Equivariant sheaves on loop spaces

TL;DR

This paper develops a dg-model for the derived category of equivariant differential forms on a scheme with group action, and compares it to a derived Hamiltonian reduction approach, advancing the understanding of equivariant sheaves on loop spaces.

Contribution

It introduces a new dg-model for equivariant differential forms and relates it to existing derived Hamiltonian reduction frameworks.

Findings

The dg-model aligns with the derived category of equivariant dg-modules.

Comparison shows equivalence with coherent sheaves on derived Hamiltonian reductions.

Provides tools for studying equivariant sheaves on loop spaces.

Abstract

Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in Arkhipov and {\O}rsted (2018a, 2018b), we define a dg-model for the derived category of dg-modules over the dg-algebra of differential forms $\Omega_X$ on $X$ equivariant with respect to the action of a derived group scheme $(G ,\Omega_G )$. We compare the obtained dg-category with the one considered in Arkhipov and Kanstrup (2015) given by coherent sheaves on the derived Hamiltonian reduction of $T^* X$.