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Quantum symmetries of noncommutative tori

David E. Evans, Corey Jones

TL;DR

This work addresses the problem of realizing non-invertible quantum symmetries on noncommutative tori by actions of unitary fusion categories. It develops a general inductive-limit method, via AT-actions, to lift finite-dimensional data to actions on AT-algebras and then identifies these algebras with noncommutative tori using $K$-theory and traces. The authors establish a no-go result ruling out many $ ext{SU}(2)_k$ cases, and then construct explicit AT-actions for Haagerup–Izumi categories on 2-tori, the adjoint subcategory of the $E_{8}$ subfactor on 3-tori, and $ ext{PSU}(2)_{15}$ on 4-tori. In each case, they identify the inductive-limit algebra with a noncommutative torus by matching ordered $K_0$-groups and traces, and in several instances realize actions of Drinfeld centers or related centers on the torus. The results broaden the landscape of quantum symmetries in noncommutative geometry and provide concrete, computable examples that connect fusion-categorical data to topological noncommutative spaces.

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.

Quantum symmetries of noncommutative tori

TL;DR

This work addresses the problem of realizing non-invertible quantum symmetries on noncommutative tori by actions of unitary fusion categories. It develops a general inductive-limit method, via AT-actions, to lift finite-dimensional data to actions on AT-algebras and then identifies these algebras with noncommutative tori using -theory and traces. The authors establish a no-go result ruling out many cases, and then construct explicit AT-actions for Haagerup–Izumi categories on 2-tori, the adjoint subcategory of the subfactor on 3-tori, and on 4-tori. In each case, they identify the inductive-limit algebra with a noncommutative torus by matching ordered -groups and traces, and in several instances realize actions of Drinfeld centers or related centers on the torus. The results broaden the landscape of quantum symmetries in noncommutative geometry and provide concrete, computable examples that connect fusion-categorical data to topological noncommutative spaces.

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 subfactor on a noncommutative 3-torus, and of on a noncommutative 4-torus.
Paper Structure (9 sections, 18 theorems, 68 equations)

This paper contains 9 sections, 18 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Actions of unitary tensor categories on C*-algebras
  3. Inductive limit actions
  4. Extending simple stationary actions
  5. Actions on noncommutative tori
  6. No-go results
  7. Actions of Haagerup-Izumi on noncommutative 2-tori
  8. Action of adjoint subcategory of $E_{8}$ quantum subgroup on a noncommutative 3-torus
  9. Action of $\text{PSU}(2)_{15}$ on a noncommutative 4-torus

Key Result

Theorem 1.1

The fusion categories $SU(2)_{k}$ with $k+2$ prime and $k+1$ not a power of $2$ admit no action on any noncommutative torus of any rank.

Theorems & Definitions (42)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • ...and 32 more