\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*{AGT correspondence}

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

\hypertarget{duality_in_string_theory}{}\paragraph*{{Duality in string theory}}\label{duality_in_string_theory}

[[!include duality in string theory -- contents]]

\hypertarget{functorial_quantum_field_theory}{}\paragraph*{{Functorial Quantum field theory}}\label{functorial_quantum_field_theory}

[[!include functorial quantum field theory - contents]]

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \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{AGT correspondence} (\hyperlink{AGT09}{AGT 09}) is a relation between

\begin{enumerate}%
\item the [[instanton]]-[[partition function]] of $SU(2)^{n+3g-3}$-[[N=2 D=4 super Yang-Mills theory]] (Nekrasov's partition function, e.g. \hyperlink{Szabo15}{Szabo 15 (2.1)})


\item the [[conformal blocks]] of [[Liouville theory]] on an $n$-punctured [[Riemann surface]] $C_{g,n}$ of [[genus]] $g$



\end{enumerate}
Here the idea is that $C_{g,n}$ is the [[super Yang-Mills theory]] obtained by [[Kaluza-Klein mechanism|compactifying]] the worldvolume [[6d (2,0)-supersymmetric QFT]] of two [[M5-branes]], see at [[N=2 D=4 super Yang-Mills theory]], the section \href{N%3D2+D%3D4+super+Yang-Mills+theory#ConstructionByCompactificationOf5Branes}{Construction by compactification}).

In particular, the [[N=2 D=4 super Yang-Mills theory]] is a [[quiver gauge theory]] and the correspondence matches the shape of its [[quiver]]-diagram to the [[genus of a surface|genus]] and punctures of the [[Riemann surface]]:

More generally, this construction yields something like a decomposition of the [[6d (2,0)-superconformal QFT]] into a [[2d SCFT]] ``with values in [[super Yang-Mills theory|4d SYM field theory]]'' (e.g. \hyperlink{Tachikawa10}{Tachikawa 10, slide 25 (33 of 54)}). Hence composition with any kind of suitable invariant of the 4d field theories yields an actual [[2d SCFT]], for instance taking the superconformal index in 4d yields a [[2d TQFT]] (\hyperlink{GPRR10}{GPRR 10}). In this picture of ``4d-SYM field theory-valued [[2d SCFT]]'' one has the following correspondences:

\begin{itemize}%
\item the [[complex structure]] in 2d is the [[coupling constants]] and [[theta angles]] etc in the 4d [[super Yang-Mills theory]];


\item the [[mapping class group]] (large [[conformal transformations]]) in 2d is the (generalized) [[S-duality]] of the 4d theory.



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

\begin{itemize}%
\item [[duality in physics]], [[duality in string theory]]


\item [[Nekrasov function]]



\end{itemize}
Wrapping the [[M5-brane]] on a [[3-manifold]] instead yields: [[3d-3d correspondence]].

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

The original articles are

\begin{itemize}%
\item [[Davide Gaiotto]], \emph{$N=2$ dualities}, JHEP08(2012)034 (\href{http://arxiv.org/abs/0904.2715}{arXiv:0904.2715})


\item Luis F. Alday, [[Davide Gaiotto]], [[Yuji Tachikawa]], \emph{Liouville Correlation Functions from Four-dimensional Gauge Theories}, Lett.Math.Phys.91:167-197, 2010 (\href{http://arxiv.org/abs/0906.3219}{arXiv:0906.3219})


\item [[Davide Gaiotto]], [[Gregory Moore]], [[Andrew Neitzke]], \emph{Wall-crossing, Hitchin Systems, and the WKB Approximation} (\href{https://arxiv.org/abs/0907.3987}{arXiv:0907.3987})



\end{itemize}
The [[2d TQFT]] obtained from this by forming the 4d index is discussed in

\begin{itemize}%
\item Abhijit Gadde, Elli Pomoni, Leonardo Rastelli, Shlomo S. Razamat, \emph{S-duality and 2d Topological QFT}, JHEP 1003:032, 2010 (\href{http://arxiv.org/abs/0910.2225}{arXiv:0910.2225})

\end{itemize}
Relation of the [[AGT-correspondence]] to the [[D=6 N=(2,0) SCFT]]

\begin{itemize}%
\item Benjamin Assel, Sakura Schafer-Nameki, Jin-Mann Wong, \emph{M5-branes on $S^2 \times M_4$: Nahm's Equations and 4d Topological Sigma-models}, J. High Energ. Phys. (2016) 2016: 120 (\href{https://arxiv.org/abs/1604.03606}{arxiv:1604.03606})

(relating to the [[moduli space of monopoles]])



\end{itemize}
and to the [[3d-3d correspondence]]:

\begin{itemize}%
\item [[Clay Cordova]], [[Daniel Jafferis]], \emph{Toda Theory From Six Dimensions}, J. High Energ. Phys. (2017) 2017: 106 (\href{https://arxiv.org/abs/1605.03997}{arxiv:1605.03997})


\item Sam van Leuven, Gerben Oling, \emph{Generalized Toda Theory from Six Dimensions and the Conifold}, J. High Energ. Phys. (2017) 2017: 50 (\href{https://arxiv.org/abs/1708.07840}{arxiv:1708.07840})



\end{itemize}
Brief surveys include

\begin{itemize}%
\item [[Yuji Tachikawa]], \emph{M5-branes, 4d gauge theory and 2d CFT}, 2010 (\href{http://member.ipmu.jp/yuji.tachikawa/transp/4d-2d-caltech.pdf}{pdf})


\item Abhijit Gadde, \emph{$\mathcal{N}= 2$ Dualities and 2d TQFT} 2012 ([[Gadde2dTQFT.pdf:file]])


\item Nikolay Bovev, \emph{New SCFTs from wrapped branes}, 2013 (\href{http://ipht.cea.fr/Meetings/Itzykson2013/Talks/bobev-Itzykson-july2013.pdf}{pdf})


\item Giulio Bonelli, \emph{Variations on AGT Correspondence}, 2013 (\href{https://indico.cern.ch/event/217384/attachments/348854/486363/Bonelli.pdf}{pdf})


\item Masato Taki, \emph{String Theory as an Attempt of PolyMathematics}, talk at 2016.4/28 iTHES-AIMR-IIS (\href{https://indico2.riken.jp/event/2215/attachments/4106/4775/Taki_invited.pdf}{pdf})



\end{itemize}
More detailed review is in

\begin{itemize}%
\item Rober Rodger, \emph{A pedagogical introduction to the AGT conjecture}, Master Thesis Utrecht (2013) (\href{http://testweb.science.uu.nl/ITF/teaching/2013/R.J.Rodger.pdf}{pdf})


\item [[Richard Szabo]], \emph{$N=2$ gauge theories, instanton moduli spaces and geometric representation theory}, Journal of Geometry and Physics Volume 109, November 2016, Pages 83-121 (\href{https://arxiv.org/abs/1507.00685}{arXiv:1507.00685})



\end{itemize}
See also

\begin{itemize}%
\item [[Alexander Belavin]], M. A. Bershtein, B. L. Feigin, A. V. Litvinov, G. M. Tarnopolsky, \emph{Instanton moduli spaces and bases in coset conformal field theory} (\href{http://arxiv.org/abs/1111.2803}{arxiv/1111.2803})


\item [[Volker Schomerus]], Paulina Suchanek, \emph{Liouville's imaginary shadow} (\href{http://arxiv.org/abs/1210.1856}{arxiv/1210.1856})


\item A.Mironov, A.Morozov, \emph{The power of Nekrasov functions} (\href{http://arxiv.org/abs/0908.2190}{arxiv/0908.2190})


\item D. Galakhov, A. Mironov, A. Morozov, \emph{S-duality as a beta-deformed Fourier transform} (\href{http://arxiv.org/abs/1205.4998}{arxiv/1205.4998})


\item A. Mironov, \emph{Spectral duality in integrable systems from AGT conjecture} (\href{http://arxiv.org/abs/1204.0913}{arxiv/1204.0913})


\item A. Belavin, V. Belavin, \emph{AGT conjecture and integrable structure of conformal field theory for $c=1$}, Nucl.Phys.B850:199-213 (2011) (\href{http://arxiv.org/abs/1102.0343}{arxiv/1102.0343})


\item A. Belavin, V. Belavin, M. Bershtein, \emph{Instantons and 2d Superconformal field theory} (\href{http://arxiv.org/abs/1106.4001}{arxiv/1106.4001})


\item Kazunobu Maruyoshi, \emph{Quantum integrable systems, matrix models, and AGT correspondence}, seminar (\href{http://db.ipmu.jp/seminar/sysimg/seminar/428.pdf}{slides pdf})


\item Giulio Bonelli, Alessandro Tanzini, \emph{Hitchin systems, N=2 gauge theories and W-gravity} (\href{http://arxiv.org/abs/0909.4031}{arxiv/0909.4031})


\item Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini, \emph{Quantum Hitchin systems via beta-deformed matrix models} (\href{http://arxiv.org/abs/1104.4016}{arxiv/1104.4016})


\item [[Oscar Chacaltana]], [[Jacques Distler]], \emph{Tinkertoys for Gaiotto Duality}, JHEP 1011:099,2010, (\href{http://arxiv.org/abs/arXiv:1008.5203}{arXiv:1008.5203})


\item Satoshi Nawata, \emph{Givental J-functions, Quantum integrable systems, AGT relation with surface operator} (\href{http://arxiv.org/abs/1408.4132}{arXiv/1408.4132})



\end{itemize}
The AGT correspondence is treated with the help of a [[Riemann-Hilbert problem]] in

\begin{itemize}%
\item G. Vartanov, [[Jörg Teschner]], \emph{Supersymmetric gauge theories, quantization of moduli spaces of flat connections, and conformal field theory} (\href{http://arxiv.org/abs/1302.3778}{arxiv/1302.3778})

\end{itemize}
category: physics

[[!redirects AGT-correspondence]]

[[!redirects AGT conjecture]] [[!redirects Class S]]



\end{document}