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Feigin-Frenkel image of Witten-Kontsevich points. (English) Zbl 1470.81059

Summary: The Witten-Kontsevich KdV tau function of topological gravity has a generalization to an arbitrary Drinfeld-Sokolov hierarchy associated to a simple complex Lie algebra. Using the Feigin-Frenkel isomorphism we describe the affine opers describing such generalized Witten-Kontsevich functions in terms of Segal-Sugawara operators associated to the Langlands dual Lie algebra. In the case where the Lie algebra is simply laced there is a second role these Segal-Sugawara operators play: their action, in the basic representation of the affine algebra associated to the Lie algebra, singles out the Witten-Kontsevich tau function within the phase space. We show that these two Langlands dual roles of Segal-Sugawara operators correspond to a duality between the first and last operator for a complete set of Segal-Sugawara operators.

MSC:

81V17 Gravitational interaction in quantum theory
70G40 Topological and differential topological methods for problems in mechanics
81T70 Quantization in field theory; cohomological methods
81T45 Topological field theories in quantum mechanics
83C45 Quantization of the gravitational field
17B81 Applications of Lie (super)algebras to physics, etc.
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
22E57 Geometric Langlands program: representation-theoretic aspects
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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