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\begin{document}

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\section*{local BRST cohomology}

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

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak


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

Given a [[Lagrangian field theory]] $(E,\mathbf{L})$ with [[field bundle]] $E \overset{fb}{\to} \Sigma$ over some [[spacetime]] $\Sigma$ and [[local Lagrangian density]] $\mathbf{L}$, then its \emph{local BV-BRST complex} (or \emph{local BRST complex}, for short) is the realization of the [[BV-BRST complex]] not on [[local observables]] $A = \tau_{\Sigma} \alpha$ given by [[functionals]] on the [[space of field histories]] $\Gamma_{\Sigma}(E)_{\delta_{EL} = 0}$ which are [[transgression of variational differential forms|transgressions]] $\tau_{\Sigma}$ of [[variational differential forms]] $\alpha \in \Omega^{\bullet, \bullet}_\Sigma(E)$ on the jet bundle, but on these variational differential forms themselves (whence ``local'', i.e. before [[transgression of variational differential forms|transgression]]).

If $s$ denotes the [[BV-BRST differential]] in a BV-resolution $\Omega^{\bullet,\bullet}_\Sigma(E)\vert_{\mathcal{E}_{BV}}$ of the restriction to the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ of the [[variational bicomplex]] $\Omega^{\bullet,\bullet}_\Sigma(E)$ with its [[total spacetime derivative]] $d$ (horizontal derivative), then the \emph{local BV-BRST cohomology} is the [[cochain cohomology]] of $s + d$, hence of the [[total complex]] of the [[double complex]] given by $s$ and $d$.

Generally, considering [[variational differential forms]] up to $d$-exact terms is the ``local'' incarnation of what under the [[integration of differential forms|integration]] involved in the [[transgression of variational differential forms|transgression]] is [[integration by parts]] and it is in this way that ``local BV-BRST cohomology'' knows about the actual BV-BRST cohomology on [[local observables]].

\hypertarget{example}{}\subsection*{{Example}}\label{example}

Consider local coordinates $(\phi^a)$ on the [[fibers]] of the [[field bundle]]. The corresponding [[antifield]] coordinates are to be denoted $\overline{\phi}_a$ and the [[BV-BRST differential]] takes them to the corresponding component

\begin{displaymath}
s(\overline{\phi}_a) = \frac{\delta_{El} L}{\delta \phi^a}
\end{displaymath}
of the [[Euler-Lagrange form]].

In degree $(p+1,0)$ the $s+d$-closed elements in vanishing [[ghost]] degree are [[pairs]] $(v,J_v)$ consisting of an [[infinitesimal symmetry of the Lagrangian]] $v$, regarded as an [[antifield]] [[density]] $v^a \overline{\phi}_a dvol_\Sigma$, together with a corresponding [[conserved current|conserved]] [[Noether's theorem|Noether current]] $J_v$:

\begin{displaymath}
\itexarray{
    \{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \}
    \\
    && \uparrow\mathrlap{-s}
    \\
    && \{ v^a \overline{\phi}_a dvol_\Sigma\}
  }
\end{displaymath}
Such pairs are $(s+d)$-exact if [[on-shell]] the infintiesimal symmetry coincides with an [[infinitesimal gauge symmetry]]. To see this, recall:

An [[infinitesimal gauge symmetry]] $v_\epsilon$ of [[gauge parameter]] $(\epsilon^\alpha)$ is a vector field on the jet bundle with components of the form

\begin{displaymath}
\mathcal{L}_{v_\epsilon} \phi^a
   \;\coloneqq\;
  R^a_\alpha \epsilon^\alpha
    +
  R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
\end{displaymath}
such that this is an [[infinitesimal symmetry of the Lagrangian]] in that

\begin{displaymath}
\begin{aligned}
    \iota_{v_\epsilon} \delta_{EL} \mathbf{L}
    & =
    v^a \frac{\delta_{EL} L}{\delta \phi^a} dvol_\Sigma
    \\
    & =
    \epsilon^\alpha
    \left(
       R^a_\alpha \frac{\delta_{EL} L}{ \delta \phi^a}
       -
       \frac{d}{d x^\mu} 
       \left(
          R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a}
       \right)
    \right)
    dvol_\Sigma
    +
    d\left(
       \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a}
    \right)
    \iota_{\partial_\mu} dvol_\Sigma
    \\
    & =
    0 + d(...)
  \end{aligned}
\end{displaymath}
for all $(\epsilon^\alpha)$.

The corresponding [[antighosts]] $\overline{c}_\alpha$ are taken by the BV-BRST differential to the antifield-preimage of the term on the left:

\begin{displaymath}
s\left(\overline{c}_\alpha\right)
  \;=\;
  R^a_\alpha \overline{\phi}_a
  -
  \frac{d}{d x^\mu} 
  \left(
    R^{a \mu}_\alpha \overline{\phi}_a
  \right)  
  \,.
\end{displaymath}
Moreover, an [[on-shell]] vanishing [[infinitesimal symmetry of the Lagrangian]] is a vector field with components of the form

\begin{displaymath}
\kappa^{a b} \frac{\delta_{EL} L}{\delta \phi^a}
\end{displaymath}
for $\kappa^{a b} = - \kappa^{b a}$ a skew-symmetric system of smooth functions on the jet bundle.

The linear combination of such an infinitesimal gauge symmetry and an on-shell vanishing infinitesimal symmetry is $(s+d)$-exact:

\begin{displaymath}
\begin{aligned}
    v^a dvol_\Sigma
    & =
    \left(
      R^a_\alpha \epsilon^\alpha
      +
      R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
      +
      \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a }
    \right)
    dvol_\Sigma
    \\
    & =
    s
    \left( 
      \epsilon^\alpha \overline{c}_\alpha 
      - 
      \tfrac{1}{2}\kappa^{a b} \overline{\phi}_a \overline{\phi}_b  
    \right) dvol_\sigma
    +
    d\left(
      \epsilon^\alpha R^{a \mu}_\alpha 
    \right)
    \iota_{\partial_\mu} dvol_\Sigma
  \end{aligned}
\end{displaymath}
(\hyperlink{BarnichBrandtHenneaux94}{Barnich-Brandt-Henneaux 94, p. 20})

It may be useful to organize this expression into the $s+d$-[[bicomplex]] like so:

\begin{displaymath}
\itexarray{
    \{K\}
     &\overset{d}{\longrightarrow}&
     \{ d K 
       + 
     \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{ \delta \phi^a}
     \} 
     &\overset{d}{\longrightarrow}& 
    \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \}
    \\
    && \mathllap{s}\uparrow && \uparrow\mathrlap{-s}
    \\
    && 
    \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a
    \iota_{\partial_\mu} dvol_\Sigma
      &\underset{d}{\longrightarrow}&
    \left\{  
      d\left(
        \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a
      \right)
      \iota_{\partial_\mu} dvol_\Sigma
      +
      \left(
        R^a_\alpha \epsilon^\alpha
        +
        R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu}
        +
        \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a }
      \right)
      \overline{\phi}_a
      \,
      dvol_\Sigma
    \right\}
    \\
    && && \uparrow\mathrlap{-s}
    \\
    &&
    &&
    \left(
       - \epsilon^\alpha \overline{c}_\alpha
       + 
      \tfrac{1}{2}\kappa^{a b } \overline{\phi}_a \overline{\phi}_b 
    \right)
    dvol_\Sigma
  }
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[variational BV-BRST bicomplex]]

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

Review includes

\begin{itemize}%
\item [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], \emph{Local BRST cohomology in gauge theories}, Phys. Rept. 338:439-569, 2000 (\href{https://arxiv.org/abs/hep-th/0002245}{arXiv:hep-th/0002245})

\end{itemize}
The general theory is discussed in

\begin{itemize}%
\item [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], \emph{Local BRST cohomology in the antifield formalism: I. General theorems}, Commun.Math.Phys.174:57-92,1995 (\href{https://arxiv.org/abs/hep-th/9405109}{arXiv:hep-th/9405109})

\end{itemize}
Details of the [[local antibracket]] are discussed in

\begin{itemize}%
\item [[Glenn Barnich]], [[Marc Henneaux]], section 2 and appendix B of \emph{Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket}, J. Math. Phys. 37 (1996) 5273-5296 (\href{https://arxiv.org/abs/hep-th/9601124}{arXiv:hep-th/9601124})

\end{itemize}
Application to [[gravity]] and/or [[Yang-Mills theory]] ([[Einstein-Yang-Mills theory]]) is discussed in

\begin{itemize}%
\item [[Glenn Barnich]], [[Friedemann Brandt]], [[Marc Henneaux]], \emph{Local BRST cohomology in Einstein--Yang--Mills theory}, Nucl.Phys. B455 (1995) 357-408 (\href{https://arxiv.org/abs/hep-th/9505173}{arXiv:hep-th/9505173})

\end{itemize}
[[!redirects local BV-BRST cohomology]]

[[!redirects local BRST complex]] [[!redirects local BRST complexes]]

[[!redirects local BV-BRST complex]] [[!redirects local BV-BRST complexes]]

[[!redirects local BV complex]] [[!redirects local BV complexes]]

[[!redirects local BV-complex]] [[!redirects local BV-complexes]]

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

[[!redirects local BV-BRST differential]] [[!redirects local BV-BRST differentials]]

[[!redirects local BV-cohomology]] [[!redirects local BV cohomology]]



\end{document}