# Classical BV formalism for group actions

April 30, 2021

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.

Keywords:

Derived algebraic geometry, derived critical locus, quotient stack, Batalin-Vilkovisky formalism