Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

TL;DR

This paper investigates the category of matrix factorizations for isolated hypersurface singularities, deriving explicit formulas for Chern characters, bilinear forms, and an analog of the Hirzebruch-Riemann-Roch theorem, including G-equivariant cases.

Contribution

It provides explicit expressions for Chern characters and bilinear forms, and establishes a Hirzebruch-Riemann-Roch type formula for matrix factorizations, extending to G-equivariant settings.

Findings

Computed the canonical bilinear form on Hochschild homology.

Derived explicit formulas for Chern characters and boundary-bulk maps.

Established an analog of the Hirzebruch-Riemann-Roch formula for matrix factorizations.

Abstract

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.