The Heisenberg algebra of a vector space and Hochschild homology

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.