Abstract
We consider the Liouville theory in fixed Euclidean AdS2 background. Expanded near the minimum of the potential the elementary field has mass squared 2 and (assuming the standard Dirichlet b.c.) corresponds to a dimension 2 operator at the boundary. We provide strong evidence for the conjecture that the boundary correlators of the Liouville field are the same as the correlators of the holomorphic stress tensor (or the Virasoro generator with the same central charge) on a half-plane or a disc restricted to the boundary. This relation was first observed at the leading semiclassical order (tree-level Witten diagrams in AdS2) in [19] and here we demonstrate its validity also at the one-loop level. We also discuss arguments that may lead to its general proof.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. D’Hoker and R. Jackiw, Space translation breaking and compactification in the Liouville theory, Phys. Rev. Lett.50 (1983) 1719 [INSPIRE].
E. D’Hoker, D.Z. Freedman and R. Jackiw, SO(2, 1) invariant quantization of the Liouville theory, Phys. Rev.D 28 (1983) 2583 [INSPIRE].
T. Inami and H. Ooguri, Dynamical breakdown of supersymmetry in two-dimensional anti-de Sitter space, Nucl. Phys.B 273 (1986) 487 [INSPIRE].
C.G. Callan Jr. and F. Wilczek, Infrared behavior at negative curvature, Nucl. Phys.B 340 (1990) 366 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
D. Carmi, L. Di Pietro and S. Komatsu, A study of quantum field theories in AdS at finite coupling, JHEP01 (2019) 200 [arXiv:1810.04185] [INSPIRE].
N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS 5 × S 5: semiclassical partition function, JHEP04 (2000) 021 [hep-th/0001204] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
A.M. Polyakov and V.S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys.B 581 (2000) 116 [hep-th/0002106] [INSPIRE].
J. Polchinski and J. Sully, Wilson loop renormalization group flows, JHEP10 (2011) 059 [arXiv:1104.5077] [INSPIRE].
N. Drukker and S. Kawamoto, Small deformations of supersymmetric Wilson loops and open spin-chains, JHEP07 (2006) 024 [hep-th/0604124] [INSPIRE].
S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2/CFT 1, Nucl. Phys.B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On non-supersymmetric generalizations of the Wilson-Maldacena loops in N = 4 SYM, Nucl. Phys.B 934 (2018) 466 [arXiv:1804.02179] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in N = 4 SYM and AdS 2/CFT 1, JHEP05 (2019) 122 [arXiv:1903.04365] [INSPIRE].
A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett.B 103 (1981) 207 [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav.18 (2001) R153 [hep-th/0104158] [INSPIRE].
Y. Nakayama, Liouville field theory: a decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].
P. Menotti and E. Tonni, Standard and geometric approaches to quantum Liouville theory on the pseudosphere, Nucl. Phys.B 707 (2005) 321 [hep-th/0406014] [INSPIRE].
H. Ouyang, Holographic four-point functions in Toda field theories in AdS 2, JHEP04 (2019) 159 [arXiv:1902.10536] [INSPIRE].
A. Strominger, AdS 2quantum gravity and string theory, JHEP01 (1999) 007 [hep-th/9809027] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2backreaction and holography, JHEP11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional nearly anti-de-Sitter space, PTEP2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS 2backreaction and holography, JHEP07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
J.-L. Gervais and A. Neveu, New quantum treatment of Liouville field theory, Nucl. Phys.B 224 (1983) 329 [INSPIRE].
P. Mansfield, Light cone quantization of the Liouville and Toda field theories, Nucl. Phys.B 222 (1983) 419 [INSPIRE].
E. Braaten, T. Curtright, G. Ghandour and C.B. Thorn, A class of conformally invariant quantum field theories, Phys. Lett.B 125 (1983) 301 [INSPIRE].
C. Ahn, C. Rim and M. Stanishkov, Exact one point function of N = 1 super-Liouville theory with boundary, Nucl. Phys.B 636 (2002) 497 [hep-th/0202043] [INSPIRE].
L.A. Takhtajan, Topics in quantum geometry of Riemann surfaces: two-dimensional quantum gravity, in Como quantum groups, Villa Monastero, Varenna, Italy (1994), pg. 541 [hep-th/9409088] [INSPIRE].
P. Menotti and E. Tonni, The tetrahedron graph in Liouville theory on the pseudosphere, Phys. Lett.B 586 (2004) 425 [hep-th/0311234] [INSPIRE].
D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Correlation functions in the CFT d/AdS d+1correspondence, Nucl. Phys.B 546 (1999) 96 [hep-th/9804058] [INSPIRE].
T. Hahn, CUBA: a library for multidimensional numerical integration, Comput. Phys. Commun.168 (2005) 78 [hep-ph/0404043] [INSPIRE].
P. Menotti and E. Tonni, Liouville field theory with heavy charges. I. The pseudosphere, JHEP06 (2006) 020 [hep-th/0602206] [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].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [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].
H. Osborn, Conformal blocks for arbitrary spins in two dimensions, Phys. Lett.B 718 (2012) 169 [arXiv:1205.1941] [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: 1904.12753
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Beccaria, M., Tseytlin, A.A. On boundary correlators in Liouville theory on AdS2. J. High Energ. Phys. 2019, 8 (2019). https://doi.org/10.1007/JHEP07(2019)008
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2019)008