Multipoint Conformal Blocks in the Comb Channel

TL;DR

This paper develops a method to compute multipoint conformal blocks in the comb channel for conformal field theories in various dimensions, expressing results in terms of hypergeometric functions and deriving explicit five-point blocks.

Contribution

It introduces a systematic approach to calculate n-point conformal blocks in the comb channel for arbitrary n in 1D and 2D, and explicitly derives the five-point block in general dimensions.

Findings

Explicit formulas for n-point conformal blocks in 1D and 2D.

Series, differential, and integral representations of hypergeometric functions.

Derived the five-point conformal block in arbitrary dimensions.

Abstract

Conformal blocks are the building blocks for correlation functions in conformal field theories. The four-point function is the most well-studied case. We consider conformal blocks for $n$-point correlation functions. For conformal field theories in dimensions $d=1$ and $d=2$, we use the shadow formalism to compute $n$-point conformal blocks, for arbitrary $n$, in a particular channel which we refer to as the comb channel. The result is expressed in terms of a multivariable hypergeometric function, for which we give series, differential, and integral representations. In general dimension $d$ we derive the $5$-point conformal block, for external and exchanged scalar operators.