Abstract
We demonstrate the presence of modular properties in partition functions of \( T\overline{T} \) deformed conformal field theories. These properties are verified explicitly for the deformed free boson. The modular features facilitate a derivation of the asymptotic density of states in these theories, which turns out to interpolate between Cardy and Hagedorn behaviours. We also point out a sub-sector of the spectrum that remains undeformed under the \( T\overline{T} \) flow. Finally, we comment on the deformation of the CFT vacuum character and its implications for the holographic dual.
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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. 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].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as a stochastic process, arXiv:1801.06895 [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, A solvable irrelevant deformation of AdS 3 /CFT 2, JHEP 12 (2017) 155 [arXiv:1707.05800] [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \) -deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \) in two-dimensional quantum field theory, hep-th/0401146 [INSPIRE].
S. Dubovsky, V. Gorbenko and M. Mirbabayi, Asymptotic fragility, near AdS 2 holography and \( T\overline{T} \), JHEP 09 (2017) 136 [arXiv:1706.06604] [INSPIRE].
S. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \) Partition Function from Topological Gravity, arXiv:1805.07386 [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] [INSPIRE].
M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, Quantisation of the effective string with TBA, JHEP 07 (2013) 071 [arXiv:1305.1278] [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, arXiv:1710.08415 [INSPIRE].
V. Shyam, Background independent holographic dual to \( T\overline{T} \) deformed CFT with large central charge in 2 dimensions, JHEP 10 (2017) 108 [arXiv:1707.08118] [INSPIRE].
M. Asrat, A. Giveon, N. Itzhaki and D. Kutasov, Holography Beyond AdS, Nucl. Phys. B 932 (2018) 241 [arXiv:1711.02690] [INSPIRE].
G. Giribet, \( T\overline{T} \) -deformations, AdS/CFT and correlation functions, JHEP 02 (2018) 114 [arXiv:1711.02716] [INSPIRE].
P. Kraus, J. Liu and D. Marolf, Cutoff AdS 3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [INSPIRE].
O. Aharony and T. Vaknin, The TT* deformation at large central charge, JHEP 05 (2018) 166 [arXiv:1803.00100] [INSPIRE].
A. Bzowski and M. Guica, The holographic interpretation of \( J\overline{T} \) -deformed CFTs, arXiv:1803.09753 [INSPIRE].
S. Chakraborty, A. Giveon, N. Itzhaki and D. Kutasov, Entanglement Beyond AdS, arXiv:1805.06286 [INSPIRE].
M. Baggio and A. Sfondrini, Strings on NS-NS Backgrounds as Integrable Deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].
A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 07 (2018) 109 [arXiv:1806.00422] [INSPIRE].
J. L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186.
R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B 493 (1997) 588 [hep-th/9609022] [INSPIRE].
S. Datta, J.R. David and S.P. Kumar, Conformal perturbation theory and higher spin entanglement entropy on the torus, JHEP 04 (2015) 041 [arXiv:1412.3946] [INSPIRE].
N.J. Iles and G.M.T. Watts, Modular properties of characters of the W 3 algebra, JHEP 01 (2016) 089 [arXiv:1411.4039] [INSPIRE].
M. Kaneko and D. Zagier, A generalized jacobi theta function and quasimodular forms, in The moduli space of curves, Springer, Heidelberg Germany (1995), pg. 165.
W.-q. Wang, W(1+infinity) algebra W(3) algebra and Friedan-Martinec-Shenker bosonization, Commun. Math. Phys. 195 (1998) 95 [q-alg/9708008] [INSPIRE].
P. Kraus and A. Maloney, A cardy formula for three-point coefficients or how the black hole got its spots, JHEP 05 (2017) 160 [arXiv:1608.03284] [INSPIRE].
J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of modular forms: Lectures at a Summer School in Nordfjordeid, Norway, Springer Science & Business Media, Berlin Germany (2008).
K. Dietz and T. Filk, On the Renormalization of String Functionals, Phys. Rev. D 27 (1983) 2944.
M. Lüscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, JHEP 07 (2004) 014 [hep-th/0406205] [INSPIRE].
M. Billó and M. Caselle, Polyakov loop correlators from D0-brane interactions in bosonic string theory, JHEP 07 (2005) 038 [hep-th/0505201] [INSPIRE].
M. Billó, M. Caselle and L. Ferro, The Partition function of interfaces from the Nambu-Goto effective string theory, JHEP 02 (2006) 070 [hep-th/0601191] [INSPIRE].
M. Billó, M. Caselle and R. Pellegrini, New numerical results and novel effective string predictions for Wilson loops, JHEP 01 (2012) 104 [Erratum ibid. 1304 (2013) 097] [arXiv:1107.4356] [INSPIRE].
E. Witten, Elliptic Genera and Quantum Field Theory, Commun. Math. Phys. 109 (1987) 525.
D. Brattan, J. Camps, R. Loganayagam and M. Rangamani, CFT dual of the AdS Dirichlet problem : Fluid/Gravity on cut-off surfaces, JHEP 12 (2011) 090 [arXiv:1106.2577] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
F. Denef, S.A. Hartnoll and S. Sachdev, Black hole determinants and quasinormal modes, Class. Quant. Grav. 27 (2010) 125001 [arXiv:0908.2657] [INSPIRE].
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].
S. Datta and J.R. David, Higher Spin Quasinormal Modes and One-Loop Determinants in the BTZ black Hole, JHEP 03 (2012) 079 [arXiv:1112.4619] [INSPIRE].
D.J. Gross, High-energy symmetries of string theory, Phys. Rev. Lett. 60 (1988) 1229.
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Datta, S., Jiang, Y. \( T\overline{T} \) deformed partition functions. J. High Energ. Phys. 2018, 106 (2018). https://doi.org/10.1007/JHEP08(2018)106
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DOI: https://doi.org/10.1007/JHEP08(2018)106