In this thesis, we investigate the higher Frobenius-Schur indicator introduced by Ng and Schauenburg and prove that it is a strong enough invariant to distinguish between any two Tambara-Yamagami fusion categories. Our method of proof is based on computation of the Frobenius-Schur indicators as Gauss sums for certain quadratic forms on finite abelian groups and relies on the classication of quadratic forms on finite abelian groups due to Wall.

As a corollary to our work, we show that the state-sum invariants of 3-manifolds associated with Tambara-Yamagami categories determine the category as long as we restrict to Tambara-Yamagami categories defined coming from groups G whose order is not a power of 2. Turaev and Vainerman proved this result under the assumption that G has odd order and they conjectured that a similar result should hold for all Tambara-Yamagami categories. Their proof used the state-sum invariant of Lens spaces L(k,1). We provide an example showing that the state-sum invariants of Lens spaces is not enough to distinguish all Tambara-Yamagami categories.