Families of Hitchin systems and N=2 theories

TL;DR

This paper explores the global behavior of families of Hitchin integrable systems related to 4d N=2 theories, focusing on degenerations, base space structures, and classifications of nilpotent orbits and symmetry groups.

Contribution

It introduces a detailed analysis of Hitchin systems over moduli spaces, including degenerations to nodal curves, vector bundle structures of Hitchin bases, and classifications of orbit-group pairs.

Findings

Hitchin systems degenerate to nodal curves with behavior encoded by nilpotent orbits and symmetry groups.

Hitchin bases form a vector bundle over the compactified moduli space.

Explicit computation of the vector bundle over ,4 moduli space.

Abstract

Motivated by the connection to 4d $\mathcal{N}=2$ theories, we study the global behavior of families of tamely-ramified $SL_N$ Hitchin integrable systems as the underlying curve varies over the Deligne-Mumford moduli space of stable pointed curves. In particular, we describe a flat degeneration of the Hitchin system to a nodal base curve and show that the behaviour of the integrable system at the node is partially encoded in a pair $(O,H)$ where $O$ is a nilpotent orbit and $H$ is a simple Lie subgroup of $F_{O}$, the flavour symmetry group associated to $O$. The family of Hitchin systems is nontrivially-fibered over the Deligne-Mumford moduli space. We prove a non-obvious result that the Hitchin bases fit together to form a vector bundle over the compactified moduli space. For the particular case of $\overline{\mathcal{M}}_{0,4}$, we compute this vector bundle explicitly. Finally, we give a classification of the allowed pairs $(O,H)$ that can arise for any given $N$.