BV, BRST, and Homological Physics Interfaces

Let $k$ be a field of characteristic $0$. A unital $\mathbb{Z}$-graded associative $k$-algebra $A={⨁}_{r\in \mathbb{Z}}{A}^r$ is graded commutative when its unit $1$ has degree $0$ and homogeneous $a,b$ satisfy $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$. A linear map $\mathrm{\Delta }:A^r\to A^{r+1}$ is a BV generator of degree $+1$ when $\mathrm{\Delta }(1)=0$, ${\mathrm{\Delta }}^2=0$, and it has order at most two in the following graded-commutator convention. For homogeneous operators $P,Q$, set $[P,Q]=PQ-(-1{)}^{\mathrm{\mid }P\mathrm{\mid }\mathrm{\mid }Q\mathrm{\mid }}QP,$ and for homogeneous $a\in A$ let $m_a(x)=ax$. The order condition is $[[[\mathrm{\Delta },m_a],m_b],m_c]=0$ for all homogeneous $a,b,c\in A$. The pair $(A,\mathrm{\Delta })$ with these data is a BV algebra in the degree-$+1$ convention. Its derived bracket is the bilinear operation of degree $+1$ $\{a,b\}=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}(\mathrm{\Delta }(ab)-(\mathrm{\Delta }a)b-(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a\mathrm{\Delta }b)$ on homogeneous elements, extended by linearity.

This item fixes the grading, commutator order condition, and bracket formula. It does not include a separate differential $Q$ or assert compatibility with one. The associated odd-skew, biderivation, and Jacobi identities are proved in The derived BV bracket is an odd Poisson bracket.

Getzler writes the second-order condition as a seven-term identity and also includes a separate differential in his larger structure. Seidel uses a degree-$-1$ operator, and Kosmann-Schwarzbach–Monterde formulate the general odd bracket first in a $\mathbb{Z}\mathrm{/}2$-graded setting. This item fixes its own degree-$+1$ commutator and bracket conventions; no result in a different grading is imported without a conversion.

Let $k$ be a characteristic-zero field, let $\mathfrak{g}$ be a Lie algebra over $k$, and let $A$ be a unital commutative $k$-algebra. A classical Lie-action BRST input is a Lie algebra homomorphism $\rho :\mathfrak{g}\to {\mathrm{End}}_k(A)$ whose values are derivations: for $x\in \mathfrak{g}$ and $a,b\in A$, $\rho (x)(ab)=\rho (x)(a)b+a\rho (x)(b),\rho (x)(1)=0.$ In particular, $A$ is a $\mathfrak{g}$-module, so its Chevalley–Eilenberg cochains are $C_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^n(\mathfrak{g},A)={\mathrm{Hom}}_k({\mathrm{\Lambda }}^n\mathfrak{g},A)(n\ge 0),$ and are zero for $n<0$. Define the degree-$+1$ classical Lie-action BRST operator by the fixed zero-based formula $\begin{array}{rl}{\displaystyle (d_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}f)(x_0,\dots ,x_n)=}& {\displaystyle \sum _{i=0}^n(-1{)}^i\rho (x_i)f(x_0,\dots ,\widehat{x_i},\dots ,x_n)}\\ {\displaystyle }& {\displaystyle +\sum _{0\le i<j\le n}(-1{)}^{i+j}f([x_i,x_j],x_0,\dots ,\widehat{x_i},\dots ,\widehat{x_j},\dots ,x_n)\mathrm{.}}\end{array}$ A hat denotes omission; the bracket is inserted first. This is exactly the Chevalley–Eilenberg differential for the stated representation, with the BRST name reserved here for this classical Lie-action model.

Its square-zero property and the resulting quotient cohomology are established in The classical Lie-action BRST differential squares to zero. This model does not claim a Koszul–Tate resolution, a BFV charge for arbitrary first-class constraints, reducible or open gauge algebras, field-dependent structure functions, or quantum BRST cohomology.

Let $(A,\mathrm{\Delta })$ be as in BV algebra with a degree-one generator. For homogeneous $a,b,c\in A$, the derived bracket has degree $\mathrm{\mid }\{a,b\}\mathrm{\mid }=\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }+1$ and satisfies $\{a,b\}=-(-1{)}^{(\mathrm{\mid }a\mathrm{\mid }+1)(\mathrm{\mid }b\mathrm{\mid }+1)}\{b,a\},$ $\{a,bc\}=\{a,b\}c+(-1{)}^{(\mathrm{\mid }a\mathrm{\mid }+1)\mathrm{\mid }b\mathrm{\mid }}b\{a,c\},$ and $\{a,\{b,c\}\}=\{\{a,b\},c\}+(-1{)}^{(\mathrm{\mid }a\mathrm{\mid }+1)(\mathrm{\mid }b\mathrm{\mid }+1)}\{b,\{a,c\}\}\mathrm{.}$ Thus it is an odd Poisson bracket, meaning an odd-skew bracket that is a derivation in each argument and satisfies the displayed odd Jacobi identity.

Let $(A,\mathrm{\Delta })$ be a BV algebra in the degree-$+1$ convention of BV algebra with a degree-one generator, with derived bracket $\{-,-\}$. For an element $S\in A^0$, the classical master equation (CME) is $\{S,S\}=0.$

For a formal version, take $A$ over $\mathbb{C}$, let $\mathrm{\hslash }$ be a central formal variable of degree $0$, and write $A[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$ for formal power series with the Cauchy product. Extend $\mathrm{\Delta }$ and $\{-,-\}$ coefficientwise. For $S_{\mathrm{\hslash }}\in A^0[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$, the formal quantum master equation (QME) in the convention used here is $\frac{1}{2}\{S_{\mathrm{\hslash }},S_{\mathrm{\hslash }}\}-i\mathrm{\hslash }\mathrm{\Delta }{S}_{\mathrm{\hslash }}=0$ as an equality in $A^1[\text{\hspace{0.17em}⁣}[\mathrm{\hslash }]\text{\hspace{0.17em}⁣}]$, checked coefficientwise. Each coefficient uses only a finite Cauchy sum, so this is a formal algebraic condition and requires no analytic convergence.

Kosmann-Schwarzbach–Monterde write a corresponding Maurer–Cartan equation as $\mathrm{\Delta }w+\frac{1}{2}\{w,w\}=0$. Where $w=i{S}_{\mathrm{\hslash }}\mathrm{/}\mathrm{\hslash }$ is formed in the Laurent-series extension, multiplying that equation by $-{\mathrm{\hslash }}^2$ gives the QME displayed here. This scalar substitution is only a normalization conversion. Neither equation defines a measure, proves a renormalization claim, or by itself establishes anomaly cancellation. The CME differential consequence is proved in A classical master solution defines a square-zero differential.

For every classical Lie-action BRST input of Classical Lie-action BRST cochain complex and every $n\ge 0$, $d_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^{n+1}{d}_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^n=0.$ Consequently, the quotient $H_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^n(\mathfrak{g},A):=\ker (d_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^n)\mathrm{/}\mathrm{im}(d_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^{n-1})$ is a well-defined vector space, with $C_{\mathrm{B}\mathrm{R}\mathrm{S}\mathrm{T}}^{-1}=0$.

Let $(A,\mathrm{\Delta })$ be as in BV algebra with a degree-one generator, and let $S\in A^0$ satisfy the classical master equation in Classical and formal quantum master equations. Then $d_S:A\to A$, $d_S(a)=\{S,a\}$, has degree $+1$ and satisfies $d_S^2=0$.