Abstract
We describe a geometric theory of Virasoro constraints in generalized Drinfeld–Sokolov hierarchies. Solutions of Drinfeld–Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of a Higgs field near infinity. String solutions for these hierarchies are defined as points having a big stabilizer under a certain Lie algebra action. We characterize principal bundles coming from string solutions as those possessing connections compatible with the Higgs field near infinity. We show that tau-functions of string solutions satisfy second-order differential equations generalizing the Virasoro constraints of 2d quantum gravity.
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Acknowledgments
First and foremost, the author would like to thank his advisor, David Ben-Zvi, without whose guidance and careful explanations this work would not be completed. The author also thanks Kevin Costello for useful discussions and the referee for pointing out a mistake in the definition of algebro-geometric solutions.
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Safronov, P. Virasoro constraints in Drinfeld–Sokolov hierarchies. Sel. Math. New Ser. 22, 27–54 (2016). https://doi.org/10.1007/s00029-015-0182-1
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DOI: https://doi.org/10.1007/s00029-015-0182-1