The Geometry of the Space of BPS Vortex-Antivortex Pairs

The gauged sigma model with target 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+,M-k-) (Sigma) of dimension k(+) + k(-) which inherits a natural Kahler metric g(L2) governing the model's low energy dynamics. This paper presents the first detailed study of g(L2), focussing on the geometry close to the boundary divisor D = partial derivative M-(k+,M-k-) (Sigma). On Sigma = S-2, rigorous estimates of g(L2) close to D are obtained which imply that M-(1,M-1) (S-2) has finite volume and is geodesically incomplete. On Sigma = R-2, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for g(L2) in the limits of small and large separation. All these results make use of a localization formula, expressing g(L2) in terms of data at the (anti)vortex positions, which is established for general M-(k+,M-k-) (Sigma). For arbitrary compact Sigma, a natural compactification of the space M-(k+,M-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,M-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.