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A bstract Conformal blocks in any number of dimensions depend on two variables z, z. Here we study their restrictions to the special “diagonal” kinematics z=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.
Hogervorst et al. (2013) studied this question.