Kaleb D. Ruscitti, Department of Pure Mathematics, University of Waterloo
"Toric degeneration of the moduli space of holomorphic SL(2,C)-bundles over a Riemann Surface"
To understand the moduli space M of semi-stable holomorphic SL(2,C) bundles over a compact Riemann surface, one can cut the surface into pairs of pants, understand the moduli space over one pair of pants, and then use Geometric Invariant Theory to glue the bundles back together. Furthermore, Hurtubise and Jeffrey showed in 2000 that if one shrinks some of the boundary loops, forming a singular curve, then the moduli space becomes a toric variety P. In 2021, Biswas and Hurtubise demonstrated that this shrinking process also induces a toric degeneration of M to P. I will explain this process and show some of the GIT computations I did for the genus two surface.
The moduli space M has a symplectic structure arising from Chern-Simons theory, and there is a pre-quantum line bundle L over M. It is a conjecture of Witten (on physical grounds) that the cohomology of this line bundle is given by the Verlinde Formula from conformal field theory. If time permits, we will discuss how the above results help work towards a proof of this conjecture and some of the remaining barriers.
MC 5417