Abstract
We classify and study defects in 2d Jackiw-Teitelboim gravity. We show these are holographically described by a deformation of the Schwarzian theory where the reparametrization mode is integrated over different coadjoint orbits of the Virasoro group. We show that the quantization of each coadjoint orbit is connected to 2d Liouville CFT between branes with insertions of Verlinde loop operators. We also propose an interpretation for the exceptional orbits. We use this perspective to solve these deformations of the Schwarzian theory, computing their partition function and correlators. In the process, we define two geometric observables: the horizon area operator Φh and the geodesic length operator L(γ). We show this procedure is structurally related to the deformation of the particle-on-a-group quantum mechanics by the addition of a chemical potential. As an example, we solve the low-energy theory of complex SYK with a U(1) symmetry and generalize to the non-abelian case.
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References
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, talk given at the Fundamental Physics Prize Symposium, November 10, 2014, https://www.youtube.com/watch?v=OQ9qN8j7EZI.
A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise, KITP seminar, February 12, 2015, http://online.kitp.ucsb.edu/online/joint98/kitaev/.
A. Kitaev, A simple model of quantum holography (part 1), talk at KITP, April 7, 2015 http://online.kitp.ucsb.edu/online/entangled15/kitaev/.
A. Kitaev, A simple model of quantum holography (part 2), talk at KITP, May 27, 2015, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/.
S. Sachdev and J. Ye, Gapless spin fluid ground state in a random, quantum Heisenberg magnet, Phys. Rev. Lett.70 (1993) 3339 [cond-mat/9212030] [INSPIRE].
J. Polchinski and V. Rosenhaus, The Spectrum in the Sachdev-Ye-Kitaev Model, JHEP04 (2016) 001 [arXiv:1601.06768] [INSPIRE].
A. Jevicki, K. Suzuki and J. Yoon, Bi-Local Holography in the SYK Model, JHEP07 (2016) 007 [arXiv:1603.06246] [INSPIRE].
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev.D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
A. Jevicki and K. Suzuki, Bi-Local Holography in the SYK Model: Perturbations, JHEP11 (2016) 046 [arXiv:1608.07567] [INSPIRE].
J.S. Cotler et al., Black Holes and Random Matrices, JHEP05 (2017) 118 [Erratum ibid.09 (2018) 002] [arXiv:1611.04650] [INSPIRE].
G. Mandal, P. Nayak and S.R. Wadia, Coadjoint orbit action of Virasoro group and two-dimensional quantum gravity dual to SYK/tensor models, JHEP11 (2017) 046 [arXiv:1702.04266] [INSPIRE].
D. Stanford and E. Witten, Fermionic Localization of the Schwarzian Theory, JHEP10 (2017) 008 [arXiv:1703.04612] [INSPIRE].
G. Turiaci and H. Verlinde, Towards a 2d QFT Analog of the SYK Model, JHEP10 (2017) 167 [arXiv:1701.00528] [INSPIRE].
D.J. Gross and V. Rosenhaus, The Bulk Dual of SYK: Cubic Couplings, JHEP05 (2017) 092 [arXiv:1702.08016] [INSPIRE].
D.J. Gross and V. Rosenhaus, All point correlation functions in SYK, JHEP12 (2017) 148 [arXiv:1710.08113] [INSPIRE].
S.R. Das, A. Jevicki and K. Suzuki, Three Dimensional View of the SYK/AdS Duality, JHEP09 (2017) 017 [arXiv:1704.07208] [INSPIRE].
S.R. Das, A. Ghosh, A. Jevicki and K. Suzuki, Space-Time in the SYK Model, JHEP07 (2018) 184 [arXiv:1712.02725] [INSPIRE].
M. Berkooz, P. Narayan and J. Simon, Chord diagrams, exact correlators in spin glasses and black hole bulk reconstruction, JHEP08 (2018) 192 [arXiv:1806.04380] [INSPIRE].
G. Gur-Ari, R. Mahajan and A. Vaezi, Does the SYK model have a spin glass phase?, JHEP11 (2018) 070 [arXiv:1806.10145] [INSPIRE].
M. Berkooz, M. Isachenkov, V. Narovlansky and G. Torrents, Towards a full solution of the large N double-scaled SYK model, JHEP03 (2019) 079 [arXiv:1811.02584] [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.126B (1983) 41 [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].
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. Alekseev and S.L. Shatashvili, From geometric quantization to conformal field theory, Commun. Math. Phys.128 (1990) 197 [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Fine Structure of Jackiw-Teitelboim Quantum Gravity, arXiv:1812.00918 [INSPIRE].
D. Anninos, D.A. Galante and D.M. Hofman, De Sitter Horizons and Holographic Liquids, JHEP07 (2019) 038 [arXiv:1811.08153] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
J. Maldacena, G.J. Turiaci and Z. Yang, Two dimensional Nearly de Sitter gravity, arXiv:1904.01911 [INSPIRE].
G.J. Turiaci, An Inelastic Bound on Chaos, JHEP07 (2019) 099 [arXiv:1901.04360] [INSPIRE].
R. Jackiw, Gauge theories for gravity on a line, Theor. Math. Phys.92 (1992) 979 [hep-th/9206093] [INSPIRE].
S. Sachdev, Bekenstein-Hawking Entropy and Strange Metals, Phys. Rev.X 5 (2015) 041025 [arXiv:1506.05111] [INSPIRE].
T.G. Mertens, G.J. Turiaci and H.L. Verlinde, Solving the Schwarzian via the Conformal Bootstrap, JHEP08 (2017) 136 [arXiv:1705.08408] [INSPIRE].
T.G. Mertens, The Schwarzian theory — origins, JHEP05 (2018) 036 [arXiv:1801.09605] [INSPIRE].
H.T. Lam, T.G. Mertens, G.J. Turiaci and H. Verlinde, Shockwave S-matrix from Schwarzian Quantum Mechanics, JHEP11 (2018) 182 [arXiv:1804.09834] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, The Schwarzian Theory — A Wilson Line Perspective, JHEP12 (2018) 022 [arXiv:1806.07765] [INSPIRE].
A. Kitaev and S.J. Suh, Statistical mechanics of a two-dimensional black hole, JHEP05 (2019) 198 [arXiv:1808.07032] [INSPIRE].
Z. Yang, The Quantum Gravity Dynamics of Near Extremal Black Holes, JHEP05 (2019) 205 [arXiv:1809.08647] [INSPIRE].
R.A. Davison, W. Fu, A. Georges, Y. Gu, K. Jensen and S. Sachdev, Thermoelectric transport in disordered metals without quasiparticles: The Sachdev-Ye-Kitaev models and holography, Phys. Rev.B 95 (2017) 155131 [arXiv:1612.00849] [INSPIRE].
W. Fu, D. Gaiotto, J. Maldacena and S. Sachdev, Supersymmetric Sachdev-Ye-Kitaev models, Phys. Rev.D 95 (2017) 026009 [Addendum ibid.D 95 (2017) 069904] [arXiv:1610.08917] [INSPIRE].
C. Peng, M. Spradlin and A. Volovich, Correlators in the \( \mathcal{N} \)= 2 Supersymmetric SYK Model, JHEP10 (2017) 202 [arXiv:1706.06078] [INSPIRE].
K. Bulycheva, A note on the SYK model with complex fermions, JHEP12 (2017) 069 [arXiv:1706.07411] [INSPIRE].
S. Sachdev, Universal theory of complex SYK and extremal charged black holes, talk at KITP, December 7, 2018, http://online.kitp.ucsb.edu/online/chord18/sachdev/pdf/Sachdev_ChOrd18_KITP.pdf.
J. Liu and Y. Zhou, Note on global symmetry and SYK model, JHEP05 (2019) 099 [arXiv:1901.05666] [INSPIRE].
P. Chaturvedi, Y. Gu, W. Song and B. Yu, A note on the complex SYK model and warped CFTs, JHEP12 (2018) 101 [arXiv:1808.08062] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Sachdev-Ye-Kitaev model as Liouville quantum mechanics, Nucl. Phys. B911 (2016) 191 [arXiv:1607.00694] [INSPIRE].
D. Bagrets, A. Altland and A. Kamenev, Power-law out of time order correlation functions in the SYK model, Nucl. Phys.B 921 (2017) 727 [arXiv:1702.08902] [INSPIRE].
D. Bagrets, Goldstone fluctuations in the SYK model, talk at KITP, October10, 2018, http://online.kitp.ucsb.edu/online/chord18/bagrets/pdf/Bagrets_ChOrd18_KITP.pdf.
U. Moitra, S.P. Trivedi and V. Vishal, Extremal and near-extremal black holes and near-CFT 1, JHEP07 (2019) 055 [arXiv:1808.08239] [INSPIRE].
A. Gaikwad, L.K. Joshi, G. Mandal and S.R. Wadia, Holographic dual to charged SYK from 3D Gravity and Chern-Simons, arXiv:1802.07746 [INSPIRE].
S. Sachdev, Universal low temperature theory of charged black holes with AdS 2horizons, J. Math. Phys.60 (2019) 052303 [arXiv:1902.04078] [INSPIRE].
D. Anninos, T. Anous and F. Denef, Disordered Quivers and Cold Horizons, JHEP12 (2016) 071 [arXiv:1603.00453] [INSPIRE].
G. Barnich, H.A. Gonzalez and P. Salgado-ReboLledó, Geometric actions for three-dimensional gravity, Class. Quant. Grav.35 (2018) 014003 [arXiv:1707.08887] [INSPIRE].
G.W. Moore and N. Seiberg, Taming the Conformal Zoo, Phys. Lett.B 220 (1989) 422 [INSPIRE].
Y. Fan, Localization and Non-Renormalization in Chern-Simons Theory, JHEP01 (2019) 065 [arXiv:1805.11076] [INSPIRE].
M. Ammon, A. Castro and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
J. Raeymaekers, Quantization of conical spaces in 3D gravity, JHEP03 (2015) 060 [arXiv:1412.0278] [INSPIRE].
E. Witten, Coadjoint Orbits of the Virasoro Group, Commun. Math. Phys.114 (1988) 1 [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].
G. Turiaci and H. Verlinde, On CFT and Quantum Chaos, JHEP12 (2016) 110 [arXiv:1603.03020] [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
N. Callebaut and H. Verlinde, Entanglement Dynamics in 2D CFT with Boundary: Entropic origin of JT gravity and Schwarzian QM, JHEP05 (2019) 045 [arXiv:1808.05583] [INSPIRE].
N. Callebaut, The gravitational dynamics of kinematic space, JHEP02 (2019) 153 [arXiv:1808.10431] [INSPIRE].
N. Drukker, D. Gaiotto and J. Gomis, The Virtue of Defects in 4D Gauge Theories and 2D CFTs, JHEP06 (2011) 025 [arXiv:1003.1112] [INSPIRE].
R. Dijkgraaf and E.P. Verlinde, Modular Invariance and the Fusion Algebra, Nucl. Phys. Proc. Suppl.5 (1988) 87 [INSPIRE].
B. Le Floch and G.J. Turiaci, AGT/ℤ2, JHEP12 (2017) 099 [arXiv:1708.04631] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
A. Goel, H.T. Lam, G.J. Turiaci and H. Verlinde, Expanding the Black Hole Interior: Partially Entangled Thermal States in SYK, JHEP02 (2019) 156 [arXiv:1807.03916] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
J. Lin, Entanglement entropy in Jackiw-Teitelboim Gravity, arXiv:1807.06575 [INSPIRE].
S.R. Coleman, Black Holes as Red Herrings: Topological Fluctuations and the Loss of Quantum Coherence, Nucl. Phys.B 307 (1988) 867 [INSPIRE].
N. Arkani-Hamed, J. Orgera and J. Polchinski, Euclidean wormholes in string theory, JHEP12 (2007) 018 [arXiv:0705.2768] [INSPIRE].
C. Jego and J. Troost, Notes on the Verlinde formula in non-rational conformal field theories, Phys. Rev.D 74 (2006) 106002 [hep-th/0601085] [INSPIRE].
J. Maldacena and X.-L. Qi, Eternal traversable wormhole, arXiv:1804.00491 [INSPIRE].
A. Gorsky and A. Johansen, Liouville theory and special coadjoint Virasoro orbits, Int. J. Mod. Phys.A 10 (1995) 785 [hep-th/9311052] [INSPIRE].
A. Blommaert, T.G. Mertens and H. Verschelde, Clocks and Rods in Jackiw-Teitelboim Quantum Gravity, arXiv:1902.11194 [INSPIRE].
T.G. Mertens, Towards Black Hole Evaporation in Jackiw-Teitelboim Gravity, JHEP07 (2019) 097 [arXiv:1903.10485] [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
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Mertens, T.G., Turiaci, G.J. Defects in Jackiw-Teitelboim quantum gravity. J. High Energ. Phys. 2019, 127 (2019). https://doi.org/10.1007/JHEP08(2019)127
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DOI: https://doi.org/10.1007/JHEP08(2019)127