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\section*{BRST complex}

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

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \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{history}{History}\dotfill \pageref*{history} \linebreak


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

What is called the ``BRST complex'' in the [[physics]] literature is the [[Lie infinity-algebroid|qDGCA]] which is the [[Chevalley-Eilenberg algebra]] of the $L_\infty$-[[Lie infinity-algebroid|algebroid]] which is the differential version in [[Lie theory]] of the $\infty$-groupoid

\begin{itemize}%
\item whose space of objects is the space of configurations/histories of a given physical system;


\item whose morphisms describe the gauge transformations between these configurations/histories;


\item whose $k$-morphisms describe the $k$-fold gauge-of-gauge transformations.



\end{itemize}
The generators of the BRST complex are called

\begin{itemize}%
\item in degree 0: \textbf{fields};


\item in degree 1: \textbf{[[ghost field]]};


\item in degree 2: \textbf{[[ghost-of-ghost fields]]};


\item etc.



\end{itemize}
The [[cochain cohomology]] of the BRST complex is called, of course, \emph{BRST cohomology}.

\hypertarget{details}{}\subsection*{{Details}}\label{details}

For details see at

\begin{itemize}%
\item \emph{[[A first idea of quantum field theory]]} the chapter \emph{\href{A+first+idea+of+quantum+field+theory#Gauge+symmetries}{Gauge symmetries}}.

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

The BRST complex described a homotopical [[quotient]] of a space by an infinitesimal [[action]]. Combined with a homotopical intersection, it is part of the [[BRST-BV complex]].

\begin{itemize}%
\item [[gauge transformation]], [[higher gauge transformation]]


\item [[ghost field]], [[ghost-of-ghost field]]


\item [[antifield]], [[antighost field]]


\item [[local BRST complex]]


\item [[variational BRST-bicomplex]]


\item [[equivariant de Rham cohomology]]



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

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

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

The idea of ``ghost'' fields was introduced in

\begin{itemize}%
\item [[Richard Feynman]], \emph{Quantum theory of gravitation} In: Acta physica polonica. vol 24, 1963, S. 697

\end{itemize}
and expanded on in

\begin{itemize}%
\item [[Richard Feynman]] in [[John Wheeler]], Klauder (eds.), \emph{Magic without Magic}, Wheeler-Festschrift (1972)

\end{itemize}
The BRST formalism originates around

\begin{itemize}%
\item [[Carlo Becchi]], A. Rouet, [[Raymond Stora]], (1976). Renormalization of gauge theories. Ann. Phys. 98: 287,

\end{itemize}
see also the references at \emph{[[BRST]]}.

A classical standard references for the Lagrangian formalism is

\begin{itemize}%
\item [[Marc Henneaux]], \emph{Lectures on the Antifield-BRST formalism for gauge theories}, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (\href{http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf}{pdf})

\end{itemize}
Similarly the bulk of the textbook

\begin{itemize}%
\item [[Marc Henneaux]], [[Claudio Teitelboim]], \emph{[[Quantization of Gauge Systems]]}

\end{itemize}
considers the Hamiltonian formulation. Chapters 17 and 18 are about the Lagrangian (``antifield'') formulation, with section 18.4 devoted to the relation between the two.

The [[L-infinity algebroid]]-structure of the [[local BRST complex]] on the [[jet bundle]] is made manifest in

\begin{itemize}%
\item [[Glenn Barnich]], \emph{A note on gauge systems from the point of view of Lie algebroids}, in P. Kielanowski, V. Buchstaber, A. Odzijewicz, M.. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (\href{https://arxiv.org/abs/1010.0899}{arXiv:1010.0899}, \href{https://doi.org/10.1063/1.3527427}{doi:/10.1063/1.3527427})

\end{itemize}
Discussion with more emphasis on the applications to quantum field theory of interest is in lecture 3 of

\begin{itemize}%
\item [[Edward Witten]], \emph{Dynamics of Quantum Field Theory} in vol II, starting page 1119, of [[Pierre Deligne]], [[Pavel Etingof]], [[Dan Freed]], L. Jeffrey, [[David Kazhdan]], [[John Morgan]], D.R. Morrison and [[Edward Witten]], eds. \emph{[[Quantum Fields and Strings]], A course for mathematicians}, 2 vols. Amer. Math. Soc. Providence 1999.

\end{itemize}
The [[perturbative quantum field theory|perturbative quantization]] of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] is discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) via [[BRST-complex]]/[[BV-formalism]] in

\begin{itemize}%
\item [[Stefan Hollands]], \emph{Renormalized Quantum Yang-Mills Fields in Curved Spacetime}, Rev. Math. Phys.20:1033-1172, 2008 (\href{https://arxiv.org/abs/0705.3340}{arXiv:0705.3340})


\item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in the functional approach to classical field theory}, Commun. Math. Phys. 314(1), 93--127 (2012) (\href{https://arxiv.org/abs/1101.5112}{arXiv:1101.5112})


\item [[Klaus Fredenhagen]], [[Kasia Rejzner]], \emph{Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory}, Commun. Math. Phys. 317(3), 697--725 (2012) (\href{https://arxiv.org/abs/1110.5232}{arXiv:1110.5232})


\item [[Katarzyna Rejzner]], \emph{Remarks on local symmetry invariance in perturbative algebraic quantum field theory} (\href{https://arxiv.org/abs/1301.7037}{arXiv:1301.7037})


\item [[Katarzyna Rejzner]], \emph{Remarks on local symmetry invariance in perturbative algebraic quantum field theory} (\href{https://arxiv.org/abs/1301.7037}{arXiv:1301.7037})


\item Mojtaba Taslimi Tehrani, \emph{Quantum BRST charge in gauge theories in curved space-time} (\href{https://arxiv.org/abs/1703.04148}{arXiv:1703.04148})



\end{itemize}
and surveyed in

\begin{itemize}%
\item [[Kasia Rejzner]], section 7 of \emph{Perturbative algebraic quantum field theory} Springer 2016

\end{itemize}
Discussion of the BRST complex of the [[bosonic string]]/for [[2d CFT]] includes

\begin{itemize}%
\item [[Graeme Segal]], p.114 and following of \emph{The definition of conformal field theory} , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) \emph{Topology, geometry and quantum field theory} , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (\href{https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf}{pdf})

\end{itemize}
Discussion of the BRST complex for the [[superstring]] (hence with the corresponding [[Lie algebroid]] being actually a [[super Lie algebroid]]) is for instance in

\begin{itemize}%
\item [[José Figueroa-O'Farrill]], Takashi Kimura, \emph{The BRST cohomology of the NSR string: vanishing and ``-ghost'` theorems}, Comm. Math. Phys. Volume 124, Number 1 (1989), 105-132. (\href{http://projecteuclid.org/euclid.cmp/1104179078}{Euclid})


\item [[Alexander Belopolsky]], \emph{De Rham Cohomology of the Supermanifolds and Superstring BRST Cohomology}, Phys.Lett. B403 (1997) 47-50 (\href{http://arxiv.org/abs/hep-th/9609220}{arXiv:hep-th/9609220})



\end{itemize}
The perspective on the BRST complex as a formal dual to a space in [[dg-geometry]] is relatively clearly stated in section 2 of

\begin{itemize}%
\item [[Kevin Costello]], \emph{Renormalisation and the Batalin-Vilkovisky formalism} (\href{http://arxiv.org/abs/0706.1533}{arXiv}).

\end{itemize}
For more along these lines see [[BV-BRST formalism]].

\hypertarget{history}{}\subsubsection*{{History}}\label{history}

\begin{itemize}%
\item [[Carlo Becchi]], \emph{BRS ``Symmetry'', prehistory and history} (\href{https://arxiv.org/abs/1107.1070}{arXiv:1107.1070})

\end{itemize}
[[!redirects BRST complexes]]

[[!redirects BRST differential]] [[!redirects BRST differentials]]

[[!redirects BRST cohomology]]

[[!redirects BRST operator]] [[!redirects BRST operators]]

[[!redirects BRST Lagrangian]]



\end{document}