Abstract
We study universal properties of the torus partition function of \( T\overline{T} \)-deformed CFTs under the assumption of modular invariance, for both the original version, referred to as the double-trace version in this paper, and the single-trace version defined as the symmetric product orbifold of double-trace \( T\overline{T} \)-deformed CFTs. In the double-trace case, we specify sparseness conditions for the light states for which the partition function at low temperatures is dominated by the vacuum when the central charge of the undeformed CFT is large. Using modular invariance, this implies a universal density of high energy states, in analogy with the behavior of holographic CFTs. For the single-trace \( T\overline{T} \) deformation, we show that modular invariance implies that the torus partition function can be written in terms of the untwisted partition function and its modular images, the latter of which can be obtained from the action of a generalized Hecke operator. The partition function and the energy of twisted states match holographic calculations in previous literature, thus providing further evidence for the conjectured holographic correspondence. In addition, we show that the single-trace partition function is universal when the central charge of the undeformed CFT is large, without needing to assume a sparse density of light states. Instead, the density of light states is shown to always saturate the sparseness condition.
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References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
A. Strominger, Black hole entropy from near horizon microstates, JHEP 02 (1998) 009 [hep-th/9712251] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes, JHEP 10 (2019) 062 [arXiv:1810.11442] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
F. Benini and E. Milan, Black Holes in 4D \( \mathcal{N} \) = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
F.A. Smirnov and A.B. Zamolodchikov, On space of integrable quantum field theories, Nucl. Phys. B 915 (2017) 363 [arXiv:1608.05499] [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field T anti-T in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
A. Cavaglià, S. Negro, I.M. Szécsényi and R. Tateo, \( T\overline{T} \)-deformed 2D Quantum Field Theories, JHEP 10 (2016) 112 [arXiv:1608.05534] [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].
B. Le Floch and M. Mezei, Solving a family of \( T\overline{T} \)-like theories, arXiv:1903.07606 [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, \( T\overline{T} \), \( J\overline{T} \), \( T\overline{J} \) and String Theory, J. Phys. A 52 (2019) 384003 [arXiv:1905.00051] [INSPIRE].
S. Frolov, \( T\overline{T} \), \( \overset{\sim }{J}J \), JT and \( \overset{\sim }{J}T \) deformations, J. Phys. A 53 (2020) 025401 [arXiv:1907.12117] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the Simplest Theory of Quantum Gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Natural Tuning: Towards A Proof of Concept, JHEP 09 (2013) 045 [arXiv:1305.6939] [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS2 holography and \( T\overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
S. Datta and Y. Jiang, \( T\overline{T} \) deformed partition functions, JHEP 08 (2018) 106 [arXiv:1806.07426] [INSPIRE].
O. Aharony et al., Modular invariance and uniqueness of \( T\overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [INSPIRE].
N. Callebaut, J. Kruthoff and H. Verlinde, \( T\overline{T} \) deformed CFT as a non-critical string, JHEP 04 (2020) 084 [arXiv:1910.13578] [INSPIRE].
A.J. Tolley, \( T\overline{T} \) deformations, massive gravity and non-critical strings, JHEP 06 (2020) 050 [arXiv:1911.06142] [INSPIRE].
G. Giribet, \( T\overline{T} \)-deformations, AdS/CFT and correlation functions, JHEP 02 (2018) 114 [arXiv:1711.02716] [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and \( T\overline{T} \) deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
B. Chen, L. Chen and P.-X. Hao, Entanglement entropy in \( T\overline{T} \)-deformed CFT, Phys. Rev. D 98 (2018) 086025 [arXiv:1807.08293] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and M. Nishida, Entanglement and Rényi entropy of multiple intervals in \( T\overline{T} \)-deformed CFT and holography, Phys. Rev. D 100 (2019) 106015 [arXiv:1906.03894] [INSPIRE].
J. Cardy, \( T\overline{T} \) deformation of correlation functions, JHEP 12 (2019) 160 [arXiv:1907.03394] [INSPIRE].
T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T2 deformation, JHEP 03 (2019) 004 [arXiv:1807.11401] [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
D.J. Gross, J. Kruthoff, A. Rolph and E. Shaghoulian, \( T\overline{T} \) in AdS2 and Quantum Mechanics, Phys. Rev. D 101 (2020) 026011 [arXiv:1907.04873] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \) and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [INSPIRE].
L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with \( T\overline{T} \), JHEP 04 (2018) 010 [arXiv:1611.03470].
M. Guica and R. Monten, \( T\overline{T} \) and the mirage of a bulk cutoff, SciPost Phys. 10 (2021) 024 [arXiv:1906.11251] [INSPIRE].
L. Apolo, S. Detournay and W. Song, TsT, \( T\overline{T} \) and black strings, JHEP 06 (2020) 109 [arXiv:1911.12359] [INSPIRE].
L. Apolo and W. Song, Strings on warped AdS3 via \( T\overline{J} \) deformations, JHEP 10 (2018) 165 [arXiv:1806.10127] [INSPIRE].
T. Araujo et al., Holographic integration of \( T\overline{T} \) & \( J\overline{T} \) via O(d, d), JHEP 03 (2019) 168 [arXiv:1811.03050] [INSPIRE].
R. Borsato and L. Wulff, Marginal deformations of WZW models and the classical Yang-Baxter equation, J. Phys. A 52 (2019) 225401 [arXiv:1812.07287] [INSPIRE].
L. Apolo and W. Song, TsT, black holes, and \( T\overline{T} \) + \( J\overline{T} \) + \( T\overline{J} \), JHEP 04 (2022) 177 [arXiv:2111.02243] [INSPIRE].
A. Hashimoto and D. Kutasov, Strings, symmetric products, \( T\overline{T} \) deformations and Hecke operators, Phys. Lett. B 806 (2020) 135479 [arXiv:1909.11118] [INSPIRE].
S. Chakraborty, A. Giveon and D. Kutasov, \( J\overline{T} \) deformed CFT2 and string theory, JHEP 10 (2018) 057 [arXiv:1806.09667] [INSPIRE].
L. Apolo and W. Song, Heating up holography for single-trace \( J\overline{T} \) deformations, JHEP 01 (2020) 141 [arXiv:1907.03745] [INSPIRE].
O. Aharony and T. Vaknin, The TT* deformation at large central charge, JHEP 05 (2018) 166 [arXiv:1803.00100] [INSPIRE].
A. Klemm and M.G. Schmidt, Orbifolds by Cyclic Permutations of Tensor Product Conformal Field Theories, Phys. Lett. B 245 (1990) 53 [INSPIRE].
R. Dijkgraaf, G.W. Moore, E.P. Verlinde and H.L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
R. Dijkgraaf, Fields, strings, matrices and symmetric products, hep-th/9912104 [INSPIRE].
P. Bantay, Symmetric products, permutation orbifolds and discrete torsion, Lett. Math. Phys. 63 (2003) 209 [hep-th/0004025] [INSPIRE].
T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, U.S.A. (1990) [https://doi.org/10.1007/978-1-4612-0999-7].
C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].
Acknowledgments
We are grateful to Alejandra Castro, Hongjie Chen, Pengxiang Hao, Wenxin Lai, and Fengjun Xu for helpful discussions. LA thanks the Asia Pacific Center for Theoretical Physics (APCTP) for hospitality during the focus program “Integrability, Duality and Related Topics”, as well as the Korea Institute for Advanced Study (KIAS) for hospitality during the “East Asia Joint Workshop on Fields and Strings 2022”, where part of this work was completed. BY thanks the Tsinghua Sanya International Mathematics Forum for hospitality during the workshop and research-in-team program on “Black holes, Quantum Chaos, and Solvable Quantum Systems”. The work of LA was supported by the Dutch Research Council (NWO) through the Scanning New Horizons programme (16SNH02). WS is supported by the national key research and development program of China No. 2020YFA0713000 and the Beijing Municipal Natural Science Foundation No. Z180003. BY is supported by NSFC Grant No. 11735001.
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Apolo, L., Song, W. & Yu, B. On the universal behavior of \( T\overline{T} \)-deformed CFTs: single and double-trace partition functions at large c. J. High Energ. Phys. 2023, 210 (2023). https://doi.org/10.1007/JHEP05(2023)210
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DOI: https://doi.org/10.1007/JHEP05(2023)210