Bifurcations and SU(n) gauge theory on 3-manifolds
by
Christopher M. Herald
University of Nevada, Reno

A rough description of the Casson invariant of a homology 3-sphere is that it is an algebraic count of flat SU(2) connections, modulo the gauge group. The flat connections are critical points of the Chern-Simons functional on the space of connections. To define the invariant (using gauge theory) more precisely, one must typically perturb the functional first before counting critical points.

Several generalizations to SU(3) have been defined. These involve counting flat SU(3) connections, but to obtain an invariant (independent of the choice of perturbation), reducible connections must be counted differently than irreducible ones. A key ingredient in the proofs that these generalizations are invariants is to show that they do not change under any of the bifurcations in the critical set which occur in generic 1-parameter deformations.

For SU(n), n > 3, there are multiple levels of reducible flat connections. In this talk I will discuss some results on the bifurcations in the flat moduli space in the SU(n) which occur during generic 1-parameter families of perturbations. I also discuss the problem of counting flat connections of the different orbit types in such a way that the result is an invariant.

Date received: February 28, 2003

Copyright © 2003 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 # cakw-20.