The Hitchin Image in Type-D

TL;DR

This paper investigates the structure of the Hitchin map image in type-D tame Hitchin systems, revealing new even and odd type constraints that influence the geometry and symmetry properties of the Hitchin base.

Contribution

It introduces the concepts of even and odd type constraints in the Hitchin image for type-D systems, extending understanding beyond the type A case and connecting to integrable systems and finite group symmetries.

Findings

Hitchin image is non-singular when even parts are present in the nilpotent orbit.

The Hitchin image always globalizes to the Hitchin base of an integrable system.

Finite group $ar{A}_b( ext{O}_H)$ encodes the size of the dual special piece.

Abstract

Motivated by their appearance as Coulomb branch geometries of Class S theories, we study the image of the local Hitchin map in tame Hitchin systems of type-D with residue in a special nilpotent orbit $\mathcal{O}_H$. We describe two important features which distinguish it from the type A case studied in arXiv:2008.01020. The first feature, which we term even type constraints, arise iff the partition label $[\mathcal{O}_H]$ has even parts. In this case, our Hitchin image is non-singular and thus different from the one studied by Baraglia and Kamgarpour. We argue that our Hitchin image always globalizes to being the Hitchin base of an integrable system. The second feature, which we term odd type constraints, is related to a particular finite group $\overline{A}_b(\mathcal{O}_H)$ being non-trivial. When this finite group is non-trivial, we have $\mid \overline{A}_b \mid$ choices for the local Hitchin base. Additionally, we also show that the finite group $\overline{A}_b(\mathcal{O}_H)$ encodes the size of the dual special piece.