The Geometry of the Space of BPS Vortex–Antivortex Pairs

The gauged sigma model with target P 1 , 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 Käh...

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Bibliographic Details
Published in:Communications in mathematical physics 2020-10, Vol.379 (2), p.723-772
Main Authors: Romão, N. M., Speight, J. M.
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Equations of state
Gas mixtures
Mathematical and Computational Physics
Mathematical Physics
Numerical analysis
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Riemann surfaces
Theoretical
Vortices
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Summary:The gauged sigma model with target P 1 , 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 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 = ∂ M ( k + , k - ) ( Σ ) . On Σ = S 2 , rigorous estimates of g L 2 close to D are obtained which imply that M ( 1 , 1 ) ( S 2 ) has finite volume and is geodesically incomplete. On Σ = 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 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 ) ( 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 Σ , and that the entropy of mixing is always positive.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-020-03824-y