\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*{volume conjecture}

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

\hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms}

[[!include manifolds and cobordisms - contents]]

\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{for__on_knot_complements}{For $SU(2)$ on knot complements}\dotfill \pageref*{for__on_knot_complements} \linebreak
\noindent\hyperlink{IdeaGenerally}{For $SU(2)$ on general 3-manifolds}\dotfill \pageref*{IdeaGenerally} \linebreak
\noindent\hyperlink{for_}{For $SU(n)$}\dotfill \pageref*{for_} \linebreak
\noindent\hyperlink{proof_strategies}{Proof strategies}\dotfill \pageref*{proof_strategies} \linebreak
\noindent\hyperlink{AsAdSCFTPlus3d3dDuality}{As combined AdS/CFT + 3d/3d duality for wrapped M5-branes}\dotfill \pageref*{AsAdSCFTPlus3d3dDuality} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{via_string_theory}{Via string theory}\dotfill \pageref*{via_string_theory} \linebreak
\noindent\hyperlink{general_2}{General}\dotfill \pageref*{general_2} \linebreak
\noindent\hyperlink{as_adscft__3d3d_duality_for_wrapped_m5branes}{As AdS/CFT + 3d/3d duality for wrapped M5-branes}\dotfill \pageref*{as_adscft__3d3d_duality_for_wrapped_m5branes} \linebreak


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

The \emph{volume conjectures} are a class of [[conjectures]] (slightly differing in generality and assumptions), saying that on a suitable (in particular: [[hyperbolic 3-manifold|hyperbolic]]) [[3-manifold]] $X$ a [[large N limit]] of [[SU(n)]]-[[Chern-Simons theory]] [[quantum observables]] ($N$-[[colored Jones polynomials]] or more generally [[RT-invariants]], [[TV-invariants]]) is [[equality|equal]] to the [[volume]] or [[complex volume]] of $X$.

For a few special cases of 3-manifolds there are explicit [[proofs]] of the volume conjecture(s). Besides this there is an abundance of [[computer experiment|numerical evidence]] for the volume conjectures, using computer algebra such as [[SnapPy]] (see also \hyperlink{Zickert07}{Zickert 07}). In fact experimentation with these numerics is what has been driving the formulation of further variants of the volume conjecture.

Hence [[experimental mathematics]] strongly suggests that the volume conjectures are true. But a conceptual explanation (let alone [[proof]]) in terms of [[quantum field theory]] has remained open (\hyperlink{Witten14}{Witten 14, bottom of p. 4}).

But an explanation via [[duality in string theory]], combining [[AdS/CFT duality]] with the [[3d/3d correspondence]] for [[wrapped brane|wrapped]] [[M5-branes]], is argued for in \hyperlink{GangKimLee14}{Gang-Kim-Lee 14, 3.2}, \hyperlink{GangKim18}{Gang-Kim 18 (21)}, see \hyperlink{AsAdSCFTPlus3d3dDuality}{below}.

\hypertarget{for__on_knot_complements}{}\subsubsection*{{For $SU(2)$ on knot complements}}\label{for__on_knot_complements}

The original \emph{volume conjecture} (also ``Kashaev's conjecture'', due to \hyperlink{Kashaev95}{Kashaev 95}, and understood in terms of the $N$-[[colored Jones polynomial]] by \hyperlink{MurakamiMurakami01}{Murakami-Murakami 01}) states that the [[large N limit]] of the $N$-[[colored Jones polynomial]] (for [[gauge group]] [[SU(2)]]) of a [[knot]] $K$ gives the simplicial [[volume]] of its [[complement]] in the [[3-sphere]] (for [[hyperbolic knots]] this is the volume of the complementary [[hyperbolic 3-manifold]])

\begin{equation}
lim_{N \to \infty}
  \left(
    \frac{
       2 \pi log
    }
    {N}
      \left\vert
        V_N(K; q = e^{\frac{2 \pi i}{N}})
      \right\vert     
  \right) 
  \;=\; 
  vol(K).
\label{KashaevConjecture}\end{equation}
Here $V_N(K; q)$ is the ratio of the values of the $N$-[[colored Jones polynomial]] of $K$ and of the [[unknot]]

\begin{displaymath}
V_N(K; q) = \frac{J_N(K; q)}{J_N(\bigcirc; q)}.
\end{displaymath}
The simplicial volume of a knot complement can be found via its unique \textbf{torus} decomposition into hyperbolic pieces and Seifert fibered pieces by a system of tori. The simplicial volume is then the sum of the hyperbolic volumes of the hyperbolic pieces of the decomposition.

If one omits the [[absolute value]] in \eqref{KashaevConjecture} then the volume conjecture instead involves the [[complex volume]] (\hyperlink{MMOTY02}{MMOTY 02, Conjecture 1.2}).

\hypertarget{IdeaGenerally}{}\subsubsection*{{For $SU(2)$ on general 3-manifolds}}\label{IdeaGenerally}

More generally, volume conjectures state [[convergence of a sequence|convergence]] of the [[Turaev-Viro invariants]] or [[Reshetikhin-Turaev invariants]] on general [[hyperbolic 3-manifolds]] to the [[volume]] or [[complex volume]], respectively.

See (\hyperlink{ChenYang15}{Chen-Yang 15})

\hypertarget{for_}{}\subsubsection*{{For $SU(n)$}}\label{for_}

Generalization from [[gauge group]] [[SU(2)]] to [[SU(n)]]: \hyperlink{ChenLiuZhu15}{Chen-Liu-Zhu 15}

\hypertarget{proof_strategies}{}\subsection*{{Proof strategies}}\label{proof_strategies}

\hypertarget{AsAdSCFTPlus3d3dDuality}{}\subsubsection*{{As combined AdS/CFT + 3d/3d duality for wrapped M5-branes}}\label{AsAdSCFTPlus3d3dDuality}

In \hyperlink{GangKimLee14b}{Gang-Kim-Lee 14b, 3.2}, \hyperlink{GangKim18}{Gang-Kim 18 (21)} it is argued that the [[volume conjecture]] for [[Chern-Simons theory]] on [[hyperbolic 3-manifolds]] $\Sigma^3$ is the combined statement of two [[dualities in string theory]]

\begin{enumerate}%
\item [[AdS/CFT duality]]


\item [[3d-3d correspondence]]



\end{enumerate}
for the situation of [[M5-branes]] [[wrapped brane|wrapped on]] $\Sigma^3$ (\hyperlink{DGKV10}{DGKV 10}):

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

\begin{itemize}%
\item [[hyperbolic manifold]]


\item [[analytically continued Chern-Simons theory]]


\item [[Borel regulator]]


\item [[membrane instanton]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Original articles include

\begin{itemize}%
\item [[Rinat Kashaev]], \emph{A Link Invariant from Quantum Dilogarithm}, Modern Physics Letters AVol. 10, No. 19, pp. 1409-1418 (1995) (\href{https://arxiv.org/abs/q-alg/9504020}{arXiv:q-alg/9504020})


\item [[Rinat Kashaev]], \emph{The Hyperbolic Volume Of Knots From The Quantum Dilogarithm} Lett. Math. Phys. 39 (1997) 269-275 (\href{https://arxiv.org/abs/q-alg/9601025}{arXiv:q-alg/9601025})


\item [[Rinat Kashaev]], O. Tirkkonen, \emph{Proof of the volume conjecture for torus knots}, Journal of Mathematical Sciences (2003) 115: 2033 (\href{http://arxiv.org/abs/math/9912210}{arXiv:math/9912210})


\item [[Hitoshi Murakami]] and [[Jun Murakami]], \emph{The Colored Jones Polynomial And The Simplicial Volume Of A Knot}, Acta Math. 186 (2001) 85-104.


\item [[Hitoshi Murakami]], [[Jun Murakami]], M. Okamoto, T. Takata, and Y. Yokota, \emph{Kashaev's Conjecture And The Chern-Simons Invariants Of Knots And Links}, Experiment. Math. 11 (2002) 427-435 (\href{https://arxiv.org/abs/math/0203119}{arXiv:math/0203119})


\item [[Hitoshi Murakami]], \emph{Asymptotic Behaviors Of The Colored Jones Polynomials Of A Torus Knot}, Internat. J. Math. 15 (2004) 547-555.



\end{itemize}
Generalization to [[Reshetikhin-Turaev construction]] on closed manifold, to the [[Turaev-Viro construction]] on [[manifolds with boundary]], and to more general [[roots of unity]] than considered before is in

\begin{itemize}%
\item [[Qingtao Chen]], [[Tian Yang]], \emph{A volume conjecture for a family of Turaev-Viro type invariants of 3-manifolds with boundary} (\href{http://arxiv.org/abs/1503.02547}{arXiv:1503.02547})


\item Dongmin Gang, Mauricio Romo, Masahito Yamazaki, \emph{All-Order Volume Conjecture for Closed 3-Manifolds from Complex Chern-Simons Theory}, Commun. Math. Phys. (2018) 359: 915. (\href{https://arxiv.org/abs/1704.00918}{arXiv:1704.00918}, \href{https://doi.org/10.1007/s00220-018-3115-y}{doi:10.1007/s00220-018-3115-y})



\end{itemize}
Generalization to [[SU(n)]]:

\begin{itemize}%
\item [[Qingtao Chen]], [[Kefeng Liu]], Shengmao Zhu, \emph{Volume conjecture for SU(n)-invariants} (\href{https://arxiv.org/abs/1511.00658}{arXiv:1511.00658})

\end{itemize}
Review includes

\begin{itemize}%
\item [[Hitoshi Murakami]], \emph{An Introduction to the Volume Conjecture} (\href{https://arxiv.org/abs/1002.0126}{arXiv:1002.0126}) In: \emph{Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory}, Contemporary Mathematics Volume 541, AMS 2011 (\href{http://dx.doi.org/10.1090/conm/541}{doi:10.1090/conm/541})


\item [[Edward Witten]], pp. 4 of \emph{Two Lectures On The Jones Polynomial And Khovanov Homology} (\href{http://arxiv.org/abs/1401.6996}{arXiv:1401.6996})


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Volume_conjecture}{Volume conjecture}}



\end{itemize}
See also

\begin{itemize}%
\item [[Walter Neumann]], \emph{Extended Bloch group and the Cheeger-Chern-Simons class}, Geom. Topol. 8 (2004) 413-474 (\href{http://arxiv.org/abs/math/0307092}{arXiv:math/0307092})


\item [[Christian Zickert]], \emph{The volume and Chern-Simons invariant of a representation}, Duke Math. J., 150 (3):489-532, 2009 (\href{http://arxiv.org/abs/0710.2049}{arXiv:0710.2049})


\item [[Walter Neumann]], \emph{Realizing arithmetic invariants of hyperbolic 3-manifolds}, Contemporary Math 541 (Amer. Math. Soc. 2011), 233--246 (\href{http://arxiv.org/abs/1108.0062}{arXiv:1108.0062})


\item Stavros Garoufalidis, [[Dylan Thurston]], [[Christian Zickert]], \emph{The complex volume of $SL(n,\mathbb{C})$-representations of 3-manifolds} (\href{http://arxiv.org/abs/1111.2828}{arXiv:1111.2828}, \href{http://projecteuclid.org/euclid.dmj/1259332507}{Euclid})



\end{itemize}
\hypertarget{via_string_theory}{}\subsubsection*{{Via string theory}}\label{via_string_theory}

\hypertarget{general_2}{}\paragraph*{{General}}\label{general_2}

Speculative discussion in terms of [[quantum field theory]] or [[string theory]] includes

\begin{itemize}%
\item [[Sergei Gukov]], \emph{Three-Dimensional Quantum Gravity, Chern-Simons Theory, And The A-Polynomial}, Commun. Math. Phys. 255 (2005) 577-627 (\href{https://arxiv.org/abs/hep-th/0306165}{arXiv:hep-th/0306165})


\item [[Robbert Dijkgraaf]], Hiroyuki Fuji, \emph{The Volume Conjecture and Topological Strings} (\href{http://arxiv.org/abs/0903.2084}{arXiv:0903.2084})


\item [[Tudor Dimofte]], [[Sergei Gukov]], \emph{Quantum Field Theory and the Volume Conjecture}, Contemporary Mathematics 541 (2011), p.41-67 (\href{http://arxiv.org/abs/1003.4808}{arxiv:1003.4808})



\end{itemize}
A conceptual explanation of the volume conjecture via [[analytically continued Chern-Simons theory]] was proposed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Analytic Continuation Of Chern-Simons Theory}, AMS/IP Stud. Adv. Math 50 (2011): 347 (\href{https://arxiv.org/abs/1001.2933}{arXiv:1001.2933})

\end{itemize}
(but it seems that as a sketch or strategy for a rigorous proof, it didn't catch on).

\hypertarget{as_adscft__3d3d_duality_for_wrapped_m5branes}{}\paragraph*{{As AdS/CFT + 3d/3d duality for wrapped M5-branes}}\label{as_adscft__3d3d_duality_for_wrapped_m5branes}

Suggestion that the statement of the [[volume conjecture]] is really [[AdS-CFT duality]] combined with the [[3d-3d correspondence]] for [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

\begin{itemize}%
\item Dongmin Gang, [[Nakwoo Kim]], Sangmin Lee, Section 3.2\_Holography of 3d-3d correspondence at Large $N$\emph{, JHEP04(2015) 091 (\href{https://arxiv.org/abs/1409.6206}{arXiv:1409.6206})}


\item Dongmin Gang, [[Nakwoo Kim]], around (21) of: \emph{Large $N$ twisted partition functions in 3d-3d correspondence and Holography}, Phys. Rev. D 99, 021901 (2019) (\href{https://arxiv.org/abs/1808.02797}{arXiv:1808.02797})



\end{itemize}
[[!redirects volume conjectures]]



\end{document}