Luu, Martin T. Feigin-Frenkel image of Witten-Kontsevich points. (English) Zbl 1470.81059 Int. Math. Res. Not. 2021, No. 7, 5520-5541 (2021). 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. Cited in 1 Document 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 Keywords:Segal-Sugawara operator PDFBibTeX XMLCite \textit{M. T. Luu}, Int. Math. Res. Not. 2021, No. 7, 5520--5541 (2021; Zbl 1470.81059) Full Text: DOI