Dimensional Reduction for Conformal Blocks

TL;DR

This paper derives a formula for expressing d-dimensional conformal blocks in terms of (d-1)-dimensional blocks, providing a new method for dimensional reduction in conformal field theories.

Contribution

It introduces a novel expansion of d-dimensional conformal blocks into (d-1)-dimensional blocks, including a closed-form formula for 3d blocks as an infinite sum.

Findings

Derived a formula for 3d conformal blocks as an infinite sum of hypergeometric functions.

Established a systematic dimensional reduction from SO(d+1,1) to SO(d,1).

Provided explicit coefficients for the expansion.

Abstract

We consider the dimensional reduction of a CFT, breaking multiplets of the d-dimensional conformal group SO(d+1,1) up into multiplets of SO(d,1). This leads to an expansion of d-dimensional conformal blocks in terms of blocks in d-1 dimensions. In particular, we obtain a formula for 3d conformal blocks as an infinite sum over 2F1 hypergeometric functions with closed-form coefficients.