# Tau functions, Hodge classes and discriminant loci on moduli spaces of Hitchin's spectral covers

## TL;DR

The paper introduces two tau functions on moduli spaces of spectral covers in Hitchin systems, linking them to divisor classes and zeros of canonical forms, advancing the understanding of geometric structures in integrable systems.

## Contribution

It defines new tau functions on Hitchin spectral cover moduli spaces and relates them to divisor classes and zeros of canonical forms, providing tools for geometric analysis.

## Key findings

- Expressed the divisor class of the Hitchin discriminant using tau functions.
- Computed the divisor of canonical 1-forms with multiple zeros.
- Linked tau functions to standard generators of the Picard group.

## Abstract

We define two tau functions, $\tau$ and $\hat{\tau}$ , on moduli spaces of spectral covers of $GL(n)$ Hitchin's systems. Analyzing the properties of $\tau$, we express the divisor class of the universal Hitchin's discriminant in terms of standard generators of the rational Picard group of the moduli spaces of spectral covers with variable base. The function $\hat{\tau}$ is used to compute the divisor of canonical 1-forms with multiple zeros.