Abstract
It was recently observed that boundary correlators of the elementary scalar field of the Liouville theory on AdS2 background are the same (up to a non-trivial proportionality coefficient) as the correlators of the chiral stress tensor of the Liouville CFT on the complex plane restricted to the real line. The same relation generalizes to the conformal abelian Toda theory: boundary correlators of Toda scalars on AdS2 are directly related to the correlation functions of the chiral \( \mathcal{W} \)-symmetry generators in the Toda CFT and thus are essentially controlled by the underlying infinite-dimensional symmetry. These may be viewed as examples of AdS2/CFT1 duality where the CFT1 is the chiral half of a 2d CFT; we shall to this as \( {\mathrm{AdS}}_2/{\mathrm{CFT}}_2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \). In this paper we demonstrate that this duality applies also to the non-abelian Toda theory containing a Liouville scalar coupled to a 2d σ-model originating from the SL(2, ℝ)/U(1) gauged WZW model. Here the Liouville scalar is again dual to the chiral stress tensor T while the other two scalars are dual to the parafermionic operators V± of the non-abelian Toda CFT. We explicitly check the duality at the next-to-leading order in the large central charge expansion by matching the chiral CFT correlators of (T, V+, V−) (computed using a free field representation) with the boundary correlators of the three Toda scalars given by the tree-level and one-loop Witten diagrams in AdS2.
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 sypersymmetry in two-dimensional Anti de Sitter space, Nucl. Phys.B 273 (1986) 487 [INSPIRE].
C.G. Callan Jr. and F. Wilczek, Iinfrared behaviour 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 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 \( \mathcal{N} \) = 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 \( \mathcal{N} \) = 4 SYM and AdS 2/CFT 1, JHEP05 (2019) 122 [arXiv:1903.04365] [INSPIRE].
S. Giombi, C. Sleight and M. Taronna, Spinning AdS Loop Diagrams: Two Point Functions, JHEP06 (2018) 030 [arXiv:1708.08404] [INSPIRE].
I. Bertan, I. Sachs and E.D. Skvortsov, Quantum ϕ 4Theory in AdS 4and its CFT Dual, JHEP02 (2019) 099 [arXiv:1810.00907] [INSPIRE].
E.Y. Yuan, Simplicity in AdS Perturbative Dynamics, arXiv:1801.07283 [INSPIRE].
J. Liu, E. Perlmutter, V. Rosenhaus and D. Simmons-Duffin, d-dimensional SYK, AdS Loops, and 6j Symbols, JHEP03 (2019) 052 [arXiv:1808.00612] [INSPIRE].
H. Ouyang, Holographic four-point functions in Toda field theories in AdS 2, JHEP04 (2019) 159 [arXiv:1902.10536] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On boundary correlators in Liouville theory on AdS 2, JHEP07 (2019) 008 [arXiv:1904.12753] [INSPIRE].
M. Beccaria and G. Landolfi, Toda theory in AdS 2and WAn-algebra structure of boundary correlators, arXiv:1906.06485 [INSPIRE].
A. Strominger, AdS 2quantum gravity and string theory, JHEP01 (1999) 007 [hep-th/9809027] [INSPIRE].
M. Hotta, Asymptotic isometry and two-dimensional anti-de Sitter gravity, gr-qc/9809035 [INSPIRE].
M. Cadoni and S. Mignemi, Asymptotic symmetries of AdS 2and conformal group in d = 1, Nucl. Phys.B 557 (1999) 165 [hep-th/9902040] [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS 2backreaction and holography, JHEP11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
K. Jensen, Chaos in AdS 2Holography, Phys. Rev. Lett.117 (2016) 111601 [arXiv:1605.06098] [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 M.V. Savelev, Black holes from nonAbelian Toda theories, Phys. Lett.B 286 (1992) 271 [hep-th/9203039] [INSPIRE].
K. Bardakci, M.J. Crescimanno and E. Rabinovici, Parafermions From Coset Models, Nucl. Phys.B 344 (1990) 344 [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev.D 44 (1991) 314 [INSPIRE].
A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett.B 103 (1981) 207 [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].
A.N. Leznov and M.V. Saveliev, Representation of zero curvature for the system of nonlinear partial differential equations \( {\chi}_{\alpha, z\overline{z}} \) = exp(kχ)αand its integrability, Lett. Math. Phys.3 (1979) 489 [INSPIRE].
A. Bilal and J.-L. Gervais, Systematic Construction of Conformal Theories with Higher Spin Virasoro Symmetries, Nucl. Phys.B 318 (1989) 579 [INSPIRE].
T.J. Hollowood and P. Mansfield, Quantum Group Structure of Quantum Toda Conformal Field Theories. 1., Nucl. Phys.B 330 (1990) 720 [INSPIRE].
V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I., JHEP11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
A.A. Tseytlin, On the Structure of the Renormalization Group Beta Functions in a Class of Two-dimensional Models, Phys. Lett.B 241 (1990) 233 [INSPIRE].
M.T. Grisaru, A. Lerda, S. Penati and D. Zanon, Renormalization Group Flows in Generalized Toda Field Theories, Nucl. Phys.B 346 (1990) 264 [INSPIRE].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].
P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept.223 (1993) 183 [hep-th/9210010] [INSPIRE].
A. Bilal, NonAbelian Toda theory: A completely integrable model for strings on a black hole background, Nucl. Phys.B 422 (1994) 258 [hep-th/9312108] [INSPIRE].
A. Bilal, Consistent string backgrounds and completely integrable 2-D field theories, Nucl. Phys. Proc. Suppl.45A (1996) 105 [hep-th/9508062] [INSPIRE].
I. Jack, D.R.T. Jones and J. Panvel, Quantum nonAbelian Toda field theories, Int. J. Mod. Phys.A 9 (1994) 3631 [hep-th/9308080] [INSPIRE].
C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in Background Fields, Nucl. Phys.B 262 (1985) 593 [INSPIRE].
A.A. Tseytlin, Sigma model approach to string theory, Int. J. Mod. Phys.A 4 (1989) 1257 [INSPIRE].
A.A. Tseytlin, On the form of the black hole solution in D = 2 theory, Phys. Lett.B 268 (1991) 175 [INSPIRE].
I. Jack, D.R.T. Jones and J. Panvel, Exact bosonic and supersymmetric string black hole solutions, Nucl. Phys.B 393 (1993) 95 [hep-th/9201039] [INSPIRE].
A.A. Tseytlin, On field redefinitions and exact solutions in string theory, Phys. Lett.B 317 (1993) 559 [hep-th/9308042] [INSPIRE].
R. Dijkgraaf, H.L. Verlinde and E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys.B 371 (1992) 269 [INSPIRE].
A.A. Tseytlin, Effective action of gauged WZW model and exact string solutions, Nucl. Phys.B 399 (1993) 601 [hep-th/9301015] [INSPIRE].
I. Bars and K. Sfetsos, Exact effective action and space-time geometry n gauged WZW models, Phys. Rev.D 48 (1993) 844 [hep-th/9301047] [INSPIRE].
A.A. Tseytlin, Conformal sigma models corresponding to gauged Wess-Zumino-Witten theories, Nucl. Phys.B 411 (1994) 509 [hep-th/9302083] [INSPIRE].
B. Hoare and A.A. Tseytlin, On the perturbative S-matrix of generalized sine-Gordon models, JHEP11 (2010) 111 [arXiv:1008.4914] [INSPIRE].
L. O’Raifeartaigh and A. Wipf, Conformally reduced WZNW theories and two-dimensional gravity, Phys. Lett.B 251 (1990) 361 [INSPIRE].
J. Balog, L. Feher, L. O’Raifeartaigh, P. Forgacs and A. Wipf, Toda Theory and W Algebra From a Gauged WZNW Point of View, Annals Phys.203 (1990) 76 [INSPIRE].
L. Feher, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories, Phys. Rept.222 (1992) 1 [INSPIRE].
C. Klimčík and A.A. Tseytlin, Exact four-dimensional string solutions and Toda like sigma models from ‘null gauged’ WZNW theories, Nucl. Phys.B 424 (1994) 71 [hep-th/9402120] [INSPIRE].
A. Gerasimov, A. Morozov, M. Olshanetsky, A. Marshakov and S.L. Shatashvili, Wess-Zumino-Witten model as a theory of free fields, Int. J. Mod. Phys.A 5 (1990) 2495 [INSPIRE].
C. Ford, Quantum parafermions in the SL(2, ℝ)/U(1) WZNW black hole model, Rept. Math. Phys.48 (2001) 67 [hep-th/0010123] [INSPIRE].
C. Kruger, Exact operator quantization of the Euclidean black hole CFT, hep-th/0411275 [INSPIRE].
I. Bakas and E. Kiritsis, Beyond the large N limit: Nonlinear W∞as symmetry of the SL(2, ℝ)/U(1) coset model, Int. J. Mod. Phys.A 7S1A (1992) 55 [hep-th/9109029] [INSPIRE].
A. Sevrin and W. Troost, Extensions of the Virasoro algebra and gauged WZW models, Phys. Lett.B 315 (1993) 304 [hep-th/9306033] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [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].
V.A. Fateev and S.L. Lukyanov, The Models of Two-Dimensional Conformal Quantum Field Theory with ZnSymmetry, Int. J. Mod. Phys.A 3 (1988) 507 [INSPIRE].
E. Braaten, T. Curtright and C.B. Thorn, An Exact Operator Solution of the Quantum Liouville Field Theory, Annals Phys.147 (1983) 365 [INSPIRE].
V. Schomerus, Lectures on branes in curved backgrounds, Class. Quant. Grav.19 (2002) 5781 [hep-th/0209241] [INSPIRE].
A. Recknagel and V. Schomerus, Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes, Cambridge Monographs on Mathematical Physics, Cambridge University Press, (2013), [https://doi.org/10.1017/CBO9780511806476].
P. Menotti and E. Tonni, The tetrahedron graph in Liouville theory on the pseudosphere, Phys. Lett.B 586 (2004) 425 [hep-th/0311234] [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: 1907.01357
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., Jiang, H. & Tseytlin, A.A. Non-abelian Toda theory on AdS2 and \( {\mathrm{AdS}}_2/{\mathrm{CFT}}_2^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \) duality. J. High Energ. Phys. 2019, 36 (2019). https://doi.org/10.1007/s13130-019-11219-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13130-019-11219-y