Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

We will discuss Witten's gauge theory approach to Jones polynomial and Khovanov homology by counting solutions to some gauge theory equations with singular boundary conditions. When we reduce these equations to 3-dimensional, we call them the extended Bogomolny equations. We develop a Donaldson-Uhlenbeck-Yau type correspondence for the moduli space of the extended Bogomolny equations on Riemann surface Σ times R^+ with Nahm pole singularity at Σ × {0} and the Teichmuller component of the stable SL(2, R) Higgs bundle, this verifies a conjecture of Gaiotto and Witten. The proof is based on an observation that the extended Bogomolny equations can be reduced to a Kazdan-Warner type equation. We will also discuss a partial correspondence for solutions with knot singularities in this program, corresponding to the non-Teichmuller components in the moduli space of stable SL(2, R) Higgs bundles. This is joint work with Rafe Mazzeo.