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.
Highlights
It is useful to go beyond four-point blocks and study higher-point conformal blocks as they are one of the central components of a potential higher-point conformal bootstrap program [16], and they provide a canonical direct-channel basis in position-space for all higher-point tree-level Witten diagrams, which are useful for investigating higher-loop effects in AdS/CFT [17]
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
The close relation between the Parisi-Sourlas SCFTd and a dimensionally reduced CFTd−2 led to the discovery of a remarkable finite series relation between ordinary d-dimensional and (d − 2)-dimensional four-point conformal blocks [37]
Summary
We begin by reviewing the relevant dimensional reduction results for fourpoint conformal blocks. We will present the generalization to all higherpoint conformal blocks. We will prove this novel proposal with help of the recently proposed Feynman rules for constructing higher-point scalar conformal blocks in arbitrary topologies, and end with various consistency checks
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