“Recursion relations for conformal blocks in four dimensions"

Most modern algorithms for computation of conformal blocks in numerical bootstrap applications are based on Zamolodchikov-like recursion relations. These relations come from the idea that conformal blocks have poles in the exchanged scaling dimension, associated to appearance of null states in the corresponding parabolic Verma module. In odd dimensions the pole is simple, the residue is another conformal block, and the recursion relation is well understood. However, in even dimensions double poles can appear, and the structure of the recursion relation is an open problem. In this talk, I will describe the surprisingly subtle solution of this problem in four dimensions. In particular, I will explain that the natural setting for this question is the principal block of the deformed parabolic BGG category O, which can be efficiently studied using Morita theory.  Based on work in progress with Colum Flynn.