\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Witten genus} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{index_theory}{}\paragraph*{{Index theory}}\label{index_theory} [[!include index theory - contents]] \hypertarget{elliptic_cohomology}{}\paragraph*{{Elliptic cohomology}}\label{elliptic_cohomology} [[!include elliptic cohomology -- contents]] \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{CharacteristicSeries}{Characteristic series}\dotfill \pageref*{CharacteristicSeries} \linebreak \noindent\hyperlink{CharacteristicSeriesInTermsOfKacWeylCharacters}{In terms of Kac-Weyl characters}\dotfill \pageref*{CharacteristicSeriesInTermsOfKacWeylCharacters} \linebreak \noindent\hyperlink{IntegralityAndModularity}{Integrality and modularity}\dotfill \pageref*{IntegralityAndModularity} \linebreak \noindent\hyperlink{ModularityForTypeIISuperstring}{For the type II superstring}\dotfill \pageref*{ModularityForTypeIISuperstring} \linebreak \noindent\hyperlink{ModularityForHeteroticString}{For the heterotic superstring}\dotfill \pageref*{ModularityForHeteroticString} \linebreak \noindent\hyperlink{RelationToSuSyQMOnLoopSpace}{Relation to Dirac operators and supersymmetric QM on loop space}\dotfill \pageref*{RelationToSuSyQMOnLoopSpace} \linebreak \noindent\hyperlink{TwistedWittenGenus}{Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models}\dotfill \pageref*{TwistedWittenGenus} \linebreak \noindent\hyperlink{as_the_global_character_of_sheaves_of_vertex_operator_algebras}{As the global character of sheaves of vertex operator algebras}\dotfill \pageref*{as_the_global_character_of_sheaves_of_vertex_operator_algebras} \linebreak \noindent\hyperlink{stolz_conjecture}{Stolz conjecture}\dotfill \pageref*{stolz_conjecture} \linebreak \noindent\hyperlink{RelationToBPSStateCounting}{Relation to BPS state counting on target space}\dotfill \pageref*{RelationToBPSStateCounting} \linebreak \noindent\hyperlink{relation_to_cayley_plane_bundles}{Relation to Cayley plane bundles}\dotfill \pageref*{relation_to_cayley_plane_bundles} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak \noindent\hyperlink{ReferencesGeneralCase}{General}\dotfill \pageref*{ReferencesGeneralCase} \linebreak \noindent\hyperlink{ReferencesRelationToKacWeylCharacters}{Relation to Kac-Weyl characters of loop group representations}\dotfill \pageref*{ReferencesRelationToKacWeylCharacters} \linebreak \noindent\hyperlink{the_stolz_conjecture}{The Stolz conjecture}\dotfill \pageref*{the_stolz_conjecture} \linebreak \noindent\hyperlink{refinement_to_the_stringorientation_of_}{Refinement to the string-orientation of $tmf$}\dotfill \pageref*{refinement_to_the_stringorientation_of_} \linebreak \noindent\hyperlink{via_index_theory_of_wouldbe_dirac_operators_on_loop_space}{Via index theory of would-be Dirac operators on loop space}\dotfill \pageref*{via_index_theory_of_wouldbe_dirac_operators_on_loop_space} \linebreak \noindent\hyperlink{via_sheaves_of_super_vertex_operator_algebras}{Via (sheaves of) super vertex operator algebras}\dotfill \pageref*{via_sheaves_of_super_vertex_operator_algebras} \linebreak \noindent\hyperlink{via_other_methods}{Via other methods}\dotfill \pageref*{via_other_methods} \linebreak \noindent\hyperlink{ReferencesTwistedCase}{Twisted case}\dotfill \pageref*{ReferencesTwistedCase} \linebreak \noindent\hyperlink{relation_to_bps_state_counting_on_target_space_2}{Relation to BPS state counting on target space}\dotfill \pageref*{relation_to_bps_state_counting_on_target_space_2} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{Witten genus} is a [[genus]] with [[coefficients]] in [[power series]] in one variable, playing the role of a universal [[elliptic genus]]. This arises (\hyperlink{Witten87}{Witten 87}) as the [[large volume limit]] of the [[partition function]] of the [[superstring]] (hence in the string worldsheet [[perturbation theory]] about constant [[worldsheet]] configurations). Specifically, for the [[type II superstring]] this reproduces the universal [[elliptic genus]] as previously introduced by [[Serge Ochanine]], while for the [[heterotic string]] it yields what is now called the Witten genus proper. Concretely, as Witten argued, this is a formal [[power series]] in string oscillation modes of the [[A-hat genus]] of the symmetric [[tensor powers]] of the [[tangent bundle]] that these modes take values in. In (\hyperlink{Witten86}{Witten 86}) it is suggested, by regarding the [[superstring]] [[sigma-model]] as [[quantum mechanics]] on the [[smooth loop space]] of its [[target space]], that the Witten genus may be thought of as the [[large volume limit]] of an $S^1$-equivariant [[A-hat genus]] on [[smooth loop space]], hence the [[index]] of the [[Dirac-Ramond operator]] in that limit. (Ever since this suggestion people have tried to make precise the concept of Dirac operator on a [[smooth loop space]] (e.g. \hyperlink{AlvarezKillingbackManganoWindey87}{Alvarez-Killingback-Mangano-Windey 87}). But notice that, by the above, only the [[formal loop space]] and the [[Dirac-Ramond operator]] really appears in the definition of the Witten genus.) A priori the [[coefficients]] of the Witten genus as a genus on [[orientation|oriented]] manifolds are [[formal power series]] over the [[rational numbers]] \begin{displaymath} w \;\colon\; M SO_\bullet \longrightarrow \mathbb{Q}[ [ q ] ] \,. \end{displaymath} In the construction from [[string theory|string physics]] this map is interpreted as sending a [[target space|target]] [[spacetime]] $X$ of the [[superstring]] to the function $w_X(q) = w_X(e^{2 \pi i \tau})$ which to each modulus $\tau \in \mathbb{C}$ characterizing a toroidal [[Riemann surface]] assigns the [[partition function]] of the [[superstring]] with [[worldsheet]] the [[torus]] $\mathbb{C}/(\mathbb{Z} + \mathbb{Z}\tau)$ and propagating on [[target space]] $X$. On manifolds with [[spin structure]] the genus refines to integral power series (via the integrality of the [[A-hat genus]] (\hyperlink{ChudnovskyChudnovsky88}{Chudnovsky-Chudnovsky 88}, \hyperlink{KreckStolz93}{Kreck-Stolz 93}, \hyperlink{Hovey91}{Hovey 91}). Moreover on manifolds with rational string structure it takes values in [[modular forms]] (\hyperlink{Zagier86}{Zagier 86}) and crucially, on manifolds with [[string structure]] it takes values in [[topological modular forms]] \begin{displaymath} \itexarray{ M String_\bullet &\longrightarrow& tmf_\bullet \\ \downarrow && \downarrow \\ \Omega_\bullet^{String, rat} &\longrightarrow& MF_\bullet \\ \downarrow && \downarrow \\ M Spin_\bullet &\longrightarrow& \mathbb{Z}[[ q ] ] \\ \downarrow && \downarrow \\ M SO_\bullet &\stackrel{w}{\longrightarrow}& \mathbb{Q}[ [ q ] ] } \,. \end{displaymath} (On the left is the image under forming [[Thom spectra]]/[[cobordism rings]] of the first stages in the [[Whitehead tower]] of $BO$, see also at \emph{[[higher spin structure]]}.) Observe here that [[topological modular forms]] are the [[coefficient]] [[ring]] of the [[E-∞ ring]] [[spectrum]] known as \emph{[[tmf]]}. By the general way in which [[genera]] (see there) tend to appear as [[decategorifications]] of [[homomorphisms]] of [[E-∞ rings]] out of a [[Thom spectrum]], this suggests that the Witten genus is the value on [[homotopy groups]] of a homomorphism of [[E-∞ rings]] of the form \begin{displaymath} \sigma \colon M String \longrightarrow tmf \end{displaymath} from the [[Thom spectrum]] of [[String bordism]] to the [[tmf]]-spectrum. This lift of the Witten genus to a universal [[orientation in generalized cohomology|orientation]] in universal [[elliptic cohomology]] indeed exists and is called the \emph{[[sigma-orientation]]}, or the \emph{[[string orientation of tmf]]}. This construction has been the central motivation behind the search for and construction of [[tmf]] (\hyperlink{Hopkins94}{Hopkins 94}). A construction of the [[string orientation of tmf]] is given in (\hyperlink{AndoHopkinsRezk}{Ando-Hopkins-Rezk 10}) and it is shown that indeed it refines the Witten genus (\hyperlink{AndoHopkinsRezk}{Ando-Hopkins-Rezk 10, prop. 15.3}). It is maybe noteworthy that [[tmf]] (and hence its universal string orientation) also arises canonically from just studying [[chromatic homotopy theory]] (see \hyperlink{tmf#MazelGee13}{Mazel-Gee 13} for a nice survey of this) a fundamental topic in [[stable homotopy theory]], hence a fundamental topic in [[mathematics]]. Therefore in the Witten genus some very fundamental pure mathematics happens to equivalently incarnate as some conjecturally very fundamental [[physics]] ([[string theory]]). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{CharacteristicSeries}{}\subsubsection*{{Characteristic series}}\label{CharacteristicSeries} The \href{genus#LogarithmAndCharacteristicSeries}{characteristic series} of the Witten genus as a [[power series]] in $z$ with [[coefficients]] in formal power series in $q$ over $\mathbb{Q}$ is \begin{displaymath} \begin{aligned} K_w(z)(q) & = \frac{z}{\exp_w(z)(q)} \\ & = \frac{z}{\sigma_L(z)(q)} \\ & = \frac{z/2}{sinh(z/2)} \prod_{n \geq 1} \frac{(1-q^n)^2}{(1-q^n e^z)(1-q^n e^{-z})} \\ & = \exp\left( \sum_{k \geq 2} G_k(q) \frac{z^k}{k!} \right) \end{aligned} \,, \end{displaymath} where \begin{itemize}% \item $\sigma_L$ is the [[Weierstrass sigma-function]] (see e.g. \hyperlink{AndoBasterra00}{Ando Basterra 00, section 5.1}); \item $G_k$ are the [[Eisenstein series]] (\hyperlink{Zagier86}{Zagier 86, equation (14)}, \hyperlink{AndoHopkinsRezk10}{Ando-Hopkins-Rezk 10, prop. 10.9}). \end{itemize} This is a [[modular form]] with respect to the variable $q$, see also the the discussion below at \emph{\hyperlink{IntegralityAndModularity}{Integrality and modularity}} . Such functions which are power series of two variables $z$ and $q$ with elliptic nature in $z$ and modular nature in $q$ are called \emph{[[Jacobi forms]]} (\hyperlink{Zagier86}{Zagier 86, p. 8}, \hyperlink{AndoFrenchGanter08}{Ando-French-Ganter 08}). There are various further ways to equivalently re-express the above in terms of other special [[modular forms]]. Here are some: \hypertarget{CharacteristicSeriesInTermsOfKacWeylCharacters}{}\paragraph*{{In terms of Kac-Weyl characters}}\label{CharacteristicSeriesInTermsOfKacWeylCharacters} The Witten genus has a close relation to the [[Kac-Weyl character]] of [[loop group representations]]. Consider of four [[irreducible representations|irreducible]] [[level]]-1 positive energy [[Spin group|Spin]]$(2k)$-[[loop group representation]] the one denoted \begin{displaymath} \tilde S_+ - \tilde S_- \in Rep(\tilde L Spin(2k)) \end{displaymath} and write its [[Kac-Weyl character]] as \begin{displaymath} \chi(\tilde S_+ - \tilde S_-) \in Rep(Spin(2k))[ [ q^{1/12} ] ] \,. \end{displaymath} Under passing to [[group characters]] this is (\hyperlink{Brylinski90}{Brylinski 90, p. 7(467)}, reviewed in \hyperlink{KL96}{KL 96, section 1.2}) equivalently \begin{displaymath} \chi(\tilde S_+ - \tilde S_-) = \frac{\prod_{1}^k \theta}{\eta^k} \,, \end{displaymath} where on the right we have the [[Jacobi theta-function]] $\theta$ divided by the [[Dedekind eta-function]] $\eta$. Comparison shows that in terms of this the exponential series of the Witten genus is equivalently (by the [[splitting principle]] the $k$-fold products are left implicit): \begin{displaymath} \exp_w = z/K_w = \eta^2 \, \chi(\tilde S_+ - \tilde S_-) \,. \end{displaymath} Notice that by the relation (see \href{equivariant+elliptic+cohomology#RelationToLoopGroupRepresentations}{here}) between [[equivariant elliptic cohomology]] and [[loop group representations]], over the complex numbers $\chi(\tilde S_+ - \tilde S_-)$ may be regarded as an element of the $Spin(2k)$-[[equivariant elliptic cohomology]] of the point (at the [[Tate curve]]). \hypertarget{IntegralityAndModularity}{}\subsubsection*{{Integrality and modularity}}\label{IntegralityAndModularity} A priori, the Witten genus has [[coefficients]] the [[power series]] ring $\mathbb{Q}[ [q] ]$ over the [[rational numbers]]. But under suitable conditions ([[quantum anomaly cancellation]]) it takes values in more interesting subrings. \hypertarget{ModularityForTypeIISuperstring}{}\paragraph*{{For the type II superstring}}\label{ModularityForTypeIISuperstring} The genus obtained from the [[type II superstring]] in the NS-R sector is a [[modular form]] for the [[congruence subgroup]] $\Gamma_2(2)$. (\href{Witten87a}{Witten 87a, below (13)}) See at \emph{\href{level+structure+on+an+elliptic+curve#RelationToSpinStructures}{congruence subgroup -- Relation to spin structures}} for more. Hence, with suitable normalization, the universal Witten-Ochanine genus takes values in the subring $MF_\bullet^{\mathbb{Q}}(\Gamma_0(2)) \hookrightarrow \mathbb{Q}[ [q] ]$ of [[modular forms]] for $\Gamma_0(2)\subset SL_2(\mathbb{Z})$ with rational coefficients (\hyperlink{Zagier86}{Zagier 86, item d) on page 2} based on \hyperlink{ChudnovskyChudnovsky88}{Chudnovsky-Chudnovsky 88}). \hypertarget{ModularityForHeteroticString}{}\paragraph*{{For the heterotic superstring}}\label{ModularityForHeteroticString} On manifolds with [[spin structure]] the heterotic string Witten genus has integral coeffcients, hence in the ring $\mathbb{Z}[ [ q ] ]$ (\hyperlink{ChudnovskyChudnovsky88}{Chudnovsky-Chudnovsky 88}, \hyperlink{Landweber88}{Landweber 88}), see also (\hyperlink{KreckStolz93}{Kreck-Stolz 93}, \hyperlink{Hovey91}{Hovey 91}). On manifolds with rational [[string structure]] (meaning spin structure and the [[first fractional Pontryagin class]] is at most [[torsion subgroup|torsion]]), then the Witten genus takes values in actual [[modular forms]] $MF_\bullet$ (\hyperlink{Zagier86}{Zagier 86, page 6}). On manifolds with actual [[string structure]], finally, the Witten genus factors through [[topological modular forms]] (\hyperlink{Hopkins94}{Hopkins 94}, \hyperlink{AndoHopkinsRezk}{Ando-Hopkins-Rezk 10}). \hypertarget{RelationToSuSyQMOnLoopSpace}{}\subsubsection*{{Relation to Dirac operators and supersymmetric QM on loop space}}\label{RelationToSuSyQMOnLoopSpace} Originally in (\hyperlink{Witten87a}{Witten 87a}) the elliptic genus was derived as the [[large volume limit]] of the [[index]] of the [[supercharge]] of the [[superstring]] [[worldsheet]] [[2d SCFT]]. Here the ``large volume limit'' is what restricts the oscillations of the string to be ``small''. But then in (\hyperlink{Witten87b}{Witten87b}) it was observed that if this supercharge -- the [[Dirac-Ramond operator]] -- would really behave like a [[Dirac operator on smooth loop space]], then the elliptic genus would be the $S^1$-equivariant [[index of a Dirac operator]], where $S^1$ acts by rigid rotationl of the parameterization of the loops, and by analogy standard formulas for equivariant indices in K-theory would imply the localization to the tangent spaces to the space of constant loops. Notice that the would-be [[Dirac operator on smooth loop space]] is what would realize the [[superstring]] quantum dynamics as [[supersymmetric quantum mechanics]] on [[smooth loop space]]. This observation was the original motivation for the study of [[supersymmetric quantum mechanics]] in (\hyperlink{Witten82}{Witten 82}, \hyperlink{Witten85}{Witten 85}) in the presence of a given [[Killing vector field]] (correspinding to the $S^1$-action on loop space ). \hypertarget{TwistedWittenGenus}{}\subsubsection*{{Heterotic (twisted) Witten genus, loop group representations and parameterized WZW models}}\label{TwistedWittenGenus} If the [[superstring]] in question is the [[heterotic string]] then generally there is a ``[[twisted differential string-structure|twist]]'' of its [[background fields]] by a [[gauge field]], hence by a $G$-[[principal bundle]] for $G$ some simply connected [[compact Lie group]] (notably [[E8]]). The partition function in this case is a ``twisted Witten genus'' (\hyperlink{Witten87}{Witten 87, equations (30), (31)}, \hyperlink{Brylinski90}{Brylinski 90}, \hyperlink{KL95}{KL 95}). The modularity condition then is no longer just that the [[tangent bundle]] has [[string structure]], but that together with the gauge bundle it has [[twisted string structure]], hence [[string{\tt \symbol{94}}c 2-group|String{\tt \symbol{94}}c]]-structure for $c$ the $G$-[[second Chern class]] (explicitly identified as such in (\hyperlink{ChenHanZhang10}{Chen-Han-Zhang 10}). An elegant formulation of twisted Witten genera (and proof of their rigidity) in terms of highest weight [[loop group representations]] is given in (\hyperlink{KL95}{KL 95}) along the lines of (\hyperlink{Brylinski90}{Brylinski 90}). In (\hyperlink{DistlerSharpe07}{Distler-Sharpe 07}), following suggestions around (\hyperlink{Ando07}{Ando 07}) this is interpreted geometrically in terms of fiberwise [[index|indices]] of [[parameterized WZW models]] [[associated bundle|associated]] to the given [[string 2-group|String]]-[[principal 2-bundle]]. What should be a concrete computation of the twisted Witten genus specifically for $G =$ [[E8]] in in (\hyperlink{Harris12}{Harris 12, section 4}). \hypertarget{as_the_global_character_of_sheaves_of_vertex_operator_algebras}{}\subsubsection*{{As the global character of sheaves of vertex operator algebras}}\label{as_the_global_character_of_sheaves_of_vertex_operator_algebras} For $U \subset \mathbb{C}$ an [[open subset]] of the [[complex plane]] then the space $\mathcal{D}^{ch}(U)$ of [[chiral differential operators]] on $U$ is naturally a [[super vertex operator algebra]]. For $X$ a [[complex manifold]] such that its [[first Chern class]] and [[second Chern class]] vanish over the [[rational numbers]], then this assignment gives a [[sheaf of vertex operator algebras]] $\mathcal{D}^{ch}_X(-)$ on $X$. Its [[cochain cohomology]] $H^\bullet(\mathcal{D}^{ch}_X)$ is itself a [[super vertex operator algebra]] and its super-[[Kac-Weyl character]] is proportional to the [[Witten genus]] $w(X)$ of $X$: \begin{displaymath} char H^\bullet(\mathcal{D}^{ch}_X)\propto w(X) \,. \end{displaymath} Physically this result is understood by observing that $\mathcal{D}^{ch}_X$is the sheaf of [[quantum observables]] of the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (see there for more on this) of which the [[Witten genus]] is (the [[large volume limit]] of) the [[partition function]]. As highlighted in (\hyperlink{Cheung10}{Cheung 10, p. 2}), there is a [[resolution]] by the [[chiral Dolbeault complex]] which gives a precise sense in which over a [[complex manifold]] the Witten genus is a stringy analog of the [[Todd genus]]. See (\hyperlink{Cheung10}{Cheung 10}) for a brief review, where furthermore the problem of generalizing of this construction to [[sheaves of vertex operator algebras]] over more general [[string structure]] manifolds is addressed. \hypertarget{stolz_conjecture}{}\subsubsection*{{Stolz conjecture}}\label{stolz_conjecture} The \emph{[[Stolz conjecture]]} due to (\hyperlink{Stolz96}{Stolz 96}) asserts that if $X$ is a [[closed manifold]] with [[String structure]] which furthermore admits a [[Riemannian metric]] with [[positive number|positive]] [[Ricci curvature]], then its Witten genus vanishes. \hypertarget{RelationToBPSStateCounting}{}\subsubsection*{{Relation to BPS state counting on target space}}\label{RelationToBPSStateCounting} By [[supersymmetry]] and by the same argument that controls the expression of the [[index of a Dirac operator]] in terms of [[supersymmetric quantum mechanics]], the Witten genus may be thought of as counting those string states on which the left moving [[supercharge]] acts trivially. In terms of the [[target space]] theory these are the [[BPS states]]. (reviews include \hyperlink{Dijkgraaf98}{Dijkgraaf 98}). Therefore the Witten genus may also be used as a generating function for BPS state counting. As such it has for instance been used in the microscopic explanation of [[Bekenstein-Hawking entropy]] of [[black holes]], see at \emph{[[black holes in string theory]]}. \hypertarget{relation_to_cayley_plane_bundles}{}\subsubsection*{{Relation to Cayley plane bundles}}\label{relation_to_cayley_plane_bundles} The rational Witten genus vanishes on total spaces of [[Cayley plane]]-[[fiber bundles]], and is indeed characterized by this property (\hyperlink{McTague10}{McTague 10}, \hyperlink{McTague11}{McTague 11}). \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[elliptic Chern character]] \end{itemize} [[!include genera and partition functions - table]] \hypertarget{References}{}\subsection*{{References}}\label{References} \hypertarget{ReferencesGeneralCase}{}\subsubsection*{{General}}\label{ReferencesGeneralCase} The original description of the Witten genus from [[string theory]] is due to \begin{itemize}% \item [[Edward Witten]], \emph{Elliptic Genera And Quantum Field Theory} , Commun.Math.Phys. 109 525 (1987) (\href{http://projecteuclid.org/euclid.cmp/1104117076}{Euclid}) \item [[Edward Witten]], \emph{The Index Of The Dirac Operator In Loop Space} Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) (\href{http://inspirehep.net/record/245523}{spire}) \end{itemize} based on insights in \begin{itemize}% \item [[Peter Landweber]], \emph{Elliptic Cohomology and Modular Forms}, in \emph{Elliptic Curves and Modular Forms in Algebraic Topology}, Lecture Notes in Mathematics Volume 1326, 1988, pp 55-68 ([[LandweberEllipticModular.pdf:file]]) \item [[Peter Landweber]], [[Douglas Ravenel]], [[Robert Stong]], \emph{Periodic cohomology theories defined by elliptic curves}, in [[Haynes Miller]] et al (eds.), \emph{The Cech centennial: A conference on homotopy theory}, June 1993, AMS (1995) (\href{http://www.math.sciences.univ-nantes.fr/~hossein/GdT-Elliptique/Landweber-Ravenel-Stong.pdf}{pdf}) \end{itemize} (That the [[partition function]] in (\hyperlink{Witten87}{Witten 87 (11)}) is indeed, after some normalization, an [[elliptic genus]] is (\hyperlink{Landweber88}{Landweber 88, theorem 3})). Rigorous proofs of the rigidity claims then appeared in \begin{itemize}% \item [[Clifford Taubes]], \emph{$S^1$ actions and elliptic genera}, Comm. Math. Phys., 122(3):455--526, 1989. \item [[Raoul Bott]], [[Clifford Taubes]], \emph{On the rigidity theorems of Witten}, J. of the Amer. Math. Soc., 2, 1989. \end{itemize} That a [[spin structure]] makes the Witten genus take values in integral series is due to \begin{itemize}% \item D.V. Chudnovsky, G.V. Chudnovsky, \emph{Elliptic modular functions and elliptic genera}, Topology, Volume 27, Issue 2, 1988, Pages 163--170 \item [[Matthias Kreck]], [[Stefan Stolz]], \emph{$\mathbb{H}P^2$-bundles and elliptic homology}, Acta Mathematica 171 (1993), 231--261. \item [[Mark Hovey]], \emph{Spin Bordism and Elliptic Homology} (1991) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.3277}{web}) \end{itemize} That it takes a rational [[string structure]] to make the elliptic genus land in [[modular forms]] was noticed in \begin{itemize}% \item [[Don Zagier]], \emph{Note on the Landweber-Stong elliptic genus} 1986 (\href{http://people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0078047/fulltext.pdf}{pdf}) \end{itemize} Surveys include \begin{itemize}% \item [[Gerald Höhn]], \emph{Complex elliptic genera and $S^1$-equivariant cobordism theory} (\href{http://arxiv.org/PS_cache/math/pdf/0405/0405232v1.pdf}{pdf}) \item Miranda Cheng, section 9 of \emph{Mathematical tools for string theorists}, lecture notes 2013 (\href{http://www.math.jussieu.fr/~chengm/lecture_notes/Thalys_School.pdf}{pdf}) \end{itemize} Further discussion of the [[Jacobi form]]-property of the Witten genus is in \begin{itemize}% \item [[Matthew Ando]], Christopher French, [[Nora Ganter]], \emph{The Jacobi orientation and the two-variable elliptic genus}, Algebraic and Geometric Topology 8 (2008) p. 493-539 (\href{http://www.msp.warwick.ac.uk/agt/2008/08-01/agt-2008-08-016s.pdf}{pdf}) \end{itemize} Further discussion of the relation to [[quantum anomalies]] and the [[Green-Schwarz mechanism]] ([[string structure]], [[string{\tt \symbol{94}}c structure]]) is in \begin{itemize}% \item [[Wolfgang Lerche]], B. Nilsson, A. Schellekens, N. Warner, \emph{Anomaly cancelling terms from the elliptic genus} (1987) (\href{http://lerche.web.cern.ch/lerche/papers/ellgenus1.pdf}{pdf}) \item [[Qingtao Chen]], [[Fei Han]], Weiping Zhang, \emph{Generalized Witten Genus and Vanishing Theorems}, JDG, 88 (2011) 1-39 (\href{http://arxiv.org/abs/1003.2325}{arXiv:1003.2325}) \end{itemize} Discussion of the Witten genus via a KO-valued Chern-character on elliptic cohomology is in \begin{itemize}% \item [[Haynes Miller]], \emph{The elliptic character and the Witten genus}, Contemporary mathematics, volume 96, 1989 (\href{http://dedekind.mit.edu/~hrm/papers/ell-char.pdf}{pdf}) \end{itemize} Relation to [[Cayley plane]]-[[fiber bundles]] is discussed in \begin{itemize}% \item [[Carl McTague]], \emph{The Cayley Plane and the Witten Genus} (\href{http://arxiv.org/abs/1006.0728}{arXiv:1006.0728}) \item [[Carl McTague]], \emph{The Cayley plane and String bordism} (\href{http://arxiv.org/abs/1111.4520}{arXiv:1111.4520}) \end{itemize} \hypertarget{ReferencesRelationToKacWeylCharacters}{}\subsubsection*{{Relation to Kac-Weyl characters of loop group representations}}\label{ReferencesRelationToKacWeylCharacters} The close relation of the Witten genus to [[Kac-Weyl characters]] of [[loop group representations]] has been highlighted and an elegant proof of rigidity of the Witten genus in these terms in \begin{itemize}% \item [[Kefeng Liu]], \emph{On elliptic genera and Theta functions}, Topology, Volume 35, Issue 3, July 1996, Pages 617--640 (\href{http://www.math.ucla.edu/~liu/Research/rigss1.pdf}{pdf}) \item [[Kefeng Liu]], \emph{On modular invariance and rigidity theorems}, J. Differential Geom. Volume 41, Number 2 (1995), 247-514 (\href{http://projecteuclid.org/euclid.jdg/1214456221}{EUCLID}, \href{http://www.math.ucla.edu/~liu/Research/loja.pdf}{pdf}) \end{itemize} along the lines of \begin{itemize}% \item [[Jean-Luc Brylinski]], \emph{Representations of loop groups, Dirac operators on loop space, and modular forms}, Topology, 29(4):461--480, 1990. \end{itemize} and further generalized to more general [[vertex operator algebra]] representations in (\hyperlink{DLM02}{DLM 02}). Review and survey of some of this is in \begin{itemize}% \item [[Kefeng Liu]], \emph{Modular forms and topology}, Proc. of the AMS Conference on the Monster and Related Topics, Contemporary Math. (1996) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.130.9779}{citeseer}) \end{itemize} \hypertarget{the_stolz_conjecture}{}\subsubsection*{{The Stolz conjecture}}\label{the_stolz_conjecture} The [[Stolz conjecture]] on the Witten genus is due to \begin{itemize}% \item [[Stephan Stolz]], \emph{A conjecture concerning positive Ricci curvature and the Witten genus}, Mathematische Annalen Volume 304, Number 1 (1996), \end{itemize} Reviews include \begin{itemize}% \item [[Anand Dessai]], \emph{Some geometric properties of the Witten genus} (\href{http://homeweb2.unifr.ch/dessaia/pub/papers/Arolla/DessaiArollaFinalRevised30June09.pdf}{pdf}) \end{itemize} \hypertarget{refinement_to_the_stringorientation_of_}{}\subsubsection*{{Refinement to the string-orientation of $tmf$}}\label{refinement_to_the_stringorientation_of_} The refinement of the Witten genus from values in [[modular forms]] to [[topological modular forms]] and further to a morphism of [[E-∞ rings]], hence to the [[string orientation of tmf]] is due to \begin{itemize}% \item [[Michael Hopkins]], \emph{Topological modular forms, the Witten Genus, and the theorem of the cube}, Proceedings of the International Congress of Mathematics, Z\"u{}rich 1994 (\href{http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf}{pdf}) \item [[Michael Hopkins]], \emph{Algebraic topology and modular forms}, Proceedings of the ICM, Beijing 2002, vol. 1, 283--309 (\href{http://arxiv.org/abs/math/0212397}{arXiv:math/0212397}) \item [[Matthew Ando]], [[Michael Hopkins]], [[Charles Rezk]], \emph{Multiplicative orientations of KO-theory and the spectrum of topological modular forms}, 2010 (\href{http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf}{pdf}) \end{itemize} see also remark 1.4 of \begin{itemize}% \item [[Paul Goerss]], \emph{Topological modular forms (after Hopkins, Miller and Lurie)} (\href{http://arxiv.org/PS_cache/arxiv/pdf/0910/0910.5130v1.pdf}{pdf}). \end{itemize} and for more on the [[sigma-orientation]] see \begin{itemize}% \item [[Matthew Ando]], \emph{The sigma orientation for analytic circle-equivariant elliptic cohomology}, Geom. Topol. 7 (2003) 91-153 (\href{http://arxiv.org/abs/math/0201092}{arXiv:math/0201092}) \item [[Matthew Ando]], Maria Basterra, \emph{The Witten genus and equivariant elliptic cohomology} (\href{http://arxiv.org/abs/math/0008192}{arXiv:0008192}) \end{itemize} \hypertarget{via_index_theory_of_wouldbe_dirac_operators_on_loop_space}{}\subsubsection*{{Via index theory of would-be Dirac operators on loop space}}\label{via_index_theory_of_wouldbe_dirac_operators_on_loop_space} Further literature emphasising the perspective of [[Dirac-Ramond operators]] as would-be [[Dirac operators on smooth loop space]] includes \begin{itemize}% \item [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, [[Paul Windey]], \emph{The Dirac-Ramond operator in string theory and loop space index theorems}, Nuclear Phys. B Proc. Suppl., 1A:189--215, 1987. Nonperturbative methods in field theory (Irvine, CA, 1987). \item [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano,[[Paul Windey]], \emph{String theory and loop space index theorems}, Comm. Math. Phys., 111(1):1--10, 1987. \item [[Jean-Luc Brylinski]], \emph{Representations of loop groups, Dirac operators on loop space, and modular forms}, Topology, 29(4):461--480, 1990. \item [[Gregory Landweber]], \emph{Dirac operators on loop space} PhD thesis (Harvard 1999) (\href{http://math.bard.edu/greg/LoopDirac.pdf}{pdf}) \item [[Orlando Alvarez]], [[Paul Windey]], \emph{Analytic index for a family of Dirac-Ramond operators}, Proc. Natl. Acad. Sci. USA, 107(11):4845--4850, 2010 \end{itemize} The observation thazt the realization of the [[Dirac-Ramond operator]] as a [[Dirac operator on smooth loop space]] would realize superstring quantum dynamics as [[supersymmetric quantum mechanics]] on [[smooth loop space]] is what inspired the observations in \begin{itemize}% \item [[Edward Witten]], \emph{Supersymmetry and Morse theory} J. Differential Geom. Volume 17, Number 4 (1982), 661-692. (\href{http://projecteuclid.org/euclid.jdg/1214437492}{Euclid}, \href{http://www.cimat.mx/~gil/docencia/2012/teoria_de_morse/witten_supersymmetry_and_morse_theory.pdf}{pdf}, \href{http://inspirehep.net/record/176416?ln=de}{spire}) \end{itemize} and \begin{itemize}% \item [[Edward Witten]], \emph{Global anomalies in string theory}, in W. Bardeen and A. White (eds.) \emph{Symposium on Anomalies}, Geometry, Topology, pp. 61--99. World Scientific, 1985 \end{itemize} \hypertarget{via_sheaves_of_super_vertex_operator_algebras}{}\subsubsection*{{Via (sheaves of) super vertex operator algebras}}\label{via_sheaves_of_super_vertex_operator_algebras} Formalization via [[super vertex operator algebras]] is discussed in \begin{itemize}% \item Hirotaka Tamanoi, \emph{Elliptic Genera and Vertex Operator Super-Algebras}, 1999 \item Chongying Dong, [[Kefeng Liu]], Xiaonan Ma, \emph{Elliptic genus and vertex operator algebras} (\href{http://arxiv.org/abs/math/0201135}{arXiv:math/0201135}) \end{itemize} and for the [[topological twist|topologically twisted]] [[2d (2,0)-superconformal QFT]] (the [[heterotic string]] with enhanced supersymmetry) via [[sheaves of vertex operator algebras]] in \begin{itemize}% \item Pokman Cheung, \emph{The Witten genus and vertex algebras} (\href{http://arxiv.org/abs/0811.1418}{arXiv:0811.1418}) \end{itemize} which is based on the detailed construction via [[chiral differential operators]] in \begin{itemize}% \item [[Vassily Gorbounov]], [[Fyodor Malikov]], [[Vadim Schechtman]], \emph{Gerbes of chiral differential operators}, Math. Res. Lett. 7(1), 55--66 (2000) (\href{http://arxiv.org/abs/math/9906117}{arXiv:math/9906117}, \href{http://arxiv.org/abs/math/0003170}{arXiv:math/0003170}, \href{http://arxiv.org/abs/math/0005201}{arXiv:math/0005201}) \end{itemize} \hypertarget{via_other_methods}{}\subsubsection*{{Via other methods}}\label{via_other_methods} In terms of [[(2,1)-dimensional Euclidean field theories and tmf]]: \begin{itemize}% \item [[Stephan Stolz]], [[Peter Teichner]], \emph{Supersymmetric field theories and generalized cohomology} in [[Hisham Sati]], [[Urs Schreiber]] (eds.), \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, Proceeding of Symposia in Pure Mathematics, volume 83, AMS (2011) (\href{http://arxiv.org/abs/1108.0189}{arXiv:1108.0189}) \end{itemize} \ldots{}and in terms of [[factorization algebras]] in \begin{itemize}% \item [[Kevin Costello]], \emph{A geometric construction of the Witten genus, II} (\href{http://arxiv.org/abs/1112.0816}{arXiv:1112.0816}) \end{itemize} \hypertarget{ReferencesTwistedCase}{}\subsubsection*{{Twisted case}}\label{ReferencesTwistedCase} The twisted Witten genus in the present of [[background gauge field]] hence for a [[twisted string structure]]/[[string{\tt \symbol{94}}c structure]] is discussed in terms of [[twisted string structures]] in \begin{itemize}% \item [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], \emph{Generalized Witten Genus and Vanishing Theorems}, Journal of Differential Geometry 88.1 (2011): 1-39. (\href{http://arxiv.org/abs/1003.2325}{arXiv:1003.2325}) \item Jianqing Yu, Bo Liu, \emph{On the Witten Rigidity Theorem for $String^c$ Manifolds}, Pacific Journal of Mathematics 266.2 (2013): 477-508. (\href{http://arxiv.org/abs/1206.5955}{arXiv:1206.5955}) \end{itemize} based on formulas from \begin{itemize}% \item [[Qingtao Chen]], [[Fei Han]], \emph{Elliptic Genera, Transgression and Loop Space Chern-Simons Forms} (\href{http://arxiv.org/abs/math/0611104}{arXiv:math/0611104}) \end{itemize} For the moment see the references at \emph{[[string{\tt \symbol{94}}c structure]]} for more on this. A geometric interpretation of this in terms of [[parameterized WZW models]] is suggested in \begin{itemize}% \item [[Jacques Distler]], [[Eric Sharpe]], section 8.5 \emph{Heterotic compactifications with principal bundles for general groups and general levels}, Adv. Theor. Math. Phys. 14:335-398, 2010 (\href{http://arxiv.org/abs/hep-th/0701244}{arXiv:hep-th/0701244}) \end{itemize} and with more emphasis on [[equivariant elliptic cohomology]] in \begin{itemize}% \item [[Matthew Ando]], \emph{Equivariant elliptic cohomology and the Fibered WZW models of Distler and Sharpe}, \href{http://www.math.ucsb.edu/~drm/GTPseminar/2007-fall.php}{talk 2007} (\href{http://www.math.ucsb.edu/~drm/GTPseminar/notes/20071026-ando/20071026-malmendier.pdf}{lecture notes pdf}) \end{itemize} An explicit computation for an [[E8]]-gauge bundle is in section 4 of \begin{itemize}% \item [[Chris Harris]], \emph{The Index Bundle for a Family of Dirac-Ramond Operators} (\href{http://arxiv.org/abs/1202.2049}{arXiv:1202.2049}) \end{itemize} \hypertarget{relation_to_bps_state_counting_on_target_space_2}{}\subsubsection*{{Relation to BPS state counting on target space}}\label{relation_to_bps_state_counting_on_target_space_2} A survey of elliptic string genera with more context within [[string theory]] and in particular with discussion of the relation to [[BPS state]] counting is \begin{itemize}% \item [[Robbert Dijkgraaf]], \emph{Fields, strings, matrices and symmetric products}, lecture notes 1998 (\href{http://www.cgtp.duke.edu/ITP99/dijkgraaf/sym.pdf}{pdf}) \end{itemize} [[!redirects Witten genera]] [[!redirects twisted Witten genus]] [[!redirects twisted Witten genera]] \end{document}