Gauge Theory and Topology Seminar: Rank three instantons, representations and sutures
Ali Daemi - Washington University in Saint Louis
ng-Mills gauge theory with gauge group SU(2) has played a significant role in the study of the topology of 3- and 4-manifolds. It is natural to ask whether we obtain more topological information by working with other choices of gauge groups such as SU(n) for higher values of n. Mariño and Moore formulated a conjecture essentially stating that there is no new information in Donaldson invariants of smooth 4-manifolds defined using SU(n) Yang-Mills gauge theory. Despite this "negative" prediction, one might still hope that there is still novel information about 3-manifolds in higher rank gauge theory.
In this talk, I will discuss a result about the topology of 3-manifolds obtained using gauge theory with respect to the Lie group SU(3): for any knot K in the 3-dimensionl sphere (or more generally an integer homology sphere) there is a non-abelian representation of the knot group of K into SU(3) such that the homotopy class of the meridian of K is mapped to a matrix with eigenvalues 1, w, w^2 with w being a primitive third root of unity. As a byproduct of the proof, we obtain a structure theorem for SU(3) Donaldson invariants of 4-manifolds, analogous to Kronheimer and Mrowka's structure theorem for SU(2) Donaldson invariants. This can be regarded as a piece of evidence supporting Mariño and Moore's conjecture. This talk is based on a recent joint work with Nobuo Iida and Chris Scaduto.