From the Tau Function of Painleve P6 Equation to Moduli Spaces

The isomonodromic tau-function of Hitchin's solution to the sixth Painleve equation can be naturally generalized for an arbitrary Riemann-Hilbert problem with quasi-permutation monodromies (the corresponding tau-function also appears in the theory of Frobenius structures on Hurwitz spaces). Such a tau-function can be interpreted as a holomorphic section of the Hodge line bundle on a Hurwitz space. Analysing the asymptotic behavior of the tau-function near the boundary of the Hurwitz spaces, we obtain new relations between divisor classes on these spaces. Similar results hold for the spaces of holomorphic 1-differentials on Riemann surfaces.