The Hitchin Image in Type-D

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 OH\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}_H$$\end{document}. We describe two important features that distinguish it from the type-A case studied in Balasubramanian et al. (Adv Theor Math Phys Ser 26(6):1585-1667, 2022. https://doi.org/10.4310/ATMP.2022.v26.n6.a2, arXiv:2008.01020 [hep-th]). The first feature, which we term even-type constraints, arises iff the partition label [OH]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\mathcal {O}_H]$$\end{document} has even parts. 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 A<overline>b(OH)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{A}_b(\mathcal {O}_H)$$\end{document} being non-trivial. This finite group parametrizes the choices for the local Hitchin base. Additionally, we also show that the finite group A<overline>b(OH)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{A}_b(\mathcal {O}_H)$$\end{document} encodes the size of the dual special piece.