A geometric picture of integrable systems
Bertrand Eynard
IPhT
Wed, Dec. 14th 2016, 11:00
Pièce 35, Bât. 774, Orme des Merisiers

\noindent ``Integrability meeting ENS/IPhT'' \\ \par Usually a Tau function is defined as a function of an infinite set of times $\tau (t_1, t_2, t_3, ...)$, and one writes either differential equations $\partial_{t_i} \tau = ...$ or difference equations $\tau (t+[x]) = ...$ Here we shall construct a Tau function starting from a spectral curve S. Locally deformations of spectral curves (i.e. tangent space of the space of spectral curves) are 1-forms. Moreover from form-cycle duality, a 1-form is dual to a cycle. In other words, the tangent space of the space of spectral curves is the space of cycles (homology of S). We argue that times are just local coordinates in the space of cycles. The idea is to reformulate integrable systems in the space of cycles in an intrinsic way, without ever chosing coordinates. Tau functions can thus be defined as functions on the space of cycles. We shall show how all usual integrability relations become very simple when written in terms of cycles. \\ \\ (Organizer: IPhT/Ivan Kostov)

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