The quantum torus as an $\mathbb E_M$-category
Lin Chen, Yifei Zhao
TL;DR
This work defines an $ extnormal{E}_M$-category $ ext{Rep}_q(reve T)$, a quantum torus parameterized by a Betti level $q$, for a 2-manifold $(M,oldsymbol{ extLambda})$ and a locally constant lattice twist. It proves a universal description of its factorization homology as twisted local systems, $igint_M ext{Rep}_q(reve T) \,\simeq \, ext{LS}_q(oldsymbol{ extGamma}_c(M, extnormal{B}^2oldsymbol{ extLambda}))$, using nonabelian Poincaré duality and trace maps. The paper analyzes the heart and $t$-structure fiberwise, and identifies a ribbon structure controlled by the quadratic form $Q$ and the symmetric form $b$ attached to $q$, including a geometric oriented lift to $ extnormal{E}_{ extnormal{BGL}_2^+}$. In the global (curve) case, it establishes a Betti-quantum Langlands-type equivalence for tori, $igint_X ext{Rep}_q(reve T) \\simeq ext{Shv}_{ ext{N},q}( ext{Bun}_T) \\simeq ext{LS}_q(oldsymbol{ extGamma}( extnormal{Sing} X^{ ext{top}}, extnormal{B}^2oldsymbol{ extLambda}))$, connecting quantum torus representations to twisted local systems on the moduli of bundles. This demonstrates how quantum tori behave in families and provides a concrete realization of the Betti geometric Langlands program for tori. The results offer a structured framework for studying families of braided- and ribbon-structured categories parametrized by a surface, with explicit global invariants and Langlands-type correspondences.
Abstract
Given an oriented $2$-manifold $M$, a locally constant sheaf of lattices $Λ$ over $M$, and a pointed morphism $q : \textsf B^2Λ\rightarrow \textsf B^4\mathbf C^{\times}$, 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.