Frobenius-Schur indicators and the mapping class group of the torus (Q2131220)

In quantum topology, a central role is played by spherical fusion categories, which are certain semisimple monoidal categories with duality. Via the Turaev-Viro construction (as extended by Barrett-Westbury) these categories give rise to so called ``topological quantum field theories'' (TQFTs), which are functors assigning vector spaces to closed surfaces and linear maps to 3-dimensional cobordisms between closed surfaces. Cobordisms are allowed to admit links in their interior colored by objects in the Drinfeld center of the category. \medskip Spherical fusion categories can be studied from a purely representation theory perspective. A fundamental tool in their study are Frobenius-Schur indicators, which depend on an object of the category \(\mathcal{C}\) as well as an object in its Drinfeld center \(\mathcal{Z}(\mathcal{C})\). These indicators have a \(SL_2(\mathbb{Z})\)-equivariance property which is essential, for instance, in the proof of the congruence subgroup conjecture. \medskip This paper gives an interpretation of Frobenius-Schur indicators of spherical fusion categories, and their \(SL_2(\mathbb{Z})\)-equivariance, in terms of Turaev-Viro theory ``with defects''. This is an extension of the TQFT in which surfaces are allowed to admit links colored with objects of \(\mathcal{C}\). It is shown that the Frobenius-Schur indicator is the scalar associated to a solid torus with a \(\mathcal{Z}(\mathcal{C})\)-colored knot in its interior and a \(\mathcal{C}\)-colored knot in its boundary and that equivariance follows from TQFT gluing properties. This extension of Turaev-Viro theory is not fully constructed though, so this paper can be seen as motivation to build such a theory.