Integral TQFT for a one-holed torus

TL;DR

This paper develops explicit formulas for Integral TQFT representations of the mapping class group of a genus one surface with boundary, enabling straightforward computation of the associated matrices and their approximations over finite fields.

Contribution

It introduces new orthogonal bases for Integral TQFT modules in arbitrary genus, facilitating explicit calculations and analysis of the representations.

Findings

Explicit formulas for TQFT representations of the genus one surface mapping class group.

Method to compute h-adic expansions and finite group approximations of the representations.

Identification of induced representations over finite fields up to isomorphism.

Abstract

We give new explicit formulas for the representations of the mapping class group of a genus one surface with one boundary component which arise from Integral TQFT. Our formulas allow one to compute the h-adic expansion of the TQFT-matrix associated to a mapping class in a straightforward way. Truncating the h-adic expansion gives an approximation of the representation by representations into finite groups. As a special case, we study the induced representations over finite fields and identify them up to isomorphism. The key technical ingredient of the paper are new bases of the Integral TQFT modules which are orthogonal with respect to the Hopf pairing. We construct these orthogonal bases in arbitrary genus, and briefly describe some other applications of them.