The quantum torus as an ${\mathbb{E}}_M$-category

Abstract

Given an oriented $2$-manifold $M$, a locally constant sheaf of lattices $\hat{I}›$ over $M$, and a pointed morphism $q:{\text{B}}^2\hat{I}›\to {\text{B}}^4{\mathbf{C}}^{\times }$, we define an ${\mathbb{E}}_M$-category ${\mathrm{Rep}}_q(\check{T})$ which we call the "quantum torus" at level $q$. We explain why this terminology is deserved and calculate the factorization homology of ${\mathrm{Rep}}_q(\check{T})$. When $M$ arises from a global complex curve, we confirm (a version of) a conjecture of Ben-Zvi and Nadler for tori.

25 pages