Integral topological quantum field theory for a one-holed torus

Patrick M. Gilmer and Gregor Masbaum

Vol. 252 (2011), No. 1, 93–112

DOI: 10.2140/pjm.2011.252.93

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 the straightforward computation of the h-adic expansion of the TQFT-matrix associated to a mapping class. 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.

Keywords

orthogonal lollipop basis, modular representations, TQFT

Mathematical Subject Classification 2000

Primary: 57R56

Milestones

Received: 23 June 2010

Revised: 22 February 2011

Accepted: 28 February 2011

Published: 8 October 2011