W-algebras: between representation theory, 2d geometry and 5d gauge theories

Prof. Dr. Gaetan Borot

Date(s) - 26/02/2021
13:30 - 14:30

video conference (zoom)

W-algebras associated to simple Lie algebras are non-linear generalisations of the Virasoro algebra, from which certain CFTs can be defined. Using a formalism of Kontsevich and Soibelman and representation theory of W-algebras via free fields (when this exist), one can construct from them systems of PDEs whose solution is computed by a “topological recursion”. Such solutions carry information about 2d (worldsheet) geometry, and “topological recursion” mean this information is reached by induction on the Euler characteristic that resembles a cutting/pasting approach. They are sometimes expressible as integrals on the moduli space of complex curves or as topological string A-model amplitudes (for certain target spaces), and gives suitable definition of topological string B-model for certain type of complex 1d singularities. I will give an overview of these connections, and describe if time permits sketch a more recent link (ongoing work) to Nekrasov partition functions in 5d gauge theories (in which W-algebra are known to appear since Alday-Gaiotto-Tachikawa proposal). This is based on joint works with Andersen, Chekhov, Orantin, Bouchard, Chidambaram, Creutzig, Noshchenko, Kramer, Schüler.