The Geometry of the Space of BPS Vortex–Antivortex Pairs

Abstract

AbstractThe gauged sigma model with target $${\mathbb {P}}^1$$ P 1 , defined on a Riemann surface $$\Sigma $$ Σ , supports static solutions in which $$k_{+}$$ k + vortices coexist in stable equilibrium with $$k_{-}$$ k - antivortices. Their moduli space is a noncompact complex manifold $${\textsf {M}}_{(k_{+},k_{-})}(\Sigma )$$ M ( k + , k - ) ( Σ ) of dimension $$k_{+}+k_{-}$$ k + + k - which inherits a natural Kähler metric $$g_{L^2}$$ g L 2 governing the model’s low energy dynamics. This paper presents the first detailed study of $$g_{L^2}$$ g L 2 , focussing on the geometry close to the boundary divisor $$D=\partial \, {\textsf {M}}_{(k_{+},k_{-})}(\Sigma )$$ D = ∂ M ( k + , k - ) ( Σ ) . On $$\Sigma =S^2$$ Σ = S 2 , rigorous estimates of $$g_{L^2}$$ g L 2 close to D are obtained which imply that $${\textsf {M}}_{(1,1)}(S^2)$$ M ( 1 , 1 ) ( S 2 ) has finite volume and is geodesically incomplete. On $$\Sigma ={\mathbb {R}}^2$$ Σ = R 2 , careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for $$g_{L^2}$$ g L 2 in the limits of small and large separation. All these results make use of a localization formula, expressing $$g_{L^2}$$ g L 2 in terms of data at the (anti)vortex positions, which is established for general $${\textsf {M}}_{(k_{+},k_{-})}(\Sigma )$$ M ( k + , k - ) ( Σ ) . For arbitrary compact $$\Sigma $$ Σ , a natural compactification of the space $${{\textsf {M}}}_{(k_{+},k_{-})}(\Sigma )$$ 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 $$\mathrm{Vol}(\mathsf{M}_{(1,1)}(S^2))$$ 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 $$\Sigma $$ Σ , and that the entropy of mixing is always positive.