Frobenius-Schur Indicators and the Mapping Class Group of the Torus

TL;DR

This paper links Frobenius-Schur indicators to Turaev-Viro invariants of solid tori, offering a geometric perspective on their SL(2,Z)-equivariance and extending 3-manifold invariants to manifolds with boundaries.

Contribution

It introduces a novel geometric interpretation of Frobenius-Schur indicators via Turaev-Viro invariants of solid tori, expanding the understanding of their symmetry properties.

Findings

Frobenius-Schur indicators can be expressed as Turaev-Viro invariants of solid tori.

Provides a geometric understanding of SL(2,Z)-equivariance of indicators.

Extends Turaev-Viro invariants to 3-manifolds with free boundaries.

Abstract

The Turaev-Viro state sum invariant can be extended to 3-manifolds with free boundaries. We use this fact to describe generalized Frobenius-Schur indicators as Turaev-Viro invariants of solid tori. This provides a geometric understanding of the $\mathrm{SL}(2,\mathbb{Z})$-equivariance of these indicators.