String/Math Seminar @ 1:10pm: Nikita Nekrasov (Simons Center) “Beneath the surface: Knizhnik-Zamolodchikov equation in four dimensions”

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Knizhnik-Zamolodchikov (KZ) equation was derived in late 1980s from Ward identities of two dimensional conformal field theory with a current algebra symmetry. It is obeyed by the WZNW conformal blocks but remarkably admits analytic continuation in spins, magnetic quantum numbers and the level. In this talk I will review the work of recent years in which this extension meets the relevant physics. I will derive equations obeyed by the expectation values of the surface defect(s) in four dimensional supersymmetric gauge theory, and demonstrate its isomorphism to the KZ equation for 4- (joint work with O.Tsymbaliuk) and 5-point (joint work with S.Jeong and Norton Lee) conformal blocks in genus zero, with infinite dimensional representations associated to the vertices. We’ll find a few surprising twists not seen in the rational conformal field theory (based on the ). If time permits I will discuss the critical level limit and the results for the eigenfunctions of quantum Gaudin and XXX models (joint work with Norton Lee).