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Topological quantum field theory

The main research topic of this line in recent years has been the so-called Chern-Simons gauge theory, one of the most important examples of a topological quantum field theory. Chern-Simons theory provides a fascinating bridge between Quantum Field Theory low-dimensional Topology, Quantum Algebra, Stochastic Analysis, and several other branches of Mathematics/Physics. The heuristic Feynman path integral of Chern-Simons theory can for instance be used to give a heuristic “definition” for a large class of highly nontrivial 3-manifold and knot invariants, which generalize the famous Jones polynomial.

These invariants can also be defined rigorously in terms of certain finite sums involving expressions from the (representation) theory of quantum groups. On the other hand some of the most important results in knot theory in recent years, like the discovery of the universal Vassiliev invariant, which is the most powerful knot invariant known till this day, were obtained by a perturbative evaluation of the Chern-Simons path integral using heuristic methods common in theoretical physics.

At the moment the (heuristic) path integral approach and the (rigorous) quantum group approach appear to be rather incompatible. In order to reconciliate them it is probably necessary to find a rigorous realization of the Chern-Simons path integral. Doing this for the original path integral expressions seems to be a very hard problem. Fortunately, the situation can be improved by applying a suitable gauge fixing. The gauge fixing we have been working with in the last 4 years is the so-called “torus gauge fixing”, which is available if the 3-manifold is of product form. We have demonstrated that for a special class of knots and links in such a manifold it is indeed possible to find a rigorous realization of the relevant (gauge fixed) path integral expressions and that the explicit values of these path integral expressions coincide with the corresponding invariants in the quantum group approach. It remains to be seen if this result can be generalized to arbitrary links.