Casimir recursion relations for general conformal blocks

TL;DR

The paper develops a general, representation-theoretic framework for spinning conformal blocks by leveraging Spin$(d)$ GT bases, $P$-functions, and a one-step Casimir recursion controlled by $6j$ symbols of Spin$(d-1)$. This unifies the computation of spinning blocks across dimensions, yielding closed-form recursions for seed blocks in general $d$ and general blocks in $d=3,4$, with explicit structures for scalars and fermions. The key innovations are the GT-based organization of descendant data, the matrom formulation of $P$-functions, and the spin-coupled recursion that mirrors the scalar Hogervorst recursion but generalized to arbitrary spins. These results enable efficient numerical computation of spinning conformal blocks and establish a concrete algorithmic path for bootstrap studies, including potential extensions to superconformal theories. Overall, the work provides a principled, universal method to obtain spinning conformal blocks via a Casimir-driven recursion and group-theoretic building blocks, with practical implications for high-precision conformal bootstrap.

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.

Figures (2)

• Figure 1: Dynkin diagrams of $\mathfrak{so}(d)$ algebras.
• Figure 2: The relationship between Young diagrams of $\mathbf{m}_d$ and $\mathbf{m}_{d-1}$. The boxes which belong to $\mathbf{m}_{d-1}$ are shaded.