Joint Mathematical Physics seminar at Chennai Mathematical Institute

Last time we covered abelian theory over S^3 using standard path integral technique. Namely, using the fact that U(1)-theory contains only quadratic kinetic term we computed the two-point function as Green function of the curl operator, and, as we saw, the corresponding Wilson correlators are given by Gauss integral formula for the linking number of pair of knots in S^3 modulo k.

This time we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity using axiomatic TQFT approach modulo some finite amount of data predicted by the connection of Chern-Simons to the conformal blocks of the underlying CFT.

Following Witten, we will show that non perturbative analysis of Chern-Simons in d=3 within Atiyah-Segal axiomatic approach to TQFT's, naturally leads to skein relations on Wilson correlators. Fix the level k. We will start with the gauge group U(1), where we recover Gauss integral formula for the linking number of a pair of knots in S^3 modulo k. Then we will turn to the non abelian case SU(N) to recover the value of Jones polynomial at the primitive (N+k)-root of unity.

We will review the definition of observables in quantum mechanics, in the Atiyah-Segal framework. Thereafter, we will review the Lagrangian field theory description of the Chern-Simons theory and discuss a variation of the standard Atiyah-Segal axioms needed to construct this theory.