Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 6

Families of Hitchin systems and $N=2$ theories

Pages: 1585 – 1667

DOI: https://dx.doi.org/10.4310/ATMP.2022.v26.n6.a2

Authors

Aswin Balasubramanian (Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey, U.S.A.)

Jacques Distler (Department of Physics, University of Texas, Austin, Tx.,U.S.A.)

Ron Donagi (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

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 $N$.

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AB’s work is supported by the US Department of Energy under the grant DE–SC0010008. JD’s work is supported by NSF grant PHY–1914679. JD would like to thank the Aspen Center for Physics (supported by NSF grant PHY–1607611) for hospitality while some of this work was conducted. During the preparation of this work, Ron Donagi was supported in part by NSF grant DMS 2001673 and by Simons HMS Collaboration grant # 390287.

Published 30 June 2023