Classical BV Antibracket and Yang–Mills Master-Action Data

Use the local field data and BRST convention on an open set $U\subseteq {\mathbb{R}}^n$ given by a finite-dimensional real Lie algebra $\mathfrak{g}$, a graded representation space $V$, an even $\mathfrak{g}$-valued gauge potential $A_{\mu }$, an odd ghost $c$ of ghost number $1$, and matter components ${\psi }^{\alpha }$ of parity ${ϵ}_{\alpha }$ and ghost number $0$. For every field or ghost coordinate ${\mathrm{\Phi }}^A$ of parity ${ϵ}_A$ and ghost number $g_A$, introduce an antifield ${\mathrm{\Phi }}_A^{\ast }$ of parity ${ϵ}_A+1(\mathrm{m}\mathrm{o}\mathrm{d}2)$ and ghost number $-1-g_A$. Thus $A^{\ast }$ and the antifields of even matter are odd of ghost number $-1$, while $c^{\ast }$ is even of ghost number $-2$. The minimal sector consists of $A,\psi ,c$ and these antifields.

For a finite-dimensional canonical coordinate algebra, take the polynomial algebra in the even coordinates tensor the exterior algebra in the odd coordinates. In the local-field notation, use finite-jet local functionals with variational derivatives only when the chosen domain makes those derivatives and integrations by parts well-defined; compact support or boundary conditions that kill surface terms are part of that domain. For homogeneous functionals $F,G$, define the canonical antibracket by

$$(F,G):=\sum _A(\frac{{\mathrm{\partial }}_{r}F}{\mathrm{\partial }{\mathrm{\Phi }}^A}\frac{{\mathrm{\partial }}_{l}G}{\mathrm{\partial }{\mathrm{\Phi }}_A^{\ast }}-\frac{{\mathrm{\partial }}_{r}F}{\mathrm{\partial }{\mathrm{\Phi }}_A^{\ast }}\frac{{\mathrm{\partial }}_{l}G}{\mathrm{\partial }{\mathrm{\Phi }}^A}),$$

with an integral over $U$ replacing the sum in a local-field expression. Left and right derivatives are the graded derivatives acting from the indicated side. Write $\mathrm{\mid }F\mathrm{\mid }\in {\mathbb{Z}}_2$ for the parity and $\mathrm{gh}(F)\in \mathbb{Z}$ for the ghost number of a homogeneous functional $F$. The bracket has parity and ghost number

$\mathrm{\mid }(F,G)\mathrm{\mid }=\mathrm{\mid }F\mathrm{\mid }+\mathrm{\mid }G\mathrm{\mid }+1(\mathrm{m}\mathrm{o}\mathrm{d}2),\mathrm{gh}((F,G))=\mathrm{gh}(F)+\mathrm{gh}(G)+1.$

For homogeneous $F,G,H$, its shifted graded-skew identity and Jacobi identity are

$(F,G)=-(-1{)}^{(\mathrm{\mid }F\mathrm{\mid }+1)(\mathrm{\mid }G\mathrm{\mid }+1)}(G,F),$

$((F,G),H)+(-1{)}^{(\mathrm{\mid }F\mathrm{\mid }+1)(\mathrm{\mid }G\mathrm{\mid }+\mathrm{\mid }H\mathrm{\mid })}((G,H),F)+(-1{)}^{(\mathrm{\mid }H\mathrm{\mid }+1)(\mathrm{\mid }F\mathrm{\mid }+\mathrm{\mid }G\mathrm{\mid })}((H,F),G)=0.$

The bracket is a graded biderivation, with the exact Leibniz identities

$(F,GH)=(F,G)H+(-1{)}^{\mathrm{\mid }G\mathrm{\mid }\mathrm{\mid }H\mathrm{\mid }}(F,H)G,(FG,H)=F(G,H)+(-1{)}^{\mathrm{\mid }F\mathrm{\mid }\mathrm{\mid }G\mathrm{\mid }}G(F,H)\mathrm{.}$

A regular finite-dimensional canonical domain has no nonzero central element of ghost number $1$: an element central against every coordinate has all coordinate derivatives zero and is a numerical constant. A formal local-functional domain used below must separately satisfy this same no-ghost-number-one-center condition.

Let a local gauge parameter be a smooth map $\lambda :U\to \mathfrak{g}$, and define its infinitesimal action on the fields by ${\delta }_{\lambda }{A}_{\mu }={\mathrm{\partial }}_{\mu }\lambda +[A_{\mu },\lambda ]$ and ${\delta }_{\lambda }\psi =-\rho (\lambda )\psi$. Write these variations as ${\delta }_{\lambda }{\mathrm{\Phi }}^i=R_{\alpha }^i[\mathrm{\Phi }]{\lambda }^{\alpha }$, where $R$ is the resulting linear differential operator in $\lambda$. Assume this declared gauge action is irreducible: if ${\delta }_{\lambda }{\mathrm{\Phi }}^i\equiv 0$ for every field configuration, then $\lambda \equiv 0$. Assume also that its Lie-algebra action closes off shell, meaning its commutator closes without using equations of motion, and that no additional gauge symmetry is included. Let $S_0[A,\psi ]$ be a specified local classical action invariant under these transformations. Write $s_0$ for the BRST operator on the minimal fields given by $s_0{A}_{\mu }={\mathrm{\partial }}_{\mu }c+[A_{\mu },c]$, $s_{0}c=-\frac{1}{2}[c,c]$, and $s_0\psi =-\rho (c)\psi$. The minimal BV functional candidate is

$S_{\mathrm{B}\mathrm{V}}:=S_0+{\int }_U{d}^{n}x{\sum }_A(-1{)}^{{ϵ}_A}{\mathrm{\Phi }}_A^{\ast }{s}_0{\mathrm{\Phi }}^A,$

where each product is in the displayed order and the same left/right derivative convention is used in the antibracket. It is a classical master action exactly when it satisfies the classical master equation $(S_{\mathrm{B}\mathrm{V}},S_{\mathrm{B}\mathrm{V}})=0$; the displayed formula alone does not assert that a particular $S_0$ satisfies that equation.

The optional nonminimal sector adds an odd antighost ${\bar{c}}^a$ of ghost number $-1$, an even auxiliary field $b^a$ of ghost number $0$, and their antifields, with the BV functional term $-{\bar{c}}_a^{\ast }{b}^a$ giving $s\bar{c}=b$ and $sb=0$. This definition makes no cohomological contractibility claim for that pair. The formal local-field notation does not define an infinite-dimensional measure, a BV Laplacian $\mathrm{\Delta }$, a path integral, or the quantum master equation.

Local field, ghost, representation, and sign conventions

Square-zero BRST operator on the minimal sector

Coordinate action of the minimal antifield coupling

Gomis, París, and Samuel, §§4.1–4.6 and 5.2

Qiu and Zabzine, §§3.1–3.2 and 4.1–4.4