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Multiplicative Hitchin systems and supersymmetric gauge theory

  • Chris ElliottEmail author
  • Vasily Pestun
Article
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Abstract

Multiplicative Hitchin systems are analogues of Hitchin’s integrable system based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is group-valued, rather than Lie algebra valued. We discuss the relationship between several occurences of these moduli spaces in geometry and supersymmetric gauge theory, with a particular focus on the case where \(C = \mathbb {CP}^1\) with a fixed framing at infinity. In this case we prove that the identification between multiplicative Higgs bundles and periodic monopoles proved by Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler spaces, and analyze the twistor rotation for the multiplicative Hitchin system. We also discuss quantization of these moduli spaces, yielding the modules for the Yangian \(Y(\mathfrak {g})\) discovered by Gerasimov, Kharchev, Lebedev and Oblezin.

Mathematics Subject Classification

14D21 53D30 

Notes

Acknowledgements

We would like to thank David Jordan, Davide Gaiotto, Dennis Gaitsgory, Nikita Nekrasov, Kevin Costello and especially Pavel Safronov for helpful conversations about this work. The calculation of twists of 5d and 6d supersymmetric gauge theories was performed independently by Dylan Butson, and we would like to thank him for sharing his forthcoming manuscript. We are also very grateful to Takuro Mochizuki for pointing out an error in an earlier version of the article. CE would like to thank the Perimeter Institute for Theoretical Physics for supporting research on this project. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. We acknowledge the support of IHÉS. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (QUASIFT Grant Agreement 677368).

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MassachusettsAmherstUSA
  2. 2.Institut des Hautes Études ScientifiquesBures-sur-YvetteFrance

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