\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*{gauged WZW model} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{wesszuminowitten_theory}{}\paragraph*{{$\infty$-Wess-Zumino-Witten theory}}\label{wesszuminowitten_theory} [[!include infinity-Wess-Zumino-Witten theory - contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{physics}{}\paragraph*{{Physics}}\label{physics} [[!include physicscontents]] \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{PartitionFunctionInEllipticCohomology}{Partition function in (equivariant) elliptic cohomology}\dotfill \pageref*{PartitionFunctionInEllipticCohomology} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{gauged WZW model} is a [[field (physics)|field]] [[theory (physics)]] which combines the [[WZW model]] with [[gauge theory]]: given a ([[simple Lie group|simple]]) [[Lie group]] $G$ and a [[subgroup]] $H \hookrightarrow G \times G$, the corresponding gauged WZW model is a 2-dimensional [[prequantum field theory]] on some [[worldvolume]] $\Sigma_2$ whose [[field (physics)|fields]] are pairs consisting of a smooth function $\Sigma_2 \to G$ and a [[Lie algebra valued 1-form]] $A \in \Omega^1(\Sigma_2, \mathfrak{h})$, with values in the [[Lie algebra]] of $H$. Where the [[Lagrangian]]/[[action functional]] of the ordinary [[WZW model]] is the sum/product of a standard [[kinetic action]] and the [[surface holonomy]] of a [[circle 2-bundle with connection]] whose [[curvature]] 3-form is the canonical 3-form $\langle \theta \wedge [\theta \wedge \theta]\rangle \in \Omega^3(G)^G$, so the action functional of the gauged WZW model is that obtained by refining this circle 2-bundle to the $H$-[[equivariant cohomology|equivariant]] [[differential cohomology]] of $G$, with [[curvature]] 3-form in [[equivariant de Rham cohomology]]. \hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition} The [[Chevalley-Eilenberg algebra]] of the [[Lie algebra]] $\mathfrak{g}$ is naturally identified with the sub-algebra of [[left invariant differential forms]] on $G$: \begin{displaymath} CE(\mathfrak{g}) \simeq \Omega^\bullet(G)^G \,. \end{displaymath} The ordinary [[WZW model]] is given by the basic [[circle 2-bundle with connection]] on $G$ whose [[curvature]] 3-form is \begin{displaymath} H = \langle \theta \wedge [\theta \wedge \theta]\rangle \in \Omega^3(G)^G \,. \end{displaymath} Now for $H \hookrightarrow G$ a [[subgroup]], write \begin{displaymath} CE(\mathfrak{g}//\mathfrak{h}) \coloneqq \Omega^\bullet(G, \mathfrak{h}^\ast[1])^G \end{displaymath} for the corresponding dg-algebra of (say) the [[Cartan model]] for [[equivariant de Rham cohomology]] on $G$. There is a canonical projection \begin{displaymath} CE(\mathfrak{g}//\mathfrak{h}) \to CE(\mathfrak{g}) \,. \end{displaymath} A curvature 3-form for the \textbf{gauged WZW model} is a 3-cocycle \begin{displaymath} \tilde H \in CE^3(\mathfrak{g}//\mathfrak{h}) \end{displaymath} in this [[equivariant de Rham cohomology]] which lifts $H \coloneqq \langle \theta \wedge [\theta \wedge \theta]\rangle$ through this projection. One finds (\hyperlink{Witten92}{Witten 92, appendix}) that in terms of the degree-2 generators $\{F^a\}$ of the [[Cartan model]] (see there) with respect to some [[basis]] $\{t_a\}$ of $\mathfrak{g}$, these lifts are of the form (\hyperlink{Witten92}{Witten 92, (A.14)}) \begin{displaymath} \tilde H = H + \lambda_a \wedge F^a \end{displaymath} where $\lambda_a \in \Omega^1(G)$ is given by (in [[matrix Lie algebra]] notation) \begin{displaymath} \lambda_a = \left\langle t_a^l \cdot (d g)g^{-1} + t_a^r \cdot g^{-1} d g \right\rangle \end{displaymath} and exist precisely if (\hyperlink{Witten92}{Witten 92, (A.16)}) for all pairs of basis elements \begin{displaymath} \langle t_a^l \cdot t_b^l - t_a^r \cdot t_b^r \rangle = 0 \,. \end{displaymath} This condition had originally been seen as a [[anomaly cancellation]]-condition of the gauged WZW model. A systematic discussion of these obstructions in [[equivariant de Rham cohomology]] is in (\hyperlink{FigueroaOFarrillStanciu94}{Figueroa-O'Farrill-Stanciu 94}). Now by [[schreiber:∞-Wess-Zumino-Witten theory]], the corresponding WZW model has as target the [[smooth groupoid]] $\tilde G//H$ such that maps into it are locally a map $g$ into $G$ together with 1-form potentials $A^a$ for the $F^a$, and the WZW term is locally a 2-form built from $d g$ and $A^a$ such that its curvature 3-form is $\tilde H$. This is the \textbf{gauged WZW model} (\hyperlink{Witten92}{Witten 92, (A.16)}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{PartitionFunctionInEllipticCohomology}{}\subsubsection*{{Partition function in (equivariant) elliptic cohomology}}\label{PartitionFunctionInEllipticCohomology} The [[partition function]] of the gauged WZW model as an [[elliptic genus]] is considered in (\hyperlink{Henningson94}{Henningsonn 94, (8)}). When done properly this should give elements in [[equivariant elliptic cohomology]], hence an [[equivariant elliptic genus]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[coset WZW model]] \item [[gauged linear sigma model]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The original articles are \begin{itemize}% \item [[Edward Witten]], Nucl. Phys. B223 (1983) 433. \item [[Edward Witten]], Commun. Math. Phys. 92 (1984) 455. \item [[Krzysztof Gawedzki]], A. Kupiainen, \emph{G/H conformal field theory from gauged WZW model} Phys. Lett. 215B, 119 (1988); \item [[Krzysztof Gawedzki]], A. Kupiainen, \emph{Coset construction from functional integrals}, Nucl. Phys. B 320 (FS), 649 (1989) \item [[Krzysztof Gawedzki]], in \emph{From Functional Integration, Geometry and Strings}, ed. by Z. Haba and J. Sobczyk (Birkhaeuser, 1989). \end{itemize} The (curvature of the)gauged WZW term was recognized/described as a [[cocycle]] in [[equivariant de Rham cohomology]] is in the appendix of \begin{itemize}% \item [[Edward Witten]], \emph{On holomorphic factorization of WZW and coset models}, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. (\href{http://projecteuclid.org/euclid.cmp/1104249222}{EUCLID}) \end{itemize} This is expanded on in \begin{itemize}% \item [[José Figueroa-O'Farrill]], S Stanciu, \emph{Gauged Wess-Zumino terms and Equivariant Cohomology}, Phys.Lett. B341 (1994) 153-159 (\href{http://arxiv.org/abs/hep-th/9407196}{arXiv:hep-th/9407196}) \item [[José de Azcárraga]], J. C. Perez Bueno, \emph{On the general structure of gauged Wess-Zumino-Witten terms} (\href{http://arxiv.org/abs/hep-th/9802192}{arXiv:hep-th/9802192}) \end{itemize} A quick review of this class of 3-cocycles in equivariant de Rham cohomology is also in section 4.1 of \begin{itemize}% \item Hugo Garcia-Compean, Pablo Paniagua, [[Bernardo Uribe]], \emph{Equivariant extensions of differential forms for non-compact Lie groups} (\href{http://arxiv.org/abs/1304.3226}{arXiv:1304.3226}) \end{itemize} which further generalizes the discussion to non-compact Lie groups. See also \begin{itemize}% \item [[Edward Witten]], \emph{The $N$ matrix model and gauged WZW models}, Nuclear Physics B Volume 371, Issues 1--2, 2 March 1992, Pages 191--245 \item Stephen-wei Chung, S.-H. Henry Tye, \emph{Chiral Gauged WZW Theories and Coset Models in Conformal Field Theory}, Phys. Rev. D47:4546-4566,1993 (\href{http://arxiv.org/abs/hep-th/9202002}{arXiv:hep-th/9202002}) \item Konstadinos Sfetsos, \emph{Gauged WZW models and Non-abelian duality}, Phys.Rev. D50 (1994) 2784-2798 (\href{http://arxiv.org/abs/hep-th/9402031}{arXiv:hep-th/9402031}) \item [[Elias Kiritsis]], \emph{Duality in gauged WZW models} (\href{http://hep.physics.uoc.gr/~kiritsis/papers2/dual.pdf}{pdf}) \end{itemize} The [[partition function]]/[[elliptic genus]] of the SU(2)/U(1) gauged WZW model is considered in \begin{itemize}% \item [[Måns Henningson]], \emph{N=2 gauged WZW models and the elliptic genus}, Nucl.Phys. B413 (1994) 73-83 (\href{http://arxiv.org/abs/hep-th/9307040}{arXiv:hep-th/9307040}) \end{itemize} Emphasis of the special case of abelian gauging in Section 2 of \begin{itemize}% \item [[Stefan Förste]], \emph{Deformations of WZW models}, Class. Quant. Grav. 21 (2004) S1517-1522 (\href{https://arxiv.org/abs/hep-th/0312202}{arXiv:hep-th/0312202}) \end{itemize} [[!redirects gauged WZW models]] [[!redirects gauged Wess-Zumino-Witten model]] [[!redirects gauged Wess-Zumino-Witten modesl]] \end{document}