This paper is devoted to families of Hitchin systems and \(\mathcal{N}=2\) theories. The authors 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, they 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. The authors 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 \(\overline{\mathcal{M}}_{0,4}\), they compute this vector bundle explicitly. Finally, they give a classification of the allowed pairs \((O,H)\) that can arise for any given \(N\).
This paper is organized as follows: Section 1 is an introduction to the subject. In Section 2, the authors review properties of the tame Hitchin system on a smooth underlying curve. In Section 3, they build a global model for the Hitchin system over \(\overline{\mathcal{M}}_{0,4}\). In particular, they note that the Hitchin bases fit together to form a nontrivial vector bundle over \(\overline{\mathcal{M}}_{0,4}\) and compute that bundle explicitly.
In Section 4, the authors use this model to take a first look at the Hitchin system on nodal curves. They also define what it means for a node to be standard or restricted. The restricted nodes are labeled by a pair, \((O, H)\), where \(O\) is a nilpotent orbit in \(sl(N)\) and \(H\) is an \(SU(l)\) or \(Sp(l)\) subgroup of \(SU(N)\). They emphasize that the pair \((O, H)\) is not a complete invariant of the singular spectral curve which covers the node. Two examples (examples 3 and 4 of this section) have the same \((O, H)=([2], SU(2))\), but the singularity structure of the spectral curve is different.
In Section 5, the authors take the results of Section 4 as motivation to build a general framework for the Hitchin system on a nodal curve such that the family of Hitchin systems on a family of smooth curves is flat in the limit as the smooth curve degenerates to the nodal one. Section 6 is devoted to flavour considerations and the Higgs branch. The possible restricted nodes are strongly constrained by physics considerations arising from the role of the flavour symmetry, as shown in this section. In Section 7, the authors provide a classification of the allowed nodal degenerations using the methods of Section 5. They also show that the results of Section 7 are compatible with those in Section 6. The paper is supported by three appendices.