The Hochschild homology of a noncommutative symmetric quotient stack

Abstract

We prove an orbifold type decomposition theorem for the Hochschild homology of the symmetric powers of a small DG category $\mathcal{A}$. In noncommutative geometry, these can be viewed as the noncommutative symmetric quotient stacks of $\mathcal{A}$. We use this decomposition to show that the total Hochschild homology of the symmetric powers of $\mathcal{A}$ is isomorphic to the symmetric algebra $S^*(\mathrm{HH}_\bullet(\mathcal{A}) \otimes t \mathbb{k}[t])$. Our methods are explicit - we construct mutually inverse homotopy equivalences of the standard Hochschild complexes involved. These explicit maps are then used to induce from the symmetric algebra onto the total Hochschild homology the structures of the Fock space for the Heisenberg algebra of $\mathcal{A}$, of a Hopf algebra, and of a free $\lambda$-ring generated by $\mathrm{HH}_\bullet(\mathcal{A})$.