The geometry of the space of BPS vortex-antivortex pairs

TL;DR

This paper investigates the geometric structure of the moduli space of vortex-antivortex pairs in a gauged sigma model, analyzing its metric, volume, and thermodynamic properties across different surfaces.

Contribution

It provides the first detailed analysis of the $L^2$ metric on the vortex-antivortex moduli space, including boundary behavior, asymptotic formulas, and a proposed compactification.

Findings

The moduli space has finite volume on $S^2$ and is geodesically incomplete.

Numerical and formal analysis suggest asymptotic behavior of the metric at large separations.

The volume and scalar curvature formulas enable thermodynamic analysis of vortex-antivortex gases.

Abstract

The gauged sigma model with target $\mathbb{P}^1$, defined on a Riemann surface $\Sigma$, supports static solutions in which $k_+$ vortices coexist in stable equilibrium with $k_-$ antivortices. Their moduli space is a noncompact complex manifold $M_{(k_+,k_-)}(\Sigma)$ of dimension $k_++k_-$ which inherits a natural K\"ahler metric $g_{L^2}$ governing the model's low energy dynamics. This paper presents the first detailed study of $g_{L^2}$, focussing on the geometry close to the boundary divisor $D=\partial M_{(k_+,k_-)}(\Sigma)$. On $\Sigma=S^2$, rigorous estimates of $g_{L^2}$ close to $D$ are obtained which imply that $M_{(1,1)}(S^2)$ has finite volume and is geodesically incomplete. On $\Sigma=\mathbb{R}^2$, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for $g_{L^2}$ in the limits of small and large separation. All these results make use of a localization formula, expressing $g_{L^2}$ in terms of data at the (anti)vortex positions, which is established for general $M_{(k_+,k_-)}(\Sigma)$. For arbitrary compact $\Sigma$, a natural compactification of the space $M_{(k_+,k_-)}(\Sigma)$ is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for $Vol(M_{(1,1)}(S^2))$, and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of $\Sigma$, and that the entropy of mixing is always positive.