Next January the 16th I will be delivering a talk about the volume of the moduli space of vortex-antivortex systems on compact manifolds at the school of mathematics, Autonomous University of Yucatan, Mexico. Talk is free and will be delivered in spanish.

Abstract

This is a talk about a problem in abelian field theory, which corresponds to the O(3)-Sigma model. In this model it is assumed that there is a matter field with a spin-like internal structure, i.e. a vector field of unit length on space. As in all field theories, there is a set of partial differential equations that determine the solution space. For the O(3)-Sigma model the set of parameters is finite and is interpreted as the position of virtual particles whose dynamics is directed by a Riemannian metric. The domain of the equations we study is a compact Riemann surface, and as a consequence of the field equations, the metric is of Kahler-type.

The geometry of this model was studied in depth by Romao and Speight, who under some assumptions found a formula for the volume of the moduli space of solutions. I will delve into the geometry of moduli space, to explain the conjecture and the progress that has been made to demonstrate it in some cases.

References

The talk will be mainly based on the paper,

The Geometry of the Space of BPS Vortex–Antivortex Pairs, N. Romao y JM Speight, Comm. in mathematical physics 379 (2020), open access at: https://link.springer.com/article/10.1007/s00220-020-03824-y

plus some contributions of my own.