It is well-known that TQFT-representations of mapping class groups of surfaces at the root of unity e^{\pi i q/p } preserve a Hermitian form whose signature depends not just on p, but also on q. The aim of this talk is to present joint work with Julien Marché where we study the asymptotic behaviour of this signature when q/p converges to an irrational number. We show that in genus 2, the asymptotics is governed by a certain modular form, and we conjecture that the signature itself also has modular properties : it should satisfy a kind of reciprocity formula similar to the classical reciprocity formula for Dedekind sums. No previous knowledge of TQFT or of modular forms will be assumed in this talk.

Café culturel  à 13:05 par Ramanujan Santharoubane