Local BRST Cohomology Modulo Total Derivatives

Let $n\ge 1$, let $U\subseteq {\mathbb{R}}^n$ be nonempty and open, let ${\mathcal{F}}_U$ be the finite-jet local algebra with nilpotent BRST derivation $s$ and commuting total derivatives $D_{\mu }$, and let $d{x}^1,\dots ,d{x}^n$ be anticommuting coordinate one-forms. For $g\in \mathbb{Z}$ and $0\le p\le n$, let ${\mathcal{A}}_U^{g,p}$ be the space of local horizontal $p$-forms of ghost number $g$, namely finite sums of $d{x}^{{\mu }_1}\wedge \cdots \wedge d{x}^{{\mu }_p}$ with coefficients in the ghost-number-$g$ part of ${\mathcal{F}}_U$. Define the horizontal differential $d_H:=d{x}^{\mu }{D}_{\mu }$; its square is $d_H^{2}a=d{x}^{\mu }\wedge d{x}^{\nu }{D}_{\mu }{D}_{\nu }a=\frac{1}{2}d{x}^{\mu }\wedge d{x}^{\nu }[D_{\mu },D_{\nu }]a=0$. Extend $s$ to horizontal forms by $s(d{x}^{\mu }\wedge a)=-d{x}^{\mu }\wedge sa$. Since $s{D}_{\mu }=D_{\mu }s$, one has $s(d_{H}a)=-d{x}^{\mu }\wedge D_{\mu }(sa)=-d_H(sa)$, hence $s{d}_H+d_{H}s=0$.

For $p=0$, set ${\mathcal{A}}_U^{g,-1}=\{0\}$. The local BRST cohomology modulo total derivatives in bidegree $(g,p)$ is

$H^{g,p}(s\mid d_H):=\frac{\{a\in {\mathcal{A}}_U^{g,p}:\text{there is }b\in {\mathcal{A}}_U^{g+1,p-1}\text{ with }sa+d_{H}b=0\}}{\{sm+d_{H}n:m\in {\mathcal{A}}_U^{g-1,p},n\in {\mathcal{A}}_U^{g,p-1}\}}\mathrm{.}$

The denominator is a subspace of the numerator: for $a=sm+d_{H}n$, choose $b=sn$ and compute $sa+d_{H}b=s^{2}m+s{d}_{H}n+d_{H}sn=0$. For $p=0$, $n=b=0$ and the same verification reduces to $s^{2}m=0$. This is cohomology of the specified local jet algebra and horizontal differential; it is not cohomology of a physical state space or Hilbert space. Interpreting $d_H$-exact top forms as equal local functionals additionally assumes compact support or boundary conditions that make the corresponding surface term vanish.

Finite-jet local algebra and BRST operator

Nilpotence and commutation with total derivatives

Barnich, Brandt, and Henneaux, §§2.3 and 4.1–4.5

Gomis, París, and Samuel, §§4.6 and 8.1