Relative local BRST cohomology

Use the four-dimensional coordinate and jet algebra and nilpotent differential $s$ fixed by the earlier Abelian local BRST definition and nilpotence lemma. For integers $r$ and $0\le p\le 4$, a compactly supported local horizontal $p$-form of ghost number $r$ is a finite sum of terms $f_{i_1\dots i_p}(x,z)d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_p},$ where each coefficient is smooth and compactly supported in $x\in {\mathbb{R}}^4$, polynomial in finitely many field-jet coordinates $z$, and homogeneous of ghost number $r$. Denote this space by ${\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$. A left derivative is used for odd jet coordinates.

For each spacetime index $\mu$, define the total derivative on a local coefficient by $D_{\mu }=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mu }}+{\sum }_{i,\alpha }{z}_{\alpha +e_{\mu }}^i\frac{{\mathrm{\partial }}_L}{\mathrm{\partial }{z}_{\alpha }^i},$ where $\alpha$ ranges over multi-indices, $i$ over the field and ghost generators, and $e_{\mu }$ is the multi-index with one derivative in direction $\mu$. The horizontal exterior derivative is $d_H=d{x}^{\mu }\wedge D_{\mu }$. It satisfies $d_H^2=0$. Extend $s$ to a homogeneous $p$-form by $s\omega =(-1{)}^p{s}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}\omega$, where $s_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ acts on its coefficient by the earlier local BRST definition; then $s^2=0$ and $s{d}_H+d_{H}s=0$.

Set ${\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}=0$ for $p<0$ or $p>4$. Evaluation of a compactly supported local four-form on any smooth off-shell configuration is a finite integral.

For a graded subcomplex ${\mathcal{K}}^{r,p}\subseteq {\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$ stable under $s$ and $d_H$, define $Z^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=\{a_p\in {\mathcal{K}}^{r,p}:s{a}_p+d_H{a}_{p-1}=0\text{ for some }{a}_{p-1}\in {\mathcal{K}}^{r+1,p-1}\}$ and $B^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=\{s{b}_p+d_H{c}_{p-1}:b_p\in {\mathcal{K}}^{r-1,p},c_{p-1}\in {\mathcal{K}}^{r,p-1}\}\mathrm{.}$ The relative local BRST cohomology is $H^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=Z^{r,p}\mathrm{/}{B}^{r,p}$. In four dimensions, $H^1(s\mathrm{\mid }{d}_H)$ is shorthand for $H^{1,4}(s\mathrm{\mid }{d}_H)$. For compactly supported coefficients, ${\int }_{{\mathbb{R}}^4}{d}_H{c}_3=0$ by the one-variable fundamental theorem of calculus in each coordinate and Fubini's theorem, so the quotient by $d_H$ identifies densities that define the same integrated local functional.

Abelian local jet BRST differential

Nilpotence and contraction of nonminimal doublets