Tau-functions on Hurwitz Spaces

Abstract

We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P1. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ‘Wirtinger tau-function’). In higher genus we compute the modulus square of the Wirtinger tau-function. In particular one gets formulas for the isomonodromic tau-functions of semisimple Frobenius manifolds connected with the Hurwitz spaces Hg,N(1,⋯,1).