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The geometry of the space of BPS vortex-antivortex pairs. (English) Zbl 1452.30006

Summary: 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 \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\) 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 \(D=\partial \, {\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). On \(\Sigma =S^2\), rigorous estimates of \(g_{L^2}\) close to \(D\) are obtained which imply that \({\mathsf{M}}_{(1,1)}(S^2)\) 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 \({\mathsf{M}}_{(k_+,k_-)}(\Sigma)\). For arbitrary compact \(\Sigma \), a natural compactification of the space \({{\mathsf{M}}}_{(k_+,k_-)}(\Sigma)\) 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 \(\text{Vol}(\mathsf{M}_{(1,1)}(S^2))\), 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.

MSC:

30C10 Polynomials and rational functions of one complex variable
30F60 Teichmüller theory for Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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