The Geometry of the Space of BPS Vortex\u2013Antivortex Pairs

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 {textsf {M}}_{(k_{+},k_{-})}(Sigma ) of dimension k_{+}+k_{-} which inherits a natural Kähler 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 , {textsf {M}}_{(k_{+},k_{-})}(Sigma ). On Sigma =S^2, rigorous estimates of g_{L^2} close to D are obtained which imply that {textsf {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 {textsf {M}}_{(k_{+},k_{-})}(Sigma ). For arbitrary compact Sigma , a natural compactification of the space {{textsf {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 mathrm{Vol}(mathsf{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.