The Heisenberg algebra of a vector space and Hochschild homology

TL;DR

This paper decategorifies the Heisenberg 2-category using Hochschild homology and extends the Heisenberg algebra action to all smooth proper noncommutative varieties, with explicit computations for orbifold cohomology.

Contribution

It generalizes the Heisenberg algebra action to noncommutative varieties within a new geometric framework, using Hochschild homology and decategorification techniques.

Findings

Decategorification of the Heisenberg 2-category via Hochschild homology.

Extension of Heisenberg algebra actions to noncommutative varieties.

Explicit computation of actions on Chen-Ruan orbifold cohomology.

Abstract

We decategorify the Heisenberg 2-category of Gyenge-Koppensteiner-Logvinenko using Hochschild homology. We use this to generalise the Heisenberg algebra action of Grojnowski and Nakajima to all smooth and proper noncommutative varieties in the noncommutative geometry setting proposed by Kontsevich and Soibelman. For ordinary commutative varieties, we compute the resulting action on Chen-Ruan orbifold cohomology. As tools, we prove results about Heisenberg algebras of a graded vector space which might be of independent interest.