OKC/Nordita High-Energy Physics Theory seminar

Integrable system constructed from the geometry of a spectral curve

by Bertrand Eynard (IPHT Saclay)

132:028 ()


One usual way of defining an integrable system is in terms of a Tau-function obeying Hirota equations. The Tau-function (example KdV) is usually defined as a function of an infinite set of times $t=(t_0,t_1,t_2,t_3,...)$. Here instead we shall define Tau as a function on the moduli space of spectral curves (plane analytic curves with extra structure), and the "times" can be viewed as local coordinates (but not global in general). The tangent space (i.e. the span of all $\partial/\partial t_k$, i.e. Hamiltonians) to the moduli space of spectral curves, is isomorphic to the space of meromorphic 1-forms on the curve, and by form-cycle duality is isomorphic to a Lagrangian in the space of cycles. In other words, we reinterpret Hamiltonians as cycles, and the symplectic Poisson structure as the intersection of cycles. The topological recursion (TR) defines invariants of the spectral curve, and we show how to get a Tau-function from the TR-invariants. This is an efficient method, which gives new insights on integrable systems.