Recursion Relations for Conformal Blocks

TL;DR

The work develops a general framework for computing conformal blocks by analyzing their analytic structure as functions of the exchanged dimension $Δ$, identifying poles with null primary descendants via parabolic Verma modules. Residues at these poles factor into blocks with shifted quantum numbers, enabling recursion relations that quickly determine CBs in several cases, including external scalars, one external vector, and one external conserved current. A large-$Δ$ Casimir-based analysis provides the leading asymptotics and fixes initial data, while explicit residue formulas connect left and right OPE data to the recursion. The approach highlights a deep link between conformal representation theory and CB computation, yielding insights into odd vs even spacetime dimensions and laying groundwork for systematic generalizations and unitarity constraints in CFTs.

Abstract

In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension $Δ$ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current.

Figures (11)

• Figure 1: Radial coordinates of arXiv:1303.1111.
• Figure 2: The conformal representation ${\mathcal{H}}_{\mathcal{O}}$ becomes reducible for some special values $\Delta^\star_A$ of the conformal dimension of ${\mathcal{O}}$. The descendant state ${\mathcal{O}}_A$ becomes a primary and the representation ${\mathcal{H}}_{{\mathcal{O}}_A}$ only contains null states.
• Figure 3: The picture shows the traceless and symmetric part of ${\mathcal{H}}_{\mathcal{O}}$. Each sequence of arrows creates a descendants of ${\mathcal{O}}$. When the conformal dimension of ${\mathcal{O}}$ takes a value $\Delta^\star_{A}$, one descendant placed at a colored dot becomes a primary state. There are three types of primary descendants labeled by an integer $n$ that counts the dots from the left to the right.
• Figure 4: The root system of $\mathfrak{so}(5)$. The dashed roots belong to $\mathfrak{so}(3)\subset \mathfrak{so}(5)$.
• Figure 5: Integral weights of $\mathfrak{so}(5)$.