Feynman path integrals and 3-manifold quantum invariants

Atle Hahn - Bonn University

*Abstract*

The study of the heuristic Chern-Simons path integral by E.
Witten inspired (at least) two general approaches to quantum topology
in dimension 3. Firstly, the pertubation approach based on the CS path
integral in the Lorentz gauge and, secondly, the "quantum group
approach" by Reshetikhin/Turaev. While for the first approach, the
relation to the CS path integral is obvious, for the second approach
it is not. In
particular, it is not clear if/how one can derive the relevant
R-matrices or quantum 6j-symbols directly from the CS path integral.
In my talk which summarizes the results of a recent preprint, I will
sketch a strategy that should lead to a clarification of this issue in
the special case where the base manifold is of product form. This
strategy is based on the "torus gauge fixing" procedure of the partition
function of CS models.
I will show that the formulas of Blau/Thompson can be generalized to
Wilson lines and that the evaluation of the expectation values of these
Wilson lines leads to the same state sum expressions in terms of which
Turaev's shadow invariant is defined.

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