The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition

Abstract: In this series of lectures, we (re)view the "geometric method" that reconstructs, from a geometric object: the "spectral curve", an integrable system, and in particular its Tau function, Baker-Akhiezer functions and "current amplitudes", all having an interpretation as CFT conformal blocks. The construction identifies Hamiltonians with cycles on the curve, and times with periods (integrals of forms over cycles). All the integrable structure is formulated in terms of homology of contours, the phase space is a space of cycles where the symplectic form is the intersection, the Hirota operator is a degree 2 second-kind cycle, a Sato shift is an addition of a 3rd kind cycle, the Hirota equations amount to saying that merging 3rd kind cycles (monopoles) yields a 2nd kind cycle (dipole). The lecture is divided into 3 parts: 1) classical case, perturbative: the spectral curve is a ramified cover of a base Riemann surface -- with some additional structure -- and the integrable system is defined as a formal power series of a small "dispersion" parameter $\epsilon$. 2) dispersive classical case, non perturbative: the spectral curve is defined not as a ramified cover (which would be a bundle with discrete fiber), but as a vector bundle -- whose dispersionless limit consists in chosing a finite set of vectors in each fiber. 3) non-commutative case, and perturbative. The spectral curve is here a "non-commutative" surface, whose geometry will be defined in lecture III. 4) the full non-commutative dispersionless theory is under development is not presented in these lectures.