Casimir recursion relations for general conformal blocks

TL;DR

This paper derives recursion relations for general spinning conformal blocks using special functions related to Spin(d) representations, enabling efficient computation of these blocks in conformal field theory.

Contribution

It introduces a new recursion relation for spinning conformal blocks based on 6j symbols, generalizing previous methods and simplifying calculations.

Findings

Recursion relations expressed via 6j symbols of Spin(d-1)

Closed-form recursion for d=3,4 and seed blocks

Potential for efficient numerical computation of conformal blocks

Abstract

We study the structure of series expansions of general spinning conformal blocks. We find that the terms in these expansions are naturally expressed by means of special functions related to matrix elements of Spin(d) representations in Gelfand-Tsetlin basis, of which the Gegenbauer polynomials are a special case. We study the properties of these functions and explain how they can be computed in practice. We show how the Casimir equation in Dolan-Osborn coordinates leads to a simple one-step recursion relation for the coefficients of the series expansion of general spinning conformal block. The form of this recursion relation is determined by 6j symbols of Spin(d-1). In particular, it can be written down in closed form in d=3, d=4, for seed blocks in general dimensions, or in any other situation when the required 6j symbols can be computed. We work out several explicit examples and briefly discuss how our recursion relation can be used for efficient numerical computation of general conformal blocks.