Geometry of Spectral Curves and All Order Dispersive Integrable System
We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2012 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
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Інститут математики НАН України
2012
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/149186 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Geometry of Spectral Curves and All Order Dispersive Integrable System / G. Borot, B. Eynard // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 81 назв. — англ. |
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Borot, G. Eynard, B. 2019-02-19T18:22:25Z 2019-02-19T18:22:25Z 2012 Geometry of Spectral Curves and All Order Dispersive Integrable System / G. Borot, B. Eynard // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 81 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H70; 14H42; 30Fxx DOI: http://dx.doi.org/10.3842/SIGMA.2012.100 https://nasplib.isofts.kiev.ua/handle/123456789/149186 We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between ''correlators'', the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry. We thank O. Babelon, M. Berg`ere, M. Bertola, B. Dubrovin, D. Korotkin, M. Mulase, J.M. Munoz Porras, N. Orantin, F. Plaza Martin, E. Previato, A. Raimondo, B. Safnuk for fruitful discussions, T. Grava and S. Romano for enlightening discussions concerning dispersionless hierarchies, their dispersive deformations and the role of Whitham equations, and I. Krichever for careful reading, valuable discussions and for pointing out references. This work is partly supported by the ANR project Grandes Matrices Al´eatoires by the European Science Foundation through the Misgam program, by the Qu´ebec government with the FQRNT, by the Fonds Europ´een S16905 (UE7 - CONFRA), by the Swiss NSF (no 200021-43434) and the ERC AG CONFRA. B.E. thanks the CERN, and G.B. thanks the SISSA for their hospitality while this work was pursued. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Geometry of Spectral Curves and All Order Dispersive Integrable System Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Geometry of Spectral Curves and All Order Dispersive Integrable System |
| spellingShingle |
Geometry of Spectral Curves and All Order Dispersive Integrable System Borot, G. Eynard, B. |
| title_short |
Geometry of Spectral Curves and All Order Dispersive Integrable System |
| title_full |
Geometry of Spectral Curves and All Order Dispersive Integrable System |
| title_fullStr |
Geometry of Spectral Curves and All Order Dispersive Integrable System |
| title_full_unstemmed |
Geometry of Spectral Curves and All Order Dispersive Integrable System |
| title_sort |
geometry of spectral curves and all order dispersive integrable system |
| author |
Borot, G. Eynard, B. |
| author_facet |
Borot, G. Eynard, B. |
| publishDate |
2012 |
| language |
English |
| container_title |
Symmetry, Integrability and Geometry: Methods and Applications |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between ''correlators'', the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.
|
| issn |
1815-0659 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/149186 |
| citation_txt |
Geometry of Spectral Curves and All Order Dispersive Integrable System / G. Borot, B. Eynard // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 81 назв. — англ. |
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2025-12-07T21:07:28Z |
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2025-12-07T21:07:28Z |
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1850885169251090432 |