The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space
We construct an asymptotic metric on the moduli space of two-centred hyperbolic monopoles by working in the point particle approximation, that is, treating well-separated monopoles as point particles with an electric, magnetic, and scalar charge and re-interpreting the dynamics of the 2-particle sys...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211988 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space. Guido Franchetti and Calum Ross. SIGMA 19 (2023), 043, 15 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862701984422821888 |
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| author | Franchetti, Guido Ross, Calum |
| author_facet | Franchetti, Guido Ross, Calum |
| citation_txt | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space. Guido Franchetti and Calum Ross. SIGMA 19 (2023), 043, 15 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We construct an asymptotic metric on the moduli space of two-centred hyperbolic monopoles by working in the point particle approximation, that is, treating well-separated monopoles as point particles with an electric, magnetic, and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the ² metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case, it is not possible to factor out the centre of mass motion. Nevertheless, we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin.
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| first_indexed | 2026-03-18T15:14:22Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211988 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T15:14:22Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
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| spelling | Franchetti, Guido Ross, Calum 2026-01-20T16:14:48Z 2023 The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space. Guido Franchetti and Calum Ross. SIGMA 19 (2023), 043, 15 pages 1815-0659 2020 Mathematics Subject Classification: 70S15; 14D21 arXiv:2302.13792 https://nasplib.isofts.kiev.ua/handle/123456789/211988 https://doi.org/10.3842/SIGMA.2023.043 We construct an asymptotic metric on the moduli space of two-centred hyperbolic monopoles by working in the point particle approximation, that is, treating well-separated monopoles as point particles with an electric, magnetic, and scalar charge and re-interpreting the dynamics of the 2-particle system as geodesic motion with respect to some metric. The corresponding analysis in the Euclidean case famously yields the negative mass Taub-NUT metric, which asymptotically approximates the ² metric on the moduli space of two Euclidean monopoles, the Atiyah-Hitchin metric. An important difference with the Euclidean case is that, due to the absence of Galilean symmetry, in the hyperbolic case, it is not possible to factor out the centre of mass motion. Nevertheless, we show that we can consistently restrict to a 3-dimensional configuration space by considering antipodal configurations. In complete parallel with the Euclidean case, the metric that we obtain is then the hyperbolic analogue of negative mass Taub-NUT. We also show how the metric obtained is related to the asymptotic form of a hyperbolic analogue of the Atiyah-Hitchin metric constructed by Hitchin. GF thanks the Simons Foundation for its support under the Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics [grant number 488631]. CR thanks Michael Singer for useful discussions about the notion of centring for hyperbolic monopoles. The work of CR was supported by the Engineering and Physical Sciences Research Council [grant number EP/V047698/1]. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space Article published earlier |
| spellingShingle | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space Franchetti, Guido Ross, Calum |
| title | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space |
| title_full | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space |
| title_fullStr | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space |
| title_full_unstemmed | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space |
| title_short | The Asymptotic Structure of the Centred Hyperbolic 2-Monopole Moduli Space |
| title_sort | asymptotic structure of the centred hyperbolic 2-monopole moduli space |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211988 |
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