\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 Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories}

\begin{quote}%
under construction


\end{quote}
\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{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

This page collects some links and other material related to the article

\begin{itemize}%
\item [[Sergei Gukov]], [[Anton Kapustin]],

\emph{Topological Quantum Field Theory, Nonlocal Operators, and Gapped Phases of Gauge Theories}

(\href{http://arxiv.org/abs/1307.4793}{arXiv:1307.4793})



\end{itemize}
The article considers the generalization of a phenomenon in [[spontaneous symmetry breaking]] in abelian ([[higher gauge theory|higher]]) [[gauge theory]], that was considered earlier in (\hyperlink{BanksSeiberg08}{Banks-Seiberg 08}), to \emph{[[nonabelian cohomology|nonabelian]]} [[higher gauge theory]], hence involving not just [[differential 1-form]] [[gauge potentials]] but 2-form gauge potentials (``nonabelian [[B-fields]]''), hence to [[principal 2-connections]].

Specifically, in (\hyperlink{BanksSeiberg08}{Banks-Seiberg 08, (2.4)-(2.9)}) is an argument saying that some properties of the low energy limit of abelian [[Yang-Mills theory|Yang-Mills like]] [[gauge theories]] in four dimensions with ([[Higgs mechanism|Higgs-like]]) [[spontaneous symmetry breaking]] to a discrete gauge [[cyclic group]] $\mathbb{Z}_p$ are described by a dual [[BF-theory]], hence a [[higher gauge theory]] represented by a [[principal 2-connection|2-connection]] instead of an ordinary [[connection on a bundle|connection]]. Here the duality is [[electric-magnetic duality]], but applied not to the 1-form [[gauge field]] but to the [[Higgs boson]]-like [[scalar field]], whose magentic dual in 4d is indeed a $(4 - (0+1) - 1 = 2)$-form [[higher gauge field]]. See at \emph{[[connection on a 2-bundle]]} for more on this.

In the nonabelian case the proposed mechanism is no longer [[electric-magnetic duality]] of a [[Higgs boson]]-like [[field (physics)|field]], but instead to condensation of [[monopoles]].

The abelian phenomenon is reviewed and expanded on in its relation to higher dimensional [[QFT with defects|defect]] [[quantum operators]] in section III. Generalization to the nonabelian case is then in section IV. An outlook on the relation to structures such as in [[geometric Langlands duality]] is in section V.

Followups include

\begin{itemize}%
\item [[Anton Kapustin]], [[Ryan Thorngren]], \emph{Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement} (\href{http://arxiv.org/abs/1308.2926}{arXiV:1308.2926})


\item [[Ryan Thorngren]], \emph{Electric-Magnetic Duality of Topological Gauge Theories from Compactification} (\href{http://arxiv.org/abs/1309.1489}{arXiv:1309.1489})



\end{itemize}

\vspace{.5em} \hrule \vspace{.5em}
\begin{quote}%
under construction


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

\noindent\hyperlink{ii_surface_operators_in_abelian_gauge_theories}{II. Surface operators in abelian gauge theories}\dotfill \pageref*{ii_surface_operators_in_abelian_gauge_theories} \linebreak
\noindent\hyperlink{TQFTsForAbelianGappedPhases}{III. TQFTs for abelian gapped phases}\dotfill \pageref*{TQFTsForAbelianGappedPhases} \linebreak
\noindent\hyperlink{a_abelian_higgs_phases}{A. Abelian Higgs phases}\dotfill \pageref*{a_abelian_higgs_phases} \linebreak
\noindent\hyperlink{b_abelian_confining_phases}{B. Abelian confining phases}\dotfill \pageref*{b_abelian_confining_phases} \linebreak
\noindent\hyperlink{c_oblique_confining_phases}{C. Oblique confining phases}\dotfill \pageref*{c_oblique_confining_phases} \linebreak
\noindent\hyperlink{d_mixed_phases}{D. Mixed phases}\dotfill \pageref*{d_mixed_phases} \linebreak
\noindent\hyperlink{iv_tqfts_for_nonabelian_gapped_phases}{IV. TQFTs for nonabelian gapped phases}\dotfill \pageref*{iv_tqfts_for_nonabelian_gapped_phases} \linebreak
\noindent\hyperlink{a_nonabelian_higgs_phases}{A. Nonabelian Higgs phases}\dotfill \pageref*{a_nonabelian_higgs_phases} \linebreak
\noindent\hyperlink{b_nonabelian_confining_phases}{B. Nonabelian confining phases}\dotfill \pageref*{b_nonabelian_confining_phases} \linebreak
\noindent\hyperlink{1_monopole_condensation}{1. Monopole condensation}\dotfill \pageref*{1_monopole_condensation} \linebreak
\noindent\hyperlink{2_nonabelian_gerbes_and_nonabelian_bfields}{2. Nonabelian gerbes and nonabelian B-fields}\dotfill \pageref*{2_nonabelian_gerbes_and_nonabelian_bfields} \linebreak
\noindent\hyperlink{3_a_tqft_for_a_nonabelian_confining_phase}{3. A TQFT for a nonabelian confining phase}\dotfill \pageref*{3_a_tqft_for_a_nonabelian_confining_phase} \linebreak
\noindent\hyperlink{c_a_tqft_for_a_nonabelian_oblique_confining_phase}{C. A TQFT for a nonabelian oblique confining phase}\dotfill \pageref*{c_a_tqft_for_a_nonabelian_oblique_confining_phase} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ii_surface_operators_in_abelian_gauge_theories}{}\subsection*{{II. Surface operators in abelian gauge theories}}\label{ii_surface_operators_in_abelian_gauge_theories}

(\ldots{})

\hypertarget{TQFTsForAbelianGappedPhases}{}\subsection*{{III. TQFTs for abelian gapped phases}}\label{TQFTsForAbelianGappedPhases}

\begin{quote}%
Here is an observation about how the idea in that section might be formalized. (Due to discussion among Domenico Fiorenza, Hisham Sati, and Urs Schreiber).


\end{quote}
The [[moduli stack]] $\mathbf{B}\mathbb{Z}_n$ of ``discrete [[gauge fields]]'' for [[gauge group]] $\mathbb{Z}_n := \mathbb{Z}/n\mathbb{Z}$ with $n \in \mathbb{N}$ may be realized as the following [[homotopy fiber]] of smooth moduli stacks of [[circle n-bundle with connection|circle connections]]

\begin{displaymath}
\itexarray{
    \mathbf{B}\mathbb{Z}_n &\to& \mathbf{B}U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}}
    \\
    \ast &\stackrel{0}{\to}& \mathbf{B}U(1)_{conn}
  }
\end{displaymath}
(in [[Smooth∞Grpd]]). By construction, an element $X \to \mathbf{B}\mathbb{Z}_n$ of this [[homotopy fiber]] is equivalently

\begin{itemize}%
\item a 1-form gauge field $\tilde F_A \colon X \to \mathbf{B}U(1)_{conn}$ with local connection 1-form $A$;


\item a 0-form gauge field $\phi \colon \tilde F_A \to 0$ trivializing $\tilde F_A$, with local component $\phi$ locally satisfying $d \phi = n A$.



\end{itemize}
This data is what motivates the discussion in the article.

The [[electric-magnetic duality]] may be formulate as a ``differential higher [[Pontryagin duality]]''

\begin{displaymath}
\mathbf{B}^2 \hat \mathbb{Z}_n \simeq [\mathbf{B}\mathbb{Z}_n, \mathbf{B}^3 U(1)_{conn}]
  \,,
\end{displaymath}
where $\hat \mathbb{Z}_n \coloneqq [\mathbb{Z}_n, U(1)] \simeq \mathbb{Z}_n$ is the [[Pontryagin dual]] group.

For this [[moduli infinity-stack|higher moduli stack]] of discrete [[higher gauge theory|higher gauge fields]] we have a local description by smooth data as before: it fits into the [[homotopy fiber sequence]]

\begin{displaymath}
\itexarray{
    \mathbf{B}^2\mathbb{Z}_n &\to& \mathbf{B}^2 U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{(-)^n}}
    \\
    \ast &\stackrel{0}{\to}& \mathbf{B}^2 U(1)_{conn}
  }
  \,.
\end{displaymath}
In analogy to the above, this now describes the space of field configurations consisting of a [[circle 2-group]]-[[connection on a 2-bundle|principal 2-connection]] $B$ and a trivialization of its $n$th power by a 1-form connection $A$, i.e. $d A - n B = 0$.

This one may also understand as the vanishing condition on the 2-form curvature of a [[connection on a 2-bundle|2-connection]]. This is the observation that drives the second part of the article.

\hypertarget{a_abelian_higgs_phases}{}\subsubsection*{{A. Abelian Higgs phases}}\label{a_abelian_higgs_phases}

\begin{itemize}%
\item [[Higgs mechanism]]

\end{itemize}
\hypertarget{b_abelian_confining_phases}{}\subsubsection*{{B. Abelian confining phases}}\label{b_abelian_confining_phases}

\begin{itemize}%
\item [[confinement]]

\end{itemize}
\hypertarget{c_oblique_confining_phases}{}\subsubsection*{{C. Oblique confining phases}}\label{c_oblique_confining_phases}

(\ldots{})

\hypertarget{d_mixed_phases}{}\subsubsection*{{D. Mixed phases}}\label{d_mixed_phases}

(\ldots{})

\hypertarget{iv_tqfts_for_nonabelian_gapped_phases}{}\subsection*{{IV. TQFTs for nonabelian gapped phases}}\label{iv_tqfts_for_nonabelian_gapped_phases}

(\ldots{})

\hypertarget{a_nonabelian_higgs_phases}{}\subsubsection*{{A. Nonabelian Higgs phases}}\label{a_nonabelian_higgs_phases}

(\ldots{})

\hypertarget{b_nonabelian_confining_phases}{}\subsubsection*{{B. Nonabelian confining phases}}\label{b_nonabelian_confining_phases}

(\ldots{})

\hypertarget{1_monopole_condensation}{}\paragraph*{{1. Monopole condensation}}\label{1_monopole_condensation}

A notce concerning the discussion of [[monopoles]] on the bottom right of p. 5.

Let $G$ be a [[connected topological space|connected]] [[Lie group]]. A [[subgroup]]

\begin{displaymath}
\Gamma_0 \hookrightarrow \pi_1(G)
\end{displaymath}
of its [[fundamental group]] defines a [[covering]] $t \colon H \to G$, exhibited by the following [[pasting diagram]] of [[homotopy pullbacks]] of [[smooth ∞-groupoids]]:

\begin{displaymath}
\itexarray{
    \pi_1(G)/\Gamma_0 &\to& H &\to& \Pi(H) &\to& \mathbf{B}\pi_1(H) \simeq \mathbf{B}\Gamma_0 
    \\
    \downarrow && \downarrow^{\mathrlap{t}} && \downarrow^{\mathrlap{\Pi(t)}} && \downarrow
    \\
    \ast &\stackrel{e}{\to}& G &\stackrel{}{\to}& \Pi(G) &\stackrel{\tau_1}{\to}& \mathbf{B} \pi_1(G)
  }
\end{displaymath}
Here $(-) \to \Pi(-)$ is the [[unit of an adjunction|unit]] of the [[shape modality]]. In the given low degree case the above diagram can be checked explicitly by standard means, but it may be worthwhile to note that the whole situation follows fully generally as by the \href{cohesive+%28infinity,1%29-topos+--+structures#GaloisTheory}{Galois theory in cohesive} (\hyperlink{SchreiberCohesive}{Schreiber, section 3.8.6}).

\hypertarget{2_nonabelian_gerbes_and_nonabelian_bfields}{}\paragraph*{{2. Nonabelian gerbes and nonabelian B-fields}}\label{2_nonabelian_gerbes_and_nonabelian_bfields}

\begin{itemize}%
\item [[connection on a 2-bundle]]


\item [[clutching construction]]



\end{itemize}
\hypertarget{3_a_tqft_for_a_nonabelian_confining_phase}{}\paragraph*{{3. A TQFT for a nonabelian confining phase}}\label{3_a_tqft_for_a_nonabelian_confining_phase}

\begin{itemize}%
\item [[BF-theory]]

\end{itemize}
\hypertarget{c_a_tqft_for_a_nonabelian_oblique_confining_phase}{}\subsubsection*{{C. A TQFT for a nonabelian oblique confining phase}}\label{c_a_tqft_for_a_nonabelian_oblique_confining_phase}

(\ldots{})

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

The argument relating aspects of the low energy limit of [[spontaneous symmetry breaking]] in ordinary [[gauge theory]] by [[electric-magnetic duality]] of the [[Higgs boson]]-like field to [[BF-theory|BF]]-type [[higher gauge theory]] is in section 2 of

\begin{itemize}%
\item [[Tom Banks]], [[Nathan Seiberg]], \emph{Symmetries and Strings in Field Theory and Gravity}, Phys. Rev. D83:084019,2011 (\href{http://arxiv.org/abs/1011.5120}{arXiv:1011.5120})

\end{itemize}
following discussion of [[D-brane]]/[[NS5-brane]] phenomena in

\begin{itemize}%
\item [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], \emph{D-brane Charges in Five-brane backgrounds}, JHEP 0110:005,2001 (\href{http://arxiv.org/abs/hep-th/0108152}{arXiv:hep-th/0108152})

\end{itemize}
(\ldots{})

The formulation of [[principal 2-connections]] via their [[higher parallel transport]] used in section IV is from

\begin{itemize}%
\item [[Urs Schreiber]], [[Konrad Waldorf]], \emph{Connections on non-abelian gerbes and their holonomy}, Theory and Applications of Categories, Vol. 28, 2013, No. 17, pp 476-540. (\href{http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html}{TAC}, \href{http://arxiv.org/abs/0808.1923}{arXiv:0808.1923}, )

\end{itemize}
expanding on

\begin{itemize}%
\item [[John Baez]], [[Urs Schreiber]], \emph{Higher gauge theory}, in A. Davydov et al. \emph{Categories in Algebra, Geometry and Mathematical Physics}, Contemp. Math. 431, AMS, Providence, Rhode Island, 2007, pp. 7-30 (\href{http://arxiv.org/abs/math/0511710}{arXiv:math/0511710}, \href{http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos+--+references/#BaezSchreiber}{web})

\end{itemize}
Some of the above comments on the formulation in [[smooth ∞-groupoids]] refer to

\begin{itemize}%
\item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
category: reference



\end{document}