Quantum symmetries of noncommutative tori

TL;DR

This paper develops a method to construct quantum symmetries, specifically fusion category actions, on noncommutative tori, expanding the understanding of symmetries in noncommutative geometry.

Contribution

It introduces a general construction technique for quantum symmetries on noncommutative tori using finite-dimensional data, and applies it to various complex categories and tori.

Findings

Constructed AT-actions of Haagerup-Izumi categories on noncommutative 2-tori

Realized actions of the even part of the $E_{8}$ subfactor on noncommutative 3-tori

Implemented $ ext{PSU}(2)_{15}$ actions on noncommutative 4-tori

Abstract

We consider the problem of building non-invertible quantum symmetries (as characterized by actions of unitary fusion categories) on noncommutative tori. We introduce a general method to construct actions of fusion categories on inductive limit C*-algberas using finite dimenionsal data, and then apply it to obtain AT-actions of arbitrary Haagerup-Izumi categories on noncommutative 2-tori, of the even part of the $E_{8}$ subfactor on a noncommutative 3-torus, and of $\text{PSU}(2)_{15}$ on a noncommutative 4-torus.