Romão, NM and Speight, JM orcid.org/0000-0002-6844-9539 (2020) The Geometry of the Space of BPS Vortex–Antivortex Pairs. Communications in Mathematical Physics, 379 (2). pp. 723-772. ISSN 0010-3616
Abstract
The gauged sigma model with target P1, 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 gL2 governing the model’s low energy dynamics. This paper presents the first detailed study of gL2, focussing on the geometry close to the boundary divisor D=∂M(k+,k−)(Σ). On Σ=S2, rigorous estimates of gL2 close to D are obtained which imply that M(1,1)(S2) has finite volume and is geodesically incomplete. On Σ=R2, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for gL2 in the limits of small and large separation. All these results make use of a localization formula, expressing gL2 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)(S2)), 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.
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Item Type: | Article |
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Copyright, Publisher and Additional Information: | This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | math.DG; math.DG; hep-th; math-ph; math.AP; math.MP; 53C80, 70S15, 35Q70 |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number EPSRC (Engineering and Physical Sciences Research Council) EP/P024688/1 |
Depositing User: | Symplectic Publications |
Date Deposited: | 11 Nov 2020 15:50 |
Last Modified: | 22 Nov 2023 09:31 |
Status: | Published |
Publisher: | Springer Nature |
Identification Number: | 10.1007/s00220-020-03824-y |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:133852 |
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