2D Chiral Algebras from Koszul Duality & Associativity

Abstract

In this thesis, we will explain how the techniques of Koszul duality and associativity can be used to completely determine certain celestial chiral algebras, characterizing collinear splitting functions not just at tree-level but to arbitrary loop order. To apply this technique, and indeed for the celestial chiral algebra to exist, the corresponding 4d massless theory must admit a local gauge-anomaly-free lift to twistor space. We will also use Koszul duality in a different context, the AdS/CFT correspondence, to determine the planar chiral algebra for a certain supersymmetric CFT. This provides a new and analytic method to completely determine the 2 and 3-point functions of 1/4-BPS operators in the duality. This thesis is based upon collaborative work and forthcoming work, with the coauthors listed explicitly at the beginning of each chapter.