Français
Integrable systems and spectral curves
Multi angle
Authors : Eynard, Bertrand (Author of the conference)
CIRM (Publisher )
37K20 - Relations with algebraic geometry, complex analysis, special functions
60B20 - Random matrices (probabilistic aspects)
CIRM (Publisher )
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Abstract :
Usually one defines a Tau function Tau(t_1,t_2,...) as a function of a family of times having to obey some equations, like Miwa-Jimbo equations, or Hirota equations.
Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.
MSC Codes : Here we shall view times as local coordinates in the moduli-space of spectral curves, and define the Tau-function of a spectral curve Tau(S), in an intrinsic way, independent of a choice of coordinates. Deformations are tangent vectors, and the tangent space is isomorphic to the space of cycles (cf Goldman bracket), so that Hamiltonians can be represented by cycles.
All the integrable system formalism can then be represented geometrically in the space of cycles: the Poisson bracket is the intersection, the conserved quantities are periods, Miwa-Jimbo equations and Seiberg-Witten equations are a mere consequence of the definition, Hirota equation is a vanishing monodromy condition, and Virasoro-W constraint are automatically satisfied by our definition, showing that our Tau-function is also a conformal block. Our definition contains KdV, KP multicomponent KP, Hitchin systems, and probably all known classical integrable systems.
37K20 - Relations with algebraic geometry, complex analysis, special functions
60B20 - Random matrices (probabilistic aspects)
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Information on the Video
Film maker : Recanzone, LucaLanguage : English Available date : 09/05/2019 Conference Date : 09/04/2019 Subseries : Research talks arXiv category : Mathematical Physics ; Analysis of PDEs Mathematical Area(s) : Mathematical Physics ; Probability & Statistics ; Analysis and its Applications Format : MP4 (.mp4) - HD Video Time : 00:53:10 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2019-04-09_Eynard.mp4 
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Information on the Event
Event Title : Chaire Jean-Morlet : Equation intégrable aux données initiales aléatoires / Jean-Morlet Chair : Integrable Equation with Random Initial DataEvent Organizers : Basor, Estelle ; Bufetov, Alexander ; Cafasso, Mattia ; Grava, Tamara ; McLaughlin, Ken Dates : 08/04/2019 - 12/04/2019 Event Year : 2019 Event URL : https://www.chairejeanmorlet.com/2104.html
Citation Data
DOI : 10.24350/CIRM.V.19515203Cite this video as: Eynard, Bertrand (2019). Integrable systems and spectral curves. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19515203 URI : http://dx.doi.org/10.24350/CIRM.V.19515203 |
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