A generalized Hirzebruch Riemann-Roch theorem

TL;DR

This paper generalizes the Hirzebruch Riemann-Roch theorem by connecting it to the Cardy condition through advanced Hodge cohomology techniques and the Hochschild-Kostant-Rosenberg map.

Contribution

It introduces a new generalized theorem linking the Hirzebruch Riemann-Roch and Cardy condition using Hochschild cohomology and Todd genus twists.

Findings

Established a generalized Riemann-Roch theorem

Connected the theorem to the Cardy condition

Utilized Hochschild-Kostant-Rosenberg map in the proof

Abstract

This short note proves a generalization of the Hirzebruch Riemann-Roch theorem equivalent to the Cardy condition described in [1]. This is done using an earlier result [4] that explicitly describes what the Mukai pairing in [1] descends to in Hodge cohomology via the Hochschild-Kostant-Rosenberg map twisted by the root Todd genus.