Equivariant categories of symplectic surfaces and fixed loci of Bridgeland moduli spaces

TL;DR

This paper explores the relationship between fixed loci of group actions on moduli spaces of stable objects in derived categories and equivariant categories, providing criteria for when these categories are equivalent to derived categories of surfaces.

Contribution

It generalizes the derived McKay correspondence by relating fixed loci of symplectic group actions to equivariant derived categories of surfaces.

Findings

Established a criterion for equivariant categories to be equivalent to derived categories of surfaces.

Generalized the derived McKay correspondence for symplectic surfaces.

Provided a framework for describing fixed loci of symplectic group actions on moduli spaces.

Abstract

Given an action of a finite group $G$ on the derived category of a smooth projective variety $X$ we relate the fixed loci of the induced $G$-action on moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an application we obtain a criterion for the equivariant category of a symplectic action on the derived category of a symplectic surface to be equivalent to the derived category of a surface. This generalizes the derived McKay correspondence, and yields a general framework for describing fixed loci of symplectic group actions on moduli spaces of stable objects on symplectic surfaces.