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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*{modular functor} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial quantum field theory}}\label{functorial_quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{central_charge_and_central_extensions}{Central charge and Central extensions}\dotfill \pageref*{central_charge_and_central_extensions} \linebreak \noindent\hyperlink{the_knizhnikzamolodchikovhitchin_connection}{The Knizhnik--Zamolodchikov-Hitchin connection}\dotfill \pageref*{the_knizhnikzamolodchikovhitchin_connection} \linebreak \noindent\hyperlink{the_verlinde_fusion_alegbra}{The Verlinde fusion alegbra}\dotfill \pageref*{the_verlinde_fusion_alegbra} \linebreak \noindent\hyperlink{TopologicalLift}{Deprojectivization, Cancelling of central charge, topological modular functor}\dotfill \pageref*{TopologicalLift} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{powers_of_the_determinant_line}{Powers of the determinant line}\dotfill \pageref*{powers_of_the_determinant_line} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{properties_2}{Properties}\dotfill \pageref*{properties_2} \linebreak \noindent\hyperlink{relation_to_equivariant_elliptic_cohomology__equivariant_}{Relation to equivariant elliptic cohomology / equivariant $tmf$}\dotfill \pageref*{relation_to_equivariant_elliptic_cohomology__equivariant_} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The local data for a [[CFT]] in [[dimension]] $d$ allows to assign to each $d$-[[dimension|dimensional]] [[cobordism]] $\Sigma$ a [[vector space]] of ``possible [[correlators]]'': those functions on the space of [[conformal structures]] on $\Sigma$ that have the correct behaviour (satisfy the conformal [[Ward identities]]) to qualify as the (chiral) [[correlator]] of a CFT. This is called a space of \emph{[[conformal blocks]]} $Bl(\Sigma)$. This assignment is [[functor|functorial]] under [[diffeomorphism]]. The corresponding functor is called a \textbf{modular functor}. (\hyperlink{Segal89}{Segal 89}, \hyperlink{Kriz03}{Kriz 03}, \hyperlink{Segal04}{Segal 04, def. 5.1}). To get an actual collection of correlators one has to choose from each space of conformal blocks $Bl(\Sigma)$ an element such that these choices glue under composition of [[cobordism]]: such that they solve the [[sewing constraints]], see for instance at \emph{[[FRS-theorem on rational 2d CFT]]}. Dually, under a [[holographic principle]] such as [[AdS3-CFT2 and CS-WZW correspondence|CS3/WZW2]] the space of [[conformal blocks]] on $\Sigma$ is equivalently the [[space of quantum states]] of the [[TQFT]] on $\Sigma$. See at \emph{[[quantization of 3d Chern-Simons theory]]} for more on this. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{CategoryOfRiemannSurfaces}\hypertarget{CategoryOfRiemannSurfaces}{} For $\Phi$ any [[finite set]] (``of lables'') write $\mathcal{S}_{\Phi}$ for the [[category]] whose [[objects]] are [[Riemann surfaces]] with [[boundary]] [[circles]] labeled by elements of $\Phi$, and whose [[morphisms]] are [[holomorphic maps]] $X \to X_{sewed}$, where $X_{sewed}$ is obtained from $X$ by [[sewing constraint|sewing]] along boundary circles carrying the same labels. \end{defn} (\hyperlink{Segal04}{Segal 04, section 4, section 5}) This category, being essentially the total [[Riemann moduli space]] is naturally a [[complex analytic stack]]. For $\Sigma$ a [[topological manifold|topological]] [[surface]] write $\mathcal{S}_\Sigma$ for the component of $\mathcal{S}$ on Riemann surfaces whose underlying topological surface is $\Sigma$. \begin{remark} \label{}\hypertarget{}{} If $\Sigma$ has at least one hole (boundary component), then the [[fundamental group]] $\pi_1(\mathcal{S}_\Sigma)$ is the [[mapping class group]] of $\Sigma$. \end{remark} \begin{defn} \label{ModularFunctor}\hypertarget{ModularFunctor}{} A \emph{modular functor} is a holomorphic functor \begin{displaymath} E \colon \mathcal{S}_\Phi \longrightarrow sFinVect \end{displaymath} (i.e. a morphism of [[complex analytic stacks]] from the [[Riemann moduli space]] to the stack of [[holomorphic vector bundles]], in general with [[super vector space]]-fibers) such that \begin{enumerate}% \item $E$ is [[strong monoidal functor|strong monoidal]]: $E(X \coprod Y) \simeq E(X)\otimes E(Y)$; \item $E$ respects seqing: if $X_\phi$ is obtained from $X$ by cutting along a circle and giving the same label $\phi \in \Phi$ to both resulting boundaries, then the [[natural transformation]] \begin{displaymath} \underset{\phi \in \Phi}{\oplus} E(X_\phi) \longrightarrow E(X) \end{displaymath} is a [[natural isomorphism]]. \item normalization: for $X = S^2$ the [[Riemann sphere]] we have $E(S^1) = 1$ (the[[tensor unit]] vector space). \end{enumerate} \end{defn} (\hyperlink{Segal04}{Segal 04, def. 5.1}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{central_charge_and_central_extensions}{}\subsubsection*{{Central charge and Central extensions}}\label{central_charge_and_central_extensions} Any modular functor defines a [[central extension]] of the [[semigroup]] of conformal [[annuli]]. These correspond precisely to [[group extensions]] of $Diff^+(S^1)$ by $\mathbb{C}^\times$. (\hyperlink{Segal04}{Segal 04, prop. 5.6}) These in turn are classified by $(c,h) \in \mathbb{C} \times \mathbb{C}/\mathbb{Z}$. (\hyperlink{Segal04}{Segal 04, prop. 5.8}) \begin{defn} \label{CentralCharge}\hypertarget{CentralCharge}{} Here in terms of standard [[2d CFT]] terminology \emph{$c$ is the \emph{central charge}} \emph{$h$ is the [[eigenvalues]] of $L_0$.} \end{defn} \hypertarget{the_knizhnikzamolodchikovhitchin_connection}{}\subsubsection*{{The Knizhnik--Zamolodchikov-Hitchin connection}}\label{the_knizhnikzamolodchikovhitchin_connection} \begin{prop} \label{TheProjFlatConnection}\hypertarget{TheProjFlatConnection}{} Given a modular functor $E$ as in def. \ref{ModularFunctor} and given a non-[[closed manifold|closed]] [[topological manifold|topological]] labelled [[surface]] $\Sigma$ with $E_\Sigma \to \mathcal{S}_\Sigma$ the resulting [[vector bundle]], then this bundle carries a canonical [[projectively flat connection]] $\nabla_\Sigma$ compatible with the [[sewing constraint|sewing operation]] of def. \ref{CategoryOfRiemannSurfaces}. \end{prop} (\hyperlink{Segal04}{Segal 04, prop. 5.4}) \begin{remark} \label{}\hypertarget{}{} When thinking of the modular functor $E$ as the functor of [[conformal blocks]] of a [[2d CFT]] then the projectively flat connection of prop. \ref{TheProjFlatConnection} would often be called the \emph{[[Knizhnik-Zamolodchikov connection]]}. Thining of $E$ dually as the functor assigning [[spaces of quantum states]] of [[Chern-Simons theory]] then it would typically be called the [[Hitchin connection]]. (see also \hyperlink{Segal04}{Segal 04, p. 44, p. 84}). \end{remark} \begin{pro} \label{FlatConnectionForVanishingCentralCharge}\hypertarget{FlatConnectionForVanishingCentralCharge}{} The connection of prop. \ref{TheProjFlatConnection} is a genuine [[flat connection]] (not projective) precisely if the central charge, \ref{CentralCharge}, vanishes. \end{pro} (\hyperlink{Segal04}{Segal 04, below prop. 5.4}) \hypertarget{the_verlinde_fusion_alegbra}{}\subsubsection*{{The Verlinde fusion alegbra}}\label{the_verlinde_fusion_alegbra} \begin{defn} \label{}\hypertarget{}{} For $\phi,\chi,\psi \in \Phi$ three labels, write $P_{\phi,\chi,\psi}$ for the three-holed sphere (``pair of pants'', ``trinion'') with inner circles labeled by $\phi$ and $\chi$ and outer circle labeled by $\psi$. For $E$ a modular functor as in def. \ref{ModularFunctor}, write \begin{displaymath} n_{\phi, \chi,\psi} \coloneqq dim(E(P_{\phi,\chi,\psi})) \end{displaymath} for the [[dimension]] of the vector space that it assigns to this surface. Then the [[free abelian group]] $\mathbb{Z}[\Phi]$ on the set of labels inherits the structure of an [[associative algebra]] via \begin{displaymath} (\phi,\chi) \mapsto \underset{\psi}{\sum} n_{\phi,\chi, \psi} \psi \,. \end{displaymath} The \emph{[[Verlinde algebra]]}. \end{defn} (\hyperlink{Segal04}{Segal 04, section 5, p. 36-37}) \hypertarget{TopologicalLift}{}\subsubsection*{{Deprojectivization, Cancelling of central charge, topological modular functor}}\label{TopologicalLift} By prop. \ref{FlatConnectionForVanishingCentralCharge} and prop. \ref{CentralChargeOfDeterminantLine}, if $E$ is a modular functor of central charge $c$ then the [[tensor product]] \begin{displaymath} \tilde E \coloneqq E \otimes Det^{\otimes c/2} \end{displaymath} with a possibly fractional power of the [[determinant line bundle]], def. \ref{PowerOfDeterminantLine}, produces a modular functor with vanishing central charge. To make sense of this however one needs to consistently define the fractional power. For that one needs to pass to surfaces equipped with a bit more structure. \begin{defn} \label{RiggedSurfaces}\hypertarget{RiggedSurfaces}{} The category of \emph{rigged surfaces} $\hat {\mathcal{S}}_\Phi$ is the [[central extension]] of that of [[smooth manifold]] [[surfaces]] such that for [[genus of a surface|genus]] $\gt 1$ it gives the universal central extension of the [[diffeomorphism group]]. For instance the category of surfaces equipped with a choice of [[universal covering space]] of the [[circle group]]-[[principal bundle]] underlying the [[determinant line bundle]] over $\mathcal{S}_\Sigma$. \end{defn} (\hyperlink{Segal04}{Segal 04, def. (5.10) and following}, also \hyperlink{BakalovKirillov}{Bakalov-Kirillov, def. 5.7.5}) \begin{prop} \label{}\hypertarget{}{} Given a [[modular functor]] $E$, def. \ref{ModularFunctor} of central charge $c$, def. \ref{CentralCharge}, then the tensor product $\tilde E \coloneqq E \otimes Det^{\otimes c/2}$ is well defined on the category $\hat S_{\phi}$ of rigged surfaces, def. \ref{RiggedSurfaces}. \end{prop} Of course if one has an extension of the diffeomorphism group by a multiple of the universal extension in def. \ref{RiggedSurfaces}, then this still trivializes the conformal anomaly for all modular functors whose central charge is a corrsponding multiple. In particular: \begin{prop} \label{SurfacesWithAtiyah2Framing}\hypertarget{SurfacesWithAtiyah2Framing}{} The category of smooth surfaces equipped with ``[[Atiyah 2-framing]]'' (hence with a trivialization of the spin lift of the double of their tangent bundle) provides an extension of the diffeomorphic group of level 12. \end{prop} (\hyperlink{Segal04}{Segal 04, p. 46}) \begin{remark} \label{}\hypertarget{}{} There is a natural functor from smooth surfaces $\Sigma$ equipped with [[3-framing]] (trivialization of $T \Sigma\oplus \underline{\mathbb{R}}$) to that equipped with [[Atiyah 2-framing]] in prop. \ref{SurfacesWithAtiyah2Framing}. \end{remark} \begin{quote}% thanks to [[Chris Schommer-Pries]] for highlighting this point. \end{quote} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{powers_of_the_determinant_line}{}\subsubsection*{{Powers of the determinant line}}\label{powers_of_the_determinant_line} \begin{defn} \label{PowerOfDeterminantLine}\hypertarget{PowerOfDeterminantLine}{} For $n \in \mathbb{Z}$ let $E = Det^{\otimes n}$ be the functor which sends a [[Riemann surface]] to the $c$th power of its [[determinant line]] (i.e. that of its [[Laplace operator]]). \end{defn} Super-line, see (\hyperlink{KrizLai13}{Kriz-Lai 13})\ldots{} \begin{defn} \label{}\hypertarget{}{} The determinant lines of def. \ref{PowerOfDeterminantLine} constitute precisely the modular functors, def. \ref{ModularFunctor}, for which $dim(E(X)) = 1$ for all $X$. \end{defn} (\hyperlink{Segal04}{Segal 04, corollary (5.17)}) \begin{prop} \label{CentralChargeOfDeterminantLine}\hypertarget{CentralChargeOfDeterminantLine}{} The central charge, def. \ref{CentralCharge}, of the determinant line $E = Det$, def. \ref{PowerOfDeterminantLine}, is \begin{displaymath} (c,h ) = (-2,0) \,. \end{displaymath} \end{prop} (\hyperlink{Segal04}{Segal 04, p.43}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[quantization of Chern-Simons theory]] \end{itemize} \hypertarget{properties_2}{}\subsection*{{Properties}}\label{properties_2} \hypertarget{relation_to_equivariant_elliptic_cohomology__equivariant_}{}\subsubsection*{{Relation to equivariant elliptic cohomology / equivariant $tmf$}}\label{relation_to_equivariant_elliptic_cohomology__equivariant_} The modular functor for $G$-[[Chern-Simons theory]] restricted to [[genus of a surface|genus-1 surfaces]] ([[elliptic curves]]) is essentially what is encoded in the universal $G$-[[equivariant elliptic cohomology]] (equivariant [[tmf]]). In fact equivariant elliptic cohomology remembers also the [[local prequantum field theory|pre-quantum]] incarnation of the modular functor as a systems of [[prequantum line bundles]] over Chern-Simons [[phase spaces]] (which are [[moduli stacks of flat connections]]) and remembers the [[quantization]]-process from there to the actual [[space of quantum states]] by forming holomorphic [[sections]]. See at \emph{\href{equivariant+elliptic+cohomology#InterpretationInQuantumFieldTheory}{equivariant elliptic cohomology -- Idea -- Interpretation in Quantum field theory}} for more on this. \hypertarget{references}{}\subsection*{{References}}\label{references} Original formulations include \begin{itemize}% \item [[Graeme Segal]], \emph{Two-dimensional conformal field theories and modular functors}, in: IXth International Congress on Mathematical Physics (Swansee 1988), Hilger, Bristol 1989, pp. 22-37 \item [[Graeme Segal]], section 5 of \emph{The definition of conformal field theory}, Topology, geometry and quantum field theory London Math. Soc. Lecture Note Ser., 308, Cambridge Univ. Press, Cambridge, (2004), 421-577 (\href{https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf}{pdf}) \item [[Igor Kriz]], \emph{On spin and modularity in conformal field theory}, Ann. Sci. ANS (4) 36 (2003), no. 1, 57112 (\href{http://www.numdam.org/item?id=ASENS_2003_4_36_1_57_0}{numdam:ASENS\_2003\_4\_36\_1\_57\_0}) \item [[Igor Kriz]], Luhang Lai, \emph{On the definition and K-theory realization of a modular functor}, (\href{http://arxiv.org/abs/1310.5174}{arxiv/1310.5174}). \end{itemize} Lectures and reviews include \begin{itemize}% \item [[Bojko Bakalov]], [[Alexander Kirillov]], chapter 5 of \emph{Lectures on tensor categories and modular functor} (\href{http://www.math.sunysb.edu/~kirillov/tensor/tensor.html}{web}, [[BakalovKirillovChapter5.pdf:file]]) \item [[Krzysztof Gawedzki]], section 5.6 of \emph{Conformal field theory: a case study} (\href{http://arxiv.org/abs/hep-th/9904145}{arXiv:hep-th/9904145}) \end{itemize} A nice review with a new concise construction is in \begin{itemize}% \item [[Eduard Looijenga]], \emph{From WZW models to Modular Functors} (\href{http://arxiv.org/abs/1009.2245}{arXiv:1009.2245}) \end{itemize} Discussion in the context of [[(2,1)-dimensional Euclidean field theories and tmf]] is in \begin{itemize}% \item [[Stefan Stolz]], [[Peter Teichner]], section 5.2 of \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} Discussion in the context of the [[cobordism hypothesis]] is in \begin{itemize}% \item [[Domenico Fiorenza]], [[Alessandro Valentino]], \emph{Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors} (\href{http://arxiv.org/abs/1409.5723}{arXiv:1409.5723}) \end{itemize} [[!redirects modular functors]] \end{document}