Six-point conformal blocks in the snowflake channel
- PDF / 812,406 Bytes
- 51 Pages / 595.276 x 841.89 pts (A4) Page_size
- 7 Downloads / 162 Views
Springer
Received: June Revised: September Accepted: October Published: November
28, 11, 12, 26,
2020 2020 2020 2020
Six-point conformal blocks in the snowflake channel
a
Département de Physique, de Génie Physique et d’Optique, Université Laval, 1045 avenue de la Médecine, Québec, 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 d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3 F2 -hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions. Keywords: Conformal Field Theory, Conformal and W Symmetry, Field Theories in Higher Dimensions ArXiv ePrint: 2004.02824
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)147
JHEP11(2020)147
Jean-François Fortin,a Wen-Jie Maa and Witold Skibab
Contents 1
2 Scalar six-point conformal blocks 2.1 M -point correlation functions from the OPE 2.2 Scalar M -point correlation functions 2.3 Scalar M -point correlation functions in the comb channel 2.4 Scalar six-point correlation functions in the snowflake channel
3 4 5 6 7
3 Sanity checks 3.1 Symmetry properties 3.2 OPE limit 3.3 Limit of unit operator
10 10 14 16
4 Discussion and conclusion
16
A Scalar five-point conformal blocks and the OPE A.1 Proof of the equivalence
22 22
B Snowflake and the OPE B.1 Proof of the snowflake B.2 An alternative form
27 27 32
C Symmetry properties C.1 Rotations of the triangle C.2 Reflections of the triangle C.3 Permutations of the dendrites
36 36 38 45
1
Introduction
The study of higher-point conformal blocks in conformal field theory (CFT) is a complicated subject without many explicit results. In a CFT, correlation functions, which are the natural observables of the theory, are given in terms of the CFT data and the conformal blocks. The CFT data, which consist of the spec
Data Loading...
