\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*{Topological Quantum Field Theories from Compact Lie Groups}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory}

[[!include infinity-Chern-Simons 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]]

This entry is about the article

\begin{itemize}%
\item [[Dan Freed]], [[Mike Hopkins]], [[Jacob Lurie]], [[Constantin Teleman]],

\emph{Topological Quantum Field Theories from Compact Lie Groups}

in P. R. Kotiuga (ed.) \emph{A celebration of the mathematical legacy of Raoul Bott} AMS (2010)

(\href{http://arxiv.org/abs/0905.0731}{arXiv:0905.0731})



\end{itemize}
on

\begin{enumerate}%
\item in sections 3 and 8; a central topic in [[higher category theory and physics]]: the [[higher category theory|abstract higher categoretic]] conception of [[path integral]] [[quantization]] of classical [[action functionals]] to [[FQFT|extended quantum field theories]], realized here for finite [[higher gauge theories]] [[Dijkgraaf-Witten theory]]-type theories (see also at [[prequantum field theory]])


\item the [[extended TQFT]]-[[quantization]] of $G$-[[Chern-Simons theory]] for abelian [[Lie groups]] $G$.



\end{enumerate}
More on the story of sections 3 and 8 is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]}, talk at \emph{Notre Dame Graduate Summer School on Topology and Field Theories} and \emph{Harvard lecture} 2012 (\href{http://www.youtube.com/watch?v=eQayYLDw1VA}{video part 1}, \href{http://www.youtube.com/watch?v=OEShrQyvmS4}{part 2}, \href{http://www.youtube.com/watch?v=nOIcdn1iUR4}{part 3} \href{http://www.youtube.com/watch?v=ZwnClYedaYM}{part 4}, \href{http://www.math.northwestern.edu/~celliott/notre_dame_notes/Lurie_notes.pdf}{pdf lecture notes} by Chris Elliott)

\end{itemize}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{preliminaries}{Preliminaries}\dotfill \pageref*{preliminaries} \linebreak
\noindent\hyperlink{the_general_abstract_notion_of_quantization_for_discrete_theories}{The general abstract notion of quantization for discrete theories}\dotfill \pageref*{the_general_abstract_notion_of_quantization_for_discrete_theories} \linebreak
\noindent\hyperlink{3dCSFullyExtended}{3d Chern-Simons as a fully extended TQFT}\dotfill \pageref*{3dCSFullyExtended} \linebreak
\noindent\hyperlink{cs_as_a_fully_extended_tqft}{CS as a fully extended TQFT}\dotfill \pageref*{cs_as_a_fully_extended_tqft} \linebreak
\noindent\hyperlink{boundary_conditions_wzw_theory}{Boundary conditions: WZW theory}\dotfill \pageref*{boundary_conditions_wzw_theory} \linebreak
\noindent\hyperlink{related_references}{Related references}\dotfill \pageref*{related_references} \linebreak


\hypertarget{preliminaries}{}\subsection*{{Preliminaries}}\label{preliminaries}

The non-toy example application that gives the paper its title is to [[Chern-Simons theory]].

The notion of quantization discussed builds on the notion of $(\infty,n)$-categories of \emph{families of $\infty$-groupoids} that appears in some of the later sections of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]} .

\end{itemize}
Together with a notion of $n$-vector spaces (the [[vertical categorification]] of [[vector space]] and [[2-vector space]]) the article sketches a general abstract formalsim making precise the notion of [[path integral]] [[quantization]] for ``finite'' theories such as [[Dijkgraaf-Witten theory]].

The development, sketching a rather grand picture, remains somewhat sketchy, though, possibly due to the fact that this is a conference proceedings. Also some of the ideas claimed to now be fully generalized have appeared elsewhere before. Notably the notion of the quantization map $Fam_n(C) \to C$ (see below) is effectively what [[John Baez]], [[Jim Dolan]] call in their program on [[groupoidification]] call \emph{degroupoidification} . The general idea underlying this, that spaces of states are computed as colimits of sections, has been made clear previously by [[Simon Willerton]] \emph{The twisted Drinfeld double of a finite group via gerbes and finite groupoids} (\href{http://arxiv.org/abs/math.QA/0503266}{arXiv:math/0503266})

\hypertarget{the_general_abstract_notion_of_quantization_for_discrete_theories}{}\subsection*{{The general abstract notion of quantization for discrete theories}}\label{the_general_abstract_notion_of_quantization_for_discrete_theories}

\begin{quote}%
Here is a summary of the general quantization aspect of the article, together with some additional remarks on how to think of all this by [[Urs Schreiber|an nLab author]].


\end{quote}
The following is the formalization of the notion of [[quantization]] for discrete theories (such as [[Dijkgraaf-Witten theory]]) as presented in the article.

Fix some $n \in \mathbb{N}$, the dimension of the [[quantum field theory]] to be described.

In

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]} .

\end{itemize}
are described the following two [[(∞,n)-category|(∞,n)-categories]]

\begin{itemize}%
\item the [[(∞,n)-category of cobordisms]] $Bord_n$: its [[k-morphism]]s are roughly $k$-dimensional [[manifold]]s with boundaries and corners;


\item for $C$ any [[(∞,n)-category]] the [[(∞,n)-category of spans]] over $C$, denoted $Fam_n(C)$,

whose

\begin{itemize}%
\item [[object]]s are [[∞-groupoid]]s $P$ equipped with functors $F_P :P \to C$


\item [[morphism]]s are [[span]]s of [[∞-groupoid]]s with [[natural transformation]]s between the corresponding functors ([[bi-brane]]s)

\begin{displaymath}
\itexarray{
  && P
  \\
  & \swarrow && \searrow
  \\
  P_{in} &&\swArrow&& P_{out}
  \\
  & {}_{\mathllap{\exp(S(-))_{in}}}\searrow
  && \swarrow_{\mathrlap{\exp(S(-))_{out}}}
  \\
  && C
  }
\end{displaymath}

\item ``and so on''.



\end{itemize}


\end{itemize}
For the application to [[quantization]] of [[sigma-model]] theories we want to be thinking of the data encoded by these $(\infty,n)$-categories as follows:

\begin{itemize}%
\item An [[k-morphism]] $\Sigma$ in $Bord_n$ is a piece of $k$-dimensional ``worldvolume'' of some extended object, whose quantum dynamics we want to describe; we may roughly think of this as a [[cospan]]

\begin{displaymath}
\itexarray{
    && \Sigma
    \\
    & \nearrow && \nwarrow
    \\
    \Sigma_{in} &&&& \Sigma_{out}
   }
\end{displaymath}
where $\Sigma_{in}$ and $\Sigma_{out}$ are pieces of the boundary of $\Sigma$. We think of $\Sigma_{in}$ as the ``incoming'' piece of the object that we want to describe, which then experiences a self-interaction as described by the topology of $\Sigma$ and comes out in the shape of $\Sigma_{out}$ (for instace $\Sigma$ might be the three-holed sphere, $\Sigma_{in}$ the disjoint unions of two of its bounding circles and $\Sigma_{out}$ the remaining one, modelling the interaction where two strings merge to a single one).


\item An [[morphism]] in $Fam_n(C)$ is to be thought of as

\begin{itemize}%
\item two \textbf{configuration spaces of fields} $P_{in}, P_{out}$ of some field theory;


\item together with an [[action functional]] on it in the form of an higher vector bundle (``[[gerbe]]'') $\exp(S_P()) : P \to C$; being the component of the [[natural transformation]] that assigns to each \emph{path} $P$ between field configuration a \emph{phase} ;



\end{itemize}


\end{itemize}
In these terms the \textbf{kinematics of a classical field theory} is a choice of $(\infty,n)$-functor

\begin{displaymath}
kin : Bord_n \to Fam_n(*)
\end{displaymath}
whereas the \textbf{dynamics of a classical field theory} -- the specificaton of an [[action functional]] on the given configuration spaces, is a lift of that to $Fam_n(C)$

\begin{displaymath}
\itexarray{
    && Fam_n(C)
    \\
    & {}^{\exp(S(-))}\nearrow & \downarrow
    \\
    Bord_n &\stackrel{kin}{\to}&
    Fam_n(*)
  }
\end{displaymath}
To illustrate this: specifically, if we consider a [[sigma-model]] [[quantum field theory]] that is induced from a \textbf{target space} geometry $X$, such that a field configuration on $\Sigma$ is a morphism $\phi : \Sigma \to X$, and with a [[gauge field|background field]] $\nabla : X \to C$, then we think of the corresponding functor

\begin{displaymath}
conf_X : Bord_n \to Fam_n(C)
\end{displaymath}
as given by homming a [[cobordism]] [[cospan]] of the form

\begin{displaymath}
\itexarray{
    && \Sigma
    \\
    & \nearrow && \nwarrow
    \\
    \Sigma_{in} &&&& \Sigma_{out}
   }
\end{displaymath}
into $X$ to produce a [[span]] of path and configuration spaces

\begin{displaymath}
\itexarray{
    && [\Sigma,X]
    \\
    & \swarrow && \searrow
    \\
    [\Sigma_{in},X] &&&& [\Sigma_{out},X]
   }
\end{displaymath}
equipped with the \textbf{transgressed background field} as the corresponding action functional

\begin{displaymath}
\itexarray{
    && [\Sigma, X]
    \\
    & \swarrow && \searrow
    \\
    [\Sigma_{in},X] &&&& [\Sigma_{out},X]
    \\
    & \searrow && \swarrow
    \\
    && C
   }
   \,.
\end{displaymath}
With that in hand, the \textbf{[[quantization]]} of the given classical field theory $\exp(S(-)) : Bord_n \to Fam_n(C)$ is its ``pushforward to the point'', given by postcomposition with a functor

\begin{displaymath}
\int : Fam_n(C) \to C
\end{displaymath}
that over objects $\exp(S(-)) : P \to C$ is given by taking $n$-categorical [[colimit]]

\begin{displaymath}
(\exp(S(-)) : P \to C) \mapsto (\lim_\to \exp(S(-)))
\end{displaymath}
which in terms of [[coend]]-notation is indeed nicely suggestively written as

\begin{displaymath}
(\exp(S(-)) : P \to C) \mapsto (\int \exp(S(-)))
  \,.
\end{displaymath}
Taking such a colimit may be thought of as forming the space of [[section]]s of the action functional $n$-vector bundle $\exp(S(-)) : P \to C$. That this is the right general idea was maybe first amplified in

\begin{itemize}%
\item [[Dan Freed]], \emph{Higher Algebraic Structures and Quantization} (\href{http://arxiv.org/abs/hep-th/9212115}{arXiv:hep-th/9212115}.

\end{itemize}
A first more categorical formulation of this is in

\begin{itemize}%
\item [[Simon Willerton]] \emph{The twisted Drinfeld double of a finite group via gerbes and finite groupoids} (\href{http://arxiv.org/abs/math.QA/0503266}{arXiv:math/0503266})

\end{itemize}
What exactly the functor $\int : Fam_n(C) \to C$ does to [[k-morphism]]s is apparently left as an exercise for the inclined reader. it requires that in $C$ limits and colimits coincide. This is the case notably for $C = Vect$.

The authors indicate in section 8 a general recursive procedure for defining higher categories of higher vector spaces, by iterating the bimodule-style definition of [[2-vector space]]s, as described there. This yields a notion $C = n Vect$, which should be the right codomain for $n$-dimensional QFTs. So we end up with a diagram

\begin{displaymath}
\itexarray{
    && Fam_n(C) &\stackrel{\int}{\to}& C
    \\
    & {}^{\exp(S(-))}\nearrow & \downarrow
    \\
    Bord_n &\stackrel{kin}{\to}&
    Fam_n(*)
  }
\end{displaymath}
whose left bit is the kinematical and dynamical input given by a classical field theory, and whose composition to to the right is supposed to give the corresponding quantum field theory, which by the logic motivating the [[cobordism hypothesis]] is a functor $Z : Bord_n \to n Vact$:

\begin{displaymath}
Z_S  = \int \exp(S(-)) : Bord_n \stackrel{\exp(S(-))}{\to}
   Fam_n(n Vect) \stackrel{\int}{\to} n Vect
   \,.
\end{displaymath}
\hypertarget{3dCSFullyExtended}{}\subsection*{{3d Chern-Simons as a fully extended TQFT}}\label{3dCSFullyExtended}

A summary of the FHLT-argument about realizing 3d [[Chern-Simons theory]] as a fully [[extended TQFT]] is given in

\begin{itemize}%
\item [[Konrad Waldorf]], \emph{Chern-Simons theory and the categorified group ring} ([[WaldorfTalbot2010.pdf:file]])

\end{itemize}
The following are some notes from a talk by [[Constantin Teleman]] on joint work with [[Dan Freed]], given at \href{http://maths-old.anu.edu.au/esi/2012/}{ESI Program on K-Theory and Quantum Fields (2012)}.

\hypertarget{cs_as_a_fully_extended_tqft}{}\subsubsection*{{CS as a fully extended TQFT}}\label{cs_as_a_fully_extended_tqft}

Goals

\begin{itemize}%
\item i) describe [[Chern-Simons theory]] for compact Lie groups as an [[extended TQFT]], generated by some structure assigned to the point


\item ii) relate to chiral [[WZW model]], also down to the point, have a formal framework for this



\end{itemize}
approach related to:

\begin{itemize}%
\item [[Kevin Walker]]: 3d Chern-Simons theory is best understood as a boundary condition for a 4d theory.


\item [[Chris Douglas]], [[Andre Henriques]]: \emph{[[conformal nets]]} and their relation to CS theory



\end{itemize}
construction is special case of [[Reshetikhin-Turaev construction]] which to a [[modular tensor category]] $\mathcal{B}$ assigns a 1-2-3 [[extended TQFT]] that assigns $\mathcal{B}$ to the [[circle]] $S^1$.

So the goal here is to extend this down to the point, to a 0-1-2-3 [[extended TQFT]], hence to an $(\infty,3)$-functor

\begin{displaymath}
Bord^{String}_{0,1,2,3} \to some\;symm\;mon.\;3\; category
\end{displaymath}
the standard choice on the right is a 3-category of (multi) [[fusion categories]], (see the reference by Douglas, Schommer-Pries and Snyder there) whose

\begin{itemize}%
\item [[objects]] are [[fusion categories]];


\item [[morphisms]] are bimodule categories;


\item [[2-morphisms]]: bimodule homomorphism functors;


\item [[n-morphisms|3-morphisms]]: [[natural transformations]] of these.



\end{itemize}
If this can be done, then

\begin{itemize}%
\item $CS(*) = T$;


\item $CS(S^1) =$ [[Drinfeld center]] of $T$ (again an MTC)



\end{itemize}
[[Witt group]] of [[modular tensor category]]: many abelian examples of CS give nontrivial classes

\textbf{Theorem} Given a [[modular tensor category]] $A$, there exists a [[symmetric monoidal (infinity,n)-category|symmetric monoidal 3-category]] $\mathcal{C}_A$ containing the [[fusion categories]] and a [[fully dualizable object]] in $X \in \mathcal{C}_A$ which [[cobordism hypothesis|generates]] a 0-1-2-3 [[extended TQFT]] whose 1-2-3 part agrees with the [[Reshetikhin-Turaev construction]] applied to $A$.

Here $\mathcal{C}_A$ and $X$ are formally constructed form $A$ by means of

\begin{itemize}%
\item $X \otimes X^\vee \simeq A$


\item $\mathcal{C}_A = FusionCat[X, X^\vee]$



\end{itemize}
\textbf{Remarks}

\begin{itemize}%
\item i) This is a theory for ``[[string structure]]'' manifolds in the sense that the [[first Stiefel-Whitney class]] $w_1$ and the [[first Pontryagin class]] $p_1$ are trivialized, but not necessarily $w_2$.


\item ii) This is the universal extension: every other one factors through it.

guess: there is some kind of an ``algebraic extension'' of fusion categories in which the equation $X \otimes X^\vee = A$ can be solved for any MTC $A$.


\item iii) a choice of cube root wil be needed to construct the theory for a [[framed manifold|framed]] 3-manifold

change of framing: $n \mapsto \times \exp(\frac{2 \pi i c n }{24})$ for $c$ a ``[[central charge]]''



\end{itemize}
because string [[bordism ring|bordism group]] in $dim = 3$ is $\mathbb{Z}_3$, in case of [[spin structure]]

\begin{displaymath}
\itexarray{
    3 & \mathbb{Z}_{24}
    \\
    2 & \mathbb{Z}_2
    \\
    1 & \mathbb{Z}_2
  }
\end{displaymath}
for spin theories one would need categorical representations of this 3-groupoid, but at the moment not known.

\textbf{Theorem} ([[Kevin Walker]], in [[Jacob Lurie]]`s language)

\begin{itemize}%
\item i) $A$ generates an invertible 4d [[extended TQFT]] for [[orientation|oriented]] manifolds;


\item ii) $A$ is a valid boundary condition over itself;



\end{itemize}
The 3d boundary theory in dim 1-2-3 is equivalent to $ReshTur(A)$ after a choice of ``bulking manifold''

\textbf{Comments} there is a symmetric monoidal 4-category whose

\begin{itemize}%
\item objects are bimodule tensor categories


\item morphisms are bialgebra categories = tensor categories $T$ with braided monoidal functors

$(B') \otimes B^{bop} \to DZ(T)$


\item 2-morphisms are bimodule categories


\item 3-morphisms are functors respecting the structure;


\item 4-morphisms are natural transformations between these.



\end{itemize}
Invertibility of a TQFT $Z$ $\Leftrightarrow$ invertibility of $Z(*)$.

so then $Z(S^1) \simeq 1$

\textbf{Lemma} [[modular tensor categories]] are invertible in this sense

(4d anomaly theory) + (bulking manifold) = (3d standalone theory)

\textbf{Result} Have a 3d TQFT defined on the full subcategory of $Bord^String_{0-1-2-3-4}$ of manifolds which bound.

So the final step in the construction of the full 0-1-2-3-4 theory is to extend from that to the full category of bordisms.

\hypertarget{boundary_conditions_wzw_theory}{}\subsubsection*{{Boundary conditions: WZW theory}}\label{boundary_conditions_wzw_theory}

general metaphor:

\begin{itemize}%
\item TQFT $\leftrightarrow$ algebra


\item boundary condition $\leftrightarrow$ bimodule



\end{itemize}
other metaphor

\begin{itemize}%
\item TQFT $Z$ determined by $Z(*)$


\item boundary condition $\alpha : 1 \to Z(*)$ (left) or $\beta : Z(*) \to 1$ (right)



\end{itemize}
For the 3d RT theory this yields for boundary conditions $DZ(T)$-algebra categories.

\textbf{Theorem} ([[Graeme Segal]])

Chiral WZW model is a conformal boundary for CS theory

\textbf{Problem} make this work down to the point

\hypertarget{related_references}{}\subsection*{{Related references}}\label{related_references}

\begin{itemize}%
\item [[Dan Freed]], \emph{[[4-3-2 8-7-6]]}

\end{itemize}
category: reference



\end{document}