Diagonal limit for conformal blocks in d dimensions

Abstract

A bstract Conformal blocks in any number of dimensions depend on two variables z , $ \overline{z} $ . Here we study their restrictions to the special “diagonal” kinematics $ z=\overline{z} $ , previously found useful as a starting point for the conformal bootstrap analysis. We show that conformal blocks on the diagonal satisfy ordinary differential equations, third-order for spin zero and fourth-order for the general case. These ODEs determine the blocks uniquely and lead to an efficient numerical evaluation algorithm. For equal external operator dimensions, we find closed-form solutions in terms of finite sums of 3 F 2 functions.