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Ron Donagi:On Integrable Systems and 3D Mirror Symmetry(Deligne-Simpson Problem)
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https://www.youtube.com/watch?v=hVaWkTPO1kw Simons Collaboration on Homological Mirror Symmetry We explore several recent developments involving Hitchin’s system. These include connections to SCFTs of Class S, extensions to families of integrable systems over the Deligne-Mumford moduli of stable curves, geometry of the base, and connections to the Deligne-Simpson problem. Ron Donagi——“On integrable Systems and 3D Mirror Symmetry”:https://people.math.harvard.edu/~auroux/schms2020notes/201204-Donagi.pdf Ron Donagi——“Families of Hitchin systems and SCFTs of class S”:https://people.math.harvard.edu/~auroux/schms2015-2020/Miami2020/ Aswin Balasubramanian, Jacques Distler, Ron Donagi——“Families of Hitchin system and N=2 theories”:https://dx.doi.org/10.4310/ATMP.2022.v26.n6.a2 Motivated by the connection to 4d 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 FO, 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 $M_{0,4}$, we compute this vector bundle explicitly. Finally, we give a classification of the allowed pairs (O,H) that can arise for any given N. (This is joint work with Aswin Balasubramanian and Jacques Distler) Aswin Balasubramanian, Jacques Distler, Ron Donagi, Carlos Perez-Pardavila——“The Hitchin Image in Type-D”:https://doi.org/10.48550/arXiv.2310.05880
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