Geometry, backgrounds, and the parameter algebra

Use

Parentheses include $\frac{1}{2}$. Indices are moved with $\eta$. The ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is the inherited smooth function, zero for $t\le 0$, one for $t\ge \epsilon$, flat at both endpoints. The switch is $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ with $P_g=\mathrm{int}\{g=1\}\ne \varnothing$. Both are fixed external functions; neither is varied, neither is a spurion, and no transformation law is assigned to them.

Fix once a nonempty open, relatively compact $W$ with

with the ordinary vector-field bracket $[\xi ,\zeta {]}^{\mu }={\xi }^{\nu }{\partial }_{\nu }{\zeta }^{\mu }-{\zeta }^{\nu }{\partial }_{\nu }{\xi }^{\mu }$. The observation $f\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$ and the switch are chosen so that (eq:UCP) holds. All of $(f,g,m,\epsilon ,T,W)$ are fixed before any parameter, ghost, or antifield is selected, and ${\mathfrak{g}}_W$ is not reduced afterwards.