Integral topological quantum field theory for a one-holed torus

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.

On Burau’s representations at roots of unity

“…This solves a particular case of a conjecture formulated by Andersen, Masbaum and Ueno in [2]; a proof of the conjecture in this case was announced by Masbaum in [2], Remark 5.9 (see also [16], p.4). We won't give further details since this is also a consequence of a stronger result recently obtained in [29].…”

Irreducible factors of modular representations of mapping class groups arising in Integral TQFT

“…See [GM2,§3] for more details about this pairing. The linked lollipop trees defining the Hopf pairing are as in the left part of Figure 3 in the case where the surface is Σ g (2c), and as in Figure 4 in the case where the surface is ∂H.…”

“…It is the purpose of this paper to study this question for mapping class group representations coming from a TQFT. Specifically, we consider the SO(3)-TQFT at a primitive p-th root of unity, where p ≥ 5 is a prime, because the theory of Integral TQFT developed in [Gi,GM1,GM2] implies that this TQFT gives rise in a natural way to modular representations of mapping class groups in positive characteristic. We remark that modular representations in characteristic different from p coming from this SO(3)-TQFT at the p-th root of unity were recently used in work of A. Reid and the second author [MRe].…”

“…We wonder about how to characterize our irreducible factors among the modular representations of the symplectic group Sp(2g, F p ) in characteristic p. As a step in this direction, in the case p = 5, we have identified our dimensions e (2) In the genus one case, where F (Σ) itself is irreducible, we have e 1 (p) = 0. In this case, we computed the modular representation F (Σ 1 (2c)) for any p and c already in [GM2], where we showed that F (Σ 1 (2c)) (which was denoted by S + p,0 (T c ) in [GM2]) is isomorphic to the space of homogeneous polynomials over F p in two variables of total degree (p − 3)/2 − c. It is wellknown that this representation of Sp(2, F p ) = SL(2, F p ) is irreducible for every c ≤ (p − 3)/2.…”

“…The study of these higher quotient representations is interesting, because they factor through finite quotients of the extended mapping class group, but approximate the TQFT-representation better and better as N increases. This is discussed in more detail in [GM2]. (4) The proper irreducible subrepresentation F odd (Σ) of F (Σ) (in the case g ≥ 2 and (g, c) = (2, 0)) is the radical of a certain Γ Σ -invariant bilinear form on the F pvector space F (Σ).…”