Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

TL;DR

This paper constructs explicit non-abelian Cauchy kernels and flat connections on the moduli space of vector bundles over Riemann surfaces, linking Higgs fields, monodromy, and the Goldman symplectic structure.

Contribution

It introduces an explicit parametrization of the moduli space using Tyurin data and constructs a non-abelian Cauchy kernel, connecting Higgs bundles with character varieties.

Findings

The Goldman symplectic form pulls back to the canonical symplectic form on the cotangent bundle.

The Liouville form's pull-back is a logarithmic form with poles along the non-abelian theta divisor.

Explicit non-abelian Cauchy kernels are constructed in terms of Tyurin data.

Abstract

We consider the moduli space of vector bundles of rank $n$ and degree $ng$ over a fixed Riemann surface of genus $g\geq 2$. We use the explicit parametrization in terms of the Tyurin data. In the moduli space there is a "non-abelian" Theta divisor, consisting of bundles with $h^1\geq 1$. On the complement of this divisor we construct a non-abelian Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which is also dependent holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the ${\rm GL}_n$ character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of affine connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by $h^1$.