The Tambara-Yamagami categories and 3-manifold invariants

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).

💡 Research Summary

The paper investigates the extent to which the state‑sum invariants derived from Tambara‑Yamagami (TY) fusion categories determine the underlying categorical data. A TY category TY(A, χ, ν) is specified by a finite abelian group A, a non‑degenerate symmetric bicharacter χ : A × A → ℂ^×, and a sign ν ∈ {±1}. Its simple objects consist of the group elements together with a single non‑invertible object m; the fusion rules, associativity constraints, and 6j‑symbols are all encoded by χ and ν.

The authors first recall the construction of Turaev‑Viro type state‑sum invariants Z_{TY}(M) for closed oriented 3‑manifolds M using a spherical structure on TY(A, χ, ν). They then focus on a family of lens spaces L(k, 1) (k ∈ ℕ) because these manifolds admit a simple triangulation that makes the computation of the state sum tractable. By assigning the TY data to the edges and tetrahedra of the triangulation, the partition function reduces to a finite sum over labelings by elements of A. The contribution of each labeling involves the bicharacter χ raised to the k‑th power, while the presence of the non‑invertible object m introduces a term multiplied by ν.

A careful analysis yields an explicit formula: \