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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Бібліографічні деталі
Дата:2012
Автори: Borot, G., Eynard, B.
Формат: Стаття
Мова:English
Опубліковано: Інститут математики НАН України 2012
Назва видання:Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:http://dspace.nbuv.gov.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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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1491862019-02-20T01:23:34Z Geometry of Spectral Curves and All Order Dispersive Integrable System Borot, G. Eynard, B. 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. 2012 Article 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 http://dspace.nbuv.gov.ua/handle/123456789/149186 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
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.
format Article
author Borot, G.
Eynard, B.
spellingShingle Borot, G.
Eynard, B.
Geometry of Spectral Curves and All Order Dispersive Integrable System
Symmetry, Integrability and Geometry: Methods and Applications
author_facet Borot, G.
Eynard, B.
author_sort Borot, G.
title Geometry of Spectral Curves and All Order Dispersive Integrable System
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
publisher Інститут математики НАН України
publishDate 2012
url http://dspace.nbuv.gov.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.
series Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv AT borotg geometryofspectralcurvesandallorderdispersiveintegrablesystem
AT eynardb geometryofspectralcurvesandallorderdispersiveintegrablesystem
first_indexed 2023-05-20T17:31:26Z
last_indexed 2023-05-20T17:31:26Z
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