Prym–Tyurin classes and tau-functions

We study the space $M_{n,g}$ of holomorphic $n$-differentials over Riemann surfaces of genus $g$ for $n>1$. We introduce a set of $n$ vector bundles over this space, which we call Prym–Tyurin vector bundles. Corresponding determinant line bundles are called Prym–Tyurin line bundles. We define a set of $n$ tau-functions on the space $M_{n,g}$ and interpret them as holomorphic sections of tensor product of certain powers of Prym–Tyurin line bungles and tautological line bundle. This allows to express the first Chern classes of Prym–Tyurin line bundles (or Prym–Tyurin classes) via the boundary classes and the first Chern class of the tautological line bundle. This is joint work with P.Zograf.