Seminar

I will  remind of the origins of the notion of isomonodromic tau function and revisit the symplectic properties of the monodromy map for  Fuchsian systems on the Riemann sphere and elucidate the role of the isomonodromic tau-function in this context. The main goal is to show how the tau function should be interpreted as a generating function for a symplectomorphism between an extension of the Kirillov-Kostant Poisson structure on one side  and  a similar symplectic extension of the Goldman Poisson bracket on the other. The construction requires to express the symplectic form in terms of Fock—Goncharov coordinates, thus effectively inverting the matrix of the (extended) quiver in explicit form.  As a corollary we prove the recent  conjecture by A.Its, O.Lisovyy and A.Prokhorov in its "strong" version while  the original "weak" version is derived from previously known results. We show also that the isomonodromic Jimbo-Miwa tau-function can be interpreted as the  generating function of the monodromy symplectomorphism.  Time permitting I will discuss how this can be extended to higher genus.