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\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*{equivariant elliptic cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak \noindent\hyperlink{MotivationFromAlgebraicTopology}{Motivation from algebraic topology}\dotfill \pageref*{MotivationFromAlgebraicTopology} \linebreak \noindent\hyperlink{InterpretationInQuantumFieldTheory}{Interpretation in Quantum field theory/String theory}\dotfill \pageref*{InterpretationInQuantumFieldTheory} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{1equivariant_elliptic_cohomology}{1-Equivariant elliptic cohomology}\dotfill \pageref*{1equivariant_elliptic_cohomology} \linebreak \noindent\hyperlink{2EquivariantEllipticCohomology}{2-Equivariant elliptic cohomology}\dotfill \pageref*{2EquivariantEllipticCohomology} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{RelationToConformalBlocks}{Relation to conformal blocks of the WZW model}\dotfill \pageref*{RelationToConformalBlocks} \linebreak \noindent\hyperlink{RelationToLoopGroupRepresentations}{Relation to loop group representations}\dotfill \pageref*{RelationToLoopGroupRepresentations} \linebreak \noindent\hyperlink{RelationToLineBundleOnBSpin}{Relation to the Chern-Simons $\infty$-line bundle on $\mathbf{B}G$}\dotfill \pageref*{RelationToLineBundleOnBSpin} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{relation_to_loop_group_representations_2}{Relation to loop group representations}\dotfill \pageref*{relation_to_loop_group_representations_2} \linebreak \noindent\hyperlink{relation_to_line_bundles}{Relation to $\infty$-line bundles}\dotfill \pageref*{relation_to_line_bundles} \linebreak \noindent\hyperlink{relation_to_superstrings_and_the_witten_genus}{Relation to superstrings and the Witten genus}\dotfill \pageref*{relation_to_superstrings_and_the_witten_genus} \linebreak \hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea} Equivariant elliptic cohomology is, or is supposed to be, an [[equivariant cohomology|equivariant]] version of [[elliptic cohomology]], hence a higher [[chromatic homotopy theory|chromatic]] analogue of [[equivariant K-theory]]. As usual in [[equivariant cohomology]], there is a ``naive'' version and refinements thereof, and typically it is these refinements that one is really interested in. The traditional motivation of these from [[algebraic topology]]/[[homotopy theory]] are indicated below in \begin{itemize}% \item \emph{\hyperlink{MotivationFromAlgebraicTopology}{Motivation from algebraic topology}} \end{itemize} Despite that motivation, the precise nature of the resulting ``genuine'' equivariant elliptic cohomology may tend to seem a bit mysterious and also a bit baroque in its technical ingredients, some of which may appear a bit unexpected in the literature. A clear conceptual picture of what equivariant elliptic cohomology is about is obtained by regarding it as encoding aspects of low dimensional [[quantum field theory]] and [[worldsheet]] [[string theory]]; this is indicated further below in \begin{itemize}% \item \emph{\hyperlink{InterpretationInQuantumFieldTheory}{Interpretation in Quantum field theory/String theory}}. \end{itemize} \hypertarget{MotivationFromAlgebraicTopology}{}\subsubsection*{{Motivation from algebraic topology}}\label{MotivationFromAlgebraicTopology} Given any [[cohomology theory]] $E$ which may be evaluated on arbitrary [[topological spaces]], then for $G$ a [[compact Lie group]] the ``naive'' $G$-[[equivariant cohomology|equivariant E-cohomology]] of the point is the $E$-cohomology of the [[classifying space]] $B G$ of $G$ (which is equivalently the [[delooping]] \begin{displaymath} B G \simeq \ast //G \end{displaymath} of $G$ regarded as an [[∞-group]], see at \emph{[[∞-action]]} for how that encodes actions on structures above it): \begin{displaymath} E_G^\bullet(\ast)_{naive} \coloneqq E^\bullet(B G) \,. \end{displaymath} In a discussion in the context of [[geometric homotopy theory]] it is clear what is ``naive'' about this definition: since $G$ has geometric structure of which $B G$ remembers only the underlying [[geometrically discrete infinity-groupoid|bare homotopy type]], one would instead want to use the something like the [[smooth stack]] $\mathbf{B}G$ (the [[moduli stack]] of $G$-[[principal bundle]]), then somehow make good sense of $\mathbf{E}^\bullet(\mathbf{B}G)$ where now $\mathbf{E}$ is some [[sheaf of spectra]] and then declare this to be the actual $G$-equivariant $E$-cohomology. The traditional argument however proceeds as follows: if $E$ is a [[complex oriented cohomology theory]] then (essentially by definition) for $G = U(1)$ the [[circle group]] then $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$ is the algebra of [[formal power series]] which one may think of as the [[algebra of functions]] on the [[formal geometry|formal]] [[neighbourhood]] of a point in some larger [[space]] $M_{S^1}$. For instance in the simpler case of [[equivariant K-theory]] this has long been well understood: here the genuine $U(1)$-equivariant cohomology of the point is the [[representation ring]] $K_{U(1)}(\ast) \simeq \mathbb{Z}[ [ t^{-1}, t] ]$ which happens to be the [[algebra of functions]] on the [[multiplicative group]]; while by [[complex oriented cohomology theory|complex orientation]] the naive equivariant cohomology $K^\bullet(B U(1)) \simeq \mathbb{Z}[ [t] ]$ is equivalently the algebra of functions on (just) the formal multiplicative group. Based on this one may want to consider an $E$-[[(∞,1)-module bundle|∞-line bundle]] over the full space $M_{S^1}$ and take the genuine $E$-equivariant cohomology to be the [[global sections]] of that. (Specifically in [[elliptic cohomology]] that space $M_{S^1}$ is equivalent to the [[elliptic curve]] $C$ that gives the theory its name, but in some sense discussed \hyperlink{InterpretationInQuantumFieldTheory}{below} the spaces $M_{S^1}$ and $C$ arise conceptually differently and it is a fairly deep coincidence that they are in fact equivalent, which one may want to remember.) In this way equivariant elliptic cohomology was defined in (\hyperlink{Grojnowski94}{Grojnowski 94}, \hyperlink{GinzburgKapranovVasserot95}{Ginzburg-Kapranov-Vasserot 95}, ), see also (\hyperlink{Ando00}{Ando 00, sections II.8, II.9}). More generally then genuine $G$-equivariant elliptic cohomology should assign to every $G$-action on some [[space]] $X$ a sheaf $\mathcal{F}$ of algebras over the $G$-equivariant cohomology of the point, and then the $G$-equivariant elliptic cohomology of $X$ should be the global sections of this. While this can be made to work, it remains maybe unclear what these spaces $M_G$ ``mean'' and what makes them related to equivariance and elliptic cohomology. Specifically, $M_G$ turns out to be essentially the [[moduli space of flat connections]] ($G$-[[principal connections]]) on the given elliptic curve (see remark \ref{ModuliSpaceOfFlatConnections} below), which suggests strong relations to [[Chern-Weil theory]] that are not apparent here. That is considerably clarified by regarding elliptic cohomology as the [[coefficients]] for [[motivic quantization|cohomological quantization]] of 3d and 2d [[quantum field theory]], to which we now turn. \hypertarget{InterpretationInQuantumFieldTheory}{}\subsubsection*{{Interpretation in Quantum field theory/String theory}}\label{InterpretationInQuantumFieldTheory} We now try to give a maybe more conceptual explanation of what genuine equivariant (and [[twisted cohomology|twisted]]) elliptic cohomology is about, when regarded over all [[elliptic curves]] (hence: ``genuine equivariant twisted [[tmf]]''). The conceptual role of plain [[elliptic cohomology]] (not equivariant) was considerably clarified when (\href{Witten+genus#Witten87}{Witten 87}) identified the [[elliptic genus]] (an element in the elliptic cohomology of a point) with the ([[large volume limit]] of) the [[partition function]] of a [[2d superconformal field theory]] -- the [[worldsheet]] [[quantum field theory]] of the ``[[superstring]]'' -- where the [[worldsheet]] [[Riemann surface]] of the string is identified with the given [[elliptic curve]]. If the [[superstring]] here is specifically the [[heterotic string]] then its dynamics and hence its [[partition function]] depends in general not just on the [[target space|target spacetime]] $X$ (of which it yields the [[elliptic genus]]) but also on a [[background field|background]] [[gauge field]] for some [[gauge group]] $G$, underlying which is a $G$-[[principal bundle]] over that spacetime. In (\href{Witten+genus#KL95}{Kefeng Liu, 95}) a succinct description of these ``twisted'' elliptic genera, twisted by a $G$-principal bundle, was given in terms of [[Kac-Weyl characters]] of [[associated bundle|associated]] [[loop group]] bundles. In (\hyperlink{DistlerSharpe07}{Distler-Sharpe 07}) the chiral [[WZW-model]] part of the [[heterotic string]] [[2d SCFT]] which emobodies the effect of this background gauge bundle was realized geometrically as a bundle of [[parameterized WZW models]] over $X$, and (\hyperlink{Ando07}{Ando 07}) highlighted (see \hyperlink{DistlerSharpe07}{Distler-Sharpe 07, section 8.5}) that this provides the string theoretic interpretation of (\href{Witten+genus#KL95}{Kefeng Liu, 95}), in particular (\hyperlink{Ando07}{Ando 07}) indicates that the corresponding twisted Witten genus lands in $G$-equivariant elliptic cohomology. Now in the special case that $X$ here is the point, then any [[parameterized WZW model]] over $X$ is just the plain single [[WZW model]], while the plain [[Witten genus]] of $X$ vanishes. So in this case the interpretation of (\hyperlink{Ando07}{Ando 07}) says that the [[partition function]] of the $G$-[[WZW model]] should be an element in the $G$-equivariant elliptic cohomology of the point. But that partition function is an element in the space of [[conformal blocks]] of the WZW-model over a [[torus]] worldsheet, hence over a complex [[elliptic curve]]. Therefore the $G$-equivariant elliptic cohomology of the point should accommodate the [[conformal blocks]] of the WZW model over the given elliptic curve. (See also below at \emph{\hyperlink{RelationToConformalBlocks}{Properties -- Relation to conformal blocks}}). Next, by the [[holographic principle]] of the [[AdS3-CFT2 and CS-WZW correspondence|3dCS/2dWZW-correspondence]], the space of conformal blocks of the WZW model on a surface is identified with the [[space of quantum states]] of [[Chern-Simons theory]] over that surface. This in turn, by the general rules of [[geometric quantization]] and specifically by the discussion at \emph{[[quantization of 3d Chern-Simons theory]]}, is the space of holomorphic sections of a [[prequantum line bundle]] over the [[moduli space of flat connections]] ($G$-[[principal connections]]) $M_G$ over the given elliptic curve. And \emph{that} is indeed what $G$-equivariant elliptic cohomology assigns to the point. In other words, universal $G$-equivariant elliptic cohomology (meaning: we vary over the [[moduli space of elliptic curves]]), hence \emph{$G$-equivariant [[tmf]]} of the point, is essentially the [[modular functor]] of [[3d Chern-Simons theory]]. This last statement appears as (\hyperlink{Lurie}{Lurie 09, remark 5.2}). But observe that actually it is a bit more: a [[modular functor]] assigns just an abstract vector space to a surface, which however is meant to be obtained by the process of [[quantization]] of [[3d Chern-Simons theory]], explicitly as the space of holomorphic sections of the [[prequantum line bundle]] (over [[phase space]], which here is the [[moduli space of flat connections]] $M_G$ on the given elliptic curve). (Beware that, while this is true over the complex numers,as discussed \href{moduli+space+of+connections#FlatConnectionsOverATorus}{here}, it is at least subtle in the algebro geometric context of elliptic cohomology, see Jacob Lurie's MO comment \href{http://mathoverflow.net/a/226716/381}{here}). Equivariant elliptic cohomology/tmf actually remembers this [[quantization]] process and not just the resulting [[space of quantum states]] in that it actually assigns to an [[elliptic curve]] $C$ and suitable [[Lie group]] $G$ that [[prequantum line bundle]] over the [[moduli space of elliptic curves]] (or equivalently its [[sheaf]] of sections). Notice that this [[local prequantum field theory|pre-quantum]] information is crucial for deep aspects in the context of [[3d Chern-Simons theory]] and the 2d [[Wess-Zumino-Witten model]]: the [[holographic principle|holographic relation]] that identifies the latter as the [[boundary field theory]] of the former (explicitly so by the [[FRS-theorem on rational 2d CFT]]) needs as input not just the quantized Chern-Simons [[3d TQFT]], which will assign an ``abstract'' vector space to a surface, but needs to know how this space arose via [[quantization]] by choosing [[polarizations]] in the form of [[conformal structures]] on the elliptic curves, such as to be actually identified with a space of [[conformal blocks]]. (In the context of the [[Reshetikhin-Turaev construction]] of the Chern-Simons [[3d TQFT]] this information is in a choice of [[equivalence of categories|equivalence]] of the given [[modular tensor category]] with the [[category of representations]] of a rational [[vertex operator algebra]]). In summary we have as a \textbf{slogan} that: \begin{itemize}% \item \emph{$G$-Equivariant $tmf$ over the point is essentially an incarnation of the [[local prequantum field theory|pre-quantum]] [[modular functor]]} of [[3d Chern-Simons theory|3d G-Chern-Simons theory]] over [[genus of a surface|genus]]-1 surfaces/[[elliptic curves]]\_, together with the [[quantization]]-process of that to the actual [[modular functor]]\_. \end{itemize} Moreover, by the above reasoning via (\hyperlink{Ando07}{Ando 07}) and using the [[AdS3-CFT2 and CS-WZW correspondence|3dCS/2dWZW holographic correspondence]] we also have the interpretation of $G$-equivariant tmf (universal $G$-equivariant elliptic cohomology) over a more general space $X$: the space of [[conformal blocks]] of a bundle of [[parameterized WZW models]] over $X$, regarded pointwise as the gauge coupling part of the twisted [[Witten genus]]. Here all the statements on the QFT/string theory side involve a parameter called the ``level'', which is the [[characteristic class]] of the [[universal Chern-Simons circle 3-bundle]] that is the [[prequantum n-bundle|prequantum 3-bundle]] governing the [[3d Chern-Simons theory]] (whose [[transgression]] to the [[moduli space of flat connections]] is the ``theta''-[[prequantum line bundle]] there). On the cohomological side this corresponds to a [[twisted cohomology|twist]] of the cohomology theory. Now with equivariant $tmf$ identified with the [[quantization of Chern-Simons theory]] in [[dimension]] 2 this way (the [[modular functor]] together with its pre-quantum origin via [[geometric quantization]]), the physical desireability of [[local quantum field theory]] (``[[extended TQFT]]'') suggests to ask for a refinement of this also to dimensions 1 and 0, such that the higher dimensional data arises by ``tracing''/[[transgression]]. There is such a [[local prequantum field theory]] refinement of 3d Chern-Simons theory, governed in dimension 0 by the [[universal Chern-Simons circle 3-bundle]] regarded as a [[prequantum n-bundle|prequantum 3-bundle]]. Indeed, the [[transgression]] of that to the [[moduli space of flat connections]] is precisely the [[prequantum bundle]] over $M_G$ that appears in the above discussion (e.g. [[schreiber:Extended higher cup-product Chern-Simons theories|FSS 12]], [[schreiber:A higher stacky perspective on Chern-Simons theory|FSS 13]]). Now that [[universal Chern-Simons circle 3-bundle]] in turn is modulated by the geometric refinement of the universal [[second Chern class]]/[[first fractional Pontryagin class]] given by a map of [[smooth infinity-stacks]] of the form $\mathbf{B}G \to \mathbf{B}^3 U(1)$. This exhibits a homomorphism of [[smooth infinity-group]] $G \to \mathbf{B}^2 U(1)$ (to the [[circle n-group|circle 3-group]]) and so one might wonder if there is a way to ``globalize'' the equivariance of equivariant elliptic cohomology (in the sense of ``[[global equivariant homotopy theory]]'') such that it may be evaluated also on [[n-group|3-groups]] such as $\mathbf{B}^2 U(1)$ and such that the homomorphism above then \emph{induces} the previous 1-equivariant data by transgression. Such a ``localization'' of equivariant elliptic cohomology seems to be just what is being vaguely hinted at in (\hyperlink{Lurie}{Lurie, section 5.1}) under the name ``\hyperlink{2EquivariantEllipticCohomology}{2-equivariant elliptic cohomology}'', we discuss this in more detail \hyperlink{2EquivariantEllipticCohomology}{below}. Hence we arrive at a refinement of the above \textbf{slogan}: \begin{itemize}% \item \emph{2-Equivariant $tmf$ over the point should be [[3d Chern-Simons theory]] in dimension 2 (hence the [[modular functor]]) [[extended TQFT|localized]] to dimensions 1 and 0 as a [[local prequantum field theory]] and including its [[higher geometric quantization]] in these dimensions 0,1, and 2}. \end{itemize} \begin{quote}% A formal systematic discussion of this story in [[schreiber:Quantization via Linear homotopy types|cohomological quantization]] is going to be in (\hyperlink{NS}{Nuiten-S.}). It essentially amounts to the discussion of diagram (\hyperlink{ESI14}{0.0.4 b)}. \end{quote} [[!include moduli of higher lines -- table]] \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{1equivariant_elliptic_cohomology}{}\subsubsection*{{1-Equivariant elliptic cohomology}}\label{1equivariant_elliptic_cohomology} Let $G$ be a [[compact Lie group]]. Write $T \hookrightarrow G$ for its [[maximal torus]] and $W$ for its [[Weyl group]]. Let $E \in CRing_\infty$ be an [[elliptic spectrum|elliptic]] [[E-∞ ring]] [[spectrum]] with [[elliptic curve]] $A \to Spec E$. \begin{defn} \label{TheModuliSpace}\hypertarget{TheModuliSpace}{} Write \begin{displaymath} A_G \coloneqq \left[\left[T,\mathbb{T}\right],A\right]/W \;\; \in Sch_{/Spec E} \end{displaymath} for the [[derived scheme]] formed from the [[character group]] of the [[maximal torus]] mapped into the given [[elliptic curve]]. \end{defn} \begin{remark} \label{}\hypertarget{}{} This $A_G$ is the [[moduli space|moduli scheme]] of [[stable bundle|semistable]] $G$-[[principal bundles]] over the dual elliptic curve $A^\vee$ (\hyperlink{GinzburgKapranovVasserot95}{Ginzburg-Kapranov-Vasserot 95, (1.4.5)}). \end{remark} \begin{remark} \label{ModuliSpaceOfFlatConnections}\hypertarget{ModuliSpaceOfFlatConnections}{} For geometry over the [[complex numbers]] and $A = \mathbb{C}/\tau$ a 2-[[torus]], the scheme $A_G$ is the [[moduli space of flat connections]] on $A$, by the discussion at \emph{\href{moduli+space+of+connections#FlatConnectionsOverATorus}{moduli space of connections -- flat connections over a torus}}. \end{remark} Wtite $L Top_G$ for the collection of [[G-CW complexes]]. Write $Orb(G)$ for the [[orbit category]] of $G$. By [[Elmendorf's theorem]] we have a [[equivalence of (∞,1)-categories]] \begin{displaymath} L Top_G \stackrel{\simeq}{\longrightarrow} PSh_\infty(Orb(G)) \,. \end{displaymath} Let the [[global points]] of the [[elliptic curve]] $A$ over $Spec E$ be equipped with an \emph{orientation} in the sense of a non-degenerate [[∞-group]] homomorphism of the form \begin{displaymath} B U(1) \longrightarrow A(Spec E) \end{displaymath} \begin{prop} \label{}\hypertarget{}{} Induced form this (\ldots{}) is an [[essential geometric morphism]] \begin{displaymath} PSh_\infty(Orb(G)) \stackrel{\overset{(-)\otimes_G A}{\longrightarrow}}{\stackrel{\overset{}{\leftarrow}}{\underset{}{\longrightarrow}}} Sh_\infty(Aff_E)_{/A_G} \end{displaymath} to the [[slice (∞,1)-topos]] over $A_G$. \end{prop} (\hyperlink{Gepner05}{Gepner 05, theorem 3}) \begin{defn} \label{}\hypertarget{}{} Let \begin{displaymath} \mathcal{O} \;\colon\; Sh_\infty(Aff_E) \longrightarrow Aff_E \simeq E Alg^{op} \end{displaymath} be the [[left adjoint]] to the [[(∞,1)-Yoneda embedding]] as discussed at \emph{[[function algebras on ∞-stacks]]}. \end{defn} \begin{prop} \label{}\hypertarget{}{} The composite \begin{displaymath} L Top_G \hookrightarrow PSh_\infty(Orb(G)) \stackrel{(-)\otimes_G A}{\longrightarrow} Sh_\infty(Aff_E)_{/A_G} \stackrel{\mathcal{O}}{\longrightarrow} E Alg^{op} \end{displaymath} takes a space with $G$-[[action]] to its $G$-equivariant elliptic cohomology spectrum. \end{prop} (\hyperlink{Gepner05}{Gepner 05, theorem 4}) \hypertarget{2EquivariantEllipticCohomology}{}\subsubsection*{{2-Equivariant elliptic cohomology}}\label{2EquivariantEllipticCohomology} \begin{quote}% under construction, tentative \end{quote} In (\hyperlink{Lurie}{Lurie, section 5.1}) is a vague mentioning of a more general perspective, where one evaluates elliptic cohomology not just on [[action groupoids]] of a [[group]], such as $B G$ but also on [[homotopy quotients]] of [[2-groups]], such as notably the [[string 2-group]], and how that gives a more conceptual picture. The following are some remarks on how to possibly realize this and at the same time refine it to geometric cohomology ([[differential cohomology]]). Tentative. Handle with care. So let $G$ be a simple, simply connected [[compact Lie group]]. Regard $\mathbf{B}G$ in [[Smooth∞Grpd]] = $Sh_\infty(SmthMfd)$. Then by the discussion at [[Lie group cohomology]] we have \begin{displaymath} \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times) \simeq H(B G, K(\mathbb{Z},4)) \simeq \mathbb{Z} \,. \end{displaymath} The [[∞-group extension]] classified by $k \in \mathbb{Z} \in \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times)$ is the [[string 2-group]] at level $k$ \begin{displaymath} \itexarray{ \mathbf{B}\mathbb{C}^\times &\longrightarrow& \mathbf{B}String_k(G) \\ && \downarrow \\ && \mathbf{B}G &\stackrel{k\mathbf{c}}{\longrightarrow}& \mathbf{B}^3 \mathbb{C}^\times } \end{displaymath} This cocycle has a [[differential cohomology]]-refinement to the [[universal Chern-Simons 3-connection]] \begin{displaymath} k \mathbf{L} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 \mathbb{C}^\times_{conn} \end{displaymath} Now given a [[torus]] $E = T^2$, regarded, for the moment, as a [[smooth manifold]], we have the [[transgression]] of the definition cocycle \begin{displaymath} \exp\left( \tfrac{i}{\hbar} \int_{E} k \mathbf{L} \right) \;\colon\; [E, \mathbf{B}G_{conn}] \stackrel{[E, k \mathbf{L}]}{\longrightarrow} [E, \mathbf{B}^3 \mathbb{C}^\times_{conn}] \stackrel{\exp\left(\tfrac{i}{\hbar} \int_{E}(-)\right)}{\longrightarrow} \mathbf{B}\mathbb{C}^\times_{conn} \end{displaymath} which now defines a $\mathbb{C}^\times$ [[bundle with connection]] on the [[moduli stack of connections]] on $E$. We can restrict to the [[moduli stack of flat connections]], the [[phase space]] of $G$-[[Chern-Simons theory]]. This is the [[Hitchin connection]]. Consider then a collection of tori $E$ parameterized trivially over some parameter space $B$. \begin{displaymath} E \times B \to B \,. \end{displaymath} Then the above yields \begin{displaymath} [(\Pi E) \times B, \mathbf{B}G] \stackrel{}{\longrightarrow} \mathbf{B}[B,\mathbb{C}^\times] \end{displaymath} hence yields a $[B,\mathbb{C}^\times]$-bundle over the [[moduli space]] of $B$-collections of flat connections on $E$. Now we want to consider this for the case that $B$ is a space in [[spectral geometry]]. To that end, pass to the larger [[(∞,1)-topos]] of [[smooth E-∞ groupoids]] over the [[complex numbers]]. Let $\mathbb{G}_m$ there denote the object which to a pair consisting of a [[smooth manifold]] $U$ and an [[E-∞ ring]] $R$ assigns \begin{displaymath} \mathbb{G}_m \;\colon\; (U, R) \mapsto GL_1(R) \otimes C^\infty(U,\mathbb{C}^\times) \end{displaymath} hence the [[tensor product]] of the [[∞-group of units]] of $R$ with the underlying [[abelian group]] of [[smooth functions]] on $X$ with values in $\mathbb{C}^\times$. Let then $A \in CAlg_\infty$ be an [[E-∞ ring]], and take now $B = Spec(A)$. Write \begin{displaymath} E \to Spec(A) \end{displaymath} for a $B$-collection of [[tori]], now taken to be an [[elliptic curve]] over $Spec(A)$. Since for a [[torus]] its [[fundamental group]] is [[isomorphic]] to its [[character group]] (via the canonical non-degenrarate [[bilinear form]] on both), we take the [[fundamental groupoid]] $\Pi(E)$ now to be \begin{displaymath} \mathbf{B}[E, \mathbb{G}_m] \,. \end{displaymath} Then since $\mathbb{C}^\times = \mathbb{G}_m$ is the [[multiplicative group]] in this context, we have now (and there is a subtlety here\ldots{}) that maps \begin{displaymath} Spec(A) \longrightarrow \mathbb{G}_m \end{displaymath} are equivalently elements in the [[∞-group of units]] of $A$. So we should get an $A$-[[(∞,1)-module bundle]] modulated by \begin{displaymath} \chi \colon [ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G ] \longrightarrow \mathbf{B}GL_1(A) \,. \end{displaymath} Forming its space of co-sections yields, by the discussion at [[Thom spectrum]], the $\chi$-[[twisted cohomology|twisted]] A-[[cohomology]] [[spectrum]] \begin{displaymath} A^\chi([ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G ]) \,. \end{displaymath} And that should be the $G$-``equivariant'' elliptic cohomology of the point. Actually the [[motivic quantization]] of $G$-[[Chern-Simons theory]]. (\ldots{}) \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{RelationToConformalBlocks}{}\subsubsection*{{Relation to conformal blocks of the WZW model}}\label{RelationToConformalBlocks} For $G$ a compact, simple and simply connected Lie group, consider the [[string 2-group]] [[∞-group extension]] \begin{displaymath} \itexarray{ \mathbf{B}^2 U(1) &\to& \mathbf{B}String \\ && \downarrow \\ && \mathbf{B}G } \,. \end{displaymath} The corresponding [[higher moduli stacks]] of [[flat ∞-connections]] on an [[elliptic curve]] $T$ form the [[∞-group extension]] \begin{displaymath} \itexarray{ [\Pi(T),\mathbf{B}^2 U(1)] &\to& [\Pi(T),\mathbf{B}String] \\ && \downarrow \\ && [\Pi(T), \mathbf{B}G] } \,. \end{displaymath} Now passing to the [[0-truncation]] turns the bottom piece into the [[moduli space of flat connections]] on the torus, which is $A_G$, def. \ref{TheModuliSpace}, remark \ref{ModuliSpaceOfFlatConnections}. By the discussion at \href{smooth+infinity-groupoid#StrucChernSimons}{smooth higher holonomy} the 0-truncation of the top left piece is $U(1)$, so under 0-truncation we should get a $U(1)$-[[principal bundle]] \begin{displaymath} \itexarray{ U(1) &\longrightarrow& \tau_0 [\Pi(T),\mathbf{B}String] \\ && \downarrow \\ && A_G } \,. \end{displaymath} This state of affairs is hinted at in (\hyperlink{Lurie}{Lurie, section 5.1}). More in detail, notice that the [[string 2-group]] extension is modulated by a map \begin{displaymath} \mathbf{c} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 U(1)_{conn} \end{displaymath} and the above circle-bundle is modulated by the [[transgression]] of that \begin{displaymath} \exp\left( \tfrac{i}{\hbar} \int_{T} \mathbf{c} \right) \;\colon\; [T, \mathbf{B}G_{conn}] \stackrel{[T,\mathbf{c}]}{\longrightarrow} [T, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(\tfrac{i}{\hbar} (-) )}{\longrightarrow} \mathbf{B}U(1)_{conn} \,. \end{displaymath} (By the discussion at [[schreiber:differential cohomology in a cohesive topos|dcct]].) By the general discussion at [[quantization of Chern-Simons theory]] and the [[holography|holograpic]] [[AdS3-CFT2 and CS-WZW correspondence|CS-WZW correspondence]], the space of [[sections]] of this line bundle is the space of [[conformal blocks]] of the [[Wess-Zumino-Witten model]] on $T$. (This statement also appears as (\hyperlink{Lurie}{Lurie, remark 5.2})). \hypertarget{RelationToLoopGroupRepresentations}{}\subsubsection*{{Relation to loop group representations}}\label{RelationToLoopGroupRepresentations} When restricting the above general construction to the [[Tate curve]], then the confromal blocks become [[loop group representations]] (when over the complex numebers, at least) (\hyperlink{Ando00}{Ando00, theorem 10.10}). In terms of differential geometry ([[schreiber:differential cohomology in a cohesive topos|dcct]]) consider the map \begin{displaymath} G \longrightarrow [S^1, \mathbf{B}G_{conn}] \end{displaymath} which locally sends a group elemnent $g$ to the constant [[principal connection]] on the circle with $g$ as its [[holonomy]]. This induces an inclusion \begin{displaymath} [S^1, G] \hookrightarrow [S^1, [S^1, \mathbf{B}G_{conn}]] \simeq [T, \mathbf{B}G_{conn}] \end{displaymath} and pulling the above WZW circle bundle back along this inclusion yields the bundle on the [[loop group]] which is the [[prequantum bundle]] whose [[geometric quantization]] yields the [[loop group representations]] of [[positive energy representations|positive energy]]. Algebraically, this corresponds to evaluating equivariant elliptic cohomology on the [[Tate curve]], this is (\hyperlink{Lurie}{Lurie, theorem 5.1}). \begin{remark} \label{}\hypertarget{}{} In the full [[derived algebraic geometry]] the space of sections of the line bundle on the moduli space has the structure of a $K((q))$-[[∞-module]], hence of an actual [[spectrum]] (\hyperlink{Lurie}{Lurie, below remark 5.4}). \end{remark} (\ldots{}) \hypertarget{RelationToLineBundleOnBSpin}{}\subsubsection*{{Relation to the Chern-Simons $\infty$-line bundle on $\mathbf{B}G$}}\label{RelationToLineBundleOnBSpin} Given an [[E-∞ ring]] $A$ with an oriented [[derived elliptic curve]] $\Sigma \to Spec(A)$ there are a priori two different $A$-[[∞-line bundles]] on $B Spin$. On the one hand there is the bundle classified by \begin{displaymath} J_A \;\colon\; B Spin \stackrel{}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S}) \longrightarrow B GL_1(A) \,, \end{displaymath} where $\mathbb{S}$ is the [[sphere spectrum]], $GL_1(-)$ the [[∞-group of units]]-construction and $J$ the [[J-homomorphism]]. (This is what appears as $\mathcal{A}_s$ in \hyperlink{Lurie}{Lurie, middle of p.38}). Notice that by (\hyperlink{AndoBlumbergGepner10}{Ando-Blumberg-Gepner 10, section 8}), for the case $A =$ [[tmf]] this is equivalently the $A$-[[∞-line bundle]] [[associated ∞-bundle|associated]] to the [[universal Chern-Simons line 3-bundle]] \begin{displaymath} A(\tfrac{1}{2}p_1) \;\colon\; B Spin \stackrel{\tfrac{1}{2}p_1}{\longrightarrow} B^4 \mathbb{Z} \stackrel{\tilde \sigma}{\longrightarrow} B GL_1(A) \,, \end{displaymath} where $\tfrac{1}{2}p_1$ is the [[first fractional Pontryagin class]] and $\tilde \sigma$ is an [[adjunct]] of the [[string orientation of tmf]]. In addition, by equivariant elliptic cohomology there is the [[theta line-bundle]] \begin{displaymath} \theta \;\colon\; Loc_{Spin}(\Sigma) \longrightarrow \mathbf{B} \mathbb{G}_m \end{displaymath} on the derived [[moduli stack of flat connections]] $Loc_{Spin}(\Sigma)$ (where in (\hyperlink{Lurie}{Lurie}) $Loc_{Spin}(\Sigma)$ is denoted $M_{Spin}$). Evaluating this bundle on [[global points]] yields the $A$-[[∞-line bundle]] \begin{displaymath} \Gamma_{Spec(A)}(\theta) \;\colon\; \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) \longrightarrow B GL_1(A) \,. \end{displaymath} So there are a priori two $A$-$\infty$-oine bundles on bare homotopy types here. But (by 2-equivariance, \hyperlink{Lurie}{Lurie, bottom of p. 38}) there is a canonical map between their base spaces \begin{displaymath} \phi \;\colon\; B Spin \longrightarrow \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma)) \,. \end{displaymath} Heuristically, this is the map that includes the trivial $Spin$-local system and its [[gauge transformations]] into the (points of the) [[moduli stack]] of all local systems. Hence the pullback of $\Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))$ yields another $A$-line bundle $\phi^\ast \Gamma_{Spec(A)}(\theta)$ over $B Spin$. These are equivalent \begin{displaymath} J_A \simeq \phi^\ast \Gamma_{Spec(A)}(\theta) \,. \end{displaymath} This is (\hyperlink{Lurie}{Lurie, theorem 5.2}). \begin{displaymath} \itexarray{ && Loc_{Spin}(\ast) \\ & \swarrow && \searrow^{\mathrlap{\phi}} \\ Loc_{Spin}(\ast) && \swArrow_{\simeq} && Loc_{Spin}(\Sigma) \\ & {}_{\mathllap{J_A}}\searrow && \swarrow_{\mathrlap{\theta}} \\ && B GL_1(A) } \end{displaymath} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[equivariant elliptic genus]] \item related but different: [[modular equivariant elliptic cohomology]] \item [[equivariant K-theory]] \item [[equivariant algebraic K-theory]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} In \begin{itemize}% \item [[Victor Ginzburg]], [[Mikhail Kapranov]], Eric Vasserot, \emph{Elliptic Algebras and Equivariant Elliptic Cohomology} (\href{http://arxiv.org/abs/q-alg/9505012}{arXiv:q-alg/9505012}) \end{itemize} a set of axioms was proposed that an equivariant elliptic cohomology theory should satisfy. See also \begin{itemize}% \item Ioanid Rosu, \emph{Equivariant Elliptic Cohomology and Rigidity}, American Journal of Mathematics 123 (2001), 647-677 (\href{http://arxiv.org/abs/math/9912089}{arXiv:math/9912089}) \end{itemize} In \begin{itemize}% \item [[David Gepner]], \emph{[[Homotopy topoi and equivariant elliptic cohomology]]}, PhD thesis, University of Illinois at Urbana-Champaign, 2005. \end{itemize} the proposal of \hyperlink{GinzburgKapranovVasserot95}{Ginzburg-Kapranov-Vasserot 95} was formalized in terms of [[geometric morphisms]] of [[(infinity,1)-toposes]]. See also \begin{itemize}% \item [[Ioanid Rosu]], \emph{Equivariant Elliptic Cohomology and Rigidity}, American Journal of Mathematics, Vol. 123, No. 4, Aug., 2001 (\href{http://www.jstor.org/stable/25099077}{JSTOR}) \item [[Jacob Lurie]], section 5.1 of \emph{[[A Survey of Elliptic Cohomology]]}, in \emph{Algebraic Topology Abel Symposia} Volume 4, 2009, pp 219-277 \item [[Daniel Berwick-Evans]], [[Arnav Tripathy]], \emph{A geometric model for complex analytic equivariant elliptic cohomology}, (\href{https://arxiv.org/abs/1805.04146}{arXiv:1805.04146}) \end{itemize} See also the references at \emph{\href{equivariant+cohomology#InComplexOrientedGeneralizedCohomologyTheory}{equivariant cohomology -- References -- Complex oriented cohomology theories}}. \hypertarget{relation_to_loop_group_representations_2}{}\subsubsection*{{Relation to loop group representations}}\label{relation_to_loop_group_representations_2} That equivariant elliptic cohomology is related to [[representations]] of [[loop groups]] as [[equivariant K-theory]] is related to the [[representation theory]] of the underlying groups had long been conjectured. The idea appears in \begin{itemize}% \item I. Grojnowski, \emph{Delocalised equivariant elliptic cohomology} (1994), in \emph{Elliptic cohomology}, volume 342 of London Math. Soc. Lecture Note Ser., pages 114--121. Cambridge Univ. Press, Cambridge, 2007 (\href{http://hopf.math.purdue.edu/Grojnowski/deloc.pdf}{pdf}) \end{itemize} based on \begin{itemize}% \item [[Eduard Looijenga]], \emph{Root systems and elliptic curves}, Invent. Math., 38(1):17--32, 1976/77. \end{itemize} took shape in \begin{itemize}% \item [[Matthew Ando]], \emph{The sigma orientation for analytic circle equivariant elliptic cohomology} . Geom. Topol., 7:91--153, 2003 (\href{http://arxiv.org/abs/math/0201092}{atrXiv:math/0201092}) \item [[Matthew Ando]], \emph{Power operations in elliptic cohomology and representations of loop groups} Transactions of the American Mathematical Society 352, 2000, pp. 5619-5666. (\href{http://www.jstor.org/stable/221905}{JSTOR}, \href{http://www.math.uiuc.edu/~mando/papers/POECLG/poeclg.pdf}{pdf}) \end{itemize} and was then further refined in section 5.2 of (\hyperlink{Lurie}{Lurie}) and in (\hyperlink{Gepner05}{Gepner 05}). More is in \begin{itemize}% \item [[Nora Ganter]], \emph{The elliptic Weyl character formula} (\href{http://arxiv.org/abs/1206.0528}{arXiv:1206.0528}) \end{itemize} \hypertarget{relation_to_line_bundles}{}\subsubsection*{{Relation to $\infty$-line bundles}}\label{relation_to_line_bundles} \begin{itemize}% \item [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], \emph{Twists of K-theory and TMF}, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], \emph{Superstrings, Geometry, Topology, and $C^*$-algebras}, Proceedings of Symposia in Pure Mathematics \href{http://www.ams.org/bookstore-getitem/item=PSPUM-81}{vol 81}, American Mathematical Society (\href{http://arxiv.org/abs/1002.3004}{arXiv:1002.3004}) \end{itemize} \hypertarget{relation_to_superstrings_and_the_witten_genus}{}\subsubsection*{{Relation to superstrings and the Witten genus}}\label{relation_to_superstrings_and_the_witten_genus} Relation to the [[Witten genus]] [[partition function]] of [[superstrings]] is discussed in \begin{itemize}% \item [[Matthew Ando]], Maria Basterra, \emph{The Witten genus and equivariant elliptic cohomology} (\href{http://arxiv.org/abs/math/0008192}{arXiv:math/0008192}) \end{itemize} and specifically in the context of [[parameterized WZW models]] 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} referring to \begin{itemize}% \item [[Jacques Distler]], [[Eric Sharpe]], section 8.5 of \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} Discussion of this in [[motivic quantization|cohomological quantization]] is in \begin{itemize}% \item [[Joost Nuiten]], [[Urs Schreiber]], some notes \end{itemize} based on \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:Quantization via Linear homotopy types]]}, talk at ESI Vienna, Feb 2014, (\href{https://dl.dropboxusercontent.com/u/12630719/factsheet.pdf}{handout pdf}) \end{itemize} A proposal for realizing this via [[(2,1)-dimensional Euclidean field theories and tmf]] is in \begin{itemize}% \item [[Stefan Stolz]], [[Peter Teichner]], \emph{Supersymmetric field theories and generalized cohomology}, in [[Hisham Sati]], [[Urs Schreiber]] (eds.), \emph{\href{http://ncatlab.org/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory#ContributionStolzTeichner}{Mathematical foundations of Quantum field theory and String theory}}, Proceedings of Symposia in Pure Mathematics, Volume 83, AMS (2011) \end{itemize} [[!redirects equivariant elliptic cohomology theory]] [[!redirects equivariant elliptic cohomology theories]] \end{document}