\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*{2d TQFT}

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

\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{classification}{Classification}\dotfill \pageref*{classification} \linebreak
\noindent\hyperlink{FiltrationOfModuliSpace}{Filtrations of the moduli space of surfaces}\dotfill \pageref*{FiltrationOfModuliSpace} \linebreak
\noindent\hyperlink{monoid_objects}{$A_\infty$-monoid objects}\dotfill \pageref*{monoid_objects} \linebreak
\noindent\hyperlink{closed_2d_quantum_field_theory}{Closed 2d quantum field theory}\dotfill \pageref*{closed_2d_quantum_field_theory} \linebreak
\noindent\hyperlink{compactified_moduli_spaces_of_riemann_surfaces}{Compactified moduli spaces of Riemann surfaces}\dotfill \pageref*{compactified_moduli_spaces_of_riemann_surfaces} \linebreak
\noindent\hyperlink{fenchelnielson_coordinates_on_moduli_space}{Fenchel-Nielson coordinates on moduli space}\dotfill \pageref*{fenchelnielson_coordinates_on_moduli_space} \linebreak
\noindent\hyperlink{openclosed_case}{Open-closed case}\dotfill \pageref*{openclosed_case} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{global}{Global}\dotfill \pageref*{global} \linebreak
\noindent\hyperlink{local}{Local}\dotfill \pageref*{local} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{2-dimensional TQFT} is a [[TQFT|topological quantum field theory]] on [[cobordisms]] of [[dimension]] 2.

\hypertarget{classification}{}\subsection*{{Classification}}\label{classification}

When formulated as an (only) ``globally'' as 1-[[functors]] on a 1-[[category of cobordisms]] (see at \emph{[[FQFT]]} for more), then 2d TQFTs have a comparatively simple classification: the [[bulk field theory]] is determined by a commutative [[Frobenius algebra]] structure on the [[finite dimensional vector space]] assigned to the circle (\hyperlink{Abrams96}{Abrams 96}).

However, such global 2d TQFTs with [[coefficients]] in [[Vect]] do not capture the 2d TQFTs of most interest in [[quantum field theory]], which instead are ``[[cohomological quantum field theories]]'' (\hyperlink{Witten91}{Witten 91}) such as the [[topological string]] [[A-model]] and [[B-model]] that participate in [[homological mirror symmetry]].

These richer 2d TQFTs are instead local TQFTs in the sense of \emph{[[extended TQFT]]}, i.e. they are [[(∞,2)-functors]] on a suitable [[(∞,2)-category of cobordisms]] (see at \emph{[[FQFT]]} for more), typically on ``non-compact'' 2-d cobordisms, meaning on those that have non-vanishing outgoing bounary. As such they are now classified by [[Calabi-Yau objects]] in an [[symmetric monoidal (infinity,2)-category]] (\hyperlink{Lurie09}{Lurie 09, section 4.2}). For coefficients in the [[(∞,2)-category]] of [[(∞,n)-vector space|(∞,2)-vector space]] (i.e. [[A-∞ algebras]] with [[(∞,1)-bimodules]] between them in the [[(∞,1)-category of chain complexes]]), these theories had been introduced under the name ``[[TCFT]]'' in (\hyperlink{Getzler92}{Getzler 92}, \hyperlink{Segal99}{Segal 99}) following ideas of [[Maxim Kontsevich]], and have been classified in (\hyperlink{Costello04}{Costello 04}), see (\hyperlink{Lurie09}{Lurie 09, theorem 4.2.11, theorem 4.2.14}).

[[!include 2d TQFT -- table]]

\hypertarget{FiltrationOfModuliSpace}{}\subsection*{{Filtrations of the moduli space of surfaces}}\label{FiltrationOfModuliSpace}

The following study of the behaviour of 2-dimensional TQFTs in terms of the [[topology]] of the [[moduli spaces]] of marked hyperbolic surfaces is due to [[Ezra Getzler]]. It provides a powerful way to read off various classification results for 2d QFTs from the [[homotopy groups]] of the corresponding [[modular operad]].

\hypertarget{monoid_objects}{}\subsubsection*{{$A_\infty$-monoid objects}}\label{monoid_objects}

Let $Core$([[FinSet]]) be the [[core]] of the [[category]] of finite sets. Under union of sets this is a [[symmetric monoidal category]]. Then for $C$ any [[monoidal category]], a [[symmetric monoidal functor]]

\begin{displaymath}
\Phi : Core(FinSet) \to C
\end{displaymath}
is a commutative [[monoid]] in $C$.

Let now $C$ be a [[category with weak equivalences]], then we can speak of a lax symmetric opmonoidal functor

\begin{displaymath}
\Phi : Core(FinSet) \to C
\end{displaymath}
if the structure maps

\begin{displaymath}
\Phi(n+m) \stackrel{\simeq}{\to} \Phi(m) \otimes \Phi(n)
\end{displaymath}
\begin{displaymath}
\phi(m) \stackrel{\simeq}{\to} \Phi(1)^{\otimes m}
\end{displaymath}
are weak equivalences.

Segal called these ``$\Delta$-objects''. Since [[Carlos Simpson]] they are called [[Segal object]]s.

There is also [[Jim Stasheff]]`s notion of an [[A-infinity algebra]], given in terms of [[associahedra]] $K_n$, which are $(n-2)$-dimensional [[polytope]]s.

There is naturally a filtration on these guys with

\begin{displaymath}
F_0 K_n \subset F_1 K_n \subset \cdots
  \,,
\end{displaymath}
where $F_0 K_n$ is the set of vertices, $F_1 K_n$ the set of edges, etc.

The collection

\begin{displaymath}
\{
    S_\bullet(K_n)
  \}
\end{displaymath}
of simplicial realizations of the $K_n$ form an [[sSet]]-[[operad]] $P$.

For $X$ a [[simplicial category]] that is symmetric monoidal, a $P$-[[algebra over an operad]] $X$ in $C$ is an $A_\infty$-monoid object

\begin{displaymath}
S_\bullet(K_n) \to C_\bullet(X^{\otimes n}, X)
\end{displaymath}
[[Saunders MacLane|MacLane]]`s [[coherence theorem]] says or uses that if $C$ is an [[n-category]], we may replace $K_m$ here by the $n$-filtration $F_n K_m$.

\hypertarget{closed_2d_quantum_field_theory}{}\subsubsection*{{Closed 2d quantum field theory}}\label{closed_2d_quantum_field_theory}

\hypertarget{compactified_moduli_spaces_of_riemann_surfaces}{}\paragraph*{{Compactified moduli spaces of Riemann surfaces}}\label{compactified_moduli_spaces_of_riemann_surfaces}

Let

\begin{displaymath}
(\Sigma, (z_1, \cdots, z_n))
\end{displaymath}
be a [[compact space|compact]] [[orientation|oriented]] surface with $n$ distinct marked points. Write

\begin{displaymath}
\mathcal{H}(\Sigma, (z_1, \cdots, z_n))
\end{displaymath}
for the [[moduli space]] of [[hyperbolic metric]]s with cusps at the $(z_i)$.

We have

\begin{displaymath}
M(\Sigma, \vec z)
  =
  \mathcal{H}(\Sigma, \vec z)/Diff_+(\Sigma, \vec z)
\end{displaymath}
and

\begin{displaymath}
M_{g,n} = \Tau_{g,n} / \Gamma^ng
  \,,
\end{displaymath}
where $\Tau_{g,n}$ is the [[Teichmüller space]] and $\Gamma$ the [[mapping class group]].

Here we can assume that the [[Euler characteristic]] $\chi(\Sigma without \{z_i\}) \lt 0$ because otherwise this moduli space is empty.

\hypertarget{fenchelnielson_coordinates_on_moduli_space}{}\paragraph*{{Fenchel-Nielson coordinates on moduli space}}\label{fenchelnielson_coordinates_on_moduli_space}

We want to parameterize Teichm\"u{}ller space by cutting surfaces into pieces with geodesic boundaries and [[Euler characteristic]] $\xi = -1$. These building blocks (of hyperbolic 2d geometry) are precisely

\begin{itemize}%
\item the 3-holed sphere;


\item the 2-holed cusp;


\item the 1-holed 2-cusp;


\item the 3-cusp



\end{itemize}
Each surface of [[genus]] $g$ with $n$ marked points will have

\begin{itemize}%
\item $2g - 2 + n$ generalized pants;


\item $3 g - 3 + n$ closed curves.



\end{itemize}
The boundary lengths $\ell_i \in \mathbb{R}_+$ and twists $t_i \in \mathbb{R}$ of these pieces for

\begin{displaymath}
1 \leq i \leq 3g-3+n
\end{displaymath}
constitute the [[Fenchel-Nielsen coordinates]] on [[Teichmüller space]] $\Tau$.

Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$

This constitutes is a real analytic [[atlas]] of Teichm\"u{}ller space. On $M$ this reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$, and these constitute a real analytic atlas of moduli space.

Allow the lengths $\ell_i$ to go to 0, but keep the angles $\theta_i$. The resulting space is a real analytic [[manifold with corners]] $\hat \Tau$ (due to [[Bill Harvey]]) and this constitutes a Borel-Serre [[bordification]] of $\Tau$.

The [[mapping class group]] $\Gamma$ still acts on $\hat \Tau$ and the quotient $\hat M$ is an [[orbifold]] with corners, inside which still sits our moduli space $M$.

Kimura-Stasheff-Voronov: add a choice of directions at each nodal point in $\Sigma$. This removes all automorphisms and hence we no longer have to deal with an [[orbifold]].

This yields the [[classifying stack]] $\mathcal{P}_{g,n}$ for $\Gamma_{g,n}$

Then the collection

\begin{displaymath}
\{
    \mathcal{P}_{g,n}
  \}
\end{displaymath}
is a [[modular operad]]: the operad that describes gluing of marked surfaces at marked points together with the informaiton on how to glue marked points of a single surface to each other.

A 2-dimensional closed [[TQFT]] is an [[algebra over an operad]] over this in a simplicial category, in the above sense.

This involves either the [[de Rham complex]] on $\mathcal{P}_{g,n}$ or $S_\bullet(\mathcal{P}_{g,n})$.

Let

\begin{displaymath}
F_k \mathcal{P}_{g,n}
  := 
  \left\{
    [\Sigma] |
    ...
  \right\}
\end{displaymath}
where $\Sigma$ has $\geq 2g-2+n-k$ spheres as components (after cutting along zero-length closed curves).

So for instance

\begin{itemize}%
\item $F_0 \mathcal{P}_{g,n}$ is the pants-decomposition;


\item $F_1 \mathcal{P}_{g,n}$ is decompositions into pants and one piece being the result of either gluing two pants to each other or of gluing two circles of a single pant to each other.

This $F_1 ..$ is a connected space, due to a theorem by Hatcher-Thurston. \textbf{Notice} This is equivalent to the familiar statement that a closed 2d TFT is a commutative [[Frobenius algebra]].


\item $F_2 \mathcal{P}_{g,n}$ is the decomposition into pieces as before together with one two-holed torus or one five-holed sphere.

This space has the space [[fundamental group]] as $\mathcal{P}_{g,n}$.

This is equivalent to the theorem by Moore and Seiberg about categorified 2-d TFT.



\end{itemize}
\begin{utheorem}
([[Ezra Getzler]])

The inclusion

\begin{displaymath}
F_k \mathcal{P}_{g,n} \hookrightarrow \mathcal{P}_{g,n}
\end{displaymath}
is $k$-[[connected]].

Here a map $X\to Y$ is $k$-connected if

\begin{itemize}%
\item $\pi_0(X) \to \pi_0(Y)$ is surjective;


\item $\pi_i(X,x) \to \pi_i(Y,f(x))$ is a bijection for $i \lt k$ and surjective $i = k$.



\end{itemize}
This means precisely that the [[mapping cone]] is $k$-connected.

\end{utheorem}
\begin{proof}
Use the cellular decomposition of moduli space $\mathcal{M}_{g,1}$ following Mumford, Thurston, Harer, Woeditch-Epstein, Penner.

\end{proof}
Some other versions of this:

\begin{displaymath}
F_k \Tau_{g,n} \to \Tau_{g,n}
\end{displaymath}
is $k$-connected.

One can also use [[Deligne-Mumford compactification]]s

\begin{displaymath}
F_k \bar \mathcal{M}_{g,n} \to \bar \mathcal{M}_{g,n}
\end{displaymath}
and this is also $k$-connected.

\hypertarget{openclosed_case}{}\subsubsection*{{Open-closed case}}\label{openclosed_case}

\ldots{}

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sewing constraint]]

\begin{itemize}%
\item [[Cardy condition]]

\end{itemize}

\item [[TQFT]]

\begin{itemize}%
\item \textbf{2d TQFT}

\begin{itemize}%
\item [[2d Chern-Simons theory]]


\item [[TCFT]]

\begin{itemize}%
\item [[A-model]], [[B-model]]

\end{itemize}

\item [[Landau-Ginzburg model]]


\item [[Levin-Wen model]]



\end{itemize}

\item [[3d TQFT]]


\item [[4d TQFT]]



\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{global}{}\subsubsection*{{Global}}\label{global}

The [[folklore]] result that global closed 2d TQFTs with coefficients in [[Vect]] are equivalent to commutative [[Frobenius algebra]] structures is proven rigorously in

\begin{itemize}%
\item [[Lowell Abrams]], \emph{Two-dimensional topological quantum eld theories and Frobenius algebras}, J. Knot Theory Ramications 5 (1996) (\href{http://home.gwu.edu/~labrams/docs/tqft.ps}{ps})

\end{itemize}
The classification result for open-closed 2d TQFTs was famously announced and sketched in

\begin{itemize}%
\item [[Greg Moore]], [[Graeme Segal]], \emph{Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002),} (\href{http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html}{web})


\item [[Calin Lazaroiu]], \emph{On the structure of open-closed topological field theory in two dimensions}, Nuclear Phys. B 603(3), 497--530 (2001), (\href{http://arxiv.org/abs/hep-th/0010269}{arXiv:hep-th/0010269})



\end{itemize}
A standard textbook is

\begin{itemize}%
\item [[Joachim Kock]], \emph{Frobenius algebras and 2D topological quantum field theory} (\href{http://mat.uab.cat/~kock/TQFT.html}{web}, \href{http://mat.uab.es/~kock/TQFT/FS.pdf}{course notes pdf})

\end{itemize}
A picture-rich description of what's going on is in

\begin{itemize}%
\item [[Aaron Lauda]], [[Hendryk Pfeiffer]], \emph{Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras} , Topology Appl. 155 (2008) 623-666. (\href{http://arxiv.org/abs/math.AT/0510664}{arXiv:math.AT/0510664})

\end{itemize}
\hypertarget{local}{}\subsubsection*{{Local}}\label{local}

The local ([[extended TQFT]]) version of 2d TQFT which captures the [[topological string]] was mathematically introduced under the name ``[[TCFT]]''.

The concept is essentially a formalization of what used to be called [[cohomological field theory]] in

\begin{itemize}%
\item [[Edward Witten]], \emph{Introduction to cohomological field theory}, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])

\end{itemize}
The definition was given independently by

\begin{itemize}%
\item [[Ezra Getzler]], \emph{Batalin-Vilkovisky algebras and two-dimensional topological field theories} , Comm. Math. Phys. 159(2), 265--285 (1994) (\href{http://arxiv.org/abs/hep-th/9212043}{arXiv:hep-th/9212043})

\end{itemize}
and

\begin{itemize}%
\item [[Graeme Segal]], \emph{Topological field theory} , (1999), Notes of lectures at Stanford university. (\href{http://www.cgtp.duke.edu/ITP99/segal/}{web}). See in particular \href{http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf}{lecture 5} (``topological field theory with cochain values'').

\end{itemize}
The classification of [[TCFT]]s (i.e. ``non-compact'' local ([[extended TQFT|extended]] 2d TQFT)) by [[Calabi-Yau A-infinity categories]] is due to

\begin{itemize}%
\item [[Kevin Costello]], \emph{Topological conformal field theories and Calabi-Yau categories} Advances in Mathematics, Volume 210, Issue 1, (2007), (\href{http://arxiv.org/abs/math/0412149}{arXiv:math/0412149})


\item [[Kevin Costello]], \emph{The Gromov-Witten potential associated to a TCFT} (\href{http://arxiv.org/abs/math/0509264}{arXiv:math/0509264})



\end{itemize}
following conjectures by [[Maxim Kontsevich]], e.g.

\begin{itemize}%
\item [[Maxim Kontsevich]], \emph{Homological algebra of mirror symmetry} , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z\"u{}rich, 1994), pages 120--139, Basel, 1995, Birkh\"a{}user.

\end{itemize}
The classification of local ([[extended TQFT|extended]]) 2d TQFT (i.e. the ``compact'' but fully local case) is spelled out in

\begin{itemize}%
\item [[Chris Schommer-Pries]], \emph{The Classification of Two-Dimensional Extended Topological Field Theories} (\href{http://arxiv.org/abs/1112.1000}{arXiv:1112.1000})

\end{itemize}
This classification is a precursor of the full [[cobordism hypothesis]]-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

\begin{itemize}%
\item [[Jacob Lurie]], section 4.2 of \emph{[[On the Classification of Topological Field Theories]]} (\href{http://arxiv.org/abs/0905.0465}{arXiv:0905.0465})

\end{itemize}
[[!redirects 2d TQFTs]]

[[!redirects 2d topological field theory]] [[!redirects 2d topological field theories]]

[[!redirects 2d topological quantum field theory]] [[!redirects 2d topological quantum field theories]]



\end{document}