The Tambara-Yamagami categories and 3-manifold invariants

Abstract

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