Abstract
Seven-point functions have two inequivalent topologies or channels. The comb channel has been computed previously and here we compute scalar conformal blocks in the extended snowflake channel in $d$ dimensions. Our computation relies on the known action of the differential operator that sets up the operator product expansion in embedding space. The scalar conformal blocks in the extended snowflake channel are obtained as a power series expansion in the conformal cross-ratios whose coefficients are a triple sum of the hypergeometric type. This triple sum factorizes into a single sum and a double sum. The single sum can be seen as originating from the comb channel and is given in terms of a ${}_3F_2$-hypergeometric function, while the double sum originates from the snowflake channel which corresponds to a Kamp\'e de F\'eriet function. We verify that our results satisfy the symmetry properties of the extended snowflake topology. Moreover, we check that the behavior of the extended snowflake conformal blocks under several limits is consistent with known results. Finally, we conjecture rules leading to a partial construction of scalar $M$-point conformal blocks in arbitrary topologies.
Highlights
With their extended spacetime symmetry group, conformal field theories (CFTs) are possibly amenable to exact nonperturbative solutions
From the operator product expansion (OPE), which expresses the product of two quasiprimary operators in terms of an infinite sum of quasiprimary operators and their descendants, conformal correlation functions can be expanded in terms of the CFT data and conformal blocks
In (1.1), T0ðMÞ is the number of inequivalent M-point topologies, which corresponds to the number of unrooted binary trees with M unlabeled leaves3; TðMÞ is the number of different ways of expressing the same full M-point correlation functions, which is given by the number of unrooted binary trees with M labeled leaves; and SM is the symmetry group of the full M-point correlation functions, which is the symmetry group of M elements
Summary
With their extended spacetime symmetry group, conformal field theories (CFTs) are possibly amenable to exact nonperturbative solutions. In (1.1), T0ðMÞ is the number of inequivalent M-point topologies, which corresponds to the number of unrooted binary trees with M unlabeled leaves; TðMÞ is the number of different ways of expressing the same full M-point correlation functions, which is given by the number of unrooted binary trees with M labeled leaves; and SM is the symmetry group of the full M-point correlation functions, which is the symmetry group of M elements Another interesting feature of higher-point conformal blocks is the appearance of extra sums, denoted by the F-function [see (2.5)], in the conformal cross-ratio power series. We continue the analysis of higher-point correlation functions by computing scalar seven-point conformal blocks in all topologies, i.e., in the comb channel and the so-called extended snowflake channel. Appendix B gives the proofs for the identities of the extended snowflake conformal blocks under the symmetry generators of H . 7jsenxotwenfdlaekde Lastly, Appendix C contains the remaining proofs related to the OPE limit and the limit of unit operator
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