ALBERT

Description: The 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 $${extsf {M}}_{(k_{+},k_{-})}(Sigma )$$ 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 $$D=partial , {extsf {M}}_{(k_{+},k_{-})}(Sigma )$$ 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 $${extsf {M}}_{(1,1)}(S^2)$$ 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 $${extsf {M}}_{(k_{+},k_{-})}(Sigma )$$ M ( k + , k - ) ( Σ ) . For arbitrary compact $$Sigma $$ Σ , a natural compactification of the space $${{extsf {M}}}_{(k_{+},k_{-})}(Sigma )$$ 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.