Classical BV formalism for group actions

Abstract

We study the derived critical locus of a function $f:[X/G]\to {\mathbb{A}}_{\mathbb{K}}^1$ on the quotient stack of a smooth affine scheme $X$ by the action of a smooth affine group scheme $G$. It is shown that $\mathrm{dCrit}(f)\simeq [Z/G]$ is a derived quotient stack for a derived affine scheme $Z$, whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.

v3: Final version accepted for publication in Communications in Contemporary Mathematics