Dimensional reduction of higher-point conformal blocks

TL;DR

This paper generalizes a known relation linking conformal blocks across dimensions to all higher-point scalar blocks, revealing a factorization property and graphical rules for their coefficients, with potential extension to spinning exchanges.

Contribution

It extends the dimensional reduction relation to all higher-point scalar conformal blocks and introduces graphical rules to determine the coefficients involved.

Findings

Coefficients obey a factorization property.

Graphical Feynman-like rules determine coefficients.

Method applicable to higher-point blocks with spinning exchanges.

Abstract

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d-2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.