Perturbative Quantum Invariants; a survey
by
Nathan Habegger
Universite de Nantes

In this talk, I will give an introduction to the still developing theory of perturbative invariants. This theory considers topological objects such as knots, links and 3-manifolds. Associated to each object of study is a `perturbative' expansion, i.e., a power series, lying in a graded vector space of `Feynman diagrams'. Arising in the theory of Topological Quantum Field Theory, or rather its perturbative version, these diagrams mesure in some sense the `topological energy' of the objects, i.e., the knottedness, linking, and complexity of the space, but what precisely they measure is a subject of much interest. These invariants are quite rich and are related to such classical invariants as the Gauss linking number, the Alexander polynomial, the Milnor [`(\mu)] invariants, and Reidemeister torsion, as well as more recent developments such as the Jones-Conway polynomial, the Casson invariant and Vassiliev invariants. Several different (rigorous) constructions of these invariants are now available, but all involve a mixture of topology in some form with the combinatorics of diagrams. This latter is by itself a subject of much interest and is tied to the theory of groups, Lie algebras, Hopf algebras and quantum groups.

Date received: May 26, 1998

Copyright © 1998 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cabo-04.