Diagonal Limit for Conformal Blocks in d Dimensions

TL;DR

This paper derives differential equations for conformal blocks restricted to diagonal kinematics in any dimension, enabling efficient computation and providing closed-form solutions for special cases.

Contribution

It establishes that diagonal conformal blocks satisfy specific ODEs, leading to a new numerical evaluation method and explicit solutions for equal external dimensions.

Findings

Diagonal conformal blocks satisfy third- and fourth-order ODEs.

The ODEs uniquely determine the blocks and facilitate efficient computation.

Closed-form solutions are obtained for equal external operator dimensions.

Abstract

Conformal blocks in any number of dimensions depend on two variables z, zbar. Here we study their restrictions to the special "diagonal" kinematics z = zbar, 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 3F2 functions.