal Riemann surface (without ghost components) at the \n$i$.th marked point and which again formally can be defined as the pull-back of the vertical cotangent line \nbundle of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ under the canonical section $\\sigma_i: \\overline{\\operatorname{\\mathcal{M}}}_r\\to\\overline{\\operatorname{\\mathcal{M}}}_{r+1}$ mapping to the $i$.th marked \npoint in the fibre. Note again that while the vertical cotangent line bundle is rather a sheaf than a true bundle since \nit becomes singular at the nodes in the fibres, the pull-backs under the canonical sections are still true line bundles \nas the marked points are different from the nodes and hence these sections avoid the singular loci. \\\\\n\nFor notational simplicity let us again restrict to the case $r=1$. Following the compactness statement in [BEHWZ], the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}_1$ consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ or $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. As we want to keep the notation as simple as possible, note that here and in what follows for product moduli spaces {\\it the first index refers to the level} and not to the genus of the curve. To be more precise, after introducing asymptotic markers as in [EGH] for orientation issues, one obtains a fibre rather than a direct product, see also [F2]. However, since all the bundles and sections we will consider do or should not depend on these asymptotic markers, we will forget about this issue in order to keep the notation as simple as possible. \\\\ \n\nOn the other hand, it directly follows from the definition of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ it is given by \n\\begin{equation*} \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}_1,\\;\\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}_2, \\end{equation*}\nwhere $\\operatorname{\\mathcal{L}}_1$, $\\operatorname{\\mathcal{L}}_2$ denotes the tautological line bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nWith this we can now introduce the notion of coherent collections of sections in (tensor products of) tautological line bundles. \\\\\n\\\\\n{\\bf Definition 1.1:} {\\it Assume that we have chosen sections $s$ in the tautological line bundles $\\operatorname{\\mathcal{L}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1$ of $\\underline{J}$-holomorphic curves with one additional marked point. Then this collection of sections $(s)$ is called} coherent {\\it if for every section $s$ in $\\operatorname{\\mathcal{L}}$ over a moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}_1$ the section $s$ agrees with the pull-back $\\pi_1^*s_1$, $\\pi_2^*s_2$ of the chosen section $s_1$, $s_2$ in the tautological line bundle $\\operatorname{\\mathcal{L}}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $\\operatorname{\\mathcal{L}}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, respectively.} \\\\\n\n\\noindent{\\bf Remark:} Since in the end we will again be interested in the zero sets of these sections, {\\it we will assume that all occuring sections are transversal to the zero section.} Furthermore, we want to assume that all the chosen sections are indeed {\\it invariant under the obvious symmetries like reordering of punctures and marked points.} In order to meet both requirements, it follows that actually need to employ {\\it multi-sections} as in [CMS], which we however want to suppress for the rest of this exposition. \\\\\n\nThe important observation is clearly that one can always find coherent collections of (transversal) sections $(s)$ by using induction on the dimension of the underlying moduli space. While for the induction start it suffices to choose a non-vanishing section in the tautological line bundle over the moduli space of orbit cylinders with one marked point, for the induction step observe that the coherency condition fixes the section on the boundary of the moduli space. Here it is important to remark that the coherency condition further ensures that two different codimension-one boundary components actually agree on their common boundary strata of higher codimension. On the other hand, we can use our assumption that every moduli space is indeed a manifold with corners to obtain the desired section by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. \\\\\n \nFor a given coherent collection of transversal sections $(s)$ we will again define for every moduli space \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^1_1 = s^{-1}(0) \\subset \\overline{\\operatorname{\\mathcal{M}}}_1. \\end{equation*}\nAs an immediate consequence of the above definition we find that $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ is a neat submanifold (with corners) of $\\overline{\\operatorname{\\mathcal{M}}}_1$, i.e., the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ are given by products $\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1}$, where $\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1} = s_1^{-1}(0)$, $\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1} = s_2^{-1}(0)$ for the section $s_1$ in $\\operatorname{\\mathcal{L}}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}$, $s_2$ in $\\operatorname{\\mathcal{L}}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, respectively. To be more precise, since we actually need to work with multi-sections rather than sections in the usual sense, the zero set is indeed a branched-labelled manifold. On the other hand, since we already suppressed the fact that our moduli spaces are indeed branched and labelled, we want to continue ignoring this technical aspect. On the other hand, we can use the above result as an induction start to obtain for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ a sequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ as in Gromov-Witten theory. \\\\\n\\\\\n{\\bf Definition 1.2:} {\\it Let $j\\in\\operatorname{\\mathbb{N}}$. Assume that for all moduli spaces we have chosen $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ are given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,0}$. Then we again call a collection of transversal sections $(s_j)$ in the j-fold tensor products $\\operatorname{\\mathcal{L}}^{\\otimes j}$ of the tautological line bundles over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$} coherent {\\it if for every section $s_j$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ the section $s_j$ agrees with the pull-back $\\pi_1^*s_{1,j}$, $\\pi_2^*s_{2,j}$ of the section $s_{1,j}$, $s_{2,j}$ in the line bundle $\\operatorname{\\mathcal{L}}^{\\otimes j}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}$, $\\operatorname{\\mathcal{L}}^{\\otimes j}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$, respectively.} \\\\\n\nWith this we will now introduce {\\it (gravitational) descendants of moduli spaces.} \\\\\n\\\\\n{\\bf Definition 1.3:} {\\it Assume that we have inductively defined a subsequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by requiring that \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ for a coherent collection of sections $s_j$ in the line bundles $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over the moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$. Then we call $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ the $j$.th (gravitational) descendant of $\\overline{\\operatorname{\\mathcal{M}}}_1$.} \\\\\n\nLet $\\operatorname{\\mathfrak{W}}^0$ be the graded Weyl algebra over $\\operatorname{\\mathbb{C}}$ of power series in the variables $\\hbar$ and $p_{\\gamma}$ with coefficients which are polynomials in the \nvariables $q_{\\gamma}$, which is obtained from the big Weyl algebra $\\operatorname{\\mathfrak{W}}$ by setting all variables $t_i$ equal to zero. In the same way define the subalgebra $\\operatorname{\\mathfrak{P}}^0$ of the Poisson algebra $\\operatorname{\\mathfrak{P}}$. Apart from the Hamiltonian $\\operatorname{\\mathbf{H}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$ counting only curves with no additional marked points,\n\\begin{equation*} \\operatorname{\\mathbf{H}}^0 = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_{g,0}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}, \\end{equation*}\nwe now want to use the chosen differential forms $\\theta_i\\in\\Omega^*(V)$, $i=1,...,N$ and the sequences $\\overline{\\operatorname{\\mathcal{M}}}^j_1 = \\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ of gravitational descendants to define sequences of new SFT Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, $(i,j)\\in\\{1,...,N\\}\\times \\operatorname{\\mathbb{N}}$, by\n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\\operatorname{ev}^*\\theta_i\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*}\n\nWe want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the neccessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. Since it can be expected that the polyfold project by Hofer and his collaborators sketched in [HWZ] will provide the required transversality theorems, we follow other papers in the field in proving everything up to transversality and state it nevertheless as a theorem. \\\\\n\\\\\n{\\bf Theorem 1.4:} {\\it Counting holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{H}}^1_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0) \\end{equation*} \n{\\it in the full SFT homology algebra with differential $D^0=[\\operatorname{\\mathbf{H}}^0,\\cdot]: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$, which commute with respect to the bracket on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$,} \n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}] = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\\\\\n{\\it Proof:} While the boundary equation $D^0\\circ D^0=0$ is well-known to follow from the identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^0]=0$, the fact that every $\\operatorname{\\mathbf{H}}^1_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$ defines an element in the homology $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0)$ follows from the identity\n\\begin{equation*} [\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^1_{i,j}]=0,\\end{equation*} \nsince this proves $\\operatorname{\\mathbf{H}}^1_{i,j}\\in\\ker D^0$. On the other hand, in order to see that any two $\\operatorname{\\mathbf{H}}^1_{i,j}$, $\\operatorname{\\mathbf{H}}^1_{k,\\ell}$ commute {\\it after passing to homology} it suffices to prove the identity\n\\begin{equation*} [\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]\\pm[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]=0 \\end{equation*} \nfor any $(i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, where the new Hamiltonian $\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}$ is defined below using descendant moduli spaces with two additional marked points. \\\\\n\nThe latter two identities directly follow from our definition of gravitational descendants of moduli spaces based on the definition of coherent sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as the identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^0]=0$ follows from the fact that the codimension-one boundary of every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ is formed by products of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, the second identity $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^1_{i,j}]=0$ follows from the fact that the codimension-one boundary of a descendant moduli space $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$. \\\\\n\nIn order to prove the third identity $[\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]\\pm[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]=0$ for every $(i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$, we slightly have to enlarge our definition of gravitational descendants in order to include moduli spaces with two additional marked points. For this observe that for every pair $j,k\\in\\operatorname{\\mathbb{N}}$ we can define decendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2$ of $\\overline{\\operatorname{\\mathcal{M}}}_2$ by setting $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2 = \\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2\\cap\\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2$, where $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2, \\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2\\subset\\overline{\\operatorname{\\mathcal{M}}}_2$ are defined in the same way as $\\overline{\\operatorname{\\mathcal{M}}}^j_1,\\overline{\\operatorname{\\mathcal{M}}}^k_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by simply forgetting the second or first additional marked point, respectively. Since the boundary of a moduli space of curves with two marked points consists of products of the form $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, it follows that the boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_2$ consists of products $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(j,0)}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$. Together with the similar result about the boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(0,k)}_2$ and using the inclusions we hence obtain that the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_2$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^k_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^k_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_{2,2}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(j,k)}_{1,2}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$. While summing over the first two products (with signs) we obtain $[\\operatorname{\\mathbf{H}}^1_{i,j},\\operatorname{\\mathbf{H}}^1_{k,\\ell}]$, summing over the latter two we get $[\\operatorname{\\mathbf{H}}^0,\\operatorname{\\mathbf{H}}^2_{(i,j),(k,\\ell)}]$, which hence sum up to zero. $\\qed$ \\\\ \n\\\\\n{\\bf Remark:} While the proof suggests that for the above algebraic relations one only has to care about the codimension-one boundary strata of the moduli spaces, it is actually even more important that the coherency condition further ensures that two different codimension-one boundary components can be glued along their common boundary strata of higher codimension. \\\\\n\nAs above we further again obtain a rational version of the above statement by expanding $\\operatorname{\\mathbf{H}}^0$ and the $\\operatorname{\\mathbf{H}}^1_{i,j}$ in powers \nof $\\hbar$. \\\\\n\\\\\n{\\bf Corollary 1.5:} {\\it Counting rational holomorphic curves with one marked point after integrating differential forms and introducing gravitational descendants defines a sequence of distinguished elements} \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^0,d^0), \\end{equation*} \n{\\it in the rational SFT homology algebra with differential $d^0=\\{\\operatorname{\\mathbf{h}}^0,\\cdot\\}: \\operatorname{\\mathfrak{P}}^0\\to\\operatorname{\\mathfrak{P}}^0$, which commute with respect to the Poisson bracket on $H_*(\\operatorname{\\mathfrak{P}}^0,d^0)$,} \n\\begin{equation*} \\{\\operatorname{\\mathbf{h}}^1_{i,j},\\operatorname{\\mathbf{h}}^1_{k,\\ell}\\} = 0,\\; (i,j),(k,\\ell)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}. \\end{equation*}\n\nSo far we have only considered the case with one additional marked point. On the other hand, the general case with $r$ additional marked points is just notationally more involved. Indeed, as we did in the proof of the above theorem we can easily define for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_r$ with $r$ additional marked points and every $r$-tuple of natural numbers $(j_1,...,j_r)$ descendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_r\\subset\\overline{\\operatorname{\\mathcal{M}}}_r$ by setting\n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_r = \\overline{\\operatorname{\\mathcal{M}}}^{(j_1,0,...,0)}_r\\cap ... \\cap \\overline{\\operatorname{\\mathcal{M}}}^{(0,...,0,j_r)}_r, \\end{equation*}\nwhere the descendant moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{(0,...,0,j_k,0,...,0)}_r\\subset\\overline{\\operatorname{\\mathcal{M}}}_r$ are defined in the same way as the one-point descendant $\\overline{\\operatorname{\\mathcal{M}}}^{j_k}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by looking at the $r$ tautological line bundles over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_r = \\overline{\\operatorname{\\mathcal{M}}}_r(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ separately and forgetting about the other points. \\\\\n\nWith this we can define the descendant Hamiltonian of SFT, which we will continue denoting by $\\operatorname{\\mathbf{H}}$, while the Hamiltonian defined in [EGH] will from now on be called {\\it primary}. In order to keep track of the descendants we will assign to every chosen differential form $\\theta_i$ now a sequence of formal variables $t_{i,j}$ with grading \n\\begin{equation*} |t_{i,j}|=2(1-j) -\\deg\\theta_i. \\end{equation*} \nThen the descendant Hamiltonian $\\operatorname{\\mathbf{H}}$ of SFT is defined by \n\\begin{equation*}\n \\operatorname{\\mathbf{H}} = \\sum_{\\Gamma^+,\\Gamma^-,I} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}}\n \\operatorname{ev}_1^*\\theta_{i_1}\\wedge...\\wedge\\operatorname{ev}_r^*\\theta_{i_r}\\; \\hbar^{g-1}t^Ip^{\\Gamma^+}q^{\\Gamma^-},\n\\end{equation*}\nwhere $p^{\\Gamma^+}=p_{\\gamma^+_1} ... p_{\\gamma^+_{n^+}}$, $q^{\\Gamma^-}=q_{\\gamma^-_1} ... q_{\\gamma^-_{n^-}}$ and $t^I=t_{i_1,j_1} ... t_{i_r,j_r}$ for $I=((i_1,j_1),...,(i_r,j_r))$. \\\\\n\nNote that expanding the Hamiltonian $\\operatorname{\\mathbf{H}}$ in powers of the formal variables $t_{i,j}$,\n\\begin{equation*} \n \\operatorname{\\mathbf{H}} = \\operatorname{\\mathbf{H}}^0 + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{H}}^1_{i,j} + o(t^2), \n\\end{equation*}\nwe get back our Hamiltonians $\\operatorname{\\mathbf{H}}^0$ and the sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ from above and it is easy to see that the primary Hamiltonian from [EGH] is recovered by setting all formal variables $t_{i,j}$ with $j>0$ equal to zero. \\\\\n\nIn the same way as it was shown for the primary Hamiltonian in [EGH], the descendant Hamiltonian continues to satisfy the master equation $[\\operatorname{\\mathbf{H}},\\operatorname{\\mathbf{H}}]=0$, which is just a generalization of the identities for $\\operatorname{\\mathbf{H}}^0$, $\\operatorname{\\mathbf{H}}^1_{i,j}$ and hence can be shown along the same lines by studying the codimension-one boundaries of descendant moduli spaces. On the other hand, expanding $\\operatorname{\\mathbf{H}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}$ in terms of powers of $\\hbar$, \n\\begin{equation*} \\operatorname{\\mathbf{H}}=\\sum_g \\hbar^{g-1}\\operatorname{\\mathbf{H}}_g, \\end{equation*} \nnote that for the rational descendant Hamiltonian $\\operatorname{\\mathbf{h}}=\\operatorname{\\mathbf{H}}_0\\in\\operatorname{\\mathfrak{P}}$ we still have $\\{\\operatorname{\\mathbf{h}},\\operatorname{\\mathbf{h}}\\}=0$. \\\\\n\n\\subsection{Invariance statement}\n\nWe now turn to the question of independence of these nice algebraic structures from the choices like contact form, cylindrical almost complex structure, abstract polyfold perturbations and, of course, the choice of the coherent collection of sections. This is the content of the following theorem, where we however again want to emphasize that the following statement is not yet a theorem in the strict mathematical sense as the analytical foundations of symplectic field theory, in particular, the neccessary transversality theorems for the Cauchy-Riemann operator, are not yet fully established. \\\\ \n\\\\\n{\\bf Theorem 1.6:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$ , abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting systems of commuting operators $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ which maps $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ to $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$.} \\\\\n\nAs above we clearly also get a rational version of the invariance statement: \\\\\n\\\\\n{\\bf Corollary 1.7:} {\\it For different choices of contact form $\\lambda^{\\pm}$, cylindrical almost complex structure $\\underline{J}^{\\pm}$, abstract polyfold perturbations and sequences of coherent collections of sections $(s^{\\pm}_j)$ the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ are isomorphic, i.e., there exists an isomorphism of the Poisson algebras $H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ and $H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$ which maps $\\operatorname{\\mathbf{h}}^{1,-}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,-},d^{0,-})$ to $\\operatorname{\\mathbf{h}}^{1,+}_{i,j}\\in H_*(\\operatorname{\\mathfrak{P}}^{0,+},d^{0,+})$.} \\\\\n\nThis theorem is an extension of the theorem in [EGH] which states that for different choices of auxiliary data the small Weyl algebras $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})$ and $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+})$ are isomorphic. On the other hand, assuming that the contact form, the cylindrical almost complex structure and also the abstract polyfold sections are fixed to have well-defined moduli spaces, the isomorphism of the homology algebras is the identity and hence the theorem states the sequence of commuting operators is indeed independent of the chosen sequences of coherent collections of sections $(s^{\\pm}_j)$, \n\\begin{equation*} \\operatorname{\\mathbf{H}}^{1,-}_{i,j} = \\operatorname{\\mathbf{H}}^{1,+}_{i,j}\\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0,D^0). \\end{equation*}\n$ $\\\\\n\nFor the proof we have to extend the proof in [EGH] to include gravitational descendants. To this end we have to study sections in the tautological line bundles over moduli spaces of holomorphic curves in symplectic manifolds with cylindrical ends. \\\\\n \nLet $(W,\\omega)$ be a symplectic manifold with cylindrical ends $(\\operatorname{\\mathbb{R}}^+\\times V^+,\\lambda^+)$ and \n$(\\operatorname{\\mathbb{R}}^-\\times V^-,\\lambda^-)$ in the sense of [BEHWZ] which is equipped with an almost complex structure \n$\\underline{J}$ which agrees with the cylindrical almost complex structures $\\underline{J}^{\\pm}$ on $\\operatorname{\\mathbb{R}}^+\\times V^+$. Then we \nstudy $\\underline{J}$-holomorphic curves in $W$ which are asymptotically cylindrical over \nchosen collections of orbits $\\Gamma^{\\pm}=\\{\\gamma^{\\pm}_1,...,\n\\gamma^{\\pm}_{n^{\\pm}}\\}$ of the Reeb vector fields $R^{\\pm}$ in $V^{\\pm}$ as the $\\operatorname{\\mathbb{R}}^{\\pm}$-factor tends \nto $\\pm\\infty$, see [BEHWZ], and denote by $\\operatorname{\\mathcal{M}}_{g,r}(\\Gamma^+,\\Gamma^-)$ the corresponding \nmoduli space of genus $g$ curves with $r$ additional marked points ([BEHWZ],[EGH]). \nPossibly after choosing abstract perturbations using polyfolds, obstruction \nbundles or domain-dependent structures, which agree with chosen abstract perturbations in the boundary as described above, we find that \n$\\operatorname{\\mathcal{M}}_{g,r}(\\Gamma^+,\\Gamma^-)$ is a weighted branched manifold of dimension equal to the Fredholm index of the Cauchy-Riemann operator for $\\underline{J}$. Note that as remarked above we will for simplicity assume that moduli space is indeed a manifold with corners, since this will be sufficient for our example and we expect that all the upcoming \nconstructions can be generalized in an appropriate way. We further extend the chosen differential forms $\\theta^{\\pm}_1,...\\theta^{\\pm}_N$ on $V^{\\pm}$ to differential forms $\\theta_1,...,\\theta_N$ on $W$ as described in [EGH]. \\\\\n\nFrom now on let $\\overline{\\operatorname{\\mathcal{M}}}_r$ denote the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)$ of holomorphic curves in $W$ for chosen collections of Reeb orbits $\\Gamma^+,\\Gamma^-$. Note in particular that there is no longer an $\\operatorname{\\mathbb{R}}$-action on the moduli space which we have to quotient out. In order to distinguish these moduli spaces in non-cylindrical manifolds from those of holomorphic curves in the cylindrical manifolds, we will use the short-hand notation $\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_r$ for moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{g,r}(\\Gamma^+,\\Gamma^-)\/\\operatorname{\\mathbb{R}}$ of holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$, respectively. Like in Gromov-Witten theory we can introduce $r$ tautological line bundles $\\operatorname{\\mathcal{L}}_1,...,\\operatorname{\\mathcal{L}}_r$, where the fibre of $\\operatorname{\\mathcal{L}}_i$ over a punctured curve $(u,z_1,...,z_r)\\in\\operatorname{\\mathcal{M}}_r$ in the noncompactified moduli space is again given by the cotangent line to the underlying closed Riemann surface at the $i$.th marked point and which formally can be defined as the pull-back of the vertical cotangent line bundle under the canonical section $\\sigma_i$ of $\\pi: \\overline{\\operatorname{\\mathcal{M}}}_{r+1}\\to\\overline{\\operatorname{\\mathcal{M}}}_r$ mapping to the $i$.th marked point in the fibre. \\\\\n\nFor notational simplicity let us again restrict to the case $r=1$. Following the compactness statement in [BEHWZ] the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}_1$ now consists of curves with one non-cylindrical level and one cylindrical level (in the sense of [BEHWZ]), whose moduli spaces can now be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ or $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of moduli spaces of strictly lower dimension, where the marked point sits on the first or the second level. Again note that here and in what follows for product moduli spaces the first index refers to the level and not to the genus of the curve. Furthermore it follows from the definition of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ it is given by \n\\begin{eqnarray*} \n \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}_1,&& \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}^+_2, \\\\\n \\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}} = \\pi_1^*\\operatorname{\\mathcal{L}}^-_1,&&\\operatorname{\\mathcal{L}}|_{\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}} = \\pi_2^*\\operatorname{\\mathcal{L}}_2, \n\\end{eqnarray*}\nwhere $\\operatorname{\\mathcal{L}}^{(-)}_1$, $\\operatorname{\\mathcal{L}}^{(+)}_2$ denotes the tautological line bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nWith this we can now introduce collections of sections in (tensor products of) tautological line bundles coherently connecting two chosen coherent collections of sections. \\\\\n\\\\\n{\\bf Definition 1.8:} {\\it Let $W$ be a symplectic manifold with cylindrical ends $V^{\\pm}$ and let $(s_{\\pm})$ be two coherent collections of sections in the tautological line bundles $\\operatorname{\\mathcal{L}}^{\\pm}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_1$ of $\\underline{J}$-holomorphic curves with one additional marked point in the cylindrical manifolds $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$. Assume that we have chosen transversal sections $s$ in the tautological line bundles $\\operatorname{\\mathcal{L}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1$ of $\\underline{J}$-holomorphic curves in the non-cylindrical manifold $W$ with one additional marked point. Then this collection of sections $(s)$ is called} coherently connecting $(s_-)$ and $(s_+)$ {\\it if for every section $s$ in $\\operatorname{\\mathcal{L}}$ over a moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}_1$ the section $s$ agrees with the pull-back $\\pi_1^*s_1$, $\\pi_1^*s^-_1$ or $\\pi_2^*s^+_2$, $\\pi_2^*s_2$ of the chosen sections $s_{1,(-)}$, $s_{2,(+)}$ in the tautological line bundles $\\operatorname{\\mathcal{L}}^{(-)}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $\\operatorname{\\mathcal{L}}^{(+)}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$, respectively.} \\\\\n\nNote that one can always find collections of sections $(s)$ coherently connecting given coherent collections of sections $(s_+)$ and $(s_-)$ as before by using induction on the dimension of the underlying moduli space. Indeed, for the induction step observe that the coherency condition again fixes the section on the boundary of the moduli space, so that the desired section can be obtained by simply extending the section from the boundary to the interior of the moduli space in an arbitrary way. \\\\\n \nFor a given coherently connecting collection of sections $(s)$ we will again define for every moduli space \n\\begin{equation*} \\overline{\\operatorname{\\mathcal{M}}}^1_1 = s^{-1}(0) \\subset \\overline{\\operatorname{\\mathcal{M}}}_1. \\end{equation*}\nAs an immediate consequence of the above definition we find that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^1_1$ are given by products \n$\\overline{\\operatorname{\\mathcal{M}}}^1_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{1,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^1_{2,1}$, where $\\overline{\\operatorname{\\mathcal{M}}}^{1,(-)}_{1,1} = s_{1,(-)}^{-1}(0)$, $\\overline{\\operatorname{\\mathcal{M}}}^{1,(+)}_{2,1} = s_{2,(+)}^{-1}(0)$ for the section $s_{1,(-)}$ in $\\operatorname{\\mathcal{L}}^{(-)}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(-)}_{1,1}$, $s_{2,(+)}$ in $\\operatorname{\\mathcal{L}}^{(+)}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{(+)}_{2,1}$, respectively. As before we can use this result as an induction start to obtain for every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$ a sequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. \\\\\n\\\\\n{\\bf Definition 1.9:} {\\it Let $j\\in\\operatorname{\\mathbb{N}}$ and let $(s_{j,\\pm})$ be two coherent collections of sections in the $j$-fold tensor products $\\operatorname{\\mathcal{L}}^{\\pm,\\otimes j}$ of the tautological line bundles over the $j-1$.st gravitational descendants $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,\\pm}_1 \\subset\\overline{\\operatorname{\\mathcal{M}}}^{\\pm}_1$ of all moduli spaces of curves in the cylindrical manifolds $\\operatorname{\\mathbb{R}}\\times V^{\\pm}$. Assume that for all moduli spaces of curves in the non-cylindrical manifold $W$ we have chosen $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ such that the components of the codimension-one boundary of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ are given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1,+}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,0}$. Then we again call a collection of transversal sections $(s_j)$ in the j-fold tensor products $\\operatorname{\\mathcal{L}}^{\\otimes j}$ of the tautological line bundles over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$} coherently connecting $(s_{j,-})$ and $(s_{j,+})$ {\\it if for every section $s_j$ the following holds: Over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_{2,1}$ of $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ the section $s_j$ agrees with the pull-back $\\pi_1^*s_{1,j}$, $\\pi_1^*s_{1,j,-}$ or $\\pi_2^*s_{2,j,+}$, $\\pi_2^*s_{2,j}$ of the section $s_{1,j,(-)}$, $s_{2,j,(+)}$ in the line bundle $\\operatorname{\\mathcal{L}}^{(-),\\otimes j}_1$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,(-)}_{1,1}$, $\\operatorname{\\mathcal{L}}^{(+),\\otimes j}_2$ over $\\overline{\\operatorname{\\mathcal{M}}}^{j-1,(+)}_{2,1}$, respectively.} \\\\\n\nWith this we can now introduce gravitational descendants of moduli spaces for symplectic manifolds with cylindrical ends. \\\\\n\\\\\n{\\bf Definition 1.10:} {\\it Assume that we have inductively defined subsequence of nested subspaces $\\overline{\\operatorname{\\mathcal{M}}}^j_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$ by requiring that \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = s_j^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ for a collection of sections $s_j$ in the line bundles $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over the moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}^{j-1}_1$ coherently connecting the coherent collections of sections $(s_{j,-})$ and $(s_{j,+})$. Then we call $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ the $j$.th (gravitational) descendant of $\\overline{\\operatorname{\\mathcal{M}}}_1$.} \\\\\n \nIn order to prove the above invariance theorem we now recall the extension of the algebraic formalism of SFT from cylindrical manifolds to symplectic cobordisms with cylindrical ends as described in [EGH]. \\\\\n\nLet $\\operatorname{\\mathfrak{D}}^0$ be the space of formal power series in the variables $\\hbar,p^+_{\\gamma}$ with coefficients which are polynomials in the \nvariables $q^-_{\\gamma}$. Elements in $\\operatorname{\\mathfrak{W}}^{0,\\pm}$ then act as differential operators from the right\/left on \n$\\operatorname{\\mathfrak{D}}^0$ via the replacements \n\\begin{equation*} \n q^+_{\\gamma}\\mapsto \\kappa_{\\gamma}\\hbar\\overleftarrow{\\frac{\\partial}{\\partial p^+_{\\gamma}}},\\;\\;\n p^-_{\\gamma}\\mapsto\\kappa_{\\gamma}\\hbar\\overrightarrow{\\frac{\\partial}{\\partial q^-_{\\gamma}}}.\n\\end{equation*}\n\nApart from the potential $\\operatorname{\\mathbf{F}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0$ counting only curves in $W$ with no additional marked points,\n\\begin{equation*} \\operatorname{\\mathbf{F}}^0 = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_{g,0}(\\Gamma^+,\\Gamma^-)\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}, \\end{equation*}\nwe now want to use the extensions $\\theta_i$, $i=1,...,N$ on $W$ of the chosen differential forms $\\theta^{\\pm}_1,...\\theta^{\\pm}_N$ on $V^{\\pm}$ and these sequences \n$\\overline{\\operatorname{\\mathcal{M}}}^j_1 = \\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)$ of gravitational descendants to define sequences of new SFT potentials $\\operatorname{\\mathbf{F}}^1_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times \\operatorname{\\mathbb{N}}$, by\n\\begin{equation*} \\operatorname{\\mathbf{F}}^1_{i,j} = \\sum_{\\Gamma^+,\\Gamma^-} \\int_{\\overline{\\operatorname{\\mathcal{M}}}^j_{g,1}(\\Gamma^+,\\Gamma^-)}\\operatorname{ev}^*\\theta_i\\; \\hbar^{g-1} p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*}\n\nFor the potential counting curves with no additional marked points we have the following identity, where we however again want to emphasize that the following statement should again be understood as a theorem up to the transversality problem in SFT. \\\\\n$ $\\\\\n{\\bf Theorem ([EGH]):} {\\it The potential $\\operatorname{\\mathbf{F}}^0\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$ satisfies the master equation}\n\\begin{equation*} e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}e^{\\operatorname{\\mathbf{F}}^0} = 0. \\end{equation*}\n\nIn [EGH] it is shown that this implies that \n\\begin{equation*}\n D^{\\operatorname{\\mathbf{F}}^0}: \\hbar^{-1}\\operatorname{\\mathfrak{D}}^0\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; \n D^{\\operatorname{\\mathbf{F}}^0}g = e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(ge^{\\operatorname{\\mathbf{F}}^0}) - (-1)^{|g|}(ge^{\\operatorname{\\mathbf{F}}^0})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}e^{-\\operatorname{\\mathbf{F}}^0} \n\\end{equation*}\nsatisfies $D^{\\operatorname{\\mathbf{F}}^0}\\circ D^{\\operatorname{\\mathbf{F}}^0} = 0$ and hence can be used to define the homology algebra \n$H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0})$. Furthermore it is shown that the maps \n\\begin{eqnarray*}\n &&F^{0,-}: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-}\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; f\\mapsto e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{f}e^{+\\operatorname{\\mathbf{F}}^0}, \\\\\n &&F^{0,+}: \\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+}\\to\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,\\; f\\mapsto e^{+\\operatorname{\\mathbf{F}}^0}\\overleftarrow{f}e^{-\\operatorname{\\mathbf{F}}^0}\n\\end{eqnarray*}\ncommute with the boundary operators,\n\\begin{equation*} F^{0,\\pm}\\circ D^{0,\\pm} = D^{\\operatorname{\\mathbf{F}}^0}\\circ F^{0,\\pm}, \\end{equation*} \nand hence descend to maps between the homology algebras\n\\begin{equation*} F^{0,\\pm}_*: H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,\\pm},D^{0,\\pm})\\to H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0}). \\end{equation*}\n\nNow assume that the contact forms $\\lambda^+$ and $\\lambda^-$ are chosen such that they define the same contact structure $(V^+,\\xi^+)=(V^-,\\xi^-)=:(V,\\xi)$ and let $W=\\operatorname{\\mathbb{R}}\\times V$ be the topologically trivial cobordism. Then in [EGH] the authors prove (up to transversality) the following fundamental theorem. \\\\\n\\\\\n{\\bf Theorem ([EGH]):} {\\it The map}\n\\begin{equation*} (F^{0,+}_*)^{-1}\\circ F^{0,-}_*: H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,-},D^{0,-})\\to H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,+},D^{0,+}) \\end{equation*}\n{\\it is an isomorphism of graded Weyl algebras.}\\\\\n \nFor the proof of the invariance statement we want to show that this map identifies the sequences $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$ on $H_*(\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{0,\\pm},D^{0,\\pm})$. In order to get the right idea for the proof, it turns out to be useful to even enlarge the picture as follows. \\\\\n\nPrecisely in the same way as for cylindrical manifolds we can define for every tuple $(j_1,...,j_r)$ of natural numbers gravitational descendants $\\overline{\\operatorname{\\mathcal{M}}}^{(j_1,...,j_r)}\\subset \\overline{\\operatorname{\\mathcal{M}}}_1$ of moduli spaces of curves in non-cylindrical manifolds with more than one additional marked point, which are collected in the {\\it descendant potential} $\\operatorname{\\mathbf{F}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$, where $\\operatorname{\\mathfrak{D}}$ is again obtained from $\\operatorname{\\mathfrak{D}}^0$ by considering coefficients which are formal powers in the graded formal variables $t_{i,j}$, $(i,j)\\in\\{1,...,N\\}\\times\\operatorname{\\mathbb{N}}$. \\\\\n\nAssuming for the moment that we have proven the fundamental identity \n\\begin{equation*} e^{\\operatorname{\\mathbf{F}}}\\overleftarrow{\\operatorname{\\mathbf{H}}^+} - \\overrightarrow{\\operatorname{\\mathbf{H}}^-}e^{\\operatorname{\\mathbf{F}}} = 0 \\end{equation*}\nand expanding the potential $\\operatorname{\\mathbf{F}}\\in\\hbar^{-1}\\operatorname{\\mathfrak{D}}$ and the two Hamiltonians $\\operatorname{\\mathbf{H}}^{\\pm}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^{\\pm}$ in \npowers of the $t$-variables,\n\\begin{equation*} \n \\operatorname{\\mathbf{F}} = \\operatorname{\\mathbf{F}}^0 + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{F}}^1_{i,j} + o(t^2), \\;\\;\n \\operatorname{\\mathbf{H}}^{\\pm} = \\operatorname{\\mathbf{H}}^{0,\\pm} + \\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j} + o(t^2), \\\\\n\\end{equation*}\nwe can deduce besides the master equation for $\\operatorname{\\mathbf{F}}^0$,\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}e^{\\operatorname{\\mathbf{F}}^0} = 0 \n\\end{equation*}\nand other identities also the identity \n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{\\operatorname{\\mathbf{F}}^0} = \n \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}, \n\\end{equation*} \nabout $\\operatorname{\\mathbf{F}}^0$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, where we used that\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}}= e^{\\operatorname{\\mathbf{F}}^0}\\cdot(1+\\sum_{i,j} t_{i,j}\\operatorname{\\mathbf{F}}^1_{i,j}) + o(t^2). \n\\end{equation*}\n\n{\\it Proof of the theorem:} Instead of proving the master equation for the full descendant potential $\\operatorname{\\mathbf{F}}$, we first show that it suffices to prove \n\\begin{equation*} \n e^{\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}} - \\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{\\operatorname{\\mathbf{F}}^0} = \n \\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}. \n\\end{equation*} \nIndeed, it is easy to see that the desired identity implies that\n\\begin{equation*} \n F^{0,+}(\\operatorname{\\mathbf{H}}^{1,+}_{i,j}) - F^{0,-}(\\operatorname{\\mathbf{H}}^{1,-}_{i,j}) = \n e^{+\\operatorname{\\mathbf{F}}^0}\\overleftarrow{\\operatorname{\\mathbf{H}}^{1,+}_{i,j}}e^{-\\operatorname{\\mathbf{F}}^0} - e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{1,-}_{i,j}}e^{+\\operatorname{\\mathbf{F}}^0} \n\\end{equation*}\nis equal to \n\\begin{equation*}\n e^{-\\operatorname{\\mathbf{F}}^0}\\overrightarrow{\\operatorname{\\mathbf{H}}^{0,-}}(e^{+\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j}) - (e^{+\\operatorname{\\mathbf{F}}^0}\\operatorname{\\mathbf{F}}^1_{i,j})\\overleftarrow{\\operatorname{\\mathbf{H}}^{0,+}}e^{-\\operatorname{\\mathbf{F}}^0} \n = D^{\\operatorname{\\mathbf{F}}^0}(\\operatorname{\\mathbf{F}}^1_{i,j}),\n\\end{equation*}\nso that, after passing to homology, we have\n\\begin{equation*} F^{0,+}_*(\\operatorname{\\mathbf{H}}^{1,+}_{i,j}) = F^{0,-}_*(\\operatorname{\\mathbf{H}}^{1,-}_{i,j}) \\in H_*(\\hbar^{-1}\\operatorname{\\mathfrak{D}}^0,D^{\\operatorname{\\mathbf{F}}^0}) \\end{equation*}\nas desired. \\\\\n\nOn the other hand, the above identity directly follows from our definition of gravitational descendants of moduli spaces based on the definition of coherently connecting sections in tautological line bundles and the compactness theorem in [BEHWZ]. Indeed, in the same way as it is shown in [EGH] that the master equation for $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$ follows from the fact that the codimension-one boundary of every moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ is formed by products of moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, the desired identity relating $\\operatorname{\\mathbf{F}}^0$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$ can be seen to follow from the fact that the codimension-one boundary of a descendant moduli space $\\overline{\\operatorname{\\mathcal{M}}}^j_1$ is given by products of the form $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}^{j,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$ and $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j,+}_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$: While the two summands involving $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}$, $\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on the left-hand-side of the equation collect all boundary components of the form $\\overline{\\operatorname{\\mathcal{M}}}^{j,-}_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}_{2,0}$, $\\overline{\\operatorname{\\mathcal{M}}}_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^{j,+}_{2,1}$, the two summands involving $\\operatorname{\\mathbf{F}}^1_{i,j}$ and $\\operatorname{\\mathbf{H}}^{0,-}$, $\\operatorname{\\mathbf{H}}^{0,+}$ on the right-hand-side of the equation collect all boundary components of the form $\\overline{\\operatorname{\\mathcal{M}}}^-_{1,0}\\times\\overline{\\operatorname{\\mathcal{M}}}^j_{2,1}$, $\\overline{\\operatorname{\\mathcal{M}}}^j_{1,1}\\times\\overline{\\operatorname{\\mathcal{M}}}^+_{2,0}$, respectively. Note that as for the master equation for $\\operatorname{\\mathbf{F}}^0$ and $\\operatorname{\\mathbf{H}}^{0,\\pm}$ the appearance of $\\operatorname{\\mathbf{F}}^0$ in the exponential follows from the fact that there corresponding curves may appear with an arbitrary number of connected components, while the curves counted for in $\\operatorname{\\mathbf{H}}^{0,\\pm}$, $\\operatorname{\\mathbf{H}}^{1,\\pm}_{i,j}$, $\\operatorname{\\mathbf{F}}^1_{i,j}$ can only appear once due to index reasons or since there is just one additional marked point. \\\\\n\nFinally, in order to see why we actually have $\\operatorname{\\mathbf{H}}^{1,-}_{i,j}=\\operatorname{\\mathbf{H}}^{1,+}_{i,j}$ on homology if we fixed $\\lambda^-=\\lambda^+=\\lambda$, $\\underline{J}^-=\\underline{J}^+=\\underline{J}$ and the abstract polyfold perturbations to have well-defined moduli spaces, observe that in this case $\\operatorname{\\mathbf{F}}^0$ just counts orbit cylinders, so that $F^{0,\\pm}$ and hence $(F^{0,\\pm})_*$ is the identity. $\\qed$ \n \n\n\\subsection{The circle bundle case}\n\nIn this subsection we briefly want to discuss the important case of circle bundles over closed symplectic manifolds, which links our constructions to gravitational descendants in Gromov-Witten theory, see also [R]. \\\\\n\nFor this recall that to any closed symplectic manifold $(M,\\omega)$ with integral symplectic form $[\\omega]\\in H^2(M,\\operatorname{\\mathbb{Z}})$ one can canonically assign a principal circle bundle $\\pi: V\\to M$ over $(M,\\omega)$ by requiring that $c_1(V)=[\\omega]$. Furthermore, it is easy to see that an $S^1$-connection form $\\lambda$ with curvature $\\omega$ on $\\pi:V\\to M$ is a contact form on the total space $V$, where the underlying contact structure agrees with the corresponding horizontal plane field $\\xi=\\ker\\lambda$, while the Reeb vector field $R$ agrees with the infinitesimal generator of the $S^1$-action. Observe that a $\\omega$-compatible almost complex structure $J$ on $M$ naturally equips $\\operatorname{\\mathbb{R}}\\times V$ with a cylindrical almost complex structure by requiring that $\\underline{J}$ maps the Reeb vector field to the $\\operatorname{\\mathbb{R}}$-direction and agrees with $J$ on the horizontal plane field $\\xi$, which is naturally identified with $TM$. \\\\\n\nSince every fibre of the circle bundle is hence a closed Reeb orbit for the contact form $\\lambda$, it follows that the space of orbits is given by $M\\times\\operatorname{\\mathbb{N}}$, where the second factor just refers to the multiplicity of the orbit. Hence, while every contact form in this class is not Morse as long as the symplectic manifold is not a point, it is still of Morse-Bott type. \\\\\n\nFollowing [EGH] the Weyl algebra $\\operatorname{\\mathfrak{W}}^0$ in this Morse-Bott case is now generated by sequences of graded formal variables $p_{\\alpha,k}$, $q_{\\alpha,k}$, $k\\in\\operatorname{\\mathbb{N}}$ assigned to cohomology classes $\\alpha$ forming a basis of $H^*(M,\\operatorname{\\mathbb{Z}})$. For circle bundles in the Morse-Bott setup we now show that the general theorem from above leads the following stronger statement. Note that in the following theorem we do {\\it not} assume that the sequences of coherent collections of sections are neccessarily $S^1$-invariant. \\\\\n\\\\\n{\\bf Theorem 1.11:} {\\it For circle bundles over symplectic manifolds, which are equipped with $S^1$-invariant contact forms, cylindrical almost complex structures (and abstract polyfold perturbations) as described above, the descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{i,j}$ define a sequence of commuting operators on $\\operatorname{\\mathfrak{W}}^0$, which is independent of the auxiliary data.} \\\\\n\\\\\n{\\it Proof:} Observing that a map $\\tilde{u}:(\\Sigma,j)\\to (\\operatorname{\\mathbb{R}}\\times V,\\underline{J})$ from a punctured Riemann sphere to the cylindrical manifold $\\operatorname{\\mathbb{R}}\\times V$, which is equipped with the canonical cylindrical almost complex structure $\\underline{J}$ defined by the $\\omega$-compatible almost complex structure $J$ on $M$, can be viewed as tuple $(h,u)$, where $u:(\\Sigma,j)\\to(M,J)$ is a $J$-holomorphic curve in $M$ and $h$ is a holomorphic section in $\\operatorname{\\mathbb{R}}\\times u^*V\\to\\Sigma$, it is easy to see that every moduli spaces studied in SFT for the contact manifold $V$ carries a natural circle bundle structure after quotienting out the natural $\\operatorname{\\mathbb{R}}$-action. It follows that $D^0=0$, so that by our first theorem the $\\operatorname{\\mathbf{H}}^1_j$ already commute as elements in $\\operatorname{\\mathfrak{W}}^0$. On the other hand, as long as the two different collections of auxiliary structures for $V$ are actually obtained as pull-backs of the corresponding auxiliary structures on $M$, it follows in the same way that the only rigid holomorphic curves in the resulting cobordisms are the orbit cylinders, so that the resulting automorphism is indeed the identity. $\\qed$ \\\\ \n \nFor $S^1$ and $S^3$ Eliashberg already pointed out in his ICM 2006 talk, see [E], that the corresponding sequences $\\operatorname{\\mathbf{h}}^1_j$ counting only genus zero curves lead to classical integrable systems, while the sequences of commuting operators $\\operatorname{\\mathbf{H}}^1_j$ provide deformation quantizations for these hierarchies. This is based on the surprising fact that the sequence $\\operatorname{\\mathbf{h}}^1_j$ of Poisson-commuting functions actually agrees with integrable system for genus zero from Gromov-Witten theory obtained using the underlying Frobenius manifold structure. In particular, for $V=S^1$ it follows that that the resulting system of Poisson-commuting functions are precisely the commuting integrals of the dispersionless KdV hierarchy,\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_j = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx,\\;\\;u(x)=\\sum_{n\\in\\operatorname{\\mathbb{N}}} p_n\\;e^{+2\\pi inx}+ q_n\\;e^{-2\\pi inx}, \\end{equation*} \nwhile in the case of the Hopf fibration $V=S^3$ over $M=S^2$ one arrives at the Poisson-commuting integrals of the continuous limit \nof the Toda lattice. \\\\\n\nIn order to see why in genus zero the SFT of the circle bundle $V$ is so closely related to the Gromov-Witten theory of its symplectic base $M$, we recall from the proof of the theorem that every $\\underline{J}$-holomorphic curve $\\tilde{u}$ can be identified with a tuple $(h,u)$, where $u$ is a $J$-holomorphic curve in $M$ and $h$ is a holomorphic section in $\\operatorname{\\mathbb{R}}\\times u^*V\\to\\Sigma$, whose poles and zeroes correspond to the positive and negative punctures with multiplicities. Since the zeroth Picard group of $S^2$ is trivial and hence every degree zero divisor is indeed a principal divisor, it follows that for every map $u$ the space of sections is isomorphic to $\\operatorname{\\mathbb{C}}$ and hence that the SFT moduli space of $\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$ is indeed a circle bundle over the corresponding Gromov-Witten moduli space of $J$-holomorphic curves in $M$. \\\\\n\nWhile this explains the close relation of SFT of circle bundles and Gromov-Witten theory in the genus zero case, the non-triviality of the Picard group for nonzero genus implies that the relation gets much more obscure when we allow for curves of arbitrary genus. Indeed, while in the case of $V=S^1$ the sequence $\\operatorname{\\mathbf{H}}^1_j$ defined by counting curves of arbitrary genus in $\\operatorname{\\mathbb{R}}\\times V$ leads to the deformation quantization of the dispersionless KdV hierarchy, in particular, a quantum integrable system, counting curves of all genera in the underlying symplectic manifold, that is, the point, leads by Witten's conjecture to the classical integrable system given by the full KdV hierarchy as proven by Kontsevich . \\\\\n\nAt the end of this subsection we again want to emphasize that the above statement crucially relies on the fact that $V$ is equipped with a $S^1$-invariant contact form, cylindrical almost complex structure and abstract polyfold perturbations. Assuming for the moment that the sequences of coherent collections of sections are also chosen to be $S^1$-invariant, note that in this case the above invariance statement can directly be deduced from the independence of the descendant Gromov-Witten potential of the auxiliary data used to define it, which essentially relies on the fact that all moduli spaces have only boundary components of codimension greater or equal to two, so that absolute rather than relative virtual classes are defined. In particular, the gravitational descendants can be defined by integrating powers of the first Chern class over the absolute moduli cycle. On the other hand, recall that for the above theorem we did {\\it not} require that the sequences of coherent collections of sections are neccessarily $S^1$-invariant. While our definition of coherent collections of sections seems to be very weak, our above theorem shows that the nice invariance property continues to hold even for a larger class of sections. \n\n\n\\section{Example: Symplectic field theory of closed geodesics} \n\n\\subsection{Symplectic field theory of a single Reeb orbit}\nWe are now going to consider a concrete example, which actually formed the starting point for the formal discussion from above. \\\\\n\nAs above consider a closed contact manifold $V$ with chosen contact form $\\lambda\\in\\Omega^1(V)$ and let $\\underline{J}$ be a compatible cylindrical almost complex structure on $\\operatorname{\\mathbb{R}}\\times V$. For any closed orbit $\\gamma$ of the corresponding Reeb vector field $R$ on $V$ the \norbit cylinder $\\operatorname{\\mathbb{R}}\\times\\gamma$ together with its branched covers are the basic examples of $\\underline{J}$-holomorphic curves in $\\operatorname{\\mathbb{R}}\\times V$. \\\\\n \nIn [F2] we prove that these {\\it orbit curves} do not contribute to the algebraic invariants of symplectic field theory as long as they do not carry additional marked points. Our proof explicitly uses that the orbit curves (over a fixed orbit) are closed under taking boundaries and gluing, which follows from the fact that orbit curves are also trivial in the sense that they have trivial contact area and that this contact area is preserved under taking boundaries and gluing. In particular, it follows, see [F2], that every algebraic invariant of symplectic field theory has a natural analog defined by counting only orbit curves. Further specifying the underlying Reeb orbit let us hence introduce the {\\it symplectic field theory of the Reeb orbit $\\gamma$:} \\\\ \n\nFor this denote by $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ be the graded Weyl subalgebra of the Weyl algebra $\\operatorname{\\mathfrak{W}}$, which is generated only by those $p$- and \n$q$-variables $p_n=p_{\\gamma^n}$, $q_n=q_{\\gamma^n}$ corresponding to Reeb orbits which are multiple covers of the fixed orbit $\\gamma$ {\\it and which are good in the sense of [BM]}. In the same way we further introduce the Poisson subalgebra $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ of $\\operatorname{\\mathfrak{P}}^0$. It will become important that the natural identification of the formal variables $p_n$ and $q_n$ does {\\it not} lead to an isomorphism of the graded algebras $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ with the corresponding graded algebras $\\operatorname{\\mathfrak{W}}^0_{S^1}$ and $\\operatorname{\\mathfrak{P}}^0_{S^1}$ for $\\gamma=V=S^1$, not only since the gradings of $p_n$ and $q_n$ are different and hence even the commutation rules may change but also that variables $p_n$ and $q_n$ may not be there since they would correspond to bad orbits. \\\\\n\nIn the same way as we introduced the (rational) Hamiltonian $\\operatorname{\\mathbf{H}}^0$ and $\\operatorname{\\mathbf{h}}^0$ as well as sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_j$ and $\\operatorname{\\mathbf{h}}^1_j$ by counting general curves in the symplectization of a contact manifold, we can define distinguished elements $\\operatorname{\\mathbf{H}}^0_{\\gamma}\\in\\hbar^{-1}\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathbf{h}}^0_{\\gamma}\\in\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, as well as sequences of descendant Hamiltonians $\\operatorname{\\mathbf{H}}^1_{\\gamma,j}$ and $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ by just counting branched covers of the orbit cylinder over $\\gamma$ with signs \n(and weights), where the preservation of the contact area under splitting and gluing of curves proves that for every theorem from above we have a version for $\\gamma$. \\\\\n\nWhile for the general part described above we have already emphasized that the theorems are not yet theorems in the strict mathematical sense since the neccessary transversality theorems for the Cauchy-Riemann operator are part of the on-going polyfold project by Hofer and his collaborators and we further used the assumption that all occuring moduli spaces are manifolds with corners, for the rest of this paper we will {\\it restrict to the rational case}, i.e., we will only be interested in the Poisson-commuting sequences $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on $H_*(\\operatorname{\\mathfrak{P}}^0_{\\gamma},d^0_{\\gamma})$, but in return solve the occuring analytical problems in all detail. In particular, we have already proven in the paper [F2] that for (rational) orbit curves the transversality problem can indeed be solved using finite-dimensional obstruction bundles instead of infinite-dimensional polybundles. In order to see why this is even neccessary, observe that while in the case when $\\gamma=V=S^1$ the Fredholm index equals the dimension of the moduli space, for general $\\gamma\\subset V$ the Fredholm index of a true branched cover is in general strictly smaller than the dimension of the moduli space of branched covers, so that transversality for the Cauchy-Riemann operator can in general not be satisfied. \\\\\n\nSo let us recall the main results about obstruction bundle transversality for orbit curves, where we refer to [F2] for all details. The first observation for orbit curves is that the cokernels of the linearized Cauchy-Riemann operators indeed fit together to give a smooth vector bundle $\\overline{\\operatorname{Coker}} \\bar{\\partial}_{\\underline{J}}$ over the compactified (nonregular) moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ of orbit curves (of constant rank). It follows that every transveral section $\\bar{\\nu}$ of this cokernel bundle leads to a compact perturbation making the Cauchy-Riemann operator transversal to the zero section in the underlying polyfold setup. \\\\\n\nIn Gromov-Witten theory we would hence obtain the contribution of the regular perturbed moduli space by integrating the Euler class of the finite-dimensional obstruction bundle over the compactified moduli space. On the other hand, passing from Gromov-Witten theory back to symplectic field theory again, we see that we just arrive at the same problem we had to face with when we wanted to define gravitational descendants in symplectic field theory. Indeed, as for the tautological line bundles, the presence of codimension-one boundary of the (nonregular) moduli spaces of branched covers implies that Euler numbers for sections in the cokernel bundles are not defined in general, since the count of zeroes depends on the compact perturbations chosen for the moduli spaces in the boundary. \\\\\n\nInstead of looking at a single moduli space, we hence again have to consider all moduli spaces at once. Replacing the tautological line bundle $\\operatorname{\\mathcal{L}}$ by the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ and considering the nonregular moduli space of branched covers instead of the regular moduli space itself, we hence now define {\\it coherent} collections of sections in the obstruction bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ as follows. \\\\\n\nFollowing the compactness statement in [BEHWZ] for the contact manifold $S^1$ the codimension-one boundary of every moduli space of branched covers $\\overline{\\operatorname{\\mathcal{M}}}$ again consists of curves with two levels (in the sense of [BEHWZ]), whose moduli spaces can be represented as products $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ of moduli spaces of strictly lower dimension, where the first index again refers to the level. On the other hand, it follows from the linear gluing result in [F2] that over the boundary component $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is given by \n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2} = \\pi_1^*\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}\\oplus\\pi_2^*\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}, \\end{equation*}\nwhere $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ denotes the cokernel bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{\\mathcal{M}}}_2$ and $\\pi_1$, $\\pi_2$ is the projection onto the first or second factor, respectively. \\\\\n\nAssuming that we have chosen sections $\\bar{\\nu}$ in the cokernel bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ over all moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}$ of branched covers, we again call this collection of sections $(\\bar{\\nu})$ {\\it coherent} if over every codimension-one boundary component $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$ of a moduli space $\\overline{\\operatorname{\\mathcal{M}}}$ the corresponding section $\\bar{\\nu}$ agrees with the pull-back $\\pi_1^*\\bar{\\nu}_1\\oplus\\pi_2^*\\bar{\\nu}_2$ of the chosen sections $\\bar{\\nu}_1$, $\\bar{\\nu}_2$ in the cokernel bundles $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_2$, respectively. \\\\\n\nSince in the end we will again be interested in the zero sets of these sections, we will again assume that all occuring sections are transversal to the zero section.\nAs before it is not hard to see that one can always find such coherent collections of (transversal) sections in the cokernel bundles by using induction on the dimension of the underlying nonregular moduli space of branched covers. Note that the latter is {\\it not} equal to the Fredholm index. \\\\\n\nIn [F2] we prove the following result about orbit curves with {\\it no} additional marked points. \\\\\n\\\\\n{\\bf Theorem ([F2]):} {\\it For the cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ over the compactification $\\overline{\\operatorname{\\mathcal{M}}}$ of every moduli space \nof branched covers over an orbit cylinder with $\\dim\\operatorname{\\mathcal{M}}-\\operatorname{rank}\\operatorname{Coker}\\bar{\\partial}_J=0$ the following holds:}\n\\begin{itemize} \n\\item {\\it For every pair $\\bar{\\nu}^0$, $\\bar{\\nu}^1$ of coherent and transversal sections in $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ the algebraic count of zeroes \n of $\\bar{\\nu}^0$ and $\\bar{\\nu}^1$ are finite and agree, so that we can define an Euler number \n $\\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J)$ for coherent sections in $\\overline{\\operatorname{Coker}}\\bar{\\partial}_J$ by} \n \\begin{equation*} \\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J) \\,:=\\, \\sharp (\\bar{\\nu}^0)^{-1}(0) \\,=\\, \\sharp (\\bar{\\nu}^1)^{-1}(0). \n \\end{equation*} \n\\item {\\it This Euler number is $\\chi(\\overline{\\operatorname{Coker}}\\bar{\\partial}_J) = 0$.} \n\\end{itemize}\n$ $\\\\\nThis theorem in turn has the following consequence.\\\\\n\\\\\n{\\bf Corollary 2.1:} { \\it For every closed Reeb orbit $\\gamma$ the Hamiltonian $\\operatorname{\\mathbf{h}}^0_{\\gamma}$ vanishes independently of the chosen coherent collection of sections $(\\bar{\\nu})$ in the cokernel bundles over all moduli spaces of branched covers,}\n\\begin{equation*} \\operatorname{\\mathbf{h}}^0 = \\operatorname{\\mathbf{h}}^{0,\\bar{\\nu}}= 0. \\end{equation*}\n{\\it In particular, the sequences of descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ already Poisson-commute as elements in $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$.}\\\\\n\nNote that the latter statement is obvious in the case $\\gamma=V=S^1$. While it directly follows from index reasons that $\\operatorname{\\mathbf{h}}^1_{S^1,j}=0$ when the string of differential forms just consists of the zero-form $1$ on $S^1$, it is shown in [E] using the results from Okounkov and Pandharipande in [OP] that for the one-form $dt$ on $S^1$ the system of Poisson commuting functions on $\\operatorname{\\mathfrak{P}}^0_{S^1}$ is given by\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{S^1,j} = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx,\\;\\;u(x)=\\sum_{n\\in\\operatorname{\\mathbb{N}}} p_n\\;e^{+2\\pi inx}+ q_n\\;e^{-2\\pi inx}, \\end{equation*} \ni.e., hence agrees with the {\\it dispersionless KdV (or Burger) integrable hierarchy.} \\\\\n \nGoing back from $\\gamma=V=S^1$ to the case of orbit curves over general Reeb orbits $\\gamma$, observe that, since for the orbit curves the evaluation map to $V$ factors through the inclusion map $\\gamma\\subset V$, it follows that it again only makes sense to consider zero- or one-forms, where we can assume without loss of generality that the zero-form agrees with $1\\in\\Omega^0(V)$ and that the integral of the one-form $\\theta\\in\\Omega^1(V)$ over the Reeb orbit is one,\n\\begin{equation*} \\int_{\\gamma}\\theta = 1. \\end{equation*}\nFor the case with no gravitational descendants, note that it follows from index reasons that the only curves to be considered are orbit cylinders with one marked point, since introducing an additional marked point adds two or one to the Fredholm index. Since orbit cylinders are always regular and their contribution hence just equals the integral of the form $\\theta$ over the closed orbit $\\gamma$, we hence get just like in the case of $\\gamma=V=S^1$ that the zeroth descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{\\gamma,0}$ vanishes if $\\deg\\theta = 0$ and \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,0} = \\oint_{S^1} \\frac{u^2(x)}{2!}\\,dx = \\sum p_n q_n \\end{equation*}\nif $\\deg\\theta = 1$ with the normalization from above. For the sum note that we only assigned formal variables $p_n$, $q_n$ to Reeb orbits which are good in the sense of [BM]. \\\\\n\nWhile the Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,0}$ hence agree with the Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,0}$ for $\\gamma=V=S^1$ up to the problem of bad orbits, since no obstruction bundles have to be considered, it is easy to see that the argument breaks down when gravitational descendants are introduced, since the underlying orbit curve then has non-zero Fredholm index $1+2(j-1)+\\deg \\theta$ and hence need not be an orbit cylinder anymore. While for the case of a one-form we can hence expect to find new integrals for the nontrivial Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,0}=\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, we first show that in the case of a zero-form not only the zeroth Hamiltonian but even the whole sequence of descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ is trivial. \\\\\n\\\\ \n{\\bf Theorem 2.2:} {\\it Let $\\gamma$ be a Reeb orbit in any contact manifold $V$ and assume that the string of differential forms on $V$ just consists of the zero-form $1\\in\\Omega^0(V)$. Then the sequence of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is trivial,}\n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,j} = 0,\\;j\\in\\operatorname{\\mathbb{N}} \\end{equation*}\n{\\it just like in the case of $\\gamma=V=S^1$.}\\\\\n\\\\\n{\\it Proof:} Since the proof of this theorem follows from completely the same arguments as the proof of our theorem in [F2] about Euler numbers of coherent sections in obstruction bundles from above, we shortly give the main idea for the proof in [F2] about orbit curves without additional marked points and then discuss its generalization to orbit curves with zero-forms and gravitational descendants. \\\\\n \nAfter proving that we can work with finite-dimensional obstruction bundles instead of infinite-dimensional polybundles, recall that the main problem lies in the presence of codimension-one boundary of the (nonregular) moduli space, so that Euler numbers of Fredholm problems are not defined in general, since the count of zeroes in general depends on the compact perturbations chosen for the moduli spaces in the boundary. In [F2] we prove the existence of the Euler number for moduli spaces of orbit curves without additional marked points by induction on the number of punctures. For the induction step we do not only use that there exist Euler numbers for the moduli spaces in the boundary, but it is further important that all these Euler numbers are in fact trivial. The vanishing of the Euler number in turn is deduced from the different parities of the Fredholm index of the Cauchy-Riemann operator and the actual dimension of the moduli space of branched covers following the idea for the vanishing of the Euler characteristic for odd-dimensional manifolds. \\\\\n\nFor the generalization to the case of additional marked points and gravitational descendants, it is clear that it still suffices to work with finite-dimensional obstruction bundles. On the other hand, recall that the only further ingredient to our proof in [F2] was that the Fredholm index and the dimension of the moduli spaces always have different parity. Hence it follows that the proof in [F2] also works for the case when $\\theta$ is a zero-form as the actual dimension of the moduli spaces is still even, while it breaks down in the case when $\\theta$ is a one-form. $\\qed$ \\\\\n\nObserve that for one-forms it is indeed no longer clear that the every Euler number has to be zero, as we for $\\gamma=V=S^1$ and $\\theta=dt$ we get nontrivial contributions from true branched covers. While at first glance the major problem seems to be the truely complicated computation of the Euler number (see [HT1], [HT2] for related results), we further have the problem that Euler numbers need no longer exist for all Fredholm problems. {\\it For the rest of this paper we will hence only be interested in the case where the chosen differential form has degree one,} $\\deg\\theta =1$.\\\\\n\nWhile for $\\gamma=V=S^1$ we actually get a unique sequence of Poisson-commuting functions, observe that for general fixed Reeb orbits $\\gamma$ in contact manifolds $V$ the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j} = \\operatorname{\\mathbf{h}}^{1,\\bar{\\nu}}_{\\gamma,j}$ may indeed depend on the chosen collection of sections in the cokernel bundles $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$. Hence the invariance statement is no longer trivial, but implies that for different choices of coherent abstract perturbations $\\bar{\\nu}^{\\pm}$ for the moduli spaces the resulting system of commuting elements $\\operatorname{\\mathbf{h}}^{1,-}_{\\gamma,j}$, $j=0,1,2,..$ and $\\operatorname{\\mathbf{h}}^{1,+}_{\\gamma,j}$, $j=0,1,2,..$ on $\\operatorname{\\mathfrak{P}}_{\\gamma}^0$ are just isomorphic, i.e., there exists an {\\it auto}morphism of the Poisson algebra $\\operatorname{\\mathfrak{P}}_{\\gamma}^0$ which identifies $\\operatorname{\\mathbf{h}}^{1,-}_{\\gamma,j}\\in \\operatorname{\\mathfrak{P}}_{\\gamma}^0$ with $\\operatorname{\\mathbf{h}}^{1,+}_{\\gamma,j}\\in \\operatorname{\\mathfrak{P}}_{\\gamma}^0$ for all $j\\in\\operatorname{\\mathbb{N}}$. \\\\\n\nThe above discussion hence shows that the computation of the symplectic field theory of a closed Reeb orbit gets much more difficult when gravitational descendants are considered. In what follows we want to determine it in the special case where the contact manifold is the unit cotangent bundle $S^*Q$ of a ($m$-dimensional) Riemannian manifold $Q$, so that every closed Reeb orbit $\\gamma$ on $V=S^*Q$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$. \\\\\n\nBefore we can state the theorem we first want to expand the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ in terms of the $p_n$- and $q_n$-variables, where set $p_n=q_{-n}$. Abbreviating $u_n(x) = q_n e^{inx}$ for every nonzero integer $n$ it follows from $u=\\sum_n u_n$ that \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{S^1,j} = \\oint_{S^1} \\frac{u^{j+2}(x)}{(j+2)!}\\,dx\n = \\oint_{S^1} \\sum \\frac{u_{n_1}(x)\\cdot ... \\cdot u_{n_{j+2}}(x)}{(j+2)!}\\, dx\n\\end{equation*}\nOn the other hand, note that the integration around the circle corresponds to selecting only those sequences of multiplicities $(n_1,...,n_{j+2})$, whose sum is equal to zero, so that \n\\begin{eqnarray*} \n\\operatorname{\\mathbf{h}}^1_{S^1,j} \\,=\\, \\sum_{n_1+...+n_{j+2} = 0} \\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!}.\n\\end{eqnarray*} \nApart from the sequence of Poisson-commuting functions for the circle, the grading of the functions given by the grading of $p_n$- and $q_n$-variables will play a central role for the upcoming theorem. For this observe that it follows from the grading conventions in symplectic field theory that the grading of the full Hamiltonian $\\operatorname{\\mathbf{H}}^0$ is $-1$, so that by $\\operatorname{\\mathbf{H}}^0 = \\sum_g \\hbar^{g-1} \\operatorname{\\mathbf{H}}^0_g$ the grading for the rational Hamiltonian $\\operatorname{\\mathbf{h}}^0=\\operatorname{\\mathbf{H}}^0_0$ is given by $|\\operatorname{\\mathbf{h}}^0| = |\\operatorname{\\mathbf{H}}^0|+|\\hbar| = -1+2(m-2)$. Since this grading has to agree with the grading of $t_j\\operatorname{\\mathbf{h}}^1_j$ with $|t_j|=2(1-j)-\\deg\\theta = 1-2j$, it follows that for every Reeb orbit $\\gamma\\subset V$ we have\n\\begin{equation*} |\\operatorname{\\mathbf{h}}^1_{\\gamma,j}| = -1+2(m-2)-1+2j = 2(m+j-3). \\end{equation*}\n\nWe already mentioned that the natural identification of the formal variables $p_n$ and $q_n$ does {\\it not} lead to an isomorphism of the graded algebras $\\operatorname{\\mathfrak{W}}^0_{\\gamma}$ and $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ with the corresponding graded algebras $\\operatorname{\\mathfrak{W}}^0_{S^1}$ and $\\operatorname{\\mathfrak{P}}^0_{S^1}$ for $\\gamma=V=S^1$, not only since the gradings of $p_n$ and $q_n$ are different and hence even the commutation rules may change but even that variables $p_n$ and $q_n$ may not be there since they would correspond to bad orbits. While for the grading of $\\gamma=V=S^1$ given by $|p_n|=|q_n|=-2$ in the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ every summand indeed has the same degree $2(m+j-3)$, passing over to a general Reeb orbit $\\gamma$ with the new grading given by $|p_n|=m-3-\\operatorname{CZ}(\\gamma^n)$, $|q_n|=m-3+\\operatorname{CZ}(\\gamma^n)$ the descendant Hamiltonian $\\operatorname{\\mathbf{h}}^1_{S^1,j}$ is no longer of pure degree, i.e., different summands of the same descendant Hamiltonian usually have different degree. While the Poisson-commuting sequence for the circle seems not to be related to the sequence of descendant Hamiltonians for general Reeb orbits $\\gamma$, we prove the following result in the case when the Reeb orbit corresponds to a closed geodesic. \\\\\n\\\\\n{\\bf Theorem 2.3:} {\\it Assume that the contact manifold is the unit cotangent bundle $V=S^*Q$ of a Riemannian manifold $Q$, so that the closed Reeb orbit $\\gamma$ corresponds to a closed geodesic $\\bar{\\gamma}$ on $Q$, and that the string of differential forms just consists of a single one-form which integrates to one around the orbit. Then the resulting system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to the system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\nWe have the following immediate corollary, which immediately follows from the behavior of the Conley-Zehnder index for multiple covers. \\\\\n\\\\\n{\\bf Corollary 2.4:} {\\it Assume that the closed geodesic $\\bar{\\gamma}$ represents a hyperbolic Reeb orbit in the unit cotangent bundle and $\\dim Q>1$. Then $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}=0$ and hence $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}=0$ for all $j>0$.} \\\\\n\nIndeed, since for hyperbolic Reeb orbits the Conley-Zehnder index $\\operatorname{CZ}(\\gamma^n)$ of $\\gamma^n$ is given by $\\operatorname{CZ}(\\gamma^n)=n\\cdot\\operatorname{CZ}(\\gamma)$, an easy computation shows that there are {\\it no} products of the above form of the desired degree. On the other hand, note that without the degree condition we would just get back the sequence of descendant Hamiltonians for the circle. Forgetting about orientation issues, in simple words we can hence say that the sequence $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ is obtained from the sequence for $\\bar{\\gamma}=Q=S^1$ by removing all summands with the wrong, that is, not maximal degree, where the latter can explicitely be computed using the formulas in [Lo] but also follows from our proof. \\\\\n\nThe proof relies on the observation that for orbit curves the gravitational descendants indeed have a geometric meaning in terms of branching conditions, which is a slight generalization of the result for the circle shown by Okounkov and Pandharipande in [OP]. Applying (and generalizing) the ideas of Cieliebak and Latschev in [CL] for relating the symplectic field theory of $V=S^*Q$ to the string topology of the underlying Riemannian manifold $Q$, we then study branched covers of the corresponding trivial half-cylinders in the cotangent bundle connecting the Reeb orbit $\\gamma$ with the underlying geodesic $\\bar{\\gamma}$ to prove that the sequence of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ is isomorphic to a sequence of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$. {\\it While the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ on the SFT side are defined using very complicated obstruction bundles over (nonregular) moduli spaces of arbitary large dimension, the key observation is that for the descendant Hamiltonians $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ on the string side we indeed only have to study obstruction bundles over discrete sets, which clearly disappear if the Fredholm index is right.} With this we get that the Poisson-commuting sequences for the closed geodesics can be computed from the sequences for the circle {\\it and} the Morse indices of the geodesic and its iterates as stated in the theorem. \\\\\n\n\\subsection{Gravitational descendants = branching conditions}\nRecall that by the above theorem from the last subsection we only have to consider the case where $\\theta$ is a one-form on $V$, where we still assume without loss of generality that the integral of $\\theta$ over $\\gamma$ is one. It follows that integrating the pullback of $\\theta$ under the evaluation map over the moduli space of orbit curves with one additional marked point and dividing out the natural $\\operatorname{\\mathbb{R}}$-action on the target $\\operatorname{\\mathbb{R}} \\times S^1\\cong\\operatorname{\\mathbb{R}}\\times\\gamma$ is equivalent to restricting to orbit curves where the additional marked point is mapped to a special point on $\\operatorname{\\mathbb{R}} \\times S^1$. In other words, in what follows we will view $h^1_{\\gamma,j}$ no longer as part of the Hamiltonian for $\\gamma$ but as part of the potential for the cylinder over $\\gamma$ equipped with a non-translation-invariant two-form. In order to save notation, $\\overline{\\operatorname{\\mathcal{M}}}_1=\\overline{\\operatorname{\\mathcal{M}}}_1(\\Gamma^+,\\Gamma^-)$ will from now on denote the corresponding moduli space. On the other hand, after introducing coherent collections $(\\bar{\\nu})$ of obstruction bundle sections, it is easy to see that the tautological line bundle $\\operatorname{\\mathcal{L}}^{\\bar{\\nu}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}$ is just the restriction of the tautological line bundle $\\operatorname{\\mathcal{L}}$ over $\\overline{\\operatorname{\\mathcal{M}}}_1$ to $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}=\\bar{\\nu}^{-1}(0)\\subset\\overline{\\operatorname{\\mathcal{M}}}_1$. \\\\\n\nFor the orbit curves we now want to give a geometric interpretation of gravitational descendants in terms of branching conditions over the special point on $\\operatorname{\\mathbb{R}} \\times S^1$. Before we state the corresponding theorem and give a rigorous proof using the stretching-of-the-neck procedure from SFT, we first informally describe a naive direct approach based on our definition of gravitational descendants from above, which should illuminate the underlying geometric ideas. \\\\\n\nRecall that if $(h,z)$ is an element in the non-compactified moduli space $\\operatorname{\\mathcal{M}}^{\\nu}_1\\subset\\overline{\\operatorname{\\mathcal{M}}}^{\\bar{\\nu}}_1$ the fibre of the canonical line bundle $\\operatorname{\\mathcal{L}}$ over $(h,z)$ is given by $\\operatorname{\\mathcal{L}}_{(h,z)}=T_z^*S$. Identifying the tangent space to the cylinder at the special point with $\\operatorname{\\mathbb{C}}$ it follows that $s(h,z)=\\frac{\\partial h}{\\partial z}(z)\\in T_z^*S$ is a section in the restriction of $\\operatorname{\\mathcal{L}}$ to $\\operatorname{\\mathcal{M}}^{\\nu}_1$. Since $s$ is a transversal section in the tautological line bundle over $\\operatorname{\\mathcal{M}}_1^{\\nu}$ if and only if it extends to a section over $\\operatorname{\\mathcal{M}}_1$ such that $s\\oplus\\nu$ is transversal to the zero section in $\\operatorname{\\mathcal{L}}\\oplus\\operatorname{Coker}\\bar{\\partial}_{\\underline{J}}$ over $\\operatorname{\\mathcal{M}}_1$, we may assume after possibly perturbing $\\nu$ that $s$ is indeed transversal. On the other hand, since $\\frac{\\partial h}{\\partial z}(z)=0$ is equivalent to saying that $z\\in S$ is a branch point of the holomorphic map $h:S\\to \\mathbb{C}\\mathbb{P}^1$, it follows that $\\operatorname{\\mathcal{M}}_1^1:=s^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1$ indeed agrees with the space of all orbit curves $(h,z)$ with one additional marked point, where $z$ is a branch point of $h$. \\\\\n\nFurther moving on to the case $j=2$ observe that a natural candidate for a generic section $s_2$ in the restriction of the product line bundle $\\operatorname{\\mathcal{L}}^{\\otimes 2}$ to $\\operatorname{\\mathcal{M}}^1_1\\subset\\operatorname{\\mathcal{M}}_1$ is given by $s_2(h,z)=\\frac{\\partial^2 h}{\\partial z^2}(z)\\in (T_z^*S)^{\\otimes 2}$, for which $\\operatorname{\\mathcal{M}}_1^2=s_2^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1^1$ agrees with the space of holomorphic maps where $z\\in S$ is now a branch point of order at least two. For general $j$ we can hence proceed by induction and define the section $s_j$ in $\\operatorname{\\mathcal{L}}^{\\otimes j}$ over $\\operatorname{\\mathcal{M}}^{j-1}_1:=s_{j-1}^{-1}(0)\\subset\\operatorname{\\mathcal{M}}_1$ by $s_j(h,z)=\\frac{\\partial^j h}{\\partial z^j}(z)$, so that $\\operatorname{\\mathcal{M}}^j_1$ agrees with the space of holomorphic maps where $z\\in S$ is a branch point of order at least $j$. \\\\\n\nIf the chosen sections $s_1,...,s_j$ over the non-compactified moduli spaces would extend in the same way to a coherent collection of sections in the tautological line bundles over the compactified moduli spaces $\\overline{\\operatorname{\\mathcal{M}}}_1^{\\bar{\\nu}}$, the above would show that in the case of orbit curves considering the $j$.th descendant moduli space is equivalent after passing to homology to requiring that the underlying additional marked point is a branch point of order $j$. In [OP] it was however shown that already for the case of the circle $\\gamma=V=S^1$ the latter assumption is not entirely true, but that one instead additionally obtains corrections from the boundary $\\overline{\\operatorname{\\mathcal{M}}}_1-\\operatorname{\\mathcal{M}}_1$. \\\\\n\nTo this end, we define a {\\it branching condition} to be a tuple of natural numbers $\\mu=(\\mu_1,...,\\mu_{\\ell(\\mu)})$ of length $\\ell(\\mu)$ and total branching order $|\\mu|=\\mu_1+...+\\mu_{\\ell(\\mu)}$. Then the moduli space $\\overline{\\operatorname{\\mathcal{M}}}^{\\mu}=\\overline{\\operatorname{\\mathcal{M}}}^{\\mu}(\\Gamma^+,\\Gamma^-)$ consists of orbit curves with $\\ell(\\mu)$ connected components, where every connected component carries one additional marked point $z_i$, which is mapped to the special point on $\\operatorname{\\mathbb{R}}\\times\\gamma$ and is a branch point of order $\\mu_i-1$ for $i=1,...,\\ell(\\mu)$. For every branching condition $\\mu=(\\mu_1,...,\\mu_{\\ell(\\mu)})$ we then define new Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,\\mu}=\\operatorname{\\mathbf{h}}^{1,\\bar{\\nu}}_{\\gamma,\\mu}$ by setting \n\\begin{equation*} \\operatorname{\\mathbf{h}}^1_{\\gamma,\\mu} = \\sum_{\\Gamma^+,\\Gamma^-} \\#\\overline{\\operatorname{\\mathcal{M}}}_1^{\\mu}(\\Gamma^+,\\Gamma^-)\\;p^{\\Gamma^+}q^{\\Gamma^-}. \\end{equation*} \n\nWith the following theorem we will prove that the abstract descendants-branching correspondence from [OP] holds for every closed Reeb orbit $\\gamma\\subset V$. \nFor every $j\\in\\operatorname{\\mathbb{N}}$ and every branching condition $\\mu$ we let $\\rho^0_{j,\\mu}$ be the number given by integrating the $j$.th power of the first Chern class of the tautological line bundle over the moduli space of connected rational curves over $\\mathbb{C}\\mathbb{P}^1$ with one marked point mapped to $0$ and $\\ell(\\mu)$ additional marked points $z_i$ mapped to $\\infty$ which are branch points of order $\\mu_i-1$, $i=1,...,\\ell(\\mu)$. \\\\ \n\\\\\n{\\bf Lemma 2.5:} {\\it Each of the descendant Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ can be written as a sum,} \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{\\gamma,j} \\;=\\; \\frac{1}{j!}\\;\\cdot\\;\\operatorname{\\mathbf{h}}^1_{\\gamma,(j+1)} \\;+\\;\\sum_{|\\mu|2$ let $H^{1,p}(h^*\\xi)\\subset C^0(h^*\\xi)$ denote the space of $H^{1,p}$-sections in $h^*\\xi$ which over every boundary component $C_k\\subset\\partial\\dot{S}$ restrict to a section in $C^0((\\bar{\\gamma}^{n_k^-})^*N)$. Furthermore we will consider the subspace $H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\subset H^{1,p}(h^*\\xi)$ consisting of all sections in $h^*\\xi$, which over every boundary circle $C_k$ restrict to sections in the subspace $TW^+(\\bar{\\gamma}^{n_k^-})\\subset C^0((\\bar{\\gamma}^{n_k^-})^*N)$. While the latter Sobolev spaces describe the normal deformations of the branched covering, we introduce similar as in [F2] for sufficiently small $d>0$ a Sobolev space with asymptotic weights $H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})$ in order to keep track of tangential deformations, where, additionally to the definitions in [F2], we impose the natural constraint that the function is real-valued over the boundary. In the same way we define the Banach spaces $L^p(\\Lambda^{(0,1)}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}} h^*\\xi)$ and $L^{p,d}(\\Lambda^{(0,1)}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$. Further we denote by $\\operatorname{\\mathcal{M}}_{0,s^-,s^+}$ the moduli space of Riemann surfaces with $s^-$ boundary circles, $s^+$ punctures and genus zero. \\\\ \n\nFollowing [F2], [BM] for the general case and [W] for the case with boundary, there exists a Banach space bundle $\\operatorname{\\mathcal{E}}$ over a Banach manifold of maps $\\operatorname{\\mathcal{B}}$ in which the Cauchy-Riemann operator $\\bar{\\partial}_J$ extends to a smooth section. In our special case it follows as in [F2] that the fibre is given by \n\\begin{equation*} \n \\operatorname{\\mathcal{E}}_{h,j} = L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi),\n\\end{equation*}\nwhile the tangent space to the Banach manifold of maps $\\operatorname{\\mathcal{B}}= \\operatorname{\\mathcal{B}}_{0,s^-}(\\Gamma)$ at $(h,j) \\in \\operatorname{\\mathcal{M}} = \\operatorname{\\mathcal{M}}_{0,s^-}(\\Gamma)$ is given by \n\\begin{equation*}\n T_{h,j}\\operatorname{\\mathcal{B}} = H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n}.\n\\end{equation*} \n\nIt follows that the linearization $D_{h,j}$ of the Cauchy-Riemann operator $\\bar{\\partial}_{\\underline{J}}$ is a linear map from $T_{h,j}\\operatorname{\\mathcal{B}}$ to $\\operatorname{\\mathcal{E}}_{h,j}$, which is surjective in the case when transversality for $\\bar{\\partial}_{\\underline{J}}$ is satisfied. In this case it follows from the implicit function theorem that $\\ker D_{h,j}=T_{h,j}\\operatorname{\\mathcal{M}}$. In order to prove that the dimension of the desired moduli space $\\operatorname{\\mathcal{M}}_{\\bar{\\Gamma}}=\\operatorname{\\mathcal{M}}(\\Gamma,\\bar{\\Gamma})=\\operatorname{ev}^{-1}(W^+(\\bar{\\Gamma}))\\subset\\operatorname{\\mathcal{M}}(\\Gamma)$ agrees with the virtual dimension expected by the Fredholm index, it remains to prove that the evaluation map $\\operatorname{ev}:\\operatorname{\\mathcal{M}}\\to\\Sigma Q^{s^-}$ is transversal to the product stable manifold $W^+(\\bar{\\Gamma})$. \\\\\n\nIn order to deal with this additional transversality problem, we introduce the Banach submanifold of maps $\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}=\\operatorname{ev}^{-1}(W^+(\\bar{\\Gamma}))\\subset\\operatorname{\\mathcal{B}}$ with tangent space \n\\begin{eqnarray*}\n T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &=&\\{v\\in T_{h,j}\\operatorname{\\mathcal{B}}: v|_{\\partial\\dot{S}}\\in TW^+(\\bar{\\Gamma})\\} \n\\end{eqnarray*} \nand view the Cauchy-Riemann operator as a smooth section in $\\operatorname{\\mathcal{E}}\\to\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$. Then we have the following nice transversality lemma. \\\\\n\\\\\n{\\bf Lemma 2.6:} {\\it Assume that $D_{h,j}: T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} \\to \\operatorname{\\mathcal{E}}_{h,j}$ is surjective. Then the linearization of the evaluation map \n$d_{h,j}\\operatorname{ev}: T_{h,j}\\operatorname{\\mathcal{M}}\\to TW^-(\\bar{\\Gamma})=C^0(\\bar{\\Gamma}^*N)\/TW^+(\\bar{\\Gamma})$ is surjective.} \\\\\n\\\\\n{\\it Proof:} Given $v_0\\in TW^-(\\bar{\\Gamma})$, choose $\\tilde{v}\\in T_{h,j}\\operatorname{\\mathcal{B}}$ such that $d_{h,j}\\operatorname{ev}\\cdot\\tilde{v}=v_0$. On the other hand, since \n$D_{h,j}: T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}} \\to \\operatorname{\\mathcal{E}}_{h,j}$ is onto, we can find $v\\in T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$ with $D_{h,j}v=D_{h,j}\\tilde{v}$, that is, \n$\\tilde{v}-v\\in\\ker D_{h,j}= T_{h,j}\\operatorname{\\mathcal{M}}$. On the other hand, since $d_{h,j}\\operatorname{ev}\\cdot v \\in TW^+(\\bar{\\Gamma})$ for all $v\\in T_{h,j}\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$ by definition, we have $d_{h,j}\\operatorname{ev}\\cdot(\\tilde{v}-v) = d_{h,j}\\operatorname{ev}\\cdot\\tilde{v} = v_0$ and the claim follows. $\\qed$ \\\\\n\nWe have seen that, instead of requiring transversality for the Cauchy-Riemann operator in the Banach space bundle over $\\operatorname{\\mathcal{B}}$ and geometric transversality for the evaluation map, it suffices to require transversality for the Cauchy-Riemann operator in the Banach space bundle over the smaller Banach manifold $\\operatorname{\\mathcal{B}}_{\\bar{\\Gamma}}$. Along the same lines as for proposition 2.1 in [F2] it can be shown that the linearized Cauchy-Riemann operator is of the form \n\\begin{eqnarray*}\n &D_{h,j}:& H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}}(h^*\\xi)\\oplus T_j\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &&\\to L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi),\\\\\n && D_{h,j} \\cdot (v_1,v_2,y) = (\\bar{\\partial} v_1+ D_j y, D_h^{\\xi} v_2),\n\\end{eqnarray*}\nwhere $\\bar{\\partial}: H^{1,p,d}_{\\operatorname{const}}(\\dot{S},\\operatorname{\\mathbb{C}}) \\to L^{p,d}(\\Lambda^{0,1}\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$ is the standard \nCauchy-Riemann operator, $D_h^{\\xi}: H^{1,p}(h^*\\xi) \\to L^p(\\Lambda^{0,1}\\dot{S}\\otimes_{j,\\underline{J}_{\\xi}}h^*\\xi)$ describes the linearization of $\\bar{\\partial}_J$ in the direction of \n$\\xi \\subset TT^*Q$ and $D_j: T_j\\operatorname{\\mathcal{M}}_{0,n} \\to L^{p,d}(T^*\\dot{S}\\otimes_{j,i}\\operatorname{\\mathbb{C}})$ describes the variation of $\\bar{\\partial}_J$ with $j\\in\\operatorname{\\mathcal{M}}_{0,n}$. \\\\\n\nIn [F2] we have shown that for branched covers of orbit cylinders the cokernels of the linearizations of the Cauchy-Riemann operator have the same dimension for every branched cover and hence fit together to give a smooth vector bundle over the nonregular moduli space of branched covers, so that we can prove transversality without waiting for the completion of the polyfold project of Hofer, Wysocki and Zehnder. The following proposition, proved in complete analogy, outlines that this still holds true for branched covers of trivial half-cylinders. \\\\\n\\\\\n{\\bf Proposition 2.7:} {\\it The cokernels of the linearizations of the Cauchy-Riemann operator fit together to give a smooth finite-dimensional vector bundle over the moduli space of branched covers of the half-cylinder.} \\\\\n\\\\\n{\\it Proof:} As in [F2] this result relies on the transversality of the standard Cauchy-Riemann operator and the super-rigidity of the trivial half-cylinder \n\\begin{equation*} \\operatorname{coker}\\bar{\\partial}=\\{0\\}\\;\\; \\textrm{and}\\;\\; \\ker D_h^{\\xi} =\\{0\\}, \\end{equation*}\nwhere the second statement is now just a linearized version of lemma 7.2 in [CL] which states that, as for orbit cylinders in the symplectizations, the branched covers of the trivial half-cylinder are characterized by the fact that they carry no energy in the sense that the action of Reeb orbits above agrees with the lenghts of the closed geodesics below. $\\qed$ \\\\\n\nIt remains to study the extension $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ of the cokernel bundle $\\operatorname{Coker}\\bar{\\partial}_{\\underline{J}}$ to the compactified moduli space. For this recall that the components of the codimension-one-boundary of the nonregular moduli space $\\overline{\\operatorname{\\mathcal{M}}}=\\overline{\\operatorname{\\mathcal{M}}}_{\\bar{\\Gamma}}$ of branched covers of the half-cylinder are either of the form $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2$, where $\\overline{\\operatorname{\\mathcal{M}}}_1=\\overline{\\operatorname{\\mathcal{M}}_1(\\Gamma_1^+,\\Gamma_1^-)\/\\operatorname{\\mathbb{R}}}$, $\\overline{\\operatorname{\\mathcal{M}}}_2=\\overline{\\operatorname{\\mathcal{M}}_2(\\Gamma_2,\\bar{\\Gamma}_2)}$ are nonregular compactified moduli spaces of branched covers of the orbit cylinder or of the trivial half-cylinder, respectively, or of the form $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$, where $\\overline{\\operatorname{\\mathcal{M}}}_0=\\overline{\\operatorname{\\mathcal{M}}_0(\\Gamma,\\bar{\\Gamma}_0)}$ is again a nonregular compactified moduli space of branched covers of the trivial half-cylinder while $S^1$ refers to the concatenation or splitting locus, which agrees with the locus where the single branch point is leaving the branched covering through the boundary. Note that for $\\bar{\\Gamma}=(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{s^-}})$ the ordered set $\\bar{\\Gamma}_0$ is either of the form \n\\begin{eqnarray*} \n\\bar{\\Gamma}_0&=&(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{k-1}},\\bar{\\gamma}^{n_k^1},\\bar{\\gamma}^{n_k^2},\\bar{\\gamma}^{n_{k+1}},...,\\bar{\\gamma}^{n_{s^-}}) \n\\;\\;\\textrm{or}\\\\ \\bar{\\Gamma}_0&=&(\\bar{\\gamma}^{n_1},...,\\bar{\\gamma}^{n_{k-1}},\\bar{\\gamma}^{n_k+n_{k+1}},\\bar{\\gamma}^{n_{k+2}},...,\\bar{\\gamma}^{n_{s^-}}),\n\\end{eqnarray*}\ncorresponding to concatenating $\\bar{\\gamma}^{n_k^1}$ and $\\bar{\\gamma}^{n_k^1}$ to get $\\bar{\\gamma}^{n_k}$ ($n_k^1+n_k^2=n_k$) or the splitting of $\\bar{\\gamma}^{n_k+n_{k+1}}$ to get $\\bar{\\gamma}^{n_k}$ and $\\bar{\\gamma}^{n_{k+1}}$. Restricting to the concatenation case, recall that the chosen special point on the simple closed Reeb orbit determines a special point on the underlying simple geodesic and that we may assume that every holomorphic curve comes equipped with asymptotic markers in the sense of [EGH] not only on the cylindrical ends but also on the boundary circles. In particular, for the concatenation and splitting processes we may assume that all multiply-covered geodesics come equipped with a parametrization by $S^1$. Denoting by $t_1, t_2\\in S^1$ the points on $\\bar{\\gamma}^{n_k^1}$, $\\bar{\\gamma}^{n_k^1}$, where we want to concatenate the two multiply-covered geodesics to get the multiply-covered geodesic $\\bar{\\gamma}^{n_k^1+n_k^2}$, we see that the coordinates must satisfy $n_k^1 t_1=n_k^2 t_2$ in order to represent the same point on the underlying simple geodesic, so that the configuration space agrees with $S^1$ by setting $t_1=n_k^2 t$, $t_2=n_k^1 t$ for $t\\in S^1$. \\\\\n\nWhile it directly follows from [F2] that over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2\\subset\\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is of the form \n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_1\\times\\overline{\\operatorname{\\mathcal{M}}}_2} = \\pi_1^*\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}\\oplus \\pi_2^*\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}, \\end{equation*}\nwhere $\\overline{\\operatorname{Coker}}^1\\bar{\\partial}_{\\underline{J}}$, $\\overline{\\operatorname{Coker}}^2\\bar{\\partial}_{\\underline{J}}$ denote the (extended) cokernel bundles over $\\overline{\\operatorname{\\mathcal{M}}}_1$, $\\overline{\\operatorname{\\mathcal{M}}}_2$, respectively, it remains to study the cokernel bundle over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$. \\\\\n\\\\\n{\\bf Proposition 2.8:} {\\it Over the boundary components $\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1\\subset \\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ is also of product form,}\n\\begin{equation*} \\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}|_{\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1} \\;=\\; \\pi_1^*\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}\\oplus \\pi_2^*\\Delta, \\end{equation*}\n{\\it where $\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}$ denotes the (extended) cokernel bundle over the moduli space $\\overline{\\operatorname{\\mathcal{M}}}_0$ and $\\Delta$ is a vector bundle over $S^1$ which is determined by the tangent spaces to the stable manifolds of the multiply-covered closed geodesics involved into the concatenation or splitting process.}\\\\ \n\\\\\n{\\it Proof:} Still restricting to the concatenation case, let $\\dot{S}_0=\\dot{S}_{01}\\cup\\dot{S}_{02}$ denote the disconnected Riemann surface of genus zero with $s^+$ punctures and $s^-+1$ boundary circles $C_1,...,C_k^1,C_k^2,...,C_{s^-}$, where we assume that $\\partial\\dot{S}_{01}=C_1\\cup...\\cup C_k^1$ and $\\partial\\dot{S}_{02}= C_k^2,...,C_{s^-}$. As before we know that the tangent spaces to the corresponding Banach manifolds of maps $\\operatorname{\\mathcal{B}}^0$, $\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}$ at a branched covering $(h_0,j_0): (\\dot{S}_0,\\partial\\dot{S}_0)\\to(\\operatorname{\\mathbb{R}}^+_0\\times S^1,\\{0\\}\\times S^1)$ are given by \n\\begin{eqnarray*}\n T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0 &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S}_0,\\operatorname{\\mathbb{C}})\\oplus H^{1,p}(h_0^*\\xi)\\oplus T_{j_0}\\operatorname{\\mathcal{M}}_{0,n},\\\\\n T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0} &=& H^{1,p,d}_{\\operatorname{const}}(\\dot{S}_0,\\operatorname{\\mathbb{C}})\\oplus H^{1,p}_{\\bar{\\Gamma}_0}(h_0^*\\xi)\\oplus T_{j_0}\\operatorname{\\mathcal{M}}_{0,n} \\\\\n &=&\\{v\\in T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma}_0)\\} \n\\end{eqnarray*} \nwhile the fibre of the corresponding Banach space bundle is given by\n\\begin{equation*} \n \\operatorname{\\mathcal{E}}^0_{h_0,j_0} = L^{p,d}(\\Lambda^{0,1}\\dot{S}_0\\otimes_{j_0,i}\\operatorname{\\mathbb{C}}) \\oplus L^p(\\Lambda^{0,1}\\dot{S}_0\\otimes_{j_0,\\underline{J}_{\\xi}}h_0^*\\xi).\n\\end{equation*}\nFor $(h_0,j_0,t)\\in\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1$ we further introduce the Banach manifold of maps $\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\subset\\operatorname{\\mathcal{B}}^*\\subset\\operatorname{\\mathcal{B}}^0$ which should consist of all branched covers of the trivial half-cylinder in $\\operatorname{\\mathcal{B}}^0$ for which the boundary circles $C_k^1, C_k^2\\cong S^1$ are concatenated at $(t_1,t_2)=(n_k^2 t, n_k^1 t)\\in C_k^1\\times C_k^2$, to give the singular Riemann surface $\\dot{S}_*$ with $s^-$ boundary circles $C_1,...,C_k^1\\cup_t C_k^2,...,C_{s^-}$ and we have\n\\begin{eqnarray*} \n T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*&=&\\{v\\in T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v_k^1(n_k^2 t)=v_k^2(n_k^1 t)\\;\\textrm{for}\\; v_k^{1,2}:=v|_{C_k^{1,2}}\\} \\\\\n T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}} &=&\\{v\\in T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*: v|_{\\partial\\dot{S}_*}\\in TW^+(\\bar{\\Gamma}_0)\\}. \n\\end{eqnarray*}\n\nThe proof of the general gluing theorem in [MDSa] suggests that over $(h_0,j_0,t)\\in\\overline{\\operatorname{\\mathcal{M}}}_0\\times S^1\\subset\\overline{\\operatorname{\\mathcal{M}}}$ the extended cokernel bundle $\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}}$ has fibre \n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}})_{h_0,j_0,t} = \\operatorname{coker} D_{h_0,j_0,t},\\;\\; D_{h_0,j_0,t}: T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\to\\operatorname{\\mathcal{E}}^0_{h_0,j_0}.\n\\end{equation*}\nBefore we describe the relation to the cokernel bundle $\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}}$ over the first factor $\\overline{\\operatorname{\\mathcal{M}}}_0$ with fibre\n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}})_{h_0,j_0} = \\operatorname{coker} D_{h_0,j_0},\\;\\; D_{h_0,j_0}: T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\to\\operatorname{\\mathcal{E}}^0_{h_0,j_0}.\n\\end{equation*}\nobserve that we still have \n\\begin{eqnarray*} \n\\operatorname{coker} D_{h_0,j_0} = \\operatorname{coker} D^{\\xi}_{h_0},&& D^{\\xi}_{h_0}: T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\to\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0},\\\\\n\\operatorname{coker} D_{h_0,j_0,t} = \\operatorname{coker} D^{\\xi}_{h_0,t},&& D^{\\xi}_{h_0,t}: T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\to\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0},\n\\end{eqnarray*}\nand $\\ker D^{\\xi}_{h_0}=\\ker D^{\\xi}_{h_0,t}=\\{0\\}$, where $T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\subset T_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}$, $T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}\\subset T_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*_{\\bar{\\Gamma}}$ and $\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0,j_0}\\subset \\operatorname{\\mathcal{E}}^0_{h_0,j_0}$ are the subspaces corresponding to normal deformations. \\\\\n\nNow observing that \n\\begin{equation*} \n TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2}) \\subset \\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}\n\\end{equation*}\nwe get from \n\\begin{eqnarray*} \n T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0} =\\{v\\in T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma}_0)\\},\\\\\n T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}} =\\{v\\in T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^*: v|_{\\partial\\dot{S}_0}\\in TW^+(\\bar{\\Gamma})\\}\\\\\n\\end{eqnarray*}\nthat $T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}\\subset T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}$ with quotient space\n\\begin{equation*} \n \\frac{T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}}{T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}} \\;=\\; \n \\frac{TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2})}{\\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}}.\n\\end{equation*}\nOn the other hand, since $\\ker D^{\\xi}_{h_0}=\\ker D^{\\xi}_{h_0,t}=\\{0\\}$ we also find for the quotient space that\n\\begin{equation*} \n \\frac{T^{\\xi}_{h_0,j_0}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}_0}}{T^{\\xi}_{h_0,j_0,t}\\operatorname{\\mathcal{B}}^0_{\\bar{\\Gamma}}} \\;=\\; \n \\frac{\\operatorname{im} D^{\\xi}_{h_0}}{\\operatorname{im} D^{\\xi}_{h_0,t}} \\;=\\; \\frac{\\operatorname{coker} D^{\\xi}_{h_0,t}}{\\operatorname{coker} D^{\\xi}_{h_0}},\n\\end{equation*}\nwhere the last equality follows from the fact that $D^{\\xi}_{h_0}$ and $D^{\\xi}_{h_0,t}$ both map to the same Banach space $\\operatorname{\\mathcal{E}}^{0,\\xi}_{h_0}$. \\\\\n\nDefining an obstruction bundle $\\Delta$ over $S^1$ by setting \n\\begin{equation*}\n \\Delta_t = \\frac{TW^+(\\bar{\\gamma}^{n_k^1})\\oplus TW^+(\\bar{\\gamma}^{n_k^2})}{\\{v\\in TW^+(\\bar{\\gamma}^{n_k}): v_k^1(n_k^2 t) = v_k^2(n_k^1 t)\\}}\n\\end{equation*}\nand putting everything together we hence found that \n\\begin{equation*} \n (\\overline{\\operatorname{Coker}}\\bar{\\partial}_{\\underline{J}})_{h_0,j_0,t} \\cong (\\overline{\\operatorname{Coker}}^0\\bar{\\partial}_{\\underline{J}})_{h_0,j_0} \\oplus \\Delta_t,\n\\end{equation*}\nas desired. $\\qed$ \\\\\n \nWith this we can prove the desired statement about $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}$. \\\\\n\\\\\n{\\bf Corollary 2.9:} {\\it We have $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}=0$.} \\\\\n\\\\\n{\\it Proof:} It follows that the obstruction bundle over the one-dimensional configuration space has rank\n\\begin{equation*} \\operatorname{rank} \\Delta = \\operatorname{Morse}(\\bar{\\gamma}^{n_k})-\\operatorname{Morse}(\\bar{\\gamma}^{n_k^1})-\\operatorname{Morse}(\\bar{\\gamma}^{n_k^2})+\\dim Q-1 \\geq 0,\n\\end{equation*}\nwhere the latter inequality can be verified as in [F2] using the multiple cover index formulas in [Lo]. When by index reasons the configuration is expected to be discrete we get a rank-one obstruction bundle over the boundary of the branched cover, which by orientability reasons must indeed be trivial. $\\qed$ \\\\\n\nOn the other hand, we want to emphasize that the proof of $\\operatorname{\\mathbf{g}}^0_{\\bar{\\gamma}}=0$ is much simpler than the proof of $\\operatorname{\\mathbf{h}}^0_{\\gamma}=0$ in [F2], which has to involve obstruction bundles of arbitrary large rank and uses induction. Besides that our proof in [F2] also holds for Reeb orbits in general contact manifolds, this does not come as surprise. Going back to the symplectic field theory of unit cotangent bundles $S^*Q$, it is already mentioned in [CL] that the SFT differential $\\operatorname{\\mathbf{D}}^0_{\\operatorname{SFT}}=\\overrightarrow{\\operatorname{\\mathbf{H}}^0}:\\AA^0[[\\hbar]]\\to\\AA^0[[\\hbar]]$ involving all moduli spaces of holomorphic curves in $\\operatorname{\\mathbb{R}}\\times S^*Q$ is much larger than the string differential $\\operatorname{\\mathbf{D}}^0_{\\operatorname{string}}=\\partial+\\Delta+\\hbar\\nabla:\\operatorname{\\mathfrak{C}}^0[[\\hbar]]\\to\\operatorname{\\mathfrak{C}}^0[[\\hbar]]$, which just involves the singular boundary operator and the string bracket and cobracket operations. \\\\\n\n\\subsection{Additional marked points and gravitational descendants}\nWe now want to understand the system of commuting operators defined for Reeb orbits by studying moduli spaces of branched covers over the cylinder over $\\gamma$ in terms of operations defined for the underlying closed geodesic $\\bar{\\gamma}$. To this end we have to extend the picture of [CL] used for computing the symplectic field theory of Reeb orbits to include additional marked points on the moduli spaces, integration of differential forms and gravitational descendants. \\\\\n\nReintroducing the sequence of formal variables $t_j$, $j\\in\\operatorname{\\mathbb{N}}$, we now consider the graded Weyl algebras $\\operatorname{\\mathfrak{W}}_{\\gamma}$, $\\operatorname{\\mathfrak{W}}_{\\bar{\\gamma}}$ of power series in $\\hbar$, the $p$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$ and $t$-variables with coefficients which are polynomials in the $q$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$. In the same way we can introduce the graded commutative algebras $\\AA_{\\gamma}$, $\\operatorname{\\mathfrak{C}}_{\\bar{\\gamma}}$ of power series in $\\hbar$, the $t$-variables with coefficients which are polynomials in the $q$-variables corresponding to multiples of $\\gamma$, $\\bar{\\gamma}$. For the expansion $\\operatorname{\\mathbf{H}}_{\\gamma}=\\operatorname{\\mathbf{H}}^0_{\\gamma}+\\sum_j t_j \\operatorname{\\mathbf{H}}^1_{\\gamma,j}+o(t^2)$ of the Hamiltonian from before, we are hence looking for an extended potential $\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}$ as well as extended string Hamiltonian $\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}$, \n\\begin{eqnarray*}\n \\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}&=&\\operatorname{\\mathbf{L}}^0_{\\gamma,\\bar{\\gamma}}+\\sum_j t_j \\operatorname{\\mathbf{L}}^1_{\\gamma,\\bar{\\gamma},j}+o(t^2), \\\\\n \\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}&=&\\operatorname{\\mathbf{G}}^0_{\\bar{\\gamma}}+\\sum_j t_j \\operatorname{\\mathbf{G}}^1_{\\bar{\\gamma},j}+o(t^2),\n\\end{eqnarray*}\nsuch that $\\overrightarrow{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}}: (\\AA_{\\gamma}[[\\hbar]],\\overrightarrow{\\operatorname{\\mathbf{H}}_{\\gamma}}) \\to (\\operatorname{\\mathfrak{C}}_{\\bar{\\gamma}}[[\\hbar]],\\overrightarrow{\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}})$ is an isomorphism of $\\operatorname{BV}_{\\infty}$-algebras. \nFor this we have to prove the extended master equation\n\\begin{equation*} \n e^{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}}\\overleftarrow{\\operatorname{\\mathbf{H}}_{\\gamma}} - \\overrightarrow{\\operatorname{\\mathbf{G}}_{\\bar{\\gamma}}}e^{\\operatorname{\\mathbf{L}}_{\\gamma,\\bar{\\gamma}}} = 0, \n\\end{equation*}\nwhile the isomorphism property again follows using the natural filtration given by the $t$-variables. \\\\\n\nSince we are only interested in the system of commuting operators $\\operatorname{\\mathbf{H}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$, which is defined by counting branched covers of orbit cylinders with at most one additional marked point, we again will only discuss the required compactness statements in the case of one additional marked point. Furthermore we will still just restrict to the rational case. In other words we will prove the following proposition, which is just a reformulation of our theorem from above. \\\\\n\\\\\n{\\bf Proposition 2.10:} {\\it The system of Poisson-commuting functions $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\gamma}$ is isomorphic to a system of Poisson-commuting functions $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$, $j\\in\\operatorname{\\mathbb{N}}$ on $\\operatorname{\\mathfrak{P}}^0_{\\bar{\\gamma}}=\\operatorname{\\mathfrak{P}}^0_{\\gamma}$, where for every $j\\in\\operatorname{\\mathbb{N}}$ the descendant Hamiltonian $\\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j}$ given by} \n\\begin{equation*} \n \\operatorname{\\mathbf{g}}^1_{\\bar{\\gamma},j} \\;=\\; \\sum \\epsilon(\\vec{n})\\frac{q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}}{(j+2)!} \n\\end{equation*}\n{\\it where the sum runs over all ordered monomials $q_{n_1}\\cdot ... \\cdot q_{n_{j+2}}$ with $n_1+...+n_{j+2} = 0$ \\textbf{and which are of degree $2(m+j-3)$}. Further $\\epsilon(\\vec{n})\\in\\{-1,0,+1\\}$ is fixed by a choice of coherent orientations in symplectic field theory and is zero if and only if one of the orbits $\\gamma^{n_1},...,\\gamma^{n_{j+2}}$ is bad.} \\\\\n\n{\\it Proof:} While the proof seems to require the definition of gravitational descendants for moduli spaces of holomorphic curves not only with punctures but also with boundary, instead of defining them recall that we have shown in the previous subsection 2.2 that the gravitational descendants can be replaced by imposing branching conditions over the special marked point on the orbit cylinder. More precisely, recall the lemma in subsection 2.2 states that we can indeed write each of the Hamiltonians $\\operatorname{\\mathbf{h}}^1_{\\gamma,j}$ as a weighted sum, \n\\begin{equation*} \n \\operatorname{\\mathbf{h}}^1_{\\gamma,j} \\;=\\; \\frac{1}{j!}\\;\\cdot\\;\\operatorname{\\mathbf{h}}^1_{\\gamma,(j)} \\;+\\;\\sum_{|\\mu| 0, \\quad B(r) \\geq B_{min} > 0, \\quad J(r) \\geq R_{min} > 0,\n\\]\nwith $R_{min}$ standing for the radius of the wormhole throat.\nNote that $r$ is a globally defined coordinate, so that the\nhorizon absence and ``flaring-out''\nconditions are satisfied automatically. \n\nIn what follows we aim to construct an asymptotically flat wormhole \non both ends, which is the case provided that\n\\[\nA(r) \\to 1 \\; , \\;\\;\\;\\; B(r) \\to 1 \\; ,\n\\;\\;\\;\\; J(r) = |r| + O(1) \\;\\;\\;\\; \\mbox{as} \\;\\;\\; r \\to \\pm \\infty \\; .\n\\]\nWe choose a specific form of the metric functions \n$A(r)$, $B(r)$ and $J(r)$ which\ndescribe a wormhole in Sec.~\\ref{sec:wormhole_example}.\nEven though $B(r)$ in eq.~\\eqref{eq:backgr_metric}\ncan be set equal to 1 by coordinate transformation,\nwe keep it arbitrary for generality.\nIn our setting the scalar field $\\pi$ is static and\ndepends on the radial coordinate only, $\\pi = \\pi(r)$ and\nhence $X = -B(r) \\pi'^2 \/2$, where prime denotes the\nderivative with respect to $r$.\n\nIn what follows we make use of background equations of motion following\nfrom action~\\eqref{eq:lagrangian}, especially \nwhen we reconstruct\nthe specific form of Lagrangian functions of beyond Horndeski theory which admits a wormhole solution in Sec.~\\ref{sec:wormhole_example}. We keep the notations used in Ref.~\\cite{wormhole1} for the equations of motion:\n\\[\n\\label{eq: background_eqs}\n {\\cal E}_A = 0, \\quad {\\cal E}_B = 0, \\quad {\\cal E}_J = 0,\n \\quad {\\cal E}_{\\pi} = 0 \\; ,\n \\]\n where $ {\\cal E}_A$ is obtained by varying the action with respect to\n $A$, etc. We give the explicit forms of\n${\\cal E}_A, {\\cal E}_B, {\\cal E}_J$ and ${\\cal E}_{\\pi}$\nin Appendix A. \n\nIn the following section we develop the linearized theory aiming to formulate a complete set of stability conditions for high energy modes\nabout \na static, spherically symmetric background~\\eqref{eq:backgr_metric}. \n\n\n\n\\section{Linearized theory}\n\\label{sec:linearized_th}\nAs it was stated above one of our priorities is to ensure stability of a wormhole solution against small perturbations. In what follows we\nconsider the\nlinearized metric\n\\begin{equation}\n\\label{eq:metric}\ng_{\\mu\\nu} = \\bar{g}_{\\mu\\nu} + h_{\\mu\\nu},\n\\end{equation}\nwhere the background $\\bar{g}_{\\mu\\nu}$ is given \nin eq.~\\eqref{eq:backgr_metric} and $h_{\\mu\\nu}$ stand for small metric perturbations, while $\\delta\\pi$ denotes perturbation about a background scalar field $\\bar{\\pi}$. \n\nTo study the behaviour of perturbations about a static and spherically-symmetric background we follow a standard approach and make use of the Regge-Wheeler formalism~\\cite{ReggeWheeler}:\nperturbations are classified into odd-parity and even-parity sectors \nbased on their behaviour under two-dimensional reflection.\nWith further expansion of perturbations into series of the spherical harmonics \n$Y_{\\ell m}(\\theta,\\phi)$ the modes with\ndifferent parity, $\\ell$ and $m$ do not mix at the linear level, so \nit is legitimate to consider them separately.\nIn the next two subsections\nwe consider odd-parity and even-parity sectors one by one. \n\n\\subsection{Odd-parity sector}\n\\label{sec:odd_sector}\nEven though ghost-like instabilities typically arise in the even-parity \nsector, we give a brief review of the stability constraints in the odd-parity sector for completeness. The contents of this subsection\nare a short summary of Sec.3.2 in Ref.~\\cite{wormhole1}. \n\nThe scalar field $\\pi$ does not obtain odd-parity perturbations, so the only source of perturbation modes in this sector is metric $g_{\\mu\\nu}$.\nThe most general form of metric \nperturbations in the odd-parity sector read~\\cite{ReggeWheeler}:\n\\[\n\\label{odd_parity}\n\\mbox{Parity~odd}\\quad \\begin{cases}\n\\begin{aligned}\n & h_{tt}=0,~~~h_{tr}=0,~~~h_{rr}=0,\\\\\n & h_{ta}=\\sum_{\\ell, m}h_{0,\\ell m}(t,r)E_{ab}\\partial^{b}Y_{\\ell m}(\\theta,\\varphi),\\\\\n & h_{ra}=\\sum_{\\ell, m}h_{1,\\ell m}(t,r)E_{ab}\\partial^{b}Y_{\\ell m}(\\theta,\\varphi),\\\\\n & h_{ab}=\\frac{1}{2}\\sum_{\\ell, m}h_{2,\\ell m}(t,r)\\left[E_{a}^{~c}\\nabla_{c}\\nabla_{b}Y_{\\ell m}(\\theta,\\varphi)+E_{b}^{~c}\\nabla_{c}\\nabla_{a}Y_{\\ell m}(\\theta,\\varphi)\\right],\n\\end{aligned}\n\\end{cases}\n\\]\nwhere $a,b = \\theta,\\varphi$, $E_{ab} = \\sqrt{\\det \\gamma}\\: \\epsilon_{ab}$, with $\\gamma_{ab} = \\mbox{diag}(1, \\;\\sin^2\\theta)$;\n$\\epsilon_{ab}$ is\ntotally antisymmetric symbol ($\\epsilon_{\\theta\\varphi} = 1$) and $\\nabla_a$ is covariant derivative on a 2-sphere. Note that perturbations with\n$\\ell=0$ do not exist and modes with $\\ell=1$ are pure gauges~\\cite{Armendariz,Kobayashi:odd}. So below in this section we consider perturbations with $\\ell \\geq 2$. \n\n\nWe make use of the gauge freedom and set $h_{2,\\ell}=0$ (Regge-Wheeler gauge), so that the quadratic action in the odd-parity sector is derived in terms of $h_{0,\\ell m}$ and $h_{1,\\ell m}$. It was explicitly shown in Refs.~\\cite{Kobayashi:odd,wormhole1} that $h_{0,\\ell m}$ is a non-dynamical DOF and one can integrate it out from the quadratic action. The resulting action for the only dynamical DOF reads \n(see Ref.~\\cite{wormhole1}\nfor a detailed derivation):\n\\[\n\\label{eq:action_odd_final}\n\\begin{aligned}\nS^{(2)}_{odd} = \\int \\mbox{d}t\\:\\mbox{d}r\\:\\sqrt{\\frac{A}{B}} J^2\n\\frac{\\ell(\\ell+1)}{2(\\ell-1)(\\ell+2)}\\cdot \\frac{B}{A}\\left[ \\frac{\\mathcal{H}^2}{A \\mathcal{G}} \\dot{Q}^2 - \\frac{B \\mathcal{H}^2}{\\mathcal{F}} (Q')^2 -\\frac{l(l+1)}{J^2}\\cdot \\mathcal{H} Q^2 - V(r) Q^2 \\right] \\; ,\n\\end{aligned}\n\\]\nwhere an overdot stands for a time derivative, $A$, $B$ and $J$ are metric functions~\\eqref{eq:backgr_metric}, $Q$ is an auxiliary field and\n\\footnote{We have introduced different notations for $\\mathcal{H}$ and $\\mathcal{G}$ in action~\\eqref{eq:action_odd_final} despite their equality for the quadratic subclass of beyond Horndeski theory~\\eqref{eq:lagrangian} as these coefficients become different as soon as one adds the cubic subclass to the set-up~\\cite{Kobayashi:odd,wormhole1}. } \n\\begin{equation}\n\\label{eq:cal_FGH}\n{\\cal F}=2 G_4 ,\n\\qquad\n{\\cal H}={\\cal G}= 2 \\left[G_4 - 2X G_{4X} + 4 X^2 F_4 \\right].\n\\end{equation} \nOnce $Q$ is known both $h_{0,\\ell m}$ and $h_{1,\\ell m}$ can be restored.\nNote that the third term in eq.~\\eqref{eq:action_odd_final} is the \nangular part of the Laplace operator, which governs stability in the angular direction. The \"potential\" $V(r)$ in eq.~\\eqref{eq:action_odd_final} reads\n\\begin{equation}\n\\label{eq:Vr}\n\\begin{aligned}\nV(r) &= \\frac{B {\\cal H}^2 }{{2\\cal F}}\n\\left[\\frac{{\\cal F}'}{{\\cal F}} \\left(2\\frac{{\\cal H}'}{{\\cal H}} -\\frac{A'}{A}+\\frac{B'}{B}+ 4\\frac{J'}{J}\\right) - \\frac{{\\cal H}'}{{\\cal H}} \\left( -\\frac{A'}{A}+3 \\frac{B'}{B}+4 \\frac{J'}{J}\\right)\\right.\\\\\n&\\left. - \\left(\\frac{A'^2}{A^2} -\\frac{A'}{A} \\frac{B'}{B} - \\frac{A''}{A} +\\frac{B''}{B} +4 \\frac{B'}{B} \\frac{J'}{J}-4 \\frac{J'^2}{J^2}+ 4 \\frac{J''}{J} \\right) - \\frac{4}{J^2 B}\\cdot \\frac{\\cal F}{\\cal H} \\right],\n\\end{aligned}\n\\end{equation}\nand governs the possible \"slow\" tachyonic instabilities, which we do not discuss in this paper. However, corresponding constraints following from the potential $V(r)$ may not be particularly restrictive~\\cite{Trincherini}.\n\n\nThe quadratic action~\\eqref{eq:action_odd_final} gives the following set of stability conditions which ensure the absence of both ghost and gradient instabilities in the odd-parity sector:\n\\begin{subequations}\n\\label{eq:stability_odd}\n\\begin{align}\n\\label{eq:stability_G}\n\\mbox{No ghosts:}\\quad &\\mathcal{G} > 0, \\\\\n\\label{eq:stability_F}\n\\mbox{No radial gradient instabilities:}\\quad &\\mathcal{F} > 0, \\\\\n\\label{eq:stability_H}\n\\mbox{No angular gradient instabilities:}\\quad &\\mathcal{H} > 0.\n\\end{align}\n\\end{subequations}\nTo have both radial and angular modes propagating at (sub)luminal speed one has to ensure that corresponding sound speeds squared are not greater than the speed of light\n\\begin{equation}\n\\label{eq:speed_odd}\nc_r^2 = \\frac{\\mathcal{G}}{\\mathcal{F}} \\leq 1, \\quad c_{\\theta}^2 = \\frac{\\mathcal{G}}{\\mathcal{H}} \\leq 1,\n\\end{equation}\nwhich is the case provided that\n\\begin{equation}\n\\label{eq:sublum_odd}\n\\mathcal{F} \\ge \\mathcal{G} > 0, \\qquad \n\\mathcal{H} \\ge \\mathcal{G} > 0 \\; .\n\\end{equation}\n\n\n\\subsection{Even-parity sector}\n\\label{sec:even_parity}\n\nWe now move on to the even-parity sector, which is the main source of stability constraints for a spherically-symmetric solution.\nWe adopt the following general form of parametrization for\nmetric perturbations:\n\\[\n\\label{eq:even_parity}\n\\mbox{Parity~even}\\quad \\begin{cases}\n\\begin{aligned}\nh_{tt}=&A(r)\\sum_{\\ell, m}H_{0,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{tr}=&\\sum_{\\ell, m}H_{1,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\\\\\nh_{rr}=&\\frac{1}{B(r)}\\sum_{\\ell, m}H_{2,\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\\\\\nh_{ta}=&\\sum_{\\ell, m}\\beta_{\\ell m}(t,r)\\partial_{a}Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{ra}=&\\sum_{\\ell, m}\\alpha_{\\ell m}(t,r)\\partial_{a}Y_{\\ell m}(\\theta,\\varphi), \\\\\nh_{ab}=&\\sum_{\\ell, m} K_{\\ell m}(t,r) g_{ab} Y_{\\ell m}(\\theta,\\varphi)+\\sum_{\\ell, m} G_{\\ell m}(t,r) \\nabla_a \\nabla_b Y_{\\ell m}(\\theta,\\varphi)\\,.\n\\end{aligned}\n\\end{cases}\n\\]\nThe scalar field $\\pi$ also acquires non-vanishing perturbations \nin the parity even sector:\n\\[\n\\label{chi}\n\\pi (t,r,\\theta,\\varphi) = \\pi(r) + \\sum_{\\ell, m}\\chi_{\\ell m}(t,r)Y_{\\ell m}(\\theta,\\varphi),\n\\]\nwhere $\\pi(r)$ is the spherically symmetric background field. As we have\nmentioned above modes with different $\\ell$ and $m$ does not get mixed at the linear level, so to simplify the expressions we usually drop both subscripts in what follows.\n\nThe next natural step is to either work with gauge-invariant variables~\\cite{Gerlach:1979rw} or to fix the gauge. Below we choose the latter option. Under the infinitesimal coordinate change $x^{\\mu} \\to x^{\\mu} + \\xi^{\\mu}$ with $\\xi^{\\mu}$ parametrized as\n\\begin{equation}\n\\label{eq:xi}\n\\xi^{\\mu} = \\Big(T_{\\ell m}(t,r), R_{\\ell m}(t,r), \\Theta_{\\ell m}(t,r) \\partial_{\\theta}, \\dfrac{\\Theta_{\\ell m}(t,r) \\partial_{\\varphi}}{\\sin^2\\theta}\\Big)\\, Y_{\\ell m}(\\theta,\\varphi )\n\\end{equation}\nthe metric perturbations in eq.~\\eqref{eq:even_parity} transform as follows\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:gauge_laws}\nH_0 &\\rightarrow H_0 + \\dfrac{2}{A} \\dot{T} - \\dfrac{A'}{A} B R, \\\\\nH_1 &\\rightarrow H_1 + \\dot{R} + T' - \\dfrac{A'}{A} T, \\\\\nH_2 &\\rightarrow H_2 + 2 B R' + B' R, \\\\\n\\beta &\\rightarrow \\beta + T + \\dot{\\Theta}, \\\\\n\\alpha &\\rightarrow \\alpha + R + \\Theta' - 2 \\dfrac{J'}{J} \\Theta,\\\\\nK &\\rightarrow K + 2 B \\dfrac{J'}{J} R, \\\\\nG &\\rightarrow G + 2 \\Theta, \\\\\n\\chi &\\rightarrow \\chi + B \\pi' R,\n\\end{aligned}\n\\end{equation}\nwhere we have dropped the arguments of functions for clarity. \nFixing the gauge amounts to choosing specific functions $T$, $R$ and $\\Theta$ in eq.~\\eqref{eq:gauge_laws} so that \nsome of the perturbations vanish. \nFor instance, \nthe original Regge--Wheeler gauge, used e.g. in Refs.~\\cite{ReggeWheeler,Armendariz}, amounts to setting $\\alpha=\\beta=G=0$. \nAnother possible gauge choice was adopted \nin Refs.~\\cite{Kobayashi:even,wormhole1} where $\\beta=K=G=0$ (let us call it a \"spherical gauge\" for brevity). Alternatively, Ref.~\\cite{Trincherini} used the Regge--Wheeler--unitary gauge with $\\beta=G=\\chi=0$. \n\n\nIn our previous work in Ref.~\\cite{wormhole1} we have already \nderived the \naction for perturbations in beyond Horndeski theory adopting \nthe spherical gauge $\\beta=K=G=0$ and found that for modes \nwith large $\\ell$ the coefficients in the action inevitably become singular at some point. This singularity is unavoidable as soon as one requires the background to be free of ghost instabilities at all points. We provide a detailed explanation in Sec.~\\ref{sec:old_gauge}.\n\nThere is another subtlety with the gauge choice where $K=0$. \nNamely, according to the \ntransformation laws~\\eqref{eq:gauge_laws} \nit is impossible to make $K$ vanishing at all points if\nthe metric function $J(r)$ is not a monotonous function, i.e. $J'(r)=0$ at some point. And this is exactly what happens in a wormhole throat, \nsince function $J(r)$ describes the profile of a wormhole \nand $J'(r)=0$ at $r=R_{min}$~\\eqref{eq:metric_bounded}. Hence, \nthe spherical gauge in Ref.~\\cite{wormhole1} is \nnot entirely convenient for checking linear stability \nof the wormhole solutions near the throat.\n\nIn this paper we aim to formulate a complete set of stability conditions \nfor modes with high momentum, which does not involve any singularities and, hence, is suitable for stability analysis throughout the whole space, where the wormhole is present. \n\n\n\nIn Sec.~\\ref{sec:old_gauge} we give a schematic derivation of the action for even-parity perturbations \nfor beyond Horndeski Lagrangian~\\eqref{eq:lagrangian} \nin the spherical gauge (see Ref.~\\cite{wormhole1} for a detailed procedure)\nand discuss its drawbacks in the context of stability analysis for spherically-symmetric solutions like wormholes. In Sec.~\\ref{sec:new_gauge} we\ncalculate the quadratic action in Regge--Wheeler--unitary gauge and find out if we can avoid singularities in the coefficients of quadratic action analogous to those in the old gauge. In result, we derive the whole set of stability conditions for modes with high $\\ell$. \n\n\n\\subsubsection{Existing results in the spherical gauge ($\\beta=K=G=0$)}\n\\label{sec:old_gauge}\nThe quadratic action for even-parity perturbations~\\eqref{eq:even_parity}\nwith $\\ell \\geq 2$ \n\\footnote{Both $\\ell = 0$ and $\\ell=1$ are somewhat special cases, which, however, do not give anything particularly new for stability analysis as compared to $\\ell \\geq 2$~\\cite{Kobayashi:even,wormhole1}.}\nand gauge conditions $\\beta=K=G=0$\nreads:\n\\begin{equation}\n\\label{eq:action_even}\n\\begin{aligned}\n&S_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \\left(H_0 \\left[ a_1 \\chi''+ a_2\n\\chi'+a_3 H_2'+j^2 a_4 \\alpha'+\\left( a_5+j^2 a_6 \\right) \\chi\\right.\\right. \\\\\n&\\left.+\\left( a_7+j^2 a_8 \\right)H_2+j^2a_9 \\alpha \\right]\n+j^2 b_1 H_1^2+H_1 \\left[b_2 {\\dot {\\chi}}'+b_3 {\\dot {\\chi}}+b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha}\\right] \\\\\n&+ c_1{\\dot H_2} {\\dot {\\chi}} + H_2 \\left[c_2 \\chi'+\\left( c_3+j^2 c_4\\right)\\chi + j^2 c_5 \\alpha \\right] + c_6 H_2^2+j^2d_1 {\\dot \\alpha}^2\n\\\\\n&\\left.+j^2 d_2 \\alpha\\chi'+j^2 d_3 \\alpha\\chi+j^2 d_4 \\alpha^2\n+ e_1{\\dot {\\chi}}^2+e_2 \\chi'^2+\\left( e_3+j^2 e_4 \\right) \\chi^2\\right),\n\\end{aligned}\n\\end{equation}\nwhere the subscripts $\\ell$, $m$ are again omitted, $j^2=\\ell(\\ell+1)$, and\nwe have integrated over $\\theta$ and $\\phi$. The\nexplicit expressions for coefficients $a_i$, $b_i$, $c_i$, $d_i$\nand $e_i$ with $\\sqrt{-g}$ included are given in Appendix B.\n\nIt follows from action~\\eqref{eq:action_even} that $H_0$ is a Lagrange multiplier and $H_1$ is also a non-dynamical DOF, which give the following constraints, respectively:\n\\begin{equation}\n\\label{eq:H0old}\n\\begin{aligned}\na_1 \\chi''+a_2 \\chi'+a_3 H_2'+\nj^2 a_4 \\alpha'+\\left( a_5+j^2 a_6 \\right) \\chi\n+\\left( a_7+j^2 a_8 \\right)H_2+j^2a_9 \\alpha = 0,\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\label{eq:H1old}\n H_1 = - \\frac{1}{2 j^2 b_1} \\left( b_2 {\\dot {\\chi}}'+b_3 {\\dot {\\chi}}+b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha}\\right).\n\\end{equation}\nIntroducing a new variable $\\psi$ as\n\\begin{equation}\n\\label{eq:H2old}\nH_2 = \\psi - \\frac{1}{a_3}\\left( a_1 \\chi' + j^2 a_4 \\alpha\\right),\n\\end{equation}\nand so that upon substitution of $H_2$ into eq.~\\eqref{eq:H0old}\nboth $\\chi''$ and $\\alpha'$ get cancelled out automatically, the resulting equation becomes algebraic for $\\alpha$ and one can express it in terms of $\\psi$ and $\\chi$ \n\\begin{equation}\n\\label{eq:alphaold}\n\\alpha = \\frac{a_3^2 \\psi' + a_3(a_7+j^2a_8) \\psi + [a_3 (a_2 - a_1') -j^2 a_1 a_8] \\chi' + a_3 (a_5 + j^2 a_6) \\chi}{j^2 \\left[ a_3 a_4' - a_3' a_4 - a_3 a_9 + a_4 (a_7 + j^2 a_8\\right]} \\;.\n\\end{equation}\nThen it becomes possible to express both $H_2$ in eq.~\\eqref{eq:H2old} and $H_1$ in eq.~\\eqref{eq:H1old} in terms \nof $\\psi$ and $\\chi$ only. Hence, the quadratic action~\\eqref{eq:action_even} can be cast in terms of two dynamical DOFs:\n\\[\n\\label{eq:even_action_final_old}\nS_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \\sqrt{\\frac{A}{B}} J^2 \\left(\n\\frac12 \\mathcal{K}_{ij} \\dot{v}^i \\dot{v}^j - \\frac12 \\mathcal{G}_{ij} v^{i\\prime} v^{j\\prime} - \\mathcal{Q}_{ij} v^i v^{j\\prime} - \\frac12 \\mathcal{M}_{ij} v^i v^j \\right),\n\\]\nwhere $i=1,2$ and $v^1=\\psi$, $v^2=\\chi$. We note that\nterms which are higher order in derivatives, like\n$\\dot{\\psi}' \\dot{\\chi}$, disappear upon integrating by parts.\n\nThe no-ghost condition amounts to requiring that $\\mathcal{K}_{ij}$ in eq.~\\eqref{eq:even_action_final_old} is positive definite:\n\\[\n\\label{eq:no_ghost_old}\n\\mathcal{K}_{11}>0, \\qquad \\det(\\mathcal{K}) > 0,\n\\]\nwhere\n\\begin{equation}\n\\label{eq:K11old}\n{\\cal K}_{11}=\\frac{8 B {\\left( 2{\\cal H} J J' +\n\\Xi \\pi' \\right)}^2 \\left[ \\ell (\\ell+1){\\cal P}_1-{\\cal F}\\right]}\n{\\ell (\\ell+1) A^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\n\\begin{equation}\n\\label{eq:detKold}\n\\det ({\\cal K})=\\frac{16 B J'^2 (\\ell-1) (\\ell+2){\\left( 2{\\cal H} J J' + \\Xi \\pi'\\right)}^2 \\left[{\\cal F}(2{\\cal P}_1-{\\cal F})\\right]}\n{\\ell (\\ell+1) A^3 J^2 \\pi'^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\nwith $\\mathcal{F}$ and $\\mathcal{H}$ are given by \\eqref{eq:cal_FGH},\n\\[\n \\begin{aligned}\n\\label{eq:Xi}\n\\Xi = 2G_{4\\pi}J^2 + 4G_{4X}BJJ'\\pi'\n -2G_{4\\pi X}BJ^2\\pi'^2\n \n - 4G_{4XX}B^2JJ'\\pi'^3 \n + 16F_{4}B^2JJ'\\pi'^3 - 4F_{4X}B^3JJ'\\pi'^5, \\\\\n \\end{aligned}\n\\]\nand\n\\begin{subequations}\n\\begin{align}\n\\label{eq:Theta}\n\\mathbf{\\Theta} &= 2 \\ell (\\ell+1) \\: J\\left(\\mathcal{H} - 2F_4 B^2 \\pi'^4\\right) +\\mathcal{P}_2,\\\\\n\\label{eq:P1}\n\\mathcal{P}_1 &= \\frac{\\sqrt{B}}{\\sqrt{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[\\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}\\right],\\\\\n\\mathcal{P}_2 &= \\frac{B(A' J - 2 A J')}{A}\n\\left(2{\\cal H} J J' + \\Xi \\pi'\\right)\n\\; .\n\\end{align}\n\\end{subequations}\nHere and in what follows we highlight $\\mathbf{\\Theta}$ in bold to emphasize that it involves $\\ell(\\ell+1)$. It follows immediately form eqs.~\\eqref{eq:K11old}-\\eqref{eq:detKold}\nthat both no-ghost conditions~\\eqref{eq:no_ghost_old} are satisfied \nprovided\n\\[\n\\label{eq:stability_even_ghost}\n2{\\cal P}_1-{\\cal F} > 0.\n\\]\n\n\nTo have no radial gradient instabilities one also has to require positive definiteness for matrix $\\mathcal{G}_{ij}$ in eq.~\\eqref{eq:even_action_final_old}:\n\\[\n\\label{eq:no_gradient_old}\n\\mathcal{G}_{11}>0, \\qquad \\det{\\mathcal{G}}>0.\n\\]\nHere\n\\begin{equation}\n\\label{eq:G11old}\n\\mathcal{G}_{11}=\\frac{4 B^{2} \\left[ \\mathcal{G} (\\ell+2)(\\ell-1) (2 \\mathcal{H} J J'+ \\Xi \\pi')^2+\\ell(\\ell+1)(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )\\right]}\n{ \\ell (\\ell+1) A \\mathcal{H}^2 \\mathbf{\\Theta} ^2},\n\\end{equation}\n\\begin{equation}\n\\label{eq:detGold}\n\\det(\\mathcal{G})=\\frac{16 B^3 J'^2 \\mathcal{G} (\\ell-1) (\\ell+2) (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}\n{ \\ell (\\ell+1) A J^2 \\pi'^2 \\mathcal{H}^2 \\mathbf{\\Theta}^2},\n\\end{equation}\nwith\n\\begin{align}\n\\label{eq:Gamma}\n& \\Gamma = \\Gamma_1 + \\frac{A'}{A} \\Gamma_2,\n\\\\\n& \\nonumber\\Gamma_{1} = 4\\left(G_{4\\pi} + 2XG_{4\\pi X} + \\frac{B\\pi'J'}{J}(G_{4X} + 2XG_{4XX}) \\right) - \\frac{16J'}{J}B\\pi'X(2F_{4} + XF_{4X}), \\nonumber\\\\\n&\\Gamma_{2} = 2B\\pi'\\left(G_{4X} - B\\pi'^2G_{4XX}\\right)\n- 8B\\pi'X(2F_{4} + XF_{4X}),\n\\nonumber\n\\end{align}\nand\n\\begin{align}\n\\label{eq:KSI}\n& \\Sigma = XF_{X} + 2F_{XX}X^2 +2\\left(\\frac{1 - BJ'^2}{J^2} - \\frac{BJ'}{J} \\frac{A'}{A}\\right)X(G_{4X} + 2XG_{4XX})\n\\nonumber\\\\& \n - \\frac{4BJ'}{J}\\left( \\frac{J'}{J } + \\frac{A'}{A}\\right)X^2(3G_{4XX} + 2XG_{4XXX}) + 2B\\pi'\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)X( \\frac{3}{2} G_{4\\pi X} + XG_{4 \\pi XX})\n \\nonumber\\\\& \n + \\frac{B^3J'(A'J + AJ')}{AJ^2}\\pi'^4(12F_{4} - 9F_{4X}B\\pi'^2 + F_{4XX}B^2\\pi'^4).\n\\end{align}\nAccording to eqs.~\\eqref{eq:G11old} and~\\eqref{eq:detGold} both conditions~\\eqref{eq:no_gradient_old} hold provided that\n\\[\n\\label{eq:stability_even_radial}\n2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B > 0.\n\\]\n\nThe corresponding sound speeds squared for perturbations propagating along the radial direction are given by the eigenvalues of matrix $(AB)^{-1}(\\mathcal{K})^{-1}\\mathcal{G}$:\n\\begin{equation}\n\\label{eq:speed_even_old}\nc_{s1}^2 = \\frac{\\mathcal{G}}{\\mathcal{F}},\n\\qquad\nc_{s2}^2 = \\frac{(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}\n{\\left(2{\\cal H} J J' + \\Xi \\pi'\\right)^2 (2{\\cal P}_1-{\\cal F})} \\; ,\n\\end{equation}\nwhere $c_{s1}^2$ coincides with the radial speed squared $c^2_r$ for odd-parity modes in eq.~\\eqref{eq:speed_odd}, which enables us\nto interpret it as a propagation speed in the radial direction of two tensor modes. \n\nPositive definiteness of matrices $\\mathcal{M}_{ij}$ and \n$\\mathcal{Q}_{ij}$ in the action~\\eqref{eq:even_action_final_old} also provide stability conditions, which ensure that angular gradient instabilities and \"slow\"\ntachyonic modes are absent. This set of conditions was not thoroughly addressed in previous works in Refs.~\\cite{Kobayashi:even,wormhole1,Trincherini} and we also omit discussing them for now. \nWe explicitly give the matrix $\\mathcal{M}_{ij}$ for high angular momentum modes\nas well as corresponding stability conditions for angular gradient instabilities in Regge-Wheeler-unitary \ngauge in Sec.~\\ref{sec:new_gauge}.\n\nLet us now show that the coefficients in the action~\\eqref{eq:even_action_final_old} become singular for modes with \nhigh momentum $\\ell$ if the no-ghost condition~\\eqref{eq:stability_even_ghost} is satisfied at all points.\nThe no-ghost condition~\\eqref{eq:stability_even_ghost} together with eq.~\\eqref{eq:P1} can be rewritten as follows:\n\\[\n\\label{eq:no_go1}\n\\frac{\\sqrt{B}}{\\sqrt{A}} \\xi' >\\frac{\\mathcal{F}}{2} \\quad \\mbox{with}\n\\quad \\xi = \\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}.\n\\]\nAs $\\mathcal{F} > 0$ due to stability constraints in the odd-parity sector~\\eqref{eq:stability_odd} it follows from eq.~\\eqref{eq:no_go1} that $\\xi'$ has to be positive.\nThen $\\xi$ is a monotonously growing function, so $\\xi=0$ at some point(s)\n\\footnote{All other options where $\\xi'>0$ but $\\xi$ does not cross zero involve either fine-tuning or special cases like $\\mathcal{F} \\to 0$ as $r \\to -\\infty$, which signals potential strong coupling in the odd-parity sector, see eq.~\\eqref{eq:action_odd_final} (see e.g. Ref.~\\cite{wormhole1} for a discussion).}.\nThe latter happens provided the numerator of $\\xi$ crosses zero,\nwhich is possible since the \ncombination $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ can take zero value, while $\\mathcal{H} >0$ due to stability in the odd-parity sector~\\eqref{eq:stability_odd}. \nLet us note here that \nin Horndeski theories $F_4 = 0$, so $\\xi$ cannot cross zero in a healthy way and, hence, the no-ghost condition~\\eqref{eq:stability_even_ghost} cannot be met at all points in this case. This is what is usually referred to as a no-go theorem in Horndeski theories, see Refs.~\\cite{Rubakov:2016zah,Olegi} for details.\n\nWhat is important for us here is that the same combination \n$\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ gives the leading contribution into $\\mathbf{\\Theta}$ in eq.~\\eqref{eq:Theta} for modes with $\\ell \\gg 1$. \nThis in turn means that as soon as\nthe no-ghost condition~\\eqref{eq:stability_even_ghost} is satisfied\nby having $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)=0$ \nat some point(s), $\\mathbf{\\Theta} \\to 0$ for high momenta modes. Thus,\nthe components of both matrices \n$\\mathcal{K}_{ij}$ and $\\mathcal{G}_{ij}$\nhit infinity for high momenta modes since they involve $\\mathbf{\\Theta}^2$ in denominators, see eqs.~\\eqref{eq:K11old},~\\eqref{eq:detKold},~\\eqref{eq:G11old} and ~\\eqref{eq:detGold}. \nThis is the problem\nof singular coefficients in the quadratic action upon $\\mathbf{\\Theta}$-crossing. \n\nThe appearance of $\\mathbf{\\Theta}$ in denominator seems to result from the gauge choice, and this choice, in particular, dictates the way non-dynamical DOFs are integrated out in action~\\eqref{eq:action_even}. Namely, the $\\ell^2$ part of $\\mathbf{\\Theta}$ comes from \nthe combination (see Appendix B for definitions of $a_i$)\n$$j^2 a_4 a_8 = j^2 \\frac A2 \\cdot \\mathcal{H} \\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right) \\;,$$\nwhich arises in the denominator of eq.~\\eqref{eq:alphaold} and originates from the field redefinition~\\eqref{eq:H2old}. \n\nThe fact that $\\mathbf{\\Theta}$-crossing\nis directly related to ensuring stability at the linearized level makes the old gauge unsuitable for construction of a completely stable wormhole solution.\n\nIn the following section we rederive the quadratic action for\nbeyond Horndeski Lagrangian~\\eqref{eq:lagrangian} \nin a Regge-Wheeler-unitary gauge and pay special attention to potential divergencies in the coefficients of the action due to $\\mathbf{\\Theta}$ crossing zero. So we keep track of $\\mathbf{\\Theta}$ and ensure the coefficients in the quadratic action do not hit singularities, so that we end up with \na regular quadratic action.\n\n\n\n\n\\subsubsection{Original calculations in the Regge-Wheeler-unitary gauge ($\\beta=G=\\chi=0$)}\n\\label{sec:new_gauge}\n\nThe quadratic action for even-parity modes in the Regge-Wheeler-unitary gauge reads:\n\\begin{equation}\n\\label{eq:action_even_RW}\n\\begin{aligned}\n&S_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \n\\left(H_0 \\left[ a_3 H_2' + \\left( a_7 + j^2 a_8 \\right)H_2 + j^2 a_4 \\alpha' + j^2 a_9 \\alpha + a_{13} K'' + a_{14} K'\\right.\\right. \\\\\n&\\left.\\left. \n+ (a_{15} + j^2 a_{16}) K \\right]\n+j^2 b_1 H_1^2 + H_1 \\left[b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha} + b_{8}\\dot{K}' + b_{9}\\dot{K} \\right] + c_{8} \\dot{H}_2 \\dot{K} + c_6 H_2^2 \\right. \\\\\n&\\left. + H_2 \\left[ j^2 c_5 \\alpha + c_{11} K' + (c_{12} +j^2 c_{13})K \\right] + j^2d_1 {\\dot \\alpha}^2 +j^2 d_4 \\alpha^2 + j^2 d_{7} \\alpha K' + p_{8}\\dot{K}^2 + p_{9}K'^2\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere the notations are similar to those in action~\\eqref{eq:action_even}\nand the coefficients $a_i$, $b_i$, $c_i$, $d_i$ and $p_i$ are given in Appendix B. In full analogy with the spherical gauge case above variation of the action~\\eqref{eq:action_even_RW} w.r.t. $H_0$ and $H_1$ gives two constraint equations, respectively:\n\\begin{equation}\n\\label{eq:H0_RW}\na_3 H_2' +\\left( a_7+j^2 a_8 \\right)H_2 + j^2 a_4 \\alpha'\n+ j^2 a_9 \\alpha + a_{13}K'' + a_{14}K' +(a_{15}+j^2 a_{16})K = 0,\n\\end{equation}\n\\begin{equation}\n\\label{eq:H1_RW}\nH_1 = - \\frac{1}{2 j^2 b_1} \\left( b_4 {\\dot H_2}+j^2b_5 {\\dot \\alpha} \n + b_8 \\dot{K}' +b_9 \\dot{K} \\right).\n\\end{equation}\nThis time the constraint~\\eqref{eq:H0_RW} contains not only the first derivatives of $H_2$, $\\alpha$ and $K$ but also $K''$. To simplify things\nlet us introduce an axillary field $\\psi$ as follows (cf. eq.~\\eqref{eq:H2old})\n\\begin{equation}\n\\label{eq:alpha_shifted}\n\\alpha = \\psi - \\dfrac{a_{13}}{j^2 a_4} K'. \n\\end{equation}\nLet us note here that $a_{13}= J^2 \\cdot a_4$ (see Appendix B), so the redefinition above is regular.\nUpon substitution of $\\alpha$ and $\\alpha'$ from eq.~\\eqref{eq:alpha_shifted} into the constraint~\\eqref{eq:H0_RW}, one can see that not only $K''$ get canceled out but also the resulting factor in front of $K'$ is automatically vanishing (see Appendix B for the explicit form of $a_i$):\n\\[\na_{14} - a_{13}' +\\frac{a_{13}}{a_4}(a_4' - a_9) = 0.\n\\]\nHence, the resulting equation is algebraic for $K$ and reads\n\\begin{equation}\n\\label{eq:K}\nK = - \\dfrac{1}{a_{15} +j^2 a_{16}} \\left[ a_3 H_2' +(a_7+j^2 a_8) H_2 +j^2 a_4 \\psi' +j^2 a_9 \\psi \\right].\n\\end{equation}\nNow by substituting $\\alpha$ from eq.~\\eqref{eq:alpha_shifted} and $K$ from eq.~\\eqref{eq:K} into the second constraint~\\eqref{eq:H1_RW} one can express $H_1$ in terms of $H_2$ and $\\psi$. Therefore, by making use of both constraint equations~\\eqref{eq:H0_RW}-\\eqref{eq:H1_RW} and field redefinition~\\eqref{eq:alpha_shifted} one can cast the quadratic action~\\eqref{eq:action_even_RW} in terms of $H_2$ and $\\psi$:\n\\begin{equation}\n\\label{eq:unconstrained_action_RW}\n\\begin{aligned}\nS_{even}^{(2)} = &\\int \\mbox{d}t\\:\\mbox{d}r \\left(\ng_{4(i)}\\dot{\\psi}'\\:^2 + g_{5(i)}\\psi''\\:^2 \n+f_{4(i)}\\dot{H_2}'^2 + f_{5(i)} H_2''^2\n+ h_{7(i)} \\dot{H_2}' \\dot{\\psi}' + h_{8(i)} H_2'' \\psi''\n\\right.\\\\ \n&\\left.\n+g_{1(i)}\\dot{\\psi}^2 + g_{2(i)}\\psi'\\:^2 + g_{3(i)}\\psi^2\n + f_{1(i)}\\dot{H_2}^2 + f_{2(i)}H_2'^2 + f_{3(i)}H_2^2 \n+ h_{5(i)}\\dot{H_2} \\dot{\\psi}' \n\\right.\\\\ \n&\\left.\n+ h_{6(i)} H_2' \\psi'' + h_{1(i)}\\dot{H_2} \\dot{\\psi} + h_{2(i)} H_2' \\psi' + h_{3(i)} H_2 \\psi' + h_{4(i)} H_2 \\psi\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere $g_{j(i)}$, $f_{j(i)}$ and $h_{j(i)}$ are some combinations of the initial coefficients $a_i$, $b_i$, $c_i$, $d_i$ and $p_i$ in eq.~\\eqref{eq:action_even_RW} and we give explicitly only those attributed to terms with four derivatives (see the first line of eq.~\\eqref{eq:unconstrained_action_RW}): \n\\begin{equation}\n\\label{eq:action_coef1}\n\\begin{aligned}\n& g_{4(i)} = \\frac{2\\ell(\\ell+1)B^{3\/2} J^2 }{(\\ell+2)(\\ell-1)A^{1\/2}}\\frac{\\mathcal{H}^2}{\\mathcal{F}}, \\qquad \\qquad\ng_{5(i)}= - g_{4(i)} \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}}, \\\\\n& f_{4(i)} = \\frac{B^{3\/2} J^2}{2 \\ell(\\ell+1)(\\ell+2)(\\ell-1) A^{1\/2}} \\frac{(2\\mathcal{H} J J' + \\Xi \\pi')^2}{\\mathcal{F}}, \\quad\nf_{5(i)} = - f_{4(i)} \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}}, \\\\\n& h_7 = - \\frac{2 B^{3\/2} J^2}{(\\ell+2)(\\ell-1) A^{1\/2}} \\frac{\\mathcal{H}(2\\mathcal{H} J J' + \\Xi \\pi')}{\\mathcal{F}}, \\qquad \\qquad\nh_8 = - h_7 \\cdot A B \\frac{\\mathcal{G}}{\\mathcal{F}},\n\\end{aligned}\n\\end{equation}\nwhere we made use of Appendix B to substitute $a_i$, $b_i$, etc. The rest of coefficients in eq.~\\eqref{eq:unconstrained_action_RW} are irrelevant at this point, however, we should note that all of them are regular.\nNote that according to\nthe unconstrained action~\\eqref{eq:unconstrained_action_RW} \nboth $\\psi$ and $H_2$ are naively described by the fourth order equations. Let us now show that in fact both DOFs are described by the second order equations upon making suitable field redefinitions.\n\nIt immediately follows from eq.~\\eqref{eq:action_coef1} that the following relations hold\n\\begin{equation}\n\\begin{aligned}\n&4\\; g_{4(i)} \\cdot f_{4(i)} - h_7^2 = 0, \\\\\n&4\\;g_{5(i)} \\cdot f_{5(i)} - h_8^2 = 0,\n\\end{aligned}\n\\end{equation}\nwhich means that the corresponding terms $\\dot{\\psi}'^2$, $\\dot{H_2}'^2$, $\\dot{H_2}'\\dot{\\psi}'$ and $\\psi''^2$, $H_2''^2$, $\\psi''H_2''$ get combined into perfect squares in eq.~\\eqref{eq:unconstrained_action_RW}. Hence, upon a field redefinition\n\\begin{equation}\n\\label{eq:tildePsi}\n\\begin{aligned}\n& \\psi = \\tilde{\\psi} + \\frac{(2\\mathcal{H} J J' + \\Xi \\pi')}{2\\ell(\\ell+1) \\; \\mathcal{H}} \\tilde{H_2}, \\\\\n& H_2 = \\tilde{H_2}.\n\\end{aligned}\n\\end{equation}\nthe terms $\\dot{\\tilde{H_2}}'^2$, $\\tilde{H_2}''^2$, $\\dot{\\tilde{H_2}}' \\dot{\\psi}'$ and $\\tilde{H_2}'' \\psi''$ vanish and the action~\\eqref{eq:unconstrained_action_RW} reduces to the following:\n\\begin{equation}\n\\label{eq:unconstrained_action_tildePsi}\n\\begin{aligned}\nS_{even}^{(2)} = &\\int \\mbox{d}t\\:\\mbox{d}r \\left(\ng_4\\; \\dot{\\tilde\\psi}'\\:^2 + g_5 \\;\\tilde\\psi''\\:^2 + g_1\\; \\dot{\\tilde\\psi}^2 + g_{2} \\;\\tilde\\psi'\\:^2 + g_{3} \\;\\tilde\\psi^2\n+ f_1 \\;\\dot{\\tilde{H_2}}^2 + f_2 \\;\\tilde{H_2}'^2 \\right.\\\\ \n&\\left.+ f_{3} \\;\\tilde{H_2}^2 \n+ h_5\\; \\dot{\\tilde{H_2}} \\dot{\\tilde\\psi}' + h_6\\; \\tilde{H_2}' \\tilde\\psi'' + h_1\\; \\dot{\\tilde{H_2}} \\dot{\\tilde\\psi} + h_2\\; \\tilde{H_2}' \\tilde\\psi' + h_{3}\\; \\tilde{H_2} \\tilde\\psi'+ h_{4}\\; \\tilde{H_2} \\tilde\\psi\n\\right),\n\\end{aligned}\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:coef_tildePsi}\n\\begin{aligned}\n& g_4 = \\frac{2 \\ell(\\ell+1) B^{3\/2} J^2 }{(\\ell^2+\\ell-2) A^{1\/2}} \\frac{\\mathcal{H}^2}{\\mathcal{F}}, \n\\qquad \\qquad\ng_5 = - \\frac{2 \\ell(\\ell+1) A^{1\/2} B^{5\/2} J^2}{\\ell^2 + \\ell -2} \\frac{\\mathcal{H}^3}{\\mathcal{F}^2}, \\\\\n& \nh_5 = - \\frac{B^{1\/2} J}{(\\ell^2+\\ell-2) A^{1\/2}} \\frac{\\mathcal{H} \\mathbf{\\Theta}}{\\mathcal{F}},\n\\qquad \\qquad\nh_6 = \\frac{A^{1\/2}B^{3\/2} J}{(\\ell^2+\\ell-2)} \\frac{\\mathcal{H}^2 \\mathbf{\\Theta}}{\\mathcal{F}^2},\\\\\n&f_1 = \\frac{1}{8\\ell(\\ell+1)(\\ell^2+\\ell-2)(AB)^{1\/2}} \\frac{\\mathbf{\\Theta}^2}{ \\mathcal{F}},\n\\qquad \\qquad\nf_2 = \\frac{ (AB)^{1\/2}}{8\\ell(\\ell+1)(\\ell^2+\\ell-2)}\\frac{\\mathcal{H} \\mathbf{\\Theta}^2}{ \\mathcal{F}^2},\n\\end{aligned}\n\\end{equation}\nwhile the explicit form of the rest of coefficients \n$g_j$, $f_j$ and $h_j$ is irrelevant here (they do not involve $\\mathbf{\\Theta}$ and were checked to be perfectly regular).\nNote that there are still terms with higher order derivatives \nof $\\tilde{\\psi}$ in the\naction~\\eqref{eq:unconstrained_action_tildePsi}. \nHowever, we see again\nthat terms $g_4\\; \\dot{\\tilde\\psi}'\\:^2 $, $h_5\\; \\dot{\\tilde{H_2}}\\dot{\\tilde\\psi}'$, $f_1 \\dot{\\tilde{H_2}}^2$ and $g_5 {\\tilde\\psi}''\\:^2$,\n$h_6\\; \\tilde{H_2}'{\\tilde\\psi}''$, $f_2\\;\\tilde{H_2}'^2$ get combined into perfect squares since\n\\begin{equation}\n\\label{eq:pefect_squares2}\n\\begin{aligned}\n&4\\; g_{4} \\cdot f_{1} - h_5^2 = 0, \\\\\n&4\\;g_{5} \\cdot f_{2} - h_6^2 = 0,\n\\end{aligned}\n\\end{equation}\nhence, it is possible to redefine $\\tilde{H_2}$ as\n\\begin{equation}\n\\label{eq:tildePsi2}\n\\begin{aligned}\n& \\tilde{\\psi} = \\bar{\\psi}, \\\\\n& \\tilde{H_2} = \\bar{H_2} + 4 \\ell(\\ell+1) B J \\dfrac{\\;{\\cal H} }{\\mathbf{\\Theta}} \\; \\bar\\psi'.\n\\end{aligned}\n\\end{equation}\nThen the terms with higher derivatives vanish and the resulting action gives the second order differential equations of motion for both $\\bar{\\psi}$ and $\\bar{H_2}$ in full analogy with the case of Sec.~\\ref{sec:old_gauge}. \n\nThere is, however, a problem with redefinition~\\eqref{eq:tildePsi2}: it diverges in the vicinity of $\\mathbf{\\Theta} = 0$, which means that the final action for $\\bar{\\psi}$ and $\\bar{H_2}$\nalso involves singular coefficients and this is exactly what we tried to avoid. One possible way to proceed is to refrain from the redefinition~\\eqref{eq:tildePsi2} and use the action~\\eqref{eq:unconstrained_action_tildePsi} for further stability analysis. The advantage of the latter approach is that all coefficients in the action are regular everywhere\nincluding $\\mathbf{\\Theta}=0$, but at the same time there are still higher derivative terms \nof $\\tilde{\\psi}$, which naively signal that the corresponding system of equations of motion\nmight admit more than four solutions for $\\tilde{\\psi}$ and $\\tilde{H_2}$, i.e. there are additional pathological DOFs. \nOne should note, however, that the redefinition~\\eqref{eq:tildePsi2} is perfectly fine away from $\\mathbf{\\Theta}=0$ and it demonstrates that in fact the action~\\eqref{eq:unconstrained_action_tildePsi} gives two second order differential equations, which normally admit four solutions. So in the following section let us\nprove that even in the vicinity of $\\mathbf{\\Theta}=0$ the action~\\eqref{eq:unconstrained_action_tildePsi} describes a healthy system, giving four regular solutions ${\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)}$.\nThis legitimizes redefinition~\\eqref{eq:tildePsi2} at all points including $\\mathbf{\\Theta}=0$ since we explicitly show that potential divergencies in the final quadratic action for $\\bar{\\psi}$\nand $\\bar{H_2}$ are not inherited from the original action~\\eqref{eq:unconstrained_action_tildePsi} and\ndo not have any physics behind them but result from our field redefinition. \n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{$\\mathbf{\\Theta}$-crossing}\n\\label{sec:TT_crossing}\n\nLet us explicitly solve the linearized equations for $\\tilde{H_2}$ and $\\tilde{\\psi}$ following from the action~\\eqref{eq:unconstrained_action_tildePsi} in the vicinity of $\\mathbf{\\Theta}=0$.\nThe general form of these linearized equations reads:\n\\begin{subequations}\n\\label{eq:deltaPsiH2}\n\\begin{align}\n\\label{eq:deltaPsi}\n\\delta\\tilde{\\psi}: \\;\\; &\n 2 g_5 \\tilde{\\psi}'''' + 4 g_5' \\tilde{\\psi}''' + (2 g_5 '' - 2 g_2)\\tilde{\\psi}'' - 2 g_2' \\tilde{\\psi}' + 2 g_3 \\tilde{\\psi} + h_6 \\tilde{H_2}''' + (2 h_6' - h_2) \\tilde{H_2}'' \\nonumber\\\\\n& + (h_6'' - h_2' - h_3) \\tilde{H_2}' + (h_4 - h_3')\\tilde{H_2} + 2 g_4 \\ddot{\\tilde{\\psi}}'' + 2 g_4' \\ddot{\\tilde{\\psi}}' \n- 2 g_1 \\ddot{\\tilde{\\psi}} + h_5 \\ddot{\\tilde{H_2}}' + (h_5' - h_1) \\ddot{\\tilde{H_2}} = 0, \\\\\\nonumber\n\\\\\n\\label{eq:deltaH2}\n\\delta\\tilde{H_2}: \\;\\; &\n2 f_2 \\tilde{H_2}'' + 2 f_2 \\tilde{H_2}' - 2 f_3 \\tilde{H_2} + h_6 \\tilde{\\psi}''' + (h_6'+h_2)\\tilde{\\psi}''\n+ (h_2' - h_3)\\tilde{\\psi}' - h_4 \\tilde{\\psi} \n+ 2 f_1 \\ddot{\\tilde{H_2}} + h_5 \\ddot{\\tilde{\\psi}}' \n+h_1 \\ddot{\\tilde{\\psi}} = 0.\n\\end{align}\n\\end{subequations}\nSince we consider a static, spherically-symmetric background \nfor both $\\tilde{\\psi}(t,r)$ and \n$\\tilde{H_2}(t,r)$ we use constant energy Ansatz:\n\\begin{equation}\n\\tilde{\\psi}(t,r) = \\tilde{\\psi}(r) e^{i\\omega t}, \\qquad\n\\tilde{H_2}(t,r) = \\tilde{H_2}(r) e^{i\\omega t},\n\\end{equation}\nwhere $\\omega$ denotes frequency. Until the end of this section we generally drop the argument of both $\\tilde{\\psi}(r)$ and $\\tilde{H_2}(r)$ for brevity. \n\nThe system~\\eqref{eq:deltaPsiH2} consists of the fourth and third order differential equations, which can be combined into the system of two third order differential equations. To see this one has to take a \ncoordinate derivative of eq.~\\eqref{eq:deltaH2} and then take a linear combination\nof the resulting equation and eq.~\\eqref{eq:deltaPsi} with a factor\n\\begin{equation}\n\\label{eq:constC}\n\\mathcal{C} = \\dfrac{1}{4 \\ell(\\ell+1) B J } \\dfrac{\\mathbf{\\Theta}}{\\cal{H}},\n\\end{equation}\nso that both \n$(2 g_5 \\mathcal{C} + h_6)\\tilde{\\psi}''''$ and $(h_6 \\mathcal{C} + 2 f_2)\\tilde{H_2}'''$ vanish (see eq.~\\eqref{eq:action_coef1}).\nThen the resulting system reads as follows:\n\\begin{subequations}\n\\label{eq:deltaPsiH2new}\n\\begin{align}\n& [4 g_5' \\mathcal{C} + 2 h_6' + h_2]\\;\\tilde{\\psi}''' + [2g_5'' \\mathcal{C} - 2g_2 \\mathcal{C} +h_6'' +2 h_2' - h_3 - \\omega^2 (2 g_4 \\mathcal{C} +h_5) ]\\; \\tilde{\\psi}'' \n+ [ h_2''-2 g_2' \\mathcal{C} - h_3' - h_4 \n\\nonumber\\\\\\nonumber\n&- \\omega^2(2 g_4' \\mathcal{C} + h_5' + h_1)]\\; \\tilde{\\psi}' + [2g_3 \\mathcal{C} - h_4' - \\omega^2(2 g_1 \\mathcal{C} + h_1')] \\tilde{\\psi}\n+ [2 h_6' \\mathcal{C} - h_2 \\mathcal{C} + 4 f_2'] \\tilde{H_2}'' \n+ [h_6'' \\mathcal{C} - h_2' \\mathcal{C} \n\\nonumber\\\\\n& \n- h_3 \\mathcal{C} + 2f_2'' -2 f_3 - \\omega^2 (h_5 \\mathcal{C} + 2 f_1)] \\tilde{H_2}' - [h_3' \\mathcal{C} - h_4 \\mathcal{C} + 2 f_3' - \\omega^2 (\\mathcal{C}(h_5' - h_1) + 2 f_1')] \\tilde{H_2} = 0,\\\\\\nonumber\n\\\\\n& 2 f_2 \\tilde{H_2}'' + 2 f_2 \\tilde{H_2}' - 2 f_3 \\tilde{H_2} + h_6 \\tilde{\\psi}''' + (h_6'+h_2)\\tilde{\\psi}''\n+ (h_2' - h_3)\\tilde{\\psi}' - h_4 \\tilde{\\psi} - \\omega^2 ( 2 f_1 \\tilde{H_2} + h_5 \\tilde{\\psi}' \n+h_1\\tilde{\\psi}) = 0,\n\\end{align}\n\\end{subequations}\nwhere the leading terms with derivative \nare $\\tilde{\\psi}'''$ and $\\tilde{H_2}''$.\nLet us now prove that the system~\\eqref{eq:deltaPsiH2new} admits only four regular solutions in the vicinity of $\\mathbf{\\Theta} = 0$.\n\nWe arrange the coordinate system so that $\\mathbf{\\Theta}$ crosses zero at $r=0$ so we write\n\\begin{equation}\n\\label{eq:linear_theta}\n\\mathbf{\\Theta} = \\gamma \\cdot r + \\dots,\n\\end{equation}\nwhere $\\gamma$ is a regular constant and dots denote higher order \nin $r$. So from now on we focus on the behaviour \nof the system~\\eqref{eq:deltaPsiH2new} in the vicinity of $r=0$. \nThe coefficients $h_j$, $g_j$ and $f_j$ \nin eqs.~\\eqref{eq:deltaPsiH2new} are regular at all points and can be expanded into power-series around $r=0$: \n\\begin{equation}\n\\begin{aligned}\n\\label{eq:ansatz_hgf}\n& h_j = h_{j,0} + h_{j,1} \\cdot r + h_{j,2}\\cdot r^2 + \\dots, \\quad (j=\\overline{1,4})\\\\\n& h_5 = h_{5,1}\\cdot r + h_{5,2}\\cdot r^2 + \\dots,\\\\\n& h_6 = h_{6,1}\\cdot r + h_{6,2}\\cdot r^2 + \\dots,\\\\\n& g_j = g_{j,0} + g_{j,1}\\cdot r + g_{j,2}\\cdot r^2 + \\dots, \\quad (j=\\overline{1,5})\\\\\n& f_3 = f_{3,0} + f_{3,1}\\cdot r + f_{3,2}\\cdot r^2 + \\dots,\n\\end{aligned}\n\\end{equation}\nwhere $h_{j,k}$, $g_{j,k}$ and $f_{j,k}$ are constants and for \n$f_1$ and $f_2$ we make use of eqs.~\\eqref{eq:pefect_squares2} to express them as follows: \n\\begin{equation}\n\\label{eq:f1f2}\nf_1 = \\frac{h_5^2}{4\\; g_4}, \\qquad f_2 = \\frac{h_6^2}{4\\; g_5}.\n\\end{equation}\nLet us note the expansions for $h_5$ and $h_6$ start from linear terms according to their definitions in eqs.~\\eqref{eq:coef_tildePsi}.\n\nSo upon substitution of \nthe expansions~\\eqref{eq:linear_theta}-\\eqref{eq:f1f2} into \neqs.~\\eqref{eq:deltaPsiH2new}\nwe find four independent solutions, which to the leading order read:\n\\begin{subequations}\n\\label{eq:regular_solutions}\n\\begin{align}\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_1\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n 1 + O(r^4)\\\\\n \\dfrac{-h_{1,0}\\cdot\\omega^2-h_{4,0}}{2 f_{3,0}}+O(r) \\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_2\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r +O(r^4) \\\\\n \\dfrac{h_{2,1}-h_{3,0}}{2 f_{3,0}} + O(r)\\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_3\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r^2 + O(r^4)\\\\\n \\dfrac{h_{2,0}+h_{6,1}}{f_{3,0}} + O(r)\\\\\n \\end{array}\n \\right),\\\\\n&\\left(\n \\begin{array}{c}\n \\tilde{\\psi} \\\\\n \\tilde{H}_2 \\\\\n \\end{array}\n\\right)\n=\nc_4\\exp(i\\omega t)\\left(\n \\begin{array}{c}\n r^3 + O(r^4)\\\\\n \\dfrac{(6h_{2,0}+12h_{6,1})g_{5,0}}{2g_{5,0}f_{3,0}-h_{6,1}^2} r + O(r^2)\\\\\n \\end{array}\n \\right),\n\\end{align}\n\\end{subequations}\nwhere $c_i$ are constants.\nThe solutions~\\eqref{eq:regular_solutions} \nare indeed regular at $r=0$, i.e. when $\\mathbf{\\Theta}$ crosses zero. \nLet us note that as soon as \n$\\tilde{\\psi}$ and $\\tilde{H_2}$ are found the rest of\nmetric perturbations, namely, $K$, $\\alpha$, $H_1$ and $H_0$ can be restored from \neqs.~\\eqref{eq:tildePsi},~\\eqref{eq:H0_RW}-\\eqref{eq:alpha_shifted} \nand they turn out to be regular as well since denominators in the corresponding relations are non-zero.\n\nBy proving that solutions~\\eqref{eq:regular_solutions} behave \nregularly around $\\mathbf{\\Theta}$-crossing \nwe have shown that $\\mathbf{\\Theta}=0$ is not a special\npoint for the action~\\eqref{eq:unconstrained_action_tildePsi}. Hence, \nthe potential singularities at $\\mathbf{\\Theta}=0$ \nin the resulting action for $\\bar{\\psi}$ and\n$\\bar{H_2}$ after redefinition~\\eqref{eq:tildePsi2} should be \nattributed\nto the singularity of the redefinition upon $\\mathbf{\\Theta}$-crossing.\nKeeping this in mind, \nin the following section we explicitly adopt redefinition~\\eqref{eq:tildePsi2}\nin order to have the quadratic action for perturbations in a conventional form and \nderive a set of stability conditions for two DOFs $\\bar{\\psi}$ and $\\bar{H_2}$.\n\n\n\n\n\n\n\n\n\\subsubsection{Stability constraints in Regge-Wheeler-unitary gauge}\n\\label{sec:new_stability_constraints}\n\nLet us now make use of the redefinition~\\eqref{eq:tildePsi2} and cast \nthe action~\\eqref{eq:unconstrained_action_tildePsi} in a conventional form similar to that in the spherical gauge (cf.~\\eqref{eq:even_action_final_old}): \n\\begin{equation}\n\\label{eq:action_KGQM_new}\nS_{even}^{(2)} = \\int \\mbox{d}t\\:\\mbox{d}r \n\\left(\n\\frac12 \\bar{\\mathcal{K}}_{ij} \\dot{v}^i \\dot{v}^j - \\frac12 \\bar{\\mathcal{G}}_{ij} v^{i\\prime} v^{j\\prime} - \\bar{\\mathcal{Q}}_{ij} v^i v^{j\\prime} - \\frac12 \\bar{\\mathcal{M}}_{ij} v^i v^j\n\\right),\n\\end{equation}\nwhere $i=1,2$ with $v^1 = \\bar{H_2}$, $v^2 = \\bar{\\psi}$ and both DOFs are described by the second order differential equations.\nThe explicit expressions for $\\bar{\\mathcal{K}}_{11}$, $\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$ and $\\det\\bar{\\mathcal{G}}$ \nare as follows:\n{\\small\n\\begin{equation}\n\\label{eq:k11detk_RW}\n\\bar{\\mathcal{K}}_{11} = \n\\dfrac{ 1}{4 \\ell(\\ell+1)(\\ell +2)(\\ell-1) \\sqrt{AB}}\n\\dfrac{\\mathbf{\\Theta}^2}{\\mathcal{F}},\n\\quad\n\\det\\bar{\\mathcal{K}} = \\dfrac{\\ell(\\ell+1) B}{4(\\ell+2)(\\ell-1)A}\n\\dfrac{(2\\mathcal{H} J J' + \\Xi \\pi')^2 (2 \\mathcal{P}_1 - \\mathcal{F})}{\\mathcal{F}},\n\\end{equation}\n\\begin{equation}\n\\label{eq:g11detg_RW}\n\\bar{\\mathcal{G}}_{11} = \\dfrac{\\sqrt{AB} }{4 \\ell(\\ell+1)(\\ell +2)(\\ell-1)} \\dfrac{\\mathcal{H}\\mathbf{\\Theta}^2}{\\mathcal{F}^2}, \\quad\n\\det\\bar{\\mathcal{G}}=\\dfrac{\\ell (\\ell+1) A B^3}\n{ 4(\\ell+2) (\\ell-1)}\n\\dfrac{\\mathcal{G} (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )}{ \\mathcal{F}^2 },\n\\end{equation}}\nwhile the rest of components in both matrices are given in Appendix C.\nLet us note that $\\det\\bar{\\mathcal{K}}$ differ from its counterpart in the old gauge~\\eqref{eq:detKold} by a certain non-negative factor:\n\\begin{equation}\n\\label{eq:relationKK}\n\\det\\bar{\\mathcal{K}} = \\dfrac{\\ell^2(\\ell+1)^2 AB \\pi'^2}{64(\\ell-1)^2(\\ell+2)^2 J'^2} \\dfrac{\\mathcal{H}^2 \\mathbf{\\Theta}^2}{\\mathcal{F}^2} \\; \\det{\\mathcal{K}},\n\\end{equation}\nand the same holds for $\\det\\bar{\\mathcal{G}}$ and $\\det{\\mathcal{G}}$~\\eqref{eq:detGold}. The relation~\\eqref{eq:relationKK} serves as a cross check for our calculations in the Regge-Wheeler-unitary gauge. \nAccording to eqs.~\\eqref{eq:k11detk_RW} and~\\eqref{eq:g11detg_RW}\nno-ghost and no radial gradient instabilities constraints are still given by eqs.~\\eqref{eq:no_ghost_old} and~\\eqref{eq:no_gradient_old}, and sound speeds squared for radial perturbations are also unchanged and are given by eqs.~\\eqref{eq:speed_even_old}.\n\nAs for potentially divergent coefficients in the action~\\eqref{eq:action_KGQM_new} upon $\\mathbf{\\Theta}$-crossing\nwe see that both\n$\\det\\bar{\\mathcal{K}}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{K}}_{11}$ and $\\bar{\\mathcal{G}}_{11}$\nare non-singular \nat $\\mathbf{\\Theta}=0$ while $\\bar{\\mathcal{K}}_{22}$ and $\\bar{\\mathcal{G}}_{22}$ hit singularities (see Appendix C). \nLet us recall that $\\mathbf{\\Theta}$ inevitably crosses zero at some point(s)\nprovided that the no-ghost constraint is satisfied (see Sec.~\\ref{sec:old_gauge}).\nSo in the end we have not avoided \nsingular coefficients in the quadratic \naction~\\eqref{eq:action_KGQM_new}, but unlike the case of \naction~\\eqref{eq:even_action_final_old} in the spherical gauge we have explicitly checked that these divergencies \nappear due to our technical redefinition~\\eqref{eq:tildePsi2}\nand have no physical nature. \n\nThe significant difference between the new \nmatrices $\\bar{\\mathcal{K}}_{ij}$, $\\bar{\\mathcal{G}}_{ij}$ \nand ${\\mathcal{K}}_{ij}$, ${\\mathcal{G}}_{ij}$\nin the old gauge is that the determinants of former ones do not involve $J'$ as a common factor, cf. eqs.~\\eqref{eq:k11detk_RW},~\\eqref{eq:g11detg_RW} and eqs.~\\eqref{eq:detKold},~\\eqref{eq:detGold}. \nThis fact becomes important in stability analysis for a wormhole\nsolution, whose characteristic tunnel-like profile is encoded in the metric function\n$J(r)$, and $J'(r)=0$ at the narrowest point of the throat. The latter \nmade both $\\det\\mathcal{K}$ and $\\det\\mathcal{G}$ automatically vanishing, which made the \nstability analysis in the spherical gauge not entirely conclusive close to the center of a wormhole's throat. \n\n\nLet us now turn to matrices $\\bar{\\mathcal{Q}}_{ij}$ and $\\bar{\\mathcal{M}}_{ij}$ in action~\\eqref{eq:action_KGQM_new}. We adopt a convention where matrix $\\bar{\\mathcal{Q}}_{ij}$ has the only non-vanishing element $\\bar{\\mathcal{Q}}_{12}$, while other elements are included in matrix $\\bar{\\mathcal{M}}_{ij}$ upon integration by parts. \nIn full analogy with the odd-parity sector in Sec.~\\ref{sec:odd_sector}\nmatrices $\\bar{\\mathcal{Q}}_{ij}$ and \n$\\bar{\\mathcal{M}}_{ij}$ generally give \ntwo types of constraints for the linearized theory: one for angular gradient instabilities and the other one for \"slow\" tachyonic instabilities. In what follows we concentrate on angular gradient instabilities, so it is sufficient for us to consider only those parts \nof both $\\bar{\\mathcal{Q}}_{ij}$ and $\\bar{\\mathcal{M}}_{ij}$ which are \nproportional to $\\ell(\\ell+1)$. \nWe have found that in the leading order $\\bar{\\mathcal{Q}}_{12}$ does\nnot depend on $\\ell$\nso it does not provide any\nconstraints in the context of angular gradient instabilities, while \nthe leading order in angular momentum of both $\\bar{\\mathcal{M}}^{(\\ell^2)}_{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$ read:\n\\begin{equation}\n\\label{eq:m11_RW}\n\\bar{\\mathcal{M}}^{(\\ell^2)}_{11} = \\ell^2 \\dfrac{\\sqrt{A}}{\\sqrt{B}}\\dfrac{(\\mathcal{H} - 2 F_4 B^2 \\pi'^4)^2}{\\mathcal{F}},\n\\end{equation}\n\n{\\small\n\\begin{equation}\n\\label{eq:detM_RW}\n\\det\\bar{\\mathcal{M}}^{(\\ell^2)} = - \\dfrac{\\ell^2 \\left( \\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)^2 \\mathcal{H}^2}{16 \\mathcal{F}^2 A J^2} \n\\left(\n8 A^2 \\dfrac{\\mathcal{F} \\mathcal{G}}{\\mathcal{H}^2} + B[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2 + 4 A^{3\/2}B J^4 \\mathcal{F} \\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]\n\\right),\n\\end{equation}\n}\nwith\n\\begin{equation}\n\\mathcal{P}_4 = \\dfrac{\\mathcal{H} A' J + 2 \\mathcal{G} A J' + \\Gamma A J \\pi'}{\\mathcal{H} \\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)},\n\\end{equation}\nand the rest of matrix components are given in Appendix C.\nWe have introduced a superscript $\\ell^2$ to emphasize that this is only\na part of matrix $\\bar{\\mathcal{M}}$ that is proportional to $\\ell(\\ell+1)$.\n\nWe see that in full analogy with $\\bar{\\mathcal{K}}_{ij}$ and $\\bar{\\mathcal{G}}_{ij}$ matrix $\\bar{\\mathcal{M}}^{(\\ell^2)} _{ij}$ involves \nthe key combination $\\left( \\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ which comes from $\\mathbf{\\Theta}$ (see eq.~\\eqref{eq:Theta}) upon taking the leading order in $\\ell$ and has to cross zero in order to avoid ghost instabilities, see eq.~\\eqref{eq:no_go1}. Hence, the\nmatrix $\\bar{\\mathcal{M}}^{(\\ell^2)} _{ij}$ behaves in a similar way to \n$\\bar{\\mathcal{K}}_{ij}$ and $\\bar{\\mathcal{G}}_{ij}$\nupon $\\mathbf{\\Theta}$-crossing: $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $ is regular while\n$\\bar{\\mathcal{M}}^{(\\ell^2)}_{11}$ crosses zero. \n\nAs before to ensure that angular gradient instabilities are absent \nit is sufficient to require positive definiteness of matrix $\\bar{\\mathcal{M}}$:\n\\begin{equation}\n\\label{eq:positiveM}\n\\bar{\\mathcal{M}}^{(\\ell^2)} _{11} > 0, \\qquad \\det\\bar{\\mathcal{M}}^{(\\ell^2)} >0.\n\\end{equation}\nAccording to eq.~\\eqref{eq:m11_RW} $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ is automatically non-negative, while making the determinant positive\nrequires the following: \n\\begin{equation}\n\\label{eq:no_ang_grad}\n\\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]< - \\frac{1}{4 A^{3\/2} J^4 }\n\\left(8 \\frac{A^2}{B} \\dfrac{ \\mathcal{G}}{\\mathcal{H}^2} \n+ \\frac{[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2}{\\mathcal{F}} \\right).\n\\end{equation}\nThus, parity even modes have no angular gradient instabilities \nprovided that inequality~\\eqref{eq:no_ang_grad} is satisfied. The corresponding sound speeds squared propagating along the angular direction $c^2_{a1,2}$ are given as eigenvalues \nof matrix $(A^{-1}J^2)(\\bar{\\mathcal{K}})^{-1}\\bar{\\mathcal{M}}^{(\\ell^2)} $. We omit their explicit expressions here as they are quite cumbersome and not really illuminating.\n\nWe stress again that our stability analysis for the even-parity modes in this section remains incomplete as opposed to the odd-parity modes, see Sec.~\\ref{sec:odd_sector}. Namely, we have not addressed \n\"slow\" tachyonic instabilities, which become significant if we consider all modes, including those with low momentum. As we mentioned above the corresponding stability conditions for tachyons are associated with the remaining parts of matrices $\\bar{\\mathcal{Q}}_{ij}$ and \n$\\bar{\\mathcal{M}}_{ij}$ which are lower order in $\\ell$. \nFormulating these conditions in a concise form is challenging and\nwe leave it for future development.\n\nTo sum up, the quadratic action~\\eqref{eq:action_KGQM_new} gives the following set of stability conditions which ensure the absence of both ghost and gradient instabilities in the even-parity sector\n(see eqs.~\\eqref{eq:stability_even_ghost},~\\eqref{eq:stability_even_radial} and~\\eqref{eq:no_ang_grad}):\n\\begin{subequations}\n\\label{eq:stability_even}\n\\begin{align}\n\\label{eq:stability_K_RW}\n\\mbox{No ghosts:}\\quad &2\\frac{\\sqrt{B}}{\\sqrt{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[\\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'}\\right]-\\mathcal{F} > 0, \\\\\n\\label{eq:stability_G_RW}\n\\mbox{No radial gradient instabilities:}\\quad &2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B > 0, \\\\\n\\label{eq:stability_M_RW}\n\\mbox{No angular gradient instabilities:}\\quad & \\frac{d}{dr}\\left[ \\frac{\\sqrt{B}}{\\sqrt{A} J^3\n} \\mathcal{P}_4 \\right]< - \\frac{1}{4 A^{3\/2} J^4 }\n\\left(8 \\frac{A^2}{B} \\dfrac{ \\mathcal{G}}{\\mathcal{H}^2} \n+ \\frac{[\\mathcal{F}\\mathcal{P}_4 - (A'J - 2 A J')]^2}{\\mathcal{F}} \\right).\n\\end{align}\n\\end{subequations}\nSo the complete set of stability conditions for high momenta modes in parity odd and parity even sectors is given by eqs.~\\eqref{eq:stability_odd} and~\\eqref{eq:stability_even}.\n\n\n\\section{Wormhole beyond Horndeski: an example}\n\\label{sec:wormhole_example}\n\nIn this section we give a specific example of beyond Horndeski Lagrangian~\\eqref{eq:lagrangian} which admits a static, spherically-symmetric wormhole solution. Our main requirement to the solution is that it has to be stable against ghosts and gradient instabilities in both parity odd and parity even sectors.\nIn full analogy with our previous wormhole solution in Ref.~\\cite{wormhole1} we adopt a \"reconstruction\" procedure: we\nchoose the background metric~\\eqref{eq:backgr_metric} of a wormhole form\nand concoct the Lagrangian functions so that background equations of motion~\\eqref{eq: background_eqs} and stability conditions in eqs.~\\eqref{eq:stability_odd} and~\\eqref{eq:stability_even}\nare satisfied. We also ensure that\nthe modes in the odd sector propagate at safely subluminal speed~\\eqref{eq:sublum_odd}, \nand the same is done for the speeds of the even parity modes propagating in the radial direction~\\eqref{eq:speed_even_old}. \n\nWe begin with choosing a specific form of metric functions in~\\eqref{eq:backgr_metric}:\n\\[\n\\label{eq:AJ}\nA = 1, \\qquad J = \\tau\\log\\left[1+2\\cosh\\left(\\frac{r}{\\tau}\\right)\\right],\n\\]\nwhere the parameter $\\tau \\sim R_{min}$\nregulates the size of the wormhole\nthroat at $r=0$. Our choice for the last metric function $B(r)$ is somewhat less straightforward. There is a linear combination of the \nbackground equations of motion~\\eqref{eq: background_eqs} which involves metric functions explicitly, while all the Lagrangian functions get combined into $\\mathcal{F}$ and $\\mathcal{H}$ (see eq.~\\eqref{eq:cal_FGH}):\n\\begin{equation}\n\\label{eq:dY}\nA\\cdot\\mathcal{E}_J + J^2\\cdot\\mathcal{E}_A = -\\frac{\\mathcal{F}}{J^2}+\\frac{BA'^2}{4A^2}+\\frac{B'J'}{2J}-\\frac{A'(JB'+2BJ')}{4AJ}-\\frac{BA''}{2A}+B\\left(\\frac{J'^2}{J^2}+\\frac{J''}{J}\\right)=0,\n\\end{equation}\nwhere $\\mathcal{E}_J$ and $\\mathcal{E}_A$ are given in Appendix A and we have already set $\\mathcal{H} = 1$ for simplicity (this is a deliberate choice in our model below although not an obligatory one). Stability conditions for the odd parity sector~\\eqref{eq:stability_odd} require $\\mathcal{F} > 0$ so\naccording to eq.~\\eqref{eq:dY} we cannot freely choose all\nthree metric functions. Having fixed $A(r)$ and $J(r)$ in eq.~\\eqref{eq:AJ}\nwe take $B(r)$ as follows:\n\\begin{equation}\n\\label{eq:B}\nB(r) = 1+ \\mbox{sech}\\left(\\frac{r}{\\tau}\\right),\n\\end{equation} \nwhere $\\tau$ is the same parameter as in eq.~\\eqref{eq:AJ}. This choice for $B$ ensures that eq.~\\eqref{eq:dY} holds for $\\mathcal{F} > \\mathcal{H}$ with already fixed $\\mathcal{H}=1$ above. \n\n\nFinally, we choose static, spherically-symmetric scalar field $\\pi(r)$, which supports the throat of a wormhole, as follows:\n\\begin{equation}\n\\label{eq:pi}\n\\pi(r)=\\tanh\\left(\\frac{r}{\\tau}\\right)-1.\n\\end{equation} \nThis completes our arrangement of the background setting.\nOur choice for $\\pi(r)$, $A$, $B$ and $J$ above has the following asymptotical behaviour as $r \\to\\pm\\infty$:\n\\begin{equation}\n\\label{eq:asymp_background}\n\\pi(r) \\to 0, \\qquad A(r) = 1, \\qquad B(r) \\to 1, \\qquad J(r) \\to r,\n\\end{equation} \nwhich means that the space-time\nin our set up tends to an empty Minkowski space far away from the wormhole. \n\nLet us note that we choose the metric functions $A$, $B$ and $J$ above differently as compared to the original set of $A=B=1$ and $J=\\sqrt{\\tau^2 + r^2}$ in Ref.~\\cite{wormhole1}. This is due to the fact that the corresponding equations of motion and stability conditions in the latter case become oversimplified due to random cancellations so that we have less parameters in the model to control stability at the linearized level.\n \n\nTo find the Lagrangian functions for the devised background, \nwe choose the\nfollowing Ansatz:\n\\begin{subequations}\n\\label{ansatz}\n\\begin{align}\n\\label{F}\n& F(\\pi, X) = f_0(\\pi) + f_1(\\pi)\\cdot X + f_2(\\pi)\\cdot X^2, \\\\\n\\label{G4}\n& G_4(\\pi, X) = \\frac12 + g_{40}(\\pi) + g_{41}(\\pi) \\cdot X + g_{42}(\\pi) \\cdot X^2,\\\\\n\\label{F4}\n& F_4(\\pi, X) = f_{40}(\\pi) + f_{41}(\\pi) \\cdot X,\n\\end{align}\n\\end{subequations}\nin full analogy with Ref.~\\cite{wormhole1}, but here we have an additional term $g_{42}(\\pi)$ which will be used to satisfy the new stability condition in the even parity sector~\\eqref{eq:no_ang_grad}.\n\nThe following steps \nare almost identical to those in Ref.~\\cite{wormhole1}, however, all\nthe formulae become significantly more involved due to non-trivial $J(r)$ and $B(r)$ in eqs.~\\eqref{eq:AJ} and~\\eqref{eq:B}. \nIn Sec.\\ref{sec:wormhole_reconstruction} we schematically describe the way we find functions $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$ by making use of stability conditions~\\eqref{eq:stability_odd},~\\eqref{eq:stability_even} and background equations, but we skip some of the most lengthy intermediate steps of calculations and do not give the resulting Lagrangian functions analytically to avoid bulky expressions.\nWe present all necessary plots in Sec.~\\eqref{sec:wormhole_results} though, which show the graphs of Lagrangian functions we come up with and explicitly prove that the stability conditions are indeed satisfied. The reader that is interested in the results only\nmay safely skip the following technical section and go straight to Sec.~\\eqref{sec:wormhole_results}.\n\n\n\\subsection{Beyond Horndeski Lagrangian functions: reconstruction}\n\\label{sec:wormhole_reconstruction}\n\nWe make use of eq.~\\eqref{ansatz} and express $\\mathcal{H}$, $\\mathcal{F}$, \n$\\Xi$, $\\Sigma$ and $\\Gamma$ in terms of $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$:\n\\begin{subequations}\n\\begin{align}\n\\label{eq:D:H}\n& {\\cal H} = {\\cal G} = 1 + 2 g_{40}(\\pi) + B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 {B^2} \\pi'^4 \\cdot [4\\; f_{40}(\\pi) - 3\\; g_{42}(\\pi)] -\n B^3 \\pi'^6 \\cdot f_{41}(\\pi) ,\\\\\n %\n\\label{eq:D:F}\n& {\\cal F} = 1 + 2 g_{40}(\\pi) - B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 B^2 \\pi'^4 \\cdot g_{42}(\\pi), \\\\\n %\n\\label{eq:D:Xi}\n& \\Xi = 4 B J J' \\pi' \\left[-3 B^2 \\pi'^4 \\cdot f_{41}(\\pi) \n+ 4 B \\pi'^2 \\cdot f_{40}(\\pi)\n- 3 B \\pi'^2 \\cdot g_{42}(\\pi) \n+ g_{41}(\\pi) \\right]\\\\\\nonumber\n&\\qquad + J^2 \\left[ \\frac52 B^2 \\pi'^4 \\cdot g_{42}'(\\pi) \n- 3 B \\pi'^2 \\cdot g_{41}'(\\pi) + 2 \\cdot g_{40}'(\\pi)\\right],\\\\\n\\label{eq:D:Sigma}\n&\\Sigma=- \\frac12 B^2 \\pi'^2\\left( 6 B \\left(\\frac{A'}{A} + \\frac{J'}{J }\\right)\\frac{J'}{J } \\pi'^2 (5 B f_{41}(\\pi) \\pi'^2-4 f_{40}(\\pi)+3 g_{42}(\\pi)) \\right.\\\\\\nonumber&\\left.\n+ \\left(\\frac{A'}{A} + 4 \\frac{J'}{J }\\right)\n\\pi'(3 g_{41}'(\\pi)-5 B \\pi'^2 g_{42}'(\\pi)) - 2 g_{41}(\\pi) \\left(\\frac{A'}{A} + \\frac{J'}{J }\\right) \\frac{J'}{J} \\right) \n\\\\\\nonumber\n& + \\frac{1}{2 J^2}B \\pi'^2 \\left( 6 B g_{42}(\\pi) \\pi'^2 + 2 g_{41}(\\pi)\\right) \n -\\frac{3}{2} B^2 f_{2}(\\pi) \\pi'^4 - \\frac{1}{2}B \\pi'^2 f_{1}(\\pi),\\\\\n\\label{eq:D:Gamma}\n& \\Gamma = \\left(2 B \\pi' \\frac{A'}{A} + 4 B \\pi' \\frac{ J'}{ J}\\right)\n\\left[{4 B \\pi'^2} \\cdot f_{40}(\\pi) \n- {3 B ^2 \\pi'^4 \\cdot f_{41}(\\pi) } + { g_{41}(\\pi)} \n- {3 B \\pi'^2 \\cdot g_{42}(\\pi) } \\right] \\\\\\nonumber\n& \\qquad + 4 g_{40}'(\\pi) - 6 B \\pi'^2 \\cdot g_{41}'(\\pi) +5 B ^2 \\pi'^4 \\cdot g_{42}'(\\pi),\n\\end{align}\n\\end{subequations}\nwhere we do not explicitly substitute the coordinate dependence of $\\pi$\nfrom eq.~\\eqref{eq:pi} to keep the equations more concise, but we bear in mind that in fact we work with functions of $r$.\n\nFirst, we recall that we have already fixed $\\mathcal{H} = 1$ (and $\\mathcal{G}=1$, see eq.~\\eqref{eq:cal_FGH}) for simplicity when we chose $B$ in eq.~\\eqref{eq:B}, so now we can make use of eq.~\\eqref{eq:D:H} and express $g_{40}(\\pi)$ in terms of $f_{40}(\\pi)$, $f_{41}(\\pi)$, $g_{41}(\\pi)$ and $g_{42}(\\pi)$:\n\\begin{equation}\n\\label{eq:g40}\ng_{40}(\\pi) = - \\frac12 \\left[ B \\pi'^2 \\cdot g_{41}(\\pi) +\n \\frac12 {B^2} \\pi'^4 \\cdot [4\\; f_{40}(\\pi) - 3\\; g_{42}(\\pi)] -\n B^3 \\pi'^6 \\cdot f_{41}(\\pi) \\right].\n\\end{equation}\nThen, with $A$, $B$ and $J$ set in eqs.~\\eqref{eq:AJ},~\\eqref{eq:B} \nwe can write eq.~\\eqref{eq:dY} substituting $\\mathcal{F}$ \nfrom~\\eqref{eq:D:F} and $g_{40}(\\pi)$ from eq.~\\eqref{eq:g40}\nand express $g_{41}$ in terms of $g_{42}(\\pi)$, $f_{40}(\\pi)$ and\n$f_{41}(\\pi)$\nas follows:\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:g41}\n& g_{41}(\\pi) = \\frac{1}{J^2 } \\left(\\frac12 B^2 \\pi'^4 \\cdot f_{41}(\\pi) - {B} \\pi'^2\n(f_{40}(\\pi) - g_{42}(\\pi)) \\right) \\\\\n&- \\frac{1}{8 \\pi'^2 }\n\\left(\\frac{A'^2}{A^2} -\n \\frac{[2 B A' J' + J (A' B' + 2 B A)]}{A B J} +\n \\frac{2[ B' J J' + 2 B J'^2 + 2 B J J''-2]}{B J^2}\\right).\n \\end{aligned}\n\\end{equation}\nNote that we chose $B(r)$ in eq.~\\eqref{eq:B} so that eq.~\\eqref{eq:dY} holds for $\\mathcal{F} > \\mathcal{H}$, i.e in our set up $\\mathcal{F} > 1$. Therefore, we have ensured that the stability conditions in the odd-parity sector~\\eqref{eq:stability_odd} are satisfied everywhere and both radial and angular speed~\\eqref{eq:speed_odd} is luminal at most.\n\nLet us turn to the no-ghost condition in the even parity sector~\\eqref{eq:stability_K_RW}:\n\\[\n\\label{eq:D:no_go}\n\\frac{\\sqrt{B}}{\\sqrt{A}}\\cdot \\frac{\\mbox{d}}{\\mbox{d}r}\\left[ \\frac{\\sqrt{A}}{\\sqrt{B}}\n\\frac{J^2 \\mathcal{H}\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)}{2{\\cal H} J J' + \\Xi \\pi'} \\right] > \\frac{{\\cal F}}{2}.\n\\]\nAs we have discussed in Sec.~\\ref{sec:old_gauge} to satisfy this no-ghost constraint for all $r$\nit is necessary to make \n$\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ cross zero at some point(s). Since we have fixed $\\mathcal{H}=1$ above, we can choose\n\\[\n\\label{eq:D:F4_choice}\n2 F_4(\\pi,X) B^2 \\pi'^4 \\equiv 2 \\left(f_{40}(\\pi) + f_{41}(\\pi)\\cdot X\\right) B^2 \\pi'^4 =\nw\\cdot \\mbox{sech}\\left(\\frac{r}{\\tau}+u\\right),\n\\]\nso that $\\left(\\mathcal{H} - 2 F_4 B^2 \\pi'^4\\right)$ crosses zero twice. \nWe use eq.~\\eqref{eq:D:F4_choice} to express $f_{40}(\\pi)$ through $f_{41}(\\pi)$. In our numerical results below we take specific values of parameters $u$, $w$ and $\\tau$ involved in eq.~\\eqref{eq:D:F4_choice}, namely, \n\\[\nu = 0, \\quad w=2, \\quad \\tau =100,\n\\]\nwhere $\\tau$ still defines the size or the wormhole throat. \nWe have introduced $u$ in eq.~\\eqref{eq:D:F4_choice} to emphasize that it is possible to take $u \\neq 0$, which proves that there is no fine-tuning and it is not obligatory to have $F_4'=0$ right at the throat $r=0$. We still take $u=0$ in our calculations for the sake of simplicity.\n\nNext we turn to the denominator of the no-ghost constraint~\\eqref{eq:D:no_go}, where $\\Xi$ involves yet undefined $g_{42}(\\pi)$ and $f_{41}(\\pi)$, see eq.~\\eqref{eq:D:Xi}. The simplest option that satisfies the stability condition~\\eqref{eq:D:no_go} everywhere is to take $\\Xi = 0$, so that we can substitute $g_{40}(\\pi)$, $g_{41}(\\pi)$ and $f_{40}(\\pi)$ from eqs.~\\eqref{eq:g40},~\\eqref{eq:g41},~\\eqref{eq:D:F4_choice} and find $f_{41}(\\pi)$ in terms of $g_{42}(\\pi)$ from the\nthe resulting equation. \n\nThe constraint that governs angular gradient instabilities in the even\nparity sector~\\eqref{eq:stability_M_RW} involves the only yet unconstrained coefficient\n$\\Gamma$. To satisfy inequality~\\eqref{eq:stability_M_RW} everywhere it turned out to be sufficient to take $\\Gamma = 0$, so we make use of eq.~\\eqref{eq:D:Gamma} and\nexpress $g_{42}(\\pi)$ as it is the only function that has not been defined above. \nThus, at this step we have a closed system for $g_{40}(\\pi)$, $g_{41}(\\pi)$, $g_{42}(\\pi)$, $f_{40}(\\pi)$ and $f_{41}(\\pi)$ so this \ncompletes the definition of $G_4(\\pi,X)$\nand $F_4(\\pi,X)$ in eqs.~\\eqref{G4} and~\\eqref{F4}.\n\nIn full analogy with Ref.~\\cite{wormhole1} \nwe find $f_0(\\pi)$ and $f_1(\\pi)$\nfrom solving two background equations of \nmotion~\\eqref{eq: background_eqs}, e.g. $\\mathcal{E}_A=0$ and $\\mathcal{E}_B=0$ (recall that we have already used a linear combination of $\\mathcal{E}_A$ and $\\mathcal{E}_J$ in eq.~\\eqref{eq:dY}, while $\\mathcal{E}_{\\pi}$ is linearly dependent on the other three equations). As before the final function $f_2(\\pi)$ is used to satisfy the stability condition against\nradial gradient instabilities~\\eqref{eq:stability_G_RW} in parity even sector and simultaneously ensure that the speed $c_{s2}^2 \\leq 1$ in eq.~\\eqref{eq:speed_even_old} ($c^2_{s1}$ is already set to be subluminal above). Both requirements can be written in a compact form:\n\\[\n\\label{D:constraint_f2}\n0 < 2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B \\leq (2 {\\cal H} J J' + \\Xi \\pi' )^2 (2 {\\cal P}_1 -{\\cal F}),\n\\]\nwhere $\\Sigma$ involves $f_2(\\pi)$, see eq.~\\eqref{eq:D:Sigma}. \nThe combination on the right hand side coincides with the no-ghost condition~\\eqref{eq:stability_K_RW} and we have already insured its positivity, so we choose $\\Sigma$ appropriately to have $c_{s2}^2 \\leq 1$:\n\\[\n\\label{eq:sigma_choice}\nc^2_{s2} = \\frac{(2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B)}{(2 {\\cal H} J J' + \\Xi \\pi' )^2 (2 {\\cal P}_1 -{\\cal F})} = \n\\frac{\\cosh\\left(\\frac{r}{\\tau}\\right)}{1+\\cosh\\left(\\frac{r}{\\tau}\\right)},\n\\]\nand express $f_2(\\pi)$ by making use of eq.~\\eqref{eq:D:Sigma} and recalling that we fixed $\\Xi=0$ in our model. This completes the reconstruction of the function $F(\\pi,X)$.\n\n\n\\subsection{Beyond Horndeski Lagrangian functions: results}\n\\label{sec:wormhole_results}\n\nThe reconstructed functions $f_i(\\pi)$, $g_{4i}(\\pi)$ and $f_{4j}(\\pi)$\nentering eqs.~\\eqref{ansatz} are given in Fig.~\\ref{fig:f-g-f4} and have the following asymptotic behaviour as $r \\to \\pm \\infty$:\n\\[\n\\label{eq:fg4f4_asymp}\nf_{0}\\propto e^{-\\frac{r}{\\tau}},\\quad \nf_{1} \\propto e^{\\frac{3r}{\\tau}}, \\quad\nf_{2}\\propto e^{\\frac{7r}{\\tau}}, \\quad\ng_{40} \\propto e^{-\\frac{r}{\\tau}}, \\quad\ng_{41} \\propto e^{\\frac{3r}{\\tau}}, \\quad \ng_{42} \\propto e^{\\frac{7r}{\\tau}}, \\quad\nf_{40} \\propto e^{\\frac{7r}{\\tau}},\\quad \nf_{41} \\propto e^{\\frac{11 r}{\\tau}} \\; ,\n\\] \nor, equivalently,\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:fg4f4XXX_asymp}\n&f_{0} =\\frac{3}{\\tau^2}e^{-\\frac{r}{\\tau}},\\quad \nf_{1} \\cdot X = -\\frac{6}{\\tau^2}e^{-\\frac{r}{\\tau}}, \\quad\nf_{2} \\cdot X^2 =\\frac{1}{\\tau^2}e^{-\\frac{r}{\\tau}}, \\quad\ng_{40} = -\\frac{13}{8}e^{-\\frac{r}{\\tau}}, \\quad\ng_{41}\\cdot X = \\frac{9}{4}e^{-\\frac{r}{\\tau}}, \\\\\n&\\qquad\\qquad \ng_{42} \\cdot X^2 = -\\frac{5}{8}e^{-\\frac{r}{\\tau}}, \\quad\nf_{40} = \\frac{15}{8}e^{-\\frac{r}{\\tau}},\\quad \nf_{41} \\cdot X = -\\frac{11}{8}e^{-\\frac{r}{\\tau}} \\; ,\n\\end{aligned}\n\\end{equation}\nwhich shows that all terms equally contribute to the Lagrangian functions. Thus, the Lagrangian functions $F(\\pi,X)$, $G_4(\\pi,X)$\nand $F_4(\\pi,X)$\nhave the following asymptotics far away from the throat (recall that $\\pi(r)$ in eq.~\\eqref{eq:pi})\n\\[\n\\label{eq:FG4F4asymp}\n\\begin{aligned}\n&F(\\pi, X) = q_1\\cdot(-\\pi)^{1\/2} + q_2\\cdot (-\\pi)^{-3\/2}\\cdot X + q_3\\cdot (-\\pi)^{-7\/2}\\cdot X^2, \\\\\n&G_4(\\pi, X) = \\frac12 + q_4\\cdot(-\\pi)^{1\/2} + q_5\\cdot (-\\pi)^{-3\/2}\\cdot X + q_6 \\cdot (-\\pi)^{-7\/2}\\cdot X^2, \\\\\n&F_4(\\pi, X) = q_7 \\cdot (-\\pi)^{-7\/2} + q_8 \\cdot (-\\pi)^{-11\/2} \\cdot X,\n\\end{aligned}\n\\]\nwhere $q_i$, $i = \\overline{1,8}$ are constants, which are irrelevant at this point. \nWe see that all functions in eq.~\\eqref{eq:fg4f4XXX_asymp} decay \nexponentially with growing $r$, which means that, \nwhile our theory is still of beyond Horndeski type away from the \nwormhole, the space-time quite rapidly becomes empty Minkowski space\nwith pure Einstein gravity. \n\n\n\n\n\n\\begin{figure}[H]\\begin{center}\\hspace{-1cm}\n{\\includegraphics[width=0.5\\textwidth]{f-2.pdf}}\\hspace{2.8cm}\\hspace{-3cm}\n{\\includegraphics[width=0.5\\textwidth] {g4-2.pdf}}\n\n\\vspace{-0.2cm}\n{\\includegraphics[width=0.5\\textwidth]\n{f4-2.pdf}}\\hspace{1cm}\n\\caption{\\footnotesize{The Lagrangian functions $f_0(r)$, $f_1(r) \\cdot X$, $f_2(r)\\cdot X^2$, $g_{40}(r)$, $g_{41}(r)\\cdot X$, \n$g_{42}(r)\\cdot X^2$,\n $f_{40}(r)$ and $f_{41}(r) \\cdot X $, with the following choice of the parameters: $u=0$, $w=2$ and $\\tau = 100$. This choice\n guarantees that the size\n of the wormhole throat safely exceeds the Planck length.}} \\label{fig:f-g-f4}\n\\end{center}\\end{figure}\n\nLet us demonstrate that our wormhole solution\n~\\eqref{eq:AJ},~\\eqref{eq:B}-\\eqref{eq:pi} \nsatisfies our set of stability conditions. We have arranged the Lagrangian functions so that $\\mathcal{H}=\\mathcal{G}=1$ and \n$\\mathcal{F} > 1$ which complies with the constraints in the\nodd-parity sector~\\eqref{eq:stability_odd},~\\eqref{eq:sublum_odd} \nand gives at most luminal sound speed squared, i.e. $c^2_{\\theta}=1$ and $c^2_r < 1$~\\eqref{eq:speed_odd}. As for the parity even modes we show in Fig.~\\ref{fig:KGM} that for our set of Lagrangian functions \n$\\bar{\\mathcal{K}}_{11}$, \n$\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$, $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $ are positive everywhere. Hence, the constraints\n~\\eqref{eq:stability_even} \nare satisfied at all points.\n\n\\begin{figure}[H]\\begin{center}\n{\\includegraphics[width=0.49\\textwidth]\n{kgm22.pdf}}\n\\hspace{0.01cm}\n{\\includegraphics[width=0.49\\textwidth]\n{kgmdet.pdf}}\n\\caption{\\footnotesize{Functions $\\bar{\\mathcal{K}}_{11}$, \n$\\det\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}_{11}$, $\\det\\bar{\\mathcal{G}}$, $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)} $, which govern the stability of the \nparity even sector (we choose $\\ell = 10$ here for definiteness). We have introduced a normalizing numerical factor for both $\\bar{\\mathcal{M}}^{(\\ell^2)} _{11}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$, which enabled us to put them on the same plots with matrices $\\bar{\\mathcal{K}}$ and $\\bar{\\mathcal{G}}$.\n}}\n\\label{fig:KGM}\n\\end{center}\\end{figure}\n\n\n\nAs for the sound speeds squared in the even-parity sector~\\eqref{eq:speed_even_old} \n$c^2_{s1} = c^2_r < 1$, while $c^2_{s2}$ was fixed in eq.~\\eqref{eq:sigma_choice} and obviously does not exceed unity. \nThe situation, however, is no so bright for the angular sound speeds squared $c^2_{a1,2}$, which are given by eigenvalues of matrix \n$(A^{-1}J^2)(\\bar{\\mathcal{K}})^{-1}\\bar{\\mathcal{M}}^{(\\ell^2)}$. Even though we do not give these speeds explicitly we can do a quick check if their product is greater than unity, which would signal superluminal propagation in the angular direction. Indeed, the product of $c^2_{a1}$ and $c^2_{a2}$\nis given by\n\\[\nc^2_{a1} \\cdot c^2_{a2} = \\frac{J^4}{A^2}\\frac{\\det\\bar{\\mathcal{M}}^{(\\ell^2)}}{\\det\\bar{\\mathcal{K}}},\n\\]\nwhere $\\det\\bar{\\mathcal{K}}$ and $\\det\\bar{\\mathcal{M}}^{(\\ell^2)}$ are given in eqs.~\\eqref{eq:k11detk_RW} and~\\eqref{eq:detM_RW}, respectively.\nOur numerical check has shown that for our choice of Lagrangian functions this ratio is greater than one, meaning that either of speeds\n$c^2_{a1}$ or $c^2_{a2}$ (or both) is greater than the speed of light. This is another troublesome \ndrawback of our solution and we hope to deal with it in\nfuture studies.\n\n\\vspace{-0.2cm}\n\\section{Conclusion}\n\\label{sec:outlook}\nTo sum up, this work aimed to support the point that scalar-tensor theories of modified gravity like beyond Horndeski theories are promising candidates admitting stable Lorentzian wormholes. \nWe have formulated a set of stability conditions which eliminate \nhigh energy pathologies like ghosts and gradient instabilities.\nAlthough imperfect we have put forward a wormhole solution \nthat complies with this set of stability constraints.\nUnfortunately, we still cannot say anything definite about the tachyon\nsector, since dealing with mass matrices is technically demanding\nin this class of field theories. It remains a challenging and intriguing\nquestion whether it is possible to have a completely stable wormhole in theories like generalised Horndeski family, let alone make it realistic and observationally traceable. \n\n\\vspace{-0.2cm}\n \\section*{Acknowledgements}{}\n\nWhen this work was close to completion, Valery Rubakov untimely deceased. All the findings in this paper are the result of our fruitful and long-lasting collaboration with Professor Rubakov. Unfortunately, he did not take part in preparing this manuscript.\n\\\\ \n\\\\\nThe work of S.M. and V.V. \non Sec.~\\ref{sec:linearized_th} of this paper has been\nsupported by Russian Science Foundation grant\n19-12-00393, while the part of work on Sec.~\\ref{sec:wormhole_example}\nhas been supported by the Foundation for the Advancement of Theoretical Physics and Mathematics \"BASIS\".\n\n\n\n\\section*{Appendix A}\n\\label{app:A}\nIn this Appendix we give Einstein and scalar field equations of motion for action~\\eqref{eq:lagrangian} in the background~\\eqref{eq:backgr_metric}:\n\\[\n\\mathcal{E}_A = 0, \\qquad \\mathcal{E}_B = 0,\n\\qquad \\mathcal{E}_J = 0, \\qquad \\mathcal{E}_{\\pi} = 0,\n\\]\nwhere\n{\\small\n\\begin{eqnarray}\n\\hspace{-1cm}\n&& \\label{CalEa} {\\cal E}_A = F + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J} - (2BJ'' + J'B')\\right)G_{4}\n +\\frac{4BJ'}{J}\\left( \\frac{J'}{J } + \\frac{ X'}{X} \n + \\frac{2BJ'' + J'B'}{BJ'}\\right)XG_{4X} \n\\nonumber\\\\&&\n + \\frac{8BJ'}{J}XX'G_{4XX} -\n B\\pi'\\left(\\frac{4J'}{J } + \\frac{ X'}{X} \\right)G_{4\\pi}+ 2B\\pi'\\left(\\frac{4J'}{J } - \\frac{ X'}{X} \\right)XG_{4\\pi X}+ 4XG_{4\\pi\\pi}\n\\nonumber\\\\&&\n -\\frac{2B^2\\pi'^3}{J^2}(5B'JJ' + B(J'^2 + 2JJ''))\\pi' F_{4}\n -\\frac{16 B^3 J' \\pi'^3}{J} \\pi'' F_4 \n + \\frac{2B^3J'\\pi'^5}{J} \\left(B'\\pi' + 2B\\pi''\\right)F_{4X}\n \\nonumber\\\\&&\n - \\frac{4B^3J'\\pi'^5}{J}F_{4\\pi} , \\\\\\nonumber\n\\end{eqnarray}}\n{\\small\n\\begin{eqnarray}\n\\nonumber\n&&\\label{CalEb} {\\cal E}_B = F - 2XF_{X} + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J } - BJ'\\frac{A'}{A} \\right)G_{4}\n -\\frac{4}{J}\\left(\\frac{1 - 2BJ'^2}{J } - 2BJ'\\frac{A'}{A}\\right)XG_{4X}\n- \\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)B\\pi'G_{4\\pi}\n \\nonumber\\\\&& \n + 8\\frac{BJ'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)X^2G_{4XX} \n - 2\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right)B\\pi'XG_{4\\pi X} \n -\\frac{10B^3J'(A'J + AJ')\\pi'^4}{AJ^2} F_{4}\n \\nonumber\\\\&&\n +\\frac{2B^4J'(A'J + AJ')\\pi'^6}{AJ^2} F_{4X},\\\\\\nonumber\n\\end{eqnarray}}\n{\\small\n\\begin{eqnarray}\n&&\\label{CalEc} {\\cal E}_J = F \n -\\left(\\frac{1}{J}\\frac{\\sqrt{B}}{\\sqrt{A}}\\left(J\\frac{\\sqrt{B}}{\\sqrt{A}}A'\\right)' + \\frac{2BJ'' + J'B'}{J} \\right)G_{4} - B\\pi'\\left(\\frac{2J'}{J } + \\frac{ A'}{A } + \\frac{2BJ'' + J'B'}{BJ'} +\n 2\\frac{\\pi'' - \\frac{J''}{J'}\\pi'}{\\pi'}\\right)G_{4\\pi} \n\\nonumber\\\\&&\n + BX\\left(-\\frac{A'^2}{A^2} + \\frac{2}{J}\\frac{2BJ'' + J'B'}{B} + \\frac{A'}{A}\\frac{2BJ'' + J'B'}{BJ'} + 2\\frac{A'J' + J(A'' - \\frac{J''}{J'}A')}{JA}\\right)G_{4X} \n\\nonumber\\\\&&\n + BX'\\left(\\frac{2J'}{J } + \\frac{ A'}{A} \\right)G_{4X} + 4XG_{4\\pi \\pi} +\n 2B\\pi'\\left(\\frac{2J'}{J } + \\frac{ A'}{A } - \\frac{ X'}{X}\\right)XG_{4\\pi X} + 2B\\left(\\frac{2J'}{J } + \\frac{ A'}{A} \\right)XX'G_{4XX}\n\\nonumber\\\\&&\n-\\frac{B^2\\pi'^4}{2}\\left(-\\frac{A'^2B}{A^2} + \\frac{A'(5B'J + 2BJ')}{AJ} + 2\\frac{A''BJ + 5AB'J' + 2ABJ''}{AJ}\\right) F_{4}\n\\nonumber\\\\&&\n - \\frac{B^3(A'J + 2AJ')}{AJ} \\pi'^3 \\left( 4 \\pi'' F_{4} + \\pi'^2 F_{4\\pi} -\n \\frac{\\pi'^2(B'\\pi' + 2B\\pi'')}{2} F_{4X} \\right),\n \n\\end{eqnarray}}\nand\n\\begin{eqnarray}\n&&\\label{CalEphi} {\\cal E}_{\\pi} = \\frac{1}{J^2}\\sqrt{\\frac{B}{A}} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d}r}\\left[J^2 J' \\sqrt{AB} \\cdot\n\\mathcal{J}_H\\right] - \\mathcal{S}_H + \\mathcal{JS}_{BH},\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n&&\\mathcal{J}_{H} = \\frac{\\pi'}{J'}F_{X} + 2\\frac{\\pi'}{J'}\\left(\\frac{1 - BJ'^2}{J^2 } - \\frac{BJ'}{J}\\frac{A'}{A}\\right)G_{4X} \n - \\frac{4B\\pi'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)XG_{4XX}\n \\nonumber \\\\&&\n - 2\\left( \\frac{4}{J } + \\frac{ A'}{AJ'}\\right)XG_{4\\pi X},\\\\\\nonumber\n \\\\\n &&\n\\mathcal{S}_{H} = -F_{\\pi} -\n \\left[\\frac{B}{2}\\left( \\frac{A'}{A} \\right)^2 + \\frac{2}{J}\\left(\\frac{1 - BJ'^2}{J } - (2BJ'' + J'B')\\right)\\right]G_{4\\pi} \n + B\\left[\\frac{X'}{X}\\left(\\frac{4J'}{J } + \\frac{ A'}{A} \\right) \n\\right.\\nonumber \\\\&&\\left.\n + \\frac{4J'}{J}\\left( \\frac{J'}{J } + \\frac{ A'}{A}\\right)\\right]XG_{4\\pi X}\n\n\n+\n \\left[\\frac{B}{2}\\left(2\\frac{A'' - \\frac{J''}{J'}A'}{A} + \\frac{A'}{A}\\left(\\frac{4J'}{J } + \\frac{2BJ'' + J'B'}{BJ'}\\right)\\right)\\right]G_{4\\pi}\n\\end{eqnarray}\n\\begin{eqnarray}\n &&\n\\mathcal{JS}_{BH} = -4B^2\\pi'^2\\left[\\frac{-A'^2BJ'}{A^2J} \\pi'+ \\frac{J'(2A''BJ + 5AB'J' + 4ABJ'')}{AJ^2}\\pi'\n\\right.\n\\nonumber\\\\ &&\n\\left.\n + \\frac{A'(5B'JJ' + 3BJ'^2 + 2BJJ'')}{AJ^2} \\pi' +\n \\frac{6BJ'(A'J + AJ')}{AJ^2}\\pi''\\right]F_{4}\n\\nonumber\\\\&&\n -\\frac{6B^3J'(A'J + AJ')\\pi'^4}{AJ^2}F_{4\\pi} +\n B^3\\pi'^4\\left[-\\frac{A'^2BJ'}{A^2J}\\pi' + \\frac{J'(2A''BJ + 11AB'J' + 4ABJ'')}{AJ^2}\\pi'\n\\right.\n\\nonumber\\\\&&\n \\left.\n + \\frac{A'(11B'JJ' + 3BJ'^2 + 2BJJ'')}{AJ^2}\n \\pi' + \\frac{18BJ'(A'J + AJ')}{AJ^2}\\pi''\\right]F_{4X}\n \\nonumber\\\\&&\n + \\frac{2B^4J'(A'J + AJ')\\pi'^6}{AJ^2}F_{4 \\pi X} -\n \\frac{B^4J'(A'J + AJ')\\pi'^6(B'\\pi' + 2B\\pi'')}{AJ^2}F_{4XX}.\n\\end{eqnarray}\n\n\n\\section*{Appendix B}\nIn this Appendix we gather the explicit form of coefficients\n$a_i$, $b_i$, $c_i$, $d_i$, $e_i$ and $p_i$ entering the quadratic action in the spherical gauge~\\eqref{eq:action_even} and \nits counterpart in\nRegge-Wheeler-unitary gauge~\\eqref{eq:action_even_RW}. The coefficients \nare given in terms of combinations $\\mathcal{H}$, $\\mathcal{F}$, $\\mathcal{G}$, $\\Xi$, $\\Gamma$\nand $\\Sigma$ introduced in the main body of the text in eqs.~\\eqref{eq:cal_FGH},~\\eqref{eq:Xi},~\\eqref{eq:Gamma} and~\\eqref{eq:KSI} respectively:\n\\begin{eqnarray\na_1&=&\\sqrt{AB}\\,\\Xi,\\\\\na_2&=&\\frac{\\sqrt{AB}}{2\\pi'}\\left[\n2\\pi'\\Xi'-\\left(2\\pi''-\\frac{A'}{A}\\pi'\\right)\\Xi\n+2JJ'\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}-4{\\cal H}JJ''\n\\right.\\\\ &&\\left.\n+\\frac{2J^2}{B}\\left({\\cal E}_B-{\\cal E}_A\\right) \\right] - \n 2 \\sqrt{AB^3} \\pi'^3 \\cdot F_4 ,\\\\\na_3&=&-\\frac{\\sqrt{AB}}{2}\\left(\\pi'\\Xi+2JJ'{\\cal H}\\right),\\\\\na_4&=&\\sqrt{AB}\\,{\\cal H},\\\\\na_5&=&-\\sqrt{\\frac{A}{B}}J^2\\frac{\\partial{\\cal E}_A}{\\partial\\pi }=a_2'-a_1'',\\\\\na_6&=&-\\sqrt{\\frac{A}{B}} \\frac{1}{J\\pi'} \\left( J {\\cal H}'+J'{\\cal H}-J'{\\cal F} \\right) + \\sqrt{AB} \\pi'^2 \\left( B' \\pi' + 2 B \\pi''\\right) \\cdot F_4, \\\\\na_7&=&a_3'+\\frac{J^2}{2} \\sqrt{\\frac{A}{B}} {\\cal E}_B,\n\\end{eqnarray}\n\\begin{eqnarray}\na_8&=&-\\frac{a_4}{2B} + \\sqrt{AB^3} \\pi'^4 \\cdot F_4, \\\\\na_9&=&\\frac{\\sqrt{A}}{J}\\frac{{\\rm d}}{{\\rm d} r}\\left(\nJ\\sqrt{B}{\\cal H}\n\\right), \\\\\nb_1&=&\\frac{1}{2}\\sqrt{\\frac{B}{A}}{\\cal H},\n\\\\\nb_2&=&-2\\sqrt{\\frac{B}{A}}\\Xi,\n\\\\\nb_3&=&\\sqrt{\\frac{B}{A}}\\frac{1}{\\pi'}\n\\left[\n\\left(2\\pi''+\\frac{B'}{B}\\pi'\\right)\\Xi -2JJ'\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}\n+\\frac{2J^2}{B}{\\cal E}_A+4JJ''{\\cal H}\n\\right\n,\n\\\\\nb_4&=&\\sqrt{\\frac{B}{A}}\\left(\\pi'\\Xi+2JJ'{\\cal H}\\right),\n\\\\\nb_5&=&-2b_1,\\\\\nc_1&=&-\\frac{1}{\\sqrt{AB}}\\Xi - 4\\sqrt{\\frac{ B^3}{A}} J J' \\pi'^3 \\cdot F_4,\n\\\\\nc_2&=&-\\sqrt{AB} \\left( \\frac{A'}{2A} \\Xi+JJ'\\Gamma-\\frac{J^2 \\pi'}{X} \\Sigma \\right)\n,\n\\\\\nc_3&=&J^2\\sqrt{\\frac{A}{B}} \\frac{\\partial {\\cal E}_B}{\\partial \\pi},\\\\\nc_4&=&\\frac{1}{2}\\sqrt{\\frac{A}{B}} \\Gamma + \\sqrt{\\frac{ B^3}{A}} \\left(A' J + 2A J'\\right) \\pi'^3\n \\cdot F_4,\n\\\\\nc_5&=&\n-\\frac{1}{2}\\sqrt{AB}\\left(\\pi'\\Gamma+\\frac{A'}{A}{\\cal H}\n+\\frac{2J'}{J}{\\cal G}\\right),\n\\\\\nc_6&=&\\frac{J^2}{2} \\sqrt{\\frac{A}{B}} \\left( \\Sigma+\\frac{A'B \\pi'}{2J^2 A}\\Xi+\\frac{B\\pi'J'}{J}\\Gamma-\\frac{1}{2} {\\cal E}_B+\\frac{BJ'^2}{J^2} {\\cal G}+\\frac{A'BJ'}{JA}{\\cal H} \\right),\n\\\\\nd_1&=&b_1,\n\\\\\nd_2&=&\\sqrt{AB}\\,\\Gamma,\n\\\\\nd_3&=& \\frac{\\sqrt{AB}}{J^2}\\left[\n\\frac{2JJ'}{\\pi'}\\left(\\frac{A'}{A}-\\frac{B'}{B}\\right){\\cal H}\n-J^2\\left(\\frac{2J'}{J}-\\frac{A'}{A}\\right)\\frac{\\partial{\\cal H}}{\\partial\\pi}\n+\\frac{2}{B\\pi'}\\left({\\cal F}-{\\cal G}\\right)\\right.\n\\nonumber\\\\&&\\left.\n-\\frac{J^2}{2\\pi'}\\left(2\\pi''+\\frac{B'}{B}\\pi'\\right)\n\\left(\\Gamma_1+\\frac{2J'}{J}\\Gamma_2\\right)\n-\\frac{2J^2}{B\\pi'}({\\cal E}_A-{\\cal E}_B)-\\frac{4JJ''}{\\pi'}{\\cal H}\n\\right]\n,\n\\\\\nd_4&=&\\frac{\\sqrt{AB}}{J^2}\n\\left({\\cal G}-J^2{\\cal E}_B\\right),\n\\\\\ne_1&=&\\frac{1}{2\\sqrt{AB}}\\left[\n\\frac{J^2}{X}({\\cal E}_A-{\\cal E}_B)-\\frac{2}{\\pi'}\\Xi'+\\left(\\frac{A'}{A}\n-\\frac{X'}{X}\\right)\\frac{\\Xi}{\\pi'}\n+\\frac{2BJ'^2}{X}{\\cal F}-\\frac{2JJ'B}{X}{\\cal H}' \\right.\\\\&&\\left.\n-{\\cal H}\\frac{B^2J'^2}{JXA}\\frac{{\\rm d}}{{\\rm d} r}\\left(\\frac{J^2A}{B}\\right)+\\frac{2BJJ''}{X}{\\cal H}\n\\right] - \\frac{4}{B \\pi'^2} \\cdot \\frac{\\mbox{d}}{\\mbox{d} r}\n\\left[\\sqrt{\\frac{B^5}{A}} J J' \\pi'^4 \\cdot F_4\\right], \\nonumber \\\\\ne_2&=&-\\sqrt{AB}\\frac{J^2}{X}\\Sigma,\\\\\ne_3&=&J^2 \\sqrt{\\frac{A}{B}} \\frac{\\partial {\\cal E}_\\pi}{\\partial \\pi},\n\\end{eqnarray}\n\\begin{eqnarray}\ne_4&=&\\frac{\\sqrt{AB}J'^2}{8X}\\left(-\\frac{4{\\cal G}}{J^2}-\\frac{4({\\cal E}_A-{\\cal E}_B)}{BJ'^2}+\\frac{2A'{\\cal H}'}{AJ'^2}+\n\\frac{4{\\cal G}'}{JJ'}+\\frac{4}{BJ^2J'^2}\\left(1-\\frac{JJ'BA'}{A}\\right){\\cal F}\n\\nonumber\\right.\\\\ &&\\left.-\n\\frac{4{\\cal H}}{BJ^2J'^2}\\left(1-BJ'^2(1+\\frac{2A'J}{AJ'})+J(B'J'+2BJ'')\\right)-\\frac{2\\pi'}{J'^2}\\Gamma'\n\\nonumber\\right.\\\\ &&\\left.\n+\\frac{2\\pi'}{JJ'}\\left(-2+\\frac{A'J}{AJ'}\\right)\\frac{\\partial{\\cal H}}{\\partial\\pi}\n+\\frac{\\Xi\\pi'}{J^3J'}\\left(2-\\frac{A'J}{AJ'}\\right)\\left[\\frac{A'BJJ'}{A}-2+2BJ'^2\\right]\n\\nonumber\\right.\\\\ &&\\left.\n+\\frac{\\Gamma_1\\pi'}{2J'}\\left[\\frac{4}{J}+\\frac{A'^2BJ}{A^2}-\\frac{4A'}{AJ'}-\\frac{4BJ'^2}{J}\n-\\frac{2B'}{BJ'}+\\frac{2BJ''}{BJ'^2}-\\frac{4\\pi''}{\\pi'J'}\n\\right]\n-\\nonumber\\right.\\\\ &&\\left.\n-\n\\frac{\\Gamma_2\\pi'}{J'}\\left[\\frac{2A'}{AJ}+\\frac{A'^2}{A^2J'}(1-BJ'^2)-\\frac{4J'}{J^2}(1-BJ'^2)+\\frac{2B'}{BJ}+\\frac{4\\pi''}{\\pi'J}\n\\right]\\right) - e_4^{BH},\n\\\\\ne_4^{BH}&=& -\\frac{\\pi'^2}{J^2 J'} \\sqrt{\\frac{B^3}{A}}\n\\left(A'' J^2 J' - 4 A J'^3 +A' J(J'^2 - J J'')\\right) \\cdot F_4\n\\\\\\nonumber&&\n- \\frac{A'J + 2AJ'}{B J^2 J' \\pi'^2} \\cdot\n\\frac{\\mbox{d}}{\\mbox{d} r} \\left[\\sqrt{\\frac{B^5}{A}} J J' \\pi'^4 \\cdot F_4\\right],\n\\\\\na_{13}&=& J^2\\; a_4,\n\\\\\na_{14}&=& \\sqrt{{A}{B}} \\;J^2 \\left[\\mathcal{H}' + \\frac12 \\left(\\frac{B'}{B} +6\\frac{J'}{J} \\right)\\mathcal{H} \\right],\n\\\\\na_{15} &=& \\sqrt{\\frac{A}{B}}\\; \\mathcal{F},\n\\\\\na_{16} &=& -\\frac12 a_{15},\n\\\\\nb_{8} &=& - 2\\sqrt{\\frac{B}{A}} J^2 \\mathcal{H},\n\\\\\nb_{9} &=& \\sqrt{\\frac{B}{A}} J^2 \\left(\\frac{A'}{A} - 2 \\frac{J'}{J} \\right) \\mathcal{H},\n\\\\\nb_{10} &=& -b_6,\n\\\\\nb_{11} &=& \\sqrt{\\frac{B}{A}} \\frac{A'}{A} \\mathcal{H},\n\\\\\nc_{8} &=& -\\frac{J^2}{\\sqrt{{A}{B}}} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4),\n\\\\\nc_{9} &=& \\frac14 \\sqrt{{A}{B}} \\left[\\Gamma \\pi' + \\left(\\frac{A'}{A} + 2 \\frac{J'}{J} \\right)\\mathcal{H} \\right],\n \\\\\nc_{10} &=& - \\frac14\\sqrt{{A}{B}} \\left[\n\\frac{J'}{J} \\left( \\Gamma + \\frac{J'}{J} \\Gamma_2\\right)\\pi' - \\frac14\\frac{A'}{A}\\left( \\Gamma_1 - 2 \\frac{J'}{J} \\Gamma_2\\right)\\pi'+\n\\right.\\\\\\nonumber&&\\left.\n\\mathcal{H} \\frac{J'}{J} \\left(\\frac{A'}{A} + 3 \\frac{J'}{J} \\right)+\n\\frac{1}{BJ^2} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4) \\right],\n\\\\\nc_{11} &=& - \n\\frac12 \\sqrt{{A}{B}} J^2 \\left[\\Gamma \\pi' + \\left(\\frac{A'}{A} + 2 \\frac{J'}{J} \\right)\\mathcal{H} \\right],\n\\\\\nc_{12} &=& - \\frac12 \\sqrt{{A}{B}} J^2 \\left[\n\\frac{1}{2 J^2}\\left(\\frac{A'}{A} + \\frac{J'}{J} \\right)\\Xi\\pi' +\n\\frac34 \\frac{J'}{J} \\left( \\Gamma_1 - 2 \\frac{J'}{J} \\Gamma_2\\right)\\pi' \n\\right.\\\\\\nonumber&&\\left.\n+ \\frac{J'}{J} \\left(\\frac{A'}{A} + \\frac{J'}{J} \\right) \\mathcal{H} +\n\\frac{1}{BJ^2} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4) \\right],\n\\end{eqnarray}\n\\begin{eqnarray}\nc_{13} &=& \\frac12 \\frac{\\sqrt{A}}{\\sqrt{B}} (\\mathcal{H} - 2 F_4 B^2 \\pi'^4),\n\\\\\nd_{5} &=& - \\frac{\\sqrt{{A}{B}}}{J^2} \\mathcal{H},\n\\\\\nd_{6} &=& 2\\frac{\\sqrt{{A}{B}}J'}{J^3} \\mathcal{H},\n\\\\\nd_{7} &=& a_4,\n\\\\\np_{8} &=& -\\frac{J^2}{2 \\sqrt{{A}{B}}} \\mathcal{F},\n\\\\\np_{9} &=& \\frac12 \\sqrt{{A}{B}} J^2 \\mathcal{H},\n\\end{eqnarray} \n\n\n\n\n\n\\section*{Appendix C}\nHere we put the rest of components of matrices $\\bar{\\mathcal{K}}$, $\\bar{\\mathcal{G}}$ and $\\bar{\\mathcal{M}}^{(\\ell^2)}$ entering the quadratic\naction~\\eqref{eq:action_KGQM_new} for completeness:\n\\begin{equation}\n\\bar{\\mathcal{K}}_{12} = \\bar{\\mathcal{K}}_{21} = \\frac{B^{1\/2}}{2(\\ell+2)(\\ell-1) A^{1\/2}} \\left[2(\\ell+2)(\\ell-1) (2\\mathcal{H} J J' + \\Xi \\pi') -\n\\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H} B J} \\frac{\\mathbf{\\Theta}}{\\mathcal{F}} \\right],\n\\end{equation}\n\\begin{equation}\n\\bar{\\mathcal{K}}_{22} = \\frac{4 \\ell(\\ell+1) A^{1\/2} B^{1\/2}}{\\mathbf{\\Theta}^2} \\left[\\ell(\\ell+1) \\frac{B}{A} (2 \\mathcal{P}_1 -\\mathcal{F}) (2\\mathcal{H} J J' + \\Xi \\pi')^2\n+ (\\ell+2)(\\ell-1) \\mathcal{F}\\; \\bar{\\mathcal{K}}_{12}^2 \\right],\n\\end{equation}\n\n\n\\begin{equation}\n\\bar{\\mathcal{G}}_{12} = \\bar{\\mathcal{G}}_{21} = AB\\frac{\\mathcal{G}}{\\mathcal{F}} \\bar{\\mathcal{K}}_{12}\n\\end{equation}\n\n\\begin{equation*}\n\\bar{\\mathcal{G}}_{22} = \\frac{4\\ell(\\ell+1) A^{1\/2} B^{1\/2}}{\\mathbf{\\Theta}^2} \\frac{\\mathcal{G}}{\\mathcal{H}} \\left[ \\ell(\\ell+1) B^2 (2 J^2 \\Gamma \\mathcal{H} \\Xi \\pi'^2 - \\mathcal{G} \\Xi^2 \\pi'^2-4 J^4 \\Sigma \\mathcal{H}^2\/B )\n+ \\frac{(\\ell+2)(\\ell-1)\\mathcal{F}^2}{AB \\;\\mathcal{G}} \\bar{\\mathcal{G}}_{12}^2\\right],\n\\end{equation*}\n\n\\begin{equation}\n\\bar{\\mathcal{M}}^{(\\ell^2)}_{12} = \\bar{\\mathcal{M}}^{(\\ell^2)}_{21} = - \\ell(\\ell+1) \\frac{ \\mathcal{H} (\\mathcal{H} - 2 F_4 B^2 \\pi'^2) } {2 \\mathcal{F} J} \\frac{A^{1\/2}}{B^{1\/2}} \\left[ 2 \\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H}^2 J} + \\frac{B(A'J - 2 A J')}{A} - \\frac{B}{A} \\mathcal{F} \\mathcal{P}_4\n \\right],\n\\end{equation}\n\n\\begin{multline}\n \\bar{\\mathcal{M}}^{(\\ell^2)}_{22} = -\\ell(\\ell+1) \\;\\frac{ \\mathcal{H}^2}{\\mathcal{F}} \\frac{A^{1\/2} B^{1\/2}}{J^2} \\left[ 2\\frac{ \\mathcal{F} \\mathcal{G} }{\\mathcal{H}^2} + \\frac{B^{1\/2}}{A^{1\/2}} J^4\\; \\mathcal{F} \n\\left[\\frac{B^{1\/2}}{A^{1\/2}J^3} \\mathcal{P}_4\\right]' +\\frac{1}{B} \\left(\\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H}^2 J}\\right)^2\n \\right] \\\\\n - \\frac{2}{(\\mathcal{H} - 2 F_4 B^2 \\pi'^2)} \\frac{\\left[\\mathcal{H}^2 B J^2 \\right]'}{\\mathcal{H} J^2} \\bar{\\mathcal{M}}_{12}^{(\\ell^2)}.\n\\end{multline}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCancers evolve through waves of mutation and clonal expansion~\\cite{Nowell1976}. Darwinian selection operates on the increased variation within the tumour, favouring clones with increased fitness, according to microenvironmental and therapeutic pressures~\\cite{Fearon1990, Stratton2009, Aparicio2013, Beerenwinkel2015}. As a consequence of this evolutionary process, tumours are generally genetically heterogeneous~\\cite{Gerlinger2012, Nik-Zainal2012a} and consist of related populations of cancer cells (\\emph{clones}) with distinct genotypes, which encode the evolutionary history of each cell population~\\cite{Nik-Zainal2012a}.\nThis genetic heterogeneity is important clinically because it can confound the molecular profiling of biopsies, and increased variation may equip tumours with more avenues to escape treatment, leading to worse prognosis~\\cite{Schwarz2015}. \n\n\\paragraph{The clonal deconvolution problem}\nIdentifying clones and their proportions is a difficult task~\\cite{Beerenwinkel2015}, aggravated by the fact that cancer genomics data generally come from bulk sequencing experiments, which profile a mixture of cells from different clones. \nClones are related to each other and can be thought of as nodes in a phylogenetic tree that describes tumour development. The root of the tree corresponds to a normal, non-mutated cell; every other node is a cancer clone with a distinct complement of mutations (its \\emph{genotype}). Each clone inherits the mutations of its parent and adds more to them. This encodes a subset relationship between parent and child nodes.\n\nHowever, none of this is directly observable. Instead, the data only consist of a set of mutations and their proportions (called \\emph{allele fractions}) in a collection of tumour samples (Figure~\\ref{fig:overview}).\nThe \\emph{clonal deconvolution problem} thus asks to identify the clonal genotypes, phylogeny, and clonal fractions that best explain the observed data~\\cite{ElKebir2015}.\n\n\\paragraph{Additional challenges}\n\nThe clonal deconvolution problem is further complicated by factors such as the selection of alleles during tumour evolution and the specifics of the data obtained from sequencing experiments. \nIn particular, convergent evolution and mutational loss contradict the common assumption that mutations arise only once in the phylogeny (the infinite sites assumption) and never disappear. Tumours are subjected to internal selective pressures in their microenvironment and external pressures from therapeutic interventions. In such cases, multiple tumour clones may acquire the same mutation in convergent evolution, especially if it is a hotspot mutation or it confers resistance to the treatment. \nAt the same time, mutations can be removed by several mechanisms, including loss of heterozygosity, the deletion of the chromosome fragment carrying the mutation.\nAnother challenge is that for cost-effective sequencing options like targeted amplicon sequencing, which we will use in the case studies, the depth of sequencing is not informative of the chromosomal copy-number of the tumour. This contradicts assumptions often made by previous methods.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{overview3.pdf}\n \\caption{Overview of the general clonal deconvolution problem. Sample A shows how from a normal root node, four clones evolved according to the displayed phylogeny. In this example, genotypes consist of two loci (blue and orange), which may be mutated (red star), gained or lost. The normal genotype consists of two non-mutated alleles for each locus. Clonal fractions are represented by the diameter of the node and reported as a percentage. A second sample B from the same patient would also consist of the same tree and genotypes; clonal fractions, however, may change (as in this example). These latent parameters give rise to the observed mutation and copy-number data, shown at the bottom. The allele fraction of a mutation is the proportion of that allele in the sample. The observed copy-number is the total copy number of each clone weighted by the clonal fractions. The increase of the orange mutation's allele fraction and the decrease of its observed copy-number are due to the growth of the clone with a single mutated copy.}\n \\label{fig:overview}\n\\end{figure}\n\n\\paragraph{Previous approaches}\n\nVarious methods have been proposed in the literature~\\cite{Beerenwinkel2015} to improve on manual analyses~\\cite{Gerlinger2012,Nik-Zainal2012a}.\nTo put our approach in context, it is useful to distinguish \\emph{direct} reconstructions that directly infer clonal genotypes (like CloneHD~\\cite{Fischer2014}, Clomial~\\cite{Zare2014} and BayClone~\\cite{Sengupta2015}) from \\emph{indirect} reconstructions that obtain clusters of mutations rather than full genotypes and require additional phylogenetic analysis to obtain clonal genotypes (like PyClone~\\cite{Roth2014}, SciClone~\\cite{Miller2014}, PhyloWGS~\\cite{Deshwar2015}, and BitPhylogeny~\\cite{Yuan2015}).\n\nDirect reconstructions generally aim to infer two quantities, a matrix of mutation assignments and a matrix of clonal fractions, which come together in an admixture to form the sampling model. The mutation assignments matrix associates each mutation with zero or more classes, which can be intuitively interpreted as clonal genotypes. For models that lack a phylogeny, inference may yield biologically implausible genotypes, as shown later in the benchmarking studies (section~\\ref{sec:benchmark}).\n\nOn the other hand, indirect methods cluster mutations based on their allele fractions across multiple samples. Joint phylogenetic modelling allows these clusters to become nodes of a tree, displaying at which node each mutation first appeared. Hence, the assignment of mutation clusters to nodes of a tree is generally inflexible to episodes of convergent evolution or mutational loss. \n\n\\paragraph{Latent feature models}\n\nHere we introduce Cloe, a phylogenetic latent feature model for clonal deconvolution that belongs to the category of direct reconstruction methods. \nLatent feature models discover independent features with which to describe a set of observed objects. The set of features possessed by an object determines the parameters of its distribution~\\cite{Griffiths2005}. In our context, observed objects are mutations, and latent features are clonal genotypes. \n\nLatent feature models have been previously applied to clonal deconvolutions, but maintained the assumption that features are independent~\\cite{Zare2014,Sengupta2015}.\nIn parallel, extensions to these models have been developed to relate features hierarchically, but placed features as the leaves of the tree~\\cite{Heaukulani2014}. Moreover, these tree structure only correlated the feature assignments, making such a model unsuitable for clonal deconvolutions. \n\nThe model we propose lifts the independence assumption and relates features with a latent hierarchy. In our framework features live at every node of the tree, thus encoding a noisy subset relationship in the mutation assignments. Our model differs from the phylogenetic Indian Buffet Process as the latter relates observed objects with a latent phylogeny, rather than the features~\\cite{Miller2012}.\nOur approach is more general than previously published methods, because it relies on fewer assumptions on clones and the evolutionary model: we can readily model multiple independent primary tumours, account for loss of mutations and penalise, though still allow, convergent evolution. \n\nWe validate Cloe{} on simulated data, on a controlled biological dataset, and apply it to two published clinical datasets: longitudinal samples from three chronic lymphocytic leukaemia patients~\\cite{Schuh2012}, and from an acute myeloid leukaemia case~\\cite{Griffith2015}. Cloe{} is available as an R package at \\url{https:\/\/bitbucket.org\/fm361\/cloe}.\n\n\n\\section{The Cloe{} model}\n\nOur model follows the overview of Figure~\\ref{fig:overview}. A latent phylogenetic tree influences the clonal genotypes; these, together with clonal fractions and additional nuisance parameters, describe the distribution of the data.\n\nWe observe data for \\( J \\) mutations in \\( T \\) samples, and the data are collected in two \\( J \\times T \\) matrices: \\textbf{X} for mutant read counts, and \\textbf{D} for read depths, the number of times a particular locus of the genome is covered by sequencing reads.\n\nThe phylogenetic tree is defined by a vector \\( \\mathcal{T} \\) with \\( K > 1 \\) elements, one for each clone. We consider the normal contamination as a fixed clone. Our analysis is restricted to mutations in copy-number neutral regions: each clone, including the normal, contributes exactly two copies of each allele, of which at most one can be mutated. Clonal genotypes are defined in a binary \\( J \\times K \\) matrix \\textbf{Z}, where each column \\( z_{\\cdot k} \\) represents the genotype of clone \\( k \\). The proportions of each clone in each sample are summarised in a \\( K \\times T \\) matrix \\textbf{F} formed by \\( T \\) stochastic vectors.\n\nOur goal is to infer the phylogeny~\\( \\mathcal{T} \\), clonal genotypes~\\textbf{Z}, and clonal fractions~\\textbf{F} from the posterior distribution \\( P(\\mathcal{T}, \\textbf{Z}, \\textbf{F} \\, \\vert \\, \\textbf{X}, \\textbf{D}) \\). To do this, in the following sections we develop a probabilistic model that links observed and unobserved variables, and an inference algorithm to explore the posterior distribution.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{graphical_model_cnn_2_alt.pdf}\n \\caption{An outline of the graphical model corresponding to Cloe, omitting for simplicity overlapping plates and convergent evolution relations. Legend: \\( \\gamma \\), clonal fractions hyperparameters; \\textbf{F}, clonal fractions matrix; \\( \\mathcal{T} \\), phylogeny; \\textbf{Z}, genotypes matrix; \\textbf{X}, mutant read counts; \\textbf{D}, depths; \\( s \\), Beta-binomial overdispersion parameter.}\n \\label{fig:graphical_model_outline}\n\\end{figure}\n\n\n\\subsection{Model definition}\nFor guidance, a simplified version of our model is outlined in Figure~\\ref{fig:graphical_model_outline}, while at the end of this section Figure~\\ref{fig:graphical_model_full} presents the complete model.\n\n\n\\paragraph{Phylogeny}\nFor \\( K > 1 \\) populations, we model the phylogenetic tree as a vector \\( \\mathcal{T} \\) of length \\( K \\), where \\( \\mathcal{T}_k = l \\) means that the parent of \\( k \\) is \\( l \\). The normal clone is fixed as the first entry, the root of the tree. To ensure that the graph encoded by \\( \\mathcal{T} \\) is a tree, we let each entry only take values on the previous entries. This definition is flexible as the tree can assume any shape, even allowing phylogenies with multiple primary tumours. \\( \\mathcal{T} \\) is defined by\n\\begin{align}\n \\begin{aligned}\n\\mathcal{T}_1 & = 0, \\\\ \n\\mathcal{T}_2 & = 1, \\\\\n\\mathcal{T}_k & \\sim \\mathcal{U}\\left(\\delta, k-1 \\right) \\quad\\text{for } k \\in \\{ 3 \\dots K \\},\n \\end{aligned}\n \\label{eq:tree}\n\\end{align}\nwhere \\( \\mathcal{U}(\\delta, a) \\) is a one-deflated discrete uniform distribution with values in~\\( \\{1, \\ldots, a \\}\\). The probability of drawing a 1 is \\( \\delta \\), and the probability of drawing an integer between \\( 2 \\) and \\( a \\) is uniform:\n\\begin{align}\n\\mathcal{U}(x; \\delta, a) =\n \\begin{cases}\n \\delta & \\text{ if } x = 1 \\\\\n \\frac{1 - \\delta}{a - 1} & \\text{ if } x \\in \\left\\{ 2, 3, \\dots, a \\right\\}\n \\end{cases}\n\\end{align}\nWe penalise multiple independent primary tumours (multiple children of the normal clone) by setting \\( \\delta = (2 k)^{-1} \\).\n\n\\paragraph{Genotypes}\nGenotypes are defined in a binary \\( J \\times K \\) matrix \\(\\text{\\textbf{Z}} = (z_{jk}) \\), where \\(1\\) denotes a mutation and \\(0\\) the un-mutated (\\emph{wildtype}) state, for each mutation \\( j \\) in each clone \\( k \\). \nWe fix the genotype of the normal clone to a zero vector of length \\( J \\), implying that all mutations are somatic. More generally, the normal genotype could be modified to accommodate for known germline variants. The genotype of a clone \\( k \\) for a mutation \\( j \\) is then defined as \n\\begin{equation}\nz_{jk} \\, \\vert \\, z_{j\\mathcal{T}_k}, \\mu, \\rho \\sim \\text{Bernoulli}(p_{jk})\n\\label{eq:genotypes}\n\\end{equation}\nwhere \\( \\mu \\) is the probability of mutating if the parent does not have a mutation, and\n\\( \\rho \\) is the probability of reverting to wildtype if the parent is mutated:\n\\begin{equation}\np_{jk} =\n\\begin{cases}\n\\,\\mu & \\text{ if } z_{j\\mathcal{T}_k} = 0 \\\\\n\\,1 - \\rho & \\text{ if } z_{j\\mathcal{T}_k} = 1.\n\\end{cases}\n\\label{eq:genotypes_probability}\n\\end{equation}\n\n\n\\paragraph{Clonal fractions}\nBecause the samples may be collected latitudinally, longitudinally, and at irregular intervals, we assume that clonal fractions are independent between samples. We represent clonal fractions with a \\( K \\times T \\) matrix \\( \\mathbf{F} = (\\mathbf{f}_{\\cdot t})_{t = 1, \\ldots, T} \\) composed of a collation of stochastic column vectors \\( \\mathbf{f}_{\\cdot t} \\) describing the proportions of each clone in a sample \\(t\\). Clonal fractions for a sample \\( t \\) are modelled with a symmetric Dirichlet distribution with hyperparameter \\( \\gamma_t \\):\n\\begin{equation}\n\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t \\sim \\text{Dirichlet}(\\gamma_t)\n\\end{equation}\n\n\\paragraph{Likelihood}\nGenotypes and clonal fractions come together in an admixture, their dot product representing the expected allele fractions for each mutation in each sample. \nWe model mutant reads as successful trials from a beta-binomial distribution with overdispersion parameter \\( s \\). The probability of success is a function of the expected allele fraction \\( p_{jt} = \\frac{1}{2} \\left( \\mathbf{z}_{j \\cdot} \\cdot \\mathbf{f}_{\\cdot t} \\right) \\). To capture sequencing noise at extreme values of \\( p_{jt} \\) we replace it with a function \\( e(p_{jt}) \\) that depends on the sequencing error rate \\( \\varepsilon \\) (e.g. 0.1\\%) such that\n\\begin{equation}\ne(p_{jt}) \\; =\n\\begin{cases}\n\\; \\varepsilon & \\text{ if } p_{jt} = 0 \\\\\n\\; 1 - \\varepsilon & \\text{ if } p_{jt} = 1 \\\\\n\\; p_{jt} & \\text{ otherwise.}\n\\end{cases}\n\\label{eq:seq_noise}\n\\end{equation}\n\nThe likelihood is then specified by\n\\begin{equation}\n\\label{eq:cnn_likelihood}\nx_{jt} \\, \\vert \\, d_{jt}, \\mathbf{z}_{j \\cdot}, \\mathbf{f}_{\\cdot t}, s \\;\\sim\\; \\text{Beta-binomial}(d_{jt}, e(p_{jt}), s).\n\\end{equation}\n\n\n\\paragraph{Nuisance parameters}\nWe let the beta-binomial overdispersion parameter \\( s \\) and the Dirichlet hyperparameters \\( \\bm{\\gamma} \\) have Gamma priors, whereas the mutation and reversion probabilities \\( \\mu \\) and \\( \\rho \\) are fixed.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{graphical_model_cnn_4.pdf}\n \\caption{The full graphical model of Cloe.}\n \\label{fig:graphical_model_full}\n\\end{figure}\n\n\n\n\\subsection{Penalising convergent evolution}\nOne of the risks of assuming the independence of features in this biological application is that the inferred genotypes may largely display convergent evolution. We can penalise such occurrences by altering the definition of genotypes (cfr. equations~\\ref{eq:genotypes} and \\ref{eq:genotypes_probability}).\n\nUnder the infinite sites assumption (ISA) every mutation occurs only once, so that if multiple clones possess a mutation \\( j \\), then the mutation must have appeared with their most recent common ancestor. In contrast, if the most recent common ancestor is not mutated, then the mutation must have appeared multiple times (convergent evolution).\nWe thus say that a mutation assignment \\( z_{jk} = 1 \\) conflicts with ISA if the most recent common ancestor of \\( \\left\\{ k' : z_{jk'} = 1, k' \\leq k \\right\\} \\) does not harbour mutation \\( j \\). \n\nWe include ISA checks into our model by using an indicator function \\( I(j, k, a) \\) that returns \\( 1 \\) if the most recent common ancestor of all clones \\( k' \\leq k \\) that harbour mutation \\( j \\) also possesses the mutation, when \\( z_{jk} = a \\).\nThat is, \\( I(j, k, a) = 1 \\) if ISA is satisfied by setting \\( z_{jk} = a \\), and 0 otherwise. \n\nThus we redefine the distribution of genotypes making them conditional on all previous genotypes and weighting assignments by a user-defined parameter \\( \\nu \\) if they comply with ISA, or by \\( 1 - \\nu \\) if they do not:\n\\begin{equation}\nP(z_{jk} = 1 \\, \\vert \\, \\mathcal{T}, \\mathbf{Z}_{j, < k}, z_{j\\mathcal{T}_k} = 0, \\mu, \\rho, \\nu) \\propto ( \\mu \\nu )^{I(j, k, 1)} ( \\mu ( 1 - \\nu ) )^{1 - I(j, k, 1)}\n\\label{eq:genotypes_isa_probability}\n\\end{equation}\nIn practice, only transitions that gain a mutation can clash with ISA, and the factor of \\( \\nu \\) immediately cancels out if the parental genotype is 1 at a given \\( j \\). An ISA-check at \\( j \\) is thus only warranted if the parent is not mutated at \\( j \\).\n\nThe graphical model corresponding to what has been described so far is shown in Figure~\\ref{fig:graphical_model_full}.\n\n\n\\begin{algorithm}\n \\begin{algorithmic}[1]\n \\For{\\( i \\) = 1, \\dots, \\#\\text{iterations}}\n \\For{\\( m \\) = 1, \\dots, \\#\\text{chains}}\n \\For{\\( j \\) = 1 \\dots \\( J \\)}\n \\Comment \\textbf{Z}\n \\State Propose new \\( \\mathbf{z}^{* (m)}_{j \\cdot} \\)\n \\State Accept with probability \\ref{eq:inference_z_mh_ratio}\n \\EndFor\n \\For{\\( k \\) = 3 \\dots \\( K \\)}\n \\Comment \\( \\mathcal{T} \\)\n \\State Compute \\( P(\\mathcal{T}^{*(m)}_k = l) \\) for \\( l \\in \\left\\{ 1 \\dots k - 1 \\right\\} \\) (eq.~\\ref{eq:inference_tree})\n \\State Sample new \\( \\mathcal{T}^{*(m)}_k \\) from \\( P(\\mathcal{T}^{*(m)}_k) \\)\n \\EndFor\n \\State Randomly swap two siblings\n \\State With probability 1\\% propose a swap between a node and its parent\n \\State Accept with probability~\\ref{eq:parent_swap}\n \\For{\\( t \\) = 1 \\dots \\( T \\)}\n \\Comment \\textbf{F}\n \\State Propose new \\( \\mathbf{f}^{*(m)}_{\\cdot t} \\) from eq.~\\ref{eq:inference_f_proposal}\n \\State Accept with probability \\ref{eq:inference_f_mh_ratio}\n \\EndFor\n \\Comment Nuisance parameters\n \\State Propose new \\( s^{*(m)} \\sim \\mathcal{N}(s^{(m)}, \\sigma_s) \\) and accept with probability~\\ref{eq:inference_s_mh_ratio}\n \\For{\\( t \\) = 1 \\dots \\( T \\)}\n \\State Propose new \\( \\gamma^{*(m)}_t \\sim \\mathcal{N}(\\gamma^{(m)}_t, \\sigma_\\gamma) \\) and accept with probability~\\ref{eq:inference_gamma_mh_ratio}\n \\EndFor\n \\EndFor\n \\If{\\( i \\) is a multiple of 100}\n \\Comment Chain swap\n \\State Propose a chain \\( j \\in \\{ 1 \\dots m - 1 \\} \\)\n \\State Accept the state swap between chains \\( j \\) and \\( j + 1 \\) with probability~\\ref{eq:inference_swap_mh_ratio}\n \\EndIf\n \\EndFor\n \\end{algorithmic}\n \\caption{MCMCMC sampling algorithm for Cloe}\n \\label{alg:inference}\n\\end{algorithm}\n\n\n\n\\subsection{Inference}\n\nWe are interested in the posterior distribution of the latent variables given the observed variables\n\\(P(\\mathcal{T}, \\mathbf{Z}, \\mathbf{F} \\, \\vert \\, \\mathbf{X}, \\mathbf{D})\\),\nwhich we explore by Metropolis-coupled Markov chain Monte Carlo (MCMCMC)~\\cite{Geyer1991}, integrating out the nuisance parameters \\( s\\) and \\( \\bm{\\gamma} \\).\nWe use Gibbs sampling to update the tree vector \\( \\mathcal{T} \\) and Metropolis-Hastings moves to update the other quantities.\n\nBecause the posterior landscape appears composed of high peaks separated by deep valleys (Supplementary Figure 1), we run five chains in parallel, with tempered posteriors. \nThe sampling strategy described hereafter is applied to each chain, and summarised in Algorithm~\\ref{alg:inference}.\n\n\\paragraph{Phylogeny}\nFor each \\( \\mathcal{T}_{k > 2} \\), we compute the conditional posterior of the parent assignment \\( \\mathcal{T}_k = l \\), for each \\( l < k \\):\n\\begin{equation}\nP( \\mathcal{T}_k = l \\, \\vert \\, \\dots) \\propto P(\\mathbf{Z} \\, \\vert \\, \\mathcal{T}_{-k}, \\mathcal{T}_k = l) \\,\nP(\\mathcal{T}_k = l)\n\\label{eq:inference_tree}\n\\end{equation}\nThe likelihood term amounts to tallying the genotype transitions from parent \\( l \\) to child \\( k \\), and reassessing how many transitions comply or clash with ISA for all clones. The prior, according to eq.~\\ref{eq:tree}, is equal to\n\\begin{equation}\n \\begin{aligned}\nP(\\mathcal{T}_k = l) & = \\delta^{\\bm{I}(l = 1)} \\left( \\frac{1 - \\delta}{k - 2} \\right)^{1 - \\bm{I}(l = 1)} \\\\\n& = \\left( \\frac{1}{2k} \\right)^{\\bm{I}(l = 1)} \\left( \\frac{2k - 1}{2k \\, (k - 2)} \\right)^{1 - \\bm{I}(l = 1)}\n \\end{aligned}\n \\label{eq:inference_tree_prior}\n\\end{equation}\n\nTo facilitate the exploration of the space of tree and genotypes configurations, we uniformly propose a pair of siblings and swap their position in the tree and in the genotypes and clonal fractions matrices. Prior to this swap, the siblings had access to one linear topology. This move allows the other linear topology to be explored, while leaving probabilities unaltered.\n\nIn addition, the swap between a node \\( k \\) and its parent \\( l \\) is proposed. A node \\( k \\) is chosen uniformly from \\( \\left\\{ 3 \\dots K \\right\\} \\). A tree \\( \\mathcal{T}^* \\) is created where \\( k \\) is the parent of \\( l \\), while any children of \\( k \\) remain children of \\( k \\); the same applies to \\( l \\). As with the sibling swap, this move requires rearranging the clone order in the genotypes and clonal fractions matrices. The parent swap affects genotype transitions from \\( \\mathcal{T}_l \\), the original parent of \\( l \\), to \\( k \\), and from \\( k \\) to \\( l \\). The proposal is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{Z}_{\\{ \\mathcal{T}_l, k, l \\}} \\, \\vert \\, \\mathcal{T}^*, \\mu, \\rho, \\nu) \\, P(\\mathcal{T}^*_{\\{ \\mathcal{T}_l, k, l \\}})}{P(\\mathbf{Z}_{\\{ \\mathcal{T}_l, k, l \\}} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu) \\, P(\\mathcal{T}_{\\{ \\mathcal{T}_l, k, l \\}})} \\right)\n\\label{eq:parent_swap}\n\\end{equation}\nWe perform this update with probability 0.01.\n\n\\paragraph{Genotypes}\nBecause mutations are independent, we update \\textbf{Z} by row, proposing a new row \\( \\mathbf{z}^*_{j \\cdot} \\) by flipping each bit of \\( \\mathbf{z}_{j \\cdot} \\) with probability \\( \\theta \\). The proposal is symmetric and the move is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{x}_{j \\cdot} \\, \\vert \\, \\mathbf{d}_{j \\cdot}, \\mathbf{z}^*_{j \\cdot}, \\mathbf{F}, s) \\, P(\\mathbf{z}^*_{j \\cdot} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu)}{P(\\mathbf{x}_{j \\cdot} \\, \\vert \\, \\mathbf{d}_{j \\cdot}, \\mathbf{z}_{j \\cdot}, \\mathbf{F}, s) \\, P(\\mathbf{z}_{j \\cdot} \\, \\vert \\, \\mathcal{T}, \\mu, \\rho, \\nu)} \\right)\n\\label{eq:inference_z_mh_ratio}\n\\end{equation}\nwhere the likelihood is only computed for mutation \\( j \\), and the prior refers to the sequence of transitions from the root genotype at \\( j \\) to the leaves, with appropriate penalties for convergent evolution.\n\n\n\\paragraph{Clonal fractions}\nBecause of the independence of the samples, the matrix \\textbf{F} is updated by column. A new vector \\( \\mathbf{f}^*_{\\cdot t} \\) is proposed from a Dirichlet distribution centred at the current value \\( \\mathbf{f}_{\\cdot t} \\):\n\\begin{equation}\nQ(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\mathbf{f}_{\\cdot t}) = \\text{Dirichlet}(\\psi \\, \\mathbf{f}_{\\cdot t} + \\epsilon)\n\\label{eq:inference_f_proposal}\n\\end{equation}\nwhere \\( \\psi \\) is a precision factor and \\( \\epsilon \\) a small bias to avoid sinks at 0. The proposal is accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{x}_{\\cdot t} \\, \\vert \\, \\mathbf{d}_{\\cdot t}, \\mathbf{Z}, \\mathbf{f}^*_{\\cdot t}, s) \\ P(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ Q(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\mathbf{f}^*_{\\cdot t})}{P(\\mathbf{x}_{\\cdot t} \\, \\vert \\, \\mathbf{d}_{\\cdot t}, \\mathbf{Z}, \\mathbf{f}_{\\cdot t}, s) \\ P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ Q(\\mathbf{f}^*_{\\cdot t} \\, \\vert \\, \\mathbf{f}_{\\cdot t})} \\right)\n\\label{eq:inference_f_mh_ratio}\n\\end{equation}\n\n\\paragraph{Nuisance parameters}\n\nThe remaining parameters are updated with Metropolis moves using Gaussian proposals. The Metropolis-Hastings acceptance ratios are\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{X} \\, \\vert \\, \\mathbf{D}, \\mathbf{Z}, \\mathbf{F}, s^*) \\ P(s^*)}{P(\\mathbf{X} \\, \\vert \\, \\mathbf{D}, \\mathbf{Z}, \\mathbf{F}, s) \\ P(s)} \\right) \\quad\\text{for}~s,\n\\label{eq:inference_s_mh_ratio}\n\\end{equation}\nand\n\\begin{equation}\n\\min\\left( 1, \\frac{P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma^*_t) \\ P(\\gamma^*_t)}{P(\\mathbf{f}_{\\cdot t} \\, \\vert \\, \\gamma_t) \\ P(\\gamma_t)} \\right) \\quad\\text{for}~\\gamma_t.\n\\label{eq:inference_gamma_mh_ratio}\n\\end{equation}\n\n\n\\paragraph{Temperatures and chain swaps}\n\nRegularly at user-defined intervals, a swap between two adjacent chains is proposed as a Metropolis-Hastings move~\\cite{Geyer1991}. Let \\( M \\) denote the number of parallel chains, \\( P^{(m)} \\) denote the tempered posterior of chain \\( m \\), and \\( \\omega_m \\) denote the state of chain \\( m \\). A chain \\( m \\) is selected among the first \\( M - 1 \\) chains. The swap between chains \\( m \\) and \\( m + 1 \\) is then accepted with probability\n\\begin{equation}\n\\min\\left( 1, \\frac{P^{(m)}(\\omega_{m+1}) \\ P^{(m+1)}(\\omega_m)}{P^{(m)}(\\omega_{m}) \\ P^{(m+1)}(\\omega_{m+1})} \\right).\n\\label{eq:inference_swap_mh_ratio}\n\\end{equation}\n\nThe temperature \\( \\tau_m \\) for each chain \\( m \\) is chosen according to the following scheme:\n\\begin{equation}\n\\tau_m = (1 + \\Delta T (m - 1))^{-1}\n\\end{equation}\nwhere \\( \\Delta T > 0 \\) regulates the temperature differences between chains.\n\n\n\\paragraph{Parameter estimates}\n\nMCMCMC parameter estimates are derived solely from the first, untempered chain. After discarding a certain proportion of the initial samples as burn-in, and thinning the chain by a factor of \\( i \\), thus considering every \\( i^{th} \\) sample, we obtain a maximum \\emph{a posteriori} (MAP) estimate of the parameters by selecting the chain state of the sample with the highest posterior value.\n\n\n\n\n\n\\section{Validation and benchmarks}\n\nWe extensively validated and benchmarked Cloe{} by using simulated data and a controlled experimental set-up based on mixing cell lines. \n\n\\subsection{Simulated data}\n\nWe first tested our model on 9 simulated datasets, one for each combination of number of clones (3, 4 or 5) and depth of sequencing (means: 50\\( \\times \\), 200\\( \\times \\), 1000\\( \\times \\)).\nThe genotypes were created according to a random tree and using parameters \\( \\mu = 0.5 \\), \\( \\rho = 0.05 \\), \\( \\nu = 0.9 \\); clonal fractions were \\emph{iid} draws from a symmetric Dirichlet distribution with parameter \\( \\gamma = 2 \\). All datasets contained 100 mutations and 5 samples, with depths obtained from a Poisson distribution and mutant read counts obtained from a binomial distribution. Because of the fixed size of our model, we ran Cloe{} for 3, 4, and 5 clones on each of the 9 datasets (running parameters are reported in Table~\\ref{tab:params}).\n\nWe measured Cloe's performance in several ways: we evaluated its ability to identify the right number of clones, we assessed mixing by computing the Gelman-Rubin statistic from three consecutive runs of the algorithm, and finally we measured the reconstruction error. To perform the latter step, we calculated two metrics, the normalised genotypes error \\( Z_{err} \\) and the normalised clonal fractions error \\( F_{err} \\), both defined as the sum of the absolute differences between inferred (\\( \\mathbf{Z}^* \\)) and the real (\\( \\mathbf{Z} \\)) matrices, normalised by the real matrix dimensions (ignoring the fixed genotype in \\textbf{Z}).\nTo control for equivalent solutions with permuted clones, we used permutations \\(\\sigma\\) of the columns of \\( \\mathbf{Z}^* \\) to minimise \\( Z_{err} \\): \n\\begin{equation}\nZ_{err} = \\frac{1}{J (K - 1)} \\, \\min_{\\sigma} \\left(\\sum_{j,k} \\vert z^*_{j\\sigma(k)} - z_{jk} \\vert \\right),\n\\end{equation}\n The same permutation is then used to rearrange the rows of \\( \\mathbf{F}^* \\), which is normalised by \\( J K \\). If the real and inferred matrices have different sizes, we pad the smaller one with normal clones with zero clonal fractions. In this instance \\( K \\) refers to the real number of clones. \n\n\\paragraph{MCMC mixing and effective sample size}\nThe Gelman--Rubin statistic was calculated from the log-posterior values of the untempered chains as a proxy for the multidimensional parameters. In every case, the potential scale reduction factor is within the accepted range, less than 1.1 (Figure~\\ref{fig:simulation_psrf}). Each run was started with a different random seed.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{simulations_psrf.pdf}\n \\caption{The Gelman-Rubin statistic computed from three runs of our algorithm on the simulated datasets. Row names consist of the number of clones that was used in the run, and the characteristics of the dataset (sequencing depth and the real number of clones).}\n \\label{fig:simulation_psrf}\n\\end{figure}\n\nWe computed the effective sample size (ESS) for the first run on each dataset, focussing on the cases where the sought number of clones matched the real number of clones of the dataset. Table~\\ref{tab:ess} reports the ESS of the nuisance parameters and the log-posterior computed from 10,000 post-thinning iterations. Modulating the standard deviation of the Gaussian proposals for the nuisance parameters could decrease their autocorrelation. However, the shape of the posterior space~(Supplementary Figure 1) may prevent efficient large moves.\n\n\\begin{table}\n \\centering\n \\begin{tabular}{lrrrrrrr}\n \\hline\nDataset & \\( \\gamma_1 \\) & \\( \\gamma_2 \\) & \\( \\gamma_3 \\) & \\( \\gamma_4 \\) & \\( \\gamma_5 \\) & \\( s \\) & LP \\\\\n \\hline\n50x, 3 & 194.63 & 272.20 & 474.59 & 117.33 & 134.96 & 1696.31 & 4366.40 \\\\\n200x, 3 & 178.25 & 352.60 & 505.93 & 171.06 & 151.80 & 890.28 & 3088.45 \\\\\n1000x, 3 & 143.95 & 442.33 & 446.48 & 220.48 & 153.46 & 426.54 & 1685.87 \\\\\n50x, 4 & 648.35 & 241.09 & 586.84 & 170.57 & 342.17 & 1345.90 & 2323.20 \\\\\n200x, 4 & 735.31 & 195.13 & 683.51 & 152.63 & 346.05 & 1025.92 & 1700.03 \\\\\n1000x, 4 & 1083.17 & 302.75 & 654.13 & 140.00 & 237.40 & 406.45 & 894.08 \\\\\n50x, 5 & 395.45 & 239.42 & 209.06 & 261.33 & 369.81 & 1182.60 & 1673.17 \\\\\n200x, 5 & 803.71 & 387.01 & 394.54 & 324.07 & 644.47 & 947.42 & 2401.41 \\\\\n1000x, 5 & 990.07 & 401.38 & 373.32 & 284.74 & 657.88 & 478.26 & 1805.82 \\\\\n \\hline\n \\end{tabular}\n \\caption{Effective sample size per dataset. The effective sample size was computed on the 10,000 post-thinning iterations for the first of the three replicate runs with the correct number of clones. The dataset is denoted by the average sequencing depth and the real number of clones. LP denotes the log-posterior, as a proxy for the multidimensional parameters.}\n \\label{tab:ess}\n\\end{table}\n\n\n\\paragraph{Model selection}\nAs a model selection criterion we used the log-posterior probability of the MAP sample. We were able to recover the correct size \\( K \\) for every dataset (Figure~\\ref{fig:simulation_results}, left) in two of the three runs. In one run, a higher log-posterior probability was given to the solution with 4 clones on the dataset with 50\\( \\times \\) and 5 clones (-14464.05 compared to the 5-clone solution's -14464.58; Supplementary Figure 2). In such cases, where the log-posterior probabilities of different models are approximately equal, we prefer the solution with higher log-likelihood value.\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.52]{simulations_results_3.pdf}\n }\n \\caption{Results on the simulated datasets. Left: inferred model size for every combinations of \\( K \\) and depths. Centre and right: reconstruction errors. All datasets consisted of five samples and 100 mutations.}\n \\label{fig:simulation_results}\n\\end{figure}\n\n\n\\paragraph{Reconstruction fidelity}\nTo assess our reconstructions we considered only the MAP solution for each dataset. The reconstruction error was low, with \\( Z_{err} \\leq 0.027 \\) and \\( F_{err} \\leq 0.033 \\). The largest errors were obtained at the lowest depth (Figure~\\ref{fig:simulation_results}, centre and right), suggesting that on these random datasets Cloe{} can not only discover the correct number of clones,\nbut also infer correct genotypes and clonal fractions with \\( > 96.7\\% \\) accuracy (Supplementary Figures 3 and 4).\n\n\\paragraph{Conclusion}\nThis validation with synthetic data provided proof that the model and the inference are sound, achieving good reconstructions even with several clones and low sequencing depth (Supplementary Figure 5).\n\n\\begin{table*}\n \\begin{tabular}{lr}\n\\hline\n\\textbf{Parameter} & \\textbf{Value} \\\\\n\\hline\n\\multicolumn{2}{c}{MCMCMC} \\\\\n\\hline\niterations & 40000 \\\\\nchains & 5 \\\\\n\\( \\Delta T \\) & 0.4 \\\\\nswap interval & 50 \\\\\nburn-in & 50\\% \\\\\nthinning factor & 4 \\\\\n\\hline\n\\multicolumn{2}{c}{\\textbf{Z}} \\\\\n\\hline\n\\( \\mu \\) & 0.3 \\\\\n\\( \\rho \\) & 0.1 \\\\\n\\( \\nu \\) & 0.75 \\\\\n\\( \\theta \\) (proposal) & 0.20 \\\\\n\\( \\varepsilon \\) (likelihood) & 0.005, 0.002 \\\\\n\\hline\n\\multicolumn{2}{c}{\\textbf{F}} \\\\\n\\hline\n\\( \\psi \\) (proposal) & 200 \\\\\n\\( \\epsilon \\) (proposal) & 4 \\\\\n\\hline\n\\multicolumn{2}{c}{Nuisance parameters} \\\\\n\\hline\n\\( \\gamma \\) (prior, shape) & 2 \\\\\n\\( \\gamma \\) (prior, rate) & 1 \\\\\n\\( \\sigma_\\gamma \\) (proposal) & 0.2 \\\\\n\\( s \\) (prior, shape) & 11 \\\\\n\\( s \\) (prior, rate) & 0.10 \\\\\n\\( \\sigma_s \\) (proposal) & 16 \\\\\n\\hline\n \\end{tabular}\n \\caption{Running parameters for the simulated and validation datasets. For \\( \\varepsilon \\), the sequencing error parameter, the first value refers to the simulations, the second to the validation dataset.}\n \\label{tab:params}\n\\end{table*}\n\n\n\n\n\\subsection{Controlled experimental data}\n\nBecause synthetic data may not capture the variability seen in real biological data, we tested our method on a bespoke experiment.\nIn order to mimic heterogeneous tumour samples, we genotyped five cell lines and mixed them together at known proportions. Four single-cell-diluted cancer cell lines (HD124, HD212, HD249, and HD659, from Horizon Discovery, UK) were mixed with a normal cell line (HG00131, 1000 Genomes Project, Coriell Institute, USA). All five cell lines were subjected to whole-exome sequencing (Nextera Rapid Capture Exome, Illumina, USA).\nMutational and single nucleotide polymorphism (SNP) profiles were obtained using the standard samtools workflow \\cite{samtools}. To identify copy-number neutral regions, we generated copy-number profiles with the R package CopywriteR (version 2.2.0). We created 14 mixtures, with a median tumour cellularity of 64\\% (Table~\\ref{tab:mixtures}).\n\nBecause the cell lines were unrelated, we selected a subset of mutations (heterozygous single nucleotide variants and small indels) so as to embed the cell lines into an artificial phylogeny (Figure~\\ref{fig:data_cnn}, left). We focussed on regions that were copy-number neutral across all cell lines, and measured allele fractions by targeted sequencing (Figure~\\ref{fig:data_cnn}, right) \\cite{Forshew2012}. We excluded from the final dataset mutations whose genotypes had been erroneously inferred from exome sequencing data. The final dataset included 82 mutations, of which one displayed convergent evolution and two were reversions to wildtype, and was sequenced to a median of 17260\\( \\times \\) coverage.\nTargeted sequencing data were processed with an \\emph{in house} pipeline and the known mutations were quantified from pileups with a base quality cutoff of 30. \n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.5]{data.pdf}\n }\n \\caption{The observed data for the validation dataset. On the left is the artificial phylogeny, where N denotes the normal clone, C1 to C4 are the cancer cell lines, playing in this context the role of clones. On the right are the observed mutational dynamics (allele fractions over samples) at an average depth of 17260\\( \\times \\).}\n \\label{fig:data_cnn}\n\\end{figure}\n\n\\begin{table*}\n \\centering\n \\makebox[\\textwidth][c]{\n \\begin{tabular}{rrrrrrrrrrrrrrr}\n\\hline\n\\ & M1 & M2 & M3 & M4 & M5 & M6 & M7 & M8 & M9 & M10 & M11 & M12 & M13 & M14 \\\\\n\\hline\nN & 0.26 & 0.34 & 0.22 & 0.13 & 0.37 & 0.38 & 0.72 & 0.26 & 0.13 & 0.45 & 0.65 & 0.38 & 0.53 & 0.00 \\\\\nC1 & 0.03 & 0.41 & 0.14 & 0.29 & 0.14 & 0.14 & 0.04 & 0.38 & 0.03 & 0.02 & 0.06 & 0.09 & 0.19 & 0.08 \\\\\nC2 & 0.18 & 0.00 & 0.19 & 0.25 & 0.06 & 0.04 & 0.11 & 0.05 & 0.32 & 0.18 & 0.03 & 0.28 & 0.00 & 0.30 \\\\\nC3 & 0.44 & 0.19 & 0.00 & 0.17 & 0.10 & 0.41 & 0.13 & 0.18 & 0.43 & 0.23 & 0.20 & 0.25 & 0.22 & 0.57 \\\\\nC4 & 0.10 & 0.05 & 0.44 & 0.15 & 0.33 & 0.05 & 0.00 & 0.14 & 0.10 & 0.12 & 0.07 & 0.00 & 0.06 & 0.04 \\\\\n\\hline\n \\end{tabular}\n }\n \\caption{Clonal fractions in the 14 mixtures of the validation experiment.}\n \\label{tab:mixtures}\n\\end{table*}\n\nThe large number of samples and the high depth of sequencing that we obtained afforded a sensitivity analysis, in which we varied the number of samples and the depth. Cloe{} was run on these datasets with the same parameters as for the synthetic data (Table~\\ref{tab:params}).\n\n\\paragraph{Model selection}\n\nWe ran Cloe{} for \\( K \\in \\left\\{ 3, 4, 5, 6 \\right\\} \\) and performed model selection based on the log-posterior values of the MAP estimates. In every case we were able to identify the correct number of clones (Figure~\\ref{fig:validation_k}), suggesting that either a moderate depth of sequencing or multiple samples should suffice in obtaining good estimates of the number of clones.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.75]{validation_model_selection_lp.pdf}\n \\caption{Inferred model size \\( K \\) for every combination of samples and depths in the validation dataset. The validation data consist of a mixture of five clones. Only the top solution was considered in each case.}\n \\label{fig:validation_k}\n\\end{figure}\n\n\n\\paragraph{Reconstruction fidelity}\n\nOverall, we obtained precise reconstructions for almost all depth-samples combinations. Considering for each combination only the first solution suggested by Cloe, on average 1\\% of mutation assignments were inaccurate (\\( Z_{err} \\) median 0, mean 0.013), and clonal fractions were inferred with an average error lower than 2\\% (\\( F_{err} \\) median 0.017, mean 0.019). As expected, we observed a pattern of decreasing errors as the data increase in the number of samples or in depth (Figure~\\ref{fig:validation_err5}).\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.6]{validation_norm_zf_error_k5.pdf}\n }\n \\caption{Reconstruction errors on validation data obtained running Cloe{} with \\( K = 5 \\) clones. The heatmaps show the genotypes error (left) and clonal fractions error (right) for various combinations of depth and samples.}\n \\label{fig:validation_err5}\n\\end{figure}\n\n\\paragraph{Specific low-depth cases}\nPoorer reconstructions were obtained at lower depths (\\( \\leq 60\\times \\)) for the datasets with three samples. In every case, the inferred genotypes showed a faulty separation between two expected genotypes (Supplementary Figure 6), which led to high error metrics: \\( Z_{err} \\leq 0.134 \\) and \\( F_{err} \\leq 0.065 \\). Despite the imprecise reconstruction, there is an overall good agreement with the observed data (Supplementary Figure 6 (e)).\n\nThese results could be improved by tuning the running parameters of Cloe{} for these datasets. Because the height of the posterior peaks at these levels of depth is lower than at high depth, using less tempered chains may result in higher acceptance of chain swaps, and, consequently, in a more complete exploration of the posterior space. Increasing the number of MCMCMC iterations could also prove beneficial.\n\nIt should be also noted that at low depths sampling noise may promote suboptimal parameter combinations to near-optimal. In this case, more mutations should be analysed in order to average sampling noise effects, though this may place a heavy burden on our implementation's runtime. Alternatively, one could model more data in terms of samples. If the clonal fractions are dynamic enough, meaning that most clones grow and shrink at some point in the samples, more opportunities are provided to separate clonal signals.\n\n\n\\subsection{Comparison to previous approaches}\n\\label{sec:benchmark}\n\nTo further benchmark Cloe, we compared the results of three published methods compatible with targeted sequencing on our validation dataset: BayClone \\cite{Sengupta2015} and Clomial \\cite{Zare2014}, two latent feature models, and PyClone \\cite{Roth2014}, a non-parametric model. Other methods, like PhyloWGS~\\cite{Deshwar2015} or CloneHD~\\cite{Fischer2014}, are not applicable to targeted sequencing.\n\nWe ran two tests on the validation dataset described in the last section, first using all samples and the entire depth, and then 3 samples and a depth of 100\\( \\times \\). Our method's performance on the first dataset is a perfect reconstruction of the genotypes (\\(Z_{err} = 0 \\)) and a near-perfect reconstruction of the clonal fractions (\\(F_{err} = 0.005 \\)), with a correct identification of the number of clones. With less data, there are three misassignment (\\( Z_{err} = 0.009 \\)) and the error of the clonal fractions is 0.017 (Figure~\\ref{fig:method_comparison}); again, the number of clones is correctly inferred.\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.56]{method_comparison_zf_err_160404.pdf}\n \\caption{Comparison of Cloe{} against three published methods on our validation dataset, consisting of targeted sequencing for 82 mutations in 14 samples (average depth 17260\\( \\times \\)); five clones are present in the data. \\( Z_{err} \\) denotes the reconstruction error on genotypes; \\( F_{err} \\) the error on clonal fractions. The two datasets (17260\\( \\times \\) depth and 14 samples, and 100\\( \\times \\) and 3 samples) are denoted ``high'' and ``low'', respectively. The legend refers to the number of inferred clones.}\n \\label{fig:method_comparison}\n\\end{figure}\n\nBayClone \\cite{bayclone} was run with default parameters for 45000 iterations, discarding the first 5000 and thinning the chain by a factor of 4. We tested the same model sizes as for our own method, namely 3, 4, 5, and 6. Through the log-pseudo-marginal likelihood, BayClone was able to identify \\( K = 5 \\) as the best solution on the first dataset. However, the reconstruction of the genotypes was less precise (Figure~\\ref{fig:method_comparison_z}), with \\( Z_{err} = 0.25 \\) and \\( F_{err} = 0.103 \\). Here \\( Z_{err} \\), the normalised absolute difference between inferred and real genotypes matrices, ignores the ploidy of the mutation. The reconstruction was poorer on the second dataset because of the less precise data: six clones were inferred with \\( Z_{err} = 0.287 \\), and \\( F_{err} = 0.147 \\) (Supplementary Figure 7).\n\nClomial (version 1.6.0) implements an EM algorithm, and it was run with default parameters (1000 restarts, and 100 maximum EM iterations) using model sizes of 3, 4, 5, and 6. On the first dataset, model selection with BIC (and AIC) indicated \\( K = 5 \\) as the best solution, with one misassignment (\\( Z_{err} = 0.003 \\)) and an accurate reconstruction of the clonal fractions (\\( F_{err} = 0.006 \\)). With less data, model selection was unclear as AIC, BIC and the log-likelihoods were all discordant. Using the correct model size Clomial obtained a \\( Z_{err} = 0.076 \\) and \\( F_{err} = 0.044 \\).\n\nPyClone (version 0.12.9) was run for 30000 iterations with a beta-binomial density and copy-number neutral states allowing a single mutant allele out of two (AB mode). PyClone was also provided with estimates of cellularity for each of the samples. We removed the first 3000 iterations as burn-in samples and thinned the chain by a factor of 4. The output of PyClone consists of a clustering of the observed mutations, \nwhere each cluster should roughly correspond to one of the non-root nodes of Figure~\\ref{fig:data_cnn}. Phylogenetic modelling can translate these clusters into genotypes.\nOn the full dataset, PyClone produced three clusters, as shown in Figure~\\ref{fig:method_comparison_z}. Because the cluster of stem mutations was merged with one of its two children, we were unable to interpret the results phylogenetically. Hence, we could not derive genotypes nor clonal fractions. The estimate of \\( K \\) is 4 with \\( Z_{err} = 0.241 \\). On less data, two clusters were produced, leading to an estimate of \\( K \\) of 3, with \\( Z_{err} = 0.357 \\).\n\nIn summary, our benchmark shows that Cloe{} compares favourably against similar published methods (Figure~\\ref{fig:method_comparison_z}). It is expected that the accuracy of the reconstruction would be affected by the quality of the data. Indeed every model performed more poorly on less data, however Cloe{} seemed to be affected to a lesser extent (Supplementary Figure 7).\n\n\\begin{figure}\n \\centering\n \\makebox[\\textwidth][c]{\n \\includegraphics[scale=0.56]{method_comparison_z_D17260_T14_160404.pdf}\n }\n \\caption{Comparison of the genotypes inferred by the four benchmarked methods using all the data in our validation dataset. PyClone's reconstruction is a clustering of the mutations. In this representation of the genotypes we padded solutions with the normal clone (C1) for a more direct comparison with our method. The legend refers to the proportion of mutated alleles out of two.\n }\n \\label{fig:method_comparison_z}\n\\end{figure}\n\n\n\n\n\\section{Case studies}\n\nWe show the applicability of Cloe{} to clinical data in two case studies.\n\n\n\\subsection{Chronic lymphocytic leukaemia}\nThis dataset consists of five time points for each of three chronic lymphocytic leukaemia patients \\cite{Schuh2012}. The original study identified mutations by whole-genome sequencing (WGS; average depth across the mutation loci 39\\( \\times \\)) and quantified a subset of these with deep targeted sequencing (TAR; average depth 101600\\( \\times \\)). \n\nThe authors' analysis reported evolutionary trees and clonal fractions for each of the three cases. We used this information to run Cloe{} with known clonal fractions on all mutations, prioritising information from the higher depth datasets. We interpreted these results as ground truth mutation assignments for all three patients, and scored our reconstructions to these reference parameters.\n\nWe ran Cloe{} on the reported mutations with \\( K \\in \\left\\{ 3 \\dots 7 \\right\\} \\) for each case and each experiment (WGS, low-depth, and TAR, high-depth), comparing our results with the original study, with PhyloSub's results \\cite{Jiao2014}, and with CloneHD's reconstruction of case CLL003 \\cite{Fischer2014}. We also included another dataset, which consisted of the WGS dataset with data from the higher depth TAR dataset for mutations in common.\n\nCase CLL003 displays a radical clonal shift (Supplementary Figure 8 (a) and (b)): the main clone in the early time points is replaced by a distinct new clone that appears only at the second time point and expands to become the predominant clone.\nUsing targeted sequencing data, Cloe{} obtained a very accurate reconstruction, identifying the correct number of clones, obtaining a single misassignment and average errors on clonal fractions of 1\\% (Figure~\\ref{fig:cll_error}, Supplementary Figure 9). On less data, our method opted for a solution with 4 clones that ignored the founding clone, only present in the first of five samples at a clonal fraction of 3\\%. Choosing the top solution with 5 clones recovered the correct clonal structure; on WGS data there were five incorrect mutation assignments (Supplementary Figure 10), whereas with the combined dataset only one (Supplementary Figure 11). Barring the rare founding clone, the 4-clone reconstructions are correct with one (combined data) and two (WGS data) misassignments.\n\nThe remaining cases showed more stable dynamics (Supplementary Figure 8 (c)--(f)). For CLL006, Cloe{} assigned the nine mutations of the TAR dataset to six clones without errors; three errors were observed with 18 mutations in the WGS dataset (Figure~\\ref{fig:cll_error}). Analysis of the combination of the two data types yielded an additional clone, though similar log-posterior probabilities and a higher log-likelihood were obtained by a six-clone solution. Removing clone C5 from the seven-clone solution yields a correct reconstruction (Supplementary Figure 12).\n\nFinally, for CLL077, Cloe's analysis resulted in a perfect reconstruction of the genotypes with targeted sequencing data. Two misassignments were obtained for the combined dataset, whereas four of the five clones were identified in the WGS data: the founding clone, with only four of the 20 mutations, was merged with one of its children. After the four-clone solutions, solutions with six-clones had high log-posterior probabilities. Indeed the first of these solutions is an accurate reconstruction with two misassignments and one clone repeated twice almost identically. In the middle, solutions with the expected number of clones, five, had six errors (Supplementary Figure 13).\n\n\\begin{figure}\n \\centering\n \\includegraphics[scale=0.75]{CLL_error_combined.pdf}\n \\caption{Performance metrics of Cloe{} on the CLL datasets. The correct number of clones for cases CLL003 and CLL077 is 5, whereas for CLL006 it is 6. TAR stands for targeted sequencing (average depth 101600\\( \\times \\)); WGS stands for whole-genome sequencing (average depth 39\\( \\times \\); BOTH is the WGS dataset with TAR data for shared mutations. The legend refers to the number of inferred clones. When the first solution inferred the wrong number of clones, the top solution for the correct number of clones is also shown.}\n \\label{fig:cll_error}\n\\end{figure}\n\nOverall, Cloe{} produced accurate reconstructions of the latent parameters. Higher errors were observed when an incorrect number of clones was inferred. However, even in these cases, our phylogenetic model allowed us to obtain close approximations of the ground truth. \n\nAssuming that our reconstruction of the ground truth is correct, Cloe's inference results in a better reconstruction than reported by CloneHD \\cite{Fischer2014}, both using high-depth and low-depth data: first, because of our phylogenetic modelling, we were able to identify the founding clone; second, we could confidently identify the rising clone's parent (the ambiguous green clone in \\cite{Fischer2014}). With low-depth data, Cloe{} did prefer a model with four clones, but could also provide a more accurate five-clone solution.\n\nOur results on targeted sequencing data largely agree with those obtained by PhyloSub~\\cite{Jiao2014}, with two small exceptions. For CLL003, Cloe{} predicts that clone 4 (clone \\( c \\) in \\cite{Jiao2014}, Figure 7, right) does not harbour the \\textit{IL11RA} mutation. This episode appears to be supported by the data (Supplementary Figure 14) as Cloe's reconstruction leads to closer fit with the data (sum of absolute errors on the allele fractions is 0.06 for Cloe, 0.13 for PhyloSub, for this mutation). Rather than a loss of mutation, this could be due to convergent evolution at the leaf nodes, leading to a sum of absolute errors of 0.07. For case CLL006, our reconstruction agrees with that of \\cite{Schuh2012}: five tumour clones are detected, and the \\textit{EGFR} mutation is predicted to stem from the founding clone. PhyloSub preferred to place the EGFR mutation in an additional clone after the founder, leading to a closer fit: the sum of absolute errors was 0.02 compared to Cloe's 0.07 for this mutation.\n\n\n\n\n\\subsection{Acute myeloid leukaemia}\n\nAML31 refers to a patient with acute myeloid leukaemia, whose case was studied in great depth with several sequencing experiments targeting bulk DNA at various scales, RNA and also single cells \\cite{Griffith2015}. As each layer of data refined the authors' understanding of the evolution of this tumour, seven clusters and driver mutations were identified. Integration of all sequencing data revealed over 1300 mutations curated in a ``platinum list''. The tumour genomes appeared to be devoid of copy-number aberrations. \n\nWe considered a subset of platinum-list mutations for three datasets: ALLDNA (median depth of 1841\\( \\times \\) for the primary tumour sample, 388\\( \\times \\) for the relapse), TORRENT (median depths 41\\( \\times \\) and 46.5\\( \\times \\)), and WGS (median depths 323\\( \\times \\) and 41\\( \\times \\)). For each dataset, we selected a random subset of 250 mutations, halving the number of mutant reads for hemizygous mutations, and adding reported driver mutations.\n\nCloe{} was run with \\( K \\in \\left\\{ 3, \\dots, 7 \\right\\} \\) on the datasets. Model selection on ALLDNA indicated \\( K = 5 \\) as the preferred solution, followed closely by \\( K = 6 \\) which provided a closer fit to the data (Supplementary figure 15). The inferred mutation dynamics for both models are shown in Figure~\\ref{fig:alldna_fit}. Whereas both model sizes could capture the trends in the data, the solution for \\( K = 6 \\) correctly identified two groups of mutations that rise in allele fraction in the relapse sample. \n\nOur reconstruction shows a decrease in tumour burden at relapse, a single origin for all clones, and branched evolution after the founding clone (Figure~\\ref{fig:alldna_params}). Clone 5 and its child, clone 6, become the main clones in the relapse sample, supplanting clones 3 and 4. The founding clone appears present only at very low clonal fractions.\n\nMatching our clones to the original clusters, we found a close correspondence (Table~\\ref{tab:alldna_match}), corroborating Cloe's inference. The only misassignment is \\textit{TP53} to clone 6, which in the original study required single-cell sequencing and additional time points to identify as belonging to a separate clone. Beyond the genotypes, there was also a close match between inferred and expected clonal fractions, with a maximum absolute difference of 4\\%.\n\nCluster 6 was not identified by our model. According to the original analysis, this cluster was present at less than 5.5\\% clonal fraction in the primary sample, to then disappear at relapse. Such a cluster would contribute half of its clonal fraction in allele fraction, due to the heterozygosity of the mutations. We do observe that nine of the 257 mutations were not assigned to any clone. Their average allele fraction was 2.2\\% in the primary sample and close to 0\\% in the relapse (Supplementary Figure 16). Because they did not fit the dynamics of the other clones, sequencing noise was used to fit them (eq.~\\ref{eq:seq_noise}). \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{ALLDNA_fit.pdf}\n \\caption{Observed and inferred mutation dynamics for 257 mutations from the ALLDNA dataset. Left: observed allele fractions; centre: allele fractions inferred by Cloe{} with 5 clones; right: allele fractions inferred by Cloe{} with 6 clones.}\n \\label{fig:alldna_fit}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{ALLDNA_params6.pdf}\n \\caption{Parameters inferred by Cloe{} running with 6 clones on 257 mutations from the ALLDNA dataset. Genotypes are shown on the left, where green denotes presence of a mutation; clonal fractions for each clone are shown on the right. C1 is fixed as the normal contamination.}\n \\label{fig:alldna_params}\n\\end{figure}\n\n\\begin{table}\n \\centering\n \\begin{tabular}{ccl}\n\\hline\nClone & Cluster & Drivers \\\\\n\\hline\nC2 & 1 & \\textit{DNMT3A} \\\\\nC4 & 4 & \\textit{FOXP1} \\\\\nC5 & 3 & \\textit{IDH2} \\\\\nC3 & 2 & \\textit{IDH1} \\\\\nC6 & 5 & \\textit{CXCL17}, \\textit{TP53} \\\\\n\\hline\n \\end{tabular}\n \\caption{Correspondence between Cloe's inferred clones, and the clusters in the original analysis by \\cite{Griffith2015}. While drivers are also present in the children of a clone, here we report the clone in which the mutations first appeared.}\n \\label{tab:alldna_match}\n\\end{table}\n\nOn the modest amount of data of the TORRENT dataset our model selection produced a more conservative estimate of the number of clones, preferring four clones. Using five or more clones improved the log-likelihood to the same extent. We compare here solutions for \\( K = 4 \\) and \\( K = 5 \\) (Figure~\\ref{fig:torrent_fit}).\n\nWith three tumour clones, our model matched the main trends: two large clones in the primary sample that disappear at relapse, and one growing clone. In addition, tumour content was accurately inferred: 89\\% for the primary sample and 44\\% for the relapse sample, compared to the expected values of 91\\% and 47\\%. The addition of a fourth tumour clone (\\( K = 5 \\)) allows a better disambiguation of the clones present in the primary, while the spread of allele fractions in the relapse sample makes it difficult to distinguish two rising clone.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth]{TORRENT_fit.pdf}\n \\caption{Observed and inferred mutation dynamics for 254 mutations from the TORRENT dataset. Left: observed allele fractions; centre: allele fractions inferred by Cloe{} with 4 clones; right: allele fractions inferred by Cloe{} with 5 clones.}\n \\label{fig:torrent_fit}\n\\end{figure}\n\nIdentifying seven clones including the normal in two samples with a median depth less than 45\\( \\times \\) is an arduous task. \\cite{Griffith2015} show that SciClone detects four clones up to around 100\\( \\times \\) depth using all mutations on the platinum list. While Cloe{} prefers four clones using a subset of mutations at a depth of 45\\( \\times \\), it is capable of splitting the observed dynamics further, obtaining closer approximations of the real clonal structure.\n\nFinally, for the WGS dataset, Cloe's solution with 5 clones obtained the highest posterior probability, while 6 and 7 clones obtained closer fits to the data (Supplementary Figure 17). With four tumour clones, Cloe{} identified three decreasing groups of mutations and one group that arose at relapse. This matches the observed dynamics, as the low depth at relapse accounts for a larger spread of the allele fractions that confounds the identification of two rising clones (Supplementary figure 18). Interestingly, the addition of another clone, rather than fitting this low-depth relapse data, matches a fourth group of mutations present only in the primary around 5\\% clonal fraction. These mutations overlap with the unassigned mutations in the ALLDNA dataset and the inferred clone does not harbour additional driver mutations other than \\textit{DNMT3A}, which derives from its parent.\n\nWith this case study we applied Cloe{} to a scenario with two samples, highlighting the difficulties of automatic model selection, especially when trying to identify a large number of clones with a moderate amount of data.\n\nThe running parameters for the two case studies differ from the ones listed in Table~\\ref{tab:params} in that we used five chains with \\( \\Delta T = 0.25 \\). In addition, for the AML datasets we ran 50000 iterations of our sampler with \\( \\mu = 0.2 \\), \\( \\rho = 0.04 \\), and \\( \\varepsilon = 0.001 \\).\n\n\n\n\n\\section{Discussion}\n\n\nAs tumour sequencing data grow in depth and breadth, the question of tumour heterogeneity will continue to be focal. In this study we presented Cloe, a novel latent feature model for direct clonal reconstruction. Our model discovers genotypes in the data by assigning observed mutations to latent features (clones) guided by a latent phylogeny. This phylogenetic deconvolution sets Cloe{} apart from other direct reconstruction methods \\cite{Fischer2014, Zare2014, Sengupta2015}. Compared to indirect reconstruction methods, our algorithm can handle multiple primary tumours, the loss of mutations and convergent evolution. In particular, to our knowledge this is the first method to allow and penalise convergent evolution.\n\nOur study on simulated data showed a good performance of our MCMCMC algorithm. However, tuning the MCMCMC parameters in order to correctly explore the spiked posterior landscape is not trivial. We empirically found parameters that would allow the chains to mix well. Regions of high posterior probability are quickly reached, yet finding the right peak is a slow process, complicated by each biological constraint on the model.\nMany parameters can be tuned in our model. We sought values that would work well for both simulated data and our validation data. Tuning the MCMCMC parameters to each dataset independently, thus optimising the exploration of the posterior space, might further improve results.\n\nIn our definition of the tree we assume that multiple primary tumours are less likely to occur than tumours with a single origin. If our understanding of clonal evolution were to suggest otherwise, the definition of the tree may be simplified to a discrete uniform distribution, giving equal weight to a single origin or multiple ones.\n\n\n\\paragraph{Limitations}\nThe main limitation of our method is the restriction to mutations from copy-number neutral regions. Whereas this may be amenable to certain types of cancer (e.g. mutation-driven rather than copy-number driven cancers), it may preclude the analysis of more genomically rearranged tumours.\n\nIn contrast to some models described in the literature, our method does not include the number of clones as a parameter. Instead, Cloe{} must be run for various choices of \\( K \\), and the best solution in terms of posterior probability will indicate the number of clones with good accuracy. On our simulation and validation datasets our model was indeed able to identify the correct number of clones in 58\/59 cases.\n\nAs shown in the case studies, model selection may not be trivial. We thus recommend manual review of the inferred parameters for various model sizes to ensure that the results of the inference are robust.\n\nAnalysing hundreds of mutations can result in a high computational burden. This limitation could be alleviated by preprocessing the input data, grouping mutations that exhibit similar dynamics throughout the samples. One way to do this is via a Chinese Restaurant Process with a product of binomials; mutant read counts and depths for all mutations in a cluster could then be summed and analysed as a single unit.\n\n\\paragraph{Extensions}\nWe see several avenues for future extensions.\nAt the theoretical level, future work should focus on optimising the inference and extending this framework to arbitrary copy-numbers. Also, to address the model selection problem, the phylogenetic latent feature model could be rephrased in a non-parametric perspective.\nIn terms of applications, our model could also be applied to epigenetics: by appropriately changing the likelihood function, Cloe{} could deconvolute methylation data into evolutionarily related epigenotypes.\n\nIn summary, Cloe{} is a rigorous and flexible framework for clonal deconvolution of cancer genomes that achieves high accuracy in benchmarking studies and leads to important insights into tumour evolution in clinical case studies. \n\n\n\n\n\n\\section*{Acknowledgements}\n\nWe would like to acknowledge the support of The University of Cambridge, Cancer Research UK and Hutchison Whampoa Limited. \nThis work was funded by CRUK core grant C14303\/A17197, in particular A19274 (Markowetz lab core grant).\nWe wish to thank Marta Grzelak and James Hadfield for their assistance with sequencing, and Malvina Josephidou for discussions on the CRP clustering of mutations.\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\label{intro} Power-laws are intrinsic features of the economic\nand financial data. There are numerous studies of power-law\nprobability distributions in various economic systems\n\\cite{1,2,3,4,41,42,8,5}. The key result in recent findings is\nthat the cumulative distributions of returns and trading activity\ncan be well described by a power-law asymptotic behavior,\ncharacterized by an exponent $\\lambda \\approx 3$, well outside the\nLevy stable regime $0<\\lambda <2$ \\cite {5}.\n\nThe time-correlations in the financial time series are studied\nextensively as well \\cite{5,6,7}. Gopikrishnan \\emph{et al} \\cite\n{5,6} provided empirical evidence that the long-range correlations\nfor volatility were due to the trading activity, measured by a\nnumber of transactions $N$.\n\n\nRecently we adapted the model of $1\/f$ noise based on the Brownian\nmotion of time interval between subsequent pulses, proposed in\n\\cite{9,10,11,12}, to model the share volume traded in the\nfinancial markets \\cite{13}. The idea to transfer long time\ncorrelations into the stochastic process of the time interval\nbetween trades or time series of trading activity is in\nconsistence with the detailed studies of the empirical financial\ndata \\cite{5,6} and fruitfully reproduces the spectral properties\nof the financial time series \\cite{13,131}. Further, we\ngeneralized the model defining stochastic multiplicative point\nprocess to reproduce a variety of self-affine time series\nexhibiting the power spectral density $S(f)$ scaling as a power of\nfrequency, $S(f) \\propto f^{-\\beta}$ \\cite{132}.\n\nIn this contribution we analyze the applicability of the\nstochastic multiplicative point process as a model of trading\nactivity in the financial markets. We investigate the spectral density\nand counting statistics of model trading activity in comparison\nwith empirical data from the stock exchange. The model reproduces the\nspectral properties of trading activity and explains the mechanism\nof power law distribution in the real markets.\n\n\\section{The model}\n\nWe consider a point process $I(t)$ as a sequence of the $\\delta$-type\nrandom correlated pulses,\n\\begin{equation}\n\\label{eq:1}I(t)=\\sum\\limits_ka_k\\delta (t-t_k),\n\\end{equation}\nand define the number of trades $N_j$ in the time intervals\n$\\tau_d$ as an integral of the signal, $N_j=\\int\\limits_{t_j}^{t_j+\\tau\n_d}I(t)dt$. Here $a_k$ is a contribution of one transaction.\nWhen $a_k=1$, the signal (\\ref{eq:1}) counts the\ntransactions in the financial market. When $a_k$ describes asset price\nchange during one transaction, the signal counts the price changes.\nWhen $a_k=\\bar a$ is a constant, the point process is\ncompletely described by the set of times of the events $\\{t_k\\}$\nor equivalently by the set of interevent intervals $\\{\\tau\n_k=t_{k+1}-t_k\\}$. Various stochastic models for $\\tau _k$ can be\nintroduced to define a stochastic point process. In papers\n\\cite{9,10,11,12} it has been shown analytically that the\nrelatively slow Brownian fluctuations of the interevent time $\\tau\n_k$ yield $1\/f$ fluctuations of the signal (\\ref{eq:1}). In\n\\cite{132} we have generalized the model introducing stochastic\nmultiplicative process for the interevent time $\\tau _k$,\n\\begin{equation}\n\\label{eq:5}\\tau _{k+1}=\\tau _k+\\gamma \\tau _k^{2\\mu -1}+\\tau\n_k^\\mu \\sigma \\varepsilon _k.\n\\end{equation}\nHere the interevent time $\\tau _k$ fluctuates due to the external\nrandom perturbation by a sequence of uncorrelated normally\ndistributed random variable $\\{\\varepsilon _k\\}$ with zero\nexpectation and unit variance, $\\sigma $ denotes the standard\ndeviation of the white noise and $\\gamma \\ll 1$ is a damping\nconstant. From the big variety of possible stochastic processes we\nhave chosen the multiplicative one, which yields multifractal\nintermittency and power-law probability distribution functions.\nPure multiplicativity corresponds to $\\mu=1$. Other values\nof $\\mu$ may be considered, as well.\n\nThe iterative relation (\\ref {eq:5}) can be rewritten as Langevine\nstochastic differential equation in $k$-space\n\\begin{equation}\n\\label{eq:6}\\frac{d\\tau _k}{dk}=\\gamma \\tau _k^{2\\mu -1}+\\tau\n_k^\\mu \\sigma \\xi \\left( k\\right).\n\\end{equation}\nHere we interpret $k$ as continuous variable while $\\left\\langle\n\\xi \\left( k\\right) \\xi \\left( k^{\\prime }\\right) \\right\\rangle\n=\\delta (k-k^{\\prime })$.\n\nThe steady state solution of the stationary Fokker-Planck\nequation with zero flow, corresponding to (\\ref {eq:6}), gives the\nprobability density function for $\\tau _k$ in the $k$-space (see,\ne.g., \\cite{19})\n\\begin{equation}\n\\label{eq:7}P_k(\\tau\n_k)=C\\tau_k^\\alpha=\\frac{\\alpha+1}{\\tau_{max}^{(\\alpha+1)}-\\tau_{min}^{(\\alpha+1)}}\\tau\n_k^\\alpha ,\\quad \\alpha =2\\gamma \/\\sigma ^2-2\\mu.\n\\end{equation}\n\nThe steady state solution (\\ref{eq:7}) assumes Ito convention\ninvolved in the relation between expressions (\\ref{eq:5}),\n(\\ref{eq:6}) and (\\ref{eq:7}) and the restriction for the diffusion\n$0<\\tau_{\\min }<\\tau_k<\\tau _{\\max }$.\n\nWe have already derived the formula for the power spectral density of the\nmultiplicative stochastic point process model, defined by\nEqs.~(\\ref {eq:5}) and (\\ref{eq:6}) for the interevent time\n\\cite{132},\n\\begin{equation}\nS_{\\mu }(f)=\\frac{2C\\overline{a}^{2}}{\\sqrt{\\pi }\\overline{\\tau\n}(3-2\\mu )f}\\left(\\frac{\\gamma }{\\pi f}\\right)^{\\frac{\\alpha\n}{3-2\\mu }}{\\mathop{\\mathrm{Re}}}\\int\\limits_{x_{\\min }}^{x_{\\max\n}}\\exp \\left\\{-i\\bigg(x-\\frac \\pi\n4\\bigg)\\right\\}{\\mathop{\\mathrm{erfc}}} (\\sqrt{-ix})x^{\\frac\\alpha\n{3-2\\mu }-\\frac 12}dx \\label{eq:12}\n\\end{equation}\nwhere $\\bar \\tau =\\left\\langle \\tau _k\\right\\rangle =T\/(k_{\\max\n}-k_{\\min })$ is the expectation of $\\tau _k$. Here we introduce\nthe scaled variable $x=\\pi f \\tau ^{3-2\\mu }\/\\gamma$ and\n$x_{\\min }=\\pi f \\tau _{\\min }^{3-2\\mu }\/\\gamma,\\quad x_{\\max }=\n\\pi f \\tau _{\\max }^{3-2\\mu }\/\\gamma$.\n\nExpression (\\ref{eq:12}) is appropriate for the numerical calculations\nof the power spectral density of the generalized\nmultiplicative point process defined by Eqs. (\\ref{eq:1}) and\n(\\ref{eq:5}). In the limit $\\tau _{\\min }\\rightarrow 0$ and $\\tau\n_{\\max }\\rightarrow \\infty $ we obtain an explicit expression\n\\begin{equation}\n\\label{eq:14}S_\\mu (f)=\\frac{C\\overline{a}^2}{\\sqrt{\\pi\n}\\overline{\\tau } (3-2\\mu )f}\\left(\\frac \\gamma {\\pi\nf}\\right)^{\\frac \\alpha {3-2\\mu }}\\frac{\\Gamma (\\frac 12+\\frac\n\\alpha {3-2\\mu })}{\\cos (\\frac{\\pi \\alpha }{2(3-2\\mu )})}.\n\\end{equation}\n\nEquation (\\ref{eq:14}) reveals that the multiplicative point\nprocess (\\ref{eq:5}) results in the power spectral density\n$S(f)\\sim 1\/{f^\\beta }$ with the scaling exponent\n\\begin{equation}\n\\label{eq:15}\\beta =1+\\frac{2\\gamma \/\\sigma ^2-2\\mu }{3-2\\mu }.\n\\end{equation}\n\nLet us compare our analytical results (\\ref{eq:12}) and\n(\\ref{eq:14}) with the numerical calculations of the power\nspectral density according to equations (\\ref{eq:1}) and\n(\\ref{eq:5}). In Fig.~\\ref{fig:1} we present the numerically\ncalculated power spectral density $S(f)$ of the signal $I(t)$ for\n$\\mu =0.5$ and $\\alpha =2\\gamma\/\\sigma ^2-1=0$, -0.5 and +0.5.\nNumerical results confirm that the multiplicative point process\nexhibits the power spectral density scaled as $S(f)\\sim 1\/f^\\beta\n$. Equation (\\ref{eq:12}) describes the model power spectral\ndensity very well in a wide range of parameters. The explicit\nformula (\\ref{eq:14}) gives a good approximation of power spectral\ndensity for the parameters when $\\beta \\simeq 1$. These results\nconfirm the earlier finding \\cite {9,10,11,12} that the power\nspectral density is related to the probability distribution of the\ninterevent time $\\tau _k$ and $1\/f$ noise occurs when this\ndistribution is flat, i.e., when $\\alpha =0$.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig1.eps}\n\\end{center}\n\\caption{Power spectral density $S(f)$ vs frequency $f$ calculated\nnumerically according to Eqs. (\\ref{eq:1}) and (\\ref{eq:5}) with the parameters\n$\\mu=0.5$, $\\sigma=0.02$ and different relaxations of the signal\n$\\gamma$. We restrict the diffusion of the interevent time in the\ninterval $\\tau _{min}=10^{-6}$; $\\tau _{max}=1$ with the\nreflective boundary condition at $\\tau _{min}$ and transition to\nthe white noise, $\\tau _{k+1}=\\tau _{max}+\\sigma\\varepsilon_k$,\nfor $\\tau_k>\\tau _{max}$. The straight lines represent the results given\nby the explicit formula (\\ref{eq:14}).}\n\\label{fig:1}\n\\end{figure}\n\n\nIt is likely that such a stochastic model with parameters in the\nregion $0.5\\leq \\beta \\leq 1.5$ may be adaptable for a wide\nvariety of different systems. In this paper we will discuss\napplicability of the model for the financial market.\n\nWe derived pdf of $N$ for the pure multiplicative model with $\\mu\n=1$ in \\cite{132}\n\n\\begin{equation}\nP(N)=\\frac{C'\\tau_{d}^{2+\\alpha}(1+\\gamma N)}{N^{3+\\alpha}\n(1+\\frac{\\gamma}{2}N)^{3+\\alpha}}\\sim\\left\\{\n\\begin{array}{ll}\n\\frac{1}{N^{3+\\alpha}},& N\\ll \\gamma^{-1}, \\\\\n\\frac{1}{N^{5+2\\alpha}},& N\\gg \\gamma^{-1}.\n\\end{array}\n\\right.\n\\label{eq:18}\n\\end{equation}\n\n\nProbability distribution function for $N$ obtained from the\nnumerical simulation of the model is in a good agreement with the\nanalytical result (\\ref{eq:18}).\n\n\\section{Discussion and conclusions}\n\nWe have introduced a multiplicative stochastic model for the time\nintervals between events of point process. Such a model of time\nseries has only a few parameters defining the statistical\nproperties of the system, i.e., the power-law behavior of the\ndistribution function and the scaled power spectral density of the\nsignal. The ability of the model to simulate $1\/f$ noise as well\nas to reproduce signals with the values of power spectral density\nslope $\\beta$ between 0.5 and 1.5 promises a wide variety of\napplications of the model.\n\n\\begin{figure}[tbp]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]{Fig2.eps}\n\\end{center}\n\\caption{Power spectral density $S(f)$ vs frequency $f$ calculated\nnumerically from the empirical data of 3 most liquid stocks from the\nLithuanian Stock Exchange. Straight line fits $S(f)\\propto f^{-0.7}$.}\n\\label{fig:2}\n\\end{figure}\n\nLet us present shortly the possible interpretations of the\nempirical data of the trading activity in the financial markets.\nWith a very natural assumption of transactions in the financial\nmarkets as point events we can model the number of transactions\n$N_j$ in equal time intervals $\\tau _d$ as the outcome of the\ndescribed multiplicative point process. We already know from\navailable studies \\cite{5} that the empirical data exhibit power\nspectral density in the low frequency limit with the slope $\\beta\n\\simeq 0.7$. Empirical data from the Lithuanian Stock Exchange for\nthe most liquid assets confirm the same value $\\beta \\simeq 0.7$,\n(see Fig~\\ref{fig:2}). For the pure multiplicative model with\n$\\mu =1$ this results in $\\alpha =2\\gamma \/\\sigma ^2-2\\mu\n\\simeq -0.3$. The corresponding cumulative distribution of $N$ in\nthe tail of high values (see equation (\\ref{eq:18})) has the\nexponent $\\lambda=4+2\\alpha =3.4$. This is in an excellent\nagreement with the empirical cumulative distribution exponent 3.4\ndefined in \\cite{5} for 1000 stocks of the three major US stock\nmarkets.\n\nThe numerical results confirm that the multiplicative stochastic\nmodel of the time interval between trades in the financial market\nis able to reproduce the main statistical properties of trading\nactivity $N$ and its power spectral density. The power-law\nexponents of the pdf of the interevent time, $\\alpha $, and the\ncumulative distribution of the trading activity, $\\lambda $, as\nwell as the slope of power spectral density, $\\beta$, are defined\njust by one parameter of the model $2\\gamma \/\\sigma ^2$. The model\nsuggests a simple mechanism of the power-law statistics of trading\nactivity in the financial markets.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}