Integral structures in TQFT
by
Gregor Masbaum
Institut de Mathematiques de Jussieu

According to Atiyah and Segal's axioms, a Topological Quantum Field Theory (TQFT) describes how to compute quantum invariants of 3-manifolds by cutting and pasting. It involves in particular finite-dimensional representations of surface mapping class groups.

In these lectures, we will focus on the Witten-Reshetikhin-Turaev SO(3)-TQFTs at odd primes. We will see that they admit a natural integral structure, meaning, among other things, that each mapping class group representation preserves a lattice defined over a ring of algebraic integers. To do so, we will review the skein-theoretical approach to TQFT and then give an explicit description of the integral lattices using lollipop trees. This is joint work with Patrick Gilmer.

Some applications of the integral structure will be discussed in the talks of Gilmer and myself at the conference.

Date received: May 30, 2009