The Mukai pairing, I: a categorical approach

TL;DR

This paper explores the Hochschild homology of smooth spaces, introducing a categorical framework for the Mukai pairing, and demonstrating how derived category functors induce compatible linear maps, with applications to Riemann-Roch and physics-inspired conditions.

Contribution

It develops a categorical approach to the Mukai pairing on Hochschild homology, linking integral transforms to linear maps and extending classical theorems in a new homological context.

Findings

Integral transforms induce linear maps on Hochschild homology.

Adjoint functors correspond to adjoint linear maps under the pairing.

A Chern character in Hochschild homology is defined, connecting to classical theorems.

Abstract

We study the Hochschild homology of smooth spaces, emphasizing the importance of a pairing which generalizes Mukai's pairing on the cohomology of K3 surfaces. We show that integral transforms between derived categories of spaces induce, functorially, linear maps on homology. Adjoint functors induce adjoint linear maps with respect to the Mukai pairing. We define a Chern character with values in Hochschild homology, and we discuss analogues of the Hirzebruch-Riemann-Roch theorem and the Cardy Condition from physics. This is done in the context of a 2-category which has spaces as its objects and integral kernels as its 1-morphisms.