The K\"ahler Potential of Abelian Higgs Vortices

TL;DR

This paper computes the K"ahler potential for Abelian Higgs vortices on different surfaces using multiple methods, revealing geometric insights and challenges in relating vortices to instantons.

Contribution

It introduces new calculations of the K"ahler potential for vortices on various geometries and explores connections to Liouville theory and Yang-Mills instantons.

Findings

Derived the K"ahler potential on $\

Connected the vortex moduli space to Liouville field theory.

Identified limitations in relating vortices to SU(2) Yang-Mills instantons.

Abstract

We calculate the K\"ahler potential for the Samols metric on the moduli space of Abelian Higgs vortices on $\mathbbm{R}^{2}$, in two different ways. The first uses a scaling argument. The second is related to the Polyakov conjecture in Liouville field theory. The K\"ahler potential on the moduli space of vortices on $\mathbbm{H}^{2}$ is also derived, and we are led to a geometrical reinterpretation of these vortices. Finally, we attempt to find the K\"ahler potential for vortices on $\mathbbm{R}^{2}$ in a third way by relating the vortices to SU(2) Yang-Mills instantons on $\mathbbm{R}^{2}\times S^{2}$. This approach does not give the correct result, and we offer a possible explanation for this.