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+{"text":"\\section{Introduction}\nIn his seminal paper~\\cite{Foellmer_1981},\nF{\\\"o}llmer presented a new perspective on\nIt{\\^o}'s stochastic calculus.\nThe main theorem of F{\\\"o}llmer~\\cite{Foellmer_1981}\nstates that a deterministic c\\`{a}dl\\`{a}g path satisfies the It{\\^o} formula\nprovided it has quadratic variation\nalong a given sequence of partitions. \nThis theorem enables us to construct the It{\\^o} integral \n$\\int_0^t f(X_{s-}) \\mathrm{d}X_s$ for\na sufficiently nice function $f$ and a path $X$ of quadratic variation.\nThis suggests the possibility \nof developing an analogue of the It{\\^o} calculus\nin completely analytic, probability-free situations.\nWe call this framework F{\\\"o}llmer's pathwise It{\\^o} calculus\nor, more simply, the It{\\^o}-F{\\\"o}llmer calculus.\nIt can be regarded as a deterministic counterpart of the classical It{\\^o} calculus.\n\nRecently, the It{\\^o}-F{\\\"o}llmer calculus has been receiving increasing attention \nfrom the viewpoint of its financial applications.\nIt is regarded as a useful tool to study financial theory under probability-free settings and has been used to construct financial strategies in a strictly pathwise manner \n(see, e.g., F{\\\"o}llmer and Schied~\\cite{Foellmer_Schied_2013},\nSchied~\\cite{Schied_2014}, \nDavis, Ob{\\l}{\\'o}j and Raval~\\cite{Davis_Obloj_Raval_2014},\nand Schied, Speiser, and Voloshchenko~\\cite{Schied_Speiser_Voloshchenko_2018}).\nWe expect that the It{\\^o}-F{\\\"o}llmer calculus will\nhave a growing presence\nin financial applications.\n\nThe It{\\^o}-F{\\\"o}llmer calculus can be applied to\na stochastic process having quadratic variation.\nA standard example of such a process is a semimartingale.\nHowever, it is known that \nthe class of processes possessing quadratic variation \nis strictly larger than that of semimartingales\n(see, e.g., F{\\\"o}llmer~\\cite{Foellmer_1981,Foellmer_1981b}).\nIn this sense, the It{\\^o}-F{\\\"o}llmer calculus\nenables us to extend stochastic integration theory beyond semimartingales.\n\nThere are several approaches to pathwise construction of stochastic integration.\nFirst, we mention classic studies by\nBichteler~\\cite{Bichteler_1981},\nKarandikar~\\cite{Karandikar_1995},\nand Willinger and Taqqu~\\cite{Willinger_Taqqu_1988,Willinger_Taqqu_1989}, but\nsee also Nutz~\\cite{Nutz_2012}.\nThe theory of Vovk's outer measure and typical\npaths was pioneered by Vovk~\\cite{Vovk_2008,Vovk_2011,Vovk_2012,Vovk_2015,Vovk_2016},\nand further developed by several authors, including \nPerkowski and Pr{\\\"o}mel~\\cite{Perkowski_Proemel_2016,Perkowski_Proemel_2015b},\n{\\L}ochowski~\\cite{Lochowski_2016},\n{\\L}ochowski, Perkowski, and Pr{\\\"o}mel~\\cite{Lochowski_Perkowski_Proemel_2018},\nand Bartl, Kupper, and Neufeld~\\cite{Bartl_Kupper_Neufeld_2019}.\nRusso and Vallois~\\cite{Russo_Vallois_1991,\n    Russo_Vallois_1993,\n    Russo_Vallois_1993b,\n    Russo_Vallois_1995,\n    Russo_Vallois_1996,\n    Russo_Vallois_2000,\n    Russo_Vallois_2007}\ndeveloped a theory called stochastic calculus via regularization.\nThe rough path theory,\npioneered by Lyons~\\cite{Lyons_1998},\nand its generalization have become \nimportant in stochastic calculus and its applications.\nIn addition, we refer to\nGubinelli~\\cite{Gubinelli_2004},\nGubinelli and Tindel~\\cite{Gubinelli_Tindel_2010},\nFriz and Shekhar~\\cite{Friz_Shekhar_2017}, \nand Friz and Zhang~\\cite{Friz_Zhang_2018}.\nSome studies have investigated the relation between the It{\\^o}-F{\\\"o}llmer\ncalculus and rough path theory (see, e.g.,\nPerkowski and Pr{\\\"o}mel~\\cite{Perkowski_Proemel_2016},\nand Friz and Hairer~\\cite{Friz_Hairer_2014}).\n\nAmong the various pathwise methods,\nwe consider F{\\\"o}llmer's approach to be\nsimplest and most intuitively clear.\nIt needs only elementary arguments to establish \ncalculation rules such as It{\\^o}'s formula \nwithin this framework.\nMoreover, the It{\\^o}-F{\\\"o}llmer calculus requires only a minimal \nassumption that the integrator has a quadratic variation.\nWe believe that these are advantages of this theory,\nand also that careful observation of F{\\\"o}llmer's theory\nallows us to know phenomena occurring on paths of processes\nwhen we consider semimartingales and stochastic integration.\n\nIncreasingly many works related to It{\\^o}-F{\\\"o}llmer calculus\nhave been appearing recently.\nFirst, we refer to Sondermann~\\cite{Sondermann_2006},\nSchied~\\cite{Schied_2014}, Hirai~\\cite{Hirai_2019}, and\nCont and Perkowski~\\cite{Cont_Perkowski_2019}.\nSchied~\\cite{Schied_2016},\nand Mishura and Schied~\\cite{Mishura_Schied_2016}\nconstruct deterministic continuous\npaths with nontrivial quadratic variation.\nSee also Chiu and Cont~\\cite{Chiu_Cont_2018}.\nFunctional extensions of the It{\\^o}-F{\\\"o}llmer calculus \nhave been developed by Dupire~\\cite{Dupire_2009}, \nCont and Fourni{\\'e}~\\cite{Cont_Fournie_2010,\nCont_Fournie_2010b,\nCont_Fournie_2013}, and \nAnanova and Cont~\\cite{Ananova_Cont_2017}, for example.\nExtension of the It{\\^o}-F{\\\"o}llmer calculus\nin terms of local times has been investigated in\nDavis, Ob{\\l}{\\'o}j, and Raval~\\cite{Davis_Obloj_Raval_2014},\nDavis, Ob{\\l}{\\'o}j, and Siorpaes~\\cite{Davis_Obloj_Siorpaes_2018},\n{\\L}ochowski et al.~\\cite{Lochowski_Obloj_Promel_Siorpaes_2021},\nand Hirai~\\cite{Hirai_2016,Hirai_2021}.\n\nTo our knowledge, however,\nthe It{\\^o}-F{\\\"o}llmer calculus in an infinite-dimensional setting\nhas not yet been sufficiently studied.\nStochastic integration in infinite dimensions naturally appears\nwhen we treat stochastic partial differential equations\n(see, e.g. Da Prato and Zabczyk~\\cite{DaPrato_Zabczyk_2014}).\nThese have played an important role\nin modelling term structures of interest rates or forward variances in mathematical finance,\nand also in models of statistical mechanics and quantum field theories.\nThen we aim to extend F{\\\"o}llmer's theory \nto Banach space-valued paths.\nIn this paper, we prove the It{\\^o} formula for \npaths in Banach spaces\nwith suitably defined quadratic variation.\nWe will study relations between various quadratic variations\nand prove some transformation formulae for quadratic variations \nin our second paper in this series~\\cite{Hirai_2021c}.\nWe not only generalize the state space of paths,\nbut also relax the assumption on the sequence of\npartitions along which we consider the quadratic variation.\nIn the context of the It{\\^o}-F{\\\"o}llmer calculus,\ntwo types of assumptions about a sequence of partitions are \nfrequently used.\nOne is $\\lvert \\pi_n \\rvert \\to 0$, as used in F{\\\"o}llmer~\\cite{Foellmer_1981},\nand the other is the condition $O_t^{-}(X;\\pi_n) \\to 0$,\nwhich is used in many papers handling continuous paths \nand some dealing with discontinuous paths\nsuch as Vovk~\\cite{Vovk_2015}.\nIn this paper, we introduce new conditions\nto a sequence of partitions and a c\\`{a}dl\\`{a}g path\n(Definition~\\ref{2d}),\nwhich gives a unified approach.\n\nThere have been many attempts to extend classical stochastic \ncalculus to Banach or Hilbert space-valued processes.\nExamples include Kunita~\\cite{Kunita_1970},\nMetivier~\\cite{Metivier_1972},\nPellaumail~\\cite{Pellaumail_1973},\nYor~\\cite{Yor_1974},\nGravereaux and Pellaumail~\\cite{Gravereaux_Pellaumail_1974},\nMetivier and Pistone~\\cite{Metivier_Pistone_1975},\nMeyer~\\cite{Meyer_1977a},\nMetivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b},\nGy{\\\"o}ngy and Krylov~\\cite{Gyongy_Krylov_1980,Gyongy_Krylov_1982},\nGy{\\\"o}ngy~\\cite{Gyongy_1982},\nMetivier~\\cite{Metivier_1982},\nPratelli~\\cite{Pratelli_1988},\nBrooks and Dinculeanu~\\cite{Brooks_Dinculeanu_1990},\nMikulevicius and Rozovskii~\\cite{Mikulevicius_Rozovskii_1998,Mikulevicius_Rozovskii_1999},\nDinculeanu~\\cite{Dinculeanu_2000}, and\nDe Donno and Pratelli~\\cite{DeDonno_Pratelli_2006}.\nNote that Di Girolami, Fabbri, and Russo~\\cite{DiGirolami_Fabbri_Russo_2014}\ntreats quadratic covariation of Banach space-valued processes\nwithin the framework of stochastic calculus via regularization,\nwith F{\\\"o}llmer's calculus in mind.\n\nOur method can be interpreted as a deterministic counterpart\nof these stochastic integration theories in Banach spaces.\nSome of the works listed above,\nsuch as Metivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b}\nand Dinculeanu~\\cite{Dinculeanu_2000},\ngive a proof of It{\\^o}'s formula\nin a similar manner to F{\\\"o}llmer's calculus.\nOne of the advantages of our approach \nappears in the statement of the It{\\^o} formula.\nFor a function $f$ to satisfy the It{\\^o} formula,\nwe require $f$ to be just $C^2$-class,\nwhile a stochastic approach needs some additional assumptions about \nthe boundedness of $f$ and its derivatives. \n\nNow we summarize the main result of F{\\\"o}llmer~\\cite{Foellmer_1981}. \nLet $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$ be a sequence of partitions of\n$\\mathbb{R}_{\\geq 0}$ such that $\\lvert \\pi_n \\rvert \\to 0$ as $n \\to \\infty$.\nWe say that a {c\\`{a}dl\\`{a}g} path $X \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}$ has quadratic variation along $\\Pi$\nif there exists a {c\\`{a}dl\\`{a}g} increasing function $[X,X]$ such that for all $t \\in \\mathbb{R}_{\\geq 0}$,\n\\begin{enumerate}\n    \\item $\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} (X_{s \\wedge t} - X_{r \\wedge t})^2$\n        converges to $[X,X]_t$ as $n \\to \\infty$, and\n    \\item $\\Delta [X,X]_{t} = (\\Delta X_t)^2$.\n\\end{enumerate}\nAn $\\mathbb{R}^d$-valued {c\\`{a}dl\\`{a}g} path $X = (X^1,\\dots,X^d)$\nhas quadratic variation along $\\Pi$ if\nthe real-valued path $X^i + X^j$ has quadratic variation along the same sequence\nfor each $i$ and $j$.\nF{\\\"o}llmer~\\cite{Foellmer_1981} proved that if $X$ has quadratic variation, then for any $f \\in C^2(\\mathbb{R}^d)$\nthe path $t \\mapsto f(X_t)$ satisfies It{\\^o}'s formula.\nThat is,\n\\begin{align} \\label{1.1c}\n    f(X_t)\n    & = \n    f(X_0)\n    + \\int_0^t \\langle Df (X_{s-}), \\mathrm{d}X_s \\rangle\n    + \\frac{1}{2} \\sum_{1 \\leq i,j \\leq d} \\int_0^t \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(X_{s-}) \\mathrm{d}[X^i,X^j]_s    \\\\\n    & \\quad\n    + \\sum_{0 < s \\leq t}\n        \\left\\{ \n            \\Delta f(X_s) \n            - \\sum_{1 \\leq i \\leq d} \\frac{\\partial f}{\\partial x_i}(X_{s-})\\Delta X^i_s\n            - \\frac{1}{2} \\sum_{1 \\leq i,j \\leq d}\n                \\frac{\\partial^2 f}{\\partial x_i \\partial x_j}(X_{s-})\n                \\Delta X^{i}_s \\Delta X^{j}_s  \\notag\n        \\right\\}\n\\end{align}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\nHere, the first term on the right-hand side,\nwhich we call the It{\\^o}-F{\\\"o}llmer integral along $\\Pi$,\nis defined as the limit\n\\begin{equation*}\n    \\int_0^t \\langle Df (X_{s-}), \\mathrm{d}X_s \\rangle\n    =\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\langle D f(X_r), X_{s \\wedge t} - X_{r \\wedge t} \\rangle.\n\\end{equation*}\nF{\\\"o}llmer's theorem claims that if $X$ has quadratic variation along $\\Pi$,\nthen the It{\\^o}-F{\\\"o}llmer integral above\nexists and it satisfies equation~\\eqref{1.1c}.\n\nAs stated above, \nour aim is to extend F{\\\"o}llmer's pathwise It{\\^o} formula to Banach space-valued paths.\nLet us give a brief summary of the result.\nLet $E$ be a Banach space.\nWe say that an $E$-valued c\\`{a}dl\\`{a}g path $X$\nhas tensor quadratic covariation along $(\\pi_n)$\nif there is a c\\`{a}dl\\`{a}g path $[X,X] \\colon \\mathbb{R}_{\\geq 0} \\to E \\widehat{\\otimes}_{\\pi} E$\nof finite variation such that \n\\begin{enumerate}\n    \\item the sequence $\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} (X_{s \\wedge t} - X_{r \\wedge t})^{\\otimes 2}$ \n        converges to $[X,X]_t$ in the norm topology of $E \\widehat{\\otimes}_{\\pi} E$\n        for all $t \\geq 0$, and\n    \\item the equation $\\Delta [X,Y]_t = \\Delta X_t^{\\otimes 2}$ holds for all $t \\geq 0$.\n\\end{enumerate}\nHere $E \\widehat{\\otimes}_{\\pi} E$ denotes the projective tensor product of the Banach space $E$.\nMoreover, we say that $X$ has upper scalar quadratic variation if there is an \nincreasing c\\`{a}dl\\`{a}g path $\\overline{Q}(X) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}_{\\geq 0}$\nsuch that \n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack} \\lVert X_{s \\wedge t} - X_{r \\wedge t} \\rVert^2\n    \\leq \\overline{Q}(X)_t.\n\\end{equation*} \nWe say that a sequence of partitions $(\\pi_n)$\ncontrols a path $X \\colon \\lbrack 0,\\infty \\mathclose{\\lbrack} \\to E$\nin a Banach space if it satisfies Conditions~(C1)--(C3) of Definition~\\ref{2d}.\nRoughly speaking, Conditions~(C1) and (C2) state that \n$(\\pi_n)$ reconstructs the information of the jumps of $X$.\nCondition~(C3) means that \n$(\\pi_n)$ controls the oscillation of $X$ in some sense.\nUnder these settings, our main result (Theorem~\\ref{2e}) is stated as follows.\nLet a c\\`{a}dl\\`{a}g path $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ have\nboth tensor and upper scalar quadratic variations\nalong a sequence of partitions $(\\pi_n)_{n \\in \\mathbb{N}}$,\nand let $A \\colon \\mathbb{R}_{\\geq 0} \\to F$ be a c\\`{a}dl\\`{a}g path of finite variation. \nSuppose that $(\\pi_n)$ controls $(X,A)$ and \nleft-side discretization of $(X,A)$ along $(\\pi_n)$\napproximates $(X_{-},A_{-})$ pointwise\n(see Definition~\\ref{2b} for the exact definition).\nIf $f \\in C^{1,2}(E \\times F,G)$,\nthen the composite function $f \\circ (A,X)$ satisfies\n\\begin{align*} \n& \n    f(A_t,X_t) - f(A_0,X_0)  \\\\\n    & \\qquad = \n    \\int_{0}^{t} \\langle D_a f(A_{s-},X_{s-}), \\mathrm{d}A^{\\mathrm{c}}_s \\rangle \n    + \\int_0^t \\langle D_x f (A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle   \\notag\\\\\n    & \\qquad\\quad \n    + \\frac{1}{2} \\int_{0}^{t} \\langle D_x^2 f(A_{s-},X_{s-}), \\mathrm{d}[X,X]^{\\mathrm{c}}_s \\rangle \n    + \\sum_{0 < s \\leq t} \\left\\{ \\Delta f(A_{s},X_{s}) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}. \n\\end{align*}\n\nTo conclude this section,\nwe give an outline of the remainder of this paper.\nIn Section~2, we set up the basic notation and terminology \nand state the main result of the paper.\nAs a preliminary, we recall basic properties of \nc\\`{a}dl\\`{a}g paths in a Banach space\nand those of c\\`{a}dl\\`{a}g paths of finite variation in Section~3.\nIn Section~4, we study conditions\non the sequence of partitions and the relation between them and \nc\\`{a}dl\\`{a}g paths.\nFundamental properties of quadratic variations are studied in Section~5.\nSection~6 shows the existence of \nquadratic variations for paths of finite variation. \nIn Section~7, we prove a lemma \n(Lemma~\\ref{2g}) that is essentially used in the main theorems of this paper.\nIn the last section, Section~8, \nwe finally prove the It{\\^o} formula \nfor a Banach space-valued path with quadratic variation.\nWe give a $C^{1,2}$-type It{\\^o} formula (Theorem~\\ref{2e}),\nwhich immediately leads to the standard $C^2$ formulation.\nIn that theorem, we can see an advantage of a strictly pathwise approach\nto It{\\^o} integration.\n\n\\section{Settings and the main result}\n\nIn this section, \nwe introduce the main theorem of this paper,\nnamely, the It{\\^o} formula\nwithin the framework of It{\\^o}-F{\\\"o}llmer calculus in Banach spaces.\nThe statement of the theorem will be found in Theorem~\\ref{2e}.\n\nFirst, we introduce the basic notation and terminology used in this paper.\nThe symbol $\\mathbb{N}$ denotes the set of natural numbers $\\{ 0,1,2,\\dots \\}$\nand $\\mathbb{R}$ denotes the real numbers.\nIf $A$ is a subset of $\\mathbb{R}$ and $a \\in \\mathbb{R}$, we define \n$A_{\\geq a} = \\{ x \\in A \\mid x \\geq a \\}$.\nIf $E$ and $F$ are two real Banach spaces,\n$\\mathcal{L}(E,F)$ denotes the space of bounded linear maps from $E$ to $F$.\nIn addition, given another Banach space $G$,\nwe define $\\mathcal{L}^{(2)}(E,F;G)$ as the \nspace of bounded bilinear maps from $E \\times F$ to $G$.\n\nLet $[0,\\infty \\mathclose{\\lbrack} = \\mathbb{R}_{\\geq 0} = \\{ r \\in \\mathbb{R} \\mid t \\geq 0 \\}$\nand let $E$ be a Banach space.\nA \\emph{c\\`{a}dl\\`{a}g} path in $E$\nis a function $X \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$,\nwhich is right-continuous at every $t \\geq 0$\nand has the left limit at every $t > 0$.\nThe term \\emph{right regular} is also used\nto stand for the same property.\nThe symbol $D([0,\\infty \\mathclose{\\lbrack}, E)$\nor $D(\\mathbb{R}_{\\geq 0},E)$ denotes the set of \nall c\\`{a}dl\\`{a}g paths in $E$.\nIf $X$ is an element of $D([0,\\infty \\mathclose{\\lbrack}, E)$, we define\n\\begin{equation*}\n    X(t-) = \\lim_{s \\uparrow\\uparrow t} X(s) = \\lim_{s \\to t, s < t} X_s, \\qquad\n    \\Delta X(t) = X(t) - X(t-).\n\\end{equation*}\nWe also use $X_t$, $X_{t-}$, and $\\Delta X_t$ \nto indicate the values $X(t)$, $X(t-)$, and $\\Delta X(t)$, respectively.\nNext, set\n\\begin{gather*}\n    D(X) = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid \\lVert \\Delta X_t \\rVert \\neq 0 \\}  \\\\\n    D_{\\varepsilon}(X) = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid \\lVert \\Delta X_t \\rVert \\geq \\varepsilon \\}  \\\\\n    D^{\\varepsilon}(X) = D(X) \\setminus D_{\\varepsilon}(X)\n        = \\{ t \\in \\mathbb{R}_{\\geq 0} \\mid 0 < \\lVert \\Delta X_s \\rVert < \\varepsilon \\}.\n\\end{gather*}\nWe simply write $D$, $D_{\\varepsilon}$, and $D^{\\varepsilon}$ if there is no ambiguity.\nGiven a discrete set $D \\subset [0,\\infty[$ and a c\\`{a}dl\\`{a}g path $X$, we define\n\\begin{equation*}\n    J_D(X)_t = J(D;X)_t = \\sum_{0 < s \\leq t} \\Delta X_s 1_{D}(s).\n\\end{equation*}\nThen $J_D(X)$ is a c\\`{a}dl\\`{a}g path of finite variation.\nFor abbreviation, we often write\n$J_\\varepsilon(X)$ instead of $J(D_{\\varepsilon}(X);X)$.\nThe difference of a path $X$ along an interval $I = \\mathopen{\\rbrack} r,s \\rbrack$\nis defined as $\\delta_I X = \\delta_r^s X = X_s - X_r$.\nIf we are given an additional point $t$,\nlet $\\delta_I X_t = \\delta_r^s X_t = X_{s \\wedge t} - X_{r \\wedge t}$.\n\nThroughout this paper,\nthe term `partition of $\\mathbb{R}_{\\geq 0}$' always \nmeans the set of intervals of the form\n$\\pi = \\{ \\mathopen{\\rbrack} t_i,t_{i+1} ; i \\in \\mathbb{N} \\}$\nwhich satisfies $0=t_0 < t_1 < \\dots \\to \\infty$. \nThe set of all partitions of $\\mathbb{R}_{\\geq 0}$ is \ndenoted by $\\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$ or $\\mathop{\\mathrm{Par}} (\\lbrack 0,\\infty \\mathclose{\\lbrack} )$.\nSimilarly, $\\mathop{\\mathrm{Par}}([a,b])$ indicates the set of partitions of the \nform $\\pi = \\{ \\mathopen{\\rbrack} t_i,t_{i+1} \\rbrack ; 0 \\leq i \\leq n-1 \\}$\nwith $a=t_0 < t_1 < \\dots < t_n = b$.\n\n\\begin{Def} \\label{2b}\nLet $E,F,G$ be Banach spaces, $b \\colon E \\times F \\to G$ a bounded bilinear map,\nand $(X,Y) \\in D(\\mathbb{R}_{\\geq 0},E \\times F)$.\nTwo paths $X$ and $Y$ have\n\\emph{quadratic covariation\nalong $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$ with respect to $b$}\nif there exists a $G$-valued c\\`{a}dl\\`{a}g path $Q_b(X,Y)$ of \nfinite variation such that\n\\begin{enumerate}\n    \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n    \\begin{equation*}\n        \\sum_{I \\in \\pi_n} b(\\delta_I X_t, \\delta_I Y_t) \\xrightarrow[n \\to \\infty]{\\text{in $G$}} Q_b(X,Y)_t,\n    \\end{equation*} and\n    \\item for all $t \\in \\mathbb{R}_{\\geq 0}$, the jump of $b \\circ (X,Y)$ is given by\n        \\begin{equation*}\n            \\Delta Q_b(X,Y)_t = b(\\Delta X_t,\\Delta Y_t).\n        \\end{equation*}\n\\end{enumerate}\nThen the path $Q_b(X,Y)$ is called\nthe quadratic covariation of $X$ and $Y$ with respect to $b$.\nIf $E=F$ and $X=Y$,\nwe call $Q_b(X,X)$ the quadratic variation of $X$ with respect to $b$.\n\\end{Def}\n\nThe quadratic covariation $Q_b(X,Y)$ depends\non the sequence of partition $\\Pi$.\nIf there is the possibility of confusion, we also use $Q_b^{\\Pi}(X,Y)$\nto indicate the quadratic covariation $Q_b(X,Y)$.\n\nFor convenience, we often write\n\\begin{equation*}\n    Q_b^{\\pi}(X,Y)_t = \\sum_{I \\in \\pi} b(\\delta_I X_t,\\delta_I Y_t).\n\\end{equation*}\nThe continuity of the map $(s,t) \\mapsto s \\wedge t$\nand the c\\`{a}dl\\`{a}g property of $X$ and $Y$\nimply that the map $t \\mapsto Q^{\\pi}_b (X,Y)_t$\nis c\\`{a}dl\\`{a}g. It is not, however, of finite variation\nunless $X$ and $Y$ are of finite variation.\n\nThe quadratic covariation with respect to the \nthe canonical bilinear map $\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$\nis denoted by $[X,Y]$ or $[X,Y]^{\\Pi}$,\nand it is called the \\emph{tensor quadratic covariation} of $X$ and $Y$.\nHere, $E \\widehat{\\otimes}_{\\pi} F$ is the projective tensor product \nof two Banach spaces $E$ and $F$.\nGiven two c\\`{a}dl\\`{a}g paths $X$ and $Y$,\nwe can consider two tensor quadratic covariations\n$[X,Y]$ and $[Y,X]$. \nThough they are equivalent under the canonical isomorphism\n$E \\widehat{\\otimes}_{\\pi} F \\cong F \\widehat{\\otimes}_{\\pi} E$,\nthey are not equal even if $E = F$.\nWe also write $[X,Y]^{\\pi} = Q_{\\otimes}^{\\pi}(X,Y)$\nand call the path $[X,Y]^{\\pi} \\colon \\mathbb{R}_{\\geq 0} \\to E \\widehat{\\otimes}_{\\pi} E$\nthe discrete tensor quadratic covariation of $X$ and $Y$ along $\\pi$.\nThe path $[X,X]$ is simply called the \\emph{tensor quadratic variation}.\n\nThe tensor quadratic variation of an $\\mathbb{R}^d$-valued path\n$X= (X^1,\\dots,X^d)$ has the following matrix representation\n\\begin{equation*}\n    [X,X]_t\n    =\n    \\begin{pmatrix}\n    [X^1,X^1]_t & \\cdots & [X^1,X^d]_t    \\\\\n    \\vdots & \\ddots & \\vdots   \\\\\n    [X^d,X^1]_t & \\cdots & [X^d,X^d]_t\n    \\end{pmatrix}\n\\in M_{d}(\\mathbb{R}) \\cong \\mathbb{R}^d \\otimes \\mathbb{R}^d = \\mathbb{R}^d \\widehat{\\otimes}_{\\pi} \\mathbb{R}^d.\n\\end{equation*}\nA c\\`{a}dl\\`{a}g path $X \\colon \\lbrack0,\\infty \\mathclose{\\lbrack} \\to \\mathbb{R}^d$\nhas tensor quadratic variation if and only if\nit has quadratic variation in the sense of \nDefinition~2.3 of Hirai~\\cite{Hirai_2019}.\n\nNow we introduce a different type of quadratic variation,\nnamely, scalar quadratic variation.\nAgain we assume that $\\Pi = (\\pi_n)$ is a sequence of partitions of $[0,\\infty \\mathclose{\\lbrack}$.\n\n\\begin{Def} \\label{2c}\nLet $E$ be a Banach space and $X$ be an $E$-valued c\\`{a}dl\\`{a}g path.\n\\begin{enumerate}\n    \\item The path $X$ has \\emph{finite upper scalar quadratic variation along $(\\pi_n)$}\n        if there is an increasing c\\`{a}dl\\`{a}g path\n        $\\overline{Q}(X) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}_{\\geq 0}$\n        such that\n        \\begin{equation*}\n            \\varlimsup_{n \\to \\infty} \\sum_{I \\in \\pi_n} \\lVert \\delta_I X_t \\rVert^2 \\leq \\overline{Q}(X)_t\n        \\end{equation*}\n        holds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n        We call such a path $\\overline{Q}(X)$ an upper scalar quadratic variation of $X$.\n    \\item A c\\`{a}dl\\`{a}g path $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ has\n        \\emph{scalar quadratic variation}\n        if there exists a real-valued c\\`{a}dl\\`{a}g increasing path\n        $Q(X)$ such that\n        \\begin{enumerate}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$,\n                \\begin{equation*}\n                    \\sum_{I \\in \\pi_n} \\lVert \\delta_I X_t \\rVert^2 \n                    \\xrightarrow[n \\to \\infty]{} Q(X)_t,\n                \\end{equation*}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$, the jump of $Q(X)$ at $t$ is given by\n                $\\Delta Q(X)_t = \\lVert \\Delta X_t \\rVert_E^2$.\n        \\end{enumerate}\n        We call the increasing path $Q(X)$ the scalar quadratic variation of $X$\n        along $(\\pi_n)$.\n\\end{enumerate}\n\\end{Def}\n\nClearly, the scalar quadratic variation $Q(X)$ is an upper scalar quadratic variation of $X$.\nIf $E$ is a Hilbert space,\nthe scalar quadratic variation coincides with the quadratic variation\n$Q_{\\langle\\phantom{x},\\phantom{x}\\rangle}(X,X)$,\nwhere $\\langle\\phantom{x},\\phantom{x}\\rangle \\colon E \\times E \\to \\mathbb{R}$\nis the inner product of $E$.\n\nIf a c\\`{a}dl\\`{a}g path\n$X = (X^1,\\dots,X^d) \\colon \\mathbb{R}_{\\geq 0} \\to \\mathbb{R}^d$\nhas tensor quadratic variation along $(\\pi_n)$,\nthen it has scalar quadratic variation given by\n\\begin{equation*}\n    Q(X)_t = \\mathop{\\mathrm{Trace}} [X,X]_t = \\sum_{1 \\leq i \\leq d} [X^i,X^i]_t.\n\\end{equation*}\nThis trace representation is still valid for \nHilbert space-valued c\\`{a}dl\\`{a}g paths.\nThis result will be proved in Hirai~\\cite{Hirai_2021c}.\n\nNext, we introduce some conditions to \na sequence of partitions and a c\\`{a}dl\\`{a}g path.\nLet $\\pi \\in \\mathop{\\mathrm{Par}}[0,\\infty[$ and $t \\in \\mathopen{\\rbrack} 0,\\infty \\mathclose{[}$.\nThe symbol $\\pi(t)$ denotes the element of $\\pi$ that contains $t$.\nBy definition, there exists only one such interval.\nThen we set\n\\begin{equation*}\n    \\overline{\\pi}(t) = \\sup \\pi(t), \\qquad \n    \\underline{\\pi}(t) = \\inf \\pi(t).\n\\end{equation*}\nHere, note that $\\pi(t) = \\mathopen{\\rbrack} \\underline{\\pi}(t),\\overline{\\pi}(t) \\rbrack$ and \n\\begin{gather*}\n    \\delta_{\\pi(s)}X_t = X(\\overline{\\pi}(s) \\wedge t) - X(\\underline{\\pi}(s) \\wedge t), \\qquad\n    \\delta_{\\pi(s)}X = X(\\overline{\\pi}(s)) - X(\\underline{\\pi}(s))\n\\end{gather*}\nhold for all $s$ and $t$ in $\\mathopen{\\rbrack} 0,\\infty \\mathclose{\\lbrack}$.\n\nLet $f \\colon S \\to E$ be a function into a Banach space and $A$ be a subset of $S$.\nThe oscillation of $f$ on the set $A$ is defined as \n\\begin{equation*}\n    \\omega(f;A) = \\sup_{x,y \\in A} \\lVert f(x) - f(y) \\rVert_E.\n\\end{equation*}\nUsing this notation, we define two kinds of oscillation of\na path $X \\in D(\\mathbb{R}_{\\geq 0},E)$ \nalong a partition $\\pi \\in \\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$ as follows.\n\\begin{gather*} \\label{2cb}\n    O^+_{t}(X;\\pi) = \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{\\rbrack} r,s \\rbrack \\cap [0,t]),  \\\\\n    O^{-}_t(X;\\pi)\n    =\n    \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{]}r,s\\mathclose{[} \\cap [0,t])\n    =\n    \\sup_{\\mathopen{]}r,s] \\in \\pi} \\omega(X;\\mathopen{[}r,s\\mathclose{[} \\cap [0,t]).\n\\end{gather*}\nThe second equality in the definition of $O^{-}_t(X;\\pi)$\nis valid because $X$ is supposed to be right continuous.\nClearly we have $O_t^-(X;\\pi) \\leq O_t^+(X;\\pi)$\nfor all $t \\geq 0$.\nIf $X$ is continuous, these two quantities coincide.\n\n\\begin{Def} \\label{2d}\nLet $E$ be a Banach space, $X \\in D(\\mathbb{R}_{\\geq 0},E)$, and $(\\pi_n)_{n \\in \\mathbb{N}}$\nbe a sequence of partitions of $\\mathbb{R}_{\\geq 0}$.\n\\begin{enumerate}\n\\item We say that $(\\pi_n)$\n    \\emph{approximates the jumps of $X$}\n    if it satisfies the following two conditions:\n\t\\begin{enumerate}\n\t\\item[(C1)] Let $t \\in [0,\\infty[$ and $\\varepsilon > 0$. \n\t\tThen there exists an $N \\in \\mathbb{N}$ such that \n        for all $n \\geq N$ and for all $I \\in \\pi_n$,\n        the set $I \\cap [0,t] \\cap D_{\\varepsilon}(X)$\n        has at most one element.\n    \\item[(C2)] Let $s \\in D(X)$ and $t \\in [s,\\infty \\mathclose{\\lbrack}$. Then\n\t\t\\begin{equation*}\n            \\lim_{n \\to \\infty} \\delta_{\\pi_n(s)} X_t \n            =\n            \\lim_{n \\to \\infty} \\left\\{ X(\\overline{\\pi_n}(s) \\wedge t) - X(\\underline{\\pi_n}(s) \\wedge t) \\right\\}\n            =\n            \\Delta X_s.\n        \\end{equation*}\n    \\end{enumerate}\n\\item The sequence $(\\pi_n)$ in $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$\n    \\emph{weakly controls the oscillation of $X$}\n    if it satisfies Condition~(C3).\n    \\begin{enumerate}\n    \\item[(C3)] For all $t \\in \\mathbb{R}_{\\geq 0}$,\n        \\begin{equation*}\n        \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} O^{+}_t(X-J_{\\varepsilon}(X);\\pi_n) = 0.\n        \\end{equation*}\n    \\end{enumerate}\n\\item The sequence $(\\pi_n)$ in $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$\n    \\emph{controls $X$}\n    if it satisfies Conditions~(C1)--(C3).\n\\item The sequence \\emph{$(\\pi_n)$ approximates $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ from the left}\n    if $\\lim_{n \\to \\infty} X(\\underline{\\pi_n}(t)) = X(t-)$ holds for all $t > 0$.\n    Then we call $(\\pi_n)$ a \\emph{left approximation sequence}\n    for $X$.\n\\end{enumerate}\n\\end{Def}\n\nUnder these assumptions, we have the following $C^{1,2}$-type It{\\^o}\nformula for Banach space-valued paths.\n\n\\begin{Th}[It{\\^o} formula] \\label{2e}\nLet $(\\pi_n)$ be a sequence in $\\mathop{\\mathrm{Par}}[0,\\infty[$.\nSuppose that a c\\`{a}dl\\`{a}g path $X$ in $E$ has tensor and finite upper quadratic variations, \nand that $A$ is a c\\`{a}dl\\`{a}g path of finite variation in $F$.\nIf $(\\pi_n)$ is both a controlling and\na left approximation sequence for $(X,A)$,\nthen for any $f \\in C^{1,2}(F \\times E, G)$ and for any $t \\in [0,\\infty[$,\nthe It{\\^o}-F{\\\"o}llmer integral\n$\\int_0^t \\langle D_x f (A_{s-},X_{s-}), \\mathrm{d}X_s \\rangle$ exists,\nand it satisfies\n\\begin{align} \\label{2f}\n    &\n    f(A_t,X_t) - f(A_0,X_0)  \\\\\n    & \\qquad = \n    \\int_{0}^{t} \\langle D_a f (A_{s-},X_{s-}), \\mathrm{d}A^{\\mathrm{c}}_s \\rangle \n    + \\int_0^t \\langle D_x f(A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle   \\notag\\\\\n    & \\qquad\\quad \n    + \\frac{1}{2} \\int_{0}^{t} \\langle D_x^2 f(A_{s-},X_{s-}), \\mathrm{d}[X,X]^{\\mathrm{c}}_s \\rangle\n    + \\sum_{0 < s \\leq t} \\left\\{ \\Delta f(A_s,X_s) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}. \\notag\n\\end{align}\n\\end{Th}\n\nHere, note that \\eqref{2f} is an equation in the Banach space $G$.\nThe It{\\^o}-F{\\\"o}llmer integral in Theorem~\\ref{2e} is defined by \n\\begin{equation*}\n    \\int_0^t \\langle D_x f(A_{s-},X_{s-}),\\mathrm{d}X_s \\rangle\n    =\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\langle f(A_{r},X_{r}),\\delta_r^s X_t \\rangle.\n\\end{equation*}\n(See Definition~\\ref{5.9b}.)\nDifferentiability of the function in Theorem~\\ref{2e}\nis understood in the sense of the Fr{\\'e}chet derivative.\nRecall that the derivatives of $f \\in C^2(U;G)$\non an open subset $U \\subset E$ satisfy\n$D f \\in C(U, \\mathcal{L}(E,G))$ and\n$D^2 f \\in C(U, \\mathcal{L}(E,\\mathcal{L}(E,G)))$.\nIn the remainder of this paper, we use the identification\n$\n    \\mathcal{L}(E,\\mathcal{L}(E,G)) \n    \\simeq \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G) \n    \\simeq \\mathcal{L}^{(2)}(E,E;G)\n$\nwithout mention.\n\nThe following lemma is essentially used to prove Theorem~\\ref{2e}.\n\n\\begin{Lem} \\label{2g}\nLet $(\\pi_n)$ be a sequence in $\\mathop{\\mathrm{Par}}[0,\\infty[$, and suppose that\n$X \\in D(\\mathbb{R}_{\\geq 0},E)$ has tensor and finite upper quadratic variations.\nMoreover, assume that $(\\pi_n)$ \ncontrols $X \\in D([0,\\infty\\mathclose{[},E)$\nand approximates $\\xi \\in D([0,\\infty\\mathclose{[},\\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G))$\nfrom the left.\nThen for all $t \\in [0,\\infty[$, we have\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\left\\langle \\xi_r, \\left( \\delta_r^s X_t \\right)^{\\otimes 2} \\right\\rangle\n    \\xrightarrow[n \\to \\infty]{}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\left\\langle \\xi_{u-}, \\mathrm{d}[X,X]_u \\right\\rangle.\n\\end{equation*}\n\\end{Lem}\n\n\\section{Remarks on Banach space-valued c\\`{a}dl\\`{a}g paths}\n\nIn this section, we review properties of c\\`{a}dl\\`{a}g paths that will be \nreferred to later.\n\nRight- and a left-continuous step functions on $[0,\\infty[$ in a Banach space $E$\nare functions of the form\n\\begin{equation*}\n    \\sum_{i \\in \\mathbb{N}} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}, \\qquad\n    1_{\\{ 0 \\}} b_0  + \\sum_{i \\in \\mathbb{N}} 1_{\\mathopen{\\rbrack} t_i,t_{i+1} \\rbrack} b_{i+1}, \n\\end{equation*}\nrespectively, where $0 = t_0 < t_1 < \\dots < t_n < \\dots \\to \\infty$\nand all $a_i,b_i$ are elements of $E$.\nAll right-continuous step functions are clearly c\\`{a}dl\\`{a}g\nand left-continuous step functions are c\\`{a}gl\\`{a}d.\nEvery right-continuous step function $f= \\sum_{i \\in \\mathbb{N}} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}$\nis strongly $\\mathcal{B}([0,\\infty \\mathclose{\\lbrack})\/\\mathcal{B}(E)$ measurable,\nbecause it is the pointwise limit of the sequence $(f_n)$ defined by\n$f_n = \\sum_{0 \\leq i \\leq n} 1_{[t_i,t_{i+1}\\mathclose{[}} a_{i}$.\n\nA c\\`{a}dl\\`{a}g path in a Banach space satisfies the following properties.\n\n\\begin{Lem} \\label{3b}\nLet $f$ be a c\\`{a}dl\\`{a}g path in a Banach space $E$.\n\\begin{enumerate}\n    \\item For every $C \\in \\mathbb{R}_{> 0}$, there are only finitely many $s$\n        satisfying $\\lVert \\Delta f_{s} \\rVert_E > C$\n        in each compact interval of $[0,\\infty \\mathclose{\\lbrack}$.\n    \\item The image $f(I)$ of any compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$ is relatively compact in $E$.\n    \\item Suppose that every jump of $f$ is smaller than $C > 0$\n        on a compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$.\n        Then for all $\\varepsilon > 0$, there exists a $\\delta > 0$\n        such that $\\lVert f(s) - f(u) \\rVert_E < C + \\varepsilon$ holds\n        for any $s,u \\in I$ satisfying $\\lvert s - u \\rvert < \\delta$.\n    \\item The path $f$ is the uniform limit of some sequence\n        of right-continuous step functions on each compact interval.\n    \\item For any $t > 0$ and $\\varepsilon > 0$,\n        there is a partition $\\pi \\in \\mathop{\\mathrm{Par}}[0,t]$ that satisfies\n        $O^-(f,\\pi) < \\varepsilon$.\n\\end{enumerate}\n\\end{Lem}\n\nFor an analogue of Proposition~\\ref{3b}\nabout c\\`{a}dl\\`{a}g paths in arbitrary separable complete metric spaces, see Billingsley~\\cite[122]{Billingsley_1999}.\n\nNext, recall that a function $f \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$ is\nof bounded variation on a compact interval $I \\subset [0,\\infty \\mathclose{\\lbrack}$ if\n\\begin{equation*}\n    V(f;I)\n    :=\n    \\sup_{\\pi \\in \\mathop{\\mathrm{Par}} I}\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi} \\lVert f(s) - f(r) \\rVert_E < \\infty.\n\\end{equation*}\nFor convenience, set $V(f;\\emptyset) = 0$ and $V(f;[a,a]) = 0$\nfor every $a \\in [0,\\infty \\mathclose{\\lbrack}$.\nThe function $f$ is of finite variation \nif it has bounded variation on every compact subinterval of $[0,\\infty \\mathclose{\\lbrack}$.\nThe set of the paths of finite variation \nin $E$ is denoted by $FV(\\mathbb{R}_{\\geq 0},E)$\nor $FV(\\lbrack 0,\\infty \\mathclose{\\lbrack},E)$.\nWe define the total variation path $V(f)$ of a function $f$\nof $FV(\\mathbb{R}_{\\geq 0},E)$ by $V(f)_t = V(f;[0,t])$.\nClearly, $V(f)$ is increasing and satisfies $V(f)_0 = 0$.\n\nWe list basic properties of a path of finite variation below.\nSee Dinculeanu~\\cite[{\\S}18]{Dinculeanu_2000} for proofs.\n\n\\begin{Lem} \\label{3c}\nLet $f \\colon [0,\\infty \\mathclose{\\lbrack} \\to E$ be a c\\`{a}dl\\`{a}g path of finite variation in a Banach space.\n\\begin{enumerate}\n    \\item If three numbers $a,b,c \\in [0,\\infty \\mathclose{\\lbrack}$ satisfy $a \\leq b \\leq c$, then\n        \\begin{equation*}\n            V(f,[a,c]) = V(f,[a,b]) + V(f,[b,c]).\n        \\end{equation*}\n    \\item The total variation path $V(f)$ is c\\`{a}dl\\`{a}g.\n    \\item The jump of $V(f)$ at $t \\geq 0$ is given by $\\Delta V(f)(t) = \\lVert \\Delta f(t) \\rVert$.\n    \\item The family $(\\Delta f(s))_{s \\in [0,t]}$ is absolutely summable for all $t \\geq 0$.\n    \\item The function $f^{\\mathrm{d}}$ defined by\n        \\begin{equation*}\n            f^{\\mathrm{d}}(t) = \\sum_{0 < s \\leq t} \\Delta f(s).\n        \\end{equation*}\n        is again a c\\`{a}dl\\`{a}g path of finite variation.\n\\end{enumerate}\n\\end{Lem}\n\nNote that the summation in (v) of Proposition~\\ref{3c}\nis defined in the following sense.\nLet $D$ be the set of all finite subsets of $\\mathopen{\\rbrack} 0,t \\rbrack$.\nWe regard $D$ as a directed set with the order defined by inclusion.\nThen the net $(\\sum_{s \\in d} \\Delta f(s))_{d \\in D}$ \nconverges in $E$, and we define\n\\begin{equation*}\n    \\sum_{0 < s \\leq t} \\Delta f(s) = \\lim_{d} \\sum_{s \\in d} \\Delta f(s).\n\\end{equation*}\nThe function $f^{\\mathrm{d}}$ defined in Proposition~\\ref{3c}\nis called the discontinuous part of $f$.\nWe also define $f^{\\mathrm{c}} = f - f^{\\mathrm{d}}$\nand call this the continuous part of $f$.\n\nLet $\\mathcal{I}$ be the set of all bounded intervals of \nthe form $\\mathopen{\\rbrack} a,b \\rbrack$ or $[0,a]$,\nwhere $a$ are $b$ are nonnegative real numbers. \nThen $\\mathcal{I}$ is a semiring of subsets of $[0,\\infty \\mathclose{\\lbrack}$\ngenerating the Borel $\\sigma$-algebra \n$\\mathcal{B}([0,\\infty \\mathclose{\\lbrack})$.\nGiven an $f \\in D(\\mathbb{R}_{\\geq 0},E)$,\ndefine\n\\begin{equation*}\n    \\mu_{f}(\\mathopen{\\rbrack} a,b \\rbrack) = f(b) - f(a), \\quad \n    \\mu_{f}([0,a]) = f(a)\n\\end{equation*}\nfor any two real numbers satisfying $0 \\leq a \\leq b$.\nIf $f$ has finite variation,\nthe function $\\mu_f \\colon \\mathcal{I} \\to E$\ncan be uniquely extended to a $\\sigma$-additive \nmeasure defined on the $\\delta$-ring\ngenerated by $\\mathcal{I}$.\nRefer to Theorem~18.19 of Dinculeanu~\\cite[208]{Dinculeanu_2000}\nfor a proof.\n\nBecause there is a measure $\\mu_f$ associated with $f$,\nwe can consider the Stieltjes integral with respect to $f$.\nLet $b \\colon F \\times E \\to G$ a continuous bilinear map\nbetween Banach spaces.\nSet $L^{1}_{\\mathrm{loc}}(\\mu_f) = L^{1}_{\\mathrm{loc}}(\\lvert \\mu_f \\rvert)$,\nwhere $\\lvert \\mu_f \\rvert$ denotes the variation measure of $\\mu_f$.\nThen for each $g \\in L^{1}_{\\mathrm{loc}}(\\mu_f)$ and\nthe compact interval $I$, define the Stieltjes integral through the formula\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(g(s), \\mathrm{d}f(s))\n    =\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} (g(s), \\mu_f(\\mathrm{d}s)).\n\\end{equation*}\nNote that the integral of the left-hand side is well-defined because \n$g$ belongs to $L^{1}_{\\mathrm{loc}}(\\mu_f)$.\n\n\\section{Auxiliary results regarding sequences of partitions}\n\nIn this section, we investigate conditions on a sequence of partitions \nalong which we deal with quadratic variations and the It{\\^o}-F{\\\"o}llmer integral.\nRecall that basic notions were defined in Definition~\\ref{2d}.\n\n\\begin{Def} \\label{4b}\nLet $E$ be a Banach space, $X \\in D(\\mathbb{R}_{\\geq 0},E)$, and $(\\pi_n)_{n \\in \\mathbb{N}}$\nbe a sequence of partitions of $\\mathbb{R}_{\\geq 0}$.\n\\begin{enumerate}\n\\item We say that $(\\pi_n)$\n    \\emph{controls the oscillation of $X$}\n    if $\\lim_{n \\to \\infty}O^-_t(X;\\pi_n) = 0$ holds for all $t$.\n\\item A sequence $(\\pi_n)$ \\emph{exhausts the jumps of $X$} if\n    $D(X) \\subset \\bigcup_{n \\in \\mathbb{N}} \\bigcap_{k \\geq n} \\pi^{\\mathrm{p}}_k$.\n\\end{enumerate}\n\\end{Def}\n\n\\begin{Exm} \\label{4c}\n\\begin{enumerate}\n    \\item Let $r$ be an irrational number and $X = 1_{\\lbrack r,\\infty \\mathclose{\\lbrack}}$.\n        For each $n \\in \\mathbb{N}$, we set $\\pi_n = \\{ \\mathopen{\\rbrack} k2^{-n},(k+1)2^{-n} \\rbrack ; k \\in \\mathbb{N} \\}$.\n        Then the sequence $(\\pi_n)$ clearly satisfies $\\lvert \\pi_n \\rvert \\to 0$ as $n \\to \\infty$.\n        The sequence, however, does not control the oscillation of $X$.\n    \\item Let $X = 1_{\\lbrack 1,\\infty \\mathclose{\\lbrack}}$\n        and $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\n        Though the sequence $(\\pi_n)$ controls the oscillation of $X$,\n        it does not satisfy $\\lvert \\pi_n \\rvert \\to 0$.\n    \\item Let $X = 1_{\\lbrack 1\/2, \\infty \\mathclose{\\lbrack}}$\n        and $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\n        The sequence $(\\pi_n)$ neither controls the oscillation of $X$\n        nor satisfies $\\lvert \\pi_n \\rvert \\to 0$.\n        However, it controls $X$ in the sense of Definition~\\ref{2d}.\n\\end{enumerate}\n\\end{Exm}\n\nCondition~(i) of Definition~\\ref{4b} is characterized as follows.\n\n\\begin{Lem} \\label{4e}\nLet $X \\in D([0,\\infty\\mathclose{[},E)$, and $(\\pi_n) \\in (\\mathop{\\mathrm{Par}}[0,\\infty\\mathclose{[})^{\\mathbb{N}}$.\nThen there is an equivalence between\n\\begin{enumerate}\n    \\item the sequence $(\\pi_n)$ controls the oscillation of $X$, and\n    \\item the sequence $(\\pi_n)$ satisfies the conditions:\n        \\begin{enumerate}\n            \\item The sequence $(\\pi_n)$ exhausts the jumps of $X$.\n            \\item If $X$ is not constant on $\\mathopen{]}s,t\\mathclose{[}$, \n                then $]s,t[$ contains at least one element of $\\pi_n^{\\mathrm{p}}$ for sufficiently large $n$.\n        \\end{enumerate}\n\\end{enumerate}\n\\end{Lem}\n\nLemma~\\ref{4e} is a generalization of\nLemma~1 of Vovk~\\cite[272]{Vovk_2015}.\nNote that the condition\n`\\emph{$]s,t[$ contains at least one point of $\\pi_n^{\\mathrm{p}}$}'\nis equivalent to\n`\\emph{there is no $I \\in \\pi_n$ including $\\mathopen{\\rbrack} s,t \\mathclose{\\lbrack}$}.'\n\n\\begin{proof}\n\\emph{Step~1.1: (i) $\\Longrightarrow$ (ii)-(a).}\nLet $s \\in D(X)$ and set $\\varepsilon = \\lVert \\Delta X(s) \\rVert_{E}$.\nMoreover, fix $T > s$ arbitrarily.\nThen, by assumption, we can choose an $N \\in \\mathbb{N}$ such that\n$O_T^-(X,\\pi_n) < \\varepsilon\/2$ holds for all $n$.\nWe will show that $s = \\overline{\\pi_n}(s)$ holds for any $n \\geq N$.\nIf we had $s < \\overline{\\pi_n}(s)$, we could take an $s'$\nfrom the interval $\\mathopen{]}\\underline{\\pi_n}(s),s \\mathclose{[}$\nsuch that\n\\begin{equation*}\n    \\lVert X_{s'} - X_{s-} \\rVert_{E} \\leq O^{-}_T(X,\\pi_n) < \\frac{\\varepsilon}{2}.\n\\end{equation*}\nThen\n\\begin{equation*}\n    \\lVert X_s - X_{s'} \\rVert_{E} \n    \\geq \n    \\lVert \\Delta X_s \\rVert_{E} - \\lVert X_{s-} - X_{s'} \\rVert_{E} \n    > \\varepsilon - \\frac{\\varepsilon}{2} \n    = \n    \\frac{\\varepsilon}{2},\n\\end{equation*}\nwhich contradicts the assumption.\nThus $(\\pi_n)$ exhausts the jumps of $X$.\n\n\\emph{Step~1.2: (i) $\\Longrightarrow$ (ii)-(b).}\nAssume that $\\varepsilon := \\omega(X;\\mathopen{]} s,t \\mathclose{[}) > 0$.\nChoose an $N \\in \\mathbb{N}$ that satisfies \n$O^{-}_t(X,\\pi_n) < \\varepsilon$ for all $n \\geq N$.\nThen, for arbitrarily fixed $n \\geq N$, pick an $i$ satisfying $s \\in [t^n_i,t^n_{i+1}[$.\n$t^n_{i+1} \\in \\mathopen{]}s,t \\mathclose{[}$ remains to be proven.\nIf $t^n_{i+1} \\geq t$, we have \n\\begin{equation*}\n    O^{-}_t(X,\\pi_n) \n    \\geq\n    \\sup_{u,v \\in [t^n_i,t^n_{i+1}\\mathclose{[} \\cap [0,t]} \\lVert X_u - X_v \\rVert\n    \\geq\n    \\sup_{u,v \\in [s,t[} \\lVert X_u - X_v \\rVert\n    =\n    \\varepsilon.\n\\end{equation*}\nThis contradicts the assumption, and hence we obtain $t^n_{i+1} < t$.\n\n\\emph{Step~2: (ii) $\\Longrightarrow$ (i).}\nSuppose that $(\\pi_n)$ satisfies the conditions (ii)-(a) and (b).\nFix an $\\varepsilon > 0$ and a $t > 0$ arbitrarily.\nBecause $X$ is c\\`{a}dl\\`{a}g, we can take a sequence $0 = s_0 < s_1 < \\dots < s_N = t$\nsuch that $\\omega(X; \\mathopen{]} s_i,s_{i+1}\\mathclose{[}) < \\varepsilon\/2$\nfor all $i$ (see Lemma~\\ref{3b}).\n\nBy assumption, we can choose an $N \\in \\mathbb{N}$ satisfying\nthe following conditions.\n\\begin{enumerate}\n    \\item[1.] If $n \\geq N$, there are no $I \\in \\pi_n$ and $i \\in \\{ 0,\\dots, N \\}$\n        satisfying $\\mathopen{\\rbrack} s_i,s_{i+1} \\mathclose{\\lbrack} \\subset I$ and\n        $\\omega(X;\\mathopen{\\rbrack} s_i,s_{i+1} \\mathclose{\\lbrack}) > 0$.\n    \\item[2.] $\n        \\{ s_0,\\dots, s_N \\} \\cap D(X) \n        \\subset \\bigcap_{n \\geq N} \\pi_n^{\\mathrm{p}}\n        $.\n\\end{enumerate}\n\nLet $n \\geq N$ and $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$.\nFirst, assume that $\\omega(X; \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}) > 0$.\nBy Condition~1, we see that there are only two cases\nfor the relationship between $\\mathopen{\\rbrack} u,v \\rbrack$ and $(s_i)_{0 \\leq i \\leq N}$, as follows.\n\\begin{enumerate}\n    \\item[A.] There is a unique $i$ such that $\\mathopen{\\rbrack} u,v \\rbrack \\subset \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}$.\n    \\item[B.] There is a unique $i$ such that $s_i \\in \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$.\n\\end{enumerate}\nIn Case~A, the oscillation of $X$ on $\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$ is estimated as\n\\begin{equation*}\n    \\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t]) \\leq \\omega(X; \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}) < \\frac{\\varepsilon}{2}.\n\\end{equation*}\nOn the other hand, in Case~B, \n$X$ is continuous at $s_i \\in \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}$\nbecause of Condition~2. Therefore,\n\\begin{align*}\n    \\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t])\n    \\leq\n    \\omega(X; \\mathopen{\\rbrack} s_{i-1},s_{i}\\mathclose{\\lbrack}) + \\omega(X; \\mathopen{\\rbrack} s_i,s_{i+1}\\mathclose{\\lbrack}) \n    <\n    \\varepsilon.\n\\end{align*}\nIf $\\omega(X; \\mathopen{\\rbrack} u,v \\mathclose{\\lbrack}) > 0$, we clearly have the same estimate.\nBy the discussion above,\nwe find that $\\omega(X;\\mathopen{\\rbrack} u,v \\mathclose{\\lbrack} \\cap [0,t]) < \\varepsilon$ holds for all\n$\\mathopen{\\rbrack} u,v, \\rbrack \\in \\pi_n$, and consequently\n\\begin{equation*}\n    O^{-}_t(X;\\pi_n) = \\sup_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\omega(X;\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]) \\leq \\varepsilon\n\\end{equation*}\nfor every $n \\geq N$. This completes the proof.\n\\end{proof}\n\nWe next define the left discretization of\n$\\xi \\colon \\mathbb{R}_{\\geq 0} \\to E$ along a partition $\\pi$ by \n\\begin{equation*}\n    {^\\pi\\xi} = \\sum_{\\mathopen{]}r,s] \\in \\pi} \\xi(r) 1_{]r,s]}.\n\\end{equation*}\nIf $\\xi$ is c\\`{a}dl\\`{a}g, then the sequence $(\\pi_n)$ approximate $\\xi$ from the left\nin the sense of Definition~\\ref{2d}\nif and only if $({^{\\pi_n}\\xi})_{n \\in \\mathbb{N}}$ converges\nto $\\xi_{-}$ pointwise.\nConsider the following example.\n\n\\begin{Exm} \\label{4d}\nLet $X = 1_{\\lbrack 1\/2, \\infty \\mathclose{\\lbrack}}$\nand $\\pi_n = \\{ \\mathopen{\\rbrack} k,k+1 \\rbrack; k \\in \\mathbb{N} \\}$.\nAs we have seen in Example~\\ref{4c},\n$(\\pi_n)$ controls the path $X$.\nFor each $n \\in \\mathbb{N}$,\nthe left discretization of $X$ is given by \n${^{\\pi_n} X} = 1_{\\mathopen{\\rbrack} 1,\\infty \\mathclose{\\lbrack}}$.\nThe sequence $({^{\\pi_n} X})$ does not converge to $X$ pointwise,\nand hence $(\\pi_n)$ does not approximate $X$ from the left.\n\\end{Exm}\n\nAs we mentioned in Section~1, \ntwo types of assumptions about a sequence of partitions are \nfrequently used in the context of the It{\\^o}-F{\\\"o}llmer calculus.\nOne is `$\\lvert \\pi_n \\rvert \\to 0$' and the other is `$O_t^{-}(X;\\pi_n) \\to 0$'.\nIn the next proposition, we show that both conditions \nimply `$(\\pi_n)$ controls $X$'.\n\n\\begin{Prop} \\label{4f}\nLet $(\\pi_n)$ be a sequence of partitions of $\\mathbb{R}_{\\geq 0}$ and let $E$ be a Banach space.\n\\begin{enumerate}\n    \\item Suppose that $(\\pi_n)$ satisfies $\\lvert \\pi_n \\rvert \\to 0$. \n        Then it controls every c\\`{a}dl\\`{a}g path in $E$.\n        Moreover, it approximates every c\\`{a}dl\\`{a}g path in $E$ from the left.\n    \\item Suppose that $(\\pi_n)$ controls the oscillation of $X \\in D([0,\\infty\\mathclose{[},E)$. \n        Then it controls $X$ and approximates $X$ from the left.\n\\end{enumerate}\n\\end{Prop}\n\n\\begin{proof}\n(i)\nIf $\\lvert \\pi_n \\rvert \\to 0$, then $\\overline{\\pi_n}(t) \\to t$\nand $\\underline{\\pi_n}(t) \\to t$ hold for every $t \\geq 0$.\nThis directly implies that $(\\pi_n)$ approximates $X$ from the left.\nMoreover, we have $\\delta_{\\pi_n(t)} X_u \\to X_{t}$ for every $t,u > 0$\nwith $t \\leq u$. Hence, $(\\pi_n)$ satisfies Condition~(C2).\nCondition~(C3) follows from (iii) of Lemma~\\ref{3b}.\nCondition~(C1) remains to be shown.\nGiven an $\\varepsilon > 0$, define\n\\begin{equation*}\n    r := \\inf \\{ \\lvert u-v \\rvert \\mid u,v \\in D_{\\varepsilon}(X) \\cap [0,t], u \\neq v \\} > 0.\n\\end{equation*}\nIf $D_{\\varepsilon}(X) \\cap [0,t]$ has only one element,\nthere is nothing to do.\nOtherwise, $r$ is not zero, \nbecause $D_{\\varepsilon}(X) \\cap [0,t]$ has at most finitely many elements\n(see (i) of Lemma~\\ref{3b}).\nNow we take an $N$ satisfying $\\lvert \\pi_n \\rvert < r$ for all $n \\geq N$.\nThen for each $n \\geq N$ and $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$, \nthe set $\\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t]$ contains at most one element of $D_{\\varepsilon}$.\n\n(ii)\nAssume that $(\\pi_n)$ controls the oscillation of $X$.\nThen we see that $(\\pi_n)$ approximates $X$ from the left\nby the following estimate.\n\\begin{equation*}\n    \\lVert X_{\\underline{\\pi_n}(t)} - X_{t-} \\rVert_{E}\n    \\leq\n    \\omega(X;\\lbrack \\underline{\\pi_n}(t), \\overline{\\pi_n}(t)\\mathclose{\\lbrack} \\cap [0,t] )\n    \\leq\n    O^{-}_t(X,\\pi_n).\n\\end{equation*}\n\nNow, let us show that $(\\pi_n)$ controls $X$.\nTo obtain (C2), take a $t \\in D(X)$.\nBecause $(\\pi_n)$ exhausts the jumps of $X$ (Lemma~\\ref{4e}), \nwe have $\\overline{\\pi_n}(t) = t$ for sufficiently large $n$.\nThis combined with the fact that $(\\pi_n)$ is a left-approximation sequence\nimplies (C2).\nNext, consider (C1). \nLet $\\varepsilon > 0$ and fix an $N_{\\varepsilon} \\in \\mathbb{N}$\nso that $O^{-}_{t}(X,\\pi_n) < \\varepsilon$ holds for any $n \\geq N_{\\varepsilon}$.\nThen for every $n \\geq N_{\\varepsilon}$\nand $\\mathopen{\\rbrack} r,s] \\in \\pi_n$, the interval $\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]$ does not \ncontain any jump of $X$ that is greater than $\\varepsilon$.\nTherefore, $\\mathopen{\\rbrack} r,s \\rbrack$ possesses at most one element of $D_{\\varepsilon}(X)$.\nThis means that $(\\pi_n)$ satisfies (C1).\nAll that is left is to check Condition~(C2).\nChoose an $M_{\\varepsilon}$\nthat satisfies $O^{-}_t(X;\\pi_n) < \\varepsilon\/2$ for all $I \\in \\pi_n$ and $n \\geq M$.\nAs we just have shown, $J_{\\varepsilon\/2}(X)$ is zero on the interior of each \n$I \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$. Hence, \n\\begin{align*}\n    \\omega(X-J_{\\varepsilon\/2}(X);\\mathopen{\\rbrack} r,s \\rbrack \\cap [0,t])\n& \\leq\n    \\omega(X-J_{\\varepsilon\/2}(X);\\mathopen{\\rbrack} r,s \\mathclose{\\lbrack} \\cap [0,t]) + \\lVert \\Delta (X-J_{\\varepsilon\/2}(X))_s \\rVert_{E}  \\\\\n& \\leq \n    O^{-}_t(X;\\pi_n)\n    + \\lVert \\Delta (X-J_{\\varepsilon\/2}(X))_s \\rVert_{E}\n<\n    \\varepsilon\n\\end{align*}\nholds for all $\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$,\nwhich implies (C3).\n\\end{proof}\n\nIn the last part of this section,\nwe give an additional lemma about a sequence of partitions.\n\n\\begin{Lem} \\label{4g}\n\\begin{enumerate}\n    \\item Let $X$ be a c\\`{a}dl\\`{a}g path in a Banach space $E$.\n        If $(\\pi_n)$ approximates $\\xi$ from the left,\n        then $(\\pi_n)$ also approximates $f \\circ \\xi$ from the left for every\n        continuous function $f \\colon E \\to E'$ to an arbitrary Banach space.\n    \\item Let $X$ and $Y$ be c\\`{a}dl\\`{a}g paths in Banach spaces $E$ and $F$,\n        respectively. If $(\\pi_n)$ controls the path $(X,Y)$ in $E \\times F$,\n        then $(\\pi_n)$ controls both $X$ and $Y$.\n        Here, we regard $E \\times F$ as a Banach space endowed with the direct sum norm.\n\\end{enumerate}\n\\end{Lem}\n\n\\begin{proof}\n(i) immediately follows from the continuity of $f$.\n\nTo show (ii), suppose that $(\\pi_n)$ controls $(X,Y)$.\nIt suffices to show that $(\\pi_n)$ controls $X$.\nFirst fix $t \\in \\mathbb{R}_{\\geq 0}$ and $\\varepsilon > 0$ arbitrarily,\nand then choose an $N$ so that $D_\\varepsilon(X,Y) \\cap I \\cap [0,t]$\nhas at most one element for all $n \\geq N$ and $I \\in \\pi_n$.\nThe inclusion\n$I \\cap [0,t] \\cap D_\\varepsilon(X) \\subset D_\\varepsilon(X,Y) \\cap I \\cap [0,t]$\nimplies that the cardinality of $I \\cap [0,t] \\cap D_\\varepsilon(X)$ is \nno greater than 1.\nCondition~(C2) obviously follows from the definition of product topology.\nCondition~(C3) remains to be shown.\nFor an arbitrary $\\delta > 0$, \nchoose an $\\varepsilon_0 > 0$ satisfying\n\\begin{equation*}\n    \\sup_{\\varepsilon \\leq \\varepsilon_0} \\varlimsup_{n \\to \\infty} O^+_t((X,Y)-J_\\varepsilon(X,Y);\\pi_n) < \\frac{\\delta}{2}.\n\\end{equation*}\nSet $\\varepsilon_1 = \\varepsilon_0 \\wedge (\\delta\/2)$.\nGiven $\\varepsilon \\leq \\varepsilon_1$,\nwe can take $M_{\\varepsilon} > 0$ such that\n\\begin{enumerate}\n    \\item[(a)] $I \\cap [0,t] \\cap D_{\\varepsilon}(X,Y)$ has at most one element\n        for all $I \\in \\pi_n$ and $n \\geq M_{\\varepsilon}$.\n    \\item[(b)] $\\sup_{n \\geq M_{\\varepsilon}} O^+_t((X,Y)-J_{\\varepsilon}(X,Y);\\pi_n) < \\delta\/2$.\n\\end{enumerate}\nIf $n \\geq M_{\\varepsilon}$ and $I \\in \\pi_n$, then for any $u,v \\in I$, we have\n\\begin{align*}\n    & \n    \\lVert (X-J_\\varepsilon(X))_u - (X-J_\\varepsilon(X))_v \\rVert_{E}  \\\\\n    & \\qquad \\leq \n    \\lVert ( X-J(D_\\varepsilon(X,Y),X) )_u - ( X-J (D_\\varepsilon(X,Y),X) )_v \\rVert_{E}  \\\\\n    & \\qquad\\quad \n    + \\lVert ( J(D_\\varepsilon(X,Y),X) - J(D_\\varepsilon(X),X) )_u - ( J(D_\\varepsilon(X,Y),X) - J (D_\\varepsilon(X),X) )_v \\rVert_{E}  \\\\\n    & \\qquad \\leq\n    \\sup_{n \\geq M_{\\varepsilon}} O^+_t((X,Y)-J_{\\varepsilon}(X,Y);\\pi_n) + \\varepsilon   \\\\\n    & \\qquad \\leq\n    \\frac{\\delta}{2} + \\frac{\\delta}{2} = \\delta.\n\\end{align*}\nHere, note that the second inequality holds by Condition~(a) above.\nThus, we get\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} O^+_t(X-J_{\\varepsilon}(X);\\pi_n)\n    \\leq\n    \\sup_{n \\geq M_{\\varepsilon}} O^+_t(X-J_{\\varepsilon}(X);\\pi_n)\n    \\leq\n    \\delta.\n\\end{equation*}\nfor arbitrary $\\varepsilon \\leq \\varepsilon_1$.\nThis implies (C3) for $X$.\n\\end{proof}\n\n\\section{Properties of quadratic variations}\nWe defined quadratic variations treated in this paper \nin Section~2.\nThis section is devoted to studying their basic properties.\nThroughout this section, suppose that\nwe are given a sequence $\\Pi = (\\pi_n)_{n \\in \\mathbb{N}}$\nof elements of $\\mathop{\\mathrm{Par}} \\lbrack 0,\\infty \\mathclose{\\lbrack}$.\n\nFirst, we give some examples of quadratic variations.\n\n\\begin{Exm} \\label{5b}\nLet $A$ be a path of finite variation in a Banach space $E$.\nIf $(\\pi_n)$ satisfies $\\lvert \\pi_n \\rvert \\to \\to 0$,\nthen $A$ has tensor and scalar quadratic variations given by \n\\begin{equation*}\n    [A,A]_t = \\sum_{0 < s \\leq t} (\\Delta A_s)^{\\otimes 2}, \\qquad \n    Q(A)_t = \\sum_{0 < s \\leq t} \\lVert \\Delta A_s \\rVert^2.\n\\end{equation*}\nThis result will be proved later in this section.\n\\end{Exm}\n\n\\begin{Exm} \\label{5c}\nSuppose that $x \\colon [0,\\infty \\mathclose{\\lbrack} \\to \\mathbb{R}$\nhave quadratic variation along $(\\pi_n)$.\nLet $a$ be any element of a Banach space $E$ and \nset $X(t) = x(t) a$.\nThen $X$ have tensor and scalar quadratic variation\nand they have the following expression.\n\\begin{equation*}\n    [X,X]_t = [x,x]_t a \\otimes a, \\qquad \n    Q(A)_t = [x,x]_t \\lVert a \\rVert^2.\n\\end{equation*}\nFor the construction of \na real continuous path of nontrivial quadratic variation,\nrefer to Schied~\\cite{Schied_2016} and Mishura and Schied~\\cite{Mishura_Schied_2016}.\n\\end{Exm}\n\nNext, we treat an example from the theory of stochastic processes.\n\n\\begin{Exm} \\label{5d}\nLet $(\\Omega,\\mathcal{F},(\\mathcal{F}_t)_{t \\geq 0},P)$ be\na filtered probability space satisfying the usual condition.\nConsider a semimartingale $X = (X_t)_{t \\geq 0}$ \nin a separable Hilbert space $H$.\nMoreover, let $\\pi = (\\tau^n_k)_{n \\in \\mathbb{N}}$ be an \nincreasing sequence of bounded stopping times such that \n$\\tau^n_k \\to \\infty$ as $k \\to \\infty$\nand $\\sup_{k} (\\tau^{n+1}_k - \\tau^n_k) \\to 0$ as $n \\to \\infty$\nalmost surely.\nThen the process $[X,X]^{\\pi}_t$ converges to \nthe quadratic variation process $[X,X]$ uniformly in probability.\nBy passing to an appropriate subsequence, \nwe see that almost all paths have quadratic variation \nalong the subsequence\n(see Metivier and Pellaumail~\\cite{Metivier_Pellaumail_1980b} for details). \n\\end{Exm}\n\nNow we consider the transpose of quadratic covariation.\nLet $b \\in \\mathcal{L}^{(2)}(E,F;G)$ and define the transpose $^tb \\colon F \\times E \\to G$\nof $b$ as $^tb(y,x) = b(x,y)$.\nThen $X \\in D(\\mathbb{R}_{\\geq 0},E)$\nand $Y \\in D(\\mathbb{R}_{\\geq 0},F)$ have quadratic covariation with respect to $b$\nif and only if $Y$ and $X$ do with respect to the transpose $^tb$.\n\nRecall that a $d$-dimensional c\\`{a}dl\\`{a}g path \n$X = (X_1,\\dots, X_d)$ has tensor quadratic variation \nif and only if $X_i$ and $X_j$ have quadratic covariation \nfor each $i$ and $j$.\nThis characterization is generalized to quadratic covariation \nwith respect to a bilinear map in Banach spaces.\n\n\\begin{Prop} \\label{5.4c}\nGiven a family of bilinear maps\n$(b_{ij})_{1 \\leq i,j \\leq 2} \\in \\prod_{1 \\leq i,j \\leq 2} \\mathcal{L}^{(2)}(E_i,F_j;G_{ij})$,\ndefine a continuous bilinear map $\\mathbf{b} \\colon (E_1 \\times E_2) \n\\times (F_1 \\times F_2) \\to \\prod_{1 \\leq i,j \\leq 2} G_{i,j}$ as\n\\begin{equation*}\n    \\mathbf{b}((x_1,x_2),(y_1,y_2)) = (b_{ij}(x_i,y_i))_{1 \\leq i,j \\leq 2}.\n\\end{equation*}\nThen, $(X_1,X_2) \\in D(\\mathbb{R}_{\\geq 0}, E_1 \\times E_2)$\nand $(Y_1,Y_2) \\in D(\\mathbb{R}_{\\geq 0}, F_1 \\times F_2)$\nhave quadratic covariation with respect to $\\mathbf{b}$\nif and only if $X_i$ and $Y_j$ have quadratic covariation\nwith respect to $b_{ij}$ for every $i$ and $j$.\nIn this case, we have\n\\begin{equation} \\label{5.4cb}\n    Q_{\\mathbf{b}}((X_1,X_2),(Y_1,Y_2)) = (Q_{b_{ij}}(X_i,X_j))_{i,j \\in \\{1, 2 \\}}.\n\\end{equation}\n\\end{Prop}\n\nUsing the matrix notation, we can also express equation~\\eqref{5.4cb} as follows.\n\\begin{equation*}\n    Q_{\\mathbf{b}}((X_1,X_2),(Y_1,Y_2)) =\n    \\begin{pmatrix}\n        Q_{b_{11}}(X_1,Y_1) & Q_{b_{12}}(X_1,Y_2)  \\\\\n        Q_{b_{21}}(X_2,Y_1) & Q_{b_{22}}(X_2,Y_2)\n    \\end{pmatrix}.\n\\end{equation*}\n\n\\begin{proof}\nThe equation\n\\begin{align*}\n    \\sum_{\\mathopen{]}r,s] \\in \\pi_n} \\mathbf{b} \\biggl( \\Bigl( \\delta_r^s X_1(t), \\delta_r^s X_2(t) \\Bigr), \\Bigl( \\delta_r^s Y_1(t), \\delta_r^s Y_2(t) \\Bigr) \\biggr)\n    & = \\left( \\sum_{\\mathopen{]}r,s] \\in \\pi_n}  b_{ij} \\bigl( \\delta_r^s X_i(t),\\delta_r^s Y_i(t) \\bigr) \\right)_{i,j \\in \\{1,2 \\}}.\n\\end{align*}\nimplies the equivalence of convergence of discrete quadratic covariations.\nMoreover, the equation\n\\begin{align*}\n    \\mathbf{b} \\left( \\Delta (X_1,X_2)_t,\\Delta (Y_1,Y_2)_t \\right)\n    & = \\left( b_{ij}( \\Delta X_i(t),\\Delta Y_j(t) ) \\right)_{i,j \\in \\{1,2\\}}.\n\\end{align*}\nguarantees the equivalence of the jump conditions.\n\\end{proof}\n\nApplying Proposition~\\ref{5.4c} to\nthe canonical bilinear map $\\otimes \\colon E_i \\times E_j \\to E_i \\widehat{\\otimes}_{\\pi} E_j$,\nwe obtain the following corollary.\n\n\\begin{Cor} \\label{5.4d}\nLet $(X_1,X_2) \\in D(\\mathbb{R}_{\\geq 0},E_1 \\times E_2)$.\nThen $(X_1,X_2)$ has tensor quadratic variation if and only if\n$X_i$ and $X_j$ have tensor quadratic covariation\nwith respect to the canonical bilinear map\n$\\otimes \\colon E_i \\times E_j \\to E_i \\widehat{\\otimes}_{\\pi} E_j$ for each $i,j \\in \\{1,2 \\}$.\n\\end{Cor}\n\nNext, we show the bilinear property of quadratic covariation.\n\n\\begin{Prop} \\label{5.4e}\nLet $X_1,X_2 \\in D(\\mathbb{R}_{\\geq 0},E)$,\n$Y_1,Y_2 \\in D(\\mathbb{R}_{\\geq 0},F)$.\nSuppose that $X_i$ and $Y_j$ have quadratic covariation with respect to a bounded bilinear\nmap $b \\colon E \\times F \\to G$ for each $i,j \\in \\{ 1,2 \\}$.\nThen, $X_1 + X_2$ and $Y_1 + Y_2$ have\nquadratic covariation with respect to $b$, and\n$Q_b(X_1+X_2,Y_1+Y_2)$ satisfies \n\\begin{equation*}\n    Q_b(X_1+X_2,Y_1+Y_2) = \\sum_{ i,j \\in \\{1,2\\} } Q_b(X_i,Y_j).\n\\end{equation*}\n\\end{Prop}\n\n\\begin{proof}\nAccording to the bilinear property of $b$, we see that\n\\begin{equation*}\n    \\sum_{I \\in \\pi_n} b(\\delta_I (X_1+X_2)_t,\\delta_I (Y_1+Y_2)_t)\n    =\n    \\sum_{1 \\leq i,j \\leq 2} \\sum_{I \\in \\pi_n} b(\\delta_I(X_i)_t,\\delta_I (Y_j)_t).\n\\end{equation*}\nEach term on the right-hand side converges to $Q_b(X_i,Y_j)$.\nTherefore, the left-hand side converges to $\\sum_{ij} Q_b(X_i,Y_j)$.\nAgain by bilinearity, we obtain\n\\begin{align*}\n    \\Delta \\left( \\sum_{i,j \\in \\{1,2\\}} Q_b(X_i,Y_j) \\right)_t\n    & = \\sum_{i,j \\in \\{1,2\\}} \\Delta Q_b(X_i,Y_j)_t   \\\\\n    & = \\sum_{i,j \\in \\{1,2\\}} b(\\Delta (X_i)_t, \\Delta (Y_j)_t)   \\\\\n    & = b\\left( \\Delta (X_1 + X_2)_t, \\Delta (Y_1 + Y_2)_t \\right).\n\\end{align*}\nHence, $\\sum_{ij} Q_b(X_i,Y_j)$ is the quadratic covariation of $X_1 + X_2$ and $Y_1 + Y_2$\nwith respect to $b$.\n\\end{proof}\n\n\\begin{Cor} \\label{5.4eb} \nLet $X_1,X_2 \\in D(\\mathbb{R}_{\\geq 0},E)$\nand $Y_1,Y_2 \\in D(\\mathbb{R}_{\\geq 0},F)$.\nSuppose that $X_i$ and $Y_j$ have tensor quadratic covariation\nfor every $i,j \\in \\{ 1,2 \\}$.\nThen, $X_1 + X_2$ and $Y_1 + Y_2$ have\ntensor quadratic covariation and satisfy\n\\begin{equation*}\n    [X_1+X_2,Y_1+Y_2] = [X_1,Y_1] + [X_1,Y_2] + [X_2,Y_1] + [X_2,Y_2].\n\\end{equation*}\n\\end{Cor}\n\nNext, we investigate quadratic variations of a path of finite variation.\nFor convenience, we introduce the following notation.\nLet $D \\subset \\mathbb{R}_{\\geq 0}$.\nThen we define functions\n$e^1_D$ and $e^2_D$ from $\\mathcal{P}(\\mathbb{R}_{\\geq 0})$ to $\\{ 0,1 \\}$ as \n\\begin{equation*}\n    e^1_D(A) =\n    \\begin{cases}\n        1 & \\text{if} \\quad A \\cap D \\neq \\emptyset  \\\\\n        0 & \\text{if} \\quad A \\cap D = \\emptyset\n    \\end{cases}\n\\end{equation*}\nand $e^2_D = 1 - e^1_D$.\nHere, $\\mathcal{P}(\\mathbb{R}_{\\geq 0})$ denotes the \npower set of $\\mathbb{R}_{\\geq 0}$.\n\n\\begin{Prop} \\label{5.6b}\nLet $A \\in FV(\\mathbb{R}_{\\geq 0},E)$\nand $X \\in D(\\mathbb{R}_{\\geq 0},F)$.\nIf $(\\pi_n) \\in \\mathop{\\mathrm{Par}}[0,\\infty \\mathclose{\\lbrack}^{\\mathbb{N}}$ controls both $X$ and $A$,\nthen they have quadratic covariation with respect to every $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nMoreover, the quadratic covariation is \n\\begin{equation*}\n    Q_b(A,X)_t = \\sum_{0 < s \\leq t} b(\\Delta A_s,\\Delta X_s).\n\\end{equation*}\n\\end{Prop}\n\n\\begin{proof}\nLet $b \\in \\mathcal{L}^{(2)}(E,F;G)$,\nand set $D = D(X)$, $D_{\\varepsilon} = D_{\\varepsilon}(X)$,\nand $D^{\\varepsilon} = D^{\\varepsilon}(X)$.\nMoreover, fix $t \\in \\mathbb{R}_{\\geq 0}$ arbitrarily.\nThen we have\n\\begin{align} \\label{5.6c}\n    & \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t) - \\sum_{0 < u \\leq t} b \\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}   \\\\\n    & \\qquad \\leq \\left\\lVert \\sum_{I \\in \\pi_n}\n        b(\\delta_I A_t,\\delta_I X_t)e^1_{D_{\\varepsilon}}(I) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u, \\Delta X_u \\right) \\right\\rVert_{G}\n        + \\left\\lVert \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G}  \\notag\n\\end{align}\nfor any $t \\in \\mathbb{R}_{\\geq 0}$.\nWe will observe the behaviour of each term of the right-hand side of equation~\\eqref{5.6c}.\n\nBecause $(\\pi_n)$ controls $X$,\nthere exists an $N_1$ such that $D_{\\varepsilon} \\cap [0,t] \\cap I$\ncontains at most one point for all $n \\geq N_1$ and $I \\in \\pi_n$.\nIf $n \\geq N_1$, we have\n\\begin{align*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e^1_{D_{\\varepsilon}}(I)\n    & = \\sum_{u \\in D_{\\varepsilon}} b\\left( \\delta_{\\pi_n(u)} A_t, \\delta_{\\pi_n(u)} X_t \\right).\n\\end{align*}\nNote that the c\\`{a}dl\\`{a}g property of $X$ implies that $D_{\\varepsilon}$ is a discrete set.\nTherefore, combining the above equality with Condition~(C2), we obtain\n\\begin{equation*}\n    \\lim_{n \\to \\infty}\n        \\sum_{u \\in D_{\\varepsilon}}\n            b \\left( \\delta_{\\pi_n(u)} A_t, \\delta_{\\pi_n(u)} X_t \\right)\n    =\n    \\sum_{u \\in D_{\\varepsilon} \\cap [0,t]}\n        b\\left( \\Delta A_u,\\Delta X_u \\right).\n\\end{equation*}\nThis observation allows us to deduce that\n\\begin{align*}\n    & \n    \\lim_{n \\to \\infty} \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)e_{D_{\\varepsilon}}(I) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right)\\right\\rVert_G  \\\\\n    & \\qquad = \n    \\left\\lVert \\sum_{u \\in D_{\\varepsilon} \\cap [0,t]} b\\left( \\Delta A_u,\\Delta X_u \\right)\n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_G   \\\\\n    & \\qquad \\leq \n    \\lVert b \\rVert \\sup_{u \\in [0,t]} \\lVert \\Delta X_u \\rVert_F \\sum_{u \\in D^{\\varepsilon} \\cap [0,t]} \\lVert \\Delta A_u \\rVert_E .\n\\end{align*}\n\nNext, we consider the second norm of the right-hand side of equation~\\eqref{5.6c},\nwhich is estimated as\n\\begin{equation*}\n    \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t)\n        e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G} \n    \\leq\n    \\lVert b \\rVert \\sum_{I \\in \\pi_n} \\lVert \\delta_I A_t \\rVert_E \\lVert \n        \\delta_I X_t \\rVert_F e^2_{D_{ \\varepsilon}}(\\mathopen{\\rbrack} r,s \\rbrack).\n\\end{equation*}\nIf $e^2_{D_{ \\varepsilon}}(I) = 1$, \n$X$ has no jumps greater than $\\varepsilon$ on $I$.\nThen\n\\begin{equation*}\n    \\left\\lVert \\delta_I X_t \\right\\rVert_F e^2_{D_{ \\varepsilon}}(I)\n= \n    \\lVert \\delta_I (X - J_{D_\\varepsilon}(X))_t \\rVert\n        e^2_{D_{ \\varepsilon}}(I)\n\\leq \n    O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n) e^2_{D_{ \\varepsilon}}(I).\n\\end{equation*}\nTherefore,\n\\begin{equation*}\n    \\left\\lVert \\sum_{I \\in \\pi_n} b(\\delta_I A_t,\\delta_I X_t) \n        e^2_{D_{ \\varepsilon}}(I) \\right\\rVert_{G}  \\\\\n    \\leq \\lVert b \\rVert O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n) V(A)_t.\n\\end{equation*}\n\nBy the discussion above, we can deduce that\n\\begin{align*}\n    & \n    \\varlimsup_{n \\to \\infty} \\left\\lVert \\sum_{I \\in \\pi_n} \n        b(\\delta_I A_t,\\delta_I X_t ) \n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}  \\\\\n    & \\qquad \n    \\leq \n    \\lVert b \\rVert \\sup_{u \\in [0,t]} \\lVert \\Delta X_u \\rVert_F \n        \\sum_{u \\in D^{\\varepsilon} \\cap [0,t]} \\lVert \\Delta A_u \\rVert_E\n    + \\lVert b \\rVert V(A)_t \\varlimsup_{n \\to \\infty} O_t^{+}(X-J_{D_{\\varepsilon}}(X);\\pi_n).\n\\end{align*}\nLetting $\\varepsilon$ go to $0$, we obtain \n\\begin{equation*}\n    \\limsup_{n \\to \\infty} \\left\\lVert \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        b(\\delta_I A_t,\\delta_I X_t)\n        - \\sum_{0 < u \\leq t} b\\left( \\Delta A_u,\\Delta X_u \\right) \\right\\rVert_{G}  \\\\\n    = 0,\n\\end{equation*}\nwhich is the desired conclusion.\n\\end{proof}\n\nApplying Proposition~\\ref{5.6b} to the canonical \nbilinear maps $\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$\nand $\\otimes \\colon F \\times E \\to F \\widehat{\\otimes}_{\\pi} E$,\nwe get the following corollary.\n\n\\begin{Cor} \\label{5.6d}\nLet $A \\in FV(\\mathbb{R}_{\\geq 0},E)$\nand $X \\in D(\\mathbb{R}_{\\geq 0},F)$.\nIf $(\\pi_n)$ controls both $A$ and $X$,\nthey have tensor quadratic covariations\n$[A,X]$ and $[X,A]$ given by\n\\begin{equation*}\n    [A,X]_t = \\sum_{0 < s \\leq t} \\Delta A_s \\otimes \\Delta X_s, \\qquad\n    [X,A]_t = \\sum_{0 < s \\leq t} \\Delta X_s \\otimes \\Delta A_s.\n\\end{equation*}\n\\end{Cor}\n\nUsing Corollaries~\\ref{5.4eb} and \\ref{5.6d}, we obtain the following.\n\n\\begin{Cor} \\label{5.6e}\nLet $(X,A) \\in D(\\mathbb{R}_{\\geq 0},E) \\times FV(\\mathbb{R}_{\\geq 0},E)$\nand suppose that $(\\pi_n)$ controls both $X$ and $A$.\nIf $X \\colon \\mathbb{R}_{\\geq 0} \\to E$ has\ntensor quadratic variation along $(\\pi_n)$,\nthen $X+A$ has tensor quadratic variation along $(\\pi_n)$.\nThe tensor quadratic variation is expressed as\n\\begin{equation*}\n    [X+A,X+A] = [X,X] + [X,A] + [A,X] + [A,A].\n\\end{equation*} \n\\end{Cor}\n\nBy a discussion similar to the proof of Proposition~\\ref{5.6b},\nwe see that a path of finite variation has scalar quadratic variation.\n\n\\begin{Prop} \\label{5.6f}\nLet $A$ be a c\\`{a}dl\\`{a}g path of finite variation in a Banach space $E$.\nIf $(\\pi_n)$ controls $A$, then $A$ has the scalar quadratic covariation given by \n\\begin{equation*}\n    Q(A)_t = \\sum_{0 < s \\leq t} \\lVert \\Delta A_s \\rVert^2.\n\\end{equation*}\n\\end{Prop}\n\nIn the proceeding part of this paper,\nwe have used the summation\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b(X_{s \\wedge t}-X_{r \\wedge t},Y_{s \\wedge t}-Y_{r \\wedge t})\n\\end{equation*}\nto define the quadratic covariation.\nWe can also consider a different form of summation\n\\begin{equation*}\n    \\sum_{\\substack{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n \\\\ r < t}} b(X_{s}-X_{r},Y_{s}-Y_{r}),\n\\end{equation*}\nwhich is a slightly modified version the summation \nused in the original paper by F{\\\"o}llmer~\\cite{Foellmer_1981}.\nLet us investigate the relation between\nthese two summations.\n\n\\begin{Prop} \\label{5.4f}\nLet $(X,Y) \\in D(\\mathbb{R}_{\\geq 0}, E \\times F)$\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nSuppose that $(X,Y)_{\\overline{\\pi_n}(t)} \\to (X,Y)_t$\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\nThen the following two conditions are equivalent:\n\\begin{enumerate}\n    \\item The paths $X$ and $Y$ have quadratic covariation along $(\\pi_n)$ with respect to $b$.\n    \\item There exists a c\\`{a}dl\\`{a}g path $V \\in FV(\\mathbb{R}_{\\geq 0},G)$ such that\n        \\begin{enumerate}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n                \\begin{equation*}\n                    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n                    1_{\\mathopen{\\rbrack} r,\\infty\\mathclose{\\lbrack}}(t) b(X_{s}-X_{r},Y_{s}-Y_{r})\n                    \\xrightarrow[n \\to \\infty]{} V_t \\quad \\text{in $G$,}\n                \\end{equation*}\n            \\item for all $t \\in \\mathbb{R}_{\\geq 0}$\n                \\begin{equation*}\n                    \\Delta V_t = b(\\Delta X_t,\\Delta Y_t).\n                \\end{equation*}\n        \\end{enumerate}\n\\end{enumerate}\nIf these conditions are satisfied, the path $V$ coincides with the quadratic covariation\n$Q_b(X,Y)$.\n\\end{Prop}\n\n\\begin{proof}\nLet $t \\in \\mathbb{R}_{> 0}$.\nThen the difference of the two summations is estimated as follows.\n\\begin{align*}\n    & \n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b( \\delta_r^s X_t,\\delta_r^s Y_t )\n        - \\sum_{\\substack{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n \\\\ r < t}} b(\\delta_r^s X,\\delta_r^s Y)\n    \\right\\rVert_G  \\\\\n    & \\qquad =\n    \\left\\lVert \n        b \\left( \\delta_{\\pi_n(t)} X_t,\\delta_{\\pi_n(t)} Y_t \\right) \n        - b\\left( \\delta_{\\pi_n(t)} X,\\delta_{\\pi_n(t)} Y \\right) \n    \\right\\rVert_G    \\\\\n    & \\qquad \\leq\n    \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X_t-\\delta_{\\pi_n(t)} X, \\delta_{\\pi_n(t)}Y_t \\right)\n    \\right\\rVert_G\n    + \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X,\\delta_{\\pi_n(t)} Y_t-\\delta_{\\pi_n(t)} Y \\right) \n    \\right\\rVert_G \\\\\n    & \\qquad =\n    \\left\\lVert\n        b \\left( X_{t} - X_{\\overline{\\pi_n}(t)}, \\delta_{\\pi_n(t)}Y_t \\right)\n    \\right\\rVert_G\n    + \\left\\lVert\n        b \\left( \\delta_{\\pi_n(t)} X, Y_{t} - Y_{\\overline{\\pi_n}(t) } \\right) \n    \\right\\rVert_G \\\\\n    & \\qquad \\leq \n    \\lVert b \\rVert\n        \\left\\lVert X_{\\overline{\\pi_n}(t) \\wedge t} - X_t \\right\\rVert_{E}\n        \\left\\lVert \\delta_{\\pi_n(t)} Y_t \\right\\rVert_F\n    + \\lVert b \\rVert\n        \\left\\lVert \\delta_{\\pi_n(t)} X \\right\\lVert_E\n        \\left\\lVert Y_{\\overline{\\pi_n}(t) \\wedge t} - Y_t \\right\\rVert_F.\n\\end{align*}\nThis combined with the assumption implies\nthe equivalence of the conditions.\n\\end{proof}\n\nAccording to Proposition~\\ref{5.4f},\nwe see that the two definitions of quadratic covariation are equivalent\nprovided that $(\\pi_n)$ satisfies the assumption in the proposition.\nThe first definition,\nwhich is given in Definition~\\ref{2b}, is more intuitive.\nThe second one has some technical advantages because the path\n$t \\mapsto \\sum_{\\pi_n} 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t) b(\\delta_r^s X,\\delta_r^s Y)$\nis of finite variation\\footnote{Note that the path $t \\mapsto \\sum_{\\pi_n} 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t) b(\\delta_r^s X,\\delta_r^s Y)$ is \\emph{c\\`{a}gl\\`{a}d} but not c\\`{a}dl\\`{a}g.}.\n\n\n\\begin{Rem} \\label{5.5c}\nFollowing a discussion similar to that in Proposition~\\ref{5.4f},\nwe can obtain an equivalent definition of scalar quadratic variation using the summation\n$\\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\left\\lVert X_{s}-X_{r} \\right\\rVert^2 1_{\\mathopen{\\rbrack} r,\\infty \\mathclose{\\lbrack}}(t)$\nif $(\\pi_n)$ satisfies the same condition as Proposition~\\ref{5.4f}.\n\\end{Rem}\n\n\\section{Proof of Lemma \\ref{2g}}\n\nIn this section, we prove Lemma~\\ref{2g},\nwhich is essentially used to show the main theorems of this paper.\nTo prove that lemma, we prepare some additional lemmas.\n\n\\begin{Lem} \\label{5.7c}\nSuppose that $X \\in D(\\mathbb{R}_{\\geq 0},E)$ has\ntensor quadratic variation along\na controlling sequence $(\\pi_n)$.\nLet $r,s$ be two real numbers satisfying $0 \\leq r < s$.\nIf $(\\pi_n)$ approximates $1_{\\lbrack r,s \\mathclose{\\lbrack}}$\nfrom the left, then \n\\begin{equation*}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack r,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2} = [X,X]_{s \\wedge t } - [X,X]_{r \\wedge t}\n\\end{equation*}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n\\end{Lem}\n\n\\begin{proof}\nBy considering the decomposition \n\\begin{equation*}\n\\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack r,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n= \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    - \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,r \\mathclose{\\lbrack}}(u)\\left( \\delta_u^v X_t \\right)^{\\otimes 2},\n\\end{equation*}\nwe see that it suffices to show that the convergence\n\\begin{gather*}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,r \\mathclose{\\lbrack}}(u) \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    = [X,X]_{t \\wedge r}  \\\\\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} 1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u) \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    = [X,X]_{t \\wedge s}\n\\end{gather*}\nholds for all $t \\in \\mathbb{R}_{\\geq 0}$.\n\nIf $t \\leq s$, we see that\n\\begin{align*}\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n    =\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2} \n    =\n    [X,X]_{t}\n    =\n    [X,X]_{t \\wedge s}.\n\\end{align*}\n\nNext, assume $s < t$.\nTake an arbitrary $\\varepsilon > 0$.\nThen $s \\notin \\pi_n(s+\\varepsilon)$ holds\nfor sufficiently large $n$,\nbecause $(\\pi_n)$ approximates $1_{\\lbrack r,s \\mathclose{\\lbrack}}$\nfrom the left.\nThis leads to $\\overline{\\pi_n}(s) \\leq s + \\varepsilon$\nfor sufficiently large $n$,\nand hence $\\overline{\\pi_n}(s) \\to s$.\nTherefore, we obtain\n\\begin{equation*}\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X_t \\right)^{\\otimes 2}\n=\n    \\lim_{n \\to \\infty}\n        \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \n        1_{\\lbrack 0,s \\mathclose{\\lbrack}}(u)\n        \\left( \\delta_u^v X \\right)^{\\otimes 2}\n=\n    [X,X]_s\n=\n    [X,X]_{s \\wedge t}.\n\\end{equation*}\n\nThe convergence about $1_{\\lbrack 0,r \\mathclose{\\lbrack}}$ is shown\nby a similar discussion.\n\\end{proof}\n\n\\begin{Lem} \\label{5.7d}\nLet $X \\in D(\\mathbb{R}_{\\geq 0},E)$ be a path of\ntensor quadratic variation along a controlling sequence $(\\pi_n)$.\nSuppose that $\\xi \\in D(\\mathbb{R}_{\\geq 0}, \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G))$\nhas the representation\n\\begin{equation} \\label{5.7e}\n    \\xi = \\sum_{i \\geq 1} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}} a_{i},\n\\end{equation}\nwhere $0 = \\tau_0 < \\tau_1 < \\dots < \\tau_i < \\tau_{i+1} < \\dots \\to \\infty$\nand $(a_i) \\in \\mathcal{L}(E \\widehat{\\otimes}_{\\pi} E,G)^{\\mathbb{N}}$.\nIf $(\\pi_n)$ approximates $\\xi$ from the left, the Stieltjes integral of $\\xi_{-}$ with respect\nto $[X,X]$ is approximated as\n\\begin{equation} \\label{5.7f}\n    \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        \\langle \\xi_{r}, (\\delta_r^s X_t)^{\\otimes 2} \\rangle\n    = \\int_{]0,t]} \\langle \\xi_{s-},\\mathrm{d}[X,X]_s \\rangle.\n\\end{equation}\n\\end{Lem}\n\n\\begin{proof}\nFirst, note that the Stieltjes integral on the right-hand side \nof equation~\\eqref{5.7f} has the representation\n\\begin{equation*}\n    \\int_{]0,t]} \\langle \\xi_{s-},\\mathrm{d}[X,X]_s \\rangle\n    = \\sum_{i \\geq 1} \\langle a_{i}, [X,X]_{\\tau_i \\wedge t} - [X,X]_{\\tau_{i-1} \\wedge t} \\rangle.\n\\end{equation*}\nBy the bilinearity of the canonical pairing, \nwe can calculate this as\n\\begin{align*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \\langle \\xi_{r}, (\\delta_r^s X_t)^{\\otimes 2} \\rangle\n    = \n    \\sum_{i \\geq 1} \\left\\langle a_{i},\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}}(r)\n        (\\delta_r^s X_t)^{\\otimes 2} \\right\\rangle.\n\\end{align*}\nHence, it remains to show that\n\\begin{equation*}\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} 1_{\\lbrack \\tau_{i-1},\\tau_{i} \\mathclose{\\lbrack}}(r)\n        (\\delta_r^s X_t)^{\\otimes 2}\n    \\xrightarrow[n \\to \\infty]{}\n    [X,X]_{\\tau_i \\wedge t} - [X,X]_{\\tau_{i-1} \\wedge t}\n\\end{equation*}\nholds in the sense of the norm topology of $E \\widehat{\\otimes}_{\\pi} E$.\nThis has already been proved in Lemma~\\ref{5.7c}.\n\\end{proof}\n\nFinally, we start dealing with the proof of Lemma~\\ref{2g}.\n\n\\begin{proof}[Proof of Lemma \\ref{2g}]\nFix $t > 0$ and $\\varepsilon > 0$ arbitrarily.\nWe can choose\nan $h \\in D(\\mathbb{R}_{\\geq 0},L(E \\widehat{\\otimes}_{\\pi} E,G))$\nof the form \\eqref{5.7e}\nso that $\\lVert h(s) - \\xi(s) \\rVert \\leq \\varepsilon$ holds for all $s$\nand $(\\pi_n)$ controls $h$ from the left\\footnote{\n    In particular, one can take an appropriate partition\n    $\\tau = \\{ \\mathopen{\\rbrack} t_i, t_{i+1} \\rbrack ; i \\in \\mathbb{N} \\} \\in \\mathop{\\mathrm{Par}}(\\mathbb{R}_{\\geq 0})$\n    such that $I \\subset [t_i, t_{i+1}]$\n    for some $i$ if $\\xi$ is constant on $I$.\n    Now set \n    \\begin{equation*}\n        h = \\sum_{i} \\xi(t_i) \\lbrack t_i,t_{i+1} \\mathclose{\\lbrack}.\n    \\end{equation*}\n    Then the sequence $(\\pi_n)$ approximates $h$ from the left.\n}.\n\nThen,\n\\begin{align*}\n    &\n    \\left\\lVert \n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle\n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t \n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle \n                \\xi_{u-}, \\mathrm{d}\\!\\left[ X,X \\right]_u\n            \\right\\rangle\n    \\right\\rVert_G   \\notag \\\\\n    & \\qquad \\leq \n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n            \\left\\langle\n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle \n    \\right\\rVert_G \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n    \\right\\rVert_G \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                \\xi(u-), \\mathrm{d}[ X,X ]_u \n            \\right\\rangle\n    \\right\\rVert_G.\n\\end{align*}\nWe will observe the behaviour of each part of the last side.\n\nWe know from Lemma~\\ref{5.7d} that the second term converges to $0$ as $n \\to \\infty$.\nThat is,\n\\begin{equation*}\n    \\left\\lVert\n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle \n                h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n            \\right\\rangle\n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-), \\mathrm{d}[X, X]_u\n            \\right\\rangle \n    \\right\\rVert_G\n    \\xrightarrow[n \\to \\infty]{} 0.\n\\end{equation*}\nFrom the choice of $h$, we find that\n\\begin{align*}\n    \\left\\lVert\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\langle \n            \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rangle\n    - \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} \n        \\left\\langle\n            h(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rangle \n    \\right\\rVert_G\n& \\leq\n    \\varepsilon\n    \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\lVert \n            \\delta_r^s X_t\n        \\right\\rVert_E^2.\n\\end{align*}\nTherefore,\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n        \\left\\lVert \\xi(r)-h(r) \\right\\rVert\n        \\left\\lVert \n            \\delta_r^s X_t \\otimes \\delta_r^s X_t\n        \\right\\rVert\n    \\leq \\varepsilon \\overline{Q}(X)_t.\n\\end{equation*}\nOn the other hand, we have\n\\begin{align*}\n    \\left\\lVert\n        \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle\n                h(u-)-\\xi(u-), \\mathrm{d}[ X,X ]_u \n            \\right\\rangle\n    \\right\\rVert_G\n& \\leq \n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n        \\left\\lVert h(u-)-\\xi(u-) \\right\\rVert\n        \\mathrm{d}V([X,X])_t  \\\\\n& \\leq\n    \\varepsilon V([X,X])_t.\n\\end{align*}\n\nConsequently, we see that\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty}\n    \\left\\lVert \n        \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n}\n            \\left\\langle\n                \\xi(r), \\delta_r^s X_t \\otimes \\delta_r^s X_t \n            \\right\\rangle \n        - \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n            \\left\\langle \n                \\xi_{u-}, \\mathrm{d}[ X,X ]_u\n            \\right\\rangle\n    \\right\\rVert_G  \\\\\n\\leq\n    \\varepsilon \\{ \\overline{Q}(X)_t + V([X,X])_t \\}.\n\\end{equation*}\nBecause $\\varepsilon$ is chosen arbitrarily,\nwe get the desired conclusion.\n\\end{proof}\n\n\\section{The It{\\^o} formula}\n\nThis section is devoted to showing the It{\\^o} formula\nwithin our framework of the It{\\^o}-F{\\\"o}llmer calculus \nin Banach space.\nLet us begin with the definition of It{\\^o}-F{\\\"o}llmer integrals.\n\n\\begin{Def} \\label{5.9b}\nLet $(H,X) \\in D([0,\\infty\\mathclose{[},E \\times F)$\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nSuppose that a sequence of partitions $(\\pi_n)$ approximates\n$H$ from the left. We call the limit\n\\begin{equation*}\n    \\int_0^t b(H_{s-}, \\mathrm{d}X_{s})\n    = \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s})\n    := \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b\\left( H_r, \\delta_r^s X_t \\right) \\in G\n\\end{equation*}\nthe \\emph{It{\\^o}-F{\\\"o}llmer integral\nof $H$ with respect to $X$ through $b$\nalong the sequence $(\\pi_n)$}, if it exists.\nSimilarly,\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b'(\\mathrm{d}X_s, H_{s-})\n    = \\lim_{n \\to \\infty} \\sum_{\\mathopen{\\rbrack} r,s \\rbrack \\in \\pi_n} b'\\left( \\delta_r^s X_t,H_{r} \\right) \\in G\n\\end{equation*}\nis defined for a $b' \\in \\mathcal{L}^{(2)}(F,E;G)$.\n\\end{Def}\n\nIf $b$ is the canonical bilinear map\n$\\otimes \\colon E \\times F \\to E \\widehat{\\otimes}_{\\pi} F$,\nwe write\n\\begin{equation*}\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}X_s)\n    = \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} H_{s-} \\otimes \\mathrm{d}X_s.\n\\end{equation*}\n\n\\begin{Rem} \\label{5.9c}\nThe It{\\^o}-F{\\\"o}llmer integral of Definition~\\ref{5.9b}\ninherits the bilinear property from\n$b \\in \\mathcal{L}^{(2)}(E,F;G)$ in the following sense. \n\n\\begin{enumerate}\n    \\item Suppose that the following two It{\\^o}-F{\\\"o}llmer integrals exist:\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}), \\qquad \n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(K_{s-}, \\mathrm{d}X_{s}).\n        \\end{equation*}\n        Then, for every $\\alpha$, $\\beta \\in \\mathbb{R}$,\n        the It{\\^o}-F{\\\"o}llmer integral of $\\alpha H + \\beta K$\n        with respect to $X$ satisfies\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-} + K_{s-}, \\mathrm{d}X_{s})\n            =\n            \\alpha \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}) \n            + \\beta \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(K_{s-}, \\mathrm{d}X_{s}).\n        \\end{equation*}\n    \\item Suppose that the following two It{\\^o}-F{\\\"o}llmer integrals exist:\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}), \\qquad \n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}Y_{s}).\n        \\end{equation*}\n        Then, for every $\\alpha$, $\\beta \\in \\mathbb{R}$,\n        the It{\\^o}-F{\\\"o}llmer integral of $H$ with respect to \n        $\\alpha X + \\beta Y$ exists and satisfies\n        \\begin{equation*}\n            \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}(\\alpha X+ \\beta Y)_{s})\n            =\n            \\alpha \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}X_{s}) \n            + \\beta \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-}, \\mathrm{d}Y_{s}).\n        \\end{equation*}\n\\end{enumerate}\n\\end{Rem}\n\nFirst, we consider the case where\nthe integrator is a path of finite variation.\nFrom the dominated convergence theorem,\nwe can easily deduce the following proposition.\n\n\\begin{Prop} \\label{5.9d}\nLet $H \\in D(\\mathbb{R}_{\\geq 0},E)$, $A \\in FV(\\mathbb{R}_{\\geq 0},F)$,\nand $b \\in \\mathcal{L}^{(2)}(E,F;G)$.\nIf a sequence of partition $(\\pi_n)$ approximates $H$ from the left,\nwe have\n\\begin{equation*}\n    (\\mathrm{IF}) \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}A_s)\n    =\n    (\\mathrm{S}) \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} b(H_{s-},\\mathrm{d}A_s).\n\\end{equation*}\nHere, the integral of the left-hand side is the It{\\^o}-F{\\\"o}llmer integral\nby Definition~\\ref{5.9b},\nand that of the right-hand side is the usual Stieltjes integral.\n\\end{Prop}\n\nNow we start to prove our main theorem.\n\n\\begin{proof}[Proof of Theorem \\ref{2e}]\nFix $t>0$ arbitrarily and pick a compact convex set $K \\subset F \\times E$\nso that the image $A[0,t] \\times X[0,t]$ is contained in $K$\\footnote{\n    For example, we can take $K$ as the convex closed hull of $A[0,t] \\times X[0,t]$.\n}.\nWe know by Proposition~\\ref{3b} that such a $K$ actually exists.\n\n\\emph{Step 1: Convergence of the summation in the formula~\\eqref{2f}.}\nWe first confirm that the summation of jump terms converges absolutely in $G$.\nBecause $X$ has quadratic variation, we have\n\\begin{equation*}\n    \\sum_{0 < s \\leq t}\n        \\left\\lVert\n            \\langle D_x^2 f(A_{s-},X_{s-}), \\Delta X_s \\otimes \\Delta X_s \\rangle\n        \\right\\rVert_{G}\n\\leq\n    \\sup_{(a,x) \\in K}\n        \\left\\lVert D_x^2 f(a,x) \\right\\rVert\n        \\sum_{0 < s \\leq t} \\left\\lVert \\Delta X_s \\right \\rVert_{E}^2\n< \\infty.\n\\end{equation*}\nBy Taylor's theorem, we also find that\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s})  - \\langle D_a f(A_{s-},X_{s-}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\leq \n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s})  - \\langle D_a f(A_{s-},X_{s}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\quad +\n    \\sum_{0 < s \\leq t}\n    \\left\\lVert\n        \\langle D_a f(A_{s-},X_{s}), \\Delta A_{s} \\rangle - \\langle D_a f(A_{s-},X_{s-}), \\Delta A_{s} \\rangle\n    \\right\\rVert_G  \\\\\n    & \\qquad \\leq\n    2\\, \\omega(D_af(a,x);K)\n    V(A)_t < \\infty.\n\\end{align*}\nAgain by Taylor's theorem, we obtain\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n        \\left\\lVert\n            f(A_{s-},X_s) - f(A_{s-},X_{s-})  - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_{s} \\rangle\n        \\right\\rVert_G  \\\\\n    & \\qquad \\leq\n    \\sup_{(a,x) \\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t\n    <\n    \\infty.\n\\end{align*}\nTherefore,\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_s,X_s) - f(A_{s-},X_{s-})\n            - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n            - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad \\leq\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_s,X_s) - f(A_{s-},X_{s})\n            - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t}\n        \\lVert\n            f(A_{s-},X_{s}) - f(A_{s-},X_{s-})\n            - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n        \\rVert_{G}  \\\\\n    & \\qquad \\leq \n    2 \\,\\omega(D_af(a,x);K) \\, \n        V(A)_t\n    +\n    \\sup_{(a,x)\\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t,\n\\end{align*}\nand that estimate implies\n\\begin{align*}\n    &\n    \\sum_{0 < s \\leq t}\n    \\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s-})\n        - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n    \\rVert_{G}  \\\\\n    & \\qquad \\leq\n    \\sum_{0 < s \\leq t}\n    \\lVert\n        f(A_s,X_s) - f(A_{s-},X_{s-})\n        - \\langle D_x f(A_{s-},X_{s-}),\\Delta X_s \\rangle\n        - \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle\n    \\rVert_{G}  \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t}\n        \\lVert \\langle D_a f(A_{s-},X_{s-}),\\Delta A_s \\rangle \\rVert_G \\\\\n    & \\qquad \\leq \n    3 \\,\\omega(D_af(a,x);K) \\, \n        V(A)_t\n    +\n    \\sup_{(a,x)\\in K} \\lVert D^2_x f(a,x) \\rVert V([X,X])_t \\\\\n    & \\qquad < \\infty.\n\\end{align*}\n\nFrom these absolute convergence results,\nwe can see that equation~\\eqref{2f}\nis equivalent to the following:\n\\begin{align} \\label{5.10d}\n    & \n    f(A_t,X_t) - f(A_0,X_0) \n    -\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack}\n        \\langle D_x f (A_{s-},X_{s-}), \\mathrm{d}X_s \\rangle   \\\\\n    & \\qquad = \n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\langle D_a f (A_{s-},X_{s-}), \\mathrm{d}A_{s} \\rangle \n    +\n    \\frac{1}{2} \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \n        \\langle D_x^2 f (A_{s-},X_{s-}), \\mathrm{d}[X,X]_s \\rangle   \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{0 < s \\leq t} \n        \\left\\{ \\Delta f (A_s,X_s) - \\langle D_x f (A_{s-},X_{s-}), \\Delta X_s \\rangle \\right\\}  \\notag \\\\\n    & \\qquad\\quad - \n    \\sum_{ 0 < s \\leq t} \n        \\langle D_a f (A_{s-},X_{s-}), \\Delta A_s \\rangle\n        - \n        \\frac{1}{2} \\sum_{0 < s \\leq t} \n            \\langle D_x^2 f (A_{s-},X_{s-}), \\Delta X_s \\otimes \\Delta X_s \\rangle. \\notag\n\\end{align}\nWe will therefore prove equation~\\eqref{5.10d} instead of \\eqref{2f}.\n\n\\emph{Step 2: The Taylor expansion.}\nNow we write $D = D(A,X)$,\n$D_{\\varepsilon} = D_{\\varepsilon}(A,X)$,\nand $D^{\\varepsilon}(A,X)$.\n\nLet $\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n$ and \nconsider the first-order Taylor expansion with respect to the variable $a$\nbetween $A_{u \\wedge t}$ and $A_{v \\wedge t}$.\nThen we have \n\\begin{equation} \\label{5.10e}\n    f(A_{v \\wedge t},X_{v \\wedge t}) - f(A_{u \\wedge t},X_{v \\wedge t})\n=\n    \\langle \n        D_a f( A_{u \\wedge t}, X_{u \\wedge t} ),\\delta_u^v A_t\n    \\rangle\n    +\n    \\langle\n        r_u^v, \\delta_u^v A_t\n    \\rangle,\n\\end{equation}\nwhere\n\\begin{equation*}\n    r_u^v\n=\n    \\int_{[0,1]}\n        \\left\\{\n            D_a f \\left( A_{u \\wedge t} + \\theta_u^v A_t,X_{v \\wedge t} \\right)\n            -\n            D_a f\\left( A_{u \\wedge t},X_{u \\wedge t} \\right)\n        \\right\\}\n    \\mathrm{d}\\theta.\n\\end{equation*}\n\nNext, we consider the second-order Taylor expansion\n\\begin{align} \\label{5.10f}\n    &\n        f(A_{u \\wedge t},X_{v \\wedge t}) - f(A_{u \\wedge t},X_{u \\wedge t})  \\\\\n    & \\qquad =\n        \\langle D_x f(A_{u \\wedge t},X_{u \\wedge t}), \\delta_u^v X_t \\rangle\n        + \n        \\frac{1}{2}\n        \\langle\n            D_x^2 f(A_{u \\wedge t},X_{u \\wedge t}), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n        +\n        \\langle \n            R_u^v, (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle, \\notag\n\\end{align}\nwith $R_u^v$ in the residual term given by \n\\begin{equation*}\n    R_u^v\n=\n    \\frac{1}{2}\n    \\int_{[0,1]}\n        (1-\\theta)\n        \\left\\{ \n            D^2_x f(A_{u \\wedge t},X_{u \\wedge t} + \\theta \\delta_u^v X_t)\n            -\n            D_x^2 f(A_{u \\wedge t},X_{u \\wedge t})\n        \\right\\}\n    \\mathrm{d}\\theta.\n\\end{equation*}\n\nCombining equations~\\eqref{5.10e} and \\eqref{5.10f}, we obtain\n\\begin{align*}\n    \\delta_u^v (f \\circ (A,X))_t\n& =\n    \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    +\n    \\langle r_u^v, \\delta_u^v A_t \\rangle\n    +\n    \\langle D_x f(A_{u},X_{u}), \\delta_u^v X_t \\rangle     \\\\\n    & \\quad +\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    +\n    \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle.\n\\end{align*}\nMoreover, by summing up this equality through $\\pi_n$, \nwe see that\n\\begin{align*}\n    &\n    f(X_t)- f(X_0)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle \n            D_x f(A_{u},X_{u}),\\delta_u^v X_t\n        \\rangle    \\\\\n    & \\qquad =\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle r_u^v, \\delta_u^v A_t \\rangle  \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right), (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle.\n\\end{align*}\nUsing the notation $e^1_{D_{\\varepsilon}}$ and $e^2_{D_{\\varepsilon}}$,\nwe can transform the previous equality as\n\\begin{align} \\label{5.10g}\n    &\n    f(X_t)- f(X_0)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle \n            D_x f(A_{u},X_{u}),\\delta_u^v X_t\n        \\rangle   \\\\\n    & \\qquad =\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\delta_u^v (f \\circ (A,X))_t e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack)\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_x f(A_u,X_u), \\delta_u^v X_t \\rangle \n        e^1_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle D_a f(A_{u},X_{u}), \\delta_{u}^v A_t \\rangle e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right),\n            (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n    -\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\frac{1}{2}\n        \\langle\n            D_x^2 f \\left( A_{u},X_{u} \\right),\n            (\\delta_u^v X_t)^{\\otimes 2}\n        \\rangle\n        e^1_{D_\\varepsilon}(\\mathopen{\\rbrack} u,v \\rbrack) \\notag \\\\\n    & \\qquad\\quad +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle r_u^v, \\delta_u^v A_t \\rangle e^2_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)\n    +\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n}\n        \\langle R_u^v, (\\delta_u^v X_t)^{\\otimes 2} \\rangle\n        e^2_{D_{\\varepsilon}}(\\mathopen{\\rbrack} u,v \\rbrack)  \\notag \\\\\n    & \\qquad =\n    I_1^{(n)}(t) - I_2^{(n)}(t) + I_3^{(n)}(t) - I_4^{(n)}(t)\n    + I_5^{(n)}(t) - I_6^{(n)}(t) + I_7^{(n)}(t) + I_8^{(n)}(t). \\notag\n\\end{align}\n\n\\emph{Step 3: Behaviour of $I_1^{(n)}(t),\\dots , I_8^{(n)}(t)$ of \\eqref{5.10g}.}\nBecause $D_{\\varepsilon}$ is discrete and $(\\pi_n)$ controls $(X,A)$,\nwe can deduce that\n\\begin{align*}\n    \\lim_{n \\to \\infty} I_4^{(n)}(t)\n& = \n    \\sum_{s \\in D_{\\varepsilon} \\cap [0,t]} \n        \\langle \n            D_a f(A_{s-},X_{s-}), \\Delta A_s\n        \\rangle   \\\\\n    \\lim_{n \\to \\infty} I_2^{(n)}(t)\n& =\n    \\sum_{s \\in D_\\varepsilon \\cap [0,t]} \n        \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle    \\\\\n    \\lim_{n \\to \\infty} I_6^{(n)}(t)\n& = \n    \\frac{1}{2} \\sum_{s \\in D_\\varepsilon \\cap [0,t]} \n        \\langle \n            D_x^2 f(A_{s-},X_{s-}), (\\Delta X_s)^{\\otimes 2}\n        \\rangle.\n\\end{align*}\nIf $s \\in D$ and $s \\leq t$, we see that \n\\begin{align*}\n    \\delta_{\\pi_n(s)}f(A,X)_t\n    & =\n    \\int_0^1 \\langle D_{a,x}f((A,X)_{\\underline{\\pi_n}(s)} + \\theta \\delta_{\\pi_n(s)} (A,X)_t), \n        \\delta_{\\pi_n(s)} (A,X)_t \\rangle \\,\\mathrm{d} \\theta   \\\\\n    & \\xrightarrow[n \\to \\infty]{}\n    \\int_0^1 \\langle D_{a,x}f((A,X)_{s-} + \\theta \\Delta (A,X)_s), \n        \\Delta (A,X)_s \\rangle \\,\\mathrm{d} \\theta  \n    = \\Delta f(A,X)_s.\n\\end{align*}\nfrom the assumption and the dominated convergence theorem. Hence, \n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_1^{(n)}(t)\n    = \n    \\sum_{s \\in D_{\\varepsilon} \\cap [0,t]} \\Delta f (A_s,X_s).\n\\end{equation*}\n\nBy Lemma~\\ref{2g}, we have\n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_5^{(n)}(t)\n= \n    \\frac{1}{2} \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \n        \\langle \n            D_x^2 f(A_{s-},X_{s-}),\\mathrm{d}[X,X]_s\n        \\rangle.\n\\end{equation*}\n\nThe dominated convergence theorem gives\n\\begin{equation*}\n    \\lim_{n \\to \\infty} I_3^{(n)}(t)\n=\n    \\int_{\\mathopen{\\rbrack} 0,t \\rbrack} \\langle D_a f(A_{s-},X_{s-}) , \\mathrm{d}A_s \\rangle.\n\\end{equation*}\n\nIt remains to estimate the residual terms.\nIf $\\mathopen{\\rbrack} u,v \\rbrack \\cap D_{\\varepsilon} = \\emptyset$,\noscillation of the paths on $\\mathopen{\\rbrack} u,v \\rbrack$ is well controlled as\n\\begin{gather*}\n\\omega(X, \\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t])\n    \\leq \n        O^{+}_t(X-J(D_{\\varepsilon}(X);X);\\pi_n) + \\varepsilon,   \\\\\n\\omega(A, \\mathopen{\\rbrack} u,v \\rbrack \\cap [0,t])\n    \\leq \n        O^{+}_t(A-J(D_{\\varepsilon}(A);A);\\pi_n) + \\varepsilon.\n\\end{gather*}\nNow we write, for convenience,\n\\begin{gather*}\n    \\alpha(\\varepsilon,n) = O^{+}_t(X-J(D_{\\varepsilon}(X);X);\\pi_n) + \\varepsilon,   \\\\\n    \\beta(\\varepsilon,n) = O^{+}_t(A-J(D_{\\varepsilon}(A);A);\\pi_n) + \\varepsilon.\n\\end{gather*}\nBy the assumption that $(\\pi_n)$ controls $(A,X)$---and hence so does both $A$ and $X$---we see that\n\\begin{equation*}\n    \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} \\alpha(\\varepsilon,n) = 0,  \\qquad\n    \\varlimsup_{\\varepsilon \\downarrow\\downarrow 0} \\varlimsup_{n \\to \\infty} \\beta(\\varepsilon,n) = 0.\n\\end{equation*}\nThis implies\n\\begin{gather*}\n    I_8^{(n)}(t)\n\\leq \n    \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n)}}\n        \\lVert D_x^2 f(z) - D_x^2 f(w) \\rVert\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \\lVert \\delta_u^v X_t \\rVert_{E}^2,   \\\\\n    I_7^{(n)}(t)\n\\leq \n    \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n) + \\beta(\\varepsilon,n)}}\n        \\lVert D_a f(z) - D_a f(w) \\rVert\n    \\sum_{\\mathopen{\\rbrack} u,v \\rbrack \\in \\pi_n} \\lVert \\delta_u^v A_t \\rVert_{E}.\n\\end{gather*}\n\nConsequently,\n\\begin{align*}\n    &\n    \\varlimsup_{n \\to \\infty}\n        \\left\\lVert\n            (\\text{RHS of \\eqref{5.10d}}) - (\\text{RHS of \\eqref{5.10g}})\n        \\right\\rVert_{G}     \\\\\n    & \\qquad \\leq\n    \\left\\lVert \n        \\sum_{s \\in D^{\\varepsilon} \\cap [0,t]}\n            \\left\\{ \n                \\Delta f(A_s,X_s) - \\langle D_x f(A_{s-},X_{s-}), \\Delta X_s \\rangle\n            \\right\\}\n    \\right\\rVert       \\\\\n    & \\qquad\\quad +\n    \\left\\lVert\n        \\sum_{ s \\in D^{\\varepsilon} \\cap [0,t] } \n            \\langle D_a f(A_{s-},X_{s-}), \\Delta A_s \\rangle\n    \\right\\rVert\n    + \n    \\left\\lVert\n        \\sum_{ s \\in D^{\\varepsilon} \\cap [0,t] } \n            \\langle D_x^2 f(A_{s-},X_{s-}),  \\Delta X_s \\otimes \\Delta X_s \\rangle \n    \\right\\rVert     \\\\\n    & \\qquad\\quad +\n    \\varlimsup_{n \\to \\infty}\n        \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n)}}\n            \\lVert D_x^2 f(z) - D_x^2 f(w) \\rVert\n        \\overline{Q}(X)_t,   \\\\\n    & \\qquad\\quad +\n    \\varlimsup_{n \\to \\infty}\n        \\sup_{\\substack{z,w \\in K  \\\\ \\lvert z-w \\rvert \\leq \\alpha(\\varepsilon,n) + \\beta(\\varepsilon,n)}}\n            \\lVert D_a f(z) - D_a f(w) \\rVert\n        V(A)_t.\n\\end{align*}\nFinally, by letting $\\varepsilon \\downarrow\\downarrow 0$,\nwe obtain\n\\begin{equation*}\n    \\varlimsup_{n \\to \\infty}\n        \\left\\lVert\n            (\\text{ RHS of \\eqref{5.10d} }) - (\\text{ RHS of \\eqref{5.10g} })\n        \\right\\rVert_{G}\n    = 0.\n\\end{equation*}\nThis immediately leads the claim of the theorem.\n\\end{proof}\n\nCombining Theorem~\\ref{2e} and Corollary~\\ref{5.6d},\nwe obtain the integration by parts formula.\nNote that the existence of the It{\\^o}-F{\\\"o}llmer\nintegral $\\int_{0}^{t} A_{s-} \\otimes \\mathrm{d}X_{s}$\nfollows from Theorem \\ref{2e} and Proposition \\ref{5.9d}.\n\n\\begin{Cor}\nLet $(\\pi_n)$, $X$, and $A$ satisfy the same assumption \nas Theorem~\\ref{2e}. Then,\n\\begin{equation*}\n    A_t \\otimes X_t\n    = \n    \\int_{0}^{t} \\mathrm{d}A_s \\otimes X_{s-}\n    + \\int_{0}^{t} A_{s-} \\otimes \\mathrm{d}X_{s}\n    + [A,X]_t.\n\\end{equation*}\n\\end{Cor}\n\n\\paragraph{Acknowledgments}\nThe author thanks his supervisor Professor Jun Sekine for helpful support and encouragement.\nThe author also thanks Professor Masaaki Fukasawa for his helpful comments and discussion. \n\n\\printbibliography[heading=bibintoc]\n\nGraduate School of Engineering Science,\nOsaka University, Osaka, Japan\n\n\\emph{E-mail}: hirai@sigmath.es.osaka-u.ac.jp\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec-intro}\n\nWe address the problem of learning representations for event sequences generated by real-world users which we call \\emph{lifestream} data or lifestreams. Event sequence data is produced in many business applications, some examples being credit card transactions and click-stream data of internet site visits, and the event sequence analysis is a very common machine learning problem~\\cite{laxman2008stream},~\\cite{wiese2009credit},~\\cite{zhang2017credit},~\\cite{bigon2019prediction}. Lifestream is an event sequence that is attributed to a person and captures his\/her regular and routine actions of certain type, e.g., transactions, search queries, phone calls and messages.\n\n\n\n\n\n\nIn this paper, we present a novel \\emph{Metric Learning for Event Sequences (MeLES)} method for learning low-dimensional representations of event sequences, which copes with specific properties of lifestreams such as their discrete nature. In a broad sense, MeLES method adopts metric learning techniques~\\cite{xing2003distance},~\\cite{Hadsell:2006:DRL:1153171.1153654}. Metric learning is often used in a supervised manner for mapping high-dimensional objects to a low-dimensional embedding space. The aim of metric learning is to represent semantically similar objects (images, video, audio, etc.) closer to each other, while dissimilar ones further. Most metric learning methods are used in such applications as speech recognition~\\cite{wan2017generalized}, computer vision~\\cite{Schroff2015FaceNetAU},~\\cite{Mao2019AdvRobust} and text analysis~\\cite{reimers-2019-sentence-bert}. In these domains, metric learning is successfully applied in a supervised manner to datasets, where pairs of high-dimensional instances are labeled as the same object or different ones. \nUnlike all the previous metric learning methods, MeLES is fully self-supervised and does not require any labels. It is based on the observation that lifestream data obeys periodicity and repeatability of events in a sequence. Therefore, one can consider some convenient sub-sequences of the same lifestream as auxiliary high-dimensional representations of the same person. The idea of MeLES is that low-dimensional embeddings of such sub-sequences should be closer to each other.\n\nSelf-supervised learning approach allows us to train rich models using the internal structure of large unlabelled or partially labeled training datasets. Self-supervised learning have demonstrated effectiveness in different machine learning domains, such as Natural Language Processing (e. g. ELMO \\cite{ELMO2018}, BERT \\cite{Devlin2019BERTPO}) and computer vision \\cite{doersch2015unsupervised}.\n\n\nMeLES model trained in self-supervised manner can be used in two ways. Representations, produced by the model can be directly used as a fixed vector of features in some supervised downstream task (e. g. classification task) similarly to~\\cite{word2vec}. Alternatively, trained model can be fine-tuned~\\cite{Devlin2019BERTPO} for the specific downstream task.\n\n\n\nWe conducted experiments on two public bank transaction datasets and evaluated performance of the method on downstream classification tasks. When MeLES representations is directly used as features the method achieve strong performance comparable to the baseline methods. The fine-tuned representations achieve state-of-the-art performance on downsteam classification tasks, outperforming several other supervised methods and methods with unsupervised pre-training by a significant margin.\n\nMoreover, we show superiority of MeLES embeddings over supervised approach applied to partially labeled raw data due to insufficient amount of the target to learn a sufficiently complex model from scratch.\n\nEmbedding generation is a one-way transformation, hence it is impossible to restore exact event sequence from its representation. Therefore, the usage of embeddings leads to better privacy and data security for the end users than when working directly with the raw event data, and all this is achieved without sacrificing valuable modelling power.\n\nIn this paper, we make the following contributions. We\n\\begin{enumerate}\n    \\item adopted the ideas of metric learning to the analysis of the lifestream data in a novel, self-supervised, manner;\n    \\item proposed a specific method, called Metric Learning for Event Sequences (MeLES), to accomplish this task; \n    \\item demonstrated that the proposed MeLES method significantly outperforms other baselines for both the supervised and the semi-supervised learning scenarios on lifestream data.\n\\end{enumerate}\n\n\n\\section{Related work} \\label{sec-rel-work}\n\n\nThe metric learning approach, which underlies our MeLES method, has been widely used in different domains, including computer vision, NLP and audio domains. \n\nIn particular, metric learning approach for face recognition was initially proposed in \\cite{chopra2005learning}, where contrastive loss function was used to learn a mapping of the input data to a low-dimensional manifold using some prior knowledge of neighborhood relationships between training samples or manual labeling. Further, in~\\cite{Schroff2015FaceNetAU}, authors introduced FaceNet, a method which learns a mapping from face images to 128-dimensional embeddings using a triplet loss function based on large margin nearest neighbor classification (LMNN)~\\cite{weinberger2006distance}. In FaceNet, authors also introduced online triplet selection and hard-positive and hard-negative mining technique for training procedure.\n\n\nAlso, metric learning has been used for the speaker verification task \\cite{wan2017generalized}, where the contrast loss is defined as embedding of each utterance being similar to the centroid of all that speaker's embeddings (positive pair) and far from other speaker's centroids with the highest similarity among all false speakers (hard negative pair).\n\nFinally, in~\\cite{reimers-2019-sentence-bert}, authors proposed a fine-tuned BERT model~\\cite{Devlin2019BERTPO} that use metric learning in the form of siamese and triplet networks to train sentence embeddings for semantic textual similarity tasks using semantic proximity annotation of sentence pairs.\n\nAlthough metric learning was used in all these domains, it has not been applied to the analysis of the lifestream problems involving transactional, click-stream and other types of lifestream data, which is the focus of this paper.\n\nImportantly, previous literature applied metric learning to their domains in a supervised manner, while our MeLES method adopts the ideas of metric learning in a novel fully self-supervised manner to the event sequence domain.  \n\nThe another idea of applying self-supervised learning to sequential data has been previously proposed in Contrastive Predictive Coding (CPC) method \\cite{DBLP:journals\/corr\/abs-1807-03748}, where meaningful representations are extracted by predicting future in the latent space by using autoregressive methods. CPC representations demonstrated strong performance on four distinct domains: audio, computer vision, natural language and reinforcement learning. \n\nIn computer vision domain, there are many other different approaches to self-supervised learning that are nicely summarized in \\cite{jing2019selfsupervised}. There are several ways to define a self-supervision task (pretext task) for an image. One option is to somehow change an image and then try to restore the original image. The examples of this approach are super-resolution, image colorization and corrupted image restoration. Another option is to predict context information from the local features e. g. predict the place of image patch on the image with several missing patches.\n\nNote that almost every self-supervised learning approach can be reused for the representation learning in the form of embeddings. There are several examples of using a single set of embeddings for several downstream tasks \\cite{Song2017LearningUE}, \\cite{Zhai:2019:LUE:3292500.3330739}.\n\n\nOne of the common approaches to learn self-supervised representations is either traditional autoencoder (\\cite{rumelhart1985learning}) or variational autoencoder (\\cite{kingma2013auto}). It is widely used for images, text and audio or aggregated lifestream data (\\cite{mancisidor2019learning}). Although autoencoders has been successfully used in several domains listed above, they has not been applied to the raw lifestream data in the form of event sequences, mainly due to the challenges of defining distances between the input and the reconstructed input sequences.\n\nIn the next section, we describe how the ideas of metric learning are applied to the event sequences in a self-supervised manner.\n\n\\section{The Method} \\label{sec-method}\n\n\\subsection{Lifestream data}\n\nWe designed the method specially for the lifestream data. Lifestream data consists of discrete events per person in continuous time, for example,  behavior on websites, credit card transactions, etc. \n\nConsidering credit card transactions, each transaction have a set of attributes, either categorical or numerical including the timestamp of the transaction. An example of the sequence of three transactions with their attributes is presented in the Table \\ref{tab-tr-data}.\nMerchant type field represents the category of a merchant, such as \"airline\", \"hotel\", \"restaurant\", etc.\n\n\\begin{table}[h]\n\\caption{Data structure for a single credit card}\n\\begin{tabular}{ | m{7em} |  m{5em} m{5em} m{5em}| }\n\\hline\n\\textbf{Amount} & 230 & 5 & 40 \\\\\n\\textbf{Currency} & EUR & USD & USD \\\\\n\\textbf{Country} & France & US & US \\\\\n\\textbf{Time} & 16:40 & 20:15 & 09:30 \\\\\n\\textbf{Date} & Jun 21 & Jun 21 & Jun 22 \\\\\n\\textbf{Merchant Type} & Restaurant & Transport\\-ation & Household Appliance \\\\\n\\hline\n\\end{tabular}\n\\label{tab-tr-data}\n\\end{table}\n\nAnother example of lifestream data is click-stream: the log of internet pages  visits. The example of a click-stream log of a single user is presented in Table \\ref{tab-cs-data}.\n\n\\begin{table}[h]\n\\caption{Click-stream structure for a single user}\n\\begin{tabular}{ | m{2em} m{3em} m{7em} m{10em} | }\n\\hline\n\\textbf{Time} & \\textbf{Date} & \\textbf{Domain} & \\textbf{Referrer Domain} \\\\\n\\hline\n17:40 & Jun 21 & amazon.com & google.com \\\\\n17:41 & Jun 21 & amazon.com & amazon.com \\\\\n17:45 & Jun 21 & en.wikipedia.org & google.com \\\\\n\\hline\n\\end{tabular}\n\\label{tab-cs-data}\n\\end{table}\n\n\\subsection{General framework}\n\n\\begin{figure*}[ht]\n  \\caption{General framework}\n  \\includegraphics[scale=0.85]{figures\/arch-v2.pdf}\n  \\label{fig-arch}\n\\end{figure*}\n\nThe overview of the method is presented at Figure \\ref{fig-arch}. Given a sequence of discrete events $\\{x_t \\}^T_{t=1}$ in a given observation interval [1, T] the ultimate goal is to obtain a sequence embedding $c_t$ for the timestamp $T$ in the latent space $R^d$. To train the encoder to generate meaningful embedding $c_t$ from $\\{x_t \\}^T_{t=1}$ we apply a metric learning approach such that the distance between embeddings of the same person is small, whereas embeddings of the different persons (negative pairs) is large.\n\nOne of the difficulties with applying metric learning approach to the lifestream data is that the notion of semantic similarity as well as dissimilarity requires underlying domain knowledge and human labor-intensive labeling process to constrain positive and negative examples. \nThe key property of the lifestream data domain is periodicity and repeatability of the events in the sequence which allows us to reformulate the metric learning task in a self-supervised manner. MeLES learns low-dimensional embeddings from person sequential data, sampling positive pairs as sub-sequences of the same person sequence and negative pairs as sub-sequences from different person sequences. See section \\ref{sec-pos-pairs} for details of the positive pairs generation.\n\nEmbedding $c_t$ is generated by encoder neural network which is described in section \\ref{sec-enc-arch}. Metric learning losses are described in section \\ref{sec-ml-loss}. Positive pairs generation strategies are described in section \\ref{sec-pos-pairs}. Negative pairs sampling strategies are described in section \\ref{sec-neg-samples}.\n\nSequence embedding $c_t$ obtained by the metric learning approach is then used in various donwstream machine learning tasks as a feature vector. Also, a possible way to improve the downstream task performance is to feed a pre-trained embedding $c_t$ (e. g. the last layer of RNN) to a task-specific classification subnetwork and then jointly fine-tune the model parameters of the encoder and classifier subnetworks.\n\n\\subsection{Encoder architecture} \\label{sec-enc-arch}\n\nTo embed a sequence of events to the fixed-size vector $c_t \\in R^d$ we use approach similar to the E.T.-RNN card transaction encoder proposed in \\cite{10.1145\/3292500.3330693}. The whole encoder network consists of two conceptual parts: the event encoder and the sequence encoder subnetworks.\n\nThe event encoder $e$ takes the set of attributes of a single event $x_t$ and outputs its representation in the latent space $Z \\in R^m$: $z_t = e(x_t)$. The sequence encoder $s$ takes latent representations of the sequence of events: $ z_{1:T} = z_1, z_2, \\cdots z_T $ and outputs the representation of the whole sequence $c_t$ in the time-step $t$: $ c_t = s(z_{1:t}) $.\n\nThe event encoder consists of the several embedding layers and batch normalization\\cite{10.5555\/3045118.3045167} layer. Each embedding layer is used to encode each categorical attribute of the event. Batch normalization is applied to numerical attributes of the event. Finally, outputs of every embedding layer and batch normalization layer are concatenated to produce the latent representation $z_t$ of the single event.\n\nThe sequence of latent representations of event representations $z_{1:t}$ is passed to sequence encoder $s$ to obtain a fixed-size vector $c_t$. Several approaches can be used to encode a sequence. One possible approach is to use the recurrent network (RNN) as in \\cite{Sutskever:2014:SSL:2969033.2969173}. The other approach is to use the encoder part of the Transformer architecture presented in \\cite{DBLP:journals\/corr\/VaswaniSPUJGKP17}. In both cases the output produced for the last event can be used to represent the whole sequence of events. In case of RNN the last output $h_t$ is a representation of the sequence.\n\nEncoder, based on RNN-type architecture like GRU\\cite{cho2014learning}, allows to calculate embedding $c_{t+k}$ by updating embedding $c_t$ instead of  calculating embedding $c_{t+k}$ from the whole sequence of past events $z_{1:t}$: $c_k = rnn(c_t, z_{t+1:k})$. This option allows to reduce inference time to update already existing person embeddings with new events, occurred after the calculation of embeddings. This is possible due to the recurrent nature of RNN-like networks.\n\n\\subsection{Metric learning losses} \\label{sec-ml-loss}\n\nMetric learning loss discriminates embeddings in a way that embeddings from same class are moved closer together and embeddings from the different class are moved further. Several metric learning losses have been considered - contrastive~loss~\\cite{Hadsell:2006:DRL:1153171.1153654}, binomial deviance loss \\cite{Yi:2014:LUE:1407.4979}, triplet loss \\cite{Hoffer:2015:LUE:1412.6622}, histogram~loss~\\cite{histogram-loss} and margin~loss~\\cite{wu2017sampling}. All of this losses address the following challenge of the metric learning approach: using all pairs of samples is inefficient, for example, some of the negative pairs are already distant enough thus this pairs are not valuable for the training~(\\cite{simo2015discriminative},~\\cite{wu2017sampling},~\\cite{Schroff2015FaceNetAU}).\n\nIn the next paragraphs we will consider two kinds of losses, which are conceptually simple, and yet demonstrated strong performance on validation set in our experiments (see Table \\ref{tab-loss-type}), namely contrastive loss and margin loss.\n\nContrastive loss has a contrastive term for the negative pair of embeddings which penalizes the model only if the negative pair is not distant enough and the distance between embeddings is less than a margin $m$:  \n\\begin{equation}\n \\mathcal{L} = \\sum_{i=1}^P \\left[ (1-Y)\\dfrac{1}{2}(D_W^i)^2 +Y*\\dfrac{1}{2}\\{max(0,m-D_W^i)\\}^2 \\right],\n\\end{equation}\nwhere $P$ is the count of all pairs in a batch, $D_W^i$ - is a distance function between a i-th labeled sample pair of embeddings $X_1$ and $X_2$, \n$Y$ is a binary label assigned to a pair: $Y = 0$ means a similar pair, $Y = 1$ means dissimilar pair, $m > 0$ is a margin.\nAs proposed in \\cite{Hadsell:2006:DRL:1153171.1153654} we use euclidean distance as the distance function: $D_W^i = D(A,B) = \\sqrt{\\sum_i(A_i - B_i)^2}$.\n\nMargin loss is similar to the contrastive loss, the main difference is that there is no penalty for positive pairs which are closer than threshold in a margin loss.\n\\begin{equation}\n \\mathcal{L} = \\sum_{i=1}^P \\left[ (1-Y)max(0, D_W^i - b + m) + Y*max(0, b-D_W^i + m) \\right],\n\\end{equation}\nwhere $P$ is the count of all pairs in a batch, $D_W^i$ - is a distance function between a i-th labeled sample pair of embeddings $X_1$ and $X_2$,\n$Y$ is a binary label assigned to a pair: $Y = 0$ means a similar pair, $Y = 1$ means dissimilar pair, $m > 0$ and $b > 0$ define the bounds of a margin.\n\n\\subsection{Negative sampling} \\label{sec-neg-samples}\n\nNegative sampling is another way to address the issue that some of the negative pairs are already distant enough thus this pairs are not valuable for the training (\\cite{simo2015discriminative}, \\cite{wu2017sampling}, \\cite{Schroff2015FaceNetAU}). Hence, only part of possible negative pairs are considered during loss calculation. Note, that only current batch samples are considered. There are several possible strategies of selecting most relevant negative pairs.\n\n\\begin{enumerate}\n    \\item Random sample of negative pairs\n    \\item Hard negative mining: generate k hardest negative pairs for each positive pair.\n    \\item Distance weighted sampling, where negative samples are drawn uniformly according to their relative distance from the anchor. \\cite{wu2017sampling}\n    \\item Semi-hard sampling, where we choose the nearest to anchor negative example, from samples which further away from the anchor than the positive exemplar (\\cite{Schroff2015FaceNetAU}).\n\\end{enumerate}\n\nIn order to select negative samples, we need to compute pair-wise distance between all possible pairs of embedding vectors of a batch. For the purpose of making this procedure more computationally effective we perform normalization of the embedding vectors, i.e. project them on a hyper-sphere of unit radius. Since $D(A,B) = \\sqrt{\\sum_i(A_i - B_i)^2} = \\sqrt{\\sum_i A_i^2 + \\sum_i B_i^2 - 2\\sum_i A_i B_i} $ and $||A||= ||B||=1$, to compute the the euclidean distance we only need to compute: $\\sqrt{2 - 2(A \\cdot B)}$.\n\nTo compute the dot product between all pairs in a batch we just need to multiply the matrix of all embedding vectors of a batch by itself transposed, which is a highly optimized computational procedure in most modern deep learning frameworks. Hence, the computational complexity of the negative pair selection is $O(n^2h)$ where $h$ is the size of the output embeddings and $n$ is the size of the batch.\n\n\\subsection{Positive pairs generation} \\label{sec-pos-pairs}\n\nThe following procedure is used to create a batch during MeLES training. $N$ initial sequences are taken for batch generation. Then, $K$ sub-sequences are produced for each initial sequence.\n\nPairs of sub-sequences produced from the same sequence are considered as positive samples and pairs from different sequences are considered as negative samples. Hence, after positive pair generation each batch contains $N \\times K$ sub-sequences used as training samples. There are $K-1$ positive pairs and $(N - 1) \\times K$ negative pairs per sample in batch.\n\nThere are several possible strategies of sub-sequence generation. The simplest strategy is the random sampling without replacement. Another strategy is to produce a sub-sequence from random splitting sequence to several sub-sequences without intersection between them (see Algorithm \\ref{alg-disj-ss}). The third option is to use randomly selected slices of events with possible intersection between slices (see Algorithm \\ref{alg-slce-ss}). \n\nNote, that the order of events in generated sub-sequences is always preserved.\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\textbf{hyperparameters:} $k$ - amount of sub-sequences to be produced. \\\\\n\\textbf{input:} A sequence $S$ of length $l$. \\\\\n\\textbf{output:} $S_1,...,S_k$ - sub-sequences from $S$. \\\\\n\n\\BlankLine\nGenerate vector $inds$ of length $l$ with random integers from [1,k].\\\\\n \\For{$i\\leftarrow 1$ \\KwTo $k$}{\n $S_i = S[inds == i]$\n }\n \\caption{Disjointed sub-sequences generation strategy}\n\\label{alg-disj-ss}\n\n\\end{algorithm}\n\n\\begin{algorithm}\n\\SetAlgoLined\n\\textbf{hyperparameters:} $m, M$ - minimal and maximal possible length of sub-sequence. $k$ - amount of sub-sequences to be produced. \\\\\n\\textbf{input:} A sequence $S$ of length $l$. \\\\\n\\textbf{output:} $S_1,...,S_k$ - sub-sequences from $S$. \\\\\n\\BlankLine\n \\For{$i\\leftarrow 1$ \\KwTo $k$}{\n Generate random integer $l_i$, $m \\leqslant l_i \\leqslant \\min(M,l).$\\\\\n Generate random integer $s$, $0 \\leqslant s \\leqslant l-l_i.$\\\\\n $S_i = S[s: s + l_i]$\n }\n \\caption{Random slices sub-sample generation strategy}\n\\label{alg-slce-ss}\n\\end{algorithm}\n\n\\section{Experiments} \\label{sec-exp}\n\n\\subsection{Datasets} \\label{sec-datasets}\nIn our research we used several publicly available datasets of bank transactions.\n\\begin{enumerate}\n    \\item \\textbf{Age group prediction competition}\\footnote{https:\/\/onti.ai-academy.ru\/competition} - the task is to predict the age group of a person within 4 classes target and accuracy is used as a performance metric.\n    The dataset consists of 44M anonymized transactions representing 50k persons with a target labeled for only 30k of them (27M out of 44M transactions), for the other 20k persons (17M out of 44M transactions) label is unknown. Each transaction includes date, type (for example, grocery store, clothes, gas station, children's goods, etc.) and amount. We use all available 44M transactions for metric learning, excluding 10\\% - for the test part of the dataset, and  5\\% for the metric learning validation.\n        \n    \\item \\textbf{Gender prediction competition}\\footnote{https:\/\/www.kaggle.com\/c\/python-and-analyze-data-final-project\/} - the task is a binary classification problem of predicting the gender of a person and ROC-AUC metric is used.\n    The dataset consists of 6,8M anonymized transactions representing 15k persons, where only 8,4k of them are labeled. Each transaction is characterized by date, type (for ex. \"ATM cash deposit\"), amount and Merchant Category Code (also known as MCC).\n\\end{enumerate}\n\n\\subsection{Experiment setup}\n\nFor each dataset, we set apart 10\\% persons from the labeled part of data as the test set that we used for comparing different models.\n\nIf we do not explicitly mention alternative, in our experiments we use contrastive loss and random slices pair generation strategy.\n\nFor all methods random search on 5-fold cross-validation over the train set is used for hyper-parameter selection. The hyper-parameters with the best out-of-fold performance on train set is then chosen.\n\nThe final set of hyper-parameters used for MeLES is shown in the Table\\ref{tab-hyper}.\n\n\\begin{table}[h]\n\\caption{Hyper-parameters for MeLES training}\n\\begin{tabular}{ | m{18em} |  m{3em} | m{3em} | }\n\\hline\n& \\textbf{Age task} & \\textbf{Gender task} \\\\\n\\hline\n\\textbf{Learning rate} & 0.002 & 0.002 \\\\\n\\textbf{Number of samples in batch} & 64 & 128 \\\\\n\\textbf{Number of epochs} & 100 & 150 \\\\\n\\textbf{Number of generated sub-samples} (see Section \\ref{sec-pos-pairs}) & 5 & 5 \\\\\n\\hline\n\\end{tabular}\n\\label{tab-hyper}\n\\end{table}\n\nFor evaluation of semi-supervised\/self-supervised techniques (including MeLES), we used all transactions including unlabeled data, except for the test set, as far as those methods are suitable for partially labeled datasets, or does not require labels at all.\n\n\\subsubsection{Performance}\n\nNeural network training was performed on a single Tesla P-100 GPU card. For the training part of MeLES, the single training batch is processed in 142 millisecond. For age prediction dataset, the single training batch contains 64 unique persons with 5 sub-samples per person, i.e. 320 training samples in total, the mean number of transactions per sample is 90, hence each batch contains around 28800 transactions.\n\n\\subsubsection{Baselines} \\label{sec-baselines}\n\nWe compare our MeLES method with the following two baselines. First, we consider the Gradient Boosting Machine (GBM) method~\\cite{friedman2001} on hand-crafted features. GBM can be considered as a strong baseline in cases of tabular data with heterogeneous features. In particular, GBM-based approaches achieve state-of-the-art results in a variety of practical tasks including web search, weather forecasting, fraud detection, and many others~\\cite{wu2010adapting, Vorobev2019filter, zhang2015gradient, niu2019comparison}.\n\nSecond, we apply recently proposed Contrastive Predictive Coding (CPC)~\\cite{DBLP:journals\/corr\/abs-1807-03748}, a self-supervised learning method, which has shown an excellent performance for sequential data of such traditional domains as audio, computer vision, natural language, and reinforcement learning.\n\nGBM based model requires a large number of hand-crafted aggregate features produced from the raw transactional data. An example of an aggregate feature would be an average spending amount in some category of merchants, such as hotels of the entire transaction history.\nWe used LightGBM\\cite{NIPS2017_6907} implementation of GBM algorithm with nearly 1k hand-crafted features for the application. Please see the companion code for the details of producing hand-crafted features.\n\n\nIn addition to the mentioned baselines we compare our method with supervised learning approach where the encoder sub-network and with classification sub-network are jointly trained on the downstream task target. Note, that no pre-training is used in this case.\n\n\\subsubsection{Design choices}\n\nIn the Table\\ref{tab-enc-type}, Table\\ref{tab-loss-type}, Table\\ref{tab-pair-gen} and Table\\ref{tab-neg-sampl} we present the results of experiments on different design choices of our method.\n\nAs shown in Table\\ref{tab-enc-type}, different choices of encoder architectures show comparable performance on the downstream tasks.\n\nIt is interesting to observe that even contrastive loss that can be considered as the basic variant of metric learning loss allows to get strong results on the downstream tasks (see Table \\ref{tab-loss-type}). Our hypothesis is that an increase in the model performance on metric learning task does not always lead to an increase in performance on downstream tasks.\n\nAlso observe that hard negative mining leads to significant increase in quality on downstream tasks in comparison to random negative sampling (see Table\\ref{tab-neg-sampl}).\n\nAnother observation is that a more complex sub-sequence generation strategy (e. g. random slices) shows slightly lower performance on the downstream tasks in comparison to the random sampling of events (see Table\\ref{tab-pair-gen}).\n\n\\begin{table}[h]\n\\caption{Comparison of encoder types}\n\\begin{tabular}{ | m{10em} |  m{7em} | m{7em} | }\n\\hline\n\\textbf{Econder type} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{LSTM} & $0.620 \\pm 0.003$ & $0.870 \\pm 0.005$ \\\\\n\\textbf{GRU} & \\pmb{$0.639 \\pm 0.006$} & \\pmb{$0.871 \\pm 0.004$}  \\\\\n\\textbf{Transformer} & $0.621 \\pm 0.001$ & $0.848 \\pm 0.002$  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-enc-type}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of metric learning losses}\n\\begin{tabular}{ | m{10em} |  m{7em} | m{7em} |}\n\\hline\n\\textbf{Loss type} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Contrastive loss} & $0.639 \\pm 0.006$ & \\pmb{$0.871 \\pm 0.003$} \\\\\n\\textbf{Binomial deviance loss} & $0.535 \\pm 0.005$ & $0.853 \\pm 0.005$ \\\\\n\\textbf{Histogram loss} & \\pmb{$0.642 \\pm 0.002$} & $0.851 \\pm 0.004$ \\\\\n\\textbf{Margin loss} & $0.631 \\pm 0.003$ & \\pmb{$0.871 \\pm 0.004$} \\\\\n\\textbf{Triplet loss} & $0.610 \\pm 0.006$ & $0.855 \\pm 0.003$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab-loss-type}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of pair generation strategies}\n\\begin{tabular}{ | m{10em} |  m{7em} |  m{7em} | }\n\\hline\n\\textbf{Pair generation method} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Random samples} & $0.628 \\pm 0.003$ & $0.851 \\pm 0.004$ \\\\\n\\textbf{Random disjoint samples} & $0.608 \\pm 0.004$ & $0.836 \\pm 0.008$ \\\\\n\\textbf{Random slices} & \\pmb{$0.639 \\pm 0.006$} & \\pmb{$0.872 \\pm 0.005$}  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-pair-gen}\n\\end{table}\n\n\\begin{table}[h]\n\\caption{Comparison of negative sampling strategies}\n\\begin{tabular}{ | m{10em} |  m{7em} |  m{7em} | }\n\\hline\n\\textbf{Negative sampling strategy} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{Hard negative mining} & \\pmb{$0.637 \\pm 0.005$} & \\pmb{$0.872 \\pm 0.004$} \\\\\n\\textbf{Random negative sampling} & $0.615 \\pm 0.005$ & $0.826 \\pm 0.004$ \\\\\n\\textbf{Distance weighted sampling} & $0.620 \\pm 0.003$ & $0.867 \\pm 0.003$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab-neg-sampl}\n\\end{table}\n\n\\begin{figure}[h]\n  \\caption{Embedding dimensionality vs. quality for age prediction task}\n  \\includegraphics[width=0.46\\textwidth]{figures\/age-pred-hidden-size.png}\n  \\label{fig-emb-dim-age}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Embedding dimensionality vs. quality for gender prediciton task}\n  \\includegraphics[width=0.46\\textwidth]{figures\/gender-hidden-size.png}\n  \\label{fig-emb-dim-gender}\n\\end{figure}\n\nFigure \\ref{fig-emb-dim-age} shows that the quality on downstream task increases with the dimensionality of embedding. The best quality is achieved at size 800. Further increase in the dimensionality of embedding reduces quality.\nThe results can be interpreted as bias-variance trade-off. When embedding dimensionality is too small, too much information can be discarded (high bias). On the other hand, when embedding dimensionality is too large, too much noise is added (high variance).\n\nAt Figure \\ref{fig-emb-dim-gender} we see a similar dependency. We can find a plateau between 256 and 2048, when quality on downstream tasks does not increase. The final embedding size used in the other experiments is 256.\n\nNote, that increasing embedding size will also linearly increase the training time and the volume of consumed memory on the GPU.\n\n\\subsubsection{Embedding visualization}\n\nIn order to visualize MeLES embeddings in 2-dimensional space, we applied tSNE transformation~\\cite{maaten2008visualizing} on them. tSNE transforms high-dimensional space to low-dimensional based on local relationships between points, so neighbour vectors in high-dimensional embedding space are pushed to be close in 2-dimensional space. We colorized 2-dimensional vectors using the target values of the datasets.\n\nNote, that embeddings was learned in a fully self-supervised way from raw user transactions without any target information. Sequence of transactions represent user' behavior, thus the MeLES model captures behavioral patterns and outputs embeddings of users with similar patterns nearby.\n\ntSNE vectors from the age prediction dataset are presented in the Figure \\ref{fig-tsne-age}. We can observe 4 clusters: clusters for group '1' and '2' are on the opposite side of the cloud, clusters for groups '2' and '3' are in the middle.\n\n\n\n\\begin{figure}[ht]\n  \\caption{2D tSNE mapping of MeLES embeddings trained on age prediction task dataset, colored by age group labels}\n  \\includegraphics[width=0.46\\textwidth]{figures\/age-pred-tsne.png}\n  \\label{fig-tsne-age}\n\\end{figure}\n\n\n\\subsection{Results} \\label{sec-res}\n\n\\subsubsection{Comparison with baselines} \\label{sec-res-baselines}\n\nAs shown in Table \\ref{tab-downstream-res} our method generates sequence embeddings of lifestream data that achieve strong performance, comparable to performance on manually crafted features when used on downstream tasks. Moreover fine-tuned representations obtained by our method achieve state-of-the-art performance on both bank transactions datasets, outperforming all previous learning methods by a significant margin.\n\nFurthermore note that the usage of sequence embedding together with hand-crafted aggregate features leads to better performance than usage of only hand-crafted features or sequence embeddings, i.e. it is possible to combine different approaches to get even better model.\n\n\\begin{table*}[ht]\n\\caption{Final results on the downstream tasks}\n\\begin{tabular}{ | l |  c | c | }\n\\hline\n\\textbf{Method} & \\makecell{\\textbf{Age,} \\\\ \\textbf{Accuracy $\\pm 95\\%$}} & \\makecell{\\textbf{Gender,} \\\\ \\textbf{AUROC $\\pm 95\\%$}} \\\\\n\\hline\n\\textbf{LightGBM on hand-crafted features} & $0.626 \\pm 0.004$ & $0.875 \\pm 0.004$ \\\\\n\\textbf{LightGBM on MeLES embeddings} & $0.639 \\pm 0.006$ & $0.872 \\pm 0.005$ \\\\\n\\textbf{LightGBM on both hand-crafted features and MeLES embeddings} & \\pmb{$0.643 \\pm 0.009$} & $0.882 \\pm 0.003$ \\\\\n\\textbf{Supervised learning} & $0.631 \\pm 0.010$ & $0.871 \\pm 0.007$ \\\\\n\\textbf{MeLES fine-tuning} & \\pmb{$0.643 \\pm 0.007$} & \\pmb{$0.888 \\pm 0.002$}  \\\\\n\\textbf{LightGBM on CPC embeddings} & $0.595 \\pm 0.004$ & $0.848 \\pm 0.004$ \\\\\n\\textbf{Fine-tuned Contrastive Predictive Coding} & $0.621 \\pm 0.007$ & $0.873 \\pm 0.007$  \\\\\n\\hline\n\\end{tabular}\n\\label{tab-downstream-res}\n\\end{table*}\n\n\\subsubsection{Semi-supervised setup} \\label{sec-semi}\n\nTo evaluate our method in condition of a restricted amount of labeled data we use only part of available target labels for the semi-supervised experiment.\nAs well as in the supervised setup we compare proposed method with ligthGBM over hand-crafted features and Contrastive Predictive Coding (see Section \\ref{sec-baselines}).\nFor both embedding generation methods (MeLES and CPC) we evaluate both performance of the lightGBM on embeddings and performance of fine-tuned models.\nIn addition to this baselines we compare our method with supervised learning on the available part of the data.\n\nIn figures \\ref{fig-semi-age-0} and \\ref{fig-semi-gender-0} we compare the quality of hand-crafted features and embeddings by learning the lightGBM on top of them. Moreover, in figures \\ref{fig-semi-age-1} and \\ref{fig-semi-gender-1} one can find comparison of a single models trained on downstream tasks considered in the paper. As you can see in figures, if labeled data is limited, MeLES significantly outperforms supervised and other approaches. Also MeLES consistently outperforms CPC for different volumes of labeled data.\n\n\\begin{figure}[h]\n  \\caption{Age group prediction task quality on features for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_age_0.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-age-0}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Gender prediction task quality on features for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_gen_0.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-gender-0}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Age group prediction task quality of single model for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_age_1_wopl.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-age-1}\n\\end{figure}\n\n\\begin{figure}[h]\n  \\caption{Gender prediction task quality of single model for different dataset sizes}\n  \\includegraphics[width=0.46\\textwidth]{figures\/ss_gen_1.png}\n  \\small{The rightmost point correspond to all labels and supervised setup. X-axis is shown on a logarithmic scale.}\n  \\label{fig-semi-gender-1}\n\\end{figure}\n\n\\section{Conclusions} \\label{sec-conclusions}\n\nIn this paper, we adopted the ideas of metric learning to the analysis of the lifestream data in a novel, self-supervised, manner. As a part of this proposal, we developed the Metric Learning for Event Sequences (MeLES) method that is based on self-supervised learning. \nIn particular, the MeLES method can be used to produce embeddings of complex event sequences that can be effectively used in various downstream tasks. Also, our method can be used for pre-training in semi-supervised settings.\n\nWe also empirically demonstrate that our approach achieves strong performance results on several downstream tasks by significantly (see Section \\ref{sec-res}) outperforming both classical machine learning baselines on hand-crafted features and neural network based approaches.\nIn the semi-supervised setting, where the number of labelled data is limited, our method demonstrates even stronger results: it outperforms supervised methods by significant margins.\n\nThe proposed method of generating embeddings is convenient for production usage since almost no pre-processing is needed for complex event streams to get their compact embeddings. The pre-calculated embeddings can be easily used for different downstream tasks without performing complex and time-consuming computations on the raw event data. For some encoder architectures, such as those presented in Section~\\ref{sec-enc-arch}, it is even possible to incrementally update the already calculated embeddings when additional new lifestream data arrives.\n\n\nAnother advantage of using event sequence based embeddings, instead of the raw explicit event data, is that it is impossible to restore the exact input sequence from its embeddings. Therefore, the usage of embeddings leads to better privacy and data security for the end users than when working directly with the raw event data, and all this is achieved without sacrificing valuable information for downstream tasks.\n\n\n\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\noindent\nEven though the general structure of the configuration space of the\nheterotic\nstring remains to a large extent terra incognita, some of its important\nproperties have been uncovered. Perhaps the most interesting of these\nis the\nrecently discovered mirror symmetry of the space of (2,2)--supersymmetric\nvacua \\cite{cls}\\cite{gp} .\nIdeally questions about the space of ground states should be analyzed\nstarting from first principles, given an appropriate parametrization of\nthis manifold. Not too much progress however has\nbeen made along this avenue. Instead one proceeds somewhat indirectly.\nThe symmetry principles of string theory are used to formulate a set of\nconsistency conditions which\nare solved explicitly. Unfortunately this\nintroduces some uncertainty as to whether the part of the space of vacua\nthat has been uncovered via these constructions represents a typical slice\nof the whole space. Properties that are generic in specific\nconstructions\nmay not at all be features of the total space one is interested in\nbut instead could merely be artefacts of the techniques employed.\n\nAn example of such an artefact is furnished by the class of\nheterotic string vacua described by\n{\\bf c}omplete {\\bf i}ntersection {\\bf C}alabi--{\\bf Y}au\nmanifolds embedded in products of projective spaces (CICYs). In this class\nthe number of generations and antigenerations of the models are\nparametrized\n by the only two\nindependent Hodge numbers $(h^{(1,1)},h^{(2,1)})$ that exist on such\nmanifolds and the number of light\ngenerations of these theories is measured by the Euler number\n$\\chi=2(h^{(1,1)}-h^{(2,1)})$.\nThe results for the latter turn out to lie in the range\n$-200 \\leq \\chi \\leq 0$ \\cite{cdls}.\nIn Fig.1 the Euler number of all CICY vacua is plotted versus the\nsum of the two independent Hodge numbers \\cite{ghl}.\n\n\\vskip .1truein\n\n\\plot{2.5truein}{\\scriptsize{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{250}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{300}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{350}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{400}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{450}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{500}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{-960}{0}{\\kern-0.2pt\\lower10pt\\hbox{\\vrule height 5pt}\\lower18pt\\hbox{-960}}\n\\Place{-720}{0}{\\kern-0.2pt\\lower10pt\\hbox{\\vrule height 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This conclusion is\n reinforced by the complete construction \\cite{ls3,fkss1} of the set\nof heterotic vacua based on tensor products of minimal $N=2$ superconformal\nfield theories \\cite{g}.\nThe resulting space is again rather asymmetric even though {\\it some}\nmirror pairs appear in this construction. Fig.2 contains again a plot\nof the Euler numbers versus the sum of generations and antigenerations for\nthose theories.\n\n\\vskip .1truein\n\n\\plot{3.5truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule 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394}\n\\begin{center}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 2}~~{\\it A plot of the Euler\nnumber\nversus $(h^{(1,1)} +h^{(2,1)})$ for all ADE tensor models.}}\n\\end{center}\n\nAs it turns out, however, that the idea of an asymmetric space of ground\nstates is not correct.\nThe purpose of this review is to first describe in Part I a class of\nstring vacua whose spectra are almost\nsymmetrically distributed in a range of positive and negative Euler\nnumbers,\nsecondly to show in Part II that this is not an accident and thirdly to\npresent evidence, in Part III, that mirror symmetry is a `robust'\nproperty in that it is not an artefact of any one construction.\n\nSections 2 and 3 are devoted to the construction of a class of\nCalabi--Yau\nmanifolds which may be realized by polynomials in weighted\n$\\relax{\\rm I\\kern-.18em P}_4's$. The result of this investigation \\cite{cls} are some\n6,500 examples\n\\fnote{1}{It was shown in \\cite{ls4} and will be discussed in later\nsections\n that not all these spaces are distinct.}.\nThis class of vacua is of considerable interest because it\ninterpolates between the previously studied class the CICYs \\cite{cdls}\nmentioned above,\nwhich have negative Euler numbers and the orbifolds of tori which have\npositive Euler number.\n\nMore recently the construction of all Calabi--Yau manifolds embedded in\nweighted $\\relax{\\rm I\\kern-.18em P}_4$ and, more generally, of all Landau--Ginzburg vacua with\nan arbitrary number of fields was completed in \\cite{ks}\\cite{krsk}. The\ntotal\nclass consists of some 10,000 Landau--Ginzburg configurations, the\nresulting spectra of which have been plotted in Fig.3.\n\nThe most remarkable feature of this class of manifolds is\nimmediately apparent from Fig~3.\nIt is evident that the manifolds are very\nevenly divided between positive and negative Euler numbers the\ndistribution\nexhibitting an approximate but compelling symmetry under\n$\\chi\\to -\\chi$. This\nresonates with the observation made with regard to conformal field\ntheories\nthat the distinction between particles and antiparticles is purely one of\nconvention and the suggestion that for every Calabi--Yau manifold\\ with Euler number\n$\\chi$ there should be one with Euler number $-\\chi$.\n\n\n\\begin{center}\n\n\\plot{5.6truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule 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262}\n\\datum{ -456}{ 264}\n\\datum{ -456}{ 272}\n\\datum{ -468}{ 286}\n\\datum{ -480}{ 246}\n\\datum{ -480}{ 262}\n\\datum{ -480}{ 278}\n\\datum{ -480}{ 286}\n\\datum{ -480}{ 306}\n\\datum{ -480}{ 334}\n\\datum{ -492}{ 256}\n\\datum{ -504}{ 276}\n\\datum{ -504}{ 312}\n\\datum{ -516}{ 302}\n\\datum{ -528}{ 278}\n\\datum{ -528}{ 286}\n\\datum{ -528}{ 334}\n\\datum{ -540}{ 274}\n\\datum{ -540}{ 334}\n\\datum{ -552}{ 306}\n\\datum{ -564}{ 322}\n\\datum{ -564}{ 330}\n\\datum{ -564}{ 340}\n\\datum{ -576}{ 314}\n\\datum{ -588}{ 346}\n\\datum{ -612}{ 330}\n\\datum{ -624}{ 330}\n\\datum{ -624}{ 358}\n\\datum{ -636}{ 342}\n\\datum{ -660}{ 366}\n\\datum{ -672}{ 374}\n\\datum{ -720}{ 394}\n\\datum{ -732}{ 386}\n\\datum{ -744}{ 402}\n\\datum{ -804}{ 430}\n\\datum{ -900}{ 474}\n\\datum{ -960}{ 502}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 3}~~{\\it A plot of Euler numbers\nagainst\n ${\\bar n}_g+n_g$ for the 2997 spectra of all the LG potentials.\n}}\n\\end{center}\n\\end{center}\n\nThe richness of the list\ncan be appreciated from the fact that the number of distinct spectra in\nthis class\nis about an order of magnitude larger than in the previous constructions.\nMirror\nsymmetry emerges only when a sufficiently large subspace of the\nconfiguration space\nof the heterotic string is probed. Furthermore this class is also of\npotential\nphenomenological interest. Whereas it had previously proved very difficult\nto find Calabi--Yau manifolds\\ with $\\chi=\\pm 6$ the construction of weighted CICYs leads to\nsome tens such manifolds. These spaces are not of immediate phenomenological\ninterest since they are all simply connected. Hence the best place to seek\ninteresting models may well be among manifolds with Euler number\n$\\pm 6k$, with\n$k>1$, although it may also be possible to construct interesting\n(2,0)--models\n\\cite{dg} from the manifolds of Euler number $\\pm 6$ which have reduced\ngauge groups\ndue to an enlarged background gauge group. This however is a question for\nfuture work.\n\nWith benefit of hindsight it seems odd that no systematic construction of\nsuch\nmanifolds was previously attempted. Hypersurfaces in weighted projective\nspaces\nwere first discussed in the physics literature by Yau \\cite{yau} and\nStrominger and Witten\\cite{hitnrun} who constructed a number of examples\nwith\nlarge negative Euler number\n$-200>\\chi>-300$. Subsequently Kim, Koh and Yoon \\cite{kky} constructed a\nfew further\nexamples with Euler numbers in a similar range. A complicating feature\nof these\nexamples was the assumed need to avoid the singular sets of the embedding\n$\\relax{\\rm I\\kern-.18em P}_4$, coupled with the large values of $|\\chi|$ achieved, this led to the\nimpression that the construction was contrived and the resulting\nmanifolds very\ncomplicated. To those elucidating the connections between exactly soluble\nconformal theories and Calabi--Yau manifolds \\cite{m}\\cite{gvw} however\nit was\napparent that avoiding the\nsingular sets of the embedding space was unnecessary since the\nsingularities\non the resulting hypersurface can be resolved while maintaining the\ncondition\n$c_1=0$. This greatly increases the number of manifolds that can be\nconstructed\nin this way. In fact of the 10,000 odd examples that have been constructed\nin refs. \\cite{cls}\\cite{ks} only a small subset consisting of spaces\ndescribed by\npolynomials of Fermat\ntype do not intersect the singular sets. Resolution of singularities has the\neffect of raising the Euler number and in this way the many Euler numbers\nplotted in Fig.~3 are achieved including many values of moderate size that\nmay be of interest for model building.\n\nEven though the high degree of symmetry of this space of groundstates\nunder the flip of the sign of the Euler numbers strongly suggests a\nrelation between dual pairs there is a priori no reason as far as the\ndifferent Landau--Ginzburg potentials are concerned why this should\nbe the case. Looking more closely at the Hodge pairs of \\cite{cls} seems to\nconfuse the issue since generically one finds for a given dual pair of\nHodge numbers not two manifolds but {\\it many} and it is unclear which\nof the possible combinations (if any) are in fact related.\n\nIt is therefore of interest to ask whether a systematic procedure can be\nestablished which relates pairs of vacua as mirror partners. This is indeed\nthe case. The construction, introduced in ref. \\cite{ls4}, consists\nof a combination of an orbifolding and a nonlinear transformation with\nsomewhat unusual properties in that it involves fractional powers. Even\nthough this procedure can be formulated in the manifold picture it is\nmost easily motivated in Landau--Ginzburg language. By applying this\nconstruction to a certain class of mirror candidates it is possible to\nestablish a close connection between dual vacua. The description of this\nconstruction comprises Part II of this review.\n\nThe starting point\nis the well known equivalence of the affine D--invariant in the\nN=2 superconformal minimal series to the $\\relax{\\sf Z\\kern-.4em Z}_2$--orbifold of the diagonal\ninvariant at the same level. In section 4 this equivalence will be lifted to\nthe full vacuum described by a tensor product of $N=2$ minimal theories.\nIt is necessary to reformulate this equivalence in terms of the\nLandau--Ginzburg (LG) potential and its projectivization as a\nCalabi--Yau (CY) manifold since for most of our models\nno exactly solvable theory is known.\nThe computations in this framework then either involve orbifoldizing\nLG--potentials \\cite{v} or modding out discrete groups of CY--manifolds and\nresolving singularities \\cite{ry}\\cite{s3}.\nOnce this procedure has been formulated in the simple framework of the\ntensor series of $N=2$ minimal models it will be clear how to proceed in\ngeneral. Sections 5 and 6 extend the discussion to more general potentials\nand illustrate to what extent our LG--potentials can be viewed as orbifolds.\nAs already mentioned a general technique to find mirror pairs will emerge.\nAnother, more mathematical, aspect which lends itself to an analysis via\nfractional transformation is the strange duality of Arnold.\n\nFinally, in Part III, the emphasis is shifted from Landau--Ginzburg theories\nto their orbifolds. Even though it is possible to map via the fractional\ntransformations of Part II many classes of orbifolds of the models described\nin Part I to complete intersections it will become clear that not all\norbifolds can be understood this way, at least with the techniques\npresently available. Hence\nit is useful to consider orbifolds in their own right firstly as a\npotential pool of new models and also in order to pursue the question\nraised at the beginning of the introduction regarding the `robustness'\n the mirror\nproperty against a change of technique\n\\fnote{2}{An orbifold analysis of minimal exactly solvable models has been\n performed in ref. \\cite{gp}. The interesting result is that the\n set of all orbifolds emerging from a given diagonal theory is\n selfdual (see also ref. \\cite{alr})}.\n In ref. \\cite{kss} we have\nconstructed the complete set of Landau--Ginzburg potentials with an\narbitrary number of fields involving\nonly couplings between at most two fields (i.e. arbitrary combinations\nof Fermat, 1--Tadpole and 1--Loop polynomials) and implemented some 40 odd\nactions of phase symmetries. It turns out that for this class\nof orbifolds the mirror property improves from about eighty percent for\nthe LG--potentials\nto about 94\\%. Figure 4 contains the plot of the resulting spectra.\n\n\n\n\\begin{center}\n\\plot{6truein}{\\tiny{$\\bullet$}}\n\\nobreak\n\\Place{-960}{50}{\\vbox{\\hrule width5pt}~~50}\n\\Place{-960}{100}{\\vbox{\\hrule width5pt}~~100}\n\\Place{-960}{150}{\\vbox{\\hrule width5pt}~~150}\n\\Place{-960}{200}{\\vbox{\\hrule width5pt}~~200}\n\\Place{-960}{250}{\\vbox{\\hrule width5pt}~~250}\n\\Place{-960}{300}{\\vbox{\\hrule width5pt}~~300}\n\\Place{-960}{350}{\\vbox{\\hrule width5pt}~~350}\n\\Place{-960}{400}{\\vbox{\\hrule width5pt}~~400}\n\\Place{-960}{450}{\\vbox{\\hrule width5pt}~~450}\n\\Place{-960}{500}{\\vbox{\\hrule width5pt}~~500}\n\\Place{960}{50}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{100}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{150}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{200}{\\kern-5pt\\vbox{\\hrule width5pt}\\vphantom{0}}\n\\Place{960}{250}{\\kern-5pt\\vbox{\\hrule 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216}{ 124}\n\\datum{ 204}{ 130}\n\\datum{ 168}{ 148}\n\\datum{ 156}{ 154}\n\\datum{ 108}{ 178}\n\\datum{ 84}{ 190}\n\\datum{ 36}{ 214}\n\\datum{ -228}{ 346}\n\\datum{ 216}{ 126}\n\\datum{ 200}{ 134}\n\\datum{ 192}{ 138}\n\\datum{ 96}{ 186}\n\\datum{ 216}{ 128}\n\\datum{ 204}{ 134}\n\\datum{ 168}{ 152}\n\\datum{ 160}{ 156}\n\\datum{ 144}{ 164}\n\\datum{ 192}{ 142}\n\\datum{ 96}{ 190}\n\\datum{ 0}{ 238}\n\\datum{ -96}{ 286}\n\\datum{ 228}{ 126}\n\\datum{ 216}{ 132}\n\\datum{ 36}{ 222}\n\\datum{ 232}{ 126}\n\\datum{ 216}{ 134}\n\\datum{ 56}{ 214}\n\\datum{ 0}{ 242}\n\\datum{ -48}{ 266}\n\\datum{ 240}{ 124}\n\\datum{ 228}{ 130}\n\\datum{ 144}{ 172}\n\\datum{ 132}{ 178}\n\\datum{ 108}{ 190}\n\\datum{ 240}{ 126}\n\\datum{ 228}{ 132}\n\\datum{ 192}{ 150}\n\\datum{ 144}{ 174}\n\\datum{ 0}{ 246}\n\\datum{ 180}{ 158}\n\\datum{ 72}{ 212}\n\\datum{ 60}{ 218}\n\\datum{ 232}{ 134}\n\\datum{ 216}{ 142}\n\\datum{ 200}{ 150}\n\\datum{ 192}{ 154}\n\\datum{ 120}{ 190}\n\\datum{ 228}{ 138}\n\\datum{ 216}{ 144}\n\\datum{ 96}{ 204}\n\\datum{ 240}{ 134}\n\\datum{ 228}{ 140}\n\\datum{ 224}{ 142}\n\\datum{ 144}{ 182}\n\\datum{ 48}{ 230}\n\\datum{ 252}{ 130}\n\\datum{ 120}{ 196}\n\\datum{ 240}{ 138}\n\\datum{ 216}{ 150}\n\\datum{ 180}{ 168}\n\\datum{ 168}{ 174}\n\\datum{ 228}{ 146}\n\\datum{ 216}{ 152}\n\\datum{ 96}{ 212}\n\\datum{ 84}{ 218}\n\\datum{ 240}{ 142}\n\\datum{ 192}{ 166}\n\\datum{ 176}{ 174}\n\\datum{ 168}{ 178}\n\\datum{ 144}{ 190}\n\\datum{ 0}{ 262}\n\\datum{ 260}{ 134}\n\\datum{ 240}{ 144}\n\\datum{ 180}{ 174}\n\\datum{ 216}{ 158}\n\\datum{ 192}{ 170}\n\\datum{ 108}{ 212}\n\\datum{ 252}{ 142}\n\\datum{ 168}{ 184}\n\\datum{ 132}{ 202}\n\\datum{ 240}{ 150}\n\\datum{ 196}{ 172}\n\\datum{ 216}{ 164}\n\\datum{ 120}{ 212}\n\\datum{ 232}{ 158}\n\\datum{ 192}{ 178}\n\\datum{ 180}{ 184}\n\\datum{ -240}{ 394}\n\\datum{ 264}{ 144}\n\\datum{ 252}{ 150}\n\\datum{ -180}{ 366}\n\\datum{ 240}{ 158}\n\\datum{ 144}{ 206}\n\\datum{ 264}{ 148}\n\\datum{ 260}{ 150}\n\\datum{ 236}{ 162}\n\\datum{ 192}{ 184}\n\\datum{ -84}{ 322}\n\\datum{ 240}{ 162}\n\\datum{ 216}{ 174}\n\\datum{ 228}{ 170}\n\\datum{ 272}{ 150}\n\\datum{ 240}{ 166}\n\\datum{ 192}{ 190}\n\\datum{ 144}{ 214}\n\\datum{ 0}{ 286}\n\\datum{ 280}{ 148}\n\\datum{ 276}{ 150}\n\\datum{ 252}{ 162}\n\\datum{ 216}{ 180}\n\\datum{ 156}{ 210}\n\\datum{ 288}{ 146}\n\\datum{ 264}{ 158}\n\\datum{ 228}{ 176}\n\\datum{ 216}{ 182}\n\\datum{ 48}{ 266}\n\\datum{ 276}{ 154}\n\\datum{ 204}{ 190}\n\\datum{ 264}{ 162}\n\\datum{ 256}{ 166}\n\\datum{ 240}{ 174}\n\\datum{ 192}{ 198}\n\\datum{ 288}{ 152}\n\\datum{ 264}{ 164}\n\\datum{ 296}{ 150}\n\\datum{ 240}{ 178}\n\\datum{ 96}{ 250}\n\\datum{ 216}{ 192}\n\\datum{ 288}{ 158}\n\\datum{ 256}{ 174}\n\\datum{ 192}{ 206}\n\\datum{ 0}{ 302}\n\\datum{ 264}{ 172}\n\\datum{ 252}{ 178}\n\\datum{ 132}{ 238}\n\\datum{ 288}{ 162}\n\\datum{ 240}{ 188}\n\\datum{ 120}{ 248}\n\\datum{ 288}{ 166}\n\\datum{ 276}{ 172}\n\\datum{ 228}{ 196}\n\\datum{ 156}{ 232}\n\\datum{ 276}{ 174}\n\\datum{ 296}{ 166}\n\\datum{ 288}{ 170}\n\\datum{ 272}{ 178}\n\\datum{ 264}{ 184}\n\\datum{ 300}{ 168}\n\\datum{ 264}{ 188}\n\\datum{ 216}{ 212}\n\\datum{ 312}{ 166}\n\\datum{ 192}{ 226}\n\\datum{ 288}{ 182}\n\\datum{ 240}{ 206}\n\\datum{ 312}{ 172}\n\\datum{ 304}{ 176}\n\\datum{ 252}{ 202}\n\\datum{ -60}{ 358}\n\\datum{ 324}{ 168}\n\\datum{ 300}{ 180}\n\\datum{ 288}{ 186}\n\\datum{ 276}{ 192}\n\\datum{ 288}{ 188}\n\\datum{ 252}{ 206}\n\\datum{ 312}{ 178}\n\\datum{ 288}{ 190}\n\\datum{ 96}{ 286}\n\\datum{ 120}{ 276}\n\\datum{ 120}{ 278}\n\\datum{ 168}{ 256}\n\\datum{ 336}{ 178}\n\\datum{ 324}{ 184}\n\\datum{ 312}{ 190}\n\\datum{ 240}{ 226}\n\\datum{ 312}{ 192}\n\\datum{ 300}{ 198}\n\\datum{ 216}{ 240}\n\\datum{ 288}{ 206}\n\\datum{ 312}{ 196}\n\\datum{ 216}{ 244}\n\\datum{ -156}{ 430}\n\\datum{ 336}{ 188}\n\\datum{ 288}{ 214}\n\\datum{ 0}{ 358}\n\\datum{ 348}{ 186}\n\\datum{ 264}{ 228}\n\\datum{ 336}{ 194}\n\\datum{ 84}{ 322}\n\\datum{ 336}{ 198}\n\\datum{ 288}{ 222}\n\\datum{ 340}{ 198}\n\\datum{ 360}{ 190}\n\\datum{ 336}{ 202}\n\\datum{ 288}{ 226}\n\\datum{ 348}{ 198}\n\\datum{ 276}{ 234}\n\\datum{ 336}{ 206}\n\\datum{ 300}{ 226}\n\\datum{ 180}{ 286}\n\\datum{ 372}{ 194}\n\\datum{ 312}{ 224}\n\\datum{ 300}{ 230}\n\\datum{ 372}{ 202}\n\\datum{ 368}{ 204}\n\\datum{ 60}{ 358}\n\\datum{ 360}{ 212}\n\\datum{ 324}{ 232}\n\\datum{ 264}{ 262}\n\\datum{ 384}{ 204}\n\\datum{ 336}{ 230}\n\\datum{ 216}{ 292}\n\\datum{ 376}{ 214}\n\\datum{ 360}{ 228}\n\\datum{ 384}{ 218}\n\\datum{ 372}{ 226}\n\\datum{ 348}{ 238}\n\\datum{ 288}{ 270}\n\\datum{ 408}{ 212}\n\\datum{ 396}{ 222}\n\\datum{ 324}{ 262}\n\\datum{ 420}{ 218}\n\\datum{ 408}{ 224}\n\\datum{ 384}{ 242}\n\\datum{ 408}{ 232}\n\\datum{ 420}{ 230}\n\\datum{ 384}{ 250}\n\\datum{ 408}{ 240}\n\\datum{ 372}{ 258}\n\\datum{ -60}{ 474}\n\\datum{ 0}{ 446}\n\\datum{ 372}{ 262}\n\\datum{ 432}{ 238}\n\\datum{ 180}{ 366}\n\\datum{ 432}{ 242}\n\\datum{ 396}{ 262}\n\\datum{ 228}{ 346}\n\\datum{ 456}{ 234}\n\\datum{ 276}{ 330}\n\\datum{ 408}{ 268}\n\\datum{ 312}{ 318}\n\\datum{ 456}{ 248}\n\\datum{ 432}{ 266}\n\\datum{ 456}{ 256}\n\\datum{ 480}{ 246}\n\\datum{ 456}{ 264}\n\\datum{ 456}{ 272}\n\\datum{ 492}{ 256}\n\\datum{ 480}{ 262}\n\\datum{ 0}{ 502}\n\\datum{ 60}{ 474}\n\\datum{ 156}{ 430}\n\\datum{ 240}{ 394}\n\\datum{ 420}{ 306}\n\\datum{ 288}{ 374}\n\\datum{ 468}{ 286}\n\\datum{ 480}{ 286}\n\\datum{ 336}{ 358}\n\\datum{ 372}{ 346}\n\\datum{ 396}{ 340}\n\\datum{ 528}{ 278}\n\\datum{ 540}{ 274}\n\\datum{ 528}{ 286}\n\\datum{ 432}{ 334}\n\\datum{ 516}{ 302}\n\\datum{ 480}{ 334}\n\\datum{ 552}{ 306}\n\\datum{ 528}{ 334}\n\\datum{ 564}{ 322}\n\\datum{ 540}{ 334}\n\\datum{ 564}{ 330}\n\\datum{ 564}{ 340}\n\\datum{ 612}{ 330}\n\\datum{ 588}{ 346}\n\\datum{ 624}{ 330}\n\\datum{ 624}{ 358}\n\\datum{ 660}{ 366}\n\\datum{ 672}{ 374}\n\\datum{ 732}{ 386}\n\\datum{ 720}{ 394}\n\\datum{ 804}{ 430}\n\\datum{ 900}{ 474}\n\\datum{ 960}{ 502}\n\n\\begin{center}\n\\parbox{6.4truein}{\\noindent {\\bf Fig. 4}~~{\\it A plot of Euler\nnumbers against\n ${\\bar n}_g+n_g$ for the 1900 odd spectra of all the LG potentials\nand phase\norbifolds constructed.}}\n\\end{center}\n\\end{center}\n\n\nIt is obvious from Fig. 4 that the upper boundary of the distribution of\nspectra is the same for the orbifolds as for the complete intersection\nmanifolds. Very likely this boundary is in fact a\nproperty of the total moduli space of all\nthree--dimensional Calabi--Yau manifolds.\nIt would be interesting\nto see whether all Calabi--Yau Hodge numbers fall into the limits defined\nby the Figures 3 \\& 4. As expected the lower part of the plot has shifted\nsince\nnew theories with smaller numbers for the total number of fields have\nappeared.\nIt is to be expected that, as the construction becomes more complete, the\nstructure\nof the lower part will change again. In fact there are well known manifolds\nthat lie below the models presented here; these however involve permutation\ngroups.\n\n\n\n\\vfill \\eject\n\\part{ Construction of Landau--Ginzburg Theories and Weighted CICYs.}\n\n\\noindent\nEven though there is some overlap between the sets of string vacua\ndescribed by Landau--Ginzburg vacua on the one hand and Calabi--Yau\nmanifolds on the other it is not, at present, clear whether the former\nis contained in the latter. It is appropriate\njustified to separate the discussion of these two classes somewhat.\nIn Sections 2 and 3 the emphasis will be on the explicit construction\nof a set of CY manifolds embedded in weighted $\\relax{\\rm I\\kern-.18em P}_4$ whereas\nthe remaining sections of Part I will be concerned\n with the complete class of LG vacua.\n\n\\vskip .1truein\n\\noindent\n\\section{Complete Intersection Manifolds in Weighted $\\relax{\\rm I\\kern-.18em P}_4$.}\n\n\\noindent\nThis section contains some elements of the theory of hypersurfaces defined\nby polynomials in weighted projective spaces. An extensive discussion\nof these\nspaces can be found in \\cite{wproj}.\n\nA weighted $\\relax{\\rm I\\kern-.18em P}_4$ with weights $(k_1,k_2,k_3,k_4,k_5)$, which will be\ndenoted by\n$\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4,k_5)}$, is most easily described in terms of 5\ncomplex `homogeneous coordinates' $(z_1,z_2,z_3,z_4,z_5)$, not all zero,\nwhich are subject to the identification\n\\begin{equation}\n(z_1,...,z_5) \\simeq (\\l^{k_1}z_1,...,\\l^{k_5}z_5)\n\\lleq{pro5}\nfor all nonzero $\\l \\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}$. Thus a weighted projective space is a\ngeneralization\nof ordinary projective space and $\\relax{\\rm I\\kern-.18em P}_4 = \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,1)}$ in\nthis notation.\nIn the following, when referring to a generic weighted $\\relax{\\rm I\\kern-.18em P}_4$, we shall\nfrequently consider the weights to be understood and write $\\relax{\\rm I\\kern-.18em P}_4$ for\n$\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,...,k_5)}$.\n\nThe first point to note concerning these spaces\nis the fact that weighted projective spaces have\norbifold singularities owing to the identification (\\ref{pro5}), except for\nthe case that the weights are all unity. This is most easily seen by setting\n$z_j = \\left ( \\zeta_j \\right)^{k_j}$ so that (\\ref{pro5}) becomes\n$(\\zeta_1,...,\\zeta_5) \\simeq \\l (\\zeta_1,...,\\zeta_5)$. However in virtue\nof the definition of $\\zeta_i$ we must also identify\n$\\zeta_j \\simeq e^{2\\pi i\/k_j} \\zeta_j$. So we see that\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4,k_5)} = \\frac{\\relax{\\rm I\\kern-.18em P}_4}\n {\\relax{\\sf Z\\kern-.4em Z}_{k_1}\\times \\cdots \\times \\relax{\\sf Z\\kern-.4em Z}_{k_5}}\n\\end{equation}\nThese identifications lead to singular sets. A simple example is\n$\\relax{\\rm I\\kern-.18em P}_{(1,1,1,2,5)}$ which has weights that are mutually prime. If we\ntake in turn $\\l=-1$ and $\\l=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ with $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ a fifth root of unity in\n(\\ref{pro5}) then we see that\n\\begin{eqnarray}\n (z_1,z_2,z_3,z_4,z_5) &\\simeq & (-z_1,-z_2,-z_3,z_4,-z_5) \\nonumber \\\\\n &\\simeq &(\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_1,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_2,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^2 z_4,z_5).\\llea{ex}\nConsider now a neighborhood of the point $(0,0,0,1,0)$. We can take\ncoordinates on the neighborhood by setting $\\l = z_4^{-1\/2}$ and\nwriting $u_j = z_j\/z_4^{1\/2}$ for $j=1,2,3$ and $u_5 = z_5\/z_4^{5\/2}$.\nWe can therefore think of points in the neighborhood as corresponding\nto $(u_1,u_2,u_3,1,u_5)$. However from the first of identifications\n(\\ref{ex})\nit follows that\n\\begin{equation}\n(u_1,u_2,u_3,1,u_5) \\simeq (-u_1,-u_2,-u_3,1,-u_5)\n\\end{equation}\nso that there is a $\\relax{\\sf Z\\kern-.4em Z}_2$ identification on the space and hence on a\nneighborhood of the point and the action fixes $(0,0,0,1,0)$. In the same\nway there is a $\\relax{\\sf Z\\kern-.4em Z}_5$ action which fixes $(0,0,0,0,1)$. In this case\nthe singular set consists of points owing to the fact that the weights\nare mutually prime. Consider now $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,2,4)}$ then the\nidentification\nis\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\simeq (\\l z_1,\\l z_2,\\l z_3,\\l^2 z_4,\\l^4 z_5)\n\\end{equation}\nIf we set $\\l = z_5^{-1\/4}$ and\nchoose coordinates $w_j = z_j\/z_5^{1\/4}$, $w_4 = z_4\/z_5^{1\/2}$ then due\nto the freedom to set $\\l = -1$ in (5) we have\n\\begin{equation}\n(w_1,w_2,w_3,w_4,1) \\simeq (-w_1,-w_2,-w_3,w_4,1)\n\\end{equation}\nand we see that we have a $\\relax{\\sf Z\\kern-.4em Z}_2$ action which fixes the curve\n$(0,0,0,z_4,z_5)$.\nThere is in addition a $\\relax{\\sf Z\\kern-.4em Z}_4$ action with fixed point $(0,0,0,0,1)$ that\nlies within this curve. In general there is a fixed point for each\nweight that\nis greater than unity, a fixed curve for every pair of weights $k_i,k_j$\nwhose greatest common factor, which we denote by $(k_i,k_j)$, is\ngreater than\nunity, a fixed surface for each triple with $(k_i,k_j,k_l) > 1$ and so on.\n\nWe wish to study Calabi--Yau\\ hypersurfaces defined by polynomials in the\nhomogeneous coordinates. We require the polynomials to be transverse,\nthat is\n$p=0$ and $dp=0$ have no common solution. Given the weights of the ambient\nspace the requirement of a vanishing first Chern class fixes the degree of\nthe polynomial as in the case of CICYs. To derive the explicit\ncondition we digress briefly on the Chern classes of the submanifold\n${\\cal M}$ defined by the equation $p=0$. We denote by ${\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4}$\nand\n${\\cal T}_{\\cal M}$ the tangent spaces to the $\\relax{\\rm I\\kern-.18em P}_4$ and ${\\cal M}$ and by\n${\\cal N}$ the normal bundle of ${\\cal M}$ in $\\relax{\\rm I\\kern-.18em P}_4$. Proceeding in a\nstandard manner we have\n\\begin{equation}\n{\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4} = {\\cal T}_{\\cal M} \\oplus {\\cal N}\n\\end{equation}\nwhich allows us to calculate the Chern polynomial of ${\\cal M}$ once\nthose for\n$\\relax{\\rm I\\kern-.18em P}_4$ and ${\\cal N}$ are known. The tangent space of $\\relax{\\rm I\\kern-.18em P}_4$ may be\nthought of\nas the set of vectors\n\\begin{equation}\n{\\cal V} = {\\cal V}^j \\frac{\\partial}{\\partial z^j}\n\\end{equation}\nwhich act on bona fide functions of the homogeneous coordinates, i.e.\nfunctions\nof degree zero. From Euler's theorem for homogeneous functions of\ndegree $m$\nwe have that\n\\begin{equation}\n\\sum_j k_j z^j \\frac{\\partial}{\\partial z^j} f = m f.\\lleq{euler}\nSo when acting on functions of degree zero we see that we may regard the\n${\\cal V}^j$ as independent apart from the identification\n${\\cal V}^j \\simeq {\\cal V}^j + k_jz^j$. This reduces the dimension of the\nspace of ${\\cal V}^j$ to 4 as is necessary. It follows that\n\\begin{equation}\n{\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4} = {\\cal O}(k_1) \\oplus \\cdots \\oplus {\\cal O}(k_5)\/{\\cal O}\n\\end{equation}\nwhere ${\\cal O}(k)$ is a line bundle with $c_1=kJ$ and $J$ is the K\\\"ahler\nclass. It is also the line\nbundle whose fibre coordinates transform like a polynomial of degree $k$.\n${\\cal O}$ is the trivial bundle. Since ${\\cal O}(k)$ is one dimensional\nwe have $c({\\cal O}(k))= 1 +kJ$ and hence\n\\begin{equation}\nc({\\cal T}_{\\relax{\\rm I\\kern-.18em P}_4}) = \\prod_{j=1}^5 \\left (1 + k_j J\\right ).\n\\end{equation}\nThe defining polynomial $p$ can be regarded as a fibre coordinate on\n${\\cal N}$ so if $p$ is of degree $d$ we have ${\\cal N} = {\\cal O}(d)$ and\n$c({\\cal N}) = 1 + dJ$. Hence\n\\begin{equation}\nc({\\cal T}_{\\cal M}) = \\frac{ \\prod_{j=1}^5 \\left (1 + k_jJ \\right )}\n { \\left (1 + dJ \\right )}\n\\lleq{ctot}\nIt follows that $c_1=0$ is the condition\n\\begin{equation}\nd = \\sum_j k_j~ .\\lleq{c1}\nWe record here also an expression for $c_3$, obtained by expanding\n(\\ref{ctot})\nto third order, which is useful for the\ncomputation of the Euler number of these spaces\n\\begin{equation}\nc_3 = - \\frac{1}{3} \\left (d^3 - \\sum_{j=1}^5 k_j^3 \\right )J^3~.\n\\end{equation}\nThere are, roughly speaking, two sorts of singularities that can arise\nfor a hypersurface defined by the vanishing of a polynomial $p$. The first\nis that the locus $p=0$ intersects the $\\relax{\\sf Z\\kern-.4em Z}_n$--singularities of the ambient\nspace. This does not pose a difficulty however as these can in general be\nresolved.\n\nThe second is that the weights of a given $\\relax{\\rm I\\kern-.18em P}_4$ may prevent all\npolynomials\nof the required degree from being transverse. This is in contradistinction\nto CICYs where every configuration admits a smooth\nrepresentative (in fact almost all representatives are smooth \\cite{gh1}).\nTo see this let\n\\begin{equation}\nd_j:= d-k_j~,~j=1,\\dots,5\n\\end{equation}\nand expand $p$ in powers of $z_1$, say,\n\\begin{equation}\np=\\sum_{r=0}^{a_1} C_r(z_m)\\,z_1^r~,~~~m\\neq 1~.\\lleq{pexp}\nIt follows from Euler's Theorem (\\ref{euler}) that $p=0$ and $dp=0$ if and\nonly if there\nis a simultaneous solution of the equations\n\\begin{equation}\n\\frac{\\partial p}{\\partial z_j}=0~,~j=1,\\dots,5~.\n\\lleq{transv5}\nApplying this to (\\ref{pexp}) we have\n\\begin{eqnarray}\n\\frac{\\partial p}{\\partial z_1}&=& \\sum_{r=1}^{a_1} rC_r(z_m)\\,z_1^{r-1} \\nonumber \\\\\n\\frac{\\partial p}{\\partial z_m}&=& \\sum_{r=0}^{a_1} \\frac{\\partial C_r}{\\partial z_m}\\,z_1^r~\n\\llea{partexp}\nNow the degrees of the $C_r$ are fixed by (\\ref{pexp})\n\\begin{eqnarray}\n\\deg\\,(C_r) &=& d-rk_1=d_1-(r-1)k_1~,~{\\rm for\\ } r\\ge 1~,\\nonumber \\\\\n\\deg\\left(\\frac{\\partial C_r}{\\partial z^m}\\right) &=&d_m-rk_1~.\n\\end{eqnarray}\nand a polynomial of negative degree is understood to vanish identically.\nUnless\nat least one of the coefficients in (\\ref{partexp}) has degree\nprecisely zero\nthen equations (\\ref{transv5}) will be satisfied for $z_1=1$ and $z_m=0$\nand $p$\n will not be transverse.\nThis leads to the following neccessary condition on the weights if\n $p$ is to be\ntransverse:\n\n\\noindent\n{\\sl For each $i$ there must exist a $j$ such that $k_i|d_j$\n ($k_i$ divides $d_j$).}\n\nThis condition is quite restrictive. For example it is immediate that\nthe only\nmanifolds of the form $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,k)}[k+4]$ that it allows are those\nfor the\nfour values $k=1,2,3,4$, and the assiduous reader may care to check\nthat apart\nfrom these there only eleven other allowed cases of the form\n$\\relax{\\rm I\\kern-.18em P}_{(1,1,1,k,l)}[k+l+3]$, where $k+l+3$ indicates the degree of the\npolynomial.\n\nThis criterion however is not a sufficient condition. A\ncounterexample being furnished by the configuration\n$\\relax{\\rm I\\kern-.18em P}_{(1,2,2,2,2)}[9]$ whose most general polynomial is of the form\n\\begin{equation}\np(z_1,z_2,z_3,z_4,z_5) = z_1 \\tilde{p}(z_1,z_2,z_3,z_4,z_5).\n\\end{equation}\nTransversality\nrequires that the equations $p=0$ and $dp=0$ have no common solution.\nConsider\nthen a neighborhood $U_5$ say, on which we can take $z_5=1$. Taking the\ndifferential gives\n$dp =\\nobreak \\tilde{p}dz_1 + z_1 d\\tilde{p}$ which has a zero for $z_1=0$\nand $\\tilde{p}=0$. These equations\ngive two constraints in 4 variables and therefore always have a solution.\nSince the polynomial constraint is identically satisfied it follows that\nthis configuration cannot admit a transverse realization.\nThus there are further criteria which must be satisfied in order for a\npolynomial to be transverse. In refs. \\cite{cls} a list of\ntransverse polynomials was constructed. A little thought shows that\npolynomials of\nFermat type for which $k_i|d_i$ for each $i$ {\\sl are} transverse. These\nare of the form\n\\begin{equation}\nz_1^a + z_2^b + z_3^c + z_4^d + z_5^e~~~~~~~~~~~~~~~~~~~~~~~~~\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\end{equation}\n\n\\noindent\nto which we have appended a diagrammatic shorthand.\n\nThere are other types which are also always transverse such as\n\\begin{eqnarray}\nz_1^az_2 + z_2^b + z_3^c + z_4^d + z_5^e\n& &~~~~~~~~~~~~{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\\\\nz_1^az_2 + z_2^bz_3 + z_3^cz_1 + z_4^d + z_5^e\n& &~~~~~~~~~~~~{\\thicklines {\\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(26,0){\\oval(52,26)[t]}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}}\n\\end{eqnarray}\nParts of these expressions corresponding to connected subdiagrams can also\nbe combined together to produce yet other transverse polynomials such as\n\\begin{equation}\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\oval(52,26)[t]}\n \\put(78,0){\\line(1,0){26}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}\n{\\rm or}~~~~~~~~~~~~~\n{\\thicklines \\begin{picture}(150,30)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}\n\\end{equation}\n\nWhat is needed then is a classification of nondegenerate weighted\nhomogeneous\npolynomials in five variables. Unfortunately, there exists as yet no such\nclassification and indeed its formulation seems to be a hard problem.\n\nThere {\\it does} exist a classification of smooth polynomials in three\nvariables \\cite{agzv} and what has been done in \\cite{cls} is to extend\nthis to a partial classification of polynomials in five variables.\nThese constructions do not describe all possible\npolynomials but they do respresent a minimal extension of Arnold's\nclassification to the case of five variables.\nTable 1 contains the polynomial types implemented in \\cite{cls}\nto construct nondegenerate polynomials in five variables. By combining\nthe nineteen types listed below in the way described above one obtains\nthirty different five dimensional catastrophes.\n\n{\\scriptsize\n{\\begin{center}\n\\begin{tabular}{|| l | l | l ||}\n\\hline\n$\\#$ & Polynomial Type & Diagram \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\\n\\hline\n1 &$z_1^a$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n2 &$z_1^az_2 + z_2^b$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n3 &$z_1^az_2 + z_2^bz_1$\n&~~~~~{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(13,0){\\oval(26,26)[t]}\n \\end{picture}\n } \\\\ [.5mm]\n\\hline\n4 &$z_1^az_2 + z_2^bz_3 + z_3^c$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n5 &$z_1^az_2 + z_2^b + z_3^cz_2 + z_1^pz_3^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{6}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n6 &$z_1^az_2 + z_2^bz_3 + z_3^cz_2 + z_1^pz_3^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(39,0){\\oval(26,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n7 &$z_1^az_2 + z_2^bz_1 + z_3^cz_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(26,0){\\oval(52,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n8 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^d$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n9 &$z_1^az_2 + z_2^bz_3 + z_3^c + z_4^dz_3 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n10 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_3 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(65,0){\\oval(26,26)[t]}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n11 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_2 + z_1^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\oval(52,26)[t]}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n12 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(39,0){\\oval(78,26)[t]}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n13 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^e$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n14 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^d + z_5^ez_4 + z_3^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,7){\\circle{12}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n15 &$z_1^az_2 + z_2^bz_3 + z_3^c + z_4^dz_3 + z_5^ez_4 + z_2^pz_4^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,7){\\circle{12}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n16 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_4 + z_3^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(91,0){\\oval(26,26)[t]}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n17 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_3 + z_2^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\oval(52,26)[t]}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n18 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_2 + z_1^pz_5^q$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(65,0){\\oval(78,26)[t]}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n19 &$z_1^az_2 + z_2^bz_3 + z_3^cz_4 + z_4^dz_5 + z_5^ez_1$\n&~~~~~{\\thicklines {\\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,0){\\line(1,0){26}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\oval(104,26)[t]}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){25}}\n \\put(104,0){\\circle*{5}}\n \\end{picture}}} \\\\ [.5mm]\n\\hline\n\\end{tabular}\n\\end{center}\n}}\n\n\\begin{center}\n\\parbox{6.4truein}\n{\\noindent {\\bf Table 1.}~~\n {\\it The polynomial types that have been implemented}.}\n\\end{center}\n\nThe last problem that confronts the construction of all Calabi--Yau\\ manifolds\nin weighted projective space is the question whether the list is finite.\nAgain we compare the new situation with the one in the case\nof CICYs. There the condition of vanishing first Chern class leads to just\none configuration that can be found in the case\nof one projective space and one polynomial, namely the quintic\n$\\relax{\\rm I\\kern-.18em P}_4[5]$. For a weighted $\\relax{\\rm I\\kern-.18em P}_4$ however this condition leads to\nthe equation (\\ref{c1}).\nIt seems at first that this equation has an infinite number of solutions.\nIn fact a little thought shows that for each of the thirty polynomial types\nthere are restrictions on the range of possible weights. For the polynomials\nof Fermat type this is already known because of their relation\n\\cite{kms}\\cite{m}\\cite{gvw} to\ncertain types of conformal field theories which have all been constructed\n\\cite{ls12}\\cite{ls3}\\cite{lr}\\cite{fkss1}. Consider a Fermat polynomial\n\\begin{equation}\np = z_1^{a_1} + z_2^{a_2} + z_3^{a_3} + z_4^{a_4} + z_5^{a_5}\n\\end{equation}\nwith weights $(k_1,....,k_5)$ and degree\n\\begin{equation}\nd= k_1a_1 = k_2a_2 = k_3a_3 = k_4a_4 = k_4a_5~.\n\\end{equation}\nThe condition of vanishing first Chern class becomes\n\\begin{equation}\n1 = \\sum_{i=1}^5 \\frac{1}{a_i} \\lleq{cycon1}\nIt is possible to iteratively bound the $a_i$, which we take to be\nordered such that $a_i\\leq a_{i+1}$, in virtue of the fact that\n$\\sum_i \\frac{1}{a_i}$ is a decreasing function of all its arguments.\nThe smallest possible value for $a_1$ is 2.\nThe next step consists in finding the smallest possible\nvalue for $a_2$. Using the lower bound on $a_1$ condition (\\ref{cycon1})\nbecomes\n\\begin{equation}\n\\frac{1}{2} = \\sum_{i=2}^5 \\frac{1}{a_i},\n\\end{equation}\nfrom which it follows that the smallest possible value for\n$a_2$ is 3. Proceeding in this manner we end up with\nthe condition\n\\begin{equation}\n\\frac{1}{42} - \\frac{1}{43} = \\frac{1}{a_5}\n\\end{equation}\nfrom which we find $a_5 = 1806$ which turns out to be the highest power\nthat arises in our polynomials.\n\nAs a second example consider polynomials of type 2,\n\\begin{equation}\np = z_1^{a_1} + z_2^{a_2} + z_3^{a_3} + z_4^{a_4} + z_5^{a_5}z_4.\n\\end{equation}\nIn this case we have\n\\begin{equation}\nd= k_1a_1 = k_2a_2 = k_3a_3 = k_4a_4 = k_4 + k_5a_5.\n\\end{equation}\nand the condition of vanishing first Chern class is now\n\\begin{equation}\n1 = \\sum_{i=1}^5 \\frac{1}{a_i} - \\frac{1}{a_4a_5}\n\\lleq{c1con2}\n\nTaking again $a_1=2$ we proceed as above. Using the lower bound on $a_1$\ncondition (\\ref{c1con2}) becomes\n\\begin{equation}\n\\frac{1}{2} = \\sum_{i=2}^5 \\frac{1}{a_i} - \\frac{1}{a_4a_5}.\n\\end{equation}\n{}From this equation it follows that the smallest possible value for\n$a_2$ is 3. At the next stage we find $a_3=7$ and then $a_4=43$. Finally\nwe find the condition\n\\begin{equation}\n\\frac{1}{42\\cdot 43} = \\frac{1}{a_5} - \\frac{1}{43\\cdot a_5}\n\\end{equation}\nwhence $a_5 = 42^2 = 1764$. In a similar way we find constraints\non all other types of polynomials.\n\n\\vskip .1truein\n\\noindent\n\\section{Computation of the Spectrum}\n\nHaving constructed these 6,500 odd spaces one wants, of course, to know\n about the spectrum, especially about the number of light\ngenerations. There are several methods available to compute\nthese numbers. First there is of course the geometrical analysis that can\nbe used to compute the independent Hodge numbers of a Calabi--Yau manifold.\nAnother method that is more useful for the class of spaces at hand are\ntechniques for computing the spectrum in the framework of Landau--Ginzburg\ntheories. In those cases for which an exactly solvable theory corresponding\nto the model is known, techniques from conformal field theory are available\nas well. Unfortunately for most of the theories constructed in the previous\nsection no exactly solvable model is known and hence the tools from\nconformal field theory, even though most powerful, are not available here.\n\nThe very first step in a systematic analysis is, of course,\nthe determination\nof the number of light generations, i.e. the computation of\nthe Euler number.\n This can be done by computing the integral\nof the third Chern class using the fact that\n$\\int J^3 = 1\/\\prod k_j$ and taking into\naccount the contributions from the singularities \\cite{yau}\n\\begin{equation}\n\\chi = -\\frac{\\frac{1}{3}\\left (d^3 - \\sum k_j^3 \\right)d}{\\prod k_j}\n - \\sum_i \\frac{\\chi (S_i)}{n_i} + \\sum_i n_i \\chi (S_i)\n\\end{equation}\nwhere $\\chi (S_i)$ is the Euler number of the singular set $S_i$ and\n$n_i$ is the order of the cyclic symmetry group $\\relax{\\sf Z\\kern-.4em Z}_{n_i}$.\n\nThis can be illustrated with an example. Consider the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,4,5,5,7)}[25] ~~~~~~~~~\n{\\thicklines \\begin{picture}(150,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\put(52,0){\\circle*{5}}\n \\put(52,0){\\line(1,0){26}}\n \\put(78,0){\\circle*{5}}\n \\put(78,0){\\line(1,0){26}}\n \\put(104,0){\\circle*{5}}\n \\put(104,7){\\circle{12}}\n\n \\end{picture}}.\n\\lleq{examp}\nIn this example we have three types of singular sets; first there is a\n$\\relax{\\sf Z\\kern-.4em Z}_4$--curve $C$ with Euler number $\\chi (C) = 2$. Next there are five\n$\\relax{\\sf Z\\kern-.4em Z}_5$--points and finally the $\\relax{\\sf Z\\kern-.4em Z}_7$ leads to one additional fixed point.\nPut together with $\\chi_s = -44\\frac{5}{14}$ for the Euler number of the\nsingular space this gives $\\chi = -6$ for this space.\n\nIt is clear from this example that the geometrical technique is rather\nawkward to apply systematically as it entails a detailed analysis of the\nsingular sets $S_i$ for each manifold. These not only depend on the\ndivisibility\nproperty of the weights but also on the type of the individual catastrophe\ninvolved and therefore need to be determined on a case by case basis. This\nis not easily automated.\n\nIt is much easier to use a result of Vafa \\cite{v} on the Euler number of\n$c=9$, $N=2$ Landau--Ginzburg models. His formula specialized to the\ncase at hand yields\n\\begin{equation}\n\\chi = \\frac{1}{d} \\sum_{l,r = 0}^{d-1}~~\n (-1)^{r+l+d}\\prod_{lq_i,rq_i \\in \\relax{\\sf Z\\kern-.4em Z}} \\frac{d-k_i}{k_i},\n\\end{equation}\nwhere $q_i = k_i\/d$.\nComputing the Euler number for all 10,839 odd spaces leads to the results\nof Figure 3. As already mentioned the Euler number $-960\\leq \\chi \\leq 960$.\nAmong these\nspaces are many that lead to 2, 3 and 4 light generations.\n\nThe next question then is how the Euler number actually splits up\ninto generations and antigenerations. Again it is possible to use both,\nmanifold techniques and Landau--Ginzburg type methods. As we have already\neasy methods to compute the Euler number $\\chi = 2(h^{(1,1)} - h^{(2,1)})$\nwe only need to compute either the number of generations of the number of\nantigenerations to have the complete Hodge diamond.\n\nWe consider first the geometrical techniques and compute the number of\nantigenerations. In a projective space there would be nothing to compute since\nin this case the dimensions of this cohomology group is always one,\nits only contribution coming from the K\\\"ahler form. In the case of weighted\nprojective spaces the K\\\"ahler form of course is not the only contribution\nbecause the blow--ups of the singular sets introduce new (1,1)--forms.\nThe singular sets consist of points and\/or curves. The techniques for blowing\nup points have been discussed in \\cite{ry} and the contributions\ncoming from blowing up curves have been discussed in \\cite{s3}.\nThese are in fact the\nonly types of singularity that can arise if $p$ is transverse. In other words\nthat singular subsets of $M$ cannot have dimension 2 or 3.\nFirst we show that the embedding $\\relax{\\rm I\\kern-.18em P}_4$ cannot have singular sets of\ndimension\ngreater than or equal to 3. Recall that the $\\relax{\\rm I\\kern-.18em P}_4$ has a singular point for\neach\n$k_i>1$, a singular curve for each pair with $(k_i,k_j)>1$, a singular subset\nof dimension 2 for each triple with $(k_i,k_j,k_l)>1$ {\\it etc\\\/}. In the definition\nof $\\relax{\\rm I\\kern-.18em P}_4$ we have assumed that the weights have no common factor so there\nare no\nsingular sets of dimension 4. Consider the possibility of a singular set of\ndimension 3. Such a set would correspond to weights such that\n$$(k_2,k_3,k_4,k_5)=m>1~~{\\rm but}~~m\\not|k_1~.$$\nSince $m$ does not divide $k_1$ but does divide the other $k$'s it cannot\ndivide\nthe degree $d=\\sum_{j=1}^5k_j$. Every monomial of degree $d$ must\ntherefore contain at least one factor of $z_1$. Thus\n$p(z)=z_1\\tilde{p}(z)$ and\nso is not transverse. On the other hand singular sets of dimension 2\ncan occur\nand these will generically intersect $M$ in subsets of dimension 1.\nIt remains\nto\nshow that a singular subset of dimension 2 cannot lie within $M$. To\nthis end\nsuppose there is a fixed point set of dimension 2 for the identification\n(\\ref{pro5})\nand that this subset lies within $M$. We may choose coordinates $(x,y,z)$\nsuch that, locally, the fixed point set is $(x,y,0)$. Suppose the\nidentification is represented by a matrix $A$. Since $A$ fixes $(x,y,0)$ it\nhas the form $$A=\\pmatrix{1&0&a\\cr 0&1&b\\cr 0&0&c}~.$$\nThe three--form $dx\\wedge dy\\wedge dz$ transforms as\n$$dx\\wedge dy\\wedge dz\\longrightarrow\\det A\\,dx\\wedge dy\\wedge dz$$\nwe must have $\\det\\,A=1$ and hence $c=1$. Since $A$ is contained in a finite\ngroup $A^n=1$ for some $n$, however\n$$A^n=\\pmatrix{1&0&na\\cr 0&1&nb\\cr 0&0&1}$$\nso $a$ and $b$ must, in fact, vanish. Thus the only identification that\nfixes a two--dimensional subset is the identity.\n\nWhen resolving the singularities it needs to be checked that the blown up\nmanifolds are still Calabi--Yau manifolds. For the case of singular points\nthis\nhas been discussed in ref. \\cite{ry} whereas the case of singular curves\nwas first analyzed in ref. \\cite{s3}.\n\nIn order to resolve curves consider the action of a $\\relax{\\sf Z\\kern-.4em Z}_n$ on a weighted\nCICY leaving a curve invariant. In the three--dimensional Calabi--Yau manifold\nthe normal bundle of this curve has fibres $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2$ and therefore this discrete\ngroup induces an action on the fibres described by the matrix\n\\begin{equation}\n\\left (\\matrix{~~\\alpha^{mq}&0\\cr 0&~~\\alpha^m\\cr} \\right ) \\lleq{normact}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is an n$^{th}$ root of unity, $0\\leq m \\leq n$, and $q,n$ have\nno common divisor. This action has an isolated singularity which needs\nto be resolved. The essential point is that the singularity of\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2\/\\relax{\\sf Z\\kern-.4em Z}_n$ can be described as the singular set of the surface\n\\begin{equation}\nS :~~ z_3^n = z_1z_2^{n-q}\n\\end{equation}\nin $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_3$ and therefore can be resolved by a construction that\nis completely determined by the type $(n,q)$ of the action through the method\nof continued fractions.\nThe expansion of $\\frac{n}{q}$ in a continued fraction\n\\begin{eqnarray}\n\\frac{n}{q} &=& [b_1,...,b_s] \\nonumber \\\\\n &=& b_1 - \\frac{1}{b_2 - \\frac{\\Large 1}{ \\ddots -\n \\frac{\\Large 1}{\\Large b_s} }}\n\\end{eqnarray}\n\n\\noindent\ndetermines the numbers $b_i$ which specify uniquely the plumbing process\nwhich replaces the singularity. Furthermore, it also determines the additional\ngenerators of the cohomology, since the number\nof $\\relax{\\rm I\\kern-.18em P}_1$'s necessary to resolve the singularity is precisely the number of\nsteps needed in the evaluation of $\\frac{n}{q}= [b_1,..,b_s]$.\nThe reason for this is that the singularity is replaced by a bundle which\nis constructed of $s+1$ patches with $s$ transition functions that are\nspecified by the $b_i$'s. Each of these glueing steps introduces a sphere\nwhich in turn\nsupports a (1,1)--form. A standard shorthand notation for the geometry of\nthe blow--up is through what is called a Hirzebruch--Jung\ntree \\cite{hirz} which in the case of the blow--up of a $\\relax{\\sf Z\\kern-.4em Z}_n$ action is\njust an\nSU($s+1$) Dynkin diagram with the negative values of the $b_i's$ attached to\nthe nodes. Each node in the diagram corresponds to a sphere and the diagram\nshows which spheres intersect each other (in the case of the $\\relax{\\sf Z\\kern-.4em Z}_n$--action\nonly the neighboring spheres intersect) whereas the $b_i$ determine the\nintersection numbers.\n\nIn order to show that these blow--up procedures can be applied, we have to\nshow that the determinant (\\ref{normact}) is always 1. This can be done by\nchecking the invariance of the holomorphic threeform $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ under $\\relax{\\sf Z\\kern-.4em Z}_n$.\n\nConsider first manifolds of Fermat type. A singular curve in such a manifold\nis signalled by three weights of the ambient space that are not coprime\n\\begin{equation}\n(k_1,k_2,k_3) =n > 1\n\\end{equation}\n($n$ does not divide $k_4,k_5$). The integer $n$ defines a $\\relax{\\sf Z\\kern-.4em Z}_n$ discrete\ngroup. The action of $\\relax{\\sf Z\\kern-.4em Z}_n$ is given by\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\mapsto (z_1,z_2,z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4} z_4, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_5} z_5),\n\\end{equation}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is an $n$th root of unity. In order to show that the blown--up\nmanifold is still of Calabi--Yau type one needs to show that $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$\nis invariant. In this case the action of $\\relax{\\sf Z\\kern-.4em Z}_n$ on $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ is\n\\begin{equation}\n\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta \\mapsto \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5} \\Omega} \\def\\s{\\sigma} \\def\\th{\\theta.\n\\end{equation}\nThe condition for $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ to be invariant is, therefore, that\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5}=1,\n\\end{equation}\nwhich is always true since for Fermat type polynomials\n\\begin{equation}\n1=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^d = \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{\\sum k_i} =\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4+k_5}.\n\\end{equation}\n\nNow suppose the curve is embedded in a non--Fermat Calabi--Yau manifold.\nThen\nthe possibility exists that there are only two weights for which\n\\begin{equation}\n(k_1,k_2) =n > 1\n\\end{equation}\n($n$ does not divide $k_3,k_4,k_5$), i.e. $k_1,k_2$ are not constrained by\nthe polynomial. In this case the coordinates parametrizing the curve occur\nas\n\\begin{equation}\nz_1^{l_1}z_p,~~~z_2^{l_2}z_q.\n\\end{equation}\nThe action of $\\relax{\\sf Z\\kern-.4em Z}_n$ is given by\n\\begin{equation}\n(z_1,z_2,z_3,z_4,z_5) \\mapsto (z_1,z_2,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_2} z_3,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_4} z_4, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_5}\nz_5),\n\\end{equation}\nand the condition for $\\Omega} \\def\\s{\\sigma} \\def\\th{\\theta$ to be invariant becomes\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_3+k_4+k_5}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_p}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_q}.\n\\end{equation}\nAs before\n\\begin{equation}\nd=k_1l_1+k_p=k_2l_2+k_q=nr_1l_1+k+_p=nr_2l_2+k_q\n\\end{equation}\nhence\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^d = \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{ k_3+k_4+k_5} =\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_p}=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{k_q}\n\\end{equation}\nwhich was to be shown.\n\nReturning now to the discussion of the example $\\relax{\\rm I\\kern-.18em P}_{(4,4,5,5,7)}[25]$, we\nfind from the results of ref. \\cite{ry} that a $\\relax{\\sf Z\\kern-.4em Z}_5$--point blow--up\nof this\nmanifold contributes two (1,1)--forms, whereas a $\\relax{\\sf Z\\kern-.4em Z}_7$--point blow--up\nleads\nto three additional generators of the second cohomology. Since\nthe $\\relax{\\sf Z\\kern-.4em Z}_5$\nsingular set $\\relax{\\rm I\\kern-.18em P}_1[5]$ consists of five points and the $\\relax{\\sf Z\\kern-.4em Z}_7$ singular\nset\nconsists just of one point all point blow--ups together contribute a total\nof\nthirteen (1,1)--forms. To find out how many (1,1)--forms the blow--up of\nthe\n$\\relax{\\sf Z\\kern-.4em Z}_4$--curve contributes we need to check the induced action of this\ndiscrete group action on the normal bundle of the curve \\cite{s3}.\n\nIn our example (\\ref{examp}) this induced action\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2 \\ni (z_1,z_2) \\longmapsto (\\alpha z_1, \\alpha^3 z_2)\n\\end{equation}\nis of type $(n,q)=(4,3)$ and therefore the singularity of $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_2\/\\relax{\\sf Z\\kern-.4em Z}_4$\nis equivalent to the singular set of the surface\n\\begin{equation}\n z_3^4 = z_1z_2 .\n\\end{equation}\nThe resolution is specified by the Hirzebruch--Jung tree\n\\begin{center}\n{\\thicklines {\\begin{picture}(60,15)\n \\put(0,6){--2}\n \\put(8,0){\\circle*{5}}\n \\put(8,0){\\line(1,0){26}}\n \\put(26,6){--2}\n \\put(34,0){\\circle*{5}}\n \\put(34,0){\\line(1,0){26}}\n \\put(52,6){--2}\n \\put(60,0){\\circle*{5}}\n \\end{picture}}}\n\\end{center}\nwhich in turn is determined completely by the continued fraction\\begin{equation}\n\\frac{4}{3} = 2 - \\frac{1}{2-\\frac{1}{2}}.\n\\end{equation}\nTherefore the blow--up of the curve contributes three more (1,1)--forms.\nTaking into account the K\\\"ahler form of the ambient space then leads\nto a total of 17 (1,1)--forms for this manifold.\n\nAs is evident however the manifold technique is awkward to implement in a\nsystematic\nway and is better used as an independent check of the Landau--Ginzburg\ntype formulation of this problem by Vafa who constructs a Poincar\\'e--type\npolynomial for the $l^{\\rm th}$ twisted sector\n\\begin{equation}\nTr_l ((t\\bt)^{dJ_0})\n= t^{d\\left (Q_l+\\frac{1}{6}c_T \\right )}\n \\bt^{d\\left (-Q_l+\\frac{1}{6}c_T \\right )}\n \\prod_{lq_i\\in \\relax{\\sf Z\\kern-.4em Z}}\n \\left ( \\frac{1- (t\\bt)^{d-k_i}}{1-(t\\bt)^{k_i}}\\right)\n\\end{equation}\nwith\n\\begin{eqnarray}\n Q_l &=& \\sum_{lq_i {\\footnotesize{\\not \\in}} \\relax{\\sf Z\\kern-.4em Z}}\n \\left (lq_i - [lq_i] - \\frac{1}{2} \\right )~,\\nonumber \\\\\n \\frac{1}{6}c_T &=& \\sum_{lq_i {\\footnotesize{\\not \\in}} \\relax{\\sf Z\\kern-.4em Z}}\n \\left (\\frac{1}{2} - q_i\\right )\n\\end{eqnarray}\nwhere $t$ and $\\bt$ are formal variables, $d$ is the degree of the\nLandau--Ginzburg\npotential, the $q_i=k_i\/d$ are the normalized weights of the fields and\n$[lq_i]$ is the integer part of $lq_i$.\nExpanding this polynomial in powers in $t$ and $\\bt$ it is possible to read\noff the contributions to the various cohomology groups from the different\nsectors of the twisted LG--theory. The (2,1) forms for example are given by\nthe number of fields with charge (1,1), i.e. the coefficient of $(t\\bt)^d$.\nIn general, the number of $(p,q)$ forms are given by the coefficients of\n$t^{(3-p)d}\\bt^{qd}$ in the Poincar\\'e polynomials summed over all sectors\n$l=0,1,\\dots,d-1$.\n\n\n\\vskip .1truein\n\\noindent\n\\section{Landau--Ginzburg Vacua}\n\n\\noindent\nIn this section we briefly review the construction of all\n Landau--Ginzburg vacua based on superpotentials with an isolated\nsingularity at the origin.\nConsider a string ground state based on a Landau--Ginzburg theory\nwhich we\nassume to be $N=2$ supersymmetric since we demand $N=1$ spacetime\nsupersymmetry.\nUsing a superspace formulation in terms of the coordinates\n$(z,\\bz,\\th^+,\\bth^+,\\th^-,\\bth^-)$ the action takes the form\n\\begin{equation}\n{\\cal A}} \\def\\cB{{\\cal B}} \\def\\cC{{\\cal C}} \\def\\cD{{\\cal D} = \\int d^2zd^4\\th~K(\\Phi_i,\\bar \\Phi} \\def\\bth{\\bar \\theta_i) + \\int d^2zd^2\\th^- ~W(\\Phi_i)\n + \\int d^2zd^2\\th^+ ~W(\\bar \\Phi} \\def\\bth{\\bar \\theta_i)\n\\end{equation}\nwhere $K$ is the K\\\"ahler potential and the superpotential $W$ is a\nholomorphic function of the chiral superfields $\\Phi_i$.\nSince the ground states of the bosonic potential are the critical points\nof the superpotential of the LG theory\n we demand their existence. The type of critical points we\nneed is determined by the fact that we wish to keep the fermions in\nthe theory massless; hence we assume that the critical points are\ncompletely degenerate. Furthermore we require that all critical points\nbe isolated, since we wish to relate the finite dimensional ring of\nmonomials associated to such a singularity with the chiral ring of\nphysical states in the Landau--Ginzburg theory, in order to construct\nthe spectrum of the corresponding string vacuum.\nFinally we demand that the theory is conformally invariant; from this\nfollows, relying on some assumptions regarding the renormalization\nproperties of the theory, that the Landau--Ginzburg potential is\nquasihomogeneous. In other words, we require to be able to assign,\nto each field\n$\\Phi_i$, a weight $q_i$ such that for any non--zero complex number\n$\\l \\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}$\n\\begin{equation}\nW(\\l^{q_1}\\Phi_1,\\dots,\\l^{q_n}\\Phi_n) =\\l W(\\Phi_1,\\dots,\\Phi_n).\n\\end{equation}\nThus we have formulated the class of potentials that we need to consider:\nquasihomogeneous polynomials that have an isolated, completely\ndegenerate singularity (which we can always shift to the origin).\n\nAssociated to each of the superpotentials, $W(\\Phi_i)$ is a\nso--called catastrophe which is obtained by first truncating the\nsuperfield $\\Phi_i$ to its lowest bosonic component\n$\\phi_i(z,\\bz)$, and then going to the\nfield theoretic limit of the string by assuming $\\phi_i$ to be constant\n$\\phi_i=z_i$. Writing the weights as $q_i = k_i\/d$, we will denote by\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(k_1,k_2,\\dots,k_n)}[d]\n\\end{equation}\nthe set of all catastrophes described by the zero locus of polynomials\nof degree $d$ in variables $z_i$ of weight $k_i$.\n\nThe affine varieties described by these polynomials are not compact\nand hence it is necessary to implement a projection in order to\ncompactify these spaces. In Landau--Ginzburg language, this amounts to an\norbifolding of the theory with respect to a discrete group $\\relax{\\sf Z\\kern-.4em Z}_d$ the\norder\nof which is the degree of the LG potential \\cite{v}. The spectrum of the\norbifold theory will contain twisted states which, together with\nthe monomial ring of the potential, describe the complete spectrum of the\ncorresponding Calabi--Yau manifold. We will denote the orbifold\nof a Landau--Ginzburg theory by\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(k_1,k_2,\\dots,k_n)}[d]\n\\end{equation}\nand call it a configuration.\n\nIn manifold speak the projection should lead to a three--dimensional\nK\\\"ahler manifold, with vanishing first Chern class. For a general\nLandau--Ginzburg theory no unambiguous universal prescription for doing\nso has been found and, as we will describe in Section 4, none can exist.\nOne way to compactify amounts to simply imposing projective\nequivalence\n\\begin{equation}\n(z_1,....,z_n) \\equiv (\\l^{k_1} z_1,.....,\\l^{k_n} z_n)\n\\lleq{projn}\nwhich embeds the hypersurface described by the zero locus of the\npolynomial into a weighted projective space $\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,\\dots,k_n)}$\nwith weights $k_i$. The set of hypersurfaces of degree $d$ embedded\nin weighted projective space will be denoted by\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,\\dots,k_n)}[d].\n\\end{equation}\nFor a potential with five scaling variables this\nconstruction is completely sufficient in order to pass from the\nLandau--Ginzburg theory to a string vacuum \\cite{m}\\cite{gvw} provided\n$d=\\sum_{i=1}^5 k_i$, which is the condition that these hypersurfaces\nhave vanishing first Chern class. For more than five variables, however,\nthis type of compactification does not lead to a string vacuum.\n\nEven though the precise relation between LG theories and CY manifolds\nis not known for the most general case certain facts are known.\nSince LG theories with five variables describe a CY manifold embedded in\na 4 complex dimensional weighted projective space one might expect\ne.g. that LG potentials with 6, 7, etc., variables describe manifolds\nembedded in 5, 6, etc., dimensional weighted projective spaces. This is\nnot correct.\n\nIn fact none of the models with more than five variables is related\nto manifolds embedded in one weighted projective space. Instead they\ndescribe Calabi--Yau manifolds embedded in products of weighted\nprojective space. A simple example is\nfurnished by the LG potential in six variables\n\\begin{equation}\nW=\\Phi_1 \\Psi_1^2+\\Phi_2 \\Psi_2^2+\\sum_{i=1}^3 \\Phi_i^{12}+\n\\Phi_4^3\n\\end{equation}\nwhich corresponds to the exactly solvable model described by the\ntensor product of $N=2$ minimal theories at the levels\n\\begin{equation}\n(22^2 \\cdot 10\\cdot 1)_{D^2\\cdot A^2},\n\\end{equation}\nwhere the subscripts indicate the affine invariants chosen for the\nindividual factors. This theory belongs to the LG configuration\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(2,11,2,11,2,8)}[24]^{(3,243)}_{-480}\n\\lleq{lgform}\nand is equivalent to the weighted complete intersection Calabi--Yau (CICY)\n manifold in the configuration\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1,1,4,6)}\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr}\n \\left [\\matrix{1&12\\cr 2&0\\cr}\\right]^{(3,243)}_{-480}\n\\lleq{cyform}\ndescribed by the intersection of the zero locus of the two potentials\n\\begin{eqnarray}\np_1 &=& x_1^2y_1+x_2^2y_2 \\nonumber \\\\\np_2 &=& y_1^{12}+y_2^{12}+y_3^{12}+y_4^3+y_5^2.\n\\end{eqnarray}\nHere we have added a trivial factor $\\Phi_5^2$ to the potential and\ntaken the field theory limit via $\\phi_i(z,\\bz)=y_i$, where $\\phi_i$\n is the lowest component of the chiral superfield $\\Phi_i$. The first\ncolumn in the degree matrix (\\ref{cyform}) indicates that the first\npolynomial is of bidegree (2,1) in the coordinates $(x_i,y_j)$\nof the product of the projective line $\\relax{\\rm I\\kern-.18em P}_1$ and the weighted\nprojective space $\\relax{\\rm I\\kern-.18em P}_{(1,1,1,4,6)}$ respectively, whereas the second\ncolumn shows that the second polynomial is independent of the\nprojective line and of degree 12 in the coordinates of the\nweighted $\\relax{\\rm I\\kern-.18em P}_4$. The superscripts in (\\ref{lgform}) and (\\ref{cyform})\ndescribe the dimensions $(h^{(1,1)},h^{(2,1)})$ of the fields\ncorresponding to the cohomology groups $(H^{(1,1)},H^{(2,1)})$,\nwhereas the subscript is the Euler number of the configuration.\n\nIt should be noted however that Landau--Ginzburg potentials in six\nvariables do not describe the most general complete intersection in\nproducts of weighted spaces, simply because not all of these manifolds\ninvolve trivial factors, or put differently, quadratic monomials.\nA simple example is the manifold that corresponds to the\nLandau--Ginzburg theory\n$(1\\cdot 16^3)_{A\\cdot E_7^3}$ with LG potential\n\\begin{equation}\nW= \\sum_{i=1}^3 \\left(\\Phi_i^3 + \\Phi_i \\Psi_i^3\\right) + \\Phi_4^3.\n\\end{equation}\nThis theory describes a\ncodimension--2 Calabi--Yau manifold embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3 &0\\cr\n 1 &3\\cr}\\right].\n\\end{equation}\nThis space has 8 (1,1)--forms and 35 (2,1)--forms which correspond to\nthe possible complex deformations in the two polynomials\n$p_1,p_2$ \\cite{s1}.\n\nAssociated to this Calabi--Yau manifold in a product of ordinary\nprojective\nspaces is an auxiliary algebraic manifold in a weighted six--dimensional\nprojective space\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,3,3,3,3)}[9].\n\\end{equation}\nobtained via the na\\\"{\\i}ve compactification (\\ref{projn}) where the first\nthree\ncoordinates come from the fields $\\Psi_i$ and the last four come from\nthe\n$\\Phi_i$. We want to compute the number of complex deformations of this\nmanifold, i.e. we want to compute the number of monomials of charge 1.\n\nThe most general monomial is of the form $\\prod_i \\Phi_i \\prod \\Psi_j$.\nIt is easy to show explicitly that there are precisely 35 monomials by\nwriting them down but the following remarks may suffice. There are\nfour different types of possible monomials, depending on whether they\ncontain the fields $\\Phi_i$ not at all (I), linearly (II),\nquadratically (III) or cubically (IV). First note\nthat monomials of type (I) do not contribute to the marginal operators.\nMonomials of type (II) have to contain cubic monomials in the $\\Psi$.\nSince there are three fields $\\Psi_i$ available, we obtain a total of\n$ 40$ marginal operators of this type.\nBecause of the equation of motion nine of these are in fact in the ideal\nand we are left with 31 complex deformations of this type. Monomials\nquadratic in the $\\Phi_i's$ again do not contribute, whereas those cubic\nin the $\\Phi_i$ fields contribute the remaining 4. Indeed, there are\n20 cubic monomials in terms of the four $\\Phi_i$ but using the equations\nof motion one finds that 16 of those are in the ideal.\n\nIn other words, for the total of $60=40+20$ monomials of degree\n9 (or charge 1) it is possible to fix the coefficients of 9 of the\n40 by allowing linear field redefinitions of the first three coordinates\nand also to fix the coefficients of 16 of the 20 via linear field\nredefinitions of the last four coordinates. Hence even though the ambient\nspace is singular and the manifold hits these singularities in a $\\relax{\\rm I\\kern-.18em P}_2$\nand in a cubic surface $\\relax{\\rm I\\kern-.18em P}_3[3]$ the resolution does not contribute any\ncomplex deformations because these surfaces are simply connected.\n\nIt should be emphasized that this manifold is not the physical internal\npart\nof a string ground state, but that it plays an auxiliary role, which\nallows to\ndiscuss just one particular sector of the string vacuum, namely the\ncomplex deformations.\n\n Before turning to the problem of constructing LG configurations\nsatisfying the constraints described above, we wish to make some remarks\nregarding the validity of the\nrequirements formulated in the previous paragraph.\n\nEven though the assumptions formulated in refs. \\cite{m}\\cite{vw} and\nreviewed above seem rather reasonable, and previous work shows that the\nset of such Landau--Ginzburg theories certainly is an interesting and\nquite extensive class of models, it is clear that it is not the most\ngeneral class of (2,2) vacua. Although it provides a rather large set\nof different models\n\\fnote{2}{The rather extravagant values that have been\n mentioned in the literature as the number of possible (2,2)\n vacua are based on extrapolations that do not take into account\n the problem of overcounting that is generic to all of these\n different constructions.},\n which contains many classes of previously constructed vacua\n\\fnote{3}{Such as vacua constructed tensor models based on the ADE minimal\n models \\cite{m}\\cite{gvw}\\cite{ls12}\\cite{ls3}\\cite{fkss1}\\cite{sy}\n and \\break\n level--1 Kazama--Suzuki models \\cite{lvwg}\\cite{fiq2}\\cite{s},\n as well as G\/H LG theories \\cite{ls5} related to Kazama--Suzuki\n models of higher levels.},\nthere are known vacua which cannot be described in this framework.\n\nPerhaps the simplest example that does not fit into the classification\n above is that of the Calabi--Yau manifolds in\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_5[4~~2],\n\\end{equation}\ndescribed by the intersection of two hypersurfaces defined by a quartic\nand a quadratic\npolynomial in a five--dimensional\nprojective space $\\relax{\\rm I\\kern-.18em P}_5$ because of the purely quadratic polynomial\nthat appears as one of the constraints defining the hypersurface.\n The requirement that the singularity\nbe completely degenerate seems, in fact, to exclude a\ngreat many of the CICY manifolds,\nthe complete class of which was constructed in ref. \\cite{cdls}.\nAn important set of manifolds in that class that does not fit into\nthe present framework\neither is defined by\npolynomials of bidegree (1,4) and (1,1)\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_4\\cr}\\left[\\matrix{1&1\\cr 1&4\\cr}\\right].\n\\end{equation}\nThe superpotential $W=p_1+p_2$ is not quasihomogeneous, nor does it\nhave an isolated singularity\n\\fnote{4}{These manifolds are important in the context of possible\n phase transitions between Calabi--Yau string vacua \\cite{cgh}\n via the process of splitting and contraction introduced in\n \\cite{cdls}.}.\nThus it appears that there ought to be a generalization of the framework\ndescribed above, which allows a modified LG description of these and other\nstring vacua. This however we leave for future work.\n\n\\vskip .1truein\n\\noindent\n\\section{Transversality of Catastrophes}\n\n\\noindent\nThe most explicit way of constructing a Landau--Ginzburg vacuum is, of\ncourse,\nto exhibit a specific potential that satisfies all the conditions\nimposed\nby the requirement that it ought to describe a consistent ground state\n of the string.\n Even though much effort has\ngone into the classification of singularities of the type described in\nthe\nprevious section,\nsuch polynomials have not been classified yet. As already mentioned above\nthe mathematicians have classified\npolynomials with at most three variables \\cite{agzv}, which is two short\nof\nthe lowest number of variables\nthat is needed in order to construct a vacuum that allows a formulation of\na four--dimensional low--energy effective theory\n\\fnote{5}{The complete list of K3 representations embedded in\n$dim_{\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}} =3$ weighted projective space $\\relax{\\rm I\\kern-.18em P}_{(k_1,k_2,k_3,k_4)}$ has\nbeen obtained in \\cite{reid}.}.\n\nIn Sections 1--3 a set of potentials in five variables was described\n which represents an obvious\ngeneralization of the polynomials that appear in two--dimensional\ncatastrophes.\nAfter imposing the conditions for these theories to describe\nstring vacua, only a finite number of the infinite number of LG theories\nsurvive and all these solutions were constructed. It is clear that this\n classification of singularities is\nnot complete, even after restricting to five variables.\nIt is indeed easy to construct polynomials that are not contained in the\nclassification of \\cite{cls}, a simple one being furnished by the\nexample \\cite{orlik}\n \\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(15,10,6,1)}[45]\\ni\n\\left\\{z_1(z_1^2+z_2^3+z_3^5)+z_4^{45}+z_2^2z_3^4z_4=0\\right\\}.\n\\end{equation}\nThis polynomial is not of\nany of the types analysed discussed above but it is nevertheless transverse.\n\nKnowledge of the explicit form of the potential of a LG theory is very\nuseful information when it comes to the detailed analysis of such a\nmodel.\nIt is however not necessary if only limited knowledge, such as the\ncomputation of the spectrum of the theory, is required. In fact\nthe only ingredients necessary for the computation of the spectrum\nof a LG vacuum \\cite{v} are the anomalous dimensions of the scaling\nfields\nas well as the fact that in a configuration of weights\nthere exists a polynomial of appropriate degree with an\nisolated singularity. However, it is much easier to check whether\nthere exists such a polynomial\nin a configuration than to actually construct\nsuch a potential. The reason is a theorem by Bertini \\cite{algeom},\nwhich asserts that\nif a polynomial does have an isolated singularity on the base locus\nthen,\neven though this potential may have worse singularities away from the\nbase locus, there exists a deformation of the original polynomial that\nonly\nadmits an isolated singularity anywhere. Hence we only have to find\ncriteria\nthat guarantee at worst an isolated singularity on the base locus.\nIt is precisely this problem that was addressed in the mathematics\nliterature \\cite{f} at the same time as the explicit construction of LG\nvacua was started in ref. \\cite{cls}.\nThe main point of the argument in \\cite{f} is the following.\n\nSuppose we\nwish to check whether a polynomial in $n$ variables $z_i$ with weights\n$k_i$ has an isolated singularity, i.e. whether the condition\n\\begin{equation}\ndp=\\sum_i \\frac{\\partial p}{\\partial z_i} dz_i = 0\n\\lleq{transv}\ncan be solved at the origin $z_1=\\cdots = z_n=0$. According to\nBertini's theorem,\nthe singularities of a general element in $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(k_1,...,k_n)}[d]$\nwill lie\non the base locus, i.e., the intersection of the hypersurface and\nall the\ncomponents of the base locus, described by coordinate planes of\ndimension\n$k=1, ..., n$. Let $\\cP_k$ such a $k$--plane, which we may assume to be\ndescribed by setting to zero the coordinates $z_{k+1}=\\cdots = z_n$.\nExpand\nthe polynomials around the non--vanishing coordinates $z_1,...,z_k$\n\\begin{equation}\np(z_1,...,z_n) =\nq_0(z_1,...,z_k)~ +~ \\sum_{j=k+1}^n q_j(z_1,...,z_k)z_j + h.o.t.\n\\end{equation}\nClearly, if $q_0\\neq 0$ then $\\cP_k$ is not part of the base locus\nand hence\nthe hypersurface is transverse. If on the other hand $q_j=0$, then\n$\\cP_k$ is part of the base locus and singular points\ncan occur on the intersection of the hypersurfaces defined by\n$\\cH_j=\\{q_j=0\\}$.\nIf, however, we can arrange this intersection to be empty, then the\npotential is smooth on the base locus.\n\nThus we have found that the conditions for transversality in any\nnumber of variables is the existence for any index set\n$\\cI_k=\\{1,...,k\\}$ of\n\\begin{itemize}\n\\begin{enumerate}\n\\item{either a monomial $z_1^{a_1}\\cdots z_k^{a_k}$ of degree $d$}\n\\item{or of at least $k$ monomials $z_1^{a_1}\\cdots z_k^{a_k}z_{e_i}$\nwith distinct $e_i$.}\n\\end{enumerate}\n\\end{itemize}\n\nAssume on the other hand that neither of these conditions\nholds for\nall index sets and let $\\cI_k$ be the subset for which they fail. Then\nthe potential has the form\n\\begin{equation}\np(z_1,...,z_n) = \\sum_{j=k+1}^n q_j(z_1,...,z_k)z_j ~+~ \\cdots\n\\end{equation}\nwith at most $k-1$ non--vanishing $q_j$. In this case the intersection\nof the hypersurfaces $\\cH_j$ will be positive and hence the polynomial\n$p$ will not be transverse.\n\nAs an example for the considerable ease with which one can check whether\na given configuration allows for the existence of a\npotential with an isolated singularity, consider the polynomial of Orlik\nand Randall\n\\begin{equation}\np=z_1^3+z_1z_2^3+z_1z_3^5+z_4^{45}+z_2^2z_3^4z_4.\n\\end{equation}\nCondition (\\ref{transv}) is equivalent to the system of equations\n\\begin{equation}\n\\begin{array}{r l r l}\n 0 &=~ 3z_1^2+z_2^3+z_3^5 , & 0 &=~ 3z_1z_2^2+2z_2z_3^4z_4 \\\\\n 0 &=~ 5z_1z_3^4 + 4z_2^2z_3^3z_4, & 0 &=~ z_2^2z_3^4+45z_4^{44} .\n\\end{array}\n\\end{equation}\nwhich, on the base locus, collapses to the trivial pair of\nequations $z_2z_3=0=z_2^3+z_5^5$. Hence this configuration allows for\nsuch a polynomial. To check the system away from the base locus\nclearly is much more complicated.\n\nBy adding a fifth variable $z_5$ of weight 13 it is possible to define\na Calabi--Yau deformation class\n$\\relax{\\rm I\\kern-.18em P}_{(1,6,10,13,15)}[45]_{-72}^{(17,53)}$, a configuration not\nconsidered in \\cite{cls}.\n\n\\vskip .1truein\n\\noindent\n\\section{Finiteness Considerations}\n\n \\noindent\nThe problem of finiteness has two parts: first one has to put a\nconstraint\non the number of scaling fields that can appear in the LG theory\nand then one has to determine limits on the exponents with which the\nvariables occur in the superpotential. Both of these constraints follow\nfrom the fact that the central charge of a Landau--Ginzburg theory with\nfields of charge $q_i$\n\\begin{equation}\nc=3\\sum \\left(1-2 q_i\\right)=:\\sum c_i\n\\lleq{cc}\nhas to be $c=9$ in order to describe a string vacuum.\n\nIt should be clear that without any additional input the number of\nLandau--Ginzburg vacuum configurations that can be exhibited is infinite.\nThis is to be\nexpected simply because it is known from the construction of CICYs\n\\cite{cls} that it is often possible to rewrite a manifold in an infinite\nnumber of ways and we ought to encounter similar things in the LG\nframework. A trivial way to do this is to simply add mass terms that\ndo not contribute to the central charge. Even though trivial such\nmass terms\nare important and necessary for LG theories, not only in orbifold\nconstructions \\cite{kss} but also in order to relate them to CY\nmanifolds. Consider e.g. the codimension--four Calabi--Yau manifold\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 &0 &0\\cr\n 1 &2 &0 &0\\cr\n 0 &1 &2 &0\\cr\n 0 &0 &1 &2\\cr}\\right]\n\\lleq{boggle}\nwith the defining polynomials\n\\begin{equation}\n\\begin{array}{r l r l}\np_1 &=~ \\sum_{i=1}^2 u_i^2v_i, & p_2 &=~ \\sum_{i=1}^3 v_i^2w_i \\\\\np_3 &=~ \\sum_{i=1}^3 w_i^2x_i, & p_4 &=~ \\sum_{i=1}^3 x_i^2\n\\end{array}\n\\lleq{bogglepollies}\nthe superpotential $W=\\sum p_i$ of which lives in\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(5,5,6,6,6,4,4,4,8,8,8)}[16]_{-32}^{10}\n\\end{equation}\nand has an isolated singularity at the origin. All eleven variables\nare coupled and hence\nthis example appears to involve three fields with zero central charge\nin a nontrivial way.\n\nIt turns out, however, that the manifold (\\ref{boggle}) is equivalent\nto a manifold with nine variables. One way to see this is by making\n use of some topological identities introduced in \\cite{cdls}. First\nconsider the well--known isomorphism\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_2[2] \\equiv \\relax{\\rm I\\kern-.18em P}_1,\n\\end{equation}\nwhich allows us to rewrite the manifold above as\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &2\\cr\n 0 &0 &2\\cr}\\right].\n\\end{equation}\nUsing the surface identity \\cite{cdls}\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr} \\left[\\matrix{ 2\\cr 1\\cr}\\right]=\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr}\n\\end{equation}\napplied via the rule\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr X\\cr} \\left[\\matrix{ 2 &0\\cr\n 1 &a\\cr\n 0 &M\\cr}\\right]=\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr X\\cr} \\left[\\matrix{ a\\cr\n a\\cr\n M\\cr}\\right]\n\\end{equation}\nshows that this space is in turn equivalent to\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_1 \\cr\\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 \\cr\n 2 &0 \\cr\n 0 &2 \\cr\n 1 &2 \\cr}\\right],\n\\lleq{nonlg}\na manifold with only nine homogeneous coordinates.\nIt should be noted that the LG potential of (\\ref{nonlg}), defined\nby the sum of the two polynomials, certainly does not\nhave an isolated singularity. Furthermore it is not possible to\n even assign weights to the fields such that the\ncentral charge comes out to be nine!\nIt is thus possible, by applying topological identities, to extend\nthe applicability of Landau--Ginzburg theories to types of Calabi--Yau\nmanifolds that were hitherto completely inaccessible by the standard\nformulation.\n\nFurther insight into the problem of redundancy in the construction of\nLG potentials can be gained by an LG theoretic analysis of this\nexample. From the weights of the scaling variables in the LG configuration\nabove it is clear that the spectrum of this LG configuration remains\nthe same if the last three coordinates are set to zero. In the\npotential\n\\begin{equation}\nW=\\sum_{i=1}^2 \\left(u_i^2v_i+v_i^2w_i+w_i^2x_i+x_i^2\\right) +\n (v_3^2w_3+w_3^2x_3+x_3^2)\n\\lleq{bogglelg}\n described by the CICY polynomials (\\ref{bogglepollies}), these variables\ncannot be set to zero because they are coupled to other fields; hence it\nseems impossible to reduce the number of fields. Consider however the\nfollowing change of variables\n\\begin{equation}\ny_i=x_i+\\frac{1}{2}w_i^2,~~i=1,2,3.\n\\end{equation}\n It follows from these transformations that the potential defined by\nby (\\ref{bogglelg}) is equivalent to\n\\begin{equation}\nW=\\sum_{i=1}^2\\left(u_i^2v_i+v_i^2w_i-\n \\frac{1}{4}w_i^4\\right)+v_3^2w_3-\\frac{1}{4}w_3^4.\n\\end{equation}\n Adding a trivial factor and splitting this potential into three\nseparate polynomials\n\\begin{equation}\np_1=u_1^2v_1+u_2^2v_2,~~~p_2=v_1^2w_1+v_2^2w_2+v_3^2w_3,~~~\np_3=-\\frac{1}{4}\\left(w_1^4+w_2^4+w_3^4\\right)+x^2\n\\end{equation}\nwe see that the original model is equivalent to a weighted\n configuration\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_2 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,2)}\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &4\\cr}\\right],\n\\end{equation}\nwhich again describes manifolds with nine variables.\nThus the original configuration (\\ref{boggle}) is in fact equivalent\nto two different (weighted) CICY representations\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_1\\cr \\relax{\\rm I\\kern-.18em P}_1 \\cr\\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 \\cr\n 2 &0 \\cr\n 0 &2 \\cr\n 1 &2 \\cr}\\right]\n{}~~\\equiv ~~\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_2 \\cr \\relax{\\rm I\\kern-.18em P}_2\\cr}\n\\left[\\matrix{2 &0 &0 &0\\cr\n 1 &2 &0 &0\\cr\n 0 &1 &2 &0\\cr\n 0 &0 &1 &2\\cr}\\right]\n{}~~\\equiv ~~\n\\matrix{\\relax{\\rm I\\kern-.18em P}_1 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_2 \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,2)}\\cr}\n\\left[\\matrix{2 &0 &0\\cr\n 1 &2 &0\\cr\n 0 &1 &4\\cr}\\right].\n\\end{equation}\n\nTo summarize the last few paragraphs, we have shown two things: first\nthat by adding trivial factors and coupling them to the remaining fields\nwe can give a Landau--Ginzburg description of a\nlarger class of Calabi--Yau manifolds than previously thought\npossible. Furthermore we can use topological identities to obtain\nan LG formulation of CY manifolds, which do not admit a\ncanonical LG potential at all.\nIncidentally we have also shown that it is possible to relate complete\nintersection manifolds embedded in products of projective spaces to\n weighted complete intersection manifolds\nembedded in products of weighted projective space.\n\nSimilarly the number of fields can grow without bound if we not only\nallow\nfields that do not contribute to the central charge but also fields\nwith\na negative contribution. Again such fields provide redundant\ndescriptions\nof simpler LG theories; they are nevertheless important for the LG\/CY\nrelation and\noccur in the constructions of splitting and contraction introduced in\nref. \\cite{cdls}. Even though these\nconstructions were discussed in \\cite{cdls} only in the context of\nCalabi--Yau\nmanifolds embedded in products of ordinary projective spaces they\nreadily generalize to the more general framework of weighted projective\nspaces.\n\nIn special circumstances, the splitting or contraction process does not\nchange\nthe spectrum of the theory and hence it provides another tool to relate\n LG potentials with at most nine variables manifolds with more than\nnine\nhomogeneous coordinates. Consider e.g. the manifold embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0\\cr 1&6\\cr}\\right]\n_{-252}^{(2,128)}\n\\end{equation}\nwhich is described by the zero locus of the two polynomials\n\\begin{eqnarray}\np_1&=&x_1^2y_1+x_2^2y_2 \\nonumber \\\\\np_2&=&y_1^6+y_2^6+y_3^6+y_4^6+y_5^2\n\\end{eqnarray}\nthat can be described by a superpotential $W=p_1+p_2$ defining a\nLandau--Ginzburg theory in\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(5,5,2,2,2,2,6)}[12]^{(2,128)}_{-252}.\n\\end{equation}\nThis manifold can be rewritten via an ineffective split in an infinite\nsequence of different representations as\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0\\cr 1&6\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0&0\\cr 1&1&0\\cr 0&1&6\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}\\cr}\n\\left[\\matrix{2&0&0&0\\cr 1&1&0&0\\cr 0&1&1&0\\cr 0&0&1&6}\\right]\n\\longrightarrow\n\\cdots\n\\end{equation}\nwhich are described by LG potentials in the alternating classes\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7}_{(5,5,2,2,2,2,6)}[12] \\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9}_{(1,1,10,10,2,2,2,2,6)}[12] \\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 11}_{(5,5,2,2,10,10,2,2,2,2,6)}[12] \\longrightarrow \\cdots\n\\end{equation}\ni.e. the infinite sequence belongs to the configurations\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7+4k}_{(5,5,2,2,10,10,2,2,10,10,...,2,2,10,10,2,2,6)}[12]\n\\end{equation}\nwhere the part $(2,2,10,10)$ occurs $k$ times, and\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 5+4k}_{(1,1,10,10,2,2,10,10,2,2,...10,10,2,2,2,2,6)}[12].\n\\end{equation}\nwhere $(10,10,2,2)$ occurs $k$ times.\n\nThe construction above easily generalizes to a number of examples\nwhich all belong to a class of spaces discussed in ref. \\cite{s3}.\nConsider manifolds embedded in\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill\\cr \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0\\cr k_1&k\\cr}\\right],\n\\end{equation}\nwhere $k=k_1+k_3+k_4+k_5$. These spaces can be split\ninto the infinite sequences\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0\\cr k_1&k\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0&0\\cr 1&1&0\\cr 0&k_1&k\\cr}\\right]\n\\longrightarrow\n\\matrix{\\relax{\\rm I\\kern-.18em P}_{(1,1)} \\hfill \\cr \\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill\\cr\\relax{\\rm I\\kern-.18em P}_{(1,1)}\\hfill \\cr\n \\relax{\\rm I\\kern-.18em P}_{(k_1,k_1,k_3,k_4,k_5)}\\cr}\n\\left[\\matrix{2&0&0&0\\cr 1&1&0&0\\cr 0&1&1&0\\cr 0&0&k_1&k}\\right]\n\\longrightarrow\n\\cdots\n\\end{equation}\nIf the weights are such that $k\/k_i$ is an integer,\nthen it is easy to write down the tensor model that corresponds to\nit (but this is not essential).\nIn such models the levels $l_i$ of the tensor model\n $l_1^2\\cdot l_3\\cdot l_4\\cdot l_5$\nin terms of the weights are given by\n\\begin{equation}\nl_1=l_2=\\frac{2k}{k_1}-2,~~~l_i=\\frac{k}{k_i}-2,~i=3,4,5\n\\end{equation}\nand the corresponding LG potentials live in\n\\begin{eqnarray}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7}_{(k-k_1,k-k_1,2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\longrightarrow\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9}_{(k_1,k_1,2(k-k_1),2(k-k_1),2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\longrightarrow \\cdots\n\\end{eqnarray}\ni.e. they belong to the sequences\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 7+4p}_{((k-k_1),(k-k_1),2k_1,2k_1,2(k-k_1),2(k-k_1),....,\n 2k_1,2k_1,2k_3,2k_4,2k_5)}[2k]\n\\end{equation}\nwhere the part $(2k_1,2k_1,2(k-k_1),2(k-k_1))$ occurs $p$ times, and\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star 9+4k}_{(k_1,k_1,2(k-k_1),2(k-k_1),\n2k_1,2k_1,...,2k_3,2k_4,2k_5)}[2k].\n\\end{equation}\nwhere $(2(k-k_1),2(k-k_1),2k_1,2k_1)$ occurs $p$ times.\n\nAll these models are constructed in such a way that they have central\ncharge\nnine, but in contrast to the example discussed previously,\nthere now appear fields with negative central charge.\nIn the case at hand, however, these dangerous fields\nonly occur in a {\\it coupled} subpart of the theory; the smallest\nsubsystem which involves these fields and which one can isolate is in\nfact a theory with positive central charge. In the series of splits\njust described e.g., the fields with negative central charge that occur\nin the\nfirst split always appear in the subsystem described by the\nconfigurations\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(k_1,2(k-k_1),2k_1)}[2k]\n\\end{equation}\nwith potentials of the form\n\\begin{equation}\nx^2y +yz+z^{2k\/k_1}.\n\\end{equation}\nThus the contribution to the central charge of this sector becomes\n\\begin{equation}\nc=3\\left[\\left(1-\\frac{k_1}{k}\\right)+\\left(1-\\frac{2(k-k_1)}{k}\\right)+\n \\left(1-\\frac{2k_1}{k}\\right)\\right],\n\\lleq{ccc}\nwhich is always positive. This formula suggests that it ought\nto be possible to dispense with the variables $y,z$ altogether, as\ntheir total contribution to the central charge adds up to zero, and\nthat this theory is equivalent to that of a single monomial of degree\n$k\/k_1$.\n\nMore generally one may consider the Landau--Ginzburg theory defined by\nthe potential\n\\begin{equation}\nx_1^ax_2+x_2x_3+\\cdots + x_{n-1}x_n + x_n^b.\n\\lleq{triv}\n From the central charge\n\\begin{equation}\nc= \\left\\{ \\begin{array}{l l}\n 6\\left(1-\\frac{1}{a}\\right)\\left(1-\\frac{1}{b}\\right),\n & n~{\\rm even} \\\\\n 3\\left(1-\\frac{2}{ab}\\right), &n~{\\rm odd}\n \\end{array} \\right\\},\n\\end{equation} as well as from the dimension of the chiral ring\n\\begin{equation}\n{\\rm dim}~R_n = \\prod \\left(\\frac{1}{q_i}-1\\right)\n = \\left\\{ \\begin{array}{l l}\n ab-b-1,&n~{\\rm even} \\\\\n ab-1, &n~{\\rm odd}\n \\end{array} \\right\\},\n\\end{equation}\nwe expect this theory to be equivalent to\n\\begin{equation}\nx_1^ax_2+x_2x_3+\\cdots x_{n-1}x_n + x_n^b =\n\\left\\{ \\begin{array}{l l}\n x_1^ax_n+x_n^b, &n~{\\rm even} \\\\\n x_1^{ab}, &n~{\\rm odd}\n \\end{array} \\right\\}.\n\\lleq{ident}\n\nSuch relations are supported by the identification of rather different\n LG configurations such as\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(1,1,1,6,6,6,3,3,3,3,3)}[9] \\equiv \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^{\\star}_{(1,1,1,3,3)}[9]\n\\end{equation}\nas well as many other identifications which are rather nontrivial\nin the context of the associated manifolds.\n\nIt follows from the above considerations that we have to assume,\n in order to avoid redundant\nreconstructions of LG theories, that the central charge\nof all scaling fields of the potential should be positive.\n In order to relate the potentials to manifolds, we may then add\none or several trivial factors or more complicated theories with zero\ncentral charge.\n\nUsing the above results we derive in the following more detailed\nfiniteness conditions. Observe first that from eq. (\\ref{cc})\nwritten as\n\\begin{equation}\n\\sum_{i=1}^r q_i=\\left( {r\\over 2}-{c\\over 6}\\right):=\\hat c\\label{cbed}\n\\end{equation}\nwe obtain $r>c\/3$.\n\nNow let $p$ be a polynomial of degree $d$ in $r$ variables.\nFor the index set $\\cI_1$ the conditions for transversality\nimply the existence of $n_i\\in \\relax{\\rm I\\kern-.18em N}^+$, $i=1,\\ldots,r$ and of a map\n$j:\\cI_r \\rightarrow \\cI_r$ such that for all $i$ there exists\n$j(i)$ such that\n\\begin{equation}\nq_i={{1-q_{j(i)}}\\over n_i}.\n\\lleq{qs1}\nLet us first see how many non-trivial fields can occur at most.\nFields which have charge $q_i\\le 1\/3$\ncontribute $c_i\\ge 1$ to the conformal anomaly.\nNow consider fields with larger charge. Since we assume $c_i>0$, they are\nin the range $1\/3<q_i<1\/2$.\nAmong these fields the transversality condition (1.) cannot hold,\nbecause two of them are not enough and three of them are too many\nfields in order to form a monomial of charge one. Transversality\ncondition (2.) implies that each of them has to occur together with a\npartner field $z_{j(i)}$. These pairs contribute to the conformal\nanomaly according to\n(\\ref{qs1},\\ref{cc}) $c_i+c_{j(i)}>2$, so we\ncan conclude that $r\\le c$.\n\nIn order to construct all transversal\nLG potentials for a given $c$,\nwe choose a specific $r$ in the range obtained above and consider\nall possible maps $j$ of which there are $r^r$.\nWithout restricting on the generality, we may\n{\\sl then} assume the $n_i$ to be ordered\n$n_1\\le \\cdots \\le n_r$.\nStarting with (\\ref{cbed}) we obtain via eq. (\\ref{qs1}) and the\npositivity of the charges a bound $n_1<r\/c$. Now we choose $n_1$ in the\nallowed range and use (\\ref{qs1}) in order to eliminate the $q_1$, if\nnecessary in favour of the $q_2,\\ldots,q_r$. This yields\nan equation of the general form $(p=2)$\n\\begin{equation}\n\\sum_{i=p}^r w^{(p)}_i q_i=\\hat c^{(p)}.\\label{cbed2}\n\\end{equation}\nIf $\\hat c^{(p)}\\neq 0$, eq. (\\ref{cbed2}) either is in contradiction\nwith the positivity of the charges or one can derive a finite bound for\n$n_{p}$. Assume $\\hat c^{(p)}>0$ and let $\\cI_+$ be the\nindices of the positive $w_i^{(p)}$; then one has\n$n_{p}<(\\sum_{i\\in \\cI_+} w_i^{(p)})\/\\hat c^{(p)}$; likewise if\n$\\hat c^{(p)}<0$ we have\n$n_{p}<(\\sum_{i\\in\\cI_-} w_i^{(p)})\/\\hat c^{(p)}$.\n\nConsider the case $\\hat c^{(p)}=0$.\nIf the $w_i^{(p)}$ are indefinite\nwe get no bound from (\\ref{cbed2}). However we will show that the\nexistence of certain monomials, which are required by the transversality\nconditions, implies a bound for $n_p$.\nLet $\\cI_a$ denote the indices of the already bounded\n$n_i$ and $\\cI_b$ the others. The charge of the field $z_a$ with\n$a\\in \\cI_a$\nwill depend on the unknown charge of a field $z_{b(a)}$ with\n$b\\in \\cI_b$ if\nthere is a chain of indices\n$a_0=a,a_1=j(a),\\dots,a_l=j(\\ldots j(a)\\ldots)=:b(a)$ linked by the\nmap $j$. The charge of $z_a$ is given by\n\\begin{equation}\nq_a={1\\over n_a}-{1\\over n_a n_{a_1}}+\n\\cdots -{(-1)^l\\over n_a\\ldots n_{l-1}}+\n{(-1)^l q_{b(a)}\\over n_a\\ldots n_{l-1}}.\n\\lleq{qform}\nIndefiniteness of the $w_i^{(p)}$ can only occur if there are fields\n$z_a$, $a\\in \\cI_a$, with odd $l$, i.e. the last term in (\\ref{qform})\n$s_a q_{b(a)}:=(-1)^l\/(n_a\\ldots n_{a_{l-1}} q_{b(a)})$ is negative.\nCall the index set of these fields $\\cI_a^-$.\nAssume first that the transversality condition (1.) holds.\nThis implies the existence of $m_i\\in \\relax{\\rm I\\kern-.18em N}^+$ ($m_i<2 n_i$)\nsuch that $\\sum_{i\\in \\cI^-_a}\nm_i q_i=1$. For the unknown $q_i$, $i\\in \\cI_b$, we get an equation\nof the form $\\sum w_i q_i=\\epsilon$, which yields a bound for the lowest\n$n_i$, $i\\in \\cI_b$, since $w_i>0$.\nThe lowest possible value for $\\epsilon>0$ can be readily calculated from\nthe denominators occurring in (\\ref{qform}).\nIf transversality condition (2.) applies, we have $|\\cI_a^-|$ equations\nof the form $\\sum_{i\\in \\cI_a^-} m^{(j)}_i q_i=1-q_{e_j}$ which can be\nrewritten as $\\sum_{i\\in \\cI_b} w_i^{(j)} q_i=\\epsilon^{(j)}$.\nOnly if all $w_i^{(j)}$ happen to be indefinite and all\n$\\epsilon^{(j)}$ are zero we get {\\sl no bound} from this condition.\n Assuming this to be true we have\n\\begin{equation}\n\\sum_{i\\in \\cI_b} m^{(j)} s_i q_i-s_{e_j} q_{b(e_j)}=0,\n\\lleq{g1}\nwhere $s_{e_j}:=1$ and $b(e_j):=b$ if $e_j\\in \\cI_b$. Note that\n$\\sum_i m^{(j)}_i\\ge 2$ in order to avoid quadratic mass terms.\nNow we can rewrite (\\ref{cbed2}) in the form\n\\begin{equation}\n\\sum s_i q_{b(i)}=0.\n\\lleq{g2}\nIf one uses now (\\ref{g1}) and $\\sum_i m^{(j)}_i\\ge 2$\nin order to eliminate the negative $s_i$, one finds\n $\\sum_i w_i q_i\\le 0$ with $w_i>0$, which is in contradiction with\nthe positivity of the charges, hence we get a bound in any case.\n\nThis procedure of restricting the bound for $n_{p}$, given\n$n_i,\\dots,n_{p-1}$,\nwas implemented in a computer program. It allows all\nconfigurations to be found without testing unnecessarily many combinations of\nthe $n_i$.\nThe actual upper bounds for the $(n_i,\\dots,n_r)$\nin the four--variable case are $(7,17,83,1805)$, and we have found $2390$\nconfigurations which allow for transversal polynomials.\nIn the five--variable case the bounds are $(4,6,14,62,923)$ and\n$5165$ configurations exist.\nBy adding a trivial mass term $z_5^2$ in the four--variable case, the\nconfigurations mentioned so far lead to three--dimensional\nCalabi--Yau manifolds described by a one polynomial constraint\nin a four--dimensional weighted projective space.\n\nThe same figures for the six--variable case and seven--variable case are\n$(3,3,5,11,41,482)$, $2567$ and $(2,2,3,4,8,26,242)$, $669$ respectively,\nleading to a total of $3236$ combinations.\nLikewise for eight-- and nine--variable potentials the bounds\nbecome $(2,2,2,2,2,3,3,5,14)$ with $47$ examples and $(2,2,2,2,2,2,2,2,2)$\nwith\n$1$ example respectively. The lists with all models can be found in\n\\cite{ks}.\n\n\n\\vskip .1truein\n\\noindent\n\\section{Results and Comparisons}\n\n\\noindent\nWe have constructed 10,839 Landau--Ginzburg theories with\n2997 different spectra, i.e. pairs of generations and antigenerations.\nThe massless spectrum is very rough information about a theory and\nit is likely that the degeneracy is lifted to a large degree when\nadditional information, such as the number of singlets and\/or the\nYukawa couplings, becomes available. We expect the situation to be\nvery similar to the class of CICYs \\cite{cdls}, which only leads to\nsome 250\ndifferent spectra \\cite{ghl}, but for which a detailed analysis of the\nYukawa couplings \\cite{ch} shows that it contains several thousand\ndistinct theories.\n\nIt is clear however that there is in fact some redundancy in this\nclass of Landau--Ginzburg theories even beyond the one discussed in\nthe previous sections. In the list there appear, for instance,\n two theories with\nspectrum $(h^{(1,1)},h^{(2,1)},\\chi)=(2,122,-240)$ involving five variables\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,1,7)}[14] \\ni \\{z_1^7+z_2^7+z_3^7+z_4^{14}+z_5^2=0\\}\n\\end{equation}\nand\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,3)}[7] \\ni \\{y_1^7+y_2^7+y_3^7+y_4^7+y_4y_5^2=0\\}.\n\\end{equation}\nUsing the fractional transformations introduced in ref. \\cite{ls4}\nit is\neasy to show that these two models are equivalent, even though\nthis is not obvious by just looking at the potentials. To see this\nconsider first the orbifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,1,7)}[14] {\\large \/} \\relax{\\sf Z\\kern-.4em Z}_2: [~0~0~0~1~1]\n\\end{equation}\nof the first model where the action indicated by\n$\\relax{\\sf Z\\kern-.4em Z}_2:~[~0~0~0~1~1] $\nmeans that the first three coordinates remain invariant, whereas the\nlatter\ntwo variables are to be multiplied with the generator of the cyclic\n$\\relax{\\sf Z\\kern-.4em Z}_2$, $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma=-1$.\nSince the action of this $\\relax{\\sf Z\\kern-.4em Z}_2$ on the weighted projective space\nis part\nof the projective equivalence, the orbifold is of course isomorphic\nto the\noriginal model. On the other hand the fractional transformation\n \\begin{equation}\nz_i=y_i,~i=1,2,3;~~~ z_4=y_4^{1\/2},~~~ z_5=y_4^{1\/2}y_5\n\\end{equation}\nassociated with this symmetry \\cite{ls4} defines a 1--1 coordinate\ntransformation on the orbifold, which transforms the first theory into\nthe second; these are therefore equivalent as well.\n\nSimilarly the equivalences\n\\begin{eqnarray}\n\\relax{\\rm I\\kern-.18em P}_{(2,2,2,3,9)}[18]_{-216}^{(4,112)}&=&\\relax{\\rm I\\kern-.18em P}_{(1,1,1,3,3)}[9] \\nonumber\n\\\\\n\\relax{\\rm I\\kern-.18em P}_{(2,6,6,7,21)}[42]_{-96}^{(17,65)}&=&\\relax{\\rm I\\kern-.18em P}_{(1,3,3,7,7)}[21]\\nonumber\n\\\\\n\\relax{\\rm I\\kern-.18em P}_{(2,5,14,14,35)}[70]_{-64}^{(27,59)} &=&\\relax{\\rm I\\kern-.18em P}_{(1,5,7,7,15)}[35]\n\\end{eqnarray}\ncan be shown, as well as a number of others.\n\nIt should be noted that even though we now have constructed\nLandau--Ginzburg potentials with an arbitrary number of scaling fields,\nthe basic range of the spectrum has not changed as compared with the\nresults of \\cite{cls} where it was found that the spectra of all\n6000 odd\ntheories constructed there lead to Euler numbers which fall in the\nrange\n\\fnote{6}{This should be compared with the result for the complete\n intersection Calabi--Yau manifolds where\n\\break $-200 \\leq \\chi \\leq 0$ \\cite{cdls}. See Figure 1.}\n\\begin{equation}\n-960 \\leq \\chi \\leq 960.\n\\end{equation}\nIn fact not only\ndo all the LG spectra fall into this range, all known Calabi--Yau\nspectra\nand all the spectra from exactly solvable tensor models are contained\nin this range as well! This suggests that perhaps the spectra of all\nstring vacua based on $c=9$\nwill be found within this range. To put it differently, we conjecture that\nthe Euler numbers of all Calabi--Yau manifolds are contained in the\nrange $-960 \\leq \\chi \\leq 960$.\n\nSimilarly to the results in \\cite{cls}, the Hodge pairs do not pair up\ncompletely.\nIn fact the mirror symmetry of the space of Landau--Ginzburg vacua is just\nabout $77\\%$.\nIt thus appears that orbifolding is an essential ingredient\nin the construction of a mirror--symmetric slice of the configuration space\nof the heterotic string. It is in fact easy to produce examples of\norbifolds whose spectrum does not appear in our list of LG vacua.\nAn example of a mirror pair is furnished by the orbifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,2)}[6]^{(1,103)}_{-204} {\\LARGE \/} \\relax{\\sf Z\\kern-.4em Z}_6:\n\\left[\\matrix{3&2&1&0&0\\cr}\\right]\n\\end{equation}\nwhich has the spectrum $(11,23,-24)$ and the space\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,1,1,1,2)}[6]^{(1,103)}_{-204} {\\LARGE \/} \\relax{\\sf Z\\kern-.4em Z}_6\\times \\relax{\\sf Z\\kern-.4em Z}_3 :\n\\left[\\matrix{3&2&1&0&0\\cr\n 0&0&1&0&2\\cr}\\right]\n\\end{equation}\nwhich has the mirror--flipped spectrum $(23,11,24)$.\nThe space of LG orbifolds that has been constructed so far \\cite{kss}\nis indeed much closer to being mirror--symmetric than the space of\nLG theories itself. Even though the construction in \\cite{kss} is\nincomplete, about 94\\% of the Hodge numbers pair up.\n\nBy now there exist many different prescriptions to construct string\nvacua and we would like to put the LG framework into the context of\nother left--right--symmetric constructions.\nAmong the more prominent ones other than Calabi--Yau manifolds, which\nare obviously closely related to LG theories, there are constructions which\nhave traditionally been called, somewhat misleadingly,\norbifolds (what is meant are\norbifolds of tori) \\cite{dhvw}, free--fermion constructions\n\\cite{freeferm}, lattice constructions \\cite{lls}, and interacting\nexactly solvable models \\cite{g}\\cite{kasu}.\n\nNone of these classes are known completely, even though much effort\nhas gone into the exploration of some of them. Because powerful\ncomputational tools are available, toroidal orbifolds have been analysed\nin some detail \\cite{orbis}\\cite{orbiyuk} and much attention has\nfocused on the explicit construction of exactly solvable theories in\nthe\ncontext of tensor models via $N=2$ superconformal minimal theories\n\\cite{ls12}\\cite{ls3}\\cite{fkss1} and Kazama--Suzuki models\n\\cite{fiq2}\\cite{lvwg}\\cite{ls5}\\cite{s}\\cite{fksv}\n\\cite{aafn} as well\nas their in depth analysis \\cite{lvwg}\\cite{s2}\\cite{fks}\\cite{exyuk}.\nIn the context of\n(2,2) vacua, orbifolds lead only to some tens of distinct models, whereas\nthe known\nclasses of exactly solvable theories lead only to a few hundred models\nwith distinct spectra. Similar results have been obtained so far via the\ncovariant--lattice approach \\cite{lsw} and hence it is obvious that these\nconstructions do\nnot exhaust the configuration space of heterotic string by far. The\nclass of Landau--Ginzburg string vacua thus appears as a rather\nextensive source of (2,2)--symmetric models.\n\nFinally we should remark that the (2,2) Landau--Ginzburg theories\nwe have constructed here can be used to build a probably much\nlarger class\nof (2,0) models along the lines described in \\cite{dg}, using an\nappropriate adaptation of the work of \\cite{cdgp}\\cite{dave}\n\\cite{klth}\\cite{anamaria} in order to determine the instantons\non which the existence of a certain split of a vector bundle\nhas to be checked.\n\n\n\\vfill \\eject\n\n\\part{ Mirror Pairs via Fractional Transformations.}\n\nEven though the emerging mirror symmetry of the construction discussed in\nPart I is clearly very suggestive there is a priori no reason why models\nwith mirror flipped spectra should be related at all. After all,\nHodge numbers do not classify manifolds topologically and knowing\nthe spectrum of a physical theory is only a very first step in the analysis\nof its properties. It turns out that the detailed analysis of the models\nrests both mathematically and physically on the computation of the Yukawa\ncouplings. This, in general, is a difficult business. It is therefore\ngratifying that it is possible to proceed via a different line of thought\nto relate different models, namely via a direct map between different\ntheories. In particular cases this rather general map turns out to be the\nmirror map.\n\n\\vskip .1truein\n\\noindent\n\\section{ From A--Models to D--Models via Orbit Construction}\n\n\\noindent\nIt is well known from the ADE--classification of partition functions of\n($N=2$)minimal models that the ${\\rm D}$--type models are equivalent to\n${\\rm A}$--type models modded out by a $\\relax{\\sf Z\\kern-.4em Z}_2$ discrete symmetry.\nAs a consequence subsets of vacua of the Heterotic String involving the\n$N=2$ minimal series are also related via orbifold constructions\n\\fnote {3}{In this article orbifold construction will be understood\n to include twisted modes in the conformal field theory or\n Landau--Ginzburg construction and blow--up modes in the\n geometric formulation.}.\nIn terms of the corresponding Landau-Ginzburg potentials of the exact\nmodels\nthis can be seen as follows. The potential corresponding to a model\nat level\n$k$ with a diagonal affine invariant is \\cite{kms}\n\\begin{equation}\n{\\rm A}_{k+1}~~~\\sim~~~\\Phi_1^{k+2} + \\Phi_2^2 \\lleq{diag}\nwhere, with hindsight, a trivial factor has been added.\nThe LG potential corresponding to the exact model at the same level\n\\fnote{4}{The ${\\rm D}$--models only exist for even level $k=2n$.}\nbut with the ${\\rm D}$--invariant is described by \\cite{m}\\cite{vw}\n\n\\begin{equation}\n{\\rm D}_{\\frac{k}{2}+2}~~\\sim~~~{\\tilde {\\Phi}}_1^{(k+2)\/2} +\n \\tilde {\\Phi}_1 \\tilde {\\Phi}_2^2. \\lleq{daff}\nBy defining the transformation of the scaling variables\n\\begin{equation}\n\\Phi_1 = \\tilde {\\Phi}_1^{1\/2}~~~ {\\rm and}~~~\\Phi_2 = \\tilde {\\Phi}_1^{1\/2}\\tilde {\\Phi}_2\n\\lleq{fraccy}\none can map the diagonal theory of (\\ref{diag}) into the nondiagonal\nmodel of\n(\\ref{daff}).\nMoreover this transformation has a constant Jacobian and therefore one\nmight naively expect that\nthey are in fact equivalent descriptions of a given theory since\ntheir path integrals are equivalent. It is clear however that this is not\nthe case as the two theories have a different spectrum:\nthe local ring of the diagonal theory consists of states\n\\begin{equation}\n{\\cal R}} \\def\\cV{{\\cal V}_{{\\rm A}_{k+1}}=\\{1,\\Phi_1, \\Phi_1^2,\\cdots,\\Phi_1^k\\}\n\\end{equation}\ni.e. the theory has $k+1$ states. The ${\\rm D}$--theory however has only\n$\\frac{k}{2}+2$ states in its spectrum\n\\begin{equation}\n{\\cal R}} \\def\\cV{{\\cal V}_{{\\rm D}_{\\frac{k}{2}+2}}=\n\\{1,\\tilde {\\Phi}_2,\\tilde {\\Phi}_1,\\cdots,\\tilde {\\Phi}_1^{\\frac{k}{2}}\\}.\n\\end{equation}\nHence these two theories are not the same. The resolution of this puzzle\ncomes\nfrom the fact that the transformation of (\\ref{fraccy}) is not a coordinate\ntransformation since it is not a bijection but is 2--1 as it stands.\nTo make it 1--1 we should identify\n\\begin{equation}\n\\Phi_i \\sim -\\Phi_i,~~~i=1,2.\n\\end{equation}\nOf the $k+1$ states of ${\\rm A}_{k+1}$--models $\\frac{k}{2}+1$ are\ninvariant with respect to the action of this $\\relax{\\sf Z\\kern-.4em Z}_2$. By including the one\ntwisted state we find precisely the states of the non--diagonal\n${\\rm D}$--theory.\n\nThis construction can immediately be applied to string compactification\nproper.\nIt may seem unlikely at first that the simple modding of a $\\relax{\\sf Z\\kern-.4em Z}_2$ should\naccount for the different behaviour that the various tensor models\nexhibit under the exchange\nof a diagonal invariant by a D--invariant. A quick look at the results\nof refs.\n\\cite{ls3}\\cite{fkss1} shows that this exchange generically changes the\nspectrum\n in some\narbitrary way without any obvious systematics. However in some special\ncases\nthe spectrum is flipped, exchanging generations and anti--generations while\nin other models the spectrum does not change at all. The reason for this\nerratic behaviour can be traced to the fact that the other coordinates\ninvolved in the construction determine the fixed point structure of the\ndiscrete group in an essential way.\n\nConsider the following example where the exchange of the affine\ninvariant does\nnot change the spectrum\n\\fnote{5}{The notation of ref. \\cite{ls3} is used.}\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,7,2,2,2)}[14]~~\\ni~~(12\\cdot 5^3)_{A_{13}\\otimes A_6^3}~~\n\\longrightarrow\n{}~~(12\\cdot 5^3)_{D_{8}\\otimes A_6^3}~~\\in~~\\relax{\\rm I\\kern-.18em P}_{(1,3,1,1,1)}[7].\n\\end{equation}\nwhere we have added one trivial factor\n\\fnote{6}{An explanation for the necessity of this trivial\n factor and when it is to be added can be found in\n \\cite{ls3}.}.\nThe reason that the spectrum does not change after the replacement of the\ndiagonal affine invariant by the D--invariant is rather obscure from the\npoint\nof view of the conformal field theory described by the tensor model\n$(12\\cdot 5^3)$ with different affine invariants but can be understood\neasily\nfrom the orbifold point of view. In the manifold picture it simply follows\nfrom the fact that the $\\relax{\\sf Z\\kern-.4em Z}_2$--action $[1,1,0,0,0]$, generated by\n\\begin{equation}\n(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\\longrightarrow\n(\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_1,\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5),\n\\end{equation}\n(where $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^2=1, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma\\neq1$) is not an action at all on the Calabi--Yau\nmanifold embedded in the ambient\nweighted projective space but it is part of the projective equivalence\ntransformation because we have\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,7,2,2,2)} = \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_5\/\\sim\n\\end{equation}\nwith $(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\\sim\n(\\l \\Phi_1,\\l^7 \\Phi_2,\\l^2 \\Phi_3,\\l^2 \\Phi_4,\\l^2 \\Phi_5)$,\nwhere $\\l\\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^*$. Therefore the $\\relax{\\sf Z\\kern-.4em Z}_2$ does not transform the\nprojective\ncoordinates at all but is trivial. In the conformal field theory picture\nthis translates into the fact that the action is actually part of the\nmodding that has to be done to implement the GSO projection. This example\nshows that not all our models are independent but that some models appear\nin different representations.\n\nA more complicated example is furnished by the pair of minimal models\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,5,1,1,2)}[10]^1_{-288}~~=~~(8^3\\cdot 3)_{A_{9}^3\\otimes A_4}\n\\longrightarrow\n(8^3\\cdot 3)_{D_{6}\\otimes A_9^2\\otimes\nA_4}~~=~~\\relax{\\rm I\\kern-.18em P}_{(2,4,1,1,2)}[10]^3_{-192}.\n\\lleq{minmod}\nThe superscript denotes the dimension of the\nsecond cohomology group $h^{(1,1)}={\\rm dim}~{\\rm H}^{(1,1)}(\\cM)$ whereas\nthe subscript denotes the Euler number. It is again not obvious from the\npoint of view of the affine invariants involved why the spectrum changes the\nway it does but the result is clear from the orbifold construction. Defining\nthe $\\relax{\\sf Z\\kern-.4em Z}_2$--action as $[1,1,0,0,0]$ it follows that in this case the map is\nnontrivial. Its fixed point set consists of two curves\n$C_1=\\relax{\\rm I\\kern-.18em P}_{(1,1,2)}[10]$ with Euler number $\\chi_{C_1}=-30$ and\n$C_2=\\relax{\\rm I\\kern-.18em P}_{(1,5,2)}[10]$ with $\\chi_{C_2}=-2$. The Euler number of the\nresolved orbit manifold is therefore\n\\begin{equation}\n\\chi=-\\frac{288}{2}-\\frac{(-30-2)}{2}+2(-30-2)=-192.\n\\end{equation}\nThe cohomology can be computed as well. Given the Euler number we only need\nto compute either the second cohomology group or the number of (2,1)--forms.\nUsing the results of \\cite{s3} it is clear that each of the curves\ncontributes\none additional generator to the second cohomology, i.e. for the resolved\nmanifold we have $h^{(1,1)}=3$ and therefore the result of (\\ref{minmod}).\n\nThese simple identifications of a class of Landau-Ginzburg potentials\nwith orbifolds of other potentials via fractional transformations can be\nconsidered as a generalization\nof the strange duality known from the exceptional singularities\nof modality one \\cite{agzv}. One of the `strangely dual pairs' appearing\nin the\nclassification of Landau--Ginzburg potentials in 3 variables with an\nisolated singularity at the origin is described by the two polynomials\n\\begin{eqnarray}\nK_{14}~&:&\\relax{\\rm I\\kern-.18em P}_{(3,12,8)}[24]~\\ni~ \\{\\Phi_1^8+\\Phi_2^2+\\Phi_3^3=0\\}\\\\\nQ_{10}~&:&\\relax{\\rm I\\kern-.18em P}_{(6,9,8)}[24]~\\ni~\\{\\tilde {\\Phi}_1^4+\\tilde {\\Phi}_1\\tilde {\\Phi}_2^2+\\tilde {\\Phi}_3^3=0\\}.\n\\end{eqnarray}\nThe dimension of the chiral ring of these singularities is 10 and 14,\nrespectively, as indicated by the subscript of $Q$ and $K$. Each of these\ncatastrophes is characterized by two triplets of numbers, the Dolgachev\nnumbers\nand the Gabrielov numbers. For the above singularity $Q_{10}$ the\ncorresponding pair of triplets is $\\cD(Q_{10})=(2,3,9)$ and\n$\\cG(Q_{10})=(3,3,4)$ for the Dolgachev and Gabrielov numbers\nrespectively whereas the singularity $K_{14}$ leads to the triplets\n$(3,3,4)$ and $(2,3,9)$ For the above pair $\\cD(Q_{10})=\\cG(K_{14})$ and\n$\\cG(Q_{10})=\\cD(K_{14})$. It is in this sense that the two polynomials are\ncalled dual\n\\fnote{7}{The precise nature of these numbers are not of\n concern here. More details can be found in\n ref. \\cite{agzv}}.\n\n{}From our previous results it is clear that an alternative way to relate\nthese two singularities flows from the description of affine\n${\\rm D}$--invariants as orbifolds of diagonal invariants.\nThe above polynomials can be viewed as the Landau--Ginzburg potentials\nof the tensor product $(1\\cdot 6)$\n\\begin{equation}\nQ_{10}\\sim (1\\cdot 6)_{A_2\\otimes D_5}~~{\\rm and}~~\nK_{14}\\sim (1\\cdot 6)_{A_2\\otimes A_7}\n\\end{equation}\nwhere we have added a trivial factor for the $K_{14}$ singularity.\nTherefore it follows that\n\\begin{equation}\nQ_{10}=K_{14}\/\\relax{\\sf Z\\kern-.4em Z}_2.\n\\end{equation}\nMore explicitly consider the chiral ring of $K_{14}$\n\\begin{equation}\n{\\cal R}_K=\\{1,\\Phi_1,\\Phi_1^2,...,\\Phi_1^6,\n \\Phi_3,\\Phi_1\\Phi_3,...,\\Phi_1^6\\Phi_3\\}.\n\\end{equation}\nThe action of the $\\relax{\\sf Z\\kern-.4em Z}_2$ is defined as $[1,0,1]$ on the fields\n$(\\Phi_1,\\Phi_2,\\Phi_3)$. The two sectors of the orbifold consist of the\neight invariant states of ${\\cal R}} \\def\\cV{{\\cal V}_K$ and of two twisted states giving the\ntotal of ten states necessary for $Q$.\n\nThis strangely dual pair can again be used as building block of\nstring compactifications. In the models constructed\nin \\cite{ls3}\\cite{fkss1}\nthis involves vacua which have $(1\\cdot 6)_{A_1\\otimes A_7}$ or\n$(1\\cdot 6)_{A_1\\otimes D_5}$ as subfactors.\n\nThe ADE models make up only a very small part of the string\ncompactifications constructed in \\cite{cls} and it is natural to ask\nwhether it is\npossible to relate spaces in this set to one another in a similar way\nand if so\nwhether it is possible to construct {\\it all} of these models as\norbifolds of\nsome basic set of polynomial types, e.g. Fermat type spaces. This is the\nquestion to which will be addressed in the following section.\n\n\\vskip .1truein\n\\noindent\n\\section{Fractional Transformations}\n\n\\noindent\nIn the notation of \\cite{cls} the transition from the diagonal affine\ninvariant to the\nnon--diagonal D--invariant can be formulated as the transition from\na Fermat\ntype polynomial with diagram\n\\begin{equation}\n{\\thicklines \\begin{picture}(120,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,7){\\circle{12}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n }\n\\Phi_1^a + \\Phi_2^b\n\\lleq{gendiag}\nto particular polynomials of the type of a tadpole\n\\begin{equation}\n{\\thicklines \\begin{picture}(120,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(26,7){\\circle{12}}\n \\end{picture}\n }\n\\tilde {\\Phi}_1^c\\tilde {\\Phi}_2 + \\tilde {\\Phi}_2^d\n\\lleq{gend}\nEquations (\\ref{diag},\\ref{daff}) clearly describe only a very small\nsubset of these polynomials and one can ask whether transformations\ngeneralizing (\\ref{fraccy}) are possible. Indeed they are.\n\nConsider the transformation\n\\begin{equation}\n\\Phi_1=\\tilde {\\Phi}_1^{c\/a} \\tilde {\\Phi}_2^{1\/a},~~~~~~~~\n\\Phi_2 = \\tilde {\\Phi}_2^{d\/b}\n\\lleq{genfraccy}\nwhich transforms the polynomial in (\\ref{gendiag}) into the one of eq.\n(\\ref{gend}).\nThe next step is to find the constraints on the weights of the fields that\nmake the Jacobian of this transformation constant. They are given by\n\\begin{equation}\nc=a~~~{\\rm and}~~~ d=b\\left[1-\\frac{1}{a}\\right].\n\\end{equation}\nAs in the case of the previous transformation however this transformation\nis not well defined yet and we have to find the discrete group which makes\nthe transformation of variables well defined. Suppose that we have a map\n\\begin{equation}\n\\Phi_1 \\longrightarrow \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma \\Phi_1 ~~~{\\rm and}~~~\n\\Phi_2 \\longrightarrow \\b \\Phi_2\n\\end{equation}\nwhere $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma, \\b$ are roots of unity. The condition that the change of variables\nin eq. (\\ref{genfraccy}) is invariant determines $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ as the generator of\n$\\relax{\\sf Z\\kern-.4em Z}_a$ and\n$\\b=\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^{a-1}$. Similarly observation regarding the inverse transformation\nleads to the group that one has to mod out by in the nondiagonal theory.\nOnly after modding out these cyclic groups does\ntransformation (\\ref{genfraccy}) become well defined.\n\nThe isomorphism can be summarized concisely with the following diagram\n\n\\vfill\n\\eject\n\\begin{eqnarray}\n& &\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{\\left(\\frac{b}{g_{ab}},\\frac{a}{g_{ab}}\\right)}\n \\left[\\frac{ab}{g_{ab}}\\right]\n \\ni \\left \\{z_1^a+z_2^b=0\\right \\}\n ~{\\Big \/}~ \\relax{\\sf Z\\kern-.4em Z}_b: \\left[\\matrix{(b-1)&1}\\right] ~~\n \\nonumber \\\\ [3ex]\n&\\sim & \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{\\left(\\frac{b^2}{h_{ab}},\\frac{a(b-1)-b}{h_{ab}}\\right)}\n \\left[\\frac{ab(b-1)}{h_{ab}}\\right]\n \\ni \\left \\{y_1^{a(b-1)\/b}+y_1y_2^b=0\\right\\}\n ~{\\Big \/}~ \\relax{\\sf Z\\kern-.4em Z}_{b-1}: \\left[\\matrix{1&(b-2)}\\right].\n\\llea{iso}\n\\vskip .1truein\n\n\\noindent\nHere $g_{ab}$ is the greatest common divisor of $a$ and $b$ and\n$h_{ab}$ is the greatest common divisor of $b^2$ and $(ab-a-b)$.\nThe action of a cyclic group $\\relax{\\sf Z\\kern-.4em Z}_b$ of order $b$ denoted by\n$[m~~n]$ indicates that the symmetry acts like\n$(z_1,z_2) \\mapsto (\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^m z_1, \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma^n z_2)$ where $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma$ is the $b^{th}$ root\nof unity.\n\n\nIt is easy to generalize this analysis to general transformations\nfrom Fermat--type polynomials to tadpole--type polynomials with $N$ points\nattached. Consider the transformation\n\\begin{equation}\n\\sum_{i=1}^N \\Phi_i^{l_i} \\longrightarrow\n\\left( \\sum_{j=1}^{N-1} \\tilde {\\Phi}_j^{n_j} \\tilde {\\Phi}_{j+1}\\right) + \\tilde {\\Phi}_N^{n_N}\n\\end{equation}\ndefined by\n\\begin{eqnarray}\n\\Phi_i &=& \\tilde {\\Phi}_i^{n_i\/l_i} \\tilde {\\Phi}_{i+1}^{1\/l_i} ~~~~~~~~i=1,..,N-1;\n \\nonumber \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{n_N\/l_n}.\n\\llea{multifrac}\nThe condition that the Jacobian of this transformation is constant\nagain leads to constraints on the exponents:\n\\begin{eqnarray}\nn_1 &=&l_1;\\\\\nn_i &=&l_i\\left[1 - \\frac{1}{l_{i-1}}\\right]~,~i=2,...,N.\n\\end{eqnarray}\nTransformation (\\ref{multifrac}) then becomes\n\\begin{eqnarray}\n\\Phi_1 &=& \\tilde {\\Phi}_1 \\tilde {\\Phi}_2^{1\/l_1}; \\\\\n\\Phi_i &=& \\tilde {\\Phi}_i^{{(l_{i-1}-1)}\/l_{i-1}} \\tilde {\\Phi}_{i+1}^{1\/l_i}\n ~~~~~~~~~~i=2,..,N-1; \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{{(l_{N-1}-1)}\/l_{N-1}}.\n\\llea{multiexp}\nAs before, this transformation is not well defined, but one can find\na discrete group under which it is invariant. Let\n\\begin{equation}\n\\Phi_i \\longrightarrow \\alpha_i \\Phi_i;\n\\lleq{scaling}\nit then follows that under this transformation\n\\begin{equation}\n\\tilde {\\Phi}_N \\longrightarrow \\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N^{l_{N-1}\/{(l_{N-1}-1)}} \\tilde {\\Phi}_N,\n\\end{equation}\ni.e. we find the constraint\n\\begin{equation}\n\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N^{l_{N-1}\/{(l_{N-1}-1)}}=1\n\\end{equation}\nfor the generator $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_N$. Plugging the solutions of this constraint\niteratively into eqs. (\\ref{multiexp}) one finds\nthat all generators $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_i$ have to satisfy $\\alpha} \\def\\b{\\beta} \\def\\e{\\epsilon} \\def\\g{\\gamma_i^{l_i} = 1~,i=1,..,N-1$\nfor the transformation of variables to be invariant under the\nidentification (\\ref{scaling}). The group by which one has to mod out is\n$\\prod_{i=1}^{N-1} \\relax{\\sf Z\\kern-.4em Z}_{l_i}$. Again one has to similarly analyze the\ninverse map. It is obvious that this generalization just amounts to\nrepeated application of the original isomorphism (\\ref{iso}).\n\nConsider e.g. the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(1,3,3,3,5)} [15]^3_{-144}=\n\\{\\Phi_1^{15}+\\Phi_2^5+\\Phi_3^5+\\Phi_4^5+\\Phi_5^3=0\\}.\n\\end{equation}\nWe will show now that the orbifold of this model with respect to two\ncyclic groups $\\relax{\\sf Z\\kern-.4em Z}_5$, the first generated by $[0,4,1,0,0]$\nthe second one by $\\relax{\\sf Z\\kern-.4em Z}_5:[0,0,4,1,0]$, is isomorphic to the manifold\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(16,60,45,39,80)}[240]^{75}_{144}~=\n \\{\\tilde {\\Phi}_1^{15}+\\tilde {\\Phi}_2^4+\\tilde {\\Phi}_2\\tilde {\\Phi}_3^4\n +\\tilde {\\Phi}_3\\tilde {\\Phi}_4^5+\\tilde {\\Phi}_5^3=0\\}.\n\\end{equation}\nConsider the first $\\relax{\\sf Z\\kern-.4em Z}_5$. The fixed points of this action are determined\nby the requirement that\n\\begin{equation}\n(\\Phi_1,\\alpha^4 \\Phi_2,\\alpha \\Phi_3,\\Phi_4,\\Phi_5)\n= c(\\Phi_1,\\Phi_2,\\Phi_3,\\Phi_4,\\Phi_5)\n\\end{equation}\nwhere $c\\in \\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}^*$. The fixed point set consists of one curve\n$\\relax{\\rm I\\kern-.18em P}_{(1,3,5)}[15]$\nwith Euler number $ \\chi= -6$ and one further fixed point $\\relax{\\rm I\\kern-.18em P}_{(3,5)}[15]$.\nThe Euler number of the resolved orbifold therefore is\n$\\chi= \\frac{-144}{5} - \\frac{1}{5}(-6+1) + 5(-6+1) - \\frac{1}{5} +5 =-48$.\nThe corresponding weighted CY--manifold of this space is described by\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,15,9,12,20)}[60]~=~\\{\\phi_1^{15}+\\phi_2^4+\\phi_2\\phi_3^5+\n \\phi_4^5+\\phi_5^3=0\\}.\n\\end{equation}\nand the coordinate transformation is defined as follows\n\\begin{equation}\n(\\Phi_1,\\Phi_2,...,\\Phi_5) =\n(\\phi_1,\\phi_2^{4\/5},\\phi_2^{1\/5}\\phi_3,\\phi_4,\\phi_5).\n\\end{equation}\n\nModding out by the other $\\relax{\\sf Z\\kern-.4em Z}_5$ leads to a fixed point set of three curves\n\\begin{equation}\n\\relax{\\rm I\\kern-.18em P}_{(4,15,20)}[60]_{15},~~~\\relax{\\rm I\\kern-.18em P}_{(15,9,20)}[60]_{16},\n ~~~\\relax{\\rm I\\kern-.18em P}_{(15,12,20)}[60]_{11}.\n\\end{equation}\nThese three curves intersect in the point $\\relax{\\rm I\\kern-.18em P}_{(15,20)}[60]=\\relax{\\rm I\\kern-.18em P}_{(3,4)}[12]$\nand the Euler number of the resolved manifold is given by\n\\begin{equation}\n\\chi = -\\frac{48}{5} - \\frac{1}{5}\\left(15 + 11 + 16 -2 \\times 5\\right) +\n 5\\left(15 +11 + 16 -2\\times 5\\right) = 144.\n\\end{equation}\nThe term $-2\\times 5$ comes from the fact that in all three curves we have\nblown up the $\\relax{\\sf Z\\kern-.4em Z}_5$ fixed point when in fact we only should have done\nso once.\n\nWe have thus established that the discrete groups of the\nLG--CY theories play a role comparable to the discrete symmetries of the\nfusion rules in conformal field theories even though for most of the models\nat hand the corresponding exact theory is not known. We have seen that\nin general there are constraints on the weights of the potentials that\ncan be identified with with orbifolds of other models. Nevertheless, in the\ntypes of potentials discussed above there was enough freedom to have\nnontrivial identifications. This is not true in general as we will show in\nthe next section.\n\n\\vskip .1truein\n\\noindent\n\\section{ Loop Potentials}\n\n\\noindent\nConsider potentials of the type\n\\begin{equation}\n{\\thicklines \\begin{picture}(80,20)\n \\put(0,0){\\circle*{5}}\n \\put(0,0){\\line(1,0){26}}\n \\put(26,0){\\circle*{5}}\n \\put(13,0){\\oval(26,26)[t]}\n \\end{picture}\n }\n\\tilde {\\Phi}_1^c\\tilde {\\Phi}_2 + \\tilde {\\Phi}_2^d\\tilde {\\Phi}_1\n\\end{equation}\nand more general polynomials of this type with an arbitrary number of\nfields. For these polynomials it is possible to find a 1--1 coordinate\ntransformation from Fermat--type polynomials, but the\ncondition that the Jacobian is constant leads to the conclusion that\nonly deformations of the original space can be obtained in this way.\n\nThe transformation of a Fermat type polynomial into a loop--type\npolynomial\n\\begin{equation}\n\\sum_{i=1}^{N} \\Phi_i^{l_i} \\longrightarrow\n \\left( \\sum_{i=1}^{N-1} \\tilde {\\Phi}_i^{n_i} \\tilde {\\Phi}_{i+1} \\right) +\n \\tilde {\\Phi}_N^{n_N}\\tilde {\\Phi}_1.\n\\end{equation}\nleads to the transformation of variables\n\\begin{eqnarray}\n\\Phi_i &=& \\tilde {\\Phi}_i^{n_i\/l_i} \\tilde {\\Phi}_{i+1}^{1\/l_i}~~~~~~~~i=1,..,N-1;\n\\nonumber \\\\\n\\Phi_N &=& \\tilde {\\Phi}_N^{n_N\/l_N} \\tilde {\\Phi}_1^{1\/l_N}.\n\\llea{loopfrac}\nThe condition that the Jacobian be constant\nleads to the constraints\n\\begin{eqnarray}\nn_1 &=& l_1\\left[1 -\\frac{1}{l_N}\\right];\\\\\nn_i &=& l_i\\left[1 -\\frac{1}{l_{i-1}}\\right],~~~~~~~~~~~~~i=2,..,N;\n\\end{eqnarray}\nso that the coordinate transformation (\\ref{loopfrac}) becomes\n\\begin{eqnarray}\n\\Phi_1 &=& \\tilde {\\Phi}_1^{{(l_N-1)}\/l_N} \\tilde {\\Phi}_2^{1\/l_1}\\\\\n\\Phi_i &=& \\tilde {\\Phi}_i^{{(l_{i-1}-1)}\/l_{i-1}} \\tilde {\\Phi}_{i+1}^{1\/l_i},\n{}~~~~~~i=2,..,N.\n\\end{eqnarray}\n{}From this it follows however that the charges of the loop fields are\nprecisely those of the fields of the Fermat type polynomial, i.e.\nthe loop--type model is a trivial deformation of the original one.\n\n\n\n\\vfill \\eject\n\n\\part{ Landau--Ginzburg Orbifolds.}\n\nEven though the class of Heterotic Vacua described in Part I is the largest\nconstructed so far it clearly does not encompass all spectra that are known.\nConsider e.g. the orbifold of the Fermat quintic in $\\relax{\\rm I\\kern-.18em P}_4[5]$ with respect\nto the action\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5: \\left[\\matrix{ 3 &1 &1 &0 &0}\\right]\n\\end{equation}\nwhich leads to a space with spectrum $(h^{(1,1)},h^{(2,1)},\\chi)=(21,17,8)$.\nNo complete intersection model with such Hodge numbers appears among the\nresults of \\cite{ks}.\nAn obvious question of course is whether one could use the fractional\ntransformation discussed in Part II to {\\it construct} the complete\nintersection representation of this orbifold. Unfortunately this yields\na singular space.\nThus it is unclear at this point whether a complete intersection of this\norbifold exist. There are many examples of this type, some of which will\nbe mentioned below.\n\nOrbifolding is important for the construction of mirrors as well because\nin many examples the weighted CICY representation of a mirror is not known\nwhereas it is easy to construct the mirror as an orbifold. A simple example\nis furnished by the manifold\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3&0\\cr 1&3\\cr}\\right].\n\\end{equation}\nthe mirror of which can easily be constructed as the orbifold with respect\nto the action\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_9\\times \\relax{\\sf Z\\kern-.4em Z}_3 ~:~ \\left[\\matrix{6~&1~&3~&2~&0~&0~&6\\cr\n 0~&0~&0~&0~&0~&2~&1\\cr}\\right]\n\\end{equation}\nwhen viewed as a symmetry on the corresponding LG theory embedded in\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(3,2,3,2,3,2,3)}[9]$.\nIt is unclear however what the CICY representation of this mirror is.\n\nA further motivation for the construction of orbifolds has been mentioned\nalready in the introduction. Namely in order to get some idea of how\n`robust' the mirror property of the configuration space is it is useful\nto implement different types of constructions. Until the proper framework\nfor mirror symmetry and its explicit form has been found this appears to be\n one feasible avenue for collecting support for the existence\n\\fnote{8}{There are of course also other reasons to look for\n new models but this is another story.}.\n\nThis last part of the review contains a brief description of the work\ndone in \\cite{kss}\nin which some 40odd types of actions have been considered\non all Landau--Ginzburg potentials that can be build from Fermat monomials,\nsingle Tadpole potentials and single Loop potentials as well as arbitrary\ncombinations thereof.\n\n\\vskip .1truein\n\\noindent\n\\section{Actions of Symmetries: General Considerations}\n\n\\noindent\nIt is useful to first discuss some general aspects that are important\nfor group actions on Landau--Ginzburg theories that have been\norbifolded with respect to the U(1)--symmetry in order to describe\nstring vacua with $N=1$ spacetime supersymmetry.\n\nAn obvious question when considering orbifolds is whether\nthere is any a priori insight into\nwhat spectra are possible for the orbifolds of a given model with respect\nto a particular set of symmetries.\nThis question is of particular interest if the goal is to produce\norbifolds with presribed spectra, say models with a small number\nof fields where the difference between the number of generations and\nantigenerations is three.\n\nEven though it is possible to formulate constraints on the\norbifold spectrum for\nparticular types of actions, we know of no constraints that hold\nin full generality, or even for arbitrary cyclic actions.\nOne very simple class of symmetries are those without fixed points.\nFor such actions there are no twisted sectors and hence\nthere exists a simple formula expressing the Euler number $\\chi_{orb}$ of the\norbifold in terms of the Euler characteristic $\\chi$ of the covering space\nand the order $|G|$ of the group\n\\begin{equation}\n\\chi_{orb} = \\frac{\\chi}{|G|}.\n\\end{equation}\nThe vast majority of actions however do have fixed points and hence the\nresult above does not apply very often.\n\nFor orbifolds with respect to cyclic groups of prime order there\nexists a generalization of this result. For such group actions it was shown\nin the first reference in \\cite{orbis} that\n\\begin{equation}\n\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g_{orb} - n^g_{orb} =\n\\left( |G|+1\\right)\\left(\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g_{inv} - n^g_{inv}\\right)\n-\\left(\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g - n^g\\right),\n\\end{equation}\nwhere $n^g_{orb}$, $n^g_{inv}$, $n^g$ are the numbers of generations\nof the orbifold theory, the invariant sector and the original LG theory,\nrespectively.\n\nConsider then the problem of constructing an orbifold with a prescribed Euler\nnumber $\\chi_{orb}$ from a given theory.\nOnly for fixed point free actions will the order of the group be completely\nspecified as $|G|=\\chi\/\\chi_{orb}$. It is important to realize that in\ngeneral the order of the group by which a theory is orbifolded does\n{\\it not} determine its spectrum -- the precise form of the action of the\nsymmetry is important.\n\nNevertheless we can derive {\\it some} constraints on the order of the\naction that we are looking for. Even though we don't know a priori what\nthe invariant\nsector of the orbifold will be we do know that its associated Euler number\nmust be an integer\n\\begin{equation}\n\\chi_{inv} = \\frac{\\chi + \\chi_{orbi}}{|G|+1}~~\\in \\relax{\\rm I\\kern-.18em N}.\n\\end{equation}\nThis simple condition does lead to restrictions for the order of the group.\nSuppose, e.g., that we wish to check whether the quintic threefold admits\na three--generation orbifold: For the deformation class of the quintic\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,1,1,1)}[5]: ~\\chi=-200\n\\end{equation}\nthe order of the discrete group in question must satisfy the constraint\n$-206\/(|G|+1) \\in \\relax{\\sf Z\\kern-.4em Z}$, implying $|G|=102$.\nHence there exists no three--generation orbifold of the quintic with respect\nto a discrete group with prime order.\nA counterexample for nonprime orders is furnished by the following theory\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,2,2,3,3,3,3)}[9]:~ (\\bar n} \\def\\bt{\\bar t} \\def\\bz{\\bar z^g,n^g,\\c)=(8, 35,-54),\n\\lleq{3gen}\nwhich corresponds to a CY theory embedded in a product of two projective\nspaces by two polynomials of bidegree $(0,3)$ and $(3,1)$ \\cite{s1}\\cite{s2}\ni.e. the Calabi--Yau manifold of this model is embedded in\nan ambient space consisting of a product of two projective spaces\n\\begin{equation}\n\\matrix{\\relax{\\rm I\\kern-.18em P}_2\\cr \\relax{\\rm I\\kern-.18em P}_3\\cr} \\left[\\matrix{3&0\\cr 1&3\\cr}\\right].\n\\end{equation}\nSuppose we are searching for three--generation orbifolds of this space\nwith $\\chi_{orb}=\\pm 6$ . If $\\chi=-6$ the constraint is not very restrictive\nand allows a number of possible groups $|G|\\in \\{2,3,5,11,19,29\\}$.\nEven though it is not known whether any of these groups lead to a\nthree--generation model it {\\it is} known that at a particular\npoint in the configuration space of (\\ref{3gen}) described by the\nsuperpotential\n\\begin{equation} W=\\sum_{i=1}^3 (\\Phi_i^3+\\Phi_i \\Psi_i^3)+\\Phi_4^3 \\end{equation}\na symmetry of order nine exists that leads to a three--generation model\n\\cite{s1}.\n\nOur interest however is not restricted to models with particular spectra\nfor reasons explicated in the introduction. Hence we wish to implement\ngeneral types of actions regardless of their fixed point structure and\norder. A general analysis of symmetries for an arbitrary Landau--Ginzburg\npotential is beyond the scope of this paper; instead we restrict our\nattention to the types of potentials that we have constructed explicitly.\nBefore we discuss these types we should remark upon a number of aspects\nconcerning actions on string vacua defined by LG--theories.\n\nIt is important to note that depending on the weights (or charges) of the\noriginal LG theory it can and does happen that actions that take rather\ndifferent forms when considered as actions\non the LG theory actually are isomorphic when viewed as action of the\nstring vacuum proper because of the U(1) projection. It is easiest to\nexplain this with an example.\nConsider the superpotential\n\\begin{equation}\nW=\\Phi_1^{18}+\\Phi_2^{18}+\\Phi_3^3+\\Phi_4^3+\\Phi_4\\Phi_5^3\n\\end{equation}\nwhich belongs to the configuration\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,6,6,4)}[18]_{-204}^9$ (here the superscript denotes the\nnumber of antigenerations and the subscript denotes the Euler number of the\nconfiguration). At this\nparticular point in moduli space we can, e.g., consider the orbifolds with\nrespect to the actions\n\\begin{eqnarray}\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{0&0&1&0&2\\cr}\\right],~~~(13, 79, -132) \\nonumber \\\\\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{1&1&1&0&0\\cr}\\right],~~~(13, 79, -132) \\nonumber \\\\\n\\relax{\\sf Z\\kern-.4em Z}_3&:&\\left[\\matrix{1&0&1&0&1\\cr}\\right],~~~(14, 44, -60) ,\\label{z3}\n\\end{eqnarray}\nwhere the notation $\\relax{\\sf Z\\kern-.4em Z}_a: [p_1~\\ldots~p_n]$ indicates that the\nfields $\\Phi_i$ transform with phases $(2\\pi i p_i\/a)$ under the\ngenerator of the $\\relax{\\sf Z\\kern-.4em Z}_a$ symmetry.\nIt is clear from the last action in (\\ref{z3}) that the order of a group\nis, in general, not suffient to determine the resulting orbifold spectrum\nbut that the specific form of the way the symmetry acts is essential.\n\nSince the first two actions lead to the same spectrum we are led to ask\nwhether the two resulting orbifolds are equivalent.\nTheories with the\nsame number of light fields need, of course, not be equivalent and to\nshow whether they are is, in general a rather involved analysis,\nentailing the transformation behaviour of the fields and the computation\nof the Yukawa couplings.\n\nIn the case at hand it is, however, very easy to check this question.\nThe first two actions only differ by the $6^{\\rm th}$ power of the\ncanonical $\\relax{\\sf Z\\kern-.4em Z}_{18}$ which is given by $\\relax{\\sf Z\\kern-.4em Z}_3:\\,[1~1~0~0~1]$.\nSince the orbifolding with respect to this group is always present\nin the construction of a LG vacuum the fist two orbifolds in\n$eq.$~(\\ref{z3}) are trivially equivalent.\n\nAnother important point is the role of trivial factors in the LG theories.\nGiven a superpotential $W_0$ with the correct central charge to define a\nHeterotic String vacuum we always have the freedom to add trivial factors\nto it\n\\begin{equation}\nW=W_0 + \\sum_i \\Phi_i^2,\n\\end{equation}\nsince neither the central charge nor the chiral ring are changed by this\noperation.\n As we restrict our attention to symmetries with unit determinant,\nwe gain, however, the possibility to cancel a negative sign of the determinant\nby giving some $\\Phi_i$ a nontrivial transformation property under a\n$\\relax{\\sf Z\\kern-.4em Z}_{2n}$. Adding a trivial factor hence changes the symmetry properties\nof the LG--potential with regards to this class of symmetries.\n\\fnote{9}{In LG theories the determinant restriction is necessary for modular\n invariance and can be avoided by introducing discrete torsion \\cite{iv}.}\nIf we wish to relate the vacuum described by the potential to a\nCalabi--Yau manifold, consideration of trivial factors becomes essential\n\\cite{ls3}.\nConsider e.g. the LG--potential\n\\begin{equation}\nW_0=\\Phi_1^{12}+\\Phi_2^{12}+\\Phi_3^6+\\Phi_4^6\n\\end{equation}\nwhich has $c=9$ and charges\n$(\\frac{1}{12},\\frac{1}{12},\\frac{1}{6},\\frac{1}{6})$ and hence is a member\nof the configuration $\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,2,2)}[12]$. Only after adding the necessary\ntrivial factor\nthis theory can be orbifolded with an action defined by\n$\\relax{\\sf Z\\kern-.4em Z}_2:[~1~0~0~0~1~]$ acting on the Fermat polynomial in\n$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,1,2,2,6)}[12]$; this action leads to the orbifold spectrum\n$(4,94,-180)$ and is not equivalent to any symmetry that\nacts only on the first four variables with determinant 1. Neglecting the\naddition of the quadratic term to the LG potential $W_0$\nwould have meant missing the above spectrum as one of the possible\norbifold results.\n\nFinally it should be noted that obviously we have to make {\\it some}\nchoice about which points in moduli space we wish to consider. Different\nmembers of a moduli space have, in general, drastically different\nsymmetry properties. An example is the well known quintic theory\nwhich we already mentioned. The most symmetric\npoint in the 101 dimensional space of complex deformations of the quintic is\ndescribed by the Fermat polynomial\n\\begin{equation} W= \\sum_i \\Phi_i^5,\n\\end{equation}\nwhich has a discrete symmetry group of order $5!\\cdot 5^4$. Any\ndeformation\nbreaks most of these symmetries but in some cases new symmetries appear\nwhich\nturn out to lead to new spectra. An example is furnished by the quintic\ndescribed by a combination of two 1--Tadpole polynomials and one Fermat\nmonomial\n\\begin{equation}\nW=\\Phi_1^5+\\Phi_2^5+\\Phi_2\\Phi_3^4+\\Phi_4^5+\\Phi_4\\Phi_5^4\n\\end{equation}\nwhich, when orbifolded with respect to the symmetry\n$\\relax{\\sf Z\\kern-.4em Z}_2: [0~0~1~0~1]$ leads to a model with spectrum $(3,59,-112)$. This\nspectrum cannot be obtained via orbifolding the Fermat quintic by any of\nits supersymmetry preserving symmetries.\n\n\\vskip .1truein\n\\noindent\n\\section{Phase Actions: Implementation and Results}\n\nConsider then a potential $W$ with $n$ order parameters\nnormalized such that the degree $d$ takes the lowest value such that\nall order\nparameters have integer weight.\nIn the following we discuss potentials of the type\n\\begin{equation}\nW= \\sum_i \\Phi_i^{a_i} + \\sum_j \\left(\\Phi_j^{e_j} + \\Phi_j \\Psi_j^{f_j}\\right)\n + \\sum_k \\left(\\Phi_k^{e_k}\\Psi_k + \\Phi_k \\Psi_k^{f_k} \\right)\n\\end{equation}\nwhich consist of Fermat parts, tadpole parts and loop parts.\n\n\\noindent\n{\\bf FERMAT POTENTIALS}:\nClearly the potential $W=\\sum_{i=1}^n \\Phi_i^{a_i}$ is\ninvariant under $\\prod_i \\relax{\\sf Z\\kern-.4em Z}_{a_i}$, i.e. the phases of the individual fields,\nacting like\n\\begin{equation}\n\\Phi_i \\longrightarrow e^{2\\pi i \\frac{m_i}{a_i}} \\Phi_i.\n\\end{equation}\nFor some divisor $a$ of ${\\rm lcm}(a_1,\\dots, a_n)$ and\n$\\frac{m_i}{a_i}=\\frac{p_i}{a}$\nwe denote such an action by\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_a:~\\left[\\matrix{p_1&p_2 &\\cdots &p_n\\cr}\\right],~~~0\\leq~ p_i \\leq~ a-1.\n\\end{equation}\nand require that $a$ divides $\\sum p_i$ in order to have determinant 1.\n\nWe have implemented such symmetries in the form\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_a:~\\left[\\matrix{(a-\\sum_l i_l)&i_1&\\cdots&i_p\n &(a-\\sum_m j_m)&j_1&\\cdots&j_q&\\cdots} \\right] \\end{equation}\nwith the obvious divisibility conditions. For small $p$ and $q$\nthese symmetries can act on a large number of spaces and therefore lead to\nmany different orbifolds, but as $p,q$ get larger\nthe number of resulting orbifolds decreases rapidly.\nWe have stopped implementation of more complicated actions when the number of\nresults for the different orbifold Hodge pairs was of the order of a few tens.\nAs already mentioned above, the precise form of the action is very important\nwhen considering symmetries with fixed points since the order itself is\nnot sufficient to determine the orbifold spectrum.\n\nMore complicated symmetries can be constructed via multiple actions\nby multiplying single actions of the type described above\n\\begin{equation}\n\\prod_c \\relax{\\sf Z\\kern-.4em Z}_{a_c} :~~\n\\left[\\matrix{(a_c-\\sum_l i_{c,l})&i_{c,1}&\\cdots&i_{c,p}\n &(a_c-\\sum_m j_{c,m})&j_{c,1}&\\cdots&j_{c,q}&\\cdots} \\right].\n\\end{equation}\nWe have considered (an incomplete set of) actions of this type with up to\nsix twists (i.e. six $\\relax{\\sf Z\\kern-.4em Z}_a$ factors). Again the precise form of the action\nis rather important.\n\n\\vskip .2truein\n\n\\noindent\n{\\bf TADPOLE AND LOOP POLYNOMIALS}:~~\nThe action of the generator of the maximal phase symmetry within a tadpole\nor loop sector is\n\\begin{equation} \\relax{\\sf Z\\kern-.4em Z}_{\\cO}: \\left[\\matrix{-f&1}\\right], \\end{equation}\nwhere $\\cO = ef$ or $ef-1$, respectively. If we want unit determinant within\none sector, we must take our generator to the $n^{\\rm th}$ power with some\n$n$ fulfilling $n(f-1)\/\\cO\\in\\relax{\\sf Z\\kern-.4em Z}$. With $\\om =gcd(f-1,\\cO)$ the action\nof the resulting subgroup can be chosen to be\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_{\\om}:~~ \\left[\\matrix{(\\om-1)&1}\\right].\n\\end{equation}\n\nOther types of actions that we have considered for superpotentials consisting\nof Fermat parts and tadpole\/loop parts involve phases acting both on the\ntadpole\/loop part as well as on a number of Fermat monomials.\nAs was the case with pure Fermat polynomials we have also implemented\nmultiple actions of the type considered above.\n\n\nWe have implemented some forty different actions of the types\ndescribed in the previous paragraphs. These symmetries lead to a large\nnumber of orbifolds not all of which are distinct however for reasons\nexplained in the previous section.\nOur computations have concentrated on the number of generations and\nanti--generations of these models and we have found some 1900 distinct\nHodge pairs. This set of spectra shows a mirror symmetry that is even\nhigher than the one exhibited by the complete intersection vacua: whereas\nabout 8\nspectra have mirror partners!\n\n\nIt is obvious from this plot that there is a large overlap between the\nresults of \\cite{cls} and the orbifolds constructed here. This might indicate\nthat the relation established in \\cite{ls4} between orbifolds of\nLandau--Ginzburg\ntheories and other Landau--Ginzburg theories is a general phenomenon and\nnot restricted to the particular classes of actions which were analysed in\n\\cite{ls4}.\n\nModels with a low number of fields are clearly\nof particular interest. There are two aspects to this question,\nas mentioned in the introduction -- low numbers for the {\\it difference}\nof\ngenerations and anti--generations\n(more precisely one wants the number 3 here)\nand low values for the {\\it total} number of generations and\nanti--generations.\nAs far as the latter are concerned the following `low--points' are the\n`highlights' among the results for phase symmetry orbifolds.\n\nThe lowest models have $\\chi=0$, more precisely the spectra (9,9,0)\nand (11,11,0). These spectra appear many times in different\norbifolds of Fermat type; an example for the first one being\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(1,\\ldots,1)}[9]\/\\relax{\\sf Z\\kern-.4em Z}_3^2: \\left[\\matrix{1&1&1&0&0&0&0&0&0\\cr\n 0&0&0&1&1&1&0&0&0\\cr}\\right] \\end{equation}\nor, even simpler,\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(4,4,4,4,4,4,3,3)}[12]\/\\relax{\\sf Z\\kern-.4em Z}_3: [1~1~1~0~0~0~0~0].\n\\end{equation}\nThe second one can be constructed e.g. as\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(4,3,3,3,3,2)}[12]\/\\relax{\\sf Z\\kern-.4em Z}_4^2: \\left[\\matrix{0&1&1&2&0&0\\cr\n 0&0&2&1&1&0\\cr}\\right].\n\\end{equation}\n\nOther examples with a total of 22 generations and anti--generations are\nthe following orbifolds of the Fermat quintic:\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5: \\left[\\matrix{0&1&2&3&4}\\right],~~\\qqd (1,21,-40)\n\\end{equation}\nand\n\\begin{equation}\n\\relax{\\sf Z\\kern-.4em Z}_5^2: \\left[\\matrix{3&1&1&0&0\\cr\n 0&3&1&1&0\\cr}\\right],~~\\qqd (21,1,40)\n\\end{equation}\n\n\\noindent\nOf particular interest, of course, are three--generation models.\nIn the list of 3112 models there are no new such models aside from the\nknown three--generation models \\cite{ks}.\n\nVia orbifolding a number of such models can be found, which all, however,\nhave\na fairly large number of generations and antigenerations. We list those\nin Table 2.\n\n\n\\begin{small}\n\\begin{center}\n\\begin{tabular}{||l|l l l r||}\n\\hline\n\\hline\n$\\#$ &Configuration &Potential &Action\n&Spectrum \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\\n\\hline\n\\hline\n1 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(9,2,5,9,2)}[27]_{-66}^{16}$\n&$\\Phi_1^3+\\Phi_2^{11}\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^3+\\Phi_4\\Phi_5^9$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~0~0~2]$ &$(18, 21, -6)$ \\hbox to0pt{\\phantom{\\Huge A}\\hss} \\\\ [2ex]\n2 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(9,2,5,3,8)}[27]_{-54}^{10}$\n&$\\Phi_1^3+\\Phi_2^{11}\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^9+\\Phi_4\\Phi_5^3$\n&$\\relax{\\sf Z\\kern-.4em Z}_2~:[0~1~1~0~2]$ &$(21, 18, 6)$ \\\\ [2ex]\n3 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(17,6,9,3,16)}[51]_{-102}^{15}$\n&$\\Phi_1^3+\\Phi_2^7\\Phi_3+\\Phi_2\\Phi_3^5+\\Phi_4^{17}+\\Phi_4\\Phi_5^3$\n&$\\relax{\\sf Z\\kern-.4em Z}_2~:[0~1~1~0~0]$ &$(31, 34, -6)$ \\\\ [2ex]\n4 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(15,15,2,9,4)}[45]_{-30}^{23}$\n& $\\Phi_1^3+\\Phi_2^3+\\Phi_2\\Phi_3^{15}+\\Phi_4^5+\\Phi_4\\Phi_5^9$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~2~0~0]$ &$(23, 20, 6)$\\\\ [2ex]\n5 &$\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(15,15,10,3,2)}[45]_{-54}^{22}$\n&$\\Phi_1^3+\\Phi_2^3+\\Phi_2\\Phi_3^3+\\Phi_4^{15}+\\Phi_4\\Phi_5^{21}$\n&$\\relax{\\sf Z\\kern-.4em Z}_3~:[1~0~2~0~0]$ &$(35, 32, 6)$ \\\\ [2ex]\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{small}\n\n{\\bf Table 2.} {\\it Three--generation orbifold models; models which are\nequivalent up to the U(1) projection are not listed separately.}\n\n\n\\noindent\nBy using the relation established in \\cite{ls4} between LG\/CY--theories\nvia fractional transformations it can be shown that the orbifold $\\#1$ in\nTable 2,\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,5,9,2,9)}[27]_{-66}^{16}\/\\relax{\\sf Z\\kern-.4em Z}_3:[~0~0~0~2~1],\n\\end{equation}\nfor which the covering model is described by the polynomial\n\\begin{equation}\nW= \\Phi_1^{11}\\Phi_2+\\Phi_1\\Phi_2^5+\\Phi_3^3+\\Phi_3\\Phi_4^9+\\Phi_5^3,\n\\end{equation}\nis isomorphic to the orbifold\n\\begin{equation}\n\\relax\\,\\hbox{$\\inbar\\kern-.3em{\\rm C}$}_{(2,5,9,3,8)}[27]_{-54}^{10}\/\\relax{\\sf Z\\kern-.4em Z}_2:[~0~0~0~1~1]\n\\end{equation}\nwhere the covering theory is described by the polynomial\n\\begin{equation}\nW= \\Phi_1^{11}\\Phi_2+\\Phi_1\\Phi_2^5+\\Phi_3^3+\\Phi_3\\Phi_4^6\n +\\Phi_4\\Phi_5^3.\n\\end{equation}\nThe latter is a theory involving a subtheory with couplings among three\nscaling fields and hence goes beyond the types of potentials we have\nimplemented.\nThis example indicates that more complicated examples than the ones\ninvestigated here are likely to yield more (perhaps more realistic)\nthree generation models.\n\nThe covering spaces of all the three generation models are described by\neither tadpole or loop type polynomials, and with our actions none of the\nFermat type polynomials leads to a three generation model.\nIt should be noted that these orbifolds exist only at particular points in\nmoduli space.\n\n\\vskip .1truein\n\\noindent\n\\section{ Conclusion}\n\n\\noindent\nIt is clear that the structure of the configuration space of the Heterotic\nString is not particularly well understood and that much remains to be done;\nit is apparent from\nthe results described above that what has been achieved so far at best is\n little more than scratching the surface. It might be hoped that once\na completely mirror symmetric part of the moduli space has been constructed\na representative part of the complete space has been uncovered.\nPursueing work\nalong the lines described above is certainly promising in this regard;\nvery likely it is possible to generate a mirror symmetric subspace by\nall orbifolds of the Landau--Ginzburg theories listed in \\cite{ks}\nor, more generally, to construct all weighted complete intersection\nCalabi--Yau\nmanifolds and their orbifolds.\n\nHaving done that it still remains to show that all potential mirror\npartners\nin this symmmetric subspace of ground states are in fact related.\nEven though many types of LG--potentials constructed in \\cite{cls} admit,\nvia fractional transformations, an interpretation as orbifolds not every\nmirror potential can be constructed in this way at present. It is therefore\n clear that a generalization of this type of mirror map is necessary.\n\nAside from the question of mirror symmetry the orbifold technique is\nextremely useful to get insight into both, the detailed structure\nof the vacua constructed via such a classification of LG--potentials and\nthe relation between these vacua.\nProperties of the ground states that are obscure from the point of view\nof the\nsuperpotentials alone or from the point of view of the partition function\nof the underlying conformal field theory (if known) become rather obvious\nin the orbifold picture.\n\n\\vskip .2truein\n\n\\noindent\n\\section*{Acknowledgement}\n\n\\noindent\nMost of the work described here has been the result of the joint efforts\nof several collaborations. I'm grateful to all the people involved, in\nparticular\nPhilip Candelas, Albrecht Klemm, Max Kreuzer and Monika Lynker.\n\n\\vfill \\eject\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs is well known the r\\^ole of symmetry in Physics is fundamental. Symmetries are usually divided in two types: spacetime and internal symmetries. \n\nIf we ignore the effect of gravity that provides spacetime curvature, Special Relativity tells us that our spacetime is the flat Minkowski space and its symmetries are given by the Poincar\\'e group. However, if no massive particle is present, the symmetry group can be larger. The largest transformation group compatible with locality is the conformal group. The conformal group is a finite-dimensional Lie group; its symmetry action is so stringent that one expects only few conformal Quantum Field Theory models to exist in four dimensions, see \\cite{BRNT} and references therein.\n\nA much richer structure comes out in lower dimensional spacetimes. On the two-dimensional Minkowski spacetime the conformal group is infinite dimensional and factors as a product of the diffeomorphism groups of the light-like (chiral) lines $x\\pm t=0$ and acts on the their compactification, namely one gets an action of ${\\mathrm {Diff}}(S^1)\\times{\\mathrm {Diff}}(S^1)$. Restricting a quantum field to a chiral line gives rise to a quantum field theory on $S^1$. \n\nWe do not dwell here on crucial r\\^ole played by Conformal Quantum Field Theory in various physical contexts, nor on its deep mathematical connections with other subjects.\nWe mention however a less emphasised r\\^ole of the subject as a laboratory for more general analysis. This is due to the variety of constructed models and the powerful methods of analysis. We are interested here in particular in Operator Algebraic methods whose effectiveness has  also been shown by recent classification results and new model constructions \\cite{KL1,KLPR,X6}.\n\nInternal symmetries are also fundamental. In particular, in Quantum Field Theory, they are basically expressed by a gauge group. Concerning the particle-antiparticle symmetry, it showed up with the discovery of the Dirac equation and is, in a sense, a derived symmetry. There are general arguments to link the particle-antiparticle symmetry with the spacetime symmetries, see \\cite{GL1}.\n\nNow there is an important higher level symmetry relating spacetime and internal symmetries: supersymmetry, see \\cite{WB}. Supersymmetry sets up a correspondence between Bose and Fermi particles. Leaving aside important physical aspects of supersymmetry, e.g. in relation to the Higgs particle, we wish to mention that the related mathematical structure has profound noncommutative geometrical implications, see \\cite{C,JLO}.\n\nIf one considers a quantum field theory which is both chiral conformal and supersymmetric, one then expects a very stringent interesting structure to show up. Superconformal models have indeed been studied by different approaches. For example, the tricritical Ising model is a basic model with a remarkable structure \\cite{FQS}.\n\nThe purpose of this paper is to initiate a general, model independent, operator algebraic study of superconformal quantum field theory on the circle and pursue this analysis to the classification of the superconformal nets in the discrete series. \n\nIn the first sections we describe the general property of a Fermi conformal net on $S^1$. Basic properties, already considered in the non-local case \\cite{DLR,LR2}, are here described in our context (Bisognano-Wichmann property, twisted Haag duality,\\dots).\n\nA Fermi net ${\\mathcal A}$ contains a local subnet ${\\mathcal A}_b$, the Bose subnet (the fixed-point under the grading symmetry) which is automatically equipped with an involutive sector ${\\sigma}$, dual to the grading. This splits the sectors of ${\\mathcal A}_b$ in two subsets, ${\\sigma}$-Bose and ${\\sigma}$-Fermi sectors, that can be studied in the standard way as ${\\mathcal A}_b$ is local.\n\nFor a general model analysis one would like to consider representations of ${\\mathcal A}$. There is an obvious extension of the notion of DHR representation to Fermi nets, that we study at the beginning. However, as we shall see, there is more general natural class of representations for Fermi net: the \\emph{general representations}. These are representations of the promotion ${\\mathcal A}^{(n)}$ of ${\\mathcal A}$ to the $n$-cover $S^{1(n)}$ of $S^1$, that restrict to DHR representations of the Bose subnet ${\\mathcal A}_b$. We shall see that a general representation is indeed  a representation of the double cover net ${\\mathcal A}^{(2)}$ and can be equivalently described as a \\emph{general soliton}. This splits the general representations in two classes: Neveu-Schwarz (DHR) and Ramond (properly associated with ${\\mathcal A}^{(2)}$). We shall later see that a representation is Neveu-Schwarz (resp. Ramond) iff it comes by $\\alpha$-extension by a ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) representation of ${\\mathcal A}_b$.\n\nHaving clarified the structure of the representations of a Fermi net, we consider the case where supersymmetry is present. In particular we consider a \\emph{supersymmetric representation} of a conformal Fermi net ${\\mathcal A}$, namely a graded general representation $\\lambda$ of ${\\mathcal A}$ where \n\\[\n\\tilde H_\\lambda \\equiv H_\\lambda - \\frac{c}{24} = Q_\\lambda^2 \\ ;\n\\]\nhere $H_\\lambda$ is the conformal Hamiltonian and $Q$ is an odd selfadjoint operator (the supercharge). In this case the McKean-Singer lemma (Appendix \\ref{MKS}) gives\n\\[\n\\Str(e^{-t\\tilde H_\\lambda}) = \\text{Fredholm index of}\\ Q_{\\lambda +}\\ , \n\\]\nfor all $t>0$, where $\\Str$ denotes the supertrace. In particular the left hand side, also called the Witten index, is an integer, the Fredholm index $\\ind(Q_{\\lambda +})$ of upper off diagonal part of $Q_\\lambda$.\n\nIf $\\lambda$ is irreducible, its restriction $\\lambda_b$ to ${\\mathcal A}_b$ has two irreducible components $\\lambda_b = \\rho\\oplus\\rho'$ and we can consider the Jones index of $\\rho$, which is the square of the Doplicher-Haag-Roberts statistical dimension $d(\\rho)$ \\cite{L5}.\n\nIf ${\\mathcal A}_b$ is modular, we can express $d(\\rho)$ by the Kac-Peterson Verlinde matrix $S$ as this is equal to the Rehren matrix $S$. By combining all this information and an argument in \\cite{KL05}, we then get the formula\n\\[\n\\ind(Q_{\\lambda +}) = \n\\frac{d(\\rho)}{\\sqrt{\\mu_{\\mathcal A}}}\\sum_{\\nu\\in\\mathfrak R}\nK(\\rho,\\nu)d(\\nu) \n\\]\nwhere $K$ is a matrix that is related to $S$, $\\mu_{\\mathcal A}$ is the $\\mu$-index \\cite{KLM} and the sum is over all Ramond irreducible sectors. This formula thus involves both the Fredholm index and the Jones index.\n\nWe then put our attention towards model analysis and aiming firstly to illustrate concrete situations where our general setting is realised.\n\nThe infinite-dimensional super-Lie algebra that governs the superconformal symmetries is the super-Virasoro algebra. It is a central extension by a central element $c$ of the Lie algebra generated by the Fourier modes $L_n$ and $G_r$ of the Bose and the Fermi stress-energy tensor. Clearly the $L_n$ generate the Virasoro algebra, thus the super-Virasoro algebra contains the Virasoro algebra; the commutation relation are given by the equations \\eqref{svirdef}. It has been studied in \\cite{GKO, FQS}.\n\nThe super-Virasoro algebra plays for superconformal fields the universal same r\\^ole that the Virasoro algebra plays for local conformal fields. Our initial task is then to construct and analyse the super-Virasoro nets. Following the work \\cite{GKO} we describe the nets; if the central charge $c$ is less than $3\/2$ we identify the Bose subnet as a coset, study the representation structure and show the modularity of the net. There is a distinguished representation with lowest weight $c\/24$ which is supersymmetric and thus provides an interesting example for our index formula. \n\nNow, if $c<3\/2$ we are in the discrete series \\cite{FQS}; analogously to the local conformal case \\cite{KL1} every superconformal net is an irreducible finite-index extension of a super-Virasoro net. We classify all such extensions. There are two series of extensions, the one given by the trivial extension and a second one given by index 2 extensions. Beside these there are six exceptional extensions that we describe explicitly.\n\\section{Fermi nets on $S^1$}\nIn this section we discuss the basic notions for a Fermi conformal net of von Neumann algebras on $S^1\\equiv \\{z\\in\\mathbb C: |z|=1\\}$, and their first implications. It is convenient to start by analysing M\\\"obius covariant nets. Note that a general discussion of M\\\"obius covariant net is contained in \\cite{DLR}, here however we focus on the Fermi case.\n\\subsection{M\\\"obius covariant Fermi nets}\n\\label{MobFermiNets}\nWe shall denote by ${\\rm\\textsf{M\\\"ob}}$ the M\\\"obius group, which is isomorphic to $SL(2,\\mathbb R)\/\\mathbb Z_2$ and acts naturally and faithfully on the circle $S^1$. The $n$-cover of ${\\rm\\textsf{M\\\"ob}}$ is denoted by ${\\rm\\textsf{M\\\"ob}}^{(n)}$, $n\\in\\mathbb N\\cup\\{\\infty\\}$. Thus ${\\rm\\textsf{M\\\"ob}}^{(2)}\\simeq SL(2,\\mathbb R)$ and ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ is the universal cover of ${\\rm\\textsf{M\\\"ob}}$.\n\nBy an interval of $S^1$ we mean, as usual, a non-empty, non-dense, open, connected subset of $S^1$ and we denote by ${\\mathcal I}$ the set of all intervals. If $I\\in{\\mathcal I}$, then also $I'\\in{\\mathcal I}$ where $I'$ is the interior of the complement of $I$. \n\nA {\\it  net ${\\mathcal A}$ of von Neumann algebras on $S^1$} is a map\n\\[\nI\\in{\\mathcal I}\\mapsto{\\mathcal A}(I)\n\\]\nfrom the set of intervals to the set of von Neumann algebras on a\n(fixed) Hilbert space ${\\mathcal H}$ which verifies the \\emph{isotony property}:\n\\[\nI_1\\subset I_2\\Rightarrow  {\\mathcal A}(I_1)\\subset{\\mathcal A}(I_2)\n\\]\nwhere $I_1 , I_2\\in{\\mathcal I}$.\n\nA \\emph{M\\\"obius covariant net} ${\\mathcal A}$ of von Neumann algebras on $S^1$ is a net of von Neumann algebras on $S^1$ such that the following properties $1-4$ hold:\n\\begin{description}\n\\item[\\textnormal{\\textsc{1. M\\\"obius covariance}}:] {\\it There is a\nstrongly continuous unitary representation $U$ of} ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ {\\it on ${\\mathcal H}$ such that}\n\\[\nU(g){\\mathcal A}(I)U(g)^*={\\mathcal A}(\\dot{g}I)\\ ,\n\\qquad g\\in {\\rm\\textsf{M\\\"ob}}^{(\\infty)},\\ I\\in{\\mathcal I} \\ ,\n\\]\n{\\it where $\\dot{g}$ is the image of $g$ in} ${\\rm\\textsf{M\\\"ob}}$ {\\it under the quotient map.}\n\\end{description}\n\\begin{description}\n\\item[$\\textnormal{\\textsc{2. Positivity of the energy}}:$]\n{\\it The generator of the rotation one-para\\-meter subgroup \n$\\theta\\mapsto U({\\rm rot}^{(\\infty)}(\\theta))$ \n(conformal Hamiltonian) is positive}, na\\-me\\-ly $U$ is a positive energy representation\n\\footnote{Here ${\\rm rot}^{(n)}(\\theta)$ is the lift to ${\\rm\\textsf{M\\\"ob}}^{(n)}$ of the $\\theta$-rotation ${\\rm rot}(\\theta)$ of ${\\rm\\textsf{M\\\"ob}}$. For shortness we sometime write ${\\rm rot}^{(n)}(\\theta)$ simply by ${\\rm rot}(\\theta)$ and $U(\\theta) = U({\\rm rot}(\\theta))$.}. \n\\end{description}\n\\begin{description}\n\\item[\\textnormal{\\textsc{3. Existence and uniqueness of the vacuum}}:]\n{\\it There exists a unit $U$-invariant vector $\\Omega$ \n(vacuum vector), unique up to a phase, and $\\Omega$ is\ncyclic for the von~Neumann algebra $\\vee_{I\\in{\\mathcal I}}{\\mathcal A}(I)$}\n\\end{description}\nGiven a M\\\"obius covariant net ${\\mathcal A}$ in $S^1$, a unitary $\\Gamma$ such that $\\Gamma\\Omega = \\Omega$, $\\Gamma{\\mathcal A}(I)\\Gamma^* ={\\mathcal A}(I)$ for all $I\\in{\\mathcal I}$ is called a gauge unitary and the adjoint net automorphism $\\gamma\\equiv{\\hbox{\\rm Ad}}\\Gamma$ a \\emph{gauge} automorphism\\footnote{The requirement $\\Gamma\\Omega =\\Omega$ (up to a phase that can be put equal to one) is equivalent to $\\Gamma U(g) = U(g)\\Gamma$ by the uniqueness of the representation $U$ in Cor. \\ref{unique}; we shall use the second formulation in the non-vacuum case.}. \n\nA $\\mathbb Z_2$-grading on ${\\mathcal A}$ is an involutive gauge automorphism $\\gamma$ of ${\\mathcal A}$. Given the grading $\\gamma$, an element $x$ of ${\\mathcal A}$ such that $\\gamma(x)=\\pm x$ is called homogeneous, indeed a Bose or Fermi element according to the $\\pm$ alternative. \nWe shall say that the degree $\\partial x$ of the homogeneous element $x$ is $0$ in the Bose case and $1$ in the Fermi case.\n\nEvery element $x$ of ${\\mathcal A}$ is uniquely the sum $x = x_0 + x_1$ with $\\partial x_k=k$, indeed \n$x_k = \\big(x  + (-1)^k \\gamma(x)\\big)\/2$.\n\nA  {\\em M\\\"obius covariant Fermi net} ${\\mathcal A}$ on $S^1$ is a $\\mathbb Z_2$-graded M\\\"obius covariant net satisfying graded locality, namely a  M\\\"obius covariant net of \nvon Neumann algebras on $S^1$ such that the following holds:\n\\begin{description}\n\\item[\\textnormal{\\textsc{4. Graded locality}}:]\n{\\it There exists a grading automorphism $\\gamma$ of ${\\mathcal A}$ such that,  if $I_1$ and $I_2$ are disjoint intervals,}\n\\[\n[x,y] = 0,\\quad x\\in{\\mathcal A}(I_1), y\\in{\\mathcal A}(I_2) \\ .\n\\]\n\\end{description}\nHere $[x,y]$ is the graded commutator with respect to the grading automorphism $\\gamma$ defined as follows: if $x,y$ are homogeneous then\n\\[\n[x,y]\\equiv xy - (-1)^{\\partial x \\cdot \\partial y}yx\n\\]\nand, for the general elements $x,y$, is  extended by linearity.\n\nNote the \\emph{Bose subnet} ${\\mathcal A}_b$, namely the $\\gamma$-fixed point subnet ${\\mathcal A}^\\gamma$ of degree zero elements, is local. Moreover, setting\n\\[\nZ\\equiv \\frac{1 - i\\Gamma}{1 - i}\n\\]\nwe have that the unitary $Z$ fixes $\\Omega$ and\n\\[\n{\\mathcal A}(I')\\subset Z{\\mathcal A}(I)'Z^*,\n\\]\n(twisted localiy w.r.t. $Z$), that is indeed equivalent to graded locality.\n\\medskip\n\n\\noindent\n{\\it Remark.} Strictly speaking a M\\\"obius covariant net is a pair $({\\mathcal A}, U)$ where ${\\mathcal A}$ is a net and $U$ is a unitary representation of ${\\rm\\textsf{M\\\"ob}}$ that satisfy the above properties. In most cases it is convenient to denote the M\\\"obius covariant net simply by ${\\mathcal A}$ and then consider the unitary representation $U$ (note that $U$ is unique once we fix the vacuum vector). However, later in this paper, it will be convenient to use the notation $({\\mathcal A}, U)$ (also in the diffeomorphism covariant case).\n\\subsection{First consequences}\nWe collect here a few first consequences of the axioms of a M\\\"obius covariant Fermi net on $S^1$. They can be mostly derived by a simple extension of the proofs in the local case, cf. \\cite{DLR,LR2}.\n\\subsubsection*{Reeh-Schlieder theorem}\n\\begin{theorem}\\label{Reeh-Schlieder} Let ${\\mathcal A}$ be a M\\\"obius covariant Fermi net on\n$S^1$.  Then $\\Omega$ is cyclic and separating for each von Neumann\nalgebra ${\\mathcal A}(I)$, $I\\in{\\mathcal I}$.\n \\end{theorem}\n \\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe cyclicity of $\\Omega$ follows exactly as in the local case. By twisted locality, then $\\Omega$ is separating too.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Bisognano-Wichmann property}\nIf $I\\in {\\mathcal I}$, we shall denote by $\\Lambda_I$ the one parameter subgroup of ${\\rm\\textsf{M\\\"ob}}$ of ``dilation associated with $I$\\ \\!'', see \\cite{BGL}. We shall use the same symbol also to denote the unique one parameter subgroup of ${\\rm\\textsf{M\\\"ob}}^{(n)}$  ($n$ finite or infinite) that project onto $\\Lambda_I$ under the quotient map. The following theorem and its corollary are proved as usual, see \\cite{DLR}.\n\\begin{theorem}\nLet $I\\in{\\mathcal I}$ and $\\Delta_I$, $J_I$ be the modular operator and the modular \nconjugation of $({\\mathcal A}(I),\\Omega)$. Then we have:\n\n$(i)$: \n\\begin{equation}\\label{BW}\n\\Delta_{I}^{it} = U(\\Lambda_I(-2\\pi t)), \\ t\\in\\mathbb R,\n\\end{equation}\n\n$(ii)$: \n$U$ extends to an (anti-)unitary representation of {\\rm ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}\\ltimes\\mathbb Z_2$}\ndetermined by\n\\[\nU(r_I)=ZJ_I,\\ I\\in{\\mathcal I},\n\\]\nacting covariantly on ${\\mathcal A}$, namely\n\\[\nU(g){\\mathcal A}(I)U(g)^*={\\mathcal A}(\\dot{g}I)\\quad g\\in\\text{\\rm ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$}\\ltimes\\mathbb Z_2\\ I\\in {\\mathcal I}\\ .\n\\]\nHere $r_I:S^1\\to S^1$ is the reflection mapping $I$ onto $I'$, see \\cite{BGL}.\n\\end{theorem}\n\\begin{corollary}\\label{unique} {\\em (Uniqueness of the M\\\"obius representation)}\nThe representation $U$ is unique.\n\\end{corollary}\n\\begin{corollary} {\\em (Additivity)}\n  Let $I$ and $I_i$ be intervals with $I\\subset\\cup_i I_i$. Then \n  ${\\mathcal A}(I)\\subset\\vee_i{\\mathcal A}(I_i)$.\n\\end{corollary}\n\\medskip\n\n\\noindent\n{\\it Remark.} If $E\\subset S^1$ is any set, we denote by ${\\mathcal A}(E)$ the von Neumann algebra generated by the ${\\mathcal A}(I)$'s as $I$ varies in the intervals $I\\in{\\mathcal I}$, $I\\subset E$. It follows easily by M\\\"obius covariance that if $I_0\\in{\\mathcal I}$ has closure $\\bar I_0$, then ${\\mathcal A}(\\bar I_0)=\\bigcap_{I\\supset {\\bar I_0}, I\\in{\\mathcal I}}{\\mathcal A}(I)$.\n\\subsubsection*{Twisted  Haag duality}\n\\begin{theorem}\nFor every  $I\\in{\\mathcal I}$, we have:\n\\[\n{\\mathcal A}(I')=Z{\\mathcal A}(I)'Z^*\n\\]\n\\end{theorem}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] As already noted twisted locality ${\\mathcal A}(I)' \\supset Z{\\mathcal A}(I')Z^*$ holds as a consequence of the Bose-Fermi commutation relations. By the Bisognano-Wichmann property, $Z{\\mathcal A}(I')Z^*$ is a von Neumann subalgebra of ${\\mathcal A}(I)'$ globally invariant under the vacuum modular group. As the vacuum vector is cyclic for ${\\mathcal A}(I')$ by the Reeh-Schlieder theorem, we have ${\\mathcal A}(I)' = Z{\\mathcal A}(I')Z^*$ by the Tomita-Takesaki modular theory.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nIn the following corollary, the grading and the graded commutator is considered on $B({\\mathcal H})$ w.r.t. ${\\hbox{\\rm Ad}}\\Gamma$.\n\\begin{corollary}\\label{td}\n${\\mathcal A}(I')= \\big\\{x\\in B({\\mathcal H}):\\ [x,y]=0\\ \\forall y\\in{\\mathcal A}(I)\\big\\}$.\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nSet $X\\equiv\\{x\\in B({\\mathcal H}):\\ [x,y]=0\\ \\forall y\\in{\\mathcal A}(I)\\}$. We have to show that $X\\subset{\\mathcal A}(I')$. \nIf $x = x = x_0 + x_1\\in X$ then $Z^*xZ =  x_0 - ix_1\\Gamma$. As $[x,y]=0$ for all $y\\in{\\mathcal A}(I)$ it follows that $Z^*xZ\\in{\\mathcal A}(I)'$ so $x\\in Z{\\mathcal A}(I)'Z^* = {\\mathcal A}(I')$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Irreducibility}\nWe shall say that ${\\mathcal A}$ is \\emph{irreducible} if the von Neumann algebra $\\vee{\\mathcal A}(I)$\ngenerated by all local algebras coincides with $B({\\mathcal H})$.\n\nThe irreducibility property is indeed equivalent to several other requirements.\n\\begin{proposition}\\label{irr} Assume all properties $1-4$ for ${\\mathcal A}$ except the uniqueness of the vacuum. The following are equivalent:\n\\begin{itemize}     \n\\item[{$(i)$}] $\\mathbb C\\Omega$ are the only $U$-invariant vectors.\n\\item[{$(ii)$}] The algebras ${\\mathcal A}(I)$, $I\\in{\\mathcal I}$, are\nfactors. In particular they are type III$_1$ factors (or ${\\rm dim}\\, {\\mathcal H} = 1$).\n\\item[{$(iii)$}] The net ${\\mathcal A}$ is irreducible.\n\\end{itemize}\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] See \\cite{DLR}. \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsubsection*{Vacuum spin-statistics relation}\n\\label{vacss} \nThe relation $U(2\\pi)=1$ holds in the local case \\cite{GL2}; in the graded local case it generalizes to\n\\[\nU(4\\pi)=1\\ ,\n\\]\nwhere $U(s) \\equiv U({\\rm rot}(s)) = e^{isL_0}$ is the rotation one-parameter unitary group, see \\cite{DLR}. Indeed we have the following.\n\\begin{proposition}\\label{vss}\nLet ${\\mathcal A}$ be M\\\"obius covariant Fermi net. Then:\n\\[\nU(2\\pi) =\\Gamma\\ .\n\\]\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy restricting to ${\\mathcal A}_b$ the identity representation of ${\\mathcal A}$ we get the direct sum $\\iota\\oplus{\\sigma}$ of the identity and an automorphism. Clearly ${\\sigma}$ has Fermi statistics because ${\\mathcal A}$ has Bose-Fermi commutation relations. Thus $U = U_\\iota \\oplus U_{\\sigma}$; by the conformal spin-statistics theorem \\cite{GL2} we then have $U(2\\pi)= U_\\iota (2\\pi) \\oplus U_{\\sigma} (2\\pi) = 1\\oplus -1 = \\Gamma$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{corollary} {\\em (Uniqueness of the grading)}\nThe grading automorphism is unique.\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nImmediate from the uniqueness of the M\\\"obius unitary representation and the spin-statistics relation $U(2\\pi) =\\Gamma$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{Fermi conformal nets on $S^1$}\nIf ${\\mathcal A}$ is a M\\\"obius covariant Fermi net on $S^1$ then, by the spin-statistics relation, the unitary representation of ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ is indeed a representation of ${\\rm\\textsf{M\\\"ob}}^{(2)}$. As ${\\rm\\textsf{M\\\"ob}}$ is naturally a subgroup of the group ${\\mathrm {Diff}}(S^1)$ of orientation preserving diffeomorphisms of $S^1$, clearly ${\\rm\\textsf{M\\\"ob}}^{(2)}$  is naturally a subgroup of ${\\mathrm {Diff}}^{(2)}(S^1)$, the 2-cover of ${\\mathrm {Diff}}(S^1)$.\n\nGiven an interval $I\\in{\\mathcal I}$, we shall denote by ${\\mathrm {Diff}}_I(S^1)$ the subgroup of diffeomorphisms $g$ of $S^1$ localised in $I$, namely $g(t)=t$ for all $t\\in I'$, and by\n${\\mathrm {Diff}}_I^{(2)}(S^1)$ the connected component of the identity of the pre-image of \n${\\mathrm {Diff}}_I(S^1)$ in ${\\mathrm {Diff}}^{(2)}(S^1)$.\n\nA \\emph{Fermi conformal net} ${\\mathcal A}$ (of von Neumann algebras) on $S^1$ is a M\\\"obius covariant Fermi net of von Neumann algebras on $S^1$ such that the following holds:\n\\begin{description}\n\\item[\\textnormal{\\textsc{5. Diffeomorphism covariance}}:]\n{\\it There exists a projective unitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ on ${\\mathcal H}$, extending the unitary representation of} ${\\rm\\textsf{M\\\"ob}}^{(2)}$, {\\it such that\n\\[\nU(g){\\mathcal A}(I)U(g)^* = {\\mathcal A}(\\dot{g}I),\\ g\\in{\\mathrm {Diff}}^{(2)}(S^1),\\ I\\in{\\mathcal I},\n\\]\nand\n\\[\nU(g)xU(g)^* = x, \\ x\\in{\\mathcal A}(I'),\\ g\\in{\\mathrm {Diff}}_I^{(2)}(S^1), \\ I\\in{\\mathcal I}\\ .\n\\]}\n\\end{description}\nHere $\\dot{g}$ denotes the image of $g$ in ${\\mathrm {Diff}}(S^1)$ under the quotient map.\n\\begin{lemma}\nFor every $g\\in{\\mathrm {Diff}}^{(2)}(S^1)$ we have $U(g)\\Gamma = \\Gamma U(g)$. In particular,\nif $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$, then $U(g)\\in{\\mathcal A}_b(I)$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nAs $\\Gamma= U(2\\pi)$, we have\n$\n\\Gamma U(g) \\Gamma^* = U(2\\pi)U(g)U(2\\pi)^*=\\chi(g) U(g)\n$ for all  $g\\in{\\mathrm {Diff}}^{(2)}(S^1)$,\nwhere $\\chi$ is a continuos scalar function on ${\\mathrm {Diff}}^{(2)}(S^1)$. As $\\Gamma^2 =1$, we have $\\chi(g)^2 =1$ for all $g$, thus $\\chi(g) =1$ because ${\\mathrm {Diff}}^{(2)}(S^1)$ is connected and $\\chi(g) =1$ is $g$ is the identity.\n\nWith $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$, then $U(g)$ commutes with $\\Gamma$, hence with $Z$. By the covariance condition $U(g)$ commutes with ${\\mathcal A}(I')$, hence with $Z{\\mathcal A}(I')Z^*$, so $U(g)\\in{\\mathcal A}(I)$ by twisted Haag duality.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nBy the above lemma, the representation of the diffeomorphism group belongs to the Bose subnet, so we may apply the uniqueness result in the local case in \\cite{CW} and get the following:\n\\begin{corollary} \\emph{(Uniqueness of the diffeomorphism representation)}\nThe projective unitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ is unique (up to a projective phase).\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWith $E = (1 +\\Gamma)\/2$ the orthogonal projection onto $\\overline{{\\mathcal A}_b \\Omega}$, clearly $U|_{E{\\mathcal H}}$ is the projective unitary representation of ${\\mathrm {Diff}}(S^1)$ associated with ${\\mathcal A}_b$, unique by \\cite{CW,W2}. As $\\Omega$ is separating for the local algebras, $U(g)E$ determines $U(g)$ if \n$g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$ for any interval $I\\in{\\mathcal I}$. By Prop. \\ref{locdiff}, as $I$ varies, ${\\mathrm {Diff}}_I(S^1)$ algebraically generates all ${\\mathrm {Diff}}(S^1)$, and $U$ is so determined up to a phase.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{DHR representations}\nThere is a natural notion of representation for a Fermi net which is the \nstraightforward extension of the notion of representation for a local net, see \\eqref{rep} here below. We shall see in later sections that is important to consider also more general representations when dealing with a Fermi net. In this section, however, we deal with the most obvious notion.\n\nIn the following we assume that ${\\mathcal A}$ be a Fermi conformal net, although certain notions and results are obviously valid in the M\\\"obius covariant case.\n\nA \\emph{representation} $\\lambda$ of ${\\mathcal A}$ is a map $I\\to\\lambda_I$ that \nassociates to an interval $I$ of $S^1$ a normal\\footnote{The normality of $\\lambda_I$ is automatic if ${\\mathcal H}_\\lambda$ is separable because ${\\mathcal A}(I)$ is a type $III$ factor.}\nrepresentation $\\lambda_I$ of ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}_{\\lambda}$ such that\n\\begin{equation}\n\\label{rep}\n\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} = \\lambda_I,\\quad I\\subset \\tilde I\\ .\n\\end{equation}\nWe shall say that a representation $\\lambda$ on ${\\mathcal H}_{\\lambda}$ \nis {\\em diffeomorphism covariant} \nif there exists a projective unitary representation $U_\\lambda$ of the universal cover \n${\\mathrm {Diff}}^{(\\infty)}(S^1)$ of ${\\mathrm {Diff}}(S^1)$ on ${\\mathcal H}_{\\lambda}$ such that\n\\[\n\\lambda_{\\dot{g}I}\\big(U(g)xU(g)^*\\big) = U_\\lambda(g)\\lambda_I(x)U_\\lambda(g)^*\\ ,\\  x\\in{\\mathcal A}(I) , \n\\forall g\\in {\\mathrm {Diff}}^{(\\infty)}(S^1)\\ .\n\\]\nHere $\\dot g$ denotes the image of $g$ in ${\\mathrm {Diff}}(S^1)$ under the quotient map.\nA M\\\"obius covariant representation is analogously defined.\n\nAs we shall later deal with more general representations, we may sometime emphasise that we are considering a representation $\\lambda$ as above, by saying that $\\lambda$ is a DHR representation, the `DHR' being however pleonastic.\n\nWe shall say that a representation $\\lambda$ on ${\\mathcal H}_{\\lambda}$ is \\emph{graded}\nif there exists a unitary $\\Gamma_{\\lambda}$ on ${\\mathcal H}_{\\lambda}$ such that\n\\[\n\\lambda_I(\\gamma(x)) = \\Gamma_{\\lambda}\\lambda_I(x)\\Gamma^*_{\\lambda},\\quad x\\in{\\mathcal A}(I) ,\n\\]\nfor all $I\\in{\\mathcal I}$. As $\\gamma$ is involutive one may then also choose a selfadjoint $\\Gamma_\\lambda$.\n\\begin{proposition}\\label{DHRgraded}\nLet $\\lambda$ be an irreducible DHR representation of the Fermi conformal net ${\\mathcal A}$. Then $\\lambda$ is graded and diffeomorphism covariant with positive energy. Moreover we may take $\\Gamma_\\lambda = U_\\lambda(2\\pi)$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe first use an argument in \\cite{KL1}. Let $I\\in{\\mathcal I}$ and $g\\in{\\mathrm {Diff}}_I^{(2)}(S^1)$. For every $x\\in{\\mathcal A}(\\tilde I)$ with $\\tilde I\\supset I$ we have $\\lambda_I(U(g))\\lambda_{\\tilde I}(x)\\lambda_I(U(g))^* = \\lambda_{\\tilde I}(U(g)xU(g))^* = \\lambda_{gI}(U(g)xU(g))^*$. As the group generated by ${\\mathrm {Diff}}_I^{(2)}(S^1)$ as $I$ varies in ${\\mathcal I}$ is the entire ${\\mathrm {Diff}}^{(2)}(S^1)$, we see every diffeomorphism is implemented in the representation $\\lambda$ by a unitary $U_\\lambda(g)$, which is equal to $\\lambda_I(U(g))$ if $g$ is localised in $I$.\n\nIt remains to show that, by multiplying $U_\\lambda(g)$ by a phase factor, we can get a continuous projective representation with positive energy.\nIn the local case, the automatic diffeomorphism covariance is proved in \\cite{DFK} and the automatic positivity of the energy in \\cite{W}. Let $U_{\\lambda_b}$ be the covariance unitary representation associated with $\\lambda_b\\equiv \\lambda |_{{\\mathcal A}_b}$. If $g$ is localised in $I$ then $U_\\lambda(g)U_{\\lambda_b}(g)^*\\in\\lambda_I\\big({\\mathcal A}_b(I)'\\cap{\\mathcal A}(I)\\big)=\\mathbb C$ (cf. Lemma \\ref{outer}), so we are done by replacing $U_\\lambda$ with $U_{\\lambda_b}$.\n\nFinally notice that, by diffeomorphism covariance, it follows that \n\\[\nU_\\lambda(2\\pi)\\lambda_I(x)U_\\lambda(2\\pi)^* = \\lambda_I(\\gamma(x))\n\\]\ndue to Prop. \\ref{vss}. So we may take $\\Gamma_\\lambda = U_\\lambda(2\\pi)$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nA \\emph{localised endomorphism} of ${\\mathcal A}$ is an endomorphism $\\rho$ of the universal $C^*$-algebra $C^*({\\mathcal A})$ such that $\\rho|_{{\\mathcal A}(I')}$ is the identity for some $I\\in{\\mathcal I}$ (then we say that $\\rho$ is localised in the interval $I$). In other words, $\\rho$ is a representation such that $\\rho_{I'}= \\iota $ and $\\rho_{\\tilde I}$ maps ${\\mathcal A}(\\tilde I)$ into itself if $\\tilde I\\supset I$. This last property is automatic by Haag duality in the local case, and we now see to hold also in the Fermi case.\n\nNext lemma shows that the grading is locally outer. This applies indeed to every gauge automorphism, \nsee \\cite{Car99,X01}.\n\\begin{lemma}\\label{outer}\nGiven any interval $I$, $\\gamma|_{{\\mathcal A}(I)}$ is an outer automorphism (unless the grading is trivial). As a consequence ${\\mathcal A}_b(I)'\\cap{\\mathcal A}(I)=\\mathbb C$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nSuppose $\\gamma|_{{\\mathcal A}(I)}$ is inner; then there exist unitaries $u\\in{\\mathcal A}(I)$ and $u'\\in{\\mathcal A}(I)'$ such that $\\Gamma =  u'u$. In fact $u$ and $u'$ are unique up to a phase factor because ${\\mathcal A}(I)$ is a factor. Now $\\Gamma$ commutes with the M\\\"obius unitary group, so $\\Delta_I^{it}u\\Omega = e^{iat}u\\Omega$ for some $a\\in\\mathbb R$. But $\\log\\Delta_I$ has no non-zero eigenvalue (see \\cite{DLR}), so $u\\Omega \\in\\mathbb C\\Omega$, thus $u$ is a scalar $\\gamma$ is the identity.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nThe following proposition generalizes to the Fermi case the well known DHR argument for the correspondence between representations and localised endomorphisms on the Minkowski space. \n\\begin{proposition}\\label{dhrendo}\nLet $\\pi$ be a representation of the Fermi conformal net ${\\mathcal A}$ on $S^1$, and suppose the Hilbert spaces ${\\mathcal H}$ and ${\\mathcal H}_\\pi$ to be separable. Given an interval $I$, there exists a localised endomorphism of ${\\mathcal A}$ unitarily equivalent to $\\pi$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nLet $\\Gamma$ be the unitary, $\\Omega$-fixing implementation of $\\gamma$. As $\\gamma|_{{\\mathcal A}(I')}$ is outer, the algebra $\\langle {\\mathcal A}(I'),\\Gamma\\rangle$ generated by ${\\mathcal A}(I')$ and $\\Gamma$ is the von Neumann algebra crossed product of ${\\mathcal A}(I')$ by $\\gamma|_{{\\mathcal A}(I')}$.\n\nNow $\\pi$ is graded and normal so, by choosing $\\Gamma_\\pi$ selfadjoint, also the algebra $\\langle \\pi_{I'}({\\mathcal A}(I')),\\Gamma_\\pi\\rangle$ generated by $\\pi_{I'}({\\mathcal A}(I'))$ and $\\Gamma_\\pi$ is naturally isomorphic to the von Neumann algebra crossed product of ${\\mathcal A}(I')$ by $\\gamma|_{{\\mathcal A}(I')}$.\n\nSo there exists an isomorphism $\\Phi:\\langle {\\mathcal A}(I'),\\Gamma\\rangle\\to\\langle \\pi_{I'}({\\mathcal A}(I')),\\Gamma_\\pi\\rangle$ such that $\\Phi|_{{\\mathcal A}(I')} =\\pi_{I'}$ and $\\Phi(\\Gamma)=\\Gamma_\\pi$, namely\n\\[\n\\Phi: x + y\\Gamma \\mapsto \\pi_{I'}(x) + \\pi_{I'}(y)\\Gamma_\\pi\\ ,\\quad x,y\\in{\\mathcal A}(I')\\ .\n\\]\nAs ${\\mathcal A}(I')$ is a type III factor, also $\\langle {\\mathcal A}(I'),\\Gamma\\rangle$ is a type III factor. Thus $\\Phi$ is spatial and there exists a unitary $U:{\\mathcal H}\\to{\\mathcal H}_\\pi$ such that $\\Phi(x) = U x U^*$ for all $x\\in{\\mathcal A}(I')$.\n\nSet $\\rho \\equiv U^*\\pi(\\cdot) U$. Clearly $\\rho$ is a representation of ${\\mathcal A}$ on ${\\mathcal H}$ that is unitarily equivalent to $\\pi$ and such that $\\rho_{I'}$ acts identically on ${\\mathcal A}(I')$. \n\nMoreover $\\rho\\cdot\\gamma = {\\hbox{\\rm Ad}}\\Gamma\\cdot \\rho$. Indeed, since $ U^*\\Gamma_\\pi= \\Gamma U^*$, we have\n\\[\n\\rho\\cdot\\gamma = {\\hbox{\\rm Ad}} U^*\\cdot\\pi\\cdot{\\hbox{\\rm Ad}}\\Gamma \n={\\hbox{\\rm Ad}} U^*\\Gamma_\\pi\\cdot\\pi  = {\\hbox{\\rm Ad}}\\Gamma U^*\\cdot\\pi ={\\hbox{\\rm Ad}}\\Gamma\\cdot\\rho \\ .\n\\]\nWith $x\\in{\\mathcal A}(I)$, we now want to show that $\\rho_I(x)\\in{\\mathcal A}(I)$. Indeed for all $y\\in{\\mathcal A}(I_0)$ with $\\bar I_0\\subset I'$ we have $[x,y]=0$, where the brackets denote the graded commutator. Therefore, choosing an interval $\\tilde I\\supset I'\\cup I_0$, we have\n\\begin{equation*}\n[\\rho_I(x),y] = [\\rho_{\\tilde I}(x),\\rho_{\\tilde I}(y)] = \\rho_{\\tilde I}([x,y]) = 0\\ .\n\\end{equation*}\nIt then follows by Cor. \\ref{td} that $\\rho_I(x)\\in{\\mathcal A}(I)$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{${\\sigma}$-Bose and ${\\sigma}$-Fermi sectors of the Bosonic subnet}\n\\label{sfb}\nAs above, let $\\gamma$ be the vacuum preserving, involutive grading automorphism of the Fermi net\n${\\mathcal A}$ on $S^1$ as above and ${\\mathcal A}_b$ the fixed-point subnet. \nWe denote by ${\\sigma}$ a representative of the sector of ${\\mathcal A}_b$ dual to $\\gamma$. Choosing \nan interval $I_0\\subset\\mathbb R$, there is a unitary \n\\[\nv\\in{\\mathcal A}(I),\\: v^*=v,\\: \\gamma(v)= -v \\ .\n\\]\nThen  we may take  ${\\sigma}\\equiv{\\hbox{\\rm Ad}} v |_{{\\mathcal A}_b}$, so ${\\sigma}$ is an automorphism of ${\\mathcal A}_b$ localised in \n$I_0$. We have $d({\\sigma})=1$ and ${\\sigma}^2 = 1$.\n\nGiven DHR endomorphisms $\\mu$ and $\\nu$ of ${\\mathcal A}_b$ \nwe denote by $\\varepsilon(\\mu,\\nu)$ the right (clockwise) statistics operator (see \\cite{R1,GL2}).\n\nThe \\emph{monodromy operator}  is given by\n\\[\nm(\\mu,\\nu)\\equiv\\varepsilon(\\mu,\\nu)\\varepsilon(\\nu,\\mu).\n\\]\nNote that if $\\mu$ is localised left to $\\nu$, then $\\varepsilon(\\nu,\\mu)=1$ \nand thus $m(\\mu,\\nu)=\\varepsilon(\\mu,\\nu)$.\n\nWe shall also set \n\\[\nK(\\mu,\\nu)\\equiv \\Phi_{\\nu}(m(\\nu,\\mu)^*)=\\Phi_{\\nu}(\\varepsilon(\\mu,\\nu)^*\\varepsilon(\\nu,\\mu)^*)\n\\]\nwhere $\\Phi_{\\nu}$ is the left inverse of $\\nu$. \nAs $\\varepsilon(\\mu,\\nu)\\in{\\hbox{Hom}}(\\mu\\nu,\\nu\\mu)$, we have $m(\\mu,\\nu)\\in {\\hbox{Hom}}(\\nu\\mu,\\nu\\mu)$, \ntherefore $K(\\mu,\\nu)\\in {\\hbox{Hom}}(\\mu,\\mu)$ and so, if $\\mu$ is irreducible, \n$K(\\mu,\\nu)$ is a complex number with modulus $\\leq 1$.\n \nLet $\\mu$ be an irreducible endomorphism localised left to ${\\sigma}$. \nAs $\\varepsilon(\\mu,{\\sigma})\\in{\\hbox{Hom}}(\\mu{\\sigma},{\\sigma}\\mu)$ and ${\\sigma}$ and $\\mu$ commute, it \nfollows that $\\varepsilon(\\mu,{\\sigma})$ is scalar. Denoting by $\\iota$ the identity \nsector, by the braiding fusion relation we have\n\\[\n1=\\varepsilon(\\mu,\\iota)=\\varepsilon(\\mu,{\\sigma}^2)\n={\\sigma}(\\varepsilon(\\mu,{\\sigma}))\\varepsilon(\\mu,{\\sigma})=\\varepsilon(\\mu,{\\sigma})\\varepsilon(\\mu,{\\sigma})\\ ,\n\\]\nthus $m(\\mu,{\\sigma})=\\varepsilon(\\mu,{\\sigma})=\\pm 1$.\n\nIf $\\mu$ is not necessarily irreducible, we shall say that $\\mu$ is \n\\emph{${\\sigma}$-Bose} \nif $m(\\mu,{\\sigma})= 1$ and that $\\mu$ is \\emph{${\\sigma}$-Fermi} \nif $m(\\mu,{\\sigma})= -1$. As we have seen, if $\\mu$ is irreducible then \n$\\mu$ is either ${\\sigma}$-Bose or ${\\sigma}$-Fermi. With $S$ Rehren matrix \\eqref{matS} we have:\n\\begin{proposition}\\label{DS}\nLet $\\rho,\\nu$ be irreducible, localized endomorphisms of ${\\mathcal A}_b$ and $\\rho'\\equiv \\rho{\\sigma}$. We have\n$S_{\\rho',\\nu} =\\pm S_{\\rho,\\nu}$ according with $\\rho$ is ${\\sigma}$-Bose or \n${\\sigma}$-Fermi. \n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy definition\n\\[\nS_{\\rho',\\nu}=S_{\\rho{\\sigma},\\nu}=\n\\frac{d(\\rho')d(\\nu)}{\\sqrt{\\mu_{\\mathcal A}}}\\Phi_{\\nu}(\\varepsilon(\\rho{\\sigma},\\nu)^*\\varepsilon(\\nu,\\rho{\\sigma})^*)\n\\]\nand we have $d(\\rho')=d(\\rho)$. We may also assume that $\\rho$, ${\\sigma}$ and $\\nu$ are localised one left to the next. We have\n\\[\nm(\\rho',\\nu)=\\varepsilon(\\rho{\\sigma},\\nu)=\\rho(\\varepsilon({\\sigma},\\nu))\\varepsilon(\\rho,\\nu) =\\pm \\varepsilon(\\rho,\\nu) = \\pm m(\\rho,\\nu)\n\\]\nas $\\varepsilon({\\sigma},\\nu)=m({\\sigma},\\nu)=\\pm 1$; thus $S_{\\rho',\\nu} =\\pm S_{\\rho,\\nu}$ where the sign depends on the ${\\sigma}$-Bose\/${\\sigma}$-Fermi alternative for $\\nu$.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nThus\n\\[\nD_{\\rho,\\nu} \\equiv S_{\\rho,\\nu} - S_{\\rho',\\nu}= \n2\\frac{d(\\rho)}{\\sqrt{\\mu_{{\\mathcal A}_b}}}\\cdot\\begin{cases} 0\\qquad &\\text{if $\\nu$ is ${\\sigma}$-Bose}\\\\\nK(\\rho,\\nu)d(\\nu)\\qquad &\\text{if $\\nu$ is ${\\sigma}$-Fermi}\n\\end{cases}\n\\]\n\\subsection{Graded tensor product} \nWe briefly recall the notion of graded tensor product of Fermi nets. With ${\\mathcal A}$ a Fermi net on $S^1$, denote by ${\\mathcal A}_f(I)$ the Fermi (i.e. degree one) subspace of ${\\mathcal A}(I)$. If $v\\in{\\mathcal A}_f(I)$ is a selfadjoint unitary then ${\\mathcal A}(I)= \\langle{\\mathcal A}_b(I),v\\rangle$ is the crossed product ${\\mathcal A}_b(I)\\rtimes \\mathbb Z_2$ with respect to the action ${\\sigma} ={\\hbox{\\rm Ad}} v$ on ${\\mathcal A}_b(I)$. If $W\\in{\\mathcal A}(I)'$ is a selfadjoint unitary, $W v$ also implements ${\\sigma}$ on ${\\mathcal A}_b(I)$ and the von Neumann algebra $\\langle{\\mathcal A}_b(I),W v\\rangle$ is also isomorphic to ${\\mathcal A}_b(I)\\rtimes \\mathbb Z_2$, namely we have an isomorphism\n\\[\na\\to a,\\quad f\\to W f, \\qquad \\partial a = 0, \\partial f = 1\\ .\n\\]\nLet now, for $i=1,2$, ${\\mathcal A}_i$ be a Fermi net on $S^1$ on the Hilbert \nspace ${\\mathcal H}_i$, and let $\\Gamma_i$ be the associated grading unitary. \nGiven an interval $I\\in{\\mathcal I}$, define the von Neumann algebra on ${\\mathcal H}_1\\otimes{\\mathcal H}_2$\n\\[\n{\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)\\equiv\n\\{a_1\\otimes a_2,\\  f_1\\otimes 1,\\  \\Gamma_1\\otimes f_2\\}''\n\\]\nwhere $a_i,f_i\\in{\\mathcal A}_i(I)$, $\\partial a_i =0$ and $\\partial f_i =1$. Namely ${\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I) $ is the direct sum\n\\[\n\\underbrace{{\\mathcal A}_{1b}(I)\\otimes{\\mathcal A}_{2b}(I) + \\Gamma_1{\\mathcal A}_{1f}(I)\\otimes{\\mathcal A}_{2f}(I)}_{\\rm Bosons}\n+ \\underbrace{{\\mathcal A}_{1f}(I)\\otimes {\\mathcal A}_{2b}(I) + \\Gamma_1 {\\mathcal A}_{1b}(I)\\otimes{\\mathcal A}_{2f}(I)}_{\\rm Fermions}\n\\]\nClearly the map $I\\to {\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)$ is isotone and satisfies graded\nlocality with respect to the grading induced by $\\Gamma_1 \\otimes \\Gamma_2$. \nThus it defines a Fermi net ${\\mathcal A}_1 \\hat\\otimes {\\mathcal A}_2$ on ${\\mathcal H}_1\\otimes {\\mathcal H}_2$. \n\nBy the previous comments, with ${\\mathcal A}(I) = 1 \n\\otimes {\\mathcal A}_2(I)$ and $W=\\Gamma_1\\otimes 1$, the von Neumann algebra $\\hat{\\mathcal A}_2(I)$ on \n${\\mathcal H}_1\\otimes{\\mathcal H}_2$ generated by $1\\otimes{\\mathcal A}_{2b}(I)$ and $\\Gamma_1\\otimes {\\mathcal A}_{2f}(I)$ \nis isomorphic to $1\\otimes{\\mathcal A}_2(I)$ and hence to ${\\mathcal A}_2(I)$. Actually, an easy \ncalculation show that \nif $a, f \\in {\\mathcal A}_2(I)$, $\\partial a = 0, \\partial f = 1$, \n$Z\\equiv \\frac{1-i\\Gamma_1\\otimes\\Gamma_2}{1-i}$ and \n $Z_2\\equiv \\frac{1-i\\Gamma_2}{1-i}$ then\n$$(1\\otimes Z_2)Z^*\\big(1\\otimes (a+f)\\big)Z(1\\otimes Z_2)^*= 1\\otimes a + \n\\Gamma_1\\otimes f.$$\nHence the unitary operator $(1\\otimes Z_2)Z^*$ on ${\\mathcal H}_1\\otimes{\\mathcal H}_2$ implements \nthe isomorphism of $1\\otimes{\\mathcal A}_2(I)$ onto $\\hat{\\mathcal A}_2(I)$ for every interval $I$. \n  \nObviously $\\hat{\\mathcal A}_1(I)\\equiv {\\mathcal A}_1(I)\\otimes 1$ is isomorphic to ${\\mathcal A}_1(I)$. Moreover \nwe have \n\\begin{equation}\n\\label{gradedtensor}\n{\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I) = \\hat{\\mathcal A}_1(I)\\vee\\hat{\\mathcal A}_2(I),\\quad [\\hat{\\mathcal A}_1(I),\\hat{\\mathcal A}_2(I)] = 0\\ ,\n\\end{equation}\nwhere the brackets denote the graded commutator corresponding.\n\nOne can check that, up to isomorphism, the graded tensor product \n${\\mathcal A}_1(I)\\hat\\otimes{\\mathcal A}_2(I)$ is the unique von Neumann algebra generated by copies\n $\\hat{\\mathcal A}_1(I)$ of ${\\mathcal A}_1(I)$ and $\\hat{\\mathcal A}_2(I)$ of ${\\mathcal A}_2(I)$, having \na grading that restricts to the grading of $\\hat{\\mathcal A}_i(I)$, $i=1,2$,\nsatisfying the relations \\eqref{gradedtensor}, and such that that ${\\mathcal A}_{1b}(I)\\vee{\\mathcal A}_{2b}(I)$ is the usual tensor product of von Neumann algebras.\n\nWe can then define the graded tensor product of DHR representations. \nIf $\\lambda_1$ and $\\lambda_2$ are DHR representations of ${\\mathcal A}_1$ and ${\\mathcal A}_2$ \nrespectively and if $\\lambda_1$ is graded, then it can be shown that there \nexists a (necessarily unique) representation $\\lambda_1 \\hat\\otimes \\lambda_2$ on \n${\\mathcal H}_{\\lambda_1} \\otimes {\\mathcal H}_{\\lambda_2}$ such that, for every interval \n$I\\subset S^1$,  \n\\[\n(\\lambda_1 \\hat\\otimes \\lambda_2)_I (x_1\\otimes a_2+\n\\Gamma_1x_1 \\otimes f_2) = {\\lambda_1}_I(x_1)\\otimes {\\lambda_2}_I(a_2)\n+\\Gamma_{\\lambda_1}{\\lambda_1}_I(x_1)\\otimes {\\lambda_2}_I(f_2)\n\\]\nwhere $x_1\\in {\\mathcal A}_1(I)$, $a_2,f_2\\in{\\mathcal A}_2(I)$, $\\partial a_2 =0$ and $\\partial \nf_2=1$. \nAnalogously we can \ndefine the graded tensor product of general representations defined below, \nprovided one of them is graded.\n\\section{Nets on $\\mathbb R$ and on a cover of $S^1$}\nBesides nets on $S^1$, it will be natural to consider nets on $\\mathbb R$ and nets on topological covers of $S^1$. Indeed the two notions are related as we shall see.\n\\subsection{Nets on $\\mathbb R$}\n\\label{netR}\nDenote by ${\\mathcal I}_{\\mathbb R}$ the family of open intervals of $\\mathbb R$, \ni.e. of open, non-empty, connected, bounded subsets of $\\mathbb R$. \n\nThe M\\\"obius group ${\\rm\\textsf{M\\\"ob}}$ can be naturally viewed as a subgroup of ${\\mathrm {Diff}}(S^1)$. We then have a corresponding inclusion ${\\rm\\textsf{M\\\"ob}}^{(n)}\\subset{\\mathrm {Diff}}^{(n)}(S^1)$ of covering groups. In the following we denote by $G$ a group that can be either the M\\\"obius group ${\\rm\\textsf{M\\\"ob}}$  or the diffeomorphism group ${\\mathrm {Diff}}(S^1)$. Analogously, we have $G^{(n)}={\\rm\\textsf{M\\\"ob}}^{(n)}$ or\n$G^{(n)}={\\mathrm {Diff}}^{(n)}(S^1)$.\nBy identifying $\\mathbb R$ with $S^1\\!\\smallsetminus\\!\\{-1\\}$ via the stereographic \nmap, we have a local action of $G$ on $\\mathbb R$\n(where $SL(2,\\mathbb R)\/\\{1,-1\\}\\simeq{\\rm\\textsf{M\\\"ob}}$ acts on $\\mathbb R$ by linear fractional transformations). See \\cite{BGL,GL5} for the definition and  a discussion about local actions. \n\nA \\emph{ $G$-covariant net on $\\mathbb R$} (of von Neumann algebras) ${\\mathcal A}$ is a isotone map \n\\[\nI\\in{\\mathcal I}_{\\mathbb R}\\mapsto{\\mathcal A}(I)\n\\]\nthat associates to each $I\\in{\\mathcal I}_{\\mathbb R}$ a von Neumann algebra ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}$ and there exists a projective, positive energy, unitary representation $U$ of $G^{(\\infty)}$ on \n${\\mathcal H}$ with\n\\[\nU(g){\\mathcal A}(I)U(g)^{-1} = {\\mathcal A}(\\dot{g} I), \\quad g\\in{\\mathcal U}_I,\n\\]\nfor every $I\\in{\\mathcal I}_{\\mathbb R}$ where ${\\mathcal U}_I$ is the connected component of the identity in $G^{(\\infty)}$ of the open set $\\{g\\in G^{(\\infty)}: gI\\in{\\mathcal I}_{\\mathbb R}\\}$.\n\nWe will further assume the irreducibility of ${\\mathcal A}$ and the existence of a vacuum vector $\\Omega$ for $U$, although this is not always necessary.\n\nNote that it would be enough to require the existence of the projective unitary representation $U$ only in a neighbourhood of the identity of $G^{(\\infty)}$, then $U$ would extend to all $G^{(\\infty)}$ by multiplicativity.\n\nSince the cohomology of the Lie algebra of ${\\rm\\textsf{M\\\"ob}}$ is trivial, \nby multiplying $U(g)$ by a phase factor (in a unique fashion), we may remove the \nprojectiveness of $U|_{{\\rm\\textsf{M\\\"ob}}^{(\\infty)}}$ and assume that the restriction of $U$ to ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$\nis a unitary representation of ${\\rm\\textsf{M\\\"ob}}^{(\\infty)}$. \n\\subsection{Nets on a cover of $S^1$}\n\\label{coverNets}\nThe group $G^{(n)}$ has a natural action on $S^{1(n)}$, $n$ finite or infinite, the one obtained by promoting the action of $G$ on $S^1$, see Sect. \\ref{top}. Here $S^{1(n)}$ denotes the $n$-cover of $S^1$. Denote by ${\\mathcal I}^{(n)}$ the family of intervals of $S^{1(n)}$, i.e. $I\\in{\\mathcal I}^{(n)}$ iff $I$ is a connected subset of $S^{1(n)}$ that projects onto a (proper) interval of the base $S^1$.\n\nA \\emph{$G$-covariant net ${\\mathcal A}$ on $S^{1(n)}$} is a isotone map \n\\[\nI\\in{\\mathcal I}^{(n)}\\mapsto{\\mathcal A}(I)\n\\]\nthat associates with each $I\\in{\\mathcal I}^{(n)}$ a von Neumann algebra ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}$, and there exists a projective\nunitary, positive energy representation $U$ of $G^{(\\infty)}$ on \n${\\mathcal H}$, implementing a  covariant action on ${\\mathcal A}$, namely \n\\[\nU(g){\\mathcal A}(I)U(g)^{-1} = {\\mathcal A}(\\dot{g}I),\\quad  I\\in {\\mathcal I}^{(n)},\\ \ng\\in G^{(\\infty)}\n\\]\nHere $\\dot{g}$ denotes the image of $g$ in $G^{(n)}$ under the quotient map.\n\nAs above, we may also assume irreducibility of the net and the existence of a vacuum vector $\\Omega$ for $U$.\n\nOf course a $G$-covariant net $\\tilde{\\mathcal A}$ on $S^{1(n)}$ determines a $G$-covariant net $\\tilde{\\mathcal A}$ on $\\mathbb R$. Indeed, with $p:S^{1(\\infty)}\\to S^1$ the covering map, every connected component of \n$p^{-1}(S^{1}\\!\\smallsetminus\\!\\{{\\rm point}\\})$\n is a copy of \n$\\mathbb R$ in $S^{1(n)}$ and we may restrict $\\tilde{\\mathcal A}$ to any of this copy; by $G$-covariance we get always the same $G$-covariant net on $\\mathbb R$, up to unitary equivalence.\n\nConversely, a $G$-covariant net ${\\mathcal A}$ on $\\mathbb R$ can be extended to a net\n$\\tilde{\\mathcal A}$ on $S^{1(\\infty)}$ by defining, for any given $I\\in{\\mathcal I}$,\n\\begin{equation}\\label{gext}\n\\tilde{\\mathcal A}(I)\\equiv U(g){\\mathcal A}(I_1)U(g)^{-1}\n\\end{equation}\nwhere $I_1\\in{\\mathcal I}_{\\mathbb R}$, $g\\in G^{(\\infty)}$, and $gI_1 = I$. Here \nthe action of $G^{(\\infty)}$ on $S^{1(\\infty)}$ is the one obtained by \npromoting the action of $G^{(\\infty)}$ on $S^1$ (see above). \nTherefore we have:\n\\begin{proposition}\\label{rtos}\nThere is a 1-1 correspondence (up to unitary equivalence) between $G$-covariant nets on $\\mathbb R$ and $G$-covariant  nets on $S^{1(\\infty)}$\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe only show that if ${\\mathcal A}$ is a $G$-covariant net on $\\mathbb R$, then formula \\eqref{gext} well defines $\\tilde{\\mathcal A}(I)$. If $g'\\in G^{(\\infty)}$ also satisfies $g'I_1 = I$, then $g$ and $g'$ have ``the same degree\", namely $h\\equiv g^{-1}g'$ maps $I_1$ onto $I_1$ and is in ${\\mathcal U}_{I_1}$. \nThen $U(h){\\mathcal A}(I_1)U(h)^{-1}= {\\mathcal A}(I_1)$, so $U(g){\\mathcal A}(I_1)U(g)^{-1}= U(g'){\\mathcal A}(I_1)U(g')^{-1}$. \nThe rest is clear.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nWith ${\\mathcal A}$ a be a $G$-covariant net on $\\mathbb R$ as above, we shall say that a unitary operator $V$ on ${\\mathcal H}$ is a gauge unitary if \n\\begin{equation}\n\\label{gauge}\nV{\\mathcal A}(I)V^* ={\\mathcal A}(I), \\ VU(g)=U(g)V,\n\\end{equation}\nfor all $I\\in{\\mathcal I}_{\\mathbb R}$ and $g$ in a suitable neighbourhood of \nthe identity of ${\\rm\\textsf{M\\\"ob}}$. Clearly the same relations \\eqref{gauge} then \nhold true for all $I\\inS^{1(\\infty)}$, $g\\in{\\rm\\textsf{M\\\"ob}}^{(\\infty)}$ for the corresponding net \non $S^{1(\\infty)}$. Now ${\\rm rot}_{2\\pi}$ is a \ncentral element of $G^{(\\infty)}$ and it is immediate to check that if $n\\in\\mathbb \nN$: \n\\[\nU(2\\pi)^n\\ \\text{is a gauge unitary}\\Leftrightarrow\\text{${\\mathcal A}$ extends to a \n$G$-covariant net on $S^{1(n)}$} ;\n\\]\nthen ${\\mathcal A}$ is covariant with respect to the corresponding action of \n$G^{(\\infty)}$ on $S^{1(n)}$. Clearly the net on $S^{1(n)}$ is the projection of the \nnet on $S^{1(\\infty)}$ by the covering map from $S^{1(\\infty)}$ to $S^{1(n)}$. In other words we have the following where $n\\in\\mathbb N$:\n\\begin{corollary}\nA $G$-covariant net ${\\mathcal A}$ on $\\mathbb R$ is the restriction of a $G$-covariant net on $S^{1(n)}$ if and only if $U(2n\\pi)$ is a gauge unitary for ${\\mathcal A}$. This is the case, in particular, if the representation $U$ of $G^{(\\infty)}$ factors through a representation of $G^{(n)}$ (i.e. $U(2n\\pi) = 1$).\n\\end{corollary}\n\\noindent\nIf ${\\mathcal A}$ is a $G$-covariant net on $\\mathbb R$ we shall denote by ${\\mathcal A}^{(\\infty)}$ its promotion to $S^{1(\\infty)}$. If $U(2\\pi)^n$ is a gauge unitary for some $n\\in\\mathbb N$, we shall denote by ${\\mathcal A}^{(n)}$ the promotion of ${\\mathcal A}$ to $S^{1(n)}$.\n\nWe now define the \\emph{promotion} of a net ${\\mathcal A}$ on $S^1$ to a net ${\\mathcal A}^{(n)}$ on $S^{1(n)}$. If ${\\mathcal A}$ is a $G$-covariant net on $S^1$, then its restriction ${\\mathcal A}_0$ to $\\mathbb R$ is a $G$-covariant net on $\\mathbb R$, and we set\n\\[\n{\\mathcal A}^{(n)}\\equiv {\\mathcal A}_0^{(n)}\\ ;\n\\]\nhere, if $n$ is finite, we have to assume that $U(2\\pi n)$ is a gauge unitary.\nWe shall be mainly interested in the case of a $G$-covariant Fermi net ${\\mathcal A}$ on $S^1$. As $U(4\\pi)=1$ in this case, we have a natural net ${\\mathcal A}^{(2)}$ on $S^{1(2)}$ associated with ${\\mathcal A}$. Of course if ${\\mathcal A}$ is local then $U(2\\pi) =1$ the net ${\\mathcal A}^{(2)}$ on $S^{1(2)}$ is defined in this case. \n\nOf course if ${\\mathcal A}^{(n)}$  is the promotion to $S^{1(n)}$ of a net ${\\mathcal A}$ on $S^1$, then  ${\\mathcal A}^{(n)}(I)={\\mathcal A}(pI)$ for any interval $I$ of $S^{1(n)}$.\n\n\n\\section{Solitons and representations of cover nets}\nWe now consider more general representations associated with a Fermi conformal net. The point is that a Fermi conformal nets lives naturally in a double cover of $S^1$ rather than on $S^1$ itself  because the $2\\pi$ rotation unitary $U(2\\pi)$ is not the identity ($U(2\\pi)=\\Gamma$, see Sect. \\ref{vacss}) but $U(4\\pi)=1$. So representations as a net on $S^{1(2)}$ come naturally into play. These representations can be equivalently viewed as a natural class of solitions.\n\\subsection{Representations of a net on $S^{1(n)}$}\nWe begin by giving the notion of representation for a net on a cover of $S^1$.\n\nLet ${\\mathcal A}$ be a $G$-covariant net of von Neumann algebras on $S^{1(n)}$ and $U$ the associated covariance unitary representation of of $G^{(\\infty)}$ (thus $U(2\\pi n)$ is a gauge unitary). A \\emph{representation} of ${\\mathcal A}$ is a map\n\\[\nI\\in{\\mathcal I}^{(n)}\\mapsto \\lambda_I\n\\]\nwhere $\\lambda_I$ is a normal representation of ${\\mathcal A}(I)$ on a fixed Hilbert space ${\\mathcal H}_\\lambda$ with the usual consistency condition $\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} = \\lambda_I$ if $\\tilde I\\supset I$. \n\nWe shall say that $\\lambda$ is \\emph{$G$-covariant} if there exists a projective unitary, positive energy representation $U_\\lambda$ of $G^{(\\infty)}$ on ${\\mathcal H}_\\lambda$ such that\n\\[\nU_\\lambda(g)\\lambda_I(x)U_\\lambda(g)^* = \\lambda_{\\dot{g}I}(U(g)xU(g)^*),\\ g\\in G^{(\\infty)},\\ I\\in{\\mathcal I}^{(n)}\\ .\n\\]\nHere $\\dot g$ denotes the image of $g\\in G^{(\\infty)}$ in $G^{(n)}$ under the quotient map, thus $\\dot{g}I$ is the projection of $gI\\inS^{1(\\infty)}$ onto $S^{1(n)}$.\n \nLet now ${\\mathcal A}$ be a $G$-covariant net on $S^1$. If $\\lambda$ is a representation of ${\\mathcal A}$, then $\\lambda$ promotes to a representation $\\lambda^{(n)}$ of ${\\mathcal A}^{(n)}$  given by\n\\[\n\\lambda^{(n)}_I \\equiv \\lambda_{p(I)},\\quad I\\in{\\mathcal I}^{(n)}\\ .\n\\]\nIf $\\lambda$ is $G$-covariant and $U_\\lambda$ is the associated unitary representation of $G^{(n)}$, then $\\lambda^{(n)}$ is also $G$-covariant with the same unitary representation $U_\\lambda$ of $G^{(n)}$.\n\\begin{lemma}\\label{prom}\nLet ${\\mathcal A}$ a $G$-covariant Fermi (resp. local) net on $S^1$ and $\\nu$ a $G$-covariant representation of ${\\mathcal A}^{(n)}$, with $U_\\nu$ the associated projective unitary representation of $G^{(n)}$.\n\nThen $\\nu$ is the promotion $\\lambda^{(n)}$ of a $G$-covariant representation $\\lambda$ of ${\\mathcal A}$ iff $U_\\nu(2\\pi)$ implements the grading (resp. the identity) in the representation $\\nu$. \n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe assume ${\\mathcal A}$ to be a Fermi net (the local case is simpler and follows by the same argument). If $\\lambda$ is a $G$-covariant representation of ${\\mathcal A}$, then $U_\\lambda(2\\pi)$ implements the grading by the spin-statistics theorem \\cite{GL2} (applied to $\\lambda |_{{\\mathcal A}_b}$). Conversely, if $\\nu$ is a $G$ covariant representation of ${\\mathcal A}^{(n)}$ and $U_\\nu(2\\pi)$ implements the grading, let $\\lambda_0$ be the restriction of $\\nu$ to a copy of $\\mathbb R$ in $S^{1(n)}$. Then $\\lambda_0$ extends by $G$-covariance to a representation of ${\\mathcal A}$ (Prop. \\ref{rtos}). \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{Solitons}\nLet ${\\mathcal A}_0$ be a net of von Neumann algebras on $\\mathbb R$ and denote by $\\bar{\\mathcal I}_{\\mathbb R}$ the family of intervals\/half-lines of $\\mathbb R$ (open connected subsets of $\\mathbb R$ different from $\\varnothing , \\mathbb R$). If $I$ is a half-line, we define ${\\mathcal A}(I)$ as the von Neumann algebras generated by the von Neumann algebras associated to the intervals contained in $I$.\n\nA \\emph{soliton} $\\lambda$ of ${\\mathcal A}_0$ a map\n\\[\nI\\in\\bar{\\mathcal I}_{\\mathbb R}\\mapsto \\lambda_I\n\\]\nwhere $\\lambda_I$ is a normal representation of the von Neumann algebra ${\\mathcal A}_0(I)$ on a fixed Hilbert space ${\\mathcal H}_\\lambda$ with the usual isotony property $\\lambda_{\\tilde I}|_{{\\mathcal A}(I)} =\\lambda_I$, $I\\subset\\tilde I$.\n\nIf ${\\mathcal A}$ is a net on $S^1$, by a soliton of ${\\mathcal A}$ we shall mean a soliton of the restriction of ${\\mathcal A}$ to $\\mathbb R$. \n\nLet ${\\mathcal A}$ be a conformal net on $S^1$. We set\n\\[\n\\text{${\\mathcal A}_0\\ \\equiv\\ $ restriction of ${\\mathcal A}$ to $\\mathbb R$}. \n\\]\nIf $\\lambda$ is a DHR representation of ${\\mathcal A}$ then, obviously, $\\lambda |_{{\\mathcal A}_0}$ is a soliton of ${\\mathcal A}_0$. We shall say that a soliton $\\lambda_0$ of ${\\mathcal A}$ is a DHR representation of ${\\mathcal A}$ if it arises in this way, namely $\\lambda_0=\\lambda |_{{\\mathcal A}_0}$ with $\\lambda$ a DHR representation of ${\\mathcal A}$ (in other words, $\\lambda |_{{\\mathcal A}_0}$ extends to a DHR representation of ${\\mathcal A}$, note that the extension is unique if strong additivity is assumed).\n\nMore generally, we get solitons of ${\\mathcal A}$ by restricting to ${\\mathcal A}_0$ representations of the cover nets ${\\mathcal A}^{(n)}$.\n\nLet ${\\mathcal A}$ be a $G$-covariant net on $S^1$ with covariance unitary representation $U$. A \\emph{$G$-covariant soliton} $\\lambda$ of ${\\mathcal A}$ is a soliton of ${\\mathcal A}_0$ such that there exists a projective unitary representation $U_\\lambda$ of $G^{(\\infty)}$ such that for every bounded interval $I$ of $\\mathbb R$ we have\n\\[\n\\lambda_{\\dot g I}(U(g)xU(g)^*) = U_\\lambda (g)\\lambda_I(x)U_\\lambda (g)^*\\ ,\\quad  g\\in{\\mathcal U}_I ,\\ x\\in{\\mathcal A}(I)\\ ,\n\\]\nwhere ${\\mathcal U}_I$ is a connected neighborhood of the identity of $G^{(\\infty)}$ as in Sect. \\ref{netR} and $\\dot g$ is the image of $g$ in $G$.\n\\begin{proposition}\\label{r-si}\nLet ${\\mathcal A}$ be a $G$-covariant net on $S^1$.\nThere is a one-to-one correspondence between\n\\begin{itemize}\n\\item[$(a)$] $G$-covariant representations of ${\\mathcal A}^{(\\infty)}$,\n\\item[$(b)$] $G$-covariant solitons of ${\\mathcal A}$.\n\\end{itemize}\nThe correspondence is given by restricting a representation of ${\\mathcal A}^{(\\infty)}$ to a copy of ${\\mathcal A}_0$.\n\nSuppose ${\\mathcal A}$ is local. Then the above restricts to a one-to-one correspondence between $(a)$: $G$-covariant representations of ${\\mathcal A}^{(n)}$ and $(b)$: $G$-covariant solitons of ${\\mathcal A}$ with $U_\\lambda(2\\pi n)$ commuting with $\\lambda$.\n\nSuppose ${\\mathcal A}$ is Fermi. Then the above restricts to a one-to-one correspondence between $(a)$: $G$-covariant representations of ${\\mathcal A}^{(n)}$ and $(b)$: $G$-covariant solitons of ${\\mathcal A}$ with $U_\\lambda(2\\pi n)$ implementing the grading ($n$ even) or commuting with $\\lambda$ ($n$ odd).\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nClearly if $\\lambda$ is a $G$-covariant representation of ${\\mathcal A}^{(\\infty)}$ then $\\lambda_0\\equiv\\lambda |_{{\\mathcal A}_0}$ is a $G$-covariant solitons of ${\\mathcal A}$. Conversely, if $\\lambda_0$ is $G$-covariant solitons of ${\\mathcal A}$ then we set\n\\[\n\\lambda_{g I}(U(g)xU(g)^*) = U_\\lambda (g)\\lambda_I(x)U_\\lambda (g)^*\\ ,\\quad  g\\in G^{(\\infty)} ,\\ x\\in{\\mathcal A}(I)\\ ,\n\\]\nwhere $I\\subset\\mathbb R$ is a bounded interval for any given copy of $\\mathbb R$ is in $S^{1(\\infty)}$ and $G^{(\\infty)}$ acts on $S^{1(\\infty)}$ as usual . By  $G$-covariance the above formula well-define a representation of ${\\mathcal A}^{(\\infty)}$.\n\nThe second part follows because by the vacuum spin-statistics relation we have $U(2\\pi)=1$ in the local case and $U(2\\pi)=\\Gamma$ in the Fermi case.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nA \\emph{graded soliton} of the Fermi net ${\\mathcal A}$ is, of course, a soliton such that the grading is unitarily implemented, namely the soliton $\\lambda$ is graded iff there exists a unitary $\\Gamma_\\lambda$ on ${\\mathcal H}_\\lambda$ such that\n\\[\n\\lambda_I(\\gamma(x)) = \\Gamma_\\lambda \\lambda(x)\\Gamma^*_\\lambda,\\quad I\\in{\\mathcal I}_{\\mathbb R},\\ x\\in{\\mathcal A}(I).\n\\]\n\\subsection{Neveu-Schwarz and Ramond representations}\nWe give now the general definition for a representation of a Fermi net. We have two formulations for this notion: as a representation of the cover net and as a soliton.\n\nLet ${\\mathcal A}$ be a Fermi net on $S^1$. A \\emph{general representation} $\\lambda$ of ${\\mathcal A}$ is a representation the cover net of ${\\mathcal A}^{(\\infty)}$ such that $\\lambda$ restricts to a DHR representation $\\lambda_b$ of the Bose subnet ${\\mathcal A}_b$.\n\nWe shall see here later on that a general representation is indeed a representation of ${\\mathcal A}^{(2)}$. We begin by noticing an automatic covariance for a general representation $\\lambda$. In this case $\\lambda$ is not always graded.\n\\begin{proposition}\\label{diffcov0}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$. Every general representation $\\lambda$ of ${\\mathcal A}$ is diffeomorphism covariant with positive energy. Indeed $U_\\lambda = U_{\\lambda_b}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe proof follows by an immediate extension of the argument proving Prop. \\ref{DHRgraded}.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nLet ${\\mathcal A}$ be a Fermi net on $S^1$. A \\emph{general soliton} $\\lambda$ of ${\\mathcal A}$ is a  diffeomorphism covariant soliton of ${\\mathcal A}$ such that $\\lambda$ restricts to a DHR representation $\\lambda_b$ of the Bose subnet ${\\mathcal A}_b$. (In other words $\\lambda|_{{\\mathcal A}_{b,0}}$ extends to a DHR representation $\\lambda_b$ of ${\\mathcal A}_b$.)\n\nBy Prop. \\ref{r-si} we have a one-to-one correspondence between\n\\begin{gather*}\n\\text{General Representations}\\\\\n\\Updownarrow\\\\\n\\text{General Solitons}\n\\end{gather*}\nWe prove now the diffeomorphism covariance of general solitons in the strong additive case. Notice that strong additivity is automatic in the completely rational case \\cite{LX}. The general proof follows by Prop. \\ref{acov} below.\n\\begin{proposition}\\label{diffcov}\nLet ${\\mathcal A}$ be a  Fermi conformal net on $S^1$. Every soliton $\\lambda$ of ${\\mathcal A}$ such that $\\lambda_b$ is a DHR representation is diffeomorphism covariant with positive energy, namely $\\lambda$ is a general soliton. Indeed $U_\\lambda = U_{\\lambda_b}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\n(Assuming strongly additive). Note that if $g\\in{\\mathrm {Diff}}(S^1)$ is localized in an interval $I$ then, up to a phase, $U_{\\lambda_b}(g) = \\lambda_b(U(g))\\in{\\mathcal A}_b(I)$.\n\nLet $I$ be an interval contained in $\\mathbb R = S^1\\smallsetminus\\{-1\\}$ a suppose that $g\\in{\\mathrm {Diff}}(S^1)$ is localised in $I$; then\n\\begin{multline*}\nU_{\\lambda_b}(g)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(g) = {\\lambda_b}_{\\tilde I}(U(g))\\lambda_{\\tilde I}(x){\\lambda_b}_{\\tilde I}(U^*(g))\n= {\\lambda}_{\\tilde I}(U(g))\\lambda_{\\tilde I}(x){\\lambda}_{\\tilde I}(U^*(g))\\\\\n= \\lambda_{\\tilde I}\\big(U(g)xU(g)^*\\big)\n= \\lambda_{g\\tilde I}\\big(U(g)xU(g)^*\\big)\\ ,\n\\quad x\\in{\\mathcal A}(\\tilde I)\\ ,\n\\end{multline*}\nfor any interval $\\tilde I\\supset I$ of $\\mathbb R$. \n\nNotice at this point that if $x\\in{\\mathcal A}(\\tilde I)$ and $v\\in{\\mathcal A}_b(\\tilde I')$ then ${\\lambda_b}_{\\tilde I'}(v)$ commutes with $\\lambda_{\\tilde I}(x)$:\n\\[\n[{\\lambda_b}_{\\tilde I'}(v),\\lambda_{\\tilde I}(x)]=0\\ .\n\\]\nIndeed, as ${\\mathcal A}$ is assumed to be strongly additive, also ${\\mathcal A}_b$ is strongly additive \\cite{X01}, see also \\cite{L03}, so it is sufficient to show the above relation with $v\\in{\\mathcal A}_b(I_0)$,  where $I_0$ is any interval with closure contained in $\\tilde I'\\smallsetminus \\{-1\\}$. Then\n\\begin{equation*}\n[{\\lambda_b}_{\\tilde I'}(v),\\lambda_{\\tilde I}(x)] =[{\\lambda_b}_{I_0}(v),\\lambda_{\\tilde I}(x)]\n= [{\\lambda}_{I_0}(v),\\lambda_{\\tilde I}(x)] \n= [{\\lambda}_{I_1}(v),\\lambda_{I_1}(x)] \n= {\\lambda}_{I_1}([v,x]) = 0\\ ,\n\\end{equation*}\nwhere we have taken an interval $I_1$ of $\\mathbb R$ containing $I_0\\cup\\tilde I$.\n\nLet now ${\\mathcal U}_I$ be a connected neighbourhood of the identity in ${\\mathrm {Diff}}(S^1)$ such that $gI$ is an interval of $\\mathbb R$ for all $g\\in{\\mathcal U}_I$. Take $g\\in{\\mathcal U}_I$ and let $\\tilde I$ be an interval of $\\mathbb R$ with $\\tilde I\\supset I\\cup gI$. Let $h\\in {\\mathrm {Diff}}(S^1)$ be a diffeomorphism localised in an interval of $\\mathbb R$ containing $\\tilde I$ such that $h|_{\\tilde I} = g|_{\\tilde I}$. Then\n\\[\nU(g) = U(h)v\n\\]\nwhere $v\\equiv U(h^{-1}g )\\in {\\mathcal A}_b(\\tilde I')$ (the Virasoro subnet is contained in the Bose subnet).\n\nWe then have with $x\\in{\\mathcal A}(\\tilde I)$:\n\\begin{multline*}\nU_{\\lambda_b}(g)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(g) =\nU_{\\lambda_b}(h){\\lambda_b}_{\\tilde I'}(v)\\lambda_{\\tilde I}(x){\\lambda_b}_{\\tilde I'}(v^*)U^*_{\\lambda_b}(h) \\\\\n= U_{\\lambda_b}(h)\\lambda_{\\tilde I}(x)U^*_{\\lambda_b}(h)\\ .\n=\\lambda_{h\\tilde I}\\big(U(h)xU(h)^*\\big)\n=\\lambda_{g\\tilde I}\\big(U(g)xU(g)^*\\big)\n\\end{multline*}\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{proposition}\\label{n1}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and $\\lambda$ an irreducible \ngeneral soliton of ${\\mathcal A}$. With $\\lambda_{b,0}\\equiv\\lambda |_{{\\mathcal A}_{b,0}}$, the following are equivalent:\n\\begin{itemize}\n\\item[$(i)$] $\\lambda$ is graded,\n\\item[$(ii)$] $\\lambda_{b,0}$ is not irreducible,\n\\item[$(iii)$] $\\lambda_{b,0}$ is direct sum of two inequivalent irreducible representations \nof ${\\mathcal A}_{b,0}$:\n\\[\n\\lambda_{b,0} \\simeq \\rho\\oplus \\rho'\n\\]\nwhere $\\rho$ is an irreducible DHR representation of ${\\mathcal A}_{b,0}$ and \n$\\rho'\\equiv \\rho{\\sigma}$ with ${\\sigma}$ is the dual involutive localised automorphism of ${\\mathcal A}_b$ as above. \\end{itemize}\nIf the above holds and $\\lambda$ (i.e. $\\rho$) has finite index\nwe then have equation for the statistics phases\n\\begin{equation}\\label{mp1}\n\\omega_{\\rho'} = - m(\\rho,{\\sigma})\\omega_\\rho\n\\end{equation}\nThus\n\\begin{equation}\\label{mp2}\nU_{\\rho'}(2\\pi) = \\mp U_\\rho(2\\pi)\n\\end{equation}\naccording to $\\rho$ is ${\\sigma}$-Bose\/${\\sigma}$-Fermi.    \n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\n$(i)\\Rightarrow (iii)$: If $\\lambda$ is graded, then the unitary $\\Gamma_\\lambda$ implementing the grading in the representation commutes with $\\lambda({\\mathcal A}_{b,0})$, so $\\lambda_{b,0}$ is reducible. Moreover, as $\\lambda({\\mathcal A}_{b,0})''= \\lambda({\\mathcal A}_0)''\\cap\\{\\Gamma_\\lambda\\}'$ , we have\n$\\lambda({\\mathcal A}_{b,0})'= \\lambda({\\mathcal A}_0)'\\vee\\{\\Gamma_\\lambda\\}''=\\{\\Gamma_\\lambda\\}''$, so $\\lambda({\\mathcal A}_{b,0})'$ is 2-dimensional. \nLet $\\rho$ one of the two irreducible components of $\\lambda_{b,0}$. As the dual canonical endomorphism associated with ${\\mathcal A}_{b,0}\\subset{\\mathcal A}_0$ is $\\iota\\oplus{\\sigma}$ (see \\cite{LR1}), the other component must be $\\rho'\\equiv\\rho{\\sigma}$, namely $\\lambda_{b,0}=\\rho\\oplus\\rho'$.\n\nIn order to show eq. \\eqref{mp2} we may assume that $\\rho$ is localized left to ${\\sigma}$, so $\\rho$ and ${\\sigma}$ commute. By using the cocycle equations for the statistics operators, we then have\n\\begin{multline*}\n\\varepsilon(\\rho',\\rho') = \\varepsilon(\\rho,\\rho')\\rho(\\varepsilon({\\sigma},\\rho')) \n= \\varepsilon(\\rho,\\rho')\\varepsilon({\\sigma},\\rho') \n= \\rho(\\varepsilon(\\rho,{\\sigma}))\\varepsilon(\\rho,\\rho)\\rho(\\varepsilon({\\sigma},{\\sigma}))\\varepsilon({\\sigma},\\rho) \\\\\n= \\varepsilon(\\rho,{\\sigma})\\varepsilon(\\rho,\\rho)\\varepsilon({\\sigma},{\\sigma})\\varepsilon({\\sigma},\\rho)\n= - m(\\rho,{\\sigma})\\varepsilon(\\rho,\\rho)\\ ,\n\\end{multline*}\nwhere we have used that $\\varepsilon({\\sigma},\\rho')$ and $\\varepsilon(\\rho,{\\sigma})$ are scalars and  $\\varepsilon({\\sigma},{\\sigma})= -1$. So, if $\\rho$ has finite index, we infer the equation \\eqref{mp1} for the statistics phases and eq. \\eqref{mp2} follows by the conformal spin-statistics theorem \\cite{GL2}.\n\nThe implication $(iii)\\Rightarrow (ii)$ is obvious. For the implication $(ii)\\Rightarrow (i)$ assume that $\\lambda$ is not graded, namely $\\gamma$ is not unitarily implemented in the representation $\\lambda$; then $\\lambda$ is irreducible by known arguments, see \\cite{Ro} (the fixed point of an irreducible $C^*$-algebra with respect to a period two outer automorphism is still irreducible).\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\noindent\n{\\it Remark.} In the above proposition the diffeomorphism covariance of $\\lambda$ is unnecessary, only the M\\\"obius covariance of $\\lambda_b$ has been used.\n\\begin{corollary}\\label{extrep}\nLet ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and $\\lambda$ an irreducible general soliton of ${\\mathcal A}$ with finite index.\nThe following alternative holds:\n\\begin{itemize}\n\\item[$(a)$] $\\lambda$ is a DHR representation of ${\\mathcal A}$. Equivalently $U_{\\lambda_b}(2\\pi)$ is not a scalar.\n\\item[$(b)$] $\\lambda$ is the restriction of a representation of ${\\mathcal A}^{(2)}$ and $\\lambda$ is not a DHR representation of ${\\mathcal A}$. Equivalently $U_{\\lambda_b}(2\\pi)$ is a scalar.\n\\end{itemize}\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nFirst note that $U_{\\lambda_b}(4\\pi) = U_{\\rho}(4\\pi)\\oplus U_{\\rho'}(4\\pi)$ is a scalar by eq. \\eqref{mp2}, so $\\lambda$ is always representation of ${\\mathcal A}^{(2)}$ by Prop. \\ref{r-si}.\n\nIf $\\lambda$ is a DHR representation of ${\\mathcal A}$ then by, Prop. \\ref{DHRgraded}, $U_\\lambda(2\\pi) =\\Gamma$ is not a scalar.\n\nAssume now that $\\lambda$ is not a DHR representation of ${\\mathcal A}$. It remains to show that if $U_\\lambda(2\\pi)$ is a scalar. \n\nIf $\\lambda$ is not graded, then by Prop. \\ref{n1} $\\lambda_b$ is irreducible; as $U_\\lambda(2\\pi)$ commutes with $\\lambda_b$, must then be a scalar.\n\nFinally, if $\\lambda$ is graded, then $\\rho$ must be a ${\\sigma}$-Fermi sector, as otherwise its $\\alpha$-induction $\\alpha^+_\\rho$ to ${\\mathcal A}$ would be a DHR representation of ${\\mathcal A}$, while $\\alpha^+_\\rho= \\lambda$ by Frobenious reciprocity. Then by eq.  \\eqref{mp2} we have $U_{\\rho'}(2\\pi) = U_\\rho(2\\pi)$, namely $U_\\lambda(2\\pi)$ is a scalar.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n{\\it Remark.} The finite index assumption in the above proposition is probably unnecessary (it has been used to make use of eq.  \\eqref{mp2}).\n\\medskip\n\n\\noindent\nIt is now convenient to use the following terminology concerning the general soliton in Corollary \\ref{extrep}. In the first case, namely if $\\lambda$ is a DHR representation of ${\\mathcal A}$, we say that $\\lambda$ is a \\emph{Neveu-Schwarz representation} of ${\\mathcal A}$. In the second case, namely if $\\lambda$ is a soliton (and not a representation of ${\\mathcal A}$) we say that $\\lambda$ is a \\emph{Ramond representation} of ${\\mathcal A}$. \n\nIf $\\lambda$ is reducible we shall say that $\\lambda$ is a Neveu-Schwarz representation of ${\\mathcal A}$ if $\\lambda$ is a DHR representation of ${\\mathcal A}$; we shall say that $\\lambda$ is a Ramond representation of ${\\mathcal A}$ if no subrepresentation of $\\lambda$ is a DHR representation of ${\\mathcal A}$.\n\nLet as above ${\\mathcal A}$ be a Fermi conformal net on $S^1$ and ${\\mathcal A}_b$ the Bose subnet. We shall now study the representations of ${\\mathcal A}_b$ in relation to the representations of ${\\mathcal A}$.\n\nGiven a representation $\\nu$ of ${\\mathcal A}_b$ we consider its $\\alpha$-induction $\\alpha_\\nu$ to ${\\mathcal A}$ (say right $\\alpha$-induction, so $\\alpha_\\nu\\equiv\\alpha^+_\\nu$). More precisely we first restrict $\\nu$ to the net ${\\mathcal A}_{b,0}$ on the real line and then consider its $\\alpha$-induction $\\alpha_\\nu$ which is a soliton of ${\\mathcal A}_0$. Note that $\\alpha_\\nu$ is defined in \\cite{LR1} assuming Haag duality on the real line (equivalent to strong additivity in the conformal case), but this assumption is unnecessary here because we are considering nets on $S^1$ that satisfy Haag duality on $S^1$, see \\cite[Sect. 3.1]{LX}.\n\\begin{proposition}\\label{acov}\nIf $\\nu$ is a DHR irreducible representation of ${\\mathcal A}_b$ then $\\alpha_\\nu$ is a general soliton of ${\\mathcal A}$, namely $\\alpha_\\nu$ is diffeomorphism covariant with positive energy. We have:\n\\begin{gather*}\n\\text{$\\nu$ is ${\\sigma}$-Bose} \\Leftrightarrow \\text{$\\alpha_\\nu$ is Neveu-Schwarz}\\\\\n\\text{$\\nu$ is ${\\sigma}$-Fermi} \\Leftrightarrow \\text{$\\alpha_\\nu$ is Ramond}\n\\end{gather*}\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nWe only sketch the proof.\nWe use the extension method by Roberts (see \\cite{Ro}), but in a covariant way.\nAs $\\nu$ is diffeomorphism covariant, there is a localized unitary cocycle $w$ with value in ${\\mathcal A}_b$ such that\n\\begin{equation}\\label{ce}\n {\\hbox{\\rm Ad}} w_g\\cdot \\nu = {\\hbox{\\rm Ad}} U(g)\\cdot \\nu\\cdot{\\hbox{\\rm Ad}} U(g^{-1}),\\quad g\\in{\\mathrm {Diff}}(S^1),\n\\end{equation}\nsee \\cite[Sect. 8]{GL1}. The cocycle $w$ reconstructs $\\nu$. Now $w$ can be seen as a cocycle with values in ${\\mathcal A}$ but with less localization properties (if $w$ is bi-localized in $I\\cup gI$ in ${\\mathcal A}_b$, it is in general only localized in $\\tilde I$ in ${\\mathcal A}$ where $\\tilde I$ is an interval containing $I\\cup gI$). Thus, in general,  $w$ can be associated with a soliton of ${\\mathcal A}$, which is $\\alpha_\\nu$. Now the covariance equation still remains true for $g$ in a neighborhood of the identity, and gives the diffeomorphism covariance of $\\alpha_\\nu$.\n\nIf $\\nu$ a DHR irreducible representation of ${\\mathcal A}_b$, its $\\alpha$-induction to ${\\mathcal A}$ is a DHR representation iff $\\nu$ has trivial monodromy with the dual canonical endomorphism $\\iota\\oplus{\\sigma}$ of ${\\mathcal A}_b$ (the restriction to ${\\mathcal A}_b$ of the identity representation of ${\\mathcal A}$), thus iff $\\nu$ has trivial monodromy with ${\\sigma}$. By definition this means that $\\nu$ is a ${\\sigma}$-Bose representation.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nAn immediate corollary is the following.\n\\begin{corollary}\nA  DHR representation $\\nu$ of ${\\mathcal A}_b$ is ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) if and only if $\\nu$ is the restriction of a Neveu-Schwarz (resp. Ramond) representation of ${\\mathcal A}$.\n\\end{corollary}\n\\noindent\nBecause of the above corollary, we shall also call Neveu-Schwarz (resp. Ramond) representation of ${\\mathcal A}_b$ a ${\\sigma}$-Bose (resp. ${\\sigma}$-Fermi) representation of ${\\mathcal A}_b$.\n\\subsection{Tensor categorical structure. Jones index}\nIn the local case, the DHR argument shows that every representation is equivalent to a localized endomorphism and a first basic consequence of this fact is that representations give rise to a tensor category because one can indeed compose localized endomorphisms and get the monoidal structure. As we have seen in Prop. \\ref{dhrendo}, we can extend the DHR argument to the case of representations of a Fermi net; this argument however makes use that DHR representations are automatically graded. In order to have a tensor structure for general representations of a Fermi net, we consider graded representations only. Note however that if $\\lambda$ is any soliton, setting where $\\lambda_\\gamma\\equiv\\gamma\\lambda\\gamma^{-1}$, then $\\lambda\\oplus\\lambda_\\gamma$ is graded.\nWe obviously have\n\\[\n{\\rm Sol}_\\gamma({\\mathcal A})\\subset{\\rm Sol}({\\mathcal A})\\ .\n\\]\nwhere ${\\rm Sol}({\\mathcal A})$ denotes the general solitons of ${\\mathcal A}$ that are localized, say, in a right half line of $\\mathbb R$ and ${\\rm Sol}_\\gamma({\\mathcal A})$ are the general solitons commuting with the grading, namely $\\lambda\\in{\\rm Sol}({\\mathcal A})$ belongs to  ${\\rm Sol}_\\gamma({\\mathcal A})$ iff $\\lambda=\\lambda^\\gamma$.\n\nNow every general representation of ${\\mathcal A}$ is equivalent to an element of ${\\rm Sol}({\\mathcal A})$. Thus, if $\\lambda_1,\\lambda_2\\in{\\rm Sol}({\\mathcal A})$ an intertwiner $T:\\lambda_1\\to\\lambda_2$ is a bounded linear operator such that\n\\[\nT\\lambda_1(x)=\\lambda_2(x)T,\\quad x\\in{\\mathcal A}_0\\ .\n\\]\nNote that $T$ is not necessarily localized in a half-line, but by twisted duality $T\\in Z{\\mathcal A}(I)Z^*$ where $I$ is a half-line where $\\lambda_1$ and $\\lambda_2$ are both localized. By further assuming that $T$ commutes with $\\Gamma$, we also have that  $T$ commutes with $Z$, so $T\\in{\\mathcal A}(I)$ and $\\partial T = 0$, thus $T\\in{\\mathcal A}_b(I)$.\n\nIt follows that ${\\rm Sol}_\\gamma({\\mathcal A})$ is a tensor category where the arrows are defined by\n\\[\n{\\rm Hom}(\\lambda_1,\\lambda_2)\\equiv T:\\lambda_1\\to\\lambda_2,\\ T\\Gamma =\\Gamma T\\ .\n\\]\n\\begin{proposition}\n$\\alpha$-induction is a surjective tensor functor \n\\[\n{\\rm Rep}({\\mathcal A}_b)\\overset{\\alpha}{\\longrightarrow}{\\rm Sol}_\\gamma({\\mathcal A})\\ .\n\\]\nwhere ${\\rm Rep}({\\mathcal A}_b)$ is the tensor category of endomorphisms of ${\\mathcal A}_b$ localized in intervals of $\\mathbb R$. The kernel of $\\alpha$ is $\\{\\iota,{\\sigma}\\}$.\n\\end{proposition}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nFirst of all note that if $\\nu\\in{\\rm Rep}({\\mathcal A}_b)$ is localized in the interval $I$, then $\\alpha_\\nu$ commutes with $\\gamma$. Indeed both $\\alpha_\\nu$ and $\\alpha^\\gamma_\\nu= \\gamma\\alpha_\\nu\\gamma$ have the same restriction to ${\\mathcal A}_b$. If $v\\in{\\mathcal A}(I)$ is a selfadjoint unitary with $\\partial v = 1$, then $\\alpha_\\nu(v) = uvu^*$ for a Bose unitary $u$ in the covariance cocycle for $\\nu$, therefore \n\\begin{equation*}\n\\alpha^\\gamma_\\nu(v) =\\gamma(\\alpha_\\nu(\\gamma(v)))= -\\gamma(\\alpha_\\nu(v))\n= -\\gamma(uvu)  = -\\gamma(u)\\gamma(v)\\gamma(u) = uvu =\\alpha_\\nu(v)\\ ,\n\\end{equation*}\nso $\\alpha=\\alpha^\\gamma_\\nu$ because ${\\mathcal A}$ is generated by ${\\mathcal A}_b$ and $v$.\n\nConversely, if $\\lambda\\in{\\rm Sol}_\\gamma({\\mathcal A})$, then $\\lambda_b$ is a localized endomorphism of ${\\mathcal A}_b$ because $\\lambda$ commutes with $\\gamma$. Now the covariance cocycle $w_g$ for $\\lambda$ has degree $0$; indeed $\\gamma(w_g)$ is also a covariance cocycle for $\\lambda$ thus $\\gamma(w_g)=w_g$ up to a phase that must be 1 by the cocycle property. Thus $w$ belongs to the Bose subnet and is the covariance cocycle for $\\lambda_b$; a calculation as above then shows that $\\lambda = \\alpha_{\\lambda_b}$.\n\nIt is now rather immediate to show that $\\alpha_\\nu$ is the identity iff $\\nu =\\iota$ or $\\nu={\\sigma}$ and that we have a corresponding surjective map of the arrows.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nSo we have a braided tensor category ${\\rm Sol}_\\gamma({\\mathcal A})$. In particular, the \\emph{Jones index} of $\\lambda$ is defined, as in the local case, see \\cite{L5,L5'}. \n\nNote however that an irreducible element of ${\\rm Sol}_\\gamma({\\mathcal A})$ may be reducible as a general representation of ${\\mathcal A}$, namely it can decomposed into non-graded subrepresentations.\n\\section{Modular and superconformal invariance}\nWe now study Fermi nets whose Bose subnets are modular and get some consequences in the supersymmetric case.\n\\subsection{Modular nets}\nWe recall here a discussion made in \\cite{KL05}.\nLet ${\\mathcal B}$ be a completely rational local conformal net of von Neumann algebras on $S^1$. Then the tensor category or representations of ${\\mathcal B}$ is modular, i.e. rational with non-degenerate unitary braiding  \\cite{KLM}.\n\nWe then have Rehren \\cite{R1} matrices defining\na unitary representation of the group $SL(2,{\\mathbb Z})$ on the space spanned by the irreducible sectors (i.e. unitary equivalence classes of representations) $\\rho$'s , in particular we have the matrices $T \\equiv \\{T_{\\lambda,\\nu}\\}$ and $S \\equiv \\{S_{\\lambda,\\nu}\\}$ where\n\\begin{equation}\\label{matS}\nS_{\\lambda,\\nu}\\equiv \n\\frac{d(\\lambda)d(\\nu)}{\\sqrt{\\sum_i d(\\rho_i)^2}}\\Phi_{\\nu}(\\varepsilon(\\nu,\\lambda)^*\\varepsilon(\\lambda,\\nu)^*) \\ .\n\\end{equation}\nHere $\\varepsilon(\\nu,\\lambda)$ is the right statistics operators between $\\lambda$\nand $\\nu$, $\\Phi_{\\nu}$ is left inverse of $\\nu$ and $d(\\lambda)$ is the \nstatistical dimension of $\\lambda$.  \n\nAssume that ${\\mathcal B}$ is diffeomorphism covariant with central charge $c$.\nFor a sector $\\rho$, we consider the specialised character\n$\\chi_\\rho(\\tau)$ for complex numbers $\\tau$ with ${\\rm Im}\\; \\tau >0$\nas follows:\n\\[\n\\chi_\\rho(\\tau)=\\Tr\\!\\big(e^{2\\pi i\\tau (L_{0,\\rho}-c\/24)}\\big).\n\\]\nHere the operator $L_{0,\\rho}$ is the conformal Hamiltonian in \nthe representation $\\rho$.  We also assume that the above trace converges, and so each eigenspace \nof $L_{0,\\rho}$ is finite dimensional.  \n\nIn many cases the $T$ and $S$ matrices give an action\nof $SL(2,{\\mathbb Z})$ on the linear span of these specialised\ncharacters through change of variables $\\tau$ as follows:\n\\begin{equation}\\label{modularity}\n\\begin{split}\n\\chi_\\rho(-1\/\\tau)&= \\sum_\\nu S_{\\rho,\\nu} \\chi_\\nu(\\tau),\\\\\n\\chi_\\rho(\\tau+1)&= \\sum_\\nu T_{\\rho,\\nu} \\chi_\\nu(\\tau).\n\\end{split}\n\\end{equation}\nThis is always the case if the Rehren matrices $S$ and $T$ coincide with the  \nso called Kac-Peterson or Verlinde matrices.\nThe Kac-Peterson-Verlinde matrix $T$ and Rehren matrix $T$ always coincide\nup to a phase by the spin-statistics theorem \\cite{GL2}.\n\nWe shall say that ${\\mathcal B}$ is \\emph{modular} if  the Rheren matrices give the modular \ntransformations of specialized character as in Eq. (\\ref{modularity}) .  \nNote that we are assuming that ${\\mathcal B}$ is completely rational, namely the $\\mu$-index is finite \n(see below) and the split property holds. However, as we are assuming\n$\\Tr(e^{-tL_{0}})<\\infty$ the split property follows, see \\cite{BDL}. \n(Strong additivity follows from  \\cite{LX}.)\n\nModularity holds in all computed rational cases, cf. \\cite{X3,X4}. \nThe $SU(N)_k$ nets and the Virasoro nets ${\\rm Vir}_c$ with $c<1$ are\nboth modular. We expect all local conformal completely rational \nnets to be modular (see \\cite{Hu} for results of similar kind). \nFurthermore, if ${\\mathcal B}\n$ is a modular local conformal net and ${\\mathcal C}\n$ an irreducible extension of ${\\mathcal B}\n$, then ${\\mathcal C}\n$ is also modular \\cite{KL05}, that allows to check \nthe modularity property in several cases.\n\nIf ${\\mathcal B}\n$ is modular, the Kac-Wakimoto formula holds\n\\begin{equation}\\label{KW1}\nd(\\rho)=\\frac{S_{\\rho,0}}{S_{0,0}}=\n\\lim_{\\tau\\to i\\infty}\\frac{\\sum_\\nu S_{\\rho,\\nu}\\chi_\\nu(\\tau)}\n{\\sum_\\nu S_{0,\\nu}\\chi_\\nu(\\tau)}=\n\\lim_{\\tau\\to 0}\\frac{\\chi_\\rho(\\tau)}{\\chi_0(\\tau)}.\n\\end{equation}\nHere we denote the vacuum sector $\\iota$ also by $0$.\n\nNow, if ${\\mathcal B}$ is a modular net, then ${\\mathcal B}$ is two-dimensional \nlog-elliptic with noncommutative area $a_0 = 2\\pi c\/24$ \\cite{KL05}, \nindeed the following asymptotic formula holds:\n\\[\n\\log\\Tr(e^{-2\\pi tL_{0,\\rho}})\\sim \\frac{\\pi c}{12}\\frac1t + \n\\frac{1}{2}\\log\\frac{d(\\rho)^2}{\\mu_{{\\mathcal B}\n}}-\n\\frac{\\pi c}{12}t\\qquad {\\rm as }\\ t\\to 0^+ \\ .\n\\]\nwhere $\\rho$ a representation of ${\\mathcal B}$ and $L_{0,\\rho}$ is the conformal \nHamiltonian in the representation $\\rho$.\n\n\\subsection{Fermi nets and modularity}\nAssume that $\\lambda$ is a graded irreducible general soliton of the Fermi conformal net ${\\mathcal A}$ \non $S^1$. Then $\\lambda$ is the restriction of a representation of the double cover ${\\mathcal A}^{(2)}$ of \n${\\mathcal A}$ and is diffeomorphism covariant. We denote by $H_{\\lambda}$ the conformal Hamiltonian of \n${\\mathcal A}$ in the representation $\\lambda$. \n\nThen $H_{\\lambda}$ is affiliated to the Virasoro subnet in the representation $\\lambda$, which is \ncontained in the Bosonic subnet, so $H_{\\lambda}$\ncommutes with $\\Gamma_{\\lambda}$ and thus respects the graded decomposition \n${\\mathcal H}_\\lambda = {\\mathcal H}_{\\lambda,+}\\oplus{\\mathcal H}_{\\lambda,-}$ given by $\\Gamma_\\lambda$; we then have a unitary equivalence\n\\[\nH_{\\lambda} \\simeq L_{0,\\rho}\\oplus L_{0,\\rho'}\n\\]\nwhere $L_{0,\\rho}$ and $L_{0,\\rho'}$ are the conformal Hamiltonians of ${\\mathcal A}_b$ \nin the  representations $\\rho$ and $\\rho'$, where $\\lambda_b = \\rho\\oplus\\rho'$. \nConsequently\n\\begin{equation}\\label{strp}\n\\Str(e^{-tH_{\\lambda}}) = \\Tr(e^{-tL_{0,\\rho}}) - \\Tr(e^{-tL_{0,\\rho'}})\n\\end{equation}\nWe also set\n\\[\n\\tilde H_{\\lambda}\\equiv H_{\\lambda} - c\/24,\\quad\\tilde L_{0,\\rho} \\equiv \nL_{0,\\rho} - c\/24\\dots\n\\]\nIf ${\\mathcal A}_b$ is modular we then have by formula\n\\begin{align}\\label{m1}\n\\Str(e^{-2\\pi t{\\tilde H_{\\lambda}}}) =&\\sum_\\nu\nS_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{\\rho,\\nu}\/t}) -\\sum_\\nu \nS_{\\rho',\\nu}\\Tr(e^{-2\\pi\\tilde L_{\\rho',\\nu}\/t})\\\\\n=& \\sum_\\nu\n(S_{\\rho,\\nu} - S_{\\rho',\\nu})\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t})\\\\\n=& \\sum_\\nu\nD_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t})\n\\end{align}\nwhere $D_{\\rho,\\nu}\\equiv S_{\\rho,\\nu} - S_{\\rho',\\nu}$ as before and, by Prop. \\ref{DS},  $D_{\\rho,\\nu}= 2S_{\\rho,\\nu}$ if $\\nu$ is ${\\sigma}$-Fermi, and zero otherwise.\n\n\\subsection{Fredholm and Jones index within superconformal invariance}\nWe shall say that a general representation $\\lambda$ of the Fermi local conformal net ${\\mathcal A}$ is \n\\emph{supersymmetric}\\footnote{A characteristic feature of supersymmetry is also that the graded derivation implemented by $Q_\\lambda$ is densely defined in a appropriate sense. We shall not need this property in this paper.}\nif $\\lambda$ is graded and the conformal Hamiltonian $H_{\\lambda}$ satisfies\n\\[\n\\tilde H_{\\lambda} \\equiv H_{\\lambda} - c\/24 = Q_{\\lambda}^2\n\\]\nwhere $Q_{\\lambda}$ is a selfadjoint on ${\\mathcal H}_{\\lambda}$ which is odd w.r.t. the grading unitary $\\Gamma_{\\lambda}$. \n\nLet then $\\lambda$ be supersymmetric. An immediate important consequence is a positive lower bound for the energy:\n\\[\nH_{\\lambda}\\geq c\/24\\ .\n\\]\nNote that by McKean-Singer lemma $\\Str(e^{-t(H_{\\lambda} - c\/24)})$ is constant thus, taking the limit as $t\\to\\infty$, we have\n\\begin{equation}\\label{Windex}\n\\Str(e^{-t(H_{\\lambda} - c\/24)}) = {\\hbox{dim}}\\ker(H_{\\lambda} - c\/24)\\ ,\n\\end{equation}\nthe multiplicity of the lowest eigenvalue $c\/24$ of $H_{\\lambda}$.\n\nLet $\\rho$ be one of the two irreducible components of $\\lambda_b$ as above. We denote by $\\mathfrak R$ the set of ${\\sigma}$-Fermi irreducible sectors of ${\\mathcal A}_b$, that correspond to the Ramond sectors of ${\\mathcal A}$, see Prop. \\ref{acov}. Note that $\\mathfrak R \\neq\\varnothing$ as otherwise ${\\sigma}$ would be a degenerate sector, which is not possible because the braided tensor category of DHR \nsectors in modular as a consequence of the complete rationality assumption.\n\nAssume now that ${\\mathcal A}_b$ is modular. We notice that formula \\eqref{m1} can be written as\n\\begin{equation}\\label{expmin}\n\\Str(e^{-2\\pi t\\tilde H_{\\lambda}}) = 2\\sum_{\\nu\\in\\mathfrak R}\nS_{\\rho,\\nu}\\Tr(e^{-2\\pi\\tilde L_{0,\\nu}\/t}) \\ .\n\\end{equation}\n\\begin{corollary} We have\n\\[\n\\sum_{\\nu\\in\\mathfrak R } S_{\\rho,\\nu}d(\\nu) = 0\n\\]\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nDivide by $\\Tr(e^{-2\\pi tL_0})$ both members of (\\ref{expmin}); \nthe statement then follows by the Kac-Wakimoto formula \\eqref{KW1}.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{sublemma}\nLet $A_{1}, A_{2},\\dots A_{n}$ be selfadjoint operators such that $\\Tr(e^{-sA_k})<\\infty$  and $\\sum_{k}c_{k}\\Tr(e^{-sA_k})$ is a constant function of $s>0$, for some scalars $c_{k}$. Then \n\\[\n\\lim_{s\\to +\\infty}\\sum_{k}c_{k}\\Tr(e^{-sA_k})=\\sum_{k}c_{k}\\Dim\\ker A_k\n\\]\n\\end{sublemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nClearly $\\lim_{s\\to +\\infty}\\Tr(e^{-sA})=\\Dim\\ker A$ if $A$ is trace class and non-negative. The finite trace condition implies that $A_k$ bounded below.\nLet $A_k = A^{+}_k + A^-_k$ with $A^{+}_k\\geq 0$ and $A^{-}_k <0$. Then the function $\\sum_{k}c_{k}\\Tr(e^{-sA^{-}_k})$ is a linear combination of exponentials of the form $e^{as}$ with $a>0$, and vanishes at infinity, thus it must be identically zero. It follows that\n\\[\n\\sum_{k}c_{k}\\Tr(e^{-sA_k})=\\sum_{k}c_{k}\\Tr(e^{-sA^+_k})\\underset{s=\\infty}{\\longrightarrow}\n\\sum_{k}c_{k}\\Dim\\ker A^+_k = \\sum_{k}c_{k}\\Dim\\ker A_k\n\\]\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\begin{lemma}\\label{ex1}\nLet ${\\mathcal A}$ be a Fermi modular net. If $\\lambda$ is supersymmetric then \n\\begin{equation}\\label{lim}\n\\Str(e^{-2\\pi t\\tilde H_\\lambda}) = 2\\sum_{\\nu\\in\\mathfrak R} S_{\\rho,\\nu}\\fn(\\nu,c\/24)\n\\end{equation}\nwhere $\\fn(\\nu,h)\\equiv\\Dim\\ker(L_{0,\\nu}- h)$.\n\\end{lemma}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nThe left hand side of  \\eqref{expmin} is constant by McKean-Singer lemma, so we have that\n$\\sum_{\\nu\\in\\mathfrak R}S_{\\rho,\\nu}\\Tr(e^{-2\\pi s\\tilde L_{0,\\nu}})$ is a constant function of \n$s>0$. \nThus  \\eqref{lim} holds by the sublemma.\n\\null\\hfill\\qed\\endtrivlist\\noindent\nWith $\\lambda$ as in the above lemma, by eq. \\eqref{Windex} we have\n\\[\n\\Str(e^{-2\\pi t\\tilde H_{\\lambda}}) = \\ind(Q_{\\lambda +}\n)\\ .\n\\]\nTherefore by Lemma \\ref{ex1} we have\n\\[\n\\ind(Q_{\\lambda +}\n) = 2\\sum_{\\nu\\in\\mathfrak R}  S_{\\rho,\\nu}\\fn(\\nu,c\/24)\n\\]\nthen, writing Rehren definition of the $S$ matrix, we have\n\\[\n\\ind(Q_{\\lambda +}\n) = 2\\frac{d(\\rho)}{\\sqrt{\\mu_{{\\mathcal A}_b}}}\\sum_{\\nu\\in\\mathfrak R}K(\\rho,\\nu)\nd(\\nu)\\fn(\\nu,c\/24)\n\\]\nwhere $\\mu_{{\\mathcal A}_b}$ is the $\\mu$-index of ${\\mathcal A}_b$. By \\cite{KLM} we have $\\mu_{{\\mathcal A}_b} = 4\\mu_{{\\mathcal A}}$  therefore:\n\\begin{theorem}\\label{FJ}\nLet ${\\mathcal A}$ be a Fermi conformal net as above and $\\lambda$ \na supersymmetric irreducible representation of ${\\mathcal A}$. Then\n\\[\n\\ind(Q_{\\lambda +}\n) = \n\\frac{d(\\rho)}{\\sqrt{\\mu_{\\mathcal A}}}\\sum_{\\nu\\in\\mathfrak R}K(\\rho,\\nu)\nd(\\nu)\\fn(\\nu,c\/24)\n\\]\nwhere $\\rho$ is one of the two irreducible components of $\\lambda_b$ and $\\mu_{\\mathcal A}$ is the $\\mu$-index of ${\\mathcal A}$.\n\\end{theorem}\n\\noindent\nIn the above formula the Fredholm index of the supercharge operator $Q_{\\lambda +}\n$ is expressed \nby a formula involving the Jones index of the Ramond representations whose lowest eigenvalue $c\/24$ modulo integers.\n\\begin{corollary} If $\\ind(Q_{\\lambda})\\neq 0$ there exists a ${\\sigma}$-Fermi sector $\\nu$ such that  $c\/24$  is an eigenvalue of $L_{0,\\nu}$.\n\\end{corollary}\n\\begin{corollary} Suppose that, in Th. \\ref{FJ}, $\\rho$ is the only Ramond sector with lowest eigenvalue $c\/24$ modulo integers. Then \n\\[\nS_{\\rho,\\rho} = \\frac{d(\\rho)^2}{\\sqrt{\\mu_{{\\mathcal A}_b}}}K(\\rho,\\rho) = \\frac12 \\ .\n\\]\n\\end{corollary}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }]\nBy Lemma \\ref{ex1} we have:\n\\[\n\\ind(Q_{\\lambda +}\n) = 2S_{\\rho,\\rho}\\fn(\\rho,c\/24)\\ .\n\\]\nOn the other hand by formula \\eqref{Windex} \n\\[\n\\ind(Q_{\\lambda +}\n) = {\\hbox{dim}}\\, {\\rm ker}(H_\\lambda - c\/24) = \\fn(\\rho,c\/24) + \\fn(\\rho',c\/24)=  \\fn(\\rho,c\/24) \n\\]\nbecause $H_\\lambda = L_{0,\\rho}\\oplus L_{0,\\rho'}$ and $c\/24$ is not in the spectrum of $L_{0,\\rho'}$, so we get our formula.\n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\section{Super-Virasoro algebra and super-Virasoro nets}\nWe now focus on model analysis and shall consider the most basic superconformal nets, namely the ones associated with the super-Virasoro algebra.\n\\subsection{Super-Virasoro algebra}\nThe super-Virasoro algebra governs the superconformal invariance \\cite{FQS,GKO}. It plays in the supersymmetric context the same r\\^ole that the Virasoro algebra  plays in the local conformal case.\n\nStrictly speaking, there are two super-Virasoro algebras. They are the super-Lie algebras generated by even \nelements $L_n$, $n\\in\\mathbb Z$, odd elements $G_r$, and a central even element $c$, \nsatisfying the relations\n\\begin{gather}\\label{svirdef}\n    [L_m , L_n] = (m-n)L_{m+n} + \\frac{c}{12}(m^3 - m)\\delta_{m+n, 0}\\\\\n    \\nonumber\n    [L_m, G_r] = (\\frac{m}{2} - r)G_{m+r}\\\\\n    \\nonumber\n    [G_r, G_s] = 2L_{r+s} + \\frac{c}{3}(r^2 - \\frac14)\\delta_{r+s,0}\n    \\end{gather}\nHere the brackets denote the super-commutator. \nIn the \\emph{Neveu-Schwarz} case $r\\in\\mathbb Z + 1\/2$, while in the \n\\emph{Ramond} case $r\\in\\mathbb Z$. We shall sometime use the term \n\\emph{super-Virasoro algebra} to indicate either the Neveu-Schwarz \nalgebra or the Ramond algebra. \n\nThe point is that, although the Neveu-Schwarz algebra and the  Ramond algebra are not isomorphic graded Lie algebras, they are representations of a same object, in a sense that we shall later see (they have the same ``isomorphic completion\").\n\nBy definition, the Neveu-Schwarz algebra and the Ramond algebra are both\nextensions of the Virasoro algebra. \n\nThe super-Virasoro algebra is equipped with the involution $L^*_n = \nL_{-n}$, $G^*_r = G_{-r}$, $c^* = c$ and we will be only interested in unitary \nrepresentations on a Hilbert space,\ni.e. representations preserving the involution. Note that unitary representations have automatically positive energy, namely $L_0\\geq 0$. Indeed we have \n\\begin{gather*}\n  L_0 =  \\frac12[G_{\\frac12}, G_{-\\frac12}] = \\frac12\\left(G_{\\frac12}G_{\\frac12}^* +\n  G_{\\frac12}^*G_{\\frac12}\\right)\\geq 0\\qquad (\\text{NS case})\\\\\n  L_0 =  \\frac12[G_0, G_0] = G_0^2 + c\/24 \\geq c\/24\n\\qquad (\\text{R case})    \n\\end{gather*}\nThe unitary, lowest weight representations of the super-Virasoro \nalgebra, namely the unitary representations of the super-Virasoro \nalgebra on a Hilbert space ${\\mathcal H}$ with a cyclic vector \n$\\xi\\in {\\mathcal H}$ satisfying\n\\[\nL_0\\xi = h\\xi, \\quad L_n\\xi = 0, \\ n>0, \\quad G_r\\xi = 0,\\ r>0,\n\\]\nare studied in \\cite{FQS,GKO}; in the NS case they are irreducible and uniquely determined \nby the values of $c$ and $h$. In the Ramond case one has to further specify the action of $G_0$ on the lowest energy subspace. It turns out that for a possible value of $c$ and $h$ there two inequivalent irreducible lowest weight representations (but for the case $c = h\/24$ when the representation is unique and graded). Not also that for $c \\neq h\/24$ the direct sum of the two inequivalent irreducible representations has a cyclic lowest energy vector and it is the unique Ramond $(c,h)$ lowest weight representation where grading is implemented.\n\nThe possible values are either $c\\geq 3\/2$, $h\\geq 0$ or\n\\begin{equation}\\label{c-values}\nc = \\frac32 \\left( 1 - \\frac{8}{m(m+2)}\\right), \\ m=2,3,\\ldots\n\\end{equation}\nand\n\\[\nh= h_{p,q}(c)\\equiv\\frac{[(m+2)p - mq]^2 - 4}{8m(m+2)} + \\frac{\\varepsilon}{8}\n\\]\nwhere $p= 1,2,\\ldots,m-1$, $q=1,2,\\ldots, m+1$ and $p-q$ is even or \nodd corresponding to the Neveu-Schwarz case ($\\varepsilon= 0$) or Ramond \ncase ($\\varepsilon = 1\/2$).\n\nNote that the Neveu-Schwarz algebra has a vacuum representation, \nnamely a irreducible representation with $0$ as eigenvalue of $L_0$, the Ramond \nalgebra has no vacuum representation. \n\\subsection{Stress-energy tensor}\n\\label{set}\nLet $c$ be an admissible value as above with $L_n$ $(n\\in\\mathbb Z)$, \n$n\\in\\mathbb Z$, $G_r$, the operators in \nthe corresponding to a Neveu-Schwarz ($r\\in\\mathbb Z + \\frac12$) or Ramond ($r\\in\\mathbb Z$)\nrepresentation. The Bose and Fermi stress-energy \ntensors are defined by\n\\begin{align}\nT_B(z) =& \\sum_{n} z^{-n-2} L_n\\\\\nT_F(z) =& \\frac12 \\sum_r z^{-r-3\/2} G_r\n\\end{align}\nnamely\n\\begin{equation}\n  \\text{Neveu-Schwarz case:}\\begin{cases}\nT_{B}(z) =& \\sum_{n\\in\\mathbb Z} z^{-n-2} L_n\\\\\nT_{F}(z) =& \\frac12 \\sum_{m\\in\\mathbb Z} z^{-m-2} G_{m+\\frac12}\n\\end{cases}\n\\end{equation}\n\\begin{equation}  \\text{Ramond case:}\\begin{cases}\nT_{B}(z) =& \\sum_{n} z^{-n-2} L_n\\\\\nT_{F}(z) =& \\frac12 \\sum_{m\\in\\mathbb Z} z^{-m-2} \\sqrt{z} G_m\n\\end{cases}\n\\end{equation}\nLet's now make a formal calculation for the (anti-)commutation relations of the Fermi stress energy tensor $T_F$. We want to show that, setting $w\\equiv z_2 \/z_1$, we have\n\\begin{equation}\\label{comrel}\n[T_F(z_1), T_F(z_2)]= \\frac12 z_1^{-1}T_B(z_1) \\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\big(w^2\\delta''(w)  + \\frac34\\delta(w)\\big)\n\\end{equation}\nboth in the Neveu-Schwarz and in the Ramond case.\n\nSetting $k= r=s\\in\\mathbb Z$ we have, say in the Ramond case, \n\\begin{align*}\n[T_F&(z_1), T_F(z_2)]= \\frac14\\sum_{r,s}[G_r, G_s]z_1^{-r-3\/2}z_2^{-s-3\/2}\\\\ \n&= \\frac12\\sum_{r,s}L_{r+s}z_1^{-r-3\/2}z_2^{-s-3\/2} \n+  \\sum_r \\frac{c}{12}\\big(r^2 - \\frac14\\big)z_1^{-r-3\/2}z_2^{r-3\/2}\\\\\n&=\\frac12\\sum_{r,k}L_{k}z_1^{-r-3\/2}z_2^{-k+r-3\/2}\n + z_1^{-3\/2}z_2^{-3\/2}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&= \\frac12 z_1^{-1}\\Big(\\sum_k L_{k}z_2^{-k-2}\\Big) w^{\\frac12}\\sum_r w^{r} + z_1^{-3}w^{-\\frac32}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&=\\frac12 z_1^{-1}T_B(z_2) w^{\\frac12}\n\\sum_r w^r + z_1^{-3} w^{-\\frac32}\\sum_r\\frac{c}{12}\\big(r^2 - \\frac14\\big)w^r\\\\\n&= \\frac12 z_1^{-1}T_B(z_2) w^{\\frac12}\\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\Big(w^2\\delta''(w)  + \\big(w-\\frac14\\big)\\delta(w)\\Big)\\\\\n&= \\frac12 z_1^{-1}T_B(z_1) \\delta(w)\n+ z_1^{-3}w^{-\\frac32}\\frac{c}{12}\n\\Big(w^2\\delta''(w)  + \\frac34\\delta(w)\\Big)\n\\end{align*}\nAs\n\\[\n\\sum_{r\\in\\mathbb Z} \\big(r^2 - \\frac14\\big)w^r \n=\\sum_{r\\in\\mathbb Z +\\frac12} \\big(r^2 - \\frac14\\big)w^r\n= w^2\\delta''(w)  + \\frac34\\delta(w)\n\\]\nthe above calculation (by using equalities as $\\delta(w)\\sqrt{w}=\\delta(w)$ and similar ones) shows that the commutation relations for the Fermi stress energy tensor $T_F$ and, analogously, the commutation relations for $T_B$ and $T_F$ are indeed the same in the Neveu-Schwarz case and in the Ramond case, namely they are representations of the same (anti-)commutation relations. This is basic reason to view the Neveu-Schwarz and Ramond algebras as different types of representations of a unique algebra. \n\nHowever the above calculation is only formal. To give it a rigorous meaning, \nand have convergent series, we have to smear the stress energy tensor with a \nsmooth test function with support in an interval. We then arrive naturally to \nconsider the net of von Neumann algebras of operators localised in intervals. \nIn the case of central charge $c<3\/2$ we shall see that Neveu-Schwarz and \nRamond representations correspond to DHR and general \nsolitons of the associated super-Virasoro net.\n\n\\subsection{Super-Virasoro nets}\nWe give here the definition of the super-Virasoro nets for all the \nallowed values of the central charge. \nWe follow the strategy adopted in \\cite{BS-M} for the case of Virasoro nets, \ncf. also \\cite{Car04,loke}. An alternative construction in the case of the \ndiscrete series ($c<3\/2)$ is outlined in Sect. \\ref{SVirnet}. \n\nLet $\\lambda$ be a unitary positive energy representation of the \nsuper-Virasoro algebra on a Hilbert space ${\\mathcal H}_\\lambda$. The \ncorresponding conformal Hamiltonian $L_0$ is the self-adjoint operator on ${\\mathcal H}_\\lambda$ \nand we denote ${\\mathcal H}_\\lambda^\\infty$ the dense subspace of smooth vectors for $L_0$, \nnamely the subspace of vectors belonging to the domain of $L_0^n$ for all \npositive integers $n$.\n\nThe operators $L_n$,\n$n\\in {\\mathbb Z}$ satisfy the linear energy bounds \n\\begin{equation}\n\\label{e-boundsB}\n\\|L_n v\\|\\leq M (1+|n|^{\\frac{3}{2}})\\|(1+L_0)v\\|, \\quad v\\in {\\mathcal \nH}_\\lambda^\\infty, \n\\end{equation}\nfor suitable constant $M>0$ depending on the central charge $c$, cf. \n\\cite{BS-M,CW}. \nMoreover from the relations \n$[G_{-r}, G_r] = 2L_{0} + \\frac{c}{3}(r^2 - \\frac14)$, \nwe find the energy bounds\n\\begin{equation}\n\\label{e-boundsF}\n\\|G_r v\\|\\leq (2+ \\frac{c}{3}r^2)^{\\frac12}\\|(1+L_0)^{\\frac12}v\\|, \\quad v\\in \n{\\mathcal H}_\\lambda^\\infty,\n\\end{equation}\nwhere $r \\in {\\mathbb Z}+1\/2$ (resp. $r \\in {\\mathbb Z}$) if $\\lambda$ is a Neveu-Schwarz \n(resp. Ramond) representation. \nWe now consider the vacuum representation of the super-Virasoro algebra with \ncentral charge $c$ and denote by ${\\mathcal H}$ the corresponding Hilbert space and \nby $\\Omega$ the vacuum vector, namely the unique (up to a phase) unit \nvector such that $L_0\\Omega=0$.  \n\nLet $f$ be a smooth function on $S^1$. It follows from the linear \nenergy bounds in Eq. (\\ref{e-boundsB}) and the fact that the Fourier \ncoefficients\n\\begin{equation}\n\\hat{f}_n=\\int_{-\\pi}^\\pi f(e^{i\\theta})e^{-in\\theta}\\frac{{\\rm d}\\theta}{2\\pi},\\;\nn\\in {\\mathbb Z},\n\\end{equation}\nare rapidly decreasing, that the smeared Bose stress-energy tensor  \n\\begin{equation}\nT_B(f)= \\sum_{n \\in {\\mathbb Z}}\\hat{f}_nL_n\n\\end{equation}\nis a well defined operator with invariant domain ${\\mathcal H}^\\infty$. Moreover, for \n$f$ real,  $T_B(f)$ is essentially self-adjoint on ${\\mathcal H}^\\infty$ (cf. \\cite{BS-M})\nand we shall denote again $T_B(f)$ its self-adjoint closure.  \n\nNow let $f$ be a smooth function on $S^1$ whose support do not contains \n$-1$. Then also the coefficients  \n\\begin{equation}\n\\hat{f}_r=\\int_{-\\pi}^\\pi f(e^{i\\theta})e^{-ir\\theta}\\frac{{\\rm d}\\theta}{2\\pi},\\;\nr\\in {\\mathbb Z} + \\frac{1}{2},\n\\end{equation}\nare rapidly decreasing and it follows from the energy bounds \nin Eq. (\\ref{e-boundsF}) that the corresponding smeared Fermi stress-energy \ntensor \n\\begin{equation}\nT_F(f)=\\frac{1}{2}\\sum_{r \\in {\\mathbb Z}+\\frac{1}{2}}\\hat{f}_rG_r\n\\end{equation}\nis also a well defined operator with invariant domain ${\\mathcal H}^\\infty$. Again, \nfor $f$ real, $T_F(f)$ is essentially self-adjoint on ${\\mathcal H}^\\infty$ \n(cf. \\cite{BS-M}) and we denote its self-adjoint closure by the same \nsymbol. \n\nAs in Sect. \\ref{netR} we identify $\\mathbb R$ with $S^1\/ \\{-1\\}$ and\nconsider the family ${\\mathcal I}_{\\mathbb R}$ of nonempty, bounded, open intervals of\n${\\mathbb R}$ as a subset of the family ${\\mathcal I}$ of intervals of $S^1$. We define\na net ${\\rm SVir}_{c}$ of von Neumann algebras on ${\\mathbb R}$ by \n\\begin{equation}\n\\label{gener1} \n{\\rm SVir}_{c}(I) \\equiv \\{ e^{iT_B(f)}, e^{iT_F(f)}:f\\in C^{\\infty}(S^1) \n\\;{\\rm real},\\, {\\rm supp}f, \\subset I\\}'', \\; I \\in {\\mathcal I}_{\\mathbb R}. \n\\end{equation} \nIsotony is clear from the definition and we have to show that the net is graded \nlocal and covariant.  \n\nWe first consider graded locality. The spectrum of $L_0$ is \ncontained in ${\\mathbb Z}\/2$ and hence the unitary operator $\\Gamma = e^{i2\\pi L_0}$ \nis an involution such that $\\Gamma \\Omega$.  \nIt is straightforward to check that if $f_1, f_2$ \nare smooth functions on $S^1$ and the support of $f_2$ does not contain $-1$ \nthen $\\Gamma T_B(f_1) = T_B(f_1)\\Gamma$ and \n$\\Gamma T_F(f_1) = -T_F(f_1) \\Gamma$.  \nHence, $\\gamma = {\\rm Ad}\\Gamma$ is a ${\\mathbb Z}_2$-grading on the net \n${\\rm SVir}_{c}$, namely \n$\\Gamma {\\rm SVir}_{c}(I) \\Gamma^* ={\\rm SVir}_{c}(I)$, for all $I\\in \n{\\mathcal I}_{\\mathbb R}$. Now let $I_1, I_2 \\in {\\mathcal I}_{\\mathbb R}$ be disjoint intervals and \nlet $f_1, f_2$ be real smooth functions on $S^1$ with support in $I_1, \nI_2$ respectively. Then the operators $T_B(f_1)$ and $T_F(f_1)$ commute\nwith $ZT_B(f_2)Z^*$ and $ZT_F(f_2)Z^*$, $Z=\\frac{1-i\\Gamma}{1-i}$, on \n${\\mathcal H}^\\infty$, cf. the (anti-)commutation relations in Sect. \\ref{set}\n(note that $Z{\\mathcal H}^\\infty={\\mathcal H}^\\infty)$. Using the energy  bounds in \nEq. (\\ref{e-boundsB}) and Eq. (\\ref{e-boundsF}) and the fact that \n$Z$ commutes with $L_0$ one can apply the argument in \n\\cite[Sect. 2]{BS-M} to show that $e^{iT_B(f_1)}$ and $e^{iT_F(f_1)}$ \ncommute with $Ze^{iT_B(f_2)}Z^*$ and $Ze^{iT_F(f_2)}Z^*$. It follows that\nthe net ${\\rm SVir}_c$ is graded local, namely \n\\begin{equation} \n{\\rm SVir}_c(I_1) \\subset Z{\\rm SVir}_c(I_2)Z^*\n\\end{equation}  \nwhenever $I_1, I_2$ are disjoint interval in ${\\mathcal I}_{\\mathbb R}$. \n\nWe now discuss covariance. The crucial fact here is that the representation \nof the Virasoro algebra on ${\\mathcal H}$ integrates to a strongly continuous \nunitary projective positive-energy representation of ${\\mathrm {Diff}}^{(\\infty)}(S^1)$ \non ${\\mathcal H}$ by \\cite{GoWa,Tol99} which factors through ${\\mathrm {Diff}}^{(2)}(S^1)$ because\n$e^{i4\\pi L_0}=1$. Hence there is a strongly continuous projective \nunitary representation $U$ of ${\\mathrm {Diff}}^{(2)}(S^1)$ on ${\\mathcal H}$ such that, \nfor all real $f \\in C^\\infty(S^1)$ and all $x \\in B({\\mathcal H})$,  \n\\begin{equation}\nU(\\exp^{(2)}(tf))xU(\\exp^{(2)}(tf))^*= \ne^{itT_B(f)}xe^{-itT_B(f)}, \n\\end{equation}\nwhere, $\\exp^{(2)}(tf)$ denotes the lift to ${\\mathrm {Diff}}^{(2)}(S^1)$ of the\none-parameter subgroup $\\exp(tf)$ of ${\\mathrm {Diff}}(S^1)$ generated by the \n(real) smooth vector field $f(e^{i\\theta})\\frac{{\\rm d}}{{\\rm d}\\theta}$. Moreover, \nif $\\theta \\to r^{(2)}(\\theta)$ is the lift to ${\\mathrm {Diff}}^{(2)}(S^1)$ of the \none-parameter subgroup of rotations in ${\\mathrm {Diff}}(S^1)$ we have  \n\\begin{equation}\nU(r^{(2)}(\\theta))= e^{i\\theta L_0},\n\\end{equation}\nfor all $\\theta$ in ${\\mathbb R}$. \nThe following properties of $U$ follow rather straightforwardly. \n\\medskip\n\n\\noindent $(1)$ The restriction of $U$ to the subgroup \n${\\rm\\textsf{M\\\"ob}}^{(2)} \\subset {\\mathrm {Diff}}^{(2)}(S^1)$ lifts to a unique strongly continuous \nunitary representation which we again denote by $U$. \nIf $\\exp^{(2)}(tf) \\in {\\rm\\textsf{M\\\"ob}}^{(2)}$ for all $t\\in {\\mathbb R}$, this \nunitary representation satisfies \n\\begin{equation}\nU(\\exp^{(2)}(tf))=e^{itT_B(f)},\\; U(\\exp^{(2)}(tf))\\Omega=\\Omega,\n\\end{equation}\nfor all $t\\in {\\mathbb R}$.   \n\n\\medskip\n\n\\noindent $(2)$  If the support of the real smooth function $f$ is contained \nin $I \\in {\\mathcal I}_{\\mathbb R}$ then \n\\begin{equation}\n\\label{covTB}\nU(g)e^{iT_B(f)}U(g)^*\\in {\\rm SVir}_c(\\dot{g}I)\n\\end{equation}\nfor all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$ such that $\\dot{g}I \\in {\\mathcal I}_{\\mathbb R}$. \n\\medskip\n\n\\noindent $(3)$ For all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$ we have \n\\begin{equation}\nU(g)\\Gamma U(g)^* = \\Gamma\n\\end{equation}\nfor all $g \\in {\\mathrm {Diff}}^{(2)}(S^1)$.\n\n\\medskip\n\nNote that property $(3)$ follows from the fact that ${\\mathrm {Diff}}^{(2)}(S^1)$ is \nconnected and $\\Gamma^2=1$. \n\nWe now consider the covariance properties of the Fermi stress-energy\ntensor $T_F$. From the commutation relations in equations (\\ref{svirdef})\nwe find,  \n\\begin{equation}\n\\label{smearedcommutator}\ni[T_B(f_1), T_F(f_2)]v = T_F(\\frac{1}{2}f'_1f_2-f'_2f_1)v, \\; v\\in \n{\\mathcal H}^\\infty\n\\end{equation}\nwhere $f_1, f_2$ are real smoot functions such that \n${\\rm supp} f_2 \\subset I$ for some $I\\in {\\mathcal I}_{\\mathbb R}$ and, for any\n$f\\in C^{\\infty}(S^1)$, \n$f'$ is defined by $f'(e^{i\\theta})=\\frac{d}{d\\theta}f(e^{\\theta})$.  \nFor any $g \\in {\\mathrm {Diff}}(S^1)$ consider the function \n$X_g:S_1 \\to {\\mathbb R}$ defined by\n\\begin{equation}\nX_g(e^{i\\theta})= -i\\frac{{\\rm d}}{{\\rm d}\\theta}\\log (ge^{i\\theta}).\n\\end{equation}\nSince $g$ is a diffeomorfism of $S^1$ preserving the orientation \nthen $X_g(z)>0$ for all $z \\in S^1$. Moreover $X_g \\in C^\\infty(S^1)$. \nAnother straightforward consequence of the definition is that \n\\begin{equation}\nX_{g_1g_2}(z)=X_{g_1}(g_2z)X_{g_2}(z). \n\\end{equation}\nAs a consequence the the family of continuous linear operators \n$\\beta(g)$, $g\\in {\\mathrm {Diff}} (S^1)$  on the Fr\\'echet space $C^\\infty(S^1)$ \ndefined by \n\\begin{equation}\n(\\beta(g)f)(z)= X_g(g^{-1}z)^{\\frac12}f(g^{-1}z)\n\\end{equation}\ngives a strongly continuous representation of ${\\mathrm {Diff}} (S^1)$ leaving \nthe real subspace of real functions invariant. Moreover if \n$f_1, f_2\\in C^\\infty(S^1)$ are real then vector valued function \n$t \\to \\beta(\\exp(tf_1))f_2$ is differentiable in $C^\\infty(S^1)$ and\n\\begin{equation}\n\\label{actionderivative}\n\\frac{d}{dt}\\beta(\\exp(tf_1))f_2|_{t=0}=\\frac{1}{2}f'_1f_2-f_1f'_2.\n\\end{equation}  \nNow let ${\\rm supp} f_2$ be a subset of some interval $I \\in I_{\\mathbb R}$ \nand let $J_I\\subset {\\mathbb R}$ be the connected component of $0$ in ${\\mathbb R}$ \nof the open set $\\{t\\in {\\mathbb R}:\\exp(tf_1)I \\in I_{\\mathbb R} \\}$. Then, for any \n$v\\in {\\mathcal H}^\\infty$ the function $J_I \\ni t \\to T_F(\\beta(\\exp(tf_1))f_2)v$ \nis differentiable in ${\\mathcal H}$ and it follows from Eq. \n(\\ref{smearedcommutator}) and Eq. (\\ref{actionderivative}) \nthat \n\\begin{equation}\n\\label{dTF} \n\\frac{d}{dt}T_F(\\beta(\\exp(tf_1))f_2)v|_{t=0}=i[T_B(f_1), T_F(f_2)]v.\n\\end{equation}\nWe now specialize to the case of M\\\"{o}bius transformations i.e. \nwe assume that $\\exp(tf_1) \\in {\\rm\\textsf{M\\\"ob}}$ for all $t\\in {\\mathbb R}$.   \nThe map $J_I \\ni t \\to v(t) \\in {\\mathcal H}$ given by \n\\begin{equation}\nv(t)=T_F(\\beta(\\exp(tf_1))f_2)U(\\exp^{(2)}(tf_1))v\n\\end{equation} \nis well defined because, $U(\\exp^{(2)}(-tf_1)){\\mathcal H}^\\infty = {\\mathcal H}^\\infty$ \nfor all $t\\in {\\mathbb R}$. Note also that $v(t)\\in {\\mathcal H}^\\infty$ for all $t\\in J_I$.\nNow using Eq. (\\ref{dTF}) and the energy bounds in Eq. \n(\\ref{e-boundsF}) it can be shown that $v(t)$ is differentiable (in the \nstrong topology of ${\\mathcal H}$) and that it satisfy the following differential \nequation on ${\\mathcal H}$ \n\\begin{equation}\n\\frac{d}{dt}v(t)=iT_B(f_1)v(t).\n\\end{equation}\nIf follows that \n\\begin{equation}\nv(t)=U(\\exp^{(2)}(tf_1))T_F(f_2)v\n\\end{equation}\nand since $v \\in {\\mathcal H}^\\infty$ was arbitrary we get, for all $t\\in J_I$, \nthe following equality of self-adjoint operators\n\\begin{equation}\nU(\\exp^{(2)}(tf_1))T_F(f_2)U(\\exp^{(2)}(-tf_1))=T_F(\\beta(\\exp(tf_1))f_2). \n\\end{equation}\nNow, if we denote by ${\\mathcal U}^{(2)}_I$ the connected component of the identity \nin ${\\rm\\textsf{M\\\"ob}}^{(2)}$\nof the open set $\\{g\\in {\\rm\\textsf{M\\\"ob}}^{(2)}: gI \\in {\\mathcal I}_{\\mathbb R} \\}$ it follows that \n\\begin{equation}\n\\label{covTF}\nU(g)T_F(f)U(g)^*=T_F(\\beta(\\dot{g})f),\n\\end{equation}\nfor any real smoot function on $S^1$ with ${\\rm supp}f \\subset I$\nand all $g\\in {\\mathcal U}^{(2)}_I$. \n\nFrom Eq. (\\ref{covTF}) and Eq. (\\ref{covTB}) we have \n\\begin{equation}\nU(g){\\rm SVir}_c (I)U(g)^* = {\\rm SVir}_c(\\dot{g}I)\\; I\\in {\\mathcal I}_{\\mathbb R},\n\\; g \\in {\\mathcal U}^{(2)}_I. \n\\end{equation} \nHence ${\\rm SVir}_c$ extends to a M\\\"{o}bius covariant  net on \n$S^1$ satisfying graded locality, see Sect. \\ref{coverNets}. Note that we \nhave not yet shown that the vacuum vector $\\Omega$ is cyclic and \nhence we still don't knew if the net satisfy all the requirements of \nProperty 3 in the definition of M\\\"{o}bius covariant Fermi nets on\n$S^1$ given in Sect. \\ref{MobFermiNets}. \nWe shall however prove the cyclicity of the vacuum as a part of the following \ntheorem. \n\n\\begin{theorem}    \n${\\rm SVir}_c$ is an irreducible Fermi conformal on $S^1$ for any of the \nallowed values \nof the central charge $c$.\n\\end{theorem}\n\\trivlist \\item[\\hskip \\labelsep{\\bf Proof.\\ }] Since $\\Omega$ is the unique (up to a phase) unit vector in \nthe kernel of $L_0$, we only have to show that $\\Omega$ is cyclic and that \nthe strongly continuous positive-energy projective representation $U$\nof ${\\mathrm {Diff}}^{(2)}(S^1)$ defined above makes the net diffeomorphism \ncovariant in the appropriate sense. We first show that $\\Omega$ \nis cyclic for the net. Let ${\\mathcal K} \\subset {\\mathcal H}$ be the closure \nof $\\bigvee_{I\\in {\\mathcal I}} {\\rm SVir}_c(I) \\Omega$. \nWe have to show that ${\\mathcal K} ={\\mathcal H}$.\nClearly $U(g){\\mathcal K}={\\mathcal K}$\nfor all $g \\in {\\rm\\textsf{M\\\"ob}}^{(2)}$. It follows that if $j\\in {\\mathbb Z} \/2$ \n$P_j$ is the orthogonal projection of ${\\mathcal H}$ onto the kernel of $L_0-k1$ \nthen $P_j {\\mathcal K} \\subset {\\mathcal K}\\cap {\\mathcal H}^\\infty$. Now let $r\\in {\\mathbb Z}+1\/2$. Since the \nsmooth functions on $S^1$ wose support does not contain the point $-1$ is \ndense in $L^2(S^1)$ we can find an interval $I\\in {\\mathcal I}_{\\mathbb R}$ and \na real smooth function $f$ with ${\\rm supp}f \\subset I$ such that \n$\\hat{f}_r \\neq 0$. Since $T_F(f)P_j {\\mathcal K} \\subset {\\mathcal K}$ we find \n\\begin{equation}\nG_rP_j{\\mathcal K}=\\frac{1}{\\hat{f}_r }P_{j-r}T_F(f)P_j{\\mathcal K} \\subset {\\mathcal K}. \n\\end{equation}\nA similar argument applies to the operators $L_n$, $n\\in {\\mathbb Z}$ \nand hence the linear span ${\\cal L}$ of the subspaces \n$P_j{\\mathcal K}$, $j\\in {\\mathbb Z} \/2$ is invariant for the \nrepresentation of the super-Virasoro algebra. Since $\\Omega \\in {\\cal L}$ \nis cyclic for the latter representation it follows that ${\\cal L}$ is \ndense in ${\\mathcal H}$. Hence ${\\mathcal K} = {\\mathcal H}$ because ${\\cal L} \\subset {\\mathcal K}$.  Hence \n${\\rm SVir}_c$ is an irreducible M\\\"{o}bius covariant Fermi net \non ${S^1}$.  \n\nTo show that ${\\rm SVir}_c$ is diffeomorphism covariant we first observe \nthat by \\cite[Sect. V.2]{loke}  for any $I\\in {\\mathcal I}$ the group generated by \ndiffeomorphisms of the form \n$\\exp(f)$ with ${\\rm supp}f \\subset I$ is dense in ${\\mathrm {Diff}}_I(S^1)$. It \nfollows that the group generated by elements of the form \n$\\exp^{(2)}(f)$ with ${\\rm supp}f \\subset I$ is dense in \n${\\mathrm {Diff}}^{(2)}_I(S^1)$. Hence, for any $I \\in {\\mathcal I}$ and any \n$g\\in \\in {\\mathrm {Diff}}^{(2)}_I(S^1)$,  $U(g) \\in {\\rm SVir}_c(I)$ and, \nby graded locality, $U(g) \\in {\\rm SVir}(I')'$ because. Now, an \nadaptation of the argument in the proof of \\cite[Proposition 3.7]{Car04}\nshows that ${\\rm SVir}_c$ is diffeomorphism covariant and the proof is \ncomplete. \n\\null\\hfill\\qed\\endtrivlist\\noindent\n\\subsection{The discrete series of super-Virasoro nets}\\label{SVirnet}\nWe shall now use the construction in \\cite{GKO} to study ${\\rm SVir}_{c}$ with $c < 3\/2$ an admissible value.\nFirst consider three real free Fermi fields in the NS representation. \nThey define a graded-local net on $S^1$.  \nThis net coincides with ${\\mathcal F}^{\\hat{\\otimes}3}={\\mathcal F}\\hat{\\otimes}{\\mathcal F}\\hat{\\otimes}{\\mathcal F}$ \nwhere ${\\mathcal F}$ is the net generated by a single real free Fermi field in the NS representation \n(cf. \\cite{Bock}) and $\\hat{\\otimes}$ denotes the graded tensor product. The net \n${\\mathcal A}_{{{\\rm SU}(2)}_2}$ embeds as a subnet of ${\\mathcal F}^{\\hat{\\otimes}3}$. Actually, from the discussion in \n\\cite[page 115]{GKO} we have\n\\[\n{\\mathcal A}_{{{\\rm SU}(2)}_2}={\\mathcal F}^{\\hat{\\otimes}3}_b\\ .\n\\] \nNow consider the conformal net ${\\mathcal F}_N$ (on the Hilbert space \n${\\mathcal H}_N$) given by ${\\mathcal F}^{\\hat{\\otimes}3}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N}$, $N$ positive integer.  \n\nConsider now the the representation of the \nsuper-Virasoro algebra on ${\\mathcal H}_N$ with central \ncharge \n\\begin{equation}\nc_N=\\frac{3}{2}\\left(1-\\frac{8}{(N+2)(N+4)}\\right),\n\\end{equation} \nconstructed in \\cite[Sect. 3]{GKO} (coset construction). Then the corresponding \nstress energy-tensors $T_B$ and $T_F$ generate a family of von Neumann algebras on ${\\mathcal H}_N$ \nas in eq. \\eqref{gener1}. Using the energy bounds in Eq. (\\ref{e-boundsB}) and Eq. \n(\\ref{e-boundsF}) it can be shown that this family defines \na Fermi subnet of ${\\mathcal F}_N$ as in eq. \\eqref{gener1} which can be identified with \nthe super-Virasoro net ${\\rm SVir}_{c_N}$.\nIn this way we obtain all the super-Virasoro nets corresponding to the discrete series.\n\n \nUsing \\cite{GKO} we can identify these super-Virasoro nets as \ncoset subnets. From the embedding  \n\\begin{equation}\n{\\mathcal A}_{{{\\rm SU}(2)}_2}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N} \\subset {\\mathcal F}_N\n\\end{equation}\nwe have the embedding\n\\begin{equation}\n{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}} \\subset {\\mathcal F}_N.\n\\end{equation}\nIt follows from Eq. (3.13) and the claim at the end of \npage 114 in \\cite[Sect. 3]{GKO} that ${\\rm SVir}_{c_N}$ is \ncontained in the coset \n\\begin{equation}\n({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})^c = ({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})'\\cap {\\mathcal F}_N.\n\\end{equation}\nMoreover it follows from the branching rules in \n\\cite[Eq. 4.15]{GKO} that these nets coincide (cf. \\cite{KL1})\nnamely \n\\begin{equation} \n{\\rm SVir}_{c_N} = ({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})^c. \n\\end{equation}\nAs a consequence the Bose subnet ${\\rm SVir}^0_{c_N}\\equiv ({\\rm SVir}_{c_N})_b$ of \nthe super-Virasoro net ${\\rm SVir}_{c_N}$ is equal to the coset\n\n\\begin{equation}\n\\label{BoseCoset}\n({\\mathcal A}_{{{\\rm SU}(2)}_{N+2}})'\\cap \\left( {\\mathcal A}_{{{\\rm SU}(2)}_2}\\otimes {\\mathcal A}_{{{\\rm SU}(2)}_N} \\right)\n\\end{equation}\nand hence, by \\cite[Corollary 3.4]{X2} and \\cite[Theorem 24]{L03} \n${\\rm SVir}^0_{c_N}\n$ is completely rational, see also \\cite[Corollary 28]{L03}. \n\nNow we look at representations. We denote $(NS)$ and $(R)$ the \nNeveu-Schwartz and Ramond representations for three Fermion fields \nrespectively. In $(NS)$ the lowest energy eigenspace is one-dimensional\n(``nondegenerate vacuum ''), whilst \nin $(R)$ it is two-dimensional (``2-fold degenerate vacuum'').\\footnote{Different Ramond \nrepresentations could be defined corresponding to different choices of the corresponding representation of the Dirac algebra of the 0-modes on the subspace of  lowest energy vectors, \ncf. page 113 and page 115 of \\cite{GKO}.} \n\nIt is almost obvious that $(NS)$ corresponds to the vacuum \nrepresentation $\\pi_{NS}$ of ${\\mathcal F}^{\\hat{\\otimes}3}$ and, \narguing as in the proof of \\cite[Lemma 4.3]{Bock}, it can be \nshown that $(R)$ corresponds to a general soliton  \n$\\pi_R$ of the latter net. \nClearly $(NS)$ and $(R)$ restrict to positive-energy \nrepresentations of ${{\\rm SU}(2)}_2$. We denote by \n$\\pi_{(N,l)}$ the representation of ${\\mathcal A}_{{{\\rm SU}(2)}_N}$ with spin $l$. \nAt level $N$ the possible values of the spin are those satisfying \n$0\\leq 2l \\leq N$. Then the following identities hold (see \\cite[page \n116]{GKO}): \n\n\\begin{eqnarray}\n\\label{NSrest} \n\\pi_{NS}|_{{\\mathcal A}_{{{\\rm SU}(2)}_2}} = \\pi_{(2,0)} \\oplus \\pi_{(2,1)} \\\\\n\\label{Rrest}\n\\pi_R|_{{\\mathcal A}_{{{\\rm SU}(2)}_2}} = \\pi_{(2,\\frac{1}{2})}.\n\\end{eqnarray}   \nNote that the restriction of $(R)$ remains irreducible because the \ngrading automorphism is not unitarily implemented, cf. Prop. \\ref{n1}. \n\nDenote by $(c_N,h_{p,q})_{NS}$, resp. $(c_N,h_{p,q})_{R}$, a \nNS, resp. R, irreducible representation of super-Virasoro algebra with central \ncharge $c_N$ and lowest energy \n$$h_{p,q}=\\frac{\\left[(N+4)p-(N+2)q\\right]^2 -4}{8(N+2)(N+4)},$$\nresp. \n$$h_{p,q}=\\frac{\\left[(N+4)p-(N+2)q\\right]^2 -4}{8(N+2)(N+4)} + \n\\frac{1}{16},$$\nwhere $p=1,2,\\dots,N+1$, $q=1,2, \\dots N+3$ and $p-q$ is even in the \nNS case and odd in the R case. \n\nAs already mentioned, in the NS case, for every value of the central charge, the lowest energy\n$h_{p,q}$ completely determines the (equivalence class of) the \nrepresentation. In contrast for a given values of the central charge \nand of the lowest energy $h_{p,q}$ there are two Ramond representations \none with \n\\[\nG_0\\Psi_{h_{p,q}}= \\sqrt{h_{p,q} - \\frac{c_N}{24}}\\Psi_{h_{p,q}}\n\\]\nand the other with\n\\[\nG_0\\Psi_{h_{p,q}}= -\\sqrt{h_{p,q} - \\frac{c_N}{24}}\\Psi_{h_{p,q}},\n\\]\nwhere $\\Psi_{h_{p,q}}$ is the lowest energy vector. These two \nrepresentations are connected by the automorphism \n$G_r \\to - G_r$ and become equivalent when restricted to the even \n(Bose) subalgebra. Accordingly $(c_N,h_{p,q})_{R}$ denotes indifferently these \ntwo representations which are clearly inequivalent when \n$h_{p,q} \\neq \\frac{c_N}{24}$. \n\nFor a given $N$ the equality $h_{p,q} = h_{p',q'}$ when \n$p-q$ and $p'-q'$ are both even or odd hold if and only \nif $p'=N+2-p$ and $q'= N+4 - q$. Note also that it may happen \nthat $h_{p,q} = h_{p',q'}$ when $p-q$ is even and $p'-q$ \nis odd. For example, if $N=2$ then $h_{2,2} = h_{1,2}= 1\/16$. \nAccordingly there are values of $N$ for which a given value \nof the lowest energy corresponds to three distinct irreducible \nrepresentations of super-Virasoro algebra: one NS representation and two\nR representations.  \n\nFrom \\cite[Section 4]{GKO} we can conclude that there exist DHR representations \n$\\pi_{h_{p,q}}^{NS}$, $p-q$ even, and general solitons \n$\\pi_{h_{p,q}}^{R}$, $p-q$ odd, of ${\\rm SVir}_{c_N}$ (associated to the representations \n$(c_N,h_{p,q})_{NS}$, resp. $(c_N,h_{p,q})_{R}$ of the of super-Virasoro algebra)\nsuch that  \n\\begin{equation}\n\\label{GKOnetsNS}\n\\left( \\pi_{NS} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\pi_{h_{p,q}}^{NS},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ even, and\n\\begin{equation}\n\\label{GKOnetsR}\n\\left( \\pi_{R} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\pi_{h_{p,q}}^{R},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ odd. \n                                                                                \nWe now denote $\\rho_{h_{p,q}}^{NS}$,  resp. $\\rho_{h_{p,q}}^{R}$, \nthe restriction of $\\pi_{h_{p,q}}^{NS}$, resp \n$\\pi_{h_{p,q}}^{R}$, to ${\\rm SVir}^0_{c_N}$. \n\nIn the representation space of $\\pi_{h_{p,q}}^{NS}$ the grading is always \nunitarily implemented and hence we have the direct sum \n$$\\rho_{h_{p,q}}^{NS}= \\rho_{h_{p,q}}^{NS+} \\oplus\\rho_{h_{p,q}}^{NS-}$$ \nof two (inequivalent) irreducible representations corresponding to the \neigenspaces with eigenvalues 1 and -1 of the grading operator \nrespectively. \n\nIn contrast in the case of $\\pi_{h_{p,q}}^{R}$ the grading automorphism \nis unitarily implemented only if $h_{p,q}=c_N\/24$. This happens  \nif and only if $N$ is even and $p=(N+2)\/2$, $q=(N+4)\/2$. \nIn this case $\\pi_{\\frac{c_N}{24}}^R$ is a supersymmetric \ngeneral representation of the Fermi conformal net ${\\rm SVir}_{c_N}$.\nMoreover we have the decomposition into irreducible (inequivalent) \nsubrepresentations $$\\rho_{\\frac{c_N}{24}}^R= \\rho_{\\frac{c_N}{24}}^{R+}\\oplus \n\\rho_{\\frac{c_N}{24}}^{R-}.$$ In the remaining cases \n$\\rho_{h_{p,q}}^{R}$ is irreducible. \n\nRestricting Eq. (\\ref{GKOnetsNS} ) and Eq. (\\ref{GKOnetsR} ) to the Bose \nelements and using Equations (\\ref{NSrest}), (\\ref{Rrest})  we get \n\\begin{equation}\n\\label{GKOnetsNSb+}\n\\left( \\pi_{(2,0)} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}\n} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{NS+},\n\\end{equation}\n\\begin{equation}\n\\label{GKOnetsNSb-}\n\\left( \\pi_{(2,1)} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{NS-},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ even, and \n\\begin{equation}\n\\label{GKOnetsRb}\n\\left( \\pi_{(2,\\frac{1}{2})} \\otimes \\pi_{(N,\\frac{1}{2}[p-1])}\\right)\n|_{{\\mathcal A}_{{{\\rm SU}(2)}_{N+2}}\\otimes {\\rm SVir}^0_{c_N}} =\n\\bigoplus_q \\pi_{(N+2, \\frac{1}{2}[q-1])} \\otimes \\rho_{h_{p,q}}^{R},\n\\end{equation}\n$1\\leq q \\leq N+3$, $p-q$ odd.     \n\nNow, recalling the identification of ${\\rm SVir}^0_{c_N}$ as a \ncoset in Eq. (\\ref{BoseCoset} ), it \nfollows from \\cite[Corollary 3.2]{X2} that every irreducible \nDHR representation of this net is equivalent to one of those considered \nbefore, namely $\\rho_{h_{p,q}}^{NS+}$, $\\rho_{h_{p,q}}^{NS-}$\nand $\\rho_{h_{p,q}}^{R}$ ($h_{p,q}\\neq c_N\/24$), $\\rho_{\\frac{c_N}{24}}^{R+}$\nand $\\rho_{\\frac{c_N}{24}}^{R-}$.  \n\\subsection{Modularity of local super-Virasoro nets}\nWe state here explicitly the modularity of the Bose subnet super-Virasoro nets for $c<3\/2$. \nIn this case the Bose super-Virasoro net can be obtained as the coset \\eqref{BoseCoset}. \nThen the Rehren $S$ and $T$ matrices as been computed by Xu in \\cite[Sect.2.2.]{X3} \n(see also Sect. \\ref{classification} below). These matrices agree with those in \\cite{GW} and \n\\cite{FSS} giving modular transformations of specialised characters. Accordingly we have \nthe following. \n\\begin{theorem} \nFor a positive even integer $N$ then ${\\rm SVir}^0_{c_N}$ is a modular conformal net. \n\\end{theorem}\n\\section{Classification of superconformal nets in the discrete series}\n\\label{classification}\nBy a \\emph{superconformal net} (of von Neumann algebras on $S^1$) \nwe shall mean a Fermi net on $S^1$ that contains\na super-Virasoro net as irreducible subnet.\nIf the central charge $c$ of a superconformal net\nis less than $3\/2$, it is of the form\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$\nfor some $m=3,4,5,\\dots$ \\cite{FQS}.  We classify all such superconformal\nnets.\n\n\\subsection{Outline of classification}\nAs above, we denote the super Virasoro net with central charge $c$\nand its Bosonic part by ${\\mathrm {SVir}}_c$ and ${\\mathrm {SVir}}^0_c$, respectively.\nWe are interested in the case $c<3\/2$.  In this case, we have\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$\nfor some $m=3,4,5,\\dots$, and we have already seen that in this case\nthe local conformal net ${\\mathrm {SVir}}^0_c$ is realised as a coset\nnet for the inclusion $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\nThis net is completely rational in the sense of \\cite{KLM}\nby \\cite{X2}.  The DHR sectors of the local conformal net ${\\mathrm {SVir}}^0_c$ is\ndescribed as follows by \\cite[Section 2.2]{X3}.\nLabel the DHR sectors of the local conformal\nnets $SU(2)_m$, $SU(2)_{m-2}$, and $SU(2)_2$ by $k=0,1,\\dots, m$,\n$j=0,1,\\dots,m-2$, and $l=0,1,2$, respectively.  Then we consider the\ntriples $(j,k,l)$ with $j-k+l$ being even.\nFor $l=0,2$, we have identification\n$$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l),$$\nthus it is enough to consider the triples $(j,k,0)$ with $j-k$\nbeing even.  Each such triple labels an irreducible DHR sector\nof the coset net ${\\mathrm {SVir}}^0_c$.  For the case $l=1$, we also have\nidentification\n$$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l),$$\nbut if we have a fixed point for this symmetry, that is,\nif $m$ is even, then the fixed point $(2\/m-1, m\/2,1)$ splits\ninto two pieces, $(2\/m-1, m\/2,1)_+$ and $(2\/m-1, m\/2,1)_-$.\nAll of these triples, with this identification\nand splitting, label all the irreducible DHR sectors of\nthe coset net ${\\mathrm {SVir}}^0_c$.  The sectors with $l=0$ and $l=1$\nare called Neveu-Schwarz and Ramond sectors, respectively.\n\nThe conformal spin of the sector $(j,k,l)$ is given by\n$$\\exp\\left(\\frac{\\pi i}{2}\\left(\\frac{j(j+2)}{m}-\n\\frac{k(k+2)}{m+2}+\\frac{l(l+2)}{4}\\right)\\right).$$\n(This also works for the case $(j,k,l)=(2\/m-1, m\/2,1)$.) \n\nFor example, if $m=3$, we have six\nirreducible DHR sectors and they are labelled with triples\n$(0,0,0)$, $(0,2,0)$, $(1,1,0)$, $(1,3,0)$, $(0,3,1)$, $(1,2,1)$.\n(This local conformal net is equal to the Virasoro net with\n$c=7\/10$.)\nFor $m=4$, we have 13 irreducible DHR sectors and they are\nlabelled with\n$(0,0,0)$, $(0,2,0)$, $(0,4,0)$, $(1,1,0)$, $(1,3,0)$,\n$(2,0,0)$, $(2,2,0)$, $(2,4,0)$, $(0,3,1)$, $(1,4,1)$,\n$(2,3,1)$, $(1,2,1)_+$, $(1,2,1)_-$, where\nthe two labels $(1,2,1)_+$, $(1,2,1)_-$ arise from the fixed\npoint $(1,2,1)$ of the symmetry of order 2.\n\nFor all $m$, the irreducible DHR sector $(m-2,m,0)$ has a\ndimension 1 and a spin $-1$.  The superconformal net\n${\\mathrm {SVir}}_c$ arises as a non-local extension of ${\\mathrm {SVir}}^0_c$\nas a crossed product by ${\\mathbb Z}_2$ using identity and this sector.\n\nLet ${\\mathcal A}$ be any superconformal net on the circle with $c<3\/2$.\nThen let ${\\mathcal B}$ be its Bosonic part.  By a similar argument to that\nin \\cite[Proposition 3.5]{KL1}, we know that the local conformal\nnet ${\\mathcal B}$ is an irreducible extension of the local conformal\nnet ${\\mathrm {SVir}}^0_c$, where $c$ is the central charge of ${\\mathcal A}$.\nBy the strategy in \\cite{KL1} based on \\cite{BEK1},\nwe know that the dual canonical endomorphism $\\theta$ of an extension\nis of the form $\\theta=\\sum_{\\lambda}Z_{0,\\lambda}\\lambda$, where $Z$ is the\nmodular invariant arising from the extension as in \\cite{BEK1}.\nCappelli \\cite{Ca} gave a list of type I modular invariants and\nconjectured that it is a complete list.  From his list,\nit is easy to guess that the dual canonical endomorphisms\nwe use for obtaining extensions are those listed\nin Table \\ref{dual-can}.\n(Cappelli also considered type II modular invariants, but they\ndo not correspond to local extensions, so we ignore them here.)\n\\begin{table}[htbp]\n\\begin{center}\n\\begin{tabular}{|c|l|l|c|}\\hline\n& $m$ & $\\theta$ & Label\n\\\\ \\hline\n(1) & any $m$ & $\\theta=(0,0,0)$ & $(A_{m-1},A_{m+1})$ \\\\ \\hline\n(2) & $m=4m'$ & $\\theta=(0,0,0)\\oplus(0,m,0)$ & $(A_{4m'-1}, D_{2m'+2})$ \n\\\\ \\hline\n(3) & $m=4m'+2$ & $\\theta=(0,0,0)\\oplus(m-2,0,0)$ & $(D_{2m'+2}, A_{4m'+3})$ \n\\\\ \\hline\n(4) & $m=10$ & $\\theta=(0,0,0)\\oplus(0,6,0)$ & $(A_9, E_6)$ \\\\ \\hline\n(5) & $m=12$ & $\\theta=(0,0,0)\\oplus(6,0,0)$ & $(E_6, A_{13})$ \\\\ \\hline\n(6) & $m=28$ & $\\theta=(0,0,0)\\oplus(0,10,0)\\oplus(0,18,0)\\oplus(0,28,0)$\n& $(A_{27},E_8)$ \\\\ \\hline\n(7) & $m=30$ & $\\theta=(0,0,0)\\oplus(10,0,0)\\oplus(18,0,0)\\oplus(28,0,0)$\n& $(E_8, A_{31})$ \\\\ \\hline\n(8) & $m=10$ & $\\theta=(0,0,0)\\oplus(0,6,0)\\oplus(8,6,0)\\oplus(8,6,0)$\n& $(D_6, E_6)$ \\\\ \\hline\n(9) & $m=12$ & $\\theta=(0,0,0)\\oplus(6,0,0)\\oplus(0,12,0)\\oplus(6,12,0)$\n& $(E_6, D_8)$ \\\\ \\hline\n\\end{tabular}\n\\caption{List of candidates of the dual canonical endomorphisms}\n\\label{dual-can}\n\\end{center}\n\\end{table}\nWe will prove that each of the dual canonical endomorphisms \nin Table \\ref{dual-can} gives a local extension of ${\\mathrm {SVir}}^0_c$ \nin a unique way and that an arbitrary such local extension of\n${\\mathrm {SVir}}^0_c$ gives one of the dual canonical endomorphisms \nin Table \\ref{dual-can}.\n\n\\subsection{Study of type I modular invariants} \nWe study type I modular invariants for the coset\nnets for the inclusions $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\n\nFirst we recall the $S$ and $T$ matrices for $SU(2)_m$.\nFor $j,k=0,1,2,\\dots,m$, we have the following.\n\\begin{eqnarray*}\nS^{(m)}_{jk}&=&\\sqrt{\\frac{2}{m+2}}\\sin \\pi\\frac{(j+1)(k+1)}{m+2},\\\\\nT^{(m)}_{jk}&=&\\delta_{jk}\\exp\\frac{\\pi i}{2}\n\\left(\\frac{(j+1)^2}{m+2}-\\frac12\\right).\n\\end{eqnarray*}\nFor odd $m$, we have no problem arising from a fixed point of\nthe order two symmetry, and in this case, the modular invariants\nhave been already classified by Gannon-Walton \\cite{GW}, which\nshows that the identity matrix is the only modular invariant.  So\nwe have no non-trivial extensions in these cases.\n\nSo we now deal with the case of even $m$ in the rest of this\nsection and put $m=2m_0$.  In this case, the\n$S$-matrix of the modular tensor category of the irreducible\nDHR-sectors of the coset net is already not so easy to\nobtain, and it has been computed by Xu \\cite{X3}.\n\nAs in the previous section, we label the irreducible DHR\nsectors of the coset net for the inclusion\n$SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$ with\ntriples $(j,k,l)$ with $j-k+l$ being even, except for the\ncase $(m\/2-1, m\/2, 1)$.  For this fixed point of the order\ntwo symmetry, we use labels $\\varphi_1=(m\/2-1, m\/2, 1)_+$, \n$\\varphi_2=(m\/2-1, m\/2, 1)_-$ to denote the two\nirreducible DHR sectors.  We also use the symbols\n$S^{(m-2)}$, $S^{(m)}$, $S^{(2)}$ and $S$ for the $S$-matrices\nfor the nets $SU(2)_{m-2}$, $SU(2)_m$, $SU(2)_2$ and the coset\nfor $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$, respectively.\nThen Xu's computation for the matrix\n$S$ in \\cite{X3} gives the following, where $n=1,2$.\n\\begin{enumerate}\n\\item $S_{(j,k,l),(j',k',l')}=2S^{(m-2)}_{jj'} \\overline{S^{(m)}_{kk'}}\nS^{(2)}_{ll'}$ for $(j,k,l), (j',k',l')\\neq \\varphi_{1,2}$.\n\\item $S_{(j,k,l),(j',k',l'),\\varphi_n}=S_{\\varphi_n,(j,k,l),(j',k',l')}=\nS^{(m-2)}_{j,m\/2-1} \\overline{S^{(m)}_{k,m\/2}}\nS^{(2)}_{l1}$ for $(j,k,l)\\neq \\varphi_{1,2}$.\n\\item $S_{\\varphi_n,\\varphi_{n'}}=\\delta_{nn'}+(S^{(m-2)}_{m\/2-1,m\/2-1}\n\\overline{S^{(m)}_{m\/2,m\/2}}S^{(2)}_{11}-1)\/2$.\n\\end{enumerate}\nThe $T$-matrix of the coset is described as follows more easily.\n\\begin{enumerate}\n\\item $T_{(j,k,l),(j',k',l')}=T^{(m-2)}_{jj'} \\overline{T^{(m)}_{kk'}}\nT^{(2)}_{ll'}$ for $(j,k,l), (j',k',l')\\neq \\varphi_{1,2}$.\n\\item $T_{(j,k,l),\\varphi_n}=T_{\\varphi_n,(j,k,l)}=\nT^{(m-2)}_{j,m\/2-1} \\overline{T^{(m)}_{k,m\/2}}\nT^{(2)}_{l1}$ for $(j,k,l)\\neq \\varphi_{1,2}$.\n\\item $T_{\\varphi_n,\\varphi_n'}=\\delta_{nn'}\nT^{(m-2)}_{m\/2-1,m\/2-1} \\overline{T^{(m)}_{m\/2,m\/2}}\nT^{(2)}_{11}$.\n\\end{enumerate}\nSuppose we have a modular invariant $Z$ for the coset\nfor $SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$.\nWe define a new matrix  $\\tilde Z_{(j,k,l),(j',k',l')}$ where\nthe triples $(j,k,l)$ and $(j',k',l')$ satisfy\n$j,j'\\in \\{0,1,\\dots,m-2\\}$,\n$k,k'\\in \\{0,1,\\dots,m\\}$,\n$l,l'\\in \\{0,1,2\\}$.  Note that we have no identification\nor splitting for the triples $(j,k,l)$ and $(j',k',l')$ here.\n\\begin{enumerate}\n\\item If $j-k+l, j'-k'+l'\\in 2{\\mathbb Z}$ and\n$(j,k,l), (j',k',l')\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(j,k,l),(j',k',l')}=Z_{(j,k,l),(j',k',l')}$.\n\\item If $j-k+l\\in 2{\\mathbb Z}$ and\n$(j,k,l)\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(j,k,l),(m\/2-1,m\/2,1)}=\nZ_{(j,k,l),\\varphi_1}+Z_{(j,k,l),\\varphi_2}$.\n\\item If $j'-k'+l'\\in 2{\\mathbb Z}$ and\n$(j',k',l')\\neq(m\/2-1,m\/2,1)$, then\nwe set $\\tilde Z_{(m\/2-1,m\/2,1),j'k'l'}=\nZ_{\\varphi_1,(j',k',l')}+Z_{\\varphi_2,j'k'l'}$.\n\\item $\\tilde Z_{(m\/2-1,m\/2,1),(m\/2-1,m\/2,1)}=\nZ_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2}$.\n\\item If $j-k+l\\notin 2{\\mathbb Z}$ or $j'-k'+l'\\notin 2{\\mathbb Z}$, then\nwe set $\\tilde Z_{(j,k,l),(j',k',l')}=0$.\n\\end{enumerate}\nThis construction is analogous to the one of ${\\cal L}^{cw}$\nin \\cite[page 178]{GW}, but now\nunlike in \\cite{GW}, our map $Z\\mapsto \\tilde Z$ may not be \ninjective because of the definition of $\\tilde Z$ involving\nthe row\/column labelled with $(m\/2-1,m\/2,1)$.\n\nFor triples $(j,k,l)$ and $(j',k',l')$ satisfying\n$j,j'\\in \\{0,1,\\dots,m-2\\}$,\n$k,k'\\in \\{0,1,\\dots,m\\}$,\n$l,l'\\in \\{0,1,2\\}$, we set as follows.\n(We do not impose the conditions $j-k+l, j'-k'+l'\\in 2{\\mathbb Z}$ here.)\n\\begin{enumerate}\n\\item $\\tilde S_{(j,k,l),(j',k',l')}=S^{(m-2)}_{jj'}\n\\overline{S^{(m)}_{kk'}} S^{(2)}_{ll'}$.\n\\item $\\tilde T_{(j,k,l),(j',k',l')}=T^{(m-2)}_{jj'}\n\\overline{T^{(m)}_{kk'}} T^{(2)}_{ll'}$.\n\\end{enumerate}\nWe also write $\\varphi=(m\/2-1,m\/2,1)$ and set\n$$I=\\{(j,k,0)\\mid j+k\\in2{\\mathbb Z}\\}\\cup\n\\{(j,k,1)\\mid j+k\\notin2{\\mathbb Z}, j+k<m-1\\},$$\nwhich gives representatives of\n$\\{(j,k,l)\\mid j-k+l\\in2{\\mathbb Z}\\}\\smallsetminus\\{\\varphi\\}$ under the\norder 2 symmetry.\n\nWe now have the following proposition.\n\\begin{proposition}\nIf the matrix $Z$ is a modular invariant for $S, T$, then\nthe matrix $\\tilde Z$ is a modular invariant for $\\tilde S, \\tilde T$.\n\\end{proposition}\n\\begin{proof}\nIt is clear that the entries of the matrix $\\tilde Z$ are\nnonnegative integers and we have $\\tilde Z_{(0,0,0), (0,0,0)}=1$.\nWe thus have to prove $\\tilde Z \\tilde S=\\tilde S \\tilde Z$ and\n$\\tilde Z \\tilde T=\\tilde T \\tilde Z$.\nThe latter is clear from the definition, so we have to prove\nthe former identity at $(j,k,l)$, $(j',k',l')$.\nWe deal with seven cases separately.\n\nCase (1).  $j-k+l\\notin2{\\mathbb Z}$, $j'-k'+l'\\notin2{\\mathbb Z}$.\nThe identity is trivial because the both hand sides are zero.\n\nCase (2).  $j-k+l\\in2{\\mathbb Z}$, $j'-k'+l'\\notin2{\\mathbb Z}$.\nThe right hand side is obviously zero by the definition of $\\tilde Z$,\nand the following three identities now prove that the left hand\nside is also zero.\n\\begin{eqnarray*}\n\\tilde S_{\\varphi,(j',k',l')}&=&0,\\\\\n\\tilde Z_{(j,k,l),(j'',k'',l'')}&=&\n\\tilde Z_{(j,k,l),(m-2-j'',m-k'',2-l'')},\\\\\n\\tilde S_{(j'',k'',l''),(j',k',l')}&=&\n-\\tilde S_{(m-2-j'',m-k'',2-l''),(j',k',l')}.\n\\end{eqnarray*}\nNote that the third identity holds because $j'-k'+l'\\notin2{\\mathbb Z}$.\n\nCase (3).  $j-k+l\\notin2{\\mathbb Z}$, $j'-k'+l'\\in2{\\mathbb Z}$.\nThis case can be proved in the same way as above.\n\nCase (4).  $j-k+l\\in2{\\mathbb Z}$, $j'-k'+l'\\in2{\\mathbb Z}$,\n$(j,k,l)\\neq \\varphi$, $(j',k',l')\\neq \\varphi$.\nThe identity $ZS=SZ$ gives the following identity.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} 2Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),(j',k',l')}+\nZ_{(j,k,l),\\varphi_1}\\tilde S_{\\varphi_1,(j',k',l')}+\nZ_{(j,k,l),\\varphi_2}\\tilde S_{\\varphi_2,(j',k',l')}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2S_{(j,k,l),(j'',k'',l'')}\nZ_{(j'',k'',l''),(j',k',l')}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_1,(j',k',l')}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_2,(j',k',l')}.\n\\end{eqnarray*}\nSince we now have\n\\begin{eqnarray*}\n\\tilde S_{(j'',k'',l''),(j',k',l')}&=&\n\\tilde S_{(m-2-j'',m-k'',2-l''),(j',k',l')},\\\\\n\\tilde S_{(j,k,l),(j'',k'',l'')}&=&\n\\tilde S_{(j,k,l),(m-2-j'',m-k'',2-l'')}\n\\end{eqnarray*}\nwe conclude $(\\tilde Z\\tilde S)_{(j,k,l),(j',k',l')}\n=(\\tilde Z\\tilde Z)_{(j,k,l),(j',k',l')}$.\n\nCase (5).  $j-k+l\\in2{\\mathbb Z}$, $(j,k,l)\\neq \\varphi$, $(j',k',l')=\\varphi$.\nThe identity $ZS=SZ$ gives the following identity for $n=1,2$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\nZ_{(j,k,l),\\varphi_1} S_{\\varphi_1,\\varphi_n}+\nZ_{(j,k,l),\\varphi_2} S_{\\varphi_2,\\varphi_n}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{(j,k,l),(j'',k'',l'')}\nZ_{(j'',k'',l''),\\varphi_n}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_1,\\varphi_n}+\n\\tilde S_{(j,k,l),\\varphi} Z_{\\varphi_2,\\varphi_n}.\n\\end{eqnarray*}\n\nAdding these two equalities for $n=1,2$, we obtain the following\nidentity, which gives $(\\tilde Z\\tilde S)_{(j,k,l),\\varphi}\n=(\\tilde Z\\tilde Z)_{(j,k,l),\\varphi}$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{(j,k,l),(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\n(Z_{(j,k,l),\\varphi_1}+Z_{(j,k,l),\\varphi_2}) \\tilde S_{\\varphi,\\varphi}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{(j,k,l),(j'',k'',l'')}\n(Z_{(j'',k'',l''),\\varphi_1}+Z_{(j'',k'',l''),\\varphi_2})+\\\\\n&&\\quad\\quad\\quad\\quad\\tilde S_{(j,k,l),\\varphi}\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2}).\n\\end{eqnarray*}\n\nCase (6).  $(j,k,l)=\\varphi$, $j'-k'+l'\\in2{\\mathbb Z}$, $(j',k',l')\\neq \\varphi$.\nThis case can be proved in the same way as above.\n\nCase (7).  $(j,k,l)=(j',k',l')=\\varphi$.\nThe identity $ZS=SZ$ gives the following identity for $n=1,2$, $n'=1,2$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} Z_{\\varphi_n,(j'',k'',l'')}\n\\tilde S_{(j'',k'',l''),\\varphi}+\nZ_{\\varphi_n,\\varphi_1} S_{\\varphi_1,\\varphi_{n'}}+\nZ_{\\varphi_n,\\varphi_2} S_{\\varphi_2,\\varphi_{n'}}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} \\tilde S_{\\varphi,(j'',k'',l'')}\nZ_{(j'',k'',l''),\\varphi_{n'}}+\nS_{\\varphi_n,\\varphi_1} Z_{\\varphi_1,\\varphi_{n'}}+\nS_{\\varphi_n,\\varphi_2} Z_{\\varphi_2,\\varphi_{n'}}.\n\\end{eqnarray*}\nAdding these four identities for $n,n'=1,2$, we obtain the following\nidentity, which gives $(\\tilde Z\\tilde S)_{\\varphi,\\varphi}\n=(\\tilde Z\\tilde Z)_{\\varphi,\\varphi}$.\n\\begin{eqnarray*}\n&&\\sum_{(j'',k'',l'')\\in I} \n2(Z_{\\varphi_1,(j'',k'',l'')}+Z_{\\varphi_2,(j'',k'',l'')})\n\\tilde S_{(j'',k'',l''),\\varphi}+\\\\\n&&\\quad\\quad\\quad\\quad\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2})\n\\tilde S_{\\varphi,\\varphi}\\\\\n&=&\\sum_{(j'',k'',l'')\\in I} 2\\tilde S_{\\varphi,(j'',k'',l'')}\n(Z_{(j'',k'',l''),\\varphi_1}+Z_{(j'',k'',l''),\\varphi_2})+\\\\\n&&\\quad\\quad\\quad\\quad\\tilde S_{\\varphi,\\varphi}\n(Z_{\\varphi_1,\\varphi_1}+Z_{\\varphi_1,\\varphi_2}+Z_{\\varphi_2,\\varphi_1}+Z_{\\varphi_2,\\varphi_2})\n\\end{eqnarray*}\n\\end{proof}\nWe now recall Gannon's parity rule \\cite[page 696]{G}.\nFor $n$ and $j$, we set $\\varepsilon_{2n}(j)$ as follows.\n\\begin{enumerate}\n\\item If $j\\equiv 0,n$ mod $2n$, then $\\varepsilon_{2n}(j)=0$.\n\\item If $j\\equiv 1,2,\\dots,n-1$ mod $2n$, then $\\varepsilon_{2n}(j)=1$.\n\\item If $j\\equiv n+1,n+2,\\dots,2n-1$ mod $2n$, then $\\varepsilon_{2n}(j)=-1$.\n\\end{enumerate}\nSet $$L=\\{n\\mid (n,8m(m+2))=1\\}.$$\nThen the parity rule is the following.\nIf we have a modular invariant $\\tilde Z$ for $\\tilde S$ and $\\tilde T$\nwith $tilde Z_{000,jkl}\\neq0$, then\nthe parity rule says that we have\n$$\\varepsilon_{2m}(n)\\varepsilon_{2m+4}(n)\\varepsilon_{8}(n)=\n\\varepsilon_{2m}(n(j+1))\\varepsilon_{2m+4}(n(k+1))\\varepsilon_{8}(n(l+1))$$\nfor all $n\\in L$.\n\nNow suppose $m_0=m\/2$ is odd.\nFor any $n_1$ with $(n_1,4m_0)=1$, choose $p\\in\\{1,-1\\}$ so\nthat $n_1 \\equiv p$ mod $4$.  Then it is easy to see that\nthere exists $n\\in L$ satisfying\n$n\\equiv n_1$ mod $4m_0$,\n$n\\equiv p$ mod $4(m_0+1)$,\n$n\\equiv p$ mod $8$, since we have $(m_0, m_0+1)=1$.\nSuppose $tilde Z_{000,jkl}\\neq0$.  Then the parity rule says\n$$\\varepsilon_{2m}(n_1)\\varepsilon_{2m+4}(p)\\varepsilon_{8}(p)=\n\\varepsilon_{2m}(n_1(j+1))\\varepsilon_{2m+4}(p(k+1))\\varepsilon_{8}(p(l+1)),$$\nwhich gives $\\varepsilon_{2m}(n_1)=\\varepsilon_{2m}(n_1(j+1))$\nfor all $n_1$ with $(n_1,2m)=1$.  If $m_0\\neq 3,5,15$, then\nwe have $j=0,m-2$ by \\cite{G}.  Using the order two symmetry\n$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l)$ and the\ndefinition of $\\tilde Z$,\nwe conclude that our dual canonical endomorphism $\\theta$ has a \ndecomposition within $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}.$\nLet ${\\cal T}_1$ be the tensor category of the representations of\n$SU(2)_m$ with the opposite braiding and ${\\cal T}_2$ be the\ntensor category of the representations of $SU(2)_2$ with the\nusual braiding.  If $\\theta$ gives a local extension for the\ntensor category generated by $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$, then\nit also gives a local extension for the tensor category\n${\\cal T}_1 \\times {\\cal T}_2$ by the description of the\nbraided tensor category having\nthe simple objects $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$ by\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1, Proof of Theorem B]{X5}.\n(Note that we have the notion\nof a local extension as a local $Q$-system in \n\\cite{LR1} for an abstract tensor category.)  Then this extension\nfor ${\\cal T}_1 \\times {\\cal T}_2$ gives a type I modular invariant\nfor the $S$- and $T$-matrices arising from ${\\cal T}_1 \\times {\\cal T}_2$.\nThen Gannon's classification of modular invariants for\n$SU(2)_m\\otimes SU(2)_2$ in \\cite[Section 7]{G} shows that\nour $\\theta$ must be one of Table \\ref{dual-can}.\n(Here we have an opposite braiding for ${\\cal T}_1$, but this\ndoes not matter since we can consider a modular invariant\n$Z_{k'l,kl'}$ instead of $Z_{kl,k'l'}$ as in \\cite[page 178]{GW}.)\n\nNext suppose $m_0=m\/2$ is even.\nFor any $n_2$ with $(n_2,4(m_0+1))=1$, choose $p\\in\\{1,-1\\}$ so\nthat $n_2 \\equiv p$ mod $4$.  Then it is easy to see that\nthere exists $n\\in L$ satisfying\n$n\\equiv p$ mod $4m_0$,\n$n\\equiv n_2$ mod $4(m_0+1)$,\n$n\\equiv p$ mod $8$, since we have $(m_0, m_0+1)=1$.\nSuppose $\\tilde Z_{(0,0,0),(j,k,l)}\\neq0$.  Then the parity rule says\n$$\\varepsilon_{2m}(p)\\varepsilon_{2m+4}(n_2)\\varepsilon_{8}(p)=\n\\varepsilon_{2m}(p(j+1))\\varepsilon_{2m+4}(n_2(k+1))\\varepsilon_{8}(p(l+1)),$$\nwhich gives $\\varepsilon_{2m+4}(n_2)=\\varepsilon_{2m+4}(n_2(k+1))$\nfor all $n_2$ with $(n_2,4(m_0+1))=1$.  If $m_0\\neq 2,4,14$, then\nwe have $k=0,m$ by \\cite{G}.  Using the order two symmetry\n$(j,k,l)\\leftrightarrow(m-2-j,m-k,2-l)$ and the\ndefinition of $\\tilde Z$ again,\nwe conclude that our dual canonical endomorphism $\\theta$ has a \ndecomposition within $\\{(j,0,l)\\mid j+l\\in2{\\mathbb Z}\\}$.\nIf $\\theta$ gives a local extension for the\ntensor category generated by $\\{(0,k,l)\\mid -k+l\\in2{\\mathbb Z}\\}$, then\nit also gives a local extension for the tensor category\nof the representations $SU(2)_{m-2}\\otimes SU(2)_2$ by\nthe description of the braided tensor category having\nthe simple objects $\\{(j,0,l)\\mid j+l\\in2{\\mathbb Z}\\}$ by\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1]{X6}.  Then this extension\ngives a type I modular invariant\nfor the $S$- and $T$-matrices arising from $SU(2)_{m-2}\\otimes SU(2)_2$,\nso Gannon's classification of modular invariants for\n$SU(2)_{m-2}\\otimes SU(2)_2$ in \\cite[Section 7]{G} again shows that\nour $\\theta$ must be one of Table \\ref{dual-can}.\n\nWe now deal with the remaining exceptional cases $m=4,6,8,10,28,30$.\nIf $\\tilde Z_{(0,0,0),(j,k,l)}\\neq0$ and $j-k+l\\in2{\\mathbb Z}$, then the\nconformal spin of $(j,k.l)$ is 1.  This condition already\ngives the restriction that the dual canonical endomorphisms\n$\\theta$ has a decomposition within\n$$\\{(j,k,0)\\mid j+l\\in2{\\mathbb Z}\\}$$\nfor $m=4,6,8,10,28$.  Again Gannon's classification of modular invariants\nin \\cite{G} shows that our $\\theta$ must be one of Table \\ref{dual-can}.\n\nThe final remaining case is $m=30$.  In this case, \nthe only sectors $(j,k,l)$ with $l=1$ and $j-k+l\\in2{\\mathbb Z}$\nhaving conformal spins 1 are\n$(3,12,1)$ and $(13,18,1)$.\nThe parity rules with $n=13, 7$\nexclude these two sectors from $\\theta$, respectively.  Thus again\nthe dual canonical endomorphism $\\theta$ has a decomposition within\n$$\\{(j,k,0)\\mid j+l\\in2{\\mathbb Z}\\}$$\nand Gannon's classification in \\cite{G} shows that our $\\theta$ must\nbe one of Table \\ref{dual-can}.\n\nWe have thus proved that if we have a local extension of\n${\\mathrm {SVir}}^0_c$, then its dual canonical endomorphism must be one of\nthose listed in Table \\ref{dual-can}.\n\n\\subsection{Classification of the Fermi extensions of the \nsuper-Vi\\-ra\\-so\\-ro net, $c<3\/2$}\n\nWe now show that each endomorphism in Table \\ref{dual-can}\nuniquely produces a superconformal net extending ${\\mathrm {SVir}}_c$.\n\n(1) We have the label $(A_{m-1}, A_{m+1})$.\nThe identity matrix obviously gives the coset net for\n$SU(2)_m\\subset SU(2)_{m-2}\\otimes SU(2)_2$ itself.  This has\na unique Fermionic extension.\n\n(2) Now we have the label $(A_{4m'-1}, D_{2m'+2})$ with $m=4m'$.\nThe irreducible DHR sectors $(0,0,0)$, $(0,4m',0)$ give the\ngroup ${\\mathbb Z}_2$.  The crossed product of the coset net by this group\ngives a local extension.  The irreducible DHR sector $(4m'-2,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe crossed product construction.\nThis $\\alpha$-induced sector gives a unique Fermionic extension\nthrough a crossed product by  ${\\mathbb Z}_2$.\n\n(3) Now we have the label $(D_{2m'+2}, A_{4m'+3})$ with $m=4m'+2$.\nThe irreducible DHR sectors $(0,0,0)$, $(4m',0,0)$ give the\ngroup ${\\mathbb Z}_2$.  The crossed product of the coset net by this group\ngives a local extension.  The irreducible DHR sector $(0,4m'+2,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe crossed product construction.\nThis again gives a unique Fermionic extension in a similar way to\ncase (2).\n\n(4) Now we have the label $(A_9, E_6)$ for $m=10$.\nThe DHR sector  $(0,0,0)\\oplus(0,6,0)$ gives a local\n$Q$-system as in \\cite[Subsection 4.2]{KL1}\nby the description of the braided tensor category having\nthe simple objects $\\{(0,k,0)\\mid k\\in2{\\mathbb Z}\\}$ as in\n\\cite[Section 4.3]{X1},\n\\cite[Proposition 2.3.1, Proof of Theorem B]{X5}.\n\nUniqueness also follows as in\n\\cite[Remark 2.5]{KL1}, using the cohomology vanishing result\nin \\cite{KL2}.\n(This is also a mirror extension in the sense of \\cite{X6}, so\nlocality of the extension also follows from \\cite{X6}.)\nThe irreducible DHR sector $(0,10,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(5) Now we have the label $(E_6, A_{13})$ for $m=12$.\nThe irreducible DHR sectors  $(0,0,0)$, $(6,0,0)$ give a local\n$Q$-system as in (4) by the description of the braided tensor\ncategory having the simple objects $\\{(j,0,0)\\mid j\\in2{\\mathbb Z}\\}$\nas in \\cite[Section 4.3]{X1}, \\cite[Proposition 2.3.1]{X5}.\nUniqueness also follows as above.\n(This is also a coset model as in \\cite{KL1}.)\nThe irreducible DHR sector $(10,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(6) Now we have the label $(A_{27}, E_8)$ for $m=28$.\nThe irreducible DHR sectors  $(0,0,0)$, $(0,10,0)$, $(0,18,0)$,\n$(0,28,0)$ give a local\n$Q$-system as in case (4).  Uniqueness also follows as above,\nusing the cohomology vanishing result in \\cite{KL2}.\n(This is again also a mirror extension in the sense of \\cite{X6}.)\nThe irreducible DHR sector $(26,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(7) Now we have the label $(E_8, A_{31})$ for $m=30$.\nThe irreducible DHR sectors  $(0,0,0)$, $(10,0,0)$, $(18,0,0)$,\n$(28,0,0)$ give a local\n$Q$-system as in (5).  Uniqueness also follows as above.\n(This is again also a coset model as in \\cite{KL1}.)\nThe irreducible DHR sector $(0,30,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(8) Now we have the label $(D_6, E_6)$ for $m=10$.\nThe irreducible DHR sectors  $(0,0,0)$, $(0,6,0)$,\n$(8,0,0)$, $(8,6,0)$ give a local\n$Q$-system, since this is a further index 2 extension of the\nlocal extension given in the above (4).\nUniqueness holds for this second extension.\nThe irreducible DHR sector $(0,10,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\n(9) Now we have the label $(E_6, D_8)$ for $m=12$.\nThe irreducible DHR sectors  $(0,0,0)$, $(6,0,0)$,\n$(0,12,0)$, $(6,12,0)$ give a local\n$Q$-system, since this is a further index 2 extension of the\nlocal extension given in the above (5).\nUniqueness holds for this second extension.\nThe irreducible DHR sector $(10,0,0)$\nhas spin $-1$, and this sector still gives an irreducible DHR sector\nof index 1 and spin $-1$ after the $\\alpha$-induction applied to\nthe local extension.  This gives a unique Fermionic extension\nas above.\n\nAll the above give the following classification theorem.\n\\begin{theorem}\\label{main}\nThe following gives a complete list of superconformal nets with\n$c<3\/2$ and the dual canonical endomorphisms for the Bosonic\npart extending ${\\mathrm {SVir}}^0_c$ in each case is given as in\nTable \\ref{dual-can}.\n\\begin{enumerate}\n\\item The super Virasoro net with\n$c=\\displaystyle\\frac{3}{2}\\left(1-\\frac{8}{m(m+2)}\\right)$,\nlabelled with $(A_{m-1}, A_{m+1})$.\n\\item Index 2 extensions of the above (1), labeled with\n$(A_{4m'-1}, D_{2m'+2})$, $(D_{2m'+2}, A_{4m'+3})$.\n\\item Six exceptionals labelled with\n$(A_9, E_6)$, $(E_6, A_{13})$, $(A_{27}, E_8)$, $(E_8, A_{31})$,\n$(D_6, E_6)$, $(E_6, D_8)$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMany analytical and practical problems require a solution to an underdetermined\nor overdetermined system of equations in the form of a generalized matrix \ninverse of a singular matrix $\\Amn$ as $\\Atinv$ satisfying\n\\begin{eqnarray}\n      \\Amn\\Atinv\\Amn & = & \\Amn,\\\\\n      \\Atinv\\Amn\\Atinv\\!\\! & = & \\Atinv.\n\\end{eqnarray}\nIn addition to these properties, a practical application may require\nthe system to be consistent with respect to a general class of \ntransformations. For example, if the behavior of the system should\nbe invariant with respect to arbitrary rotations of the Cartesian\ncoordinate frame in which it is defined then the appropriate \ngeneralized inverse should also satisfy\n\\begin{equation}\n     \\tinv{(\\Um\\Amn\\Vm)} ~ = ~ \\Vmi\\Atinv\\Umi \\label{udist}\n\\end{equation}\nwhere $\\Um$ and $\\Vm$ are arbitrary rotation matrices (or, more generally,\narbitrary orthonormal\/unitary matrices) such that $\\Um\\Umi=\\Imat$, where\n$\\Umi$ is the conjugate-transpose of $\\Um$. This condition is satisfied\nby the Moore-Penrose inverse (MP inverse), which we will denote as\n$\\Apinv$. In other applications the behavior of the system of interest is \nexpected to be invariant with respect to changes of units defined for \nthe its key state variables\/parameters. For example, if all variables\nare defined consistently in imperial units then a change to metric\nunits should have no effect on the behavior of the system, i.e., the\nsystem should be invariant to the arbitrary choice of units. This condition can\nbe expressed as \n\\begin{eqnarray}\n      \\tinv{(\\Dm\\Amn\\Em)}  & = & \\Emi\\Atinv\\Dmi\n\\end{eqnarray}\nwhere $\\Dm$ and $\\Em$ are arbitrary nonsingular diagonal\nmatrices. This condition is satisfied by a recently-derived\nunit-consistent (UC) generalized inverse~\\cite{uhlmann0}.\n\nAn unfortunate fact is that in many scientific and engineering\ndomains the MP inverse is widely used \nas the default generalized inverse without consideration\nfor the properties it preserves -- {\\em and does not preserve}.\nExamples include some forms of Channel Estimation and \nsome applications of the Relative Gain Array (RGA).\nIn particular, the RGA is used to\nmeasure process variable interactions defined by a matrix $\\Amn$\nand is defined as~\\cite{bristol}\n\\begin{equation}\n   \\mbox{RGA}(\\Amn) ~ \\doteq ~ \\Amn \\circ (\\inv{\\Amn})^{\\mbox{\\scriptsize T}}.\n\\end{equation}\nwhere $\\circ$ represents the elementwise Hadamard matrix product.\nWhen $\\Amn$ is singular (or non-square) the RGA is often \nevaluated in practice as~\\cite{changyu}\n\\begin{equation}\n    \\Amn \\circ (\\Apinv)^{\\mbox{\\scriptsize T}}\n\\end{equation}\nwithout regard for whether the relative magnitudes of interactions\nshould be invariant to the choice of units on input\/output\nvariables. In many applications involving singular system\nmatrices (likely most) unit consistency is needed \nand thus the RGA should be evaluated using the UC inverse as\n\\begin{equation}\n    \\Amn \\circ (\\Aginv)^{\\mbox{\\scriptsize T}}\n\\end{equation}\nFailure to do so can lead to erroneous pairings of controlled \nand manipulated variables due to inappropriate dependencies\non the choice of units. \n\nIn other applications there may be need for an\n$n\\times n$ generalized\ninverse to be consistent with respect to an arbitrary matrix\nsimilarity transformation:\n\\begin{equation}\n     \\tinv{(\\Sm\\Amn\\Smi)} ~ = ~ \\Sm\\Atinv\\Smi \\label{dinvcnst}\n\\end{equation}\nThe need for this condition can be seen from a standard linear model\n\\begin{equation}\n     \\yv ~ = ~ \\Amn\\cdot\\av \\label{linmod}\n\\end{equation}\nwhere the objective is to identify a vector $\\av$ of parameter values\nsatisfying the above equation for a data matrix $\\Amn$ and\na known\/desired state vector $\\yv$ defined in the same state\nspace as $\\av$. If $\\Amn$ is nonsingular\nthen there exists a unique $\\Amni$ which gives the solution\n\\begin{equation}\n     \\av ~ = ~ \\Amni\\cdot\\yv\n\\end{equation}\nIf $\\Amn$ is singular, however, then the\nMoore-Penrose inverse could be applied as\n\\begin{equation}\n     \\av ~ = ~ \\Apinv\\cdot\\yv\n\\end{equation}\nor the UC inverse could be applied as\n\\begin{equation}\n     \\av ~ = ~ \\Aginv\\cdot\\yv\n\\end{equation}\nto obtain a solution. Now \nsuppose the state space of $\\yv$ and $\\av$ is\nlinearly transformed by a nonsingular matrix $\\Sm$ as \n\\begin{eqnarray}\n     \\yyv & = & \\Sm\\yv \\\\\n     \\aav & = & \\Sm\\av\n\\end{eqnarray}\nwhere the transformation $\\Sm$ could, for example, define \ntangent-space coordinates for a robot moving on a non-planar manifold\nor, similarly, for gradient descent in a given function space.\nEq.(\\ref{linmod}) can then \nbe rewritten in the new coordinates as\n\\begin{equation}\n     \\yyv ~ = ~ \\Sm\\yv  ~ = ~ (\\Sm\\Amn\\Smi)\\cdot\\Sm\\av\n     ~ = ~ (\\Sm\\Amn\\Smi)\\cdot\\aav\n\\end{equation}\nbut for which  \n\\begin{eqnarray}\n      \\Sm\\av  &  \\neq & \\pinv{(\\Sm\\Amn\\Smi)}\\cdot\\yyv \\\\\n      \\Sm\\av  &  \\neq & \\ginv{(\\Sm\\Amn\\Smi)}\\cdot\\yyv \n\\end{eqnarray}\nIn other words, the solution in the transformed space is not\nthe same as the original solution transformed to the new\nspace -- a situation that represents failure of a fundamental\nsanity check. This is because the MP inverse and UC inverse\ndo not satisfy similarity consistency. \n\nAlthough the similarity-consistency (SC) condition of Eq.\\!~(\\ref{dinvcnst}) can be \nsatisfied using the {\\em Drazin}  inverse~\\cite{drazin}, $\\Adinv$, its use is not an option\nfor most engineering-related systems. Specifically, although the Drazin inverse\n$\\Adinv$ does have some practical applications~(\\cite{cmr76,drazinApp}), \nits use is severely limited because $\\Adinv$ may\nhave rank less than that of $\\Amn$, which is often problematic\nbecause it implies subspace information can be progressively\nand irretrievably lost. In this paper we address this limitation of\nthe Drazin inverse by defining a new generalized inverse which satisfies\nthe condition of Eq.\\!~(\\ref{dinvcnst}) while also preserving\nthe rank of the original matrix.\n\n\\section{The SC Generalized Matrix Inverse}\n\nOne way to derive a generalized inverse that satisfies\na particular consistency condition is to identify a normal\nform relative to that condition. For example, the singular\nvalue decomposition (SVD) decomposes an arbitrary \nmatrix $\\Amn$ as \n\\begin{equation}\n      \\Amn ~=~ \\Um\\Dm\\Vmi\n\\end{equation}\nwhere $\\Um$ and $\\Vm$ are unitary and $\\Dm$ is a nonnegative\ndiagonal matrix of the singular values of $\\Amn$. Critically, $\\Dm$\nis unique up to permutation, so all matrices with the same set of\nsingular values are equivalent up to left and right unitary\ntransformations. This allows the MP inverse to be derived as\n\\begin{equation}\n   \\Apinv ~ = ~ \\Vm\\tinv{\\Dm}\\Umi\n\\end{equation}\nwhere $\\tinv{\\Dm}$ simply inverts the nonzero elements of $\\Dm$.\nFrom this the UC inverse can be similarly derived from a decomposition of the form\n\\begin{equation}\n   \\Amn ~ = ~ \\Dm\\Sm\\Em\n\\end{equation}\nwhere $\\Dm$ and $\\Em$ are nonnegative diagonal\nmatrices and $\\Sm$ is unique up to left and right transformations\nby unitary diagonal matrices~\\cite{uhlmann}. The UC inverse can then be\nobtained as\n\\begin{equation}\n   \\Aginv ~ = ~ \\Emi\\pinv{\\Sm}\\Dmi\n\\end{equation}\nwhere the MP inverse can be applied because $\\Sm$ is \nunique up to unitary diagonal transformations and\nthe MP inverse is consistent with respect to general\nunitary transformations. \n\nFollowing a similar strategy, it can be recognized that the \nJordan normal form of a square matrix $\\Amn$ is\n\\begin{equation}\n   \\Amn ~ = ~ \\Pm\\Jm\\inv{\\Pm}\n\\end{equation}\nwhere $\\Pm$ is an arbitrary nonsingular matrix of the same\nsize as $\\Amn$, and $\\Jm$ is block-diagonal and unique up\nto permutation of the blocks. Thus we can derive from this\ndecomposition the following similarity-consistent (SC) inverse:\n\\begin{equation}\n   \\Asinv ~ = ~ \\Pm\\pinv{\\Jm}\\inv{\\Pm}\n\\end{equation}\nwhere the MP inverse can be applied because $\\Jm$ is \nunique up to block-diagonal permutation transformations,\nwhich are of course unitary and thus permit the MP\ninverse to be consistently applied.\n\nWhat remains is to formally show that this definition of\n$\\Asinv$ is in fact consistent with respect to similarity\ntransformations, i.e., \n\\begin{equation}\n     \\sinv{(\\Sm\\Amn\\Smi)} ~ = ~ \\Sm\\Asinv\\Smi .\n\\end{equation}\nThe Jordan decomposition of $\\Amn$ is\n\\begin{equation}\n   \\Amn ~ = ~ \\Pm\\Jm\\inv{\\Pm}\n\\end{equation}\nso the Jordan decomposition of $\\Sm\\Amn\\Smi$ must\nbe\n\\begin{eqnarray}\n   \\Sm\\Amn\\Smi & = & \\Sm(\\Pm\\Jm\\inv{\\Pm})\\Smi\\\\\n   ~ & = & (\\Sm\\Pm)\\Jm(\\inv{\\Pm}\\Smi)\\\\\n   ~ & = & (\\Sm\\Pm)\\Jm\\inv{(\\Sm\\Pm)}\n\\end{eqnarray}\nand thus the SC inverse of $\\Sm\\Amn\\Smi$ is\n\\begin{eqnarray}\n   \\sinv{(\\Sm\\Amn\\Smi)} & = & \\sinv{\\left((\\Sm\\Pm)\\Jm\\inv{(\\Sm\\Pm)}\\right)}\\\\\n   ~ & = &  (\\Sm\\Pm)\\pinv{\\Jm}(\\inv{\\Pm}\\Smi)\\\\\n   ~ & = & \\Sm(\\Pm\\pinv{\\Jm}\\inv{\\Pm})\\Smi\\\\\n   ~ & = & \\Sm\\Asinv\\Smi.\n\\end{eqnarray}\nThis establishes similarity consistency, and because the Moore-Penrose\ninverse preserves rank, the SC inverse does as well. The key difference\nbetween the SC inverse and the Drazin inverse is that the latter \nenjoys the analytically important property that:\n\\begin{equation}\n   \\Amn\\Adinv ~=~ \\Adinv\\Amn\n\\end{equation}\nwhereas product commutativity of $\\Amn$ and $\\Asinv$ \nis not a property of the SC inverse because $\\pinv{\\Jm}\\Jm$\ndoes not generally equal $\\Jm\\pinv{\\Jm}$.\n\n\\section{Example and Practical Considerations}\n\nHere we consider an illustrative numerical example\ninvolving a nilpotent matrix\n\\begin{equation}\n   \\Amn ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             4 & -1 & 2\\\\\n                             7 & -2 & 3\\\\\n                             -4 & 1 & -2\n                     \\end{tabular}\\right]\n\\end{equation}\nwith Jordan form $\\Amn=\\Pm\\Jm\\inv{\\Pm}$ where\n\\begin{equation}\n   \\Pm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             1 & -2 & -3\\\\\n                             2 & -5 & 10\\\\\n                            -1 & 2 & -2\n                     \\end{tabular}\\right]\n~ \\mbox{and} ~ ~\n   \\Jm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             0 & 1 & 0\\\\\n                             0 & 0 & 1\\\\\n                             0 & 0 &  0\n                     \\end{tabular}\\right].\n\\end{equation}\nIt can be verified that \n\\begin{equation}\n   \\Asinv ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -2 & 1 & 2\\\\\n                             10 & 0 & 15\\\\\n                             8 & -2 & 2\n                     \\end{tabular}\\right]\n~ \\mbox{and} ~ ~\n   \\Adinv ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             0 & 0 & 0\\\\\n                             0 & 0 & 0\\\\\n                             0 & 0 &  0\n                     \\end{tabular}\\right].\n\\end{equation}\nwhere the Drazin inverse of a nilpotent matrix is always\nzero, which graphically demonstrates complete loss of \nrank\\footnote{The integer matrices in this section were constructed\nusing the method of~\\cite{hanson} so that inverses are also \ninteger matrices. This allows the results to be more conveniently\nverified by direct hand calculations.}.\nWe now consider a matrix $\\Mm=\\Sm\\Amn\\Smi$ similar to $\\Amn$\nwith\n\\begin{equation}\n   \\Sm ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -1 & 4 & 0\\\\\n                             -2 & 9 & 3\\\\\n                              1 & -4 & 1\n                     \\end{tabular}\\right].\n\\end{equation}\nSimilarity consistency for the Drazin inverse is trivially\nmaintained because \n$\\dinv{\\Mm}=\\dinv{(\\Sm\\Amn\\inv{\\Sm})}=\\Sm\\dinv{\\Amn}\\inv{\\Sm}={\\bf 0}$.\nBy contrast, it can be verified that the rank of the SC inverse\n\\begin{equation}\n\\sinv{\\Mm} ~=~ \\left[\\!\\begin{tabular}{rrr}\n                             -819 & 167 & -443\\\\\n                             -2301 & 464 & -1255\\\\\n                              663 & -137 & 355\n                     \\end{tabular}\\right]\n\\end{equation}\nis the same as $\\Mm$ and that the similarity consistency condition\n$\\sinv{(\\Sm\\Amn\\inv{\\Sm})}=\\Sm\\sinv{\\Amn}\\inv{\\Sm}=\\sinv{\\Mm}$\nis satisfied. And of couse it can also be verified\nthat $\\sinv{\\Mm}$ satisfies the generalized inverse\nconditions $\\Mm\\sinv{\\Mm}\\Mm=\\Mm$ and $\\sinv{\\Mm}\\Mm\\sinv{\\Mm}=\\sinv{\\Mm}$.\nWhen this example is numerically evaluated using the Matlab {\\sf jordan(M)} \nand {\\sf pinv(M)} functions the SC inverse is found to be accurate to machine precision.\nHowever, when applied in two different high data-rate robotic control applications \nthe observed errors using the SC inverse were sufficiently large to cause the \ncontrol process to diverge rapidly\\footnote{These simulation experiments \nwere performed using variations on models for a robot arm and a \nrover examined in~\\cite{zhang}. Unit consistency was the focus of that\nwork and the two failed tests of the SC inverse were neither analyzed nor reported.}. \n\nThe numerical instability of the Jordan decomposition is well known~\\cite{golub}, \nand this problem is exacerbated when applied with the MP inverse because large errors \ncan be introduced into singular values that should be zero, which can lead to huge\nspike errors when inverted. It is not clear to what \nextent this issue can be mitigated when these two operations are used in\nsuccession to compute the SC inverse. In many practical dynamical-system applications \nthe system matrix changes relatively slowly over time, so it may be possible\nto track the evolution of its distribution of singular values to actively avoid \ncatastrophic discontinuities in the case of a singular value near the \nthreshold between zero and nonzero when the MP inverse is applied\nduring evaluation of the SC inverse. More specifically, a tracked presumed-zero \nsingular value that drifts above the MP threshold can be maintained as zero while \na tracked presumed-nonzero singular value can be maintained as nonzero even if it \ndrifts below the threshold.\n\nA potentially more general approach for mitigating some of the numerical fragility is \nto either jointly solve the Jordan+MP calculation or to relax the form of the \ndecomposition, e.g., so that the central matrix is unique only up to orthonormal\nsimilarity to a complex symmetric matrix rather than\npermutation similarity to a Jordan matrix. For example, any matrix $\\Amn$\ncan be decomposed\\!~\\cite{HJ1} as\n\\begin{equation}\n   \\Amn ~ = ~ \\Sm\\Xm\\Smi  \\label{symdecomp}\n\\end{equation}\nwhere $\\Xm$ is complex symmetric and thus\nthe SC inverse can be defined as:\n\\begin{equation}\n   \\Asinv ~ = ~ \\Sm\\pinv{\\Xm}\\Smi   \\label{symsc}.\n\\end{equation}\nThat the decomposition of\nEq.\\!~(\\ref{symsc}) is only unique up to unitary\/orthonormal \nsimilarity can be inferred from the fact that for\ncomplex symmetric $\\Ym$ and orthonormal $\\Qm$ ($\\inv{\\Qm}=\\Qmt$),\n$\\Ym=\\Qm\\Xm\\Qmt$ implies\n\\begin{eqnarray}\n   \\Asinv & = & \\Sm\\pinv{\\Xm}\\Smi\\\\\n    ~ & = & \\Sm\\pinv{(\\Qm\\Ym\\Qmt)}\\Smi \\\\\n     ~ & = &      (\\Sm\\Qm)\\pinv{\\Ym}\\inv{(\\Sm\\Qm)} \n\\end{eqnarray}\nwhere the last step is obtained by exploiting the unitary consistency property of the MP inverse.\nIt is reasonable to expect that this less rigid definition will be more amenable\nto numerically-robust evaluation than is possible using a construction requiring\nan explicit Jordan decomposition.\n\n\\section{Fixed-Rank Assumption}\n\nIn some applications it is known that the rank of the\ntime-evolving system matrix can potentially change, \ne.g., during transitions through gimbal-lock \nconfigurations, while in others\nit can be expected to remain constant. In the latter case\na fixed-rank assumption can be exploited to mitigate\nthe problem of spurious singular values when applying\nthe MP inverse to the Jordan matrix during evaluation of\nthe SC inverse. Specifically, if a fixed rank of $k$ is \nknown\/assumed then the MP inverse can be evaluated\nexplicitly using the SVD with all but the $k$\nlargest singular values set identically to zero instead of having\nto discriminate with a threshold.  \n\nThe efficacy of exploiting the fixed-rank assumption \ncan be demonstrated in a carefully controlled experiment\ninvolving a dynamical 3-dimensional system that is\nconfined to a 2-dimensional subspace such that the\n$3\\times 3$ system matrix can be assumed to have a fixed\nrank of $2$. The testing model can be further tailored \nso that the system matrix is guaranteed to never be\ndiagonalizable and thus is maximally susceptible to the \nnumerical issues discussed in the previous section. We\nachieve this by defining the Jordan form of the system\nas a function of time as\n\\begin{equation}\n   (\\Qm(t)\\Sm)\\,\\Jm\\,\\inv{(\\Qm(t)\\Sm)}\n\\end{equation}\nwith\n\\begin{equation}\n\\Jm = \\begin{array}{|ccc|}\n             1 & 1 & 0\\\\\n             0 & 1 & 0\\\\\n             0 & 0 & 0\n          \\end{array}\n~,~ ~\n\\Qm(t) = \\begin{array}{|ccc|}\n                    \\cos(t) & -\\sin(t) & 0\\\\\n                    \\sin(t)  & ~\\cos(t) & 0\\\\ \n                    0 & 0 & 1\n               \\end{array}\n\\end{equation}\nand $\\Sm$ is an arbitrary fixed nonsingular matrix. Thus\nthe system matrix is singular and has an orbit period of $2\\pi$ \nduring which it is never diagonalizable. The plot of Figure\\!~1 \nshows the maximum absolute elementwise deviation between the \ncomputed SC inverse and the ground-truth known SC inverse \nfor each of 300 uniform timesteps $t$, $0\\leq t<2\\pi$. It\nis seen that spike errors occur at a significant fraction of the \ntimesteps. These spikes are actually of magnitude $>\\!10^{14}$ \nand result from inverting spurious singular values \nthat exceed {\\sf pinv}'s zero threshold. In tests of many\nrandomly-generated $\\Sm$ matrices the number of\nspike errors over the entire 300-timestep simulation ranged from\nzero to more than $50\\%$, with Figure\\!~1 showing\nan example of the latter. \n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.5in]{SCINV-RESULT-Raw-300}\n\t\\caption{Error magnitudes (vertical axis) resulting from standard evaluation of the\n                         SC inverse from the Jordan form over 300 timesteps (horizontal axis).\n                         Clipped magnitudes are actually of size $\\approx 10^{14}$.}        \n\\end{figure}\n\n\nFigure\\!~2 shows the results of the same\nscenario but with the smallest singular value of the Jordan matrix\nalways treated as identically zero for computation of the SC inverse at\neach timestep. The results shown in Figure\\!~2 are typical of those\nobserved for the fixed-rank method in the battery of tests over many\nrandomly-generated $\\Sm$ matrices. By design it is likely that the model \nused for these tests overestimates the frequency of spike errors that can be \nexpected from standard evaluation of the SC inverse in real-world \napplications, but because any such errors will be time-correlated it likely will\nnot be practical to simply identify and discard them (e.g., apply some sort of \nheuristic outlier filter) on the assumption that they will only occur sporadically.\nIn other words, the intrinsic numerical instability of the SC inverse cannot be\nmitigated in the general case. However, the results of Figure\\!~2 suggest\nthat the problem may be effectively mitigated in a large class of applications.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=3.5in]{SCINV-RESULT-Fixed-300}\n\t\\caption{Error magnitudes (vertical axis) resulting from fixed-rank evaluation of the\n                         SC inverse from the Jordan form over 300 timesteps (horizontal axis). The mean\n                         error magnitude is $1.4420e$-$07$ and the maximum magnitude is $2.3101e$-$06$.}        \n\\end{figure}\n\n\n\n\\section{Discussion}\n\nSimilarity transformations capture the most general linear transformations that can\nbe applied to the common coordinate frame of a set of input and output \nvariables, e.g., as can arise in machine learning applications in which little is \nknown about the best space in which to represent an input-output mapping.\nIn this paper we defined a new similarity-consistent generalized matrix\ninverse that preserves\/maintains the rank of the original matrix, a property\nwhich is not preserved by the Drazin inverse. It has been noted that the\nnumerical evaluation of the SC inverse can be a challenging practical\nlimitation. One possible approach for addressing this problem is to \nrelax the structure of the decomposition from requiring similarity to a Jordan\nmatrix to similarity to a complex symmetric matrix for which uniqueness\nis only required up to orthonormal similarity, rather than permutation \nsimilarity, so the MP inverse is still applicable. This less-rigid\nformulation may be more amenable to numerically-robust evaluation,\nespecially in conjunction with the fixed-rank method.\nRegardless of the choice of decomposition, \nit would be interesting to examine whether recent\nmethods developed for evaluating the Drazin inverse can be applied\nto the SC inverse~\\cite{drazinalg,pan}. \n\nDespite its computational challenges, the SC inverse is the only \ngeneral option available for applications that demand consistency with\nrespect to arbitrary linear transformations of a common coordinate\nframe. We have provided evidence that these challenges can be \neffectively addressed in applications in which the rank of the system\ncan be assumed fixed during its time evolution, but we have also\nemphasized that intrinsic numerical instabilities represent a \nfundamental obstacle in the general case. \nThis motivates continued work toward identifying special properties\nof particular applications that can be exploited to permit the SC\ninverse to be practically applied.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}