Monodromy of the $SL(n)$ and $GL(n)$ Hitchin fibrations

TL;DR

This paper computes the monodromy groups of Hitchin fibrations for $SL(n)$ and $GL(n)$ Higgs bundles on Riemann surfaces, explicitly describing their structure using vanishing cycles and lattices.

Contribution

It explicitly constructs vanishing cycles and classifies the monodromy groups as skew-symmetric vanishing lattices, providing a complete description for any rank n.

Findings

Monodromy groups generated by Picard-Lefschetz transformations.

Complete classification of monodromy groups using vanishing lattice theory.

Determination of the image of cohomology restriction maps.

Abstract

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.