Atle Hahn

(Lisbon University)

(Lisbon University)

Seminar

Tuesday, 11 May, 2010, at 16:15-17:15, in Aud. D3 (1531-215)

Abstract:

The study of the heuristic Chern-Simons path integral by E. Witten inspired several rigorous approaches to quantum topology, e.g., the perturbative approach based on the CS path integral in the Lorentz gauge and the "quantum group approach" by Reshetikhin & Turaev. While for the first of these two approaches the relation to the CS path integral is obvious for the second approach it is not. In particular, it is not clear if/how the algebraic expressions which appear in the quantum group approach can be derived directly from the CS path integral. In my talk, I will sketch a strategy that might lead to a clarification of this issue in the special case where the base manifold M is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by Blau & Thompson for the study of the CS partition function for such manifolds. I will show that the formulas of Blau & Thompson can be generalized to Wilson lines and that at least for the simplest types of links the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which the so-called "shadow invariant" is defined. Finally, I will sketch how - using methods from Stochastic Analysis or, alternatively, a suitable discretization approach - one can obtain a rigorous realization of the path integral expressions appearing in this treatment.

Contact person: Jørgen Ellegaard Andersen