Equivalences of derived factorization categories of gauged Landau-Ginzburg models

TL;DR

This paper establishes equivalences between derived factorization categories of gauged Landau-Ginzburg models, extending known results and providing partial answers to Segal's conjecture, with applications to equivariant derived categories.

Contribution

It constructs Fourier-Mukai equivalences of derived factorization categories for gauged LG models, generalizing previous results and addressing Segal's conjecture.

Findings

Equivalences of derived factorization categories for K-equivalent gauged LG models.

Extension of Ploog's result to reductive group actions.

Partial resolution of Segal's conjecture in this context.

Abstract

For a given Fourier-Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier-Mukai equivalences of derived factorization categories of gauged Landau-Ginzburg (LG) models. As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier-Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.