Abstract
We revisit the so-called “Geodesic Witten Diagrams” (GWDs) [1], proposed to be the holographic dual configuration of scalar conformal partial waves, from the perspectives of CFT operator product expansions. To this end, we explicitly consider three point GWDs which are natural building blocks of all possible four point GWDs, discuss their gluing procedure through integration over spectral parameter, and this leads us to a direct identification with the integral representation of CFT conformal partial waves. As a main application of this general construction, we consider the holographic dual of the conformal partial waves for external primary operators with spins. Moreover, we consider the closely related “split representation” for the bulk to bulk spinning propagator, to demonstrate how ordinary scalar Witten diagram with arbitrary spin exchange, can be systematically decomposed into scalar GWDs. We also discuss how to generalize to spinning cases.
Article PDF
Similar content being viewed by others
References
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP 01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT(d)/AdS(d+1) correspondence, Nucl. Phys. B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys. B 562 (1999) 353 [hep-th/9903196] [INSPIRE].
C. Sleight and M. Taronna, Spinning Witten Diagrams, arXiv:1702.08619 [INSPIRE].
E. D’Hoker and D.Z. Freedman, Supersymmetric gauge theories and the AdS/CFT correspondence, hep-th/0201253 [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, arXiv:1601.05000.
D. Simmons-Duffin, The Conformal Bootstrap, arXiv:1602.07982 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
E. Joung and M. Taronna, Cubic interactions of massless higher spins in (A)dS: metric-like approach, Nucl. Phys. B 861 (2012) 145 [arXiv:1110.5918] [INSPIRE].
C. Sleight and M. Taronna, Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings, Phys. Rev. Lett. 116 (2016) 181602 [arXiv:1603.00022] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Towards holographic higher-spin interactions: Four-point functions and higher-spin exchange, JHEP 03 (2015) 170 [arXiv:1412.0016] [INSPIRE].
A. Castro, E. Llabrés and F. Rejon-Barrera, Geodesic Diagrams, Gravitational Interactions & OPE Structures, arXiv:1702.06128 [INSPIRE].
E. Dyer, D.Z. Freedman and J. Sully, Spinning Geodesic Witten Diagrams, arXiv:1702.06139 [INSPIRE].
M. Nishida and K. Tamaoka, Geodesic Witten diagrams with an external spinning field, arXiv:1609.04563 [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional Conformal Blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
H.-Y. Chen and J.D. Qualls, Quantum Integrable Systems from Conformal Blocks, arXiv:1605.05105 [INSPIRE].
X. Bekaert, J. Erdmenger, D. Ponomarev and C. Sleight, Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory, JHEP 11 (2015) 149 [arXiv:1508.04292] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].
C. Sleight, Interactions in Higher-Spin Gravity: a Holographic Perspective, arXiv:1610.01318 [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
J. Penedones, Writing CFT correlation functions as AdS scattering amplitudes, JHEP 03 (2011) 025 [arXiv:1011.1485] [INSPIRE].
V. Gonçalves, J. Penedones and E. Trevisani, Factorization of Mellin amplitudes, JHEP 10 (2015) 040 [arXiv:1410.4185] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1702.08818
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Chen, HY., Kuo, EJ. & Kyono, H. Anatomy of geodesic Witten diagrams. J. High Energ. Phys. 2017, 70 (2017). https://doi.org/10.1007/JHEP05(2017)070
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2017)070