Riemann-Roch for stacky matrix factorizations

TL;DR

This paper develops Riemann-Roch theorems for matrix factorizations on quotient stacks, providing explicit formulas for categorical invariants and pairings, advancing the understanding of derived categories in algebraic geometry.

Contribution

It introduces Riemann-Roch theorems for matrix factorizations on quotient stacks and constructs explicit isomorphisms and formulas for categorical invariants.

Findings

Established Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch theorems for matrix factorizations.

Constructed an explicit Hochschild-Kostant-Rosenberg type isomorphism.

Derived a formula for the categorical Chern character and the canonical pairing.

Abstract

We establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism explicit enough to yield a categorical Chern character formula. We next find an expression of the canonical pairing of Shklyarov under the isomorphism.