The Geometry of the Space of BPS Vortex–Antivortex Pairs

The gauged sigma model with target P1, defined on a Riemann surface Σ, 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−)(Σ) of dimension k++k− which inherits a natural Kähler metric gL2 governing the model’s low energy dynamics. This paper presents the first detailed study of gL2, focussing on the geometry close to the boundary divisor D=∂M(k+,k−)(Σ). On Σ=S2, rigorous estimates of gL2 close to D are obtained which imply that M(1,1)(S2) has finite volume and is geodesically incomplete. On Σ=R2, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for gL2 in the limits of small and large separation. All these results make use of a localization formula, expressing gL2 in terms of data at the (anti)vortex positions, which is established for general M(k+,k−)(Σ). For arbitrary compact Σ, a natural compactification of the space M(k+,k−)(Σ) 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)(S2)), 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 Σ, and that the entropy of mixing is always positive.