Higher-point conformal blocks in the comb channel
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Springer
Received: April 2, 2020 Accepted: June 27, 2020 Published: July 29, 2020
Jean-Fran¸cois Fortin,a Wen-Jie Maa and Witold Skibab a
D´epartement de Physique, de G´enie Physique et d’Optique, Universit´e Laval, 1045 avenue de la M´edecine, Qu´ebec, QC G1V 0A6, Canada b Department of Physics, Yale University, 217 Prospect Street, New Haven, CT 06520, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: We compute M -point conformal blocks with scalar external and exchange operators in the so-called comb configuration for any M in any dimension d. Our computation involves repeated use of the operator product expansion to increase the number of external fields. We check our results in several limits and compare with the expressions available in the literature when M = 5 for any d, and also when M is arbitrary while d = 1. Keywords: Conformal Field Theory, Conformal and W Symmetry ArXiv ePrint: 1911.11046
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)213
JHEP07(2020)213
Higher-point conformal blocks in the comb channel
Contents 1 Introduction
1
2 Higher-point conformal blocks 2.1 M -point correlation functions from the OPE 2.2 Scalar M -point correlation functions in the comb channel 2.3 OPE limits
3 3 4 6 6 7 8
4 Sanity checks 4.1 Limit of unit operator 4.2 Scalar five-point conformal blocks in comb channel 4.3 Limit d → 1
9 9 11 12
5 Discussion and conclusion
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A Notation and the differential operator
14
B Sketch of the recurrence relation (3.9)
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C Proof of (3.1) from the recurrence relation (3.9)
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D Sketch of the proof of the limit of unit operator D.1 Proof of the limit of unit operator for five-point functions
19 19
1
Introduction
Conformal blocks are essential ingredients for calculations of observables, that is correlation functions, in conformal field theories (CFTs). A CFT is completely specified by its spectrum of primary operators and its operator product expansion (OPE) coefficients. This set of numerical data is often referred to as the CFT data. Two- and three-point functions are given pretty much directly in terms of the CFT data. However, correlation functions with more than three points depend on the invariant cross-ratios and such dependence is encoded by the conformal blocks. The higher-point functions are constructed from the CFT data and appropriate conformal blocks. Even though conformal blocks are prescribed by the conformal symmetry, computing blocks is far from straightforward. Various methods for obtaining the blocks have been
–1–
JHEP07(2020)213
(d,h;p)
3 Function FM (m) 3.1 OPE differential operator 3.2 Recurrence relation
–2–
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developed over the years. These include solving the Casimir equations [1–3], the shadow formalism [4–6], weight-shifting [7, 8], integrability [9–13], utilizing the AdS/CFT correspondence [14–19], and using the OPE [20–34]. Various additional results for conformal blocks can be found in [35–58] and [59–64]. The
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