Braidings for Non-Split Tambara-Yamagami Categories over the Reals

TL;DR

This work delivers a comprehensive classification of braidings on Tambara-Yamagami categories over the real numbers, distinguishing split versus non-split, real/complex, and real/quaternionic cases. By reducing braiding data to hexagon-derived invariants and employing monoidal equivalence classifications, the authors show that braidings exist only in tightly constrained scenarios, notably when the invertible group is a power of the Klein four group with hyperbolic pairing, and they parameterize braided equivalence classes via admissible quadratic forms and a finite sign datum. They extend the analysis to split complex TY categories, providing explicit counts of braidings across rank cases, and establish a non-existence result for braidings in the complex/complex setting. Collectively, these results elucidate how time-reversal-like symmetries manifest in (2+1)D TQFTs via real and quaternionic TY categories, anchored by Gauss-sum and quadratic-form data and governed by Galois actions and automorphism-orbit structures.

Abstract

Non-split Real Tambara-Yamagami categories are a family of fusion categories over the real numbers that were recently introduced and classified by Plavnik, Sanford, and Sconce. We consider which of these categories admit braidings, and classify the resulting braided equivalence classes. We also prove some new results about the split real and split complex Tambara-Yamagami Categories. V2: Final Section removed, to appear in Transformation Groups.

Theorems & Definitions (96)

• Definition 2.1
• Definition 3.1
• Remark 3.2
• Definition 3.3
• Lemma 3.4: MR2609644
• Definition 3.5
• Remark 3.6
• Proposition 3.7
• proof
• Proposition 3.9