\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*{instanton}

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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{in_quantum_field_theory}{In quantum field theory}\dotfill \pageref*{in_quantum_field_theory} \linebreak
\noindent\hyperlink{in_string_theory}{In string theory}\dotfill \pageref*{in_string_theory} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{in_quantum_field_theory_and_specifically_yangmills_theory}{In quantum field theory and specifically Yang-Mills theory}\dotfill \pageref*{in_quantum_field_theory_and_specifically_yangmills_theory} \linebreak
\noindent\hyperlink{relation_to_skyrmions_calorons_monopoles}{Relation to Skyrmions, calorons, monopoles}\dotfill \pageref*{relation_to_skyrmions_calorons_monopoles} \linebreak
\noindent\hyperlink{in_string_theory_2}{In string theory}\dotfill \pageref*{in_string_theory_2} \linebreak


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

\hypertarget{in_quantum_field_theory}{}\subsubsection*{{In quantum field theory}}\label{in_quantum_field_theory}

In [[quantum field theory]] an \emph{instanton} is a [[configuration space|field configuration]] with a ``topological twist'': not in the connected component of the trivial field configurations. Specifically for [[gauge fields]] which mathematically are represented by [[principal connections]], an instanton is a nontrivial underlying [[principal bundle]] (or similarly non-trivial [[associated bundle|associated]] [[vector bundle]]).

The term derives from the special case of instantons on a [[sphere]] but modeled as field configurations on a [[Euclidean space]] constrained to vanish asymptotically. These look like solutions localized in [[spacetime]]: ``at an instant''.

Instantons enter the [[axial anomaly]]/[[chiral anomaly]] in the [[standard model of particle physics]] which is thought to be a source of [[baryogenesis]] in the early universe. Generally the [[QCD]] [[vacuum]] state is argued to consist of a [[superposition]] of all possible instanton sectors, see at \emph{[[QCD instanton]]}.

\hypertarget{in_string_theory}{}\subsubsection*{{In string theory}}\label{in_string_theory}

More generally, in [[string theory]] a [[brane]] which wraps a completely [[spacelike]] cycle in [[target space]] is called an \emph{instanton}, since the [[worldvolume]] of such a brane is localized in the [[time]]-direction of target space. Under passing to the [[effective quantum field theory]] of the string theory, this reproduces many instantons in the sense of quantum field theory above.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[Yang-Mills instanton]]

\begin{itemize}%
\item [[instanton in QCD]]

\end{itemize}

\item [[Coleman-De Luccia instanton]]


\item [[S-brane]]


\item [[wrapped brane]]

\begin{itemize}%
\item [[membrane instanton]]


\item [[D3-brane instanton]]


\item [[M5-brane instanton]]



\end{itemize}


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

\begin{itemize}%
\item [[Yang-Mills instanton]]


\item [[contact instanton]]


\item [[instanton sector]]


\item [[non-perturbative effect]], [[non-perturbative quantum field theory]], [[non-perturbative string theory]]


\item [[quantum tunneling]]


\item at [[thermal field theory|finite temperature]]: [[caloron]]

\begin{itemize}%
\item [[caloron correspondence]]

\end{itemize}

\item [[renormalon]]


\item [[soliton]], [[vortex]]


\item [[brane]]


\item [[Spin(7)-instanton]]


\item [[worldsheet instanton]], [[membrane instanton]]


\item \href{string+theory+FAQ#NonConvergenceOfPerturbationSeries}{string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?}



\end{itemize}
[[!include gauge field - table]]

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

\hypertarget{in_quantum_field_theory_and_specifically_yangmills_theory}{}\subsubsection*{{In quantum field theory and specifically Yang-Mills theory}}\label{in_quantum_field_theory_and_specifically_yangmills_theory}

\begin{itemize}%
\item [[Dan Freed]], [[Karen Uhlenbeck]], \emph{Instantons and four-manifolds}, Springer-Verlag, (1991)


\item Nicholas Manton, Paul M. Sutcliffe, \emph{Topological solitons}, Cambridge Monographs on Math. Physics, \href{http://books.google.com/books?id=e2tPhFdSUf8C}{gBooks}


\item [[Werner Nahm]], \emph{Self-dual monopoles and calorons}, in \emph{Group theoretical methods in physics} (Trieste, 1983), pages 189-200. Springer, Berlin (1984) (\href{http://www.springerlink.com/index/y005405w74862327.pdf}{journal})


\item [[David Tong]], \emph{TASI Lectures on Solitons} (\href{https://arxiv.org/abs/hep-th/0509216}{arXiv:hep-th/0509216}), \emph{Lecture 1: Instantons} (\href{http://www.damtp.cam.ac.uk/user/tong/tasi/instanton.pdf}{pdf})



\end{itemize}
See also literature at [[Yang-Mills instanton]].

Yang-Mills instantons on spaces other than just spheres are explicitly discussed in

\begin{itemize}%
\item [[Gabor Kunstatter]], \emph{Yang-mills theory in a multiply connected three space}, Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West), August 11-20,1981. Editor: R. Schrader, R. Seiler, D. A. Uhlenbrock, Lecture Notes in Physics, vol. 153, p.118-122 (\href{http://adsabs.harvard.edu/abs/1982LNP...153..118K}{web})

\end{itemize}
based on

\begin{itemize}%
\item [[Chris Isham]] [[Gabor Kunstatter]], Phys. Letts. v.102B, p.417, 1981. (\href{http://dx.doi.org/10.1016/0370-2693%2881%2991244-2}{doi})


\item [[Chris Isham]] [[Gabor Kunstatter]], J. Math. Phys. v.23, p.1668, 1982. (\href{http://dx.doi.org/10.1063/1.525552}{doi})



\end{itemize}
A generalization is discussed in

\begin{itemize}%
\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Instantons beyond topological theory I} (\href{http://arxiv.org/abs/hep-th/0610149}{arXiv:hep-th/0610149})


\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Instantons beyond topological theory II} (\href{http://arxiv.org/abs/0803.3302}{arXiv:hep-th/0610149})



\end{itemize}
Expositions and summaries of this are in

\begin{itemize}%
\item [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], \emph{Notes on instantons in topological field theory and beyond} (\href{http://arxiv.org/abs/hep-th/0702137}{arXiv:hep-th/0702137})


\item [[Jacques Distler]], \emph{Localized} (2006) (\href{https://golem.ph.utexas.edu/~distler/blog/archives/001030.html}{blog post})



\end{itemize}
\hypertarget{relation_to_skyrmions_calorons_monopoles}{}\subsubsection*{{Relation to Skyrmions, calorons, monopoles}}\label{relation_to_skyrmions_calorons_monopoles}

The construction of [[Skyrmions]] from [[instantons]] is due to

\begin{itemize}%
\item [[Michael Atiyah]], N S Manton, \emph{Skyrmions from instantons}, Phys. Lett. B, 222(3):438–442, 1989 ()

\end{itemize}
The relation between [[skyrmions]], [[instantons]], [[calorons]], [[solitons]] and [[monopoles]] is usefully reviewed and further developed in

\begin{itemize}%
\item [[Josh Cork]], \emph{Calorons, symmetry, and the soliton trinity}, PhD thesis, University of Leeds 2018 (\href{http://etheses.whiterose.ac.uk/22097/}{web})


\item [[Josh Cork]], \emph{Skyrmions from calorons}, J. High Energ. Phys. (2018) 2018: 137 (\href{https://arxiv.org/abs/1810.04143}{arXiv:1810.04143})



\end{itemize}
\hypertarget{in_string_theory_2}{}\subsubsection*{{In string theory}}\label{in_string_theory_2}

In [[string theory]] (for [[D-branes]]).

\begin{itemize}%
\item [[Edward Witten]], \emph{World-Sheet Corrections Via D-Instantons}, JHEP 0002:030, 2000 (\href{http://arxiv.org/abs/hep-th/9907041}{arXiv:hep-th/9907041})


\item Albion Lawrence, [[Nikita Nekrasov]], \emph{Instanton sums and five-dimensional gauge theories}, Nucl.Phys. B513 (1998) 239-265 (\href{http://arxiv.org/abs/hep-th/9706025}{arXiv:hep-th/9706025})



\end{itemize}
The study of M-brane instantons originates around

\begin{itemize}%
\item [[Katrin Becker]], [[Melanie Becker]], [[Andrew Strominger]], \emph{Fivebranes, Membranes and Non-Perturbative String Theory}, Nucl.Phys.B456:130-152,1995 (\href{http://arxiv.org/abs/hep-th/9507158}{arXiv:hep-th/9507158})

\end{itemize}
Specifically [[membrane instantons]] are further discussed in

\begin{itemize}%
\item [[Jeffrey Harvey]], [[Gregory Moore]], \emph{Superpotentials and Membrane Instantons} (\href{http://arxiv.org/abs/hep-th/9907026}{arXiv:hep-th/9907026})

\end{itemize}
and 5-brane instantons in

\begin{itemize}%
\item [[Edward Witten]], \emph{Non-Perturbative Superpotentials In String Theory}, Nucl.Phys.B474:343-360,1996 (\href{http://arxiv.org/abs/hep-th/9604030}{arXiv:hep-th/9604030})

\end{itemize}
In the context of [[F-theory]] and [[M5-brane]] instantons:

\begin{itemize}%
\item Max Kerstan, [[Timo Weigand]], \emph{Fluxed M5-instantons in F-theory} (\href{http://arxiv.org/abs/1205.4720}{arXiv:1205.4720})

\end{itemize}
[[!redirects instanton]] [[!redirects instantons]]



\end{document}