Classical BV formalism for group actions

TL;DR

This paper extends the classical Batalin-Vilkovisky (BV) formalism from finite-dimensional Lie algebra actions to more general group actions on derived critical loci, providing explicit descriptions of the associated derived stacks.

Contribution

It generalizes the BV formalism to group actions on derived critical loci, offering explicit dg-algebra descriptions and broadening the scope beyond Lie algebra actions.

Findings

Derived critical locus is equivalent to a quotient stack [Z/G] with explicit dg-algebra structure

Generalization of BV formalism from Lie algebra to group actions

Explicit description of the derived affine scheme Z and its dg-algebra

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.