Abstract
Recently proposed duality relates the critical level limit \(\widehat{k}\to -2\) of \({\text{SU}}{\left(2\right)}_{\widehat{k}}\) WZW models to a classical three-dimensional Einstein gravity on a sphere. In this paper, we propose a dimensional reduced version of this duality. The gravity side is reduced to a Jackiw-Teitelboim (JT) gravity on S2 with a non-standard boundary term, or a BF theory with SU(2) gauge symmetry. At least in low temperature limit, these two-dimensional gravity theories completely capture the original three-dimensional gravity effect. The CFT side is reduced to a certain complex Liouville quantum mechanics (LQM) with SU(2) gauge symmetry. Our proposal gives an interesting example of a holography without boundary. We also discuss a higher-spin generalization with SU(N) gauge symmetry.
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References
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
A. Strominger, The dS / CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, in the proceedings of the Strings 2001: International Conference, Mumbai, India, January 05–10 (2001) [hep-th/0106109] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [arXiv:2108.04801] [INSPIRE].
T. Kawamoto, S.-M. Ruan and T. Takayanagia, Gluing AdS/CFT, JHEP 07 (2023) 080 [arXiv:2303.01247] [INSPIRE].
Y. Hikida, T. Nishioka, T. Takayanagi and Y. Taki, Holography in de Sitter Space via Chern-Simons Gauge Theory, Phys. Rev. Lett. 129 (2022) 041601 [arXiv:2110.03197] [INSPIRE].
Y. Hikida, T. Nishioka, T. Takayanagi and Y. Taki, CFT duals of three-dimensional de Sitter gravity, JHEP 05 (2022) 129 [arXiv:2203.02852] [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
A. Almheiri and J. Polchinski, Models of AdS2 backreaction and holography, JHEP 11 (2015) 014 [arXiv:1402.6334] [INSPIRE].
K. Jensen, Chaos in AdS2 Holography, 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, PTEP 2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].
J. Engelsöy, T.G. Mertens and H. Verlinde, An investigation of AdS2 backreaction and holography, JHEP 07 (2016) 139 [arXiv:1606.03438] [INSPIRE].
A. Ghosh, H. Maxfield and G.J. Turiaci, A universal Schwarzian sector in two-dimensional conformal field theories, JHEP 05 (2020) 104 [arXiv:1912.07654] [INSPIRE].
R. Jackiw, Lower Dimensional Gravity, Nucl. Phys. B 252 (1985) 343 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP 08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
T.G. Mertens, The Schwarzian theory — origins, JHEP 05 (2018) 036 [arXiv:1801.09605] [INSPIRE].
J.M. Maldacena and A. Strominger, Statistical entropy of de Sitter space, JHEP 02 (1998) 014 [gr-qc/9801096] [INSPIRE].
M.-I. Park, Statistical entropy of three-dimensional Kerr-de Sitter space, Phys. Lett. B 440 (1998) 275 [hep-th/9806119] [INSPIRE].
M.-I. Park, Symmetry algebras in Chern-Simons theories with boundary: Canonical approach, Nucl. Phys. B 544 (1999) 377 [hep-th/9811033] [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher Spin Realization of the dS/CFT Correspondence, Class. Quant. Grav. 34 (2017) 015009 [arXiv:1108.5735] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP 07 (2012) 127 [arXiv:1205.2472] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].
E. Perlmutter, T. Prochazka and J. Raeymaekers, The semiclassical limit of WN CFTs and Vasiliev theory, JHEP 05 (2013) 007 [arXiv:1210.8452] [INSPIRE].
E. Witten, A Note On Complex Spacetime Metrics, arXiv:2111.06514 [INSPIRE].
H.-Y. Chen, Y. Hikida, Y. Taki and T. Uetoko, Complex saddles of three-dimensional de Sitter gravity via holography, Phys. Rev. D 107 (2023) L101902 [arXiv:2302.09219] [INSPIRE].
H.-Y. Chen, Y. Hikida, Y. Taki and T. Uetoko, Complex saddles of Chern-Simons gravity and dS3/CFT2 correspondence, Phys. Rev. D 108 (2023) 066005 [arXiv:2306.03330] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
G.W. Gibbons and S.W. Hawking, Cosmological Event Horizons, Thermodynamics, and Particle Creation, Phys. Rev. D 15 (1977) 2738 [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
J. Maldacena, G.J. Turiaci and Z. Yang, Two dimensional Nearly de Sitter gravity, JHEP 01 (2021) 139 [arXiv:1904.01911] [INSPIRE].
A. Goel, L.V. Iliesiu, J. Kruthoff and Z. Yang, Classifying boundary conditions in JT gravity: from energy-branes to α-branes, JHEP 04 (2021) 069 [arXiv:2010.12592] [INSPIRE].
L. Susskind, De Sitter Holography: Fluctuations, Anomalous Symmetry, and Wormholes, Universe 7 (2021) 464 [arXiv:2106.03964] [INSPIRE].
L. Susskind, Black Holes Hint towards De Sitter Matrix Theory, Universe 9 (2023) 368 [arXiv:2109.01322] [INSPIRE].
T. Kawamoto, S.-M. Ruan, Y.-K. Suzuki and T. Takayanagi, A half de Sitter holography, JHEP 10 (2023) 137 [arXiv:2306.07575] [INSPIRE].
T.G. Mertens and G.J. Turiaci, Defects in Jackiw-Teitelboim Quantum Gravity, JHEP 08 (2019) 127 [arXiv:1904.05228] [INSPIRE].
H. Maxfield and G.J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].
E. Witten, Matrix Models and Deformations of JT Gravity, Proc. Roy. Soc. Lond. A 476 (2020) 20200582 [arXiv:2006.13414] [INSPIRE].
E. Mefford and K. Suzuki, Jackiw-Teitelboim quantum gravity with defects and the Aharonov-Bohm effect, JHEP 05 (2021) 026 [arXiv:2011.04695] [INSPIRE].
A. Achucarro and P.K. Townsend, A Chern-Simons Action for Three-Dimensional anti-De Sitter Supergravity Theories, Phys. Lett. B 180 (1986) 89 [INSPIRE].
E. Witten, (2+1)-Dimensional Gravity as an Exactly Soluble System, Nucl. Phys. B 311 (1988) 46 [INSPIRE].
O. Coussaert, M. Henneaux and P. van Driel, The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant, Class. Quant. Grav. 12 (1995) 2961 [gr-qc/9506019] [INSPIRE].
L. Donnay, Asymptotic dynamics of three-dimensional gravity, PoS Modave2015 (2016) 001 [arXiv:1602.09021] [INSPIRE].
A. Achucarro, Lineal gravity from planar gravity, Phys. Rev. Lett. 70 (1993) 1037 [hep-th/9207108] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian Theory — A Wilson Line Perspective, JHEP 12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
T.G. Mertens, J. Simón and G. Wong, A proposal for 3d quantum gravity and its bulk factorization, JHEP 06 (2023) 134 [arXiv:2210.14196] [INSPIRE].
K. Isler and C.A. Trugenberger, A Gauge Theory of Two-dimensional Quantum Gravity, Phys. Rev. Lett. 63 (1989) 834 [INSPIRE].
A.H. Chamseddine and D. Wyler, Gauge Theory of Topological Gravity in (1 + 1)-Dimensions, Phys. Lett. B 228 (1989) 75 [INSPIRE].
L.V. Iliesiu, S.S. Pufu, H. Verlinde and Y. Wang, An exact quantization of Jackiw-Teitelboim gravity, JHEP 11 (2019) 091 [arXiv:1905.02726] [INSPIRE].
M.A. Vasiliev, Higher spin gauge interactions for matter fields in two-dimensions, Phys. Lett. B 363 (1995) 51 [hep-th/9511063] [INSPIRE].
A. Alekseev and S.L. Shatashvili, Path Integral Quantization of the Coadjoint Orbits of the Virasoro Group and 2D Gravity, Nucl. Phys. B 323 (1989) 719 [INSPIRE].
A. Castro, R. Gopakumar, M. Gutperle and J. Raeymaekers, Conical Defects in Higher Spin Theories, JHEP 02 (2012) 096 [arXiv:1111.3381] [INSPIRE].
J. Cotler and K. Jensen, A theory of reparameterizations for AdS3 gravity, JHEP 02 (2019) 079 [arXiv:1808.03263] [INSPIRE].
J. Cotler, K. Jensen and A. Maloney, Low-dimensional de Sitter quantum gravity, JHEP 06 (2020) 048 [arXiv:1905.03780] [INSPIRE].
T. Creutzig and Y. Hikida, Correlator correspondences for Gaiotto-Rapčák dualities and first order formulation of coset models, JHEP 12 (2021) 144 [arXiv:2109.03403] [INSPIRE].
G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B 220 (1989) 422 [INSPIRE].
J. Balog, L. Feher and L. Palla, Coadjoint orbits of the Virasoro algebra and the global Liouville equation, Int. J. Mod. Phys. A 13 (1998) 315 [hep-th/9703045] [INSPIRE].
P. Forgacs et al., Liouville and Toda Theories as Conformally Reduced WZNW Theories, Phys. Lett. B 227 (1989) 214 [INSPIRE].
K. Suzuki and Y. Taki, Correlation functions in S2 JT gravity and complex Liouville quamtum mechanics, in progress.
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
P. Di Vecchia, V.G. Knizhnik, J.L. Petersen and P. Rossi, A Supersymmetric Wess-Zumino Lagrangian in Two-Dimensions, Nucl. Phys. B 253 (1985) 701 [INSPIRE].
V.G. Kac and I.T. Todorov, Superconformal current algebras and their unitary representations, Commun. Math. Phys. 102 (1985) 337 [INSPIRE].
J. Fuchs, Superconformal Ward Identities and the WZW Model, Nucl. Phys. B 286 (1987) 455 [INSPIRE].
G. Arcioni, M. Blau and M. O’Loughlin, On the boundary dynamics of Chern-Simons gravity, JHEP 01 (2003) 067 [hep-th/0210089] [INSPIRE].
Acknowledgments
We would like to thank Yasuaki Hikida and Tadashi Takayanagi for useful comments on the draft of this paper. We are also grateful to the YITP long-term workshop QIMG2023 (workshop number YITP-T-23-01), where this work was completed. This work is supported by MEXT KAKENHI Grant-in-Aid for Transformative Research Areas (A) through the “Extreme Universe” collaboration: Grant Number 21H05187. The work of KS is also supported by JSPS KAKENHI Grant No. 23K13105. The work of YT is supported by Grant-in-Aid for JSPS Fellows No. 22KJ1971.
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Suzuki, K., Taki, Y. Dimensional reduction of the S3/WZW duality. J. High Energ. Phys. 2024, 103 (2024). https://doi.org/10.1007/JHEP03(2024)103
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DOI: https://doi.org/10.1007/JHEP03(2024)103