Equivariant Contact Darboux Quotients

Abstract

We prove an equivariant Darboux theorem for $-1$-shifted contact derived Artin stacks. Such a stack admits a smooth atlas by contact Darboux charts: the derived discriminant locus $Δ\mathrm{loc}(s)$ of a relative section $s$, extended in degree $-2$ along the relative dimension of the chart. Near a point with linearly reductive stabilizer $G$ acting on the contact line through a character $χ$, the stack is étale-locally the quotient $[Δ\mathrm{loc}_G(s)/G]$, whose degree $-2$ generators are indexed by the Lie algebra of $G$ with differential a contact moment map. The cotangent complex sees these generators as $\mathrm{Lie}\,\ker(χ)$ in degree $-2$; they vanish exactly when $\ker(χ)$ is finite, and the quotient of the discriminant locus alone is a local model at no point where they are nonzero. For a quiver with potential the contact moment map is the moment map of the doubled quiver, the classical truncation is the representation space of the Jacobi algebra, and the degree $-2$ generators are carried exactly by the locus of positive-dimensional stabilizer.