The Tambara-Yamagami categories and 3-manifold invariants

TL;DR

This paper establishes conditions under which Tambara-Yamagami categories produce identical 3-manifold invariants, linking the invariants to underlying group structures and bicharacters, with a focus on cases where the group order is odd.

Contribution

It proves that identical 3-manifold invariants from Tambara-Yamagami categories imply the categories are related by a group isomorphism, given odd group order.

Findings

Invariants determine the bicharacter hi'

Invariants determine the associator nu'

Explicit computation for lens spaces supports the main result

Abstract

We prove that if two Tambara-Yamagami categories TY(A,\chi,\nu) and TY(A',\chi',\nu') give rise to the same state sum invariants of 3-manifolds and the order of one of the groups A, A' is odd, then \nu=\nu' and there is a group isomorphism A\approx A' carrying \chi to \chi'. The proof is based on an explicit computation of the state sum invariants for the lens spaces of type (k,1).