The Mukai pairing, I: the Hochschild structure

TL;DR

This paper explores the Hochschild structure of smooth spaces and orbifolds, focusing on a pairing that generalizes Mukai's pairing on K3 surfaces, and discusses its properties and invariance under transformations.

Contribution

It introduces a Hochschild pairing generalizing Mukai's pairing, analyzes its properties without relying on HKR and Kontsevich formality, and discusses invariance under Fourier-Mukai transforms.

Findings

Establishes functoriality and adjointness of the Hochschild pairing.

Derives a formal Hirzebruch-Riemann-Roch theorem and the Cardy condition.

Shows invariance of the Hochschild structure under Fourier-Mukai transforms.

Abstract

We study the Hochschild structure of a smooth space or orbifold, emphasizing the importance of a pairing defined on Hochschild homology which generalizes a similar pairing introduced by Mukai on the cohomology of a K3 surface. We discuss those properties of the structure which can be derived without appealing to the Hochschild-Kostant-Rosenberg isomorphism and Kontsevich formality, namely: -- functoriality of homology, commutation of push-forward with the Chern character, and adjointness with respect to the generalized pairing; -- formal Hirzebruch-Riemann-Roch and the Cardy condition from physics; -- invariance of the full Hochschild structure under Fourier-Mukai transforms. Connections with homotopy theory and TQFT's are discussed in an appendix. A separate paper treats consequences of the HKR isomorphism. Applications of these results to the study of a mirror symmetric analogue of Chen-Ruan's orbifold product will be presented in a future paper.