A geometric point of view on integrable systems and tau functions

Bertrand Eynard

IPHT, CRM Montreal.

Mon, Oct. 08th 2018, 11:00-12:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

A Tau-function is usually viewed as a function of times $\tau(\vec t)$ where $\vec t=(t_0,t_1,t_2,t_3,...)$ is typically an infinite dimensional vector, and which satisfies some equations, typically Hirota equations or Miwa-Jimbo equations, but also many other equations like Seiberg-Witten or Virasoro. Miwa-Jimbo equations reflect the fact that some Hamiltonian flows commute. We propose to view the ``times'' as coordinates in a moduli space of ``spectral curves'', and reformulate the Tau function and the equations it satifies, in an intrinsic geometric and universal way in terms of spectral curves. A infinitesimal deformation is a tangent vector in the space of spectral curves, it can be viewed as a differential form, and by form-cycle suality, it can be viewed as a cycle drawn on the spectral curve. Identifying the tangent space with the space of cycles we can reinterpret Hamiltonians as cycles, and rewrite Hirota equations and all others, in terms of cycles, where they become very intuitive and simple. In a first step, we do this geometric identification in a perturbative expansion scheme, in some sense WKB-like, and we use the topological recursion. A next step (not in this talk) is a non-perturbative formulation.

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