\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*{bosonic string}

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

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

[[!include string 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_symplectic_geometry__geometric_quantization}{In symplectic geometry / geometric quantization}\dotfill \pageref*{in_symplectic_geometry__geometric_quantization} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{vacuum}{Vacuum}\dotfill \pageref*{vacuum} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesSymplecticGeometryAndGeometricQuantization}{Symplectic geometry and geometric quantization}\dotfill \pageref*{ReferencesSymplecticGeometryAndGeometricQuantization} \linebreak
\noindent\hyperlink{ReferencesMultisymplecticGeometry}{In multisymplectic geometry and higher geometric quantization}\dotfill \pageref*{ReferencesMultisymplecticGeometry} \linebreak
\noindent\hyperlink{vacuum_2}{Vacuum}\dotfill \pageref*{vacuum_2} \linebreak


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

The incarnation of the concept of [[string]] without [[worldsheet]] [[supersymmetry]] is the \emph{bosonic string} of [[bosonic string theory]]. In contrast to the [[spinning string]]/[[superstring]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_symplectic_geometry__geometric_quantization}{}\subsubsection*{{In symplectic geometry / geometric quantization}}\label{in_symplectic_geometry__geometric_quantization}

See the reference \hyperlink{ReferencesSymplecticGeometryAndGeometricQuantization}{below}

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{vacuum}{}\subsubsection*{{Vacuum}}\label{vacuum}

\begin{itemize}%
\item [[tachyon condensation]]


\item [[Sen's conjecture]]



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

\begin{itemize}%
\item [[sigma model]], [[brane]]

\begin{itemize}%
\item [[particle]], [[spinning particle]], [[superparticle]]


\item [[string]], [[spinning string]], [[superstring]]

\begin{itemize}%
\item [[type II superstring]], [[F1-brane]], [[D1-brane]], [[(p,q)-string]]


\item [[heterotic string]]



\end{itemize}

\item [[membrane]]



\end{itemize}

\item [[string theory]], [[string field theory]]


\item [[Goddard-Thorn no-ghost theorem]]


\item [[string scattering amplitude]]


\item [[string field theory]]


\item [[world sheets for world sheets]]


\item [[D-brane]], [[Chan-Paton gauge field]], [[Freed-Witten-Kapustin anomaly cancellation]]


\item [[string theory FAQ]]



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

See the references at \emph{[[string theory]]} for more.

\hypertarget{ReferencesSymplecticGeometryAndGeometricQuantization}{}\subsubsection*{{Symplectic geometry and geometric quantization}}\label{ReferencesSymplecticGeometryAndGeometricQuantization}

The ordinary [[symplectic geometry]] and ordinary [[geometric quantization]] of the bosonic string [[sigma-model]] is discussed in the following references.

The [[symplectic structure]] and [[Kähler geometry]] of [[loop space]] is discussed in

\begin{itemize}%
\item M.J. Bowick, S.G. Rajeev, \emph{String theory as the K\"a{}hler geometry of loop space} Phys. Rev. Lett. 58, 535-538 (1987)


\item M.J. Bowick, S.G. Rajeev, \emph{The holomorphic geometry of closed bosonic string theory and $Diff(S^1)/S^1$}, Nucl. Phys. B293, 348-384 (1987)


\item [[Jouko Mickelsson]], \emph{String quantization on group manifolds and the holomorphic geometry of $Diff(S^1)/S^1$} Commun. Math. Phys. 112, 653-661 (1987) (\href{http://projecteuclid.org/euclid.cmp/1104160059}{EUCLID})



\end{itemize}
with further comments in

\begin{itemize}%
\item Hendrik Grundlin, C. A. Hurst, \emph{The operator quantization of the open bosonic string: field algebra}, Communications in mathematical physics 156 (1993) (\href{http://web.maths.unsw.edu.au/~hendrik/String-CMP.pdf}{pdf})

\end{itemize}
A correction of some points in these articles is discussed in

\begin{itemize}%
\item [[Sergei Merkulov]], \emph{On the geometric quantization of bosonic string}, Class. Quantum Grav. 9 2267 (1992) (\href{http://iopscience.iop.org/0264-9381/9/10/010}{IOP})

\end{itemize}
See also

\begin{itemize}%
\item Yue Yua, Han-Ying Guoa, \emph{On the geometric quantization and BRST quantization for bosonic strings}, Physics Letters B Volume 216, Issues 1--2, (1989), Pages 68--74 (\href{http://inspirehep.net/record/279613}{web})


\item Yu-liang Liu, Su-qing Chen,Guang-jiong Ni, \emph{Geometrical quantization of bosonic string with Wess-Zumino term on genus-g Riemann surface}, Phys. Rev. D 41, 472--477 (1990)


\item A. D. Popov, \emph{Geometric quantization of strings and reparametrization invariance}, Theoretical and Mathematical Physics, Volume 83, Number 3 (1990) (\href{http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tmf&paperid=5817&option_lang=eng}{journal})



\end{itemize}
\hypertarget{ReferencesMultisymplecticGeometry}{}\subsubsection*{{In multisymplectic geometry and higher geometric quantization}}\label{ReferencesMultisymplecticGeometry}

A discussion starting systematically with the correct [[symplectic form]] obtained by transgression from an [[multisymplectic geometry|multisymplectic]] [[extended phase space]] and including the [[BRST-complex|BRST sector]] is in

\begin{itemize}%
\item Yue Yu, \emph{Symplectic geometry and geometric quantization for the open bosonic string in the BRST formalism}, Physics Letters B, Volume 216, Issue 1-2, (1989) p. 75-80.

\end{itemize}
A detailed exposition of the [[multisymplectic geometry]] of the bosonic string together with its interpretation in [[2-plectic geometry]] is in

\begin{itemize}%
\item [[John Baez]], [[Alexander Hoffnung]], [[Chris Rogers]], \emph{Categorified Symplectic Geometry and the Classical String} (\href{http://arxiv.org/abs/0808.0246/}{arXiv:0808.0246})

\end{itemize}
and the appearance of the [[string Lie 2-algebra]] as the [[Heisenberg Lie 2-algebra]] of the string [[WZW-model]] in this context is discussed in

\begin{itemize}%
\item [[John Baez]], [[Chris Rogers]], \emph{Categorified Symplectic Geometry and the String Lie 2-Algebra}. \href{http://arxiv.org/abs/0901.4721}{arXiv:0901.4721}.

\end{itemize}
\hypertarget{vacuum_2}{}\subsubsection*{{Vacuum}}\label{vacuum_2}

(\ldots{})

\begin{itemize}%
\item [[Jan Ambjørn]], Y. Makeenko, \emph{Stability of the nonperturbative bosonic string vacuum} (\href{https://arxiv.org/abs/1703.05382}{arXiv:1703.05382})

\end{itemize}
[[!redirects bosonic strings]]



\end{document}