Abstract
The \( T\overline{T} \) deformation of a 2 dimensional field theory living on a curved space- time is equivalent to coupling the undeformed field theory to 2 dimensional ‘ghost-free’ massive gravity. We derive the equivalence classically, and using a path integral formulation of the random geometries proposal, which mirrors the holographic bulk cutoff picture. We emphasize the role of the massive gravity Stückelberg fields which describe the diffeomorphism between the two metrics. For a general field theory, the dynamics of the Stückelberg fields is non-trivial, however for a CFT it trivializes and becomes equivalent to an additional pair of target space dimensions with associated curved target space geometry and dynamical worldsheet metric. That is, the \( T\overline{T} \) deformation of a CFT on curved spacetime is equivalent to a non-critical string theory in Polyakov form, with a non-zero B-field. We give a direct proof of the equivalence classically without relying on gauge fixing, and determine the explicit form for the classical Hamiltonian of the \( T\overline{T} \) deformation of an arbitrary CFT on a curved spacetime. When the QFT action is a sum of a CFT plus an operator of fixed scaling dimension, as for example in the sine-Gordon model, the equivalence to a non-critical theory string holds with a modified target space metric and modified B-field. Finally we give a stochastic path integral formulation for the general \( T\overline{T}+J\overline{T}+T\overline{J} \) deformation of a general QFT, and show that it reproduces a recent path integral proposal in the literature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
L. Castillejo, R.H. Dalitz and F.J. Dyson, Low’s scattering equation for the charged and neutral scalar theories, Phys. Rev. 101 (1956) 453 [INSPIRE].
A.B. Zamolodchikov, Expectation value of composite field \( T\overline{T} \)in two-dimensional quantum field theory, hep-th/0401146 [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. 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 random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [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. Baggio and A. Sfondrini, Strings on NS-NS backgrounds as integrable deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].
S. Frolov, \( T\overline{T} \)deformation and the light-cone gauge, arXiv:1905.07946 [INSPIRE].
S. Frolov, \( T\overline{T},\tilde{J}J, JT \)and \( \tilde{J}T \)deformations, J. Phys. A 53 (2020) 025401 [arXiv:1907.12117] [INSPIRE].
A. Sfondrini and S.J. van Tongeren, \( T\overline{T} \)deformations as T sT transformations, Phys. Rev. D 101 (2020) 066022 [arXiv:1908.09299] [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].
R. Conti, S. Negro and R. Tateo, The \( T\overline{T} \)perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [INSPIRE].
R. Conti, L. Iannella, S. Negro and R. Tateo, Generalised Born-Infeld models, Lax operators and the \( T\overline{T} \)perturbation, JHEP 11 (2018) 007 [arXiv:1806.11515] [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
E.A. Coleman, J. Aguilera-Damia, D.Z. Freedman and R.M. Soni, \( T\overline{T} \)-deformed actions and (1, 1) supersymmetry, JHEP 10 (2019) 080 [arXiv:1906.05439] [INSPIRE].
V. Rosenhaus and M. Smolkin, Integrability and renormalization under \( T\overline{T} \), arXiv:1909.02640 [INSPIRE].
M. Guica, An integrable Lorentz-breaking deformation of two-dimensional CFTs, SciPost Phys. 5 (2018) 048 [arXiv:1710.08415] [INSPIRE].
M. Guica, On correlation functions in \( J\overline{T} \)-deformed CFTs, J. Phys. A 52 (2019) 184003 [arXiv:1902.01434] [INSPIRE].
B. Le Floch and M. Mezei, Solving a family of \( T\overline{T} \)-like theories, arXiv:1903.07606 [INSPIRE].
T. Anous and M. Guica, A general definition of JTa — deformed QFTs, arXiv:1911.02031 [INSPIRE].
J. Aguilera-Damia, V.I. Giraldo-Rivera, E.A. Mazenc, I. Salazar Landea and R.M. Soni, A path integral realization of joint \( J\overline{T},T\overline{J} \)and \( T\overline{T} \)flows, arXiv:1910.06675 [INSPIRE].
R. Conti, S. Negro and R. Tateo, Conserved currents and \( T{\overline{T}}_s \)irrelevant deformations of 2D integrable field theories, JHEP 11 (2019) 120 [arXiv:1904.09141] [INSPIRE].
B. Le Floch and M. Mezei, KdV charges in \( T\overline{T} \)theories and new models with super-Hagedorn behavior, SciPost Phys. 7 (2019) 043 [arXiv:1907.02516] [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].
A. Bzowski and M. Guica, The holographic interpretation of \( J\overline{T} \)-deformed CFTs, JHEP 01 (2019) 198 [arXiv:1803.09753] [INSPIRE].
M. Guica and R. Monten, \( T\overline{T} \)and the mirage of a bulk cutoff, arXiv:1906.11251 [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].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and \( T\overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [INSPIRE].
X. Dong, E. Silverstein and G. Torroba, De Sitter holography and entanglement entropy, JHEP 07 (2018) 050 [arXiv:1804.08623] [INSPIRE].
W. Donnelly, E. LePage, Y.-Y. Li, A. Pereira and V. Shyam, Quantum corrections to finite radius holography and holographic entanglement entropy, JHEP 05 (2020) 006 [arXiv:1909.11402] [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and \( T\overline{T} \)deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
T.R. Araujo, Nonlocal charges from marginal deformations of 2D CFTs: holographic \( T\overline{T} \)and \( T\overline{J} \)and Yang-Baxter deformations, Phys. Rev. D 101 (2020) 025008 [arXiv:1909.08149] [INSPIRE].
L. Apolo and W. Song, Heating up holography for single-trace \( J\overline{T} \)deformations, JHEP 01 (2020) 141 [arXiv:1907.03745] [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].
S. Grieninger, Entanglement entropy and \( T\overline{T} \)deformations beyond antipodal points from holography, JHEP 11 (2019) 171 [arXiv:1908.10372] [INSPIRE].
H. Geng, Some information theoretic aspects of de-Sitter holography, JHEP 02 (2020) 005 [arXiv:1911.02644] [INSPIRE].
H. Geng, \( T\overline{T} \)deformation and the complexity=volume conjecture, arXiv:1910.08082 [INSPIRE].
H. Geng, S. Grieninger and A. Karch, Entropy, entanglement and swampland bounds in DS/dS, JHEP 06 (2019) 105 [arXiv:1904.02170] [INSPIRE].
A. Giveon, N. Itzhaki and D. Kutasov, \( T\overline{T} \)and LST, JHEP 07 (2017) 122 [arXiv:1701.05576] [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].
J. Cardy, \( T\overline{T} \)deformations of non-Lorentz invariant field theories, arXiv:1809.07849 [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [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].
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].
Y. Jiang, Expectation value of \( T\overline{T} \)operator in curved spacetimes, JHEP 02 (2020) 094 [arXiv:1903.07561] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Resummation of massive gravity, Phys. Rev. Lett. 106 (2011) 231101 [arXiv:1011.1232] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian structure in two space-time dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nucl. Phys. B 252 (1985) 343 [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. Dubovsky, V. Gorbenko and G. Hernández-Chifflet, \( T\overline{T} \)partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
J. Cardy, \( T\overline{T} \)deformation of correlation functions, JHEP 12 (2019) 160 [arXiv:1907.03394] [INSPIRE].
L. Freidel, Reconstructing AdS/CFT, arXiv:0804.0632 [INSPIRE].
C. de Rham, Massive gravity, Living Rev. Rel. 17 (2014) 7 [arXiv:1401.4173] [INSPIRE].
K. Hinterbichler and R.A. Rosen, Interacting Spin-2 fields, JHEP 07 (2012) 047 [arXiv:1203.5783] [INSPIRE].
N.A. Ondo and A.J. Tolley, Complete decoupling limit of ghost-free massive gravity, JHEP 11 (2013) 059 [arXiv:1307.4769] [INSPIRE].
M. Fasiello and A.J. Tolley, Cosmological stability bound in massive gravity and bigravity, JCAP 12 (2013) 002 [arXiv:1308.1647] [INSPIRE].
G. Gabadadze, K. Hinterbichler, D. Pirtskhalava and Y. Shang, Potential for general relativity and its geometry, Phys. Rev. D 88 (2013) 084003 [arXiv:1307.2245] [INSPIRE].
N. Arkani-Hamed, H. Georgi and M.D. Schwartz, Effective field theory for massive gravitons and gravity in theory space, Annals Phys. 305 (2003) 96 [hep-th/0210184] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Ghost free massive gravity in the Stückelberg language, Phys. Lett. B 711 (2012) 190 [arXiv:1107.3820] [INSPIRE].
L. Alberte and A. Khmelnitsky, Reduced massive gravity with two Stückelberg fields, Phys. Rev. D 88 (2013) 064053 [arXiv:1303.4958] [INSPIRE].
C. de Rham, A.J. Tolley and S.-Y. Zhou, Non-compact nonlinear σ-models, Phys. Lett. B 760 (2016) 579 [arXiv:1512.06838] [INSPIRE].
C. de Rham, A.J. Tolley and S.-Y. Zhou, The Λ2 limit of massive gravity, JHEP 04 (2016) 188 [arXiv:1602.03721] [INSPIRE].
C. de Rham and S. Renaux-Petel, Massive gravity on de Sitter and unique candidate for partially massless gravity, JCAP 01 (2013) 035 [arXiv:1206.3482] [INSPIRE].
C. De Rham, K. Hinterbichler and L.A. Johnson, On the (A)dS decoupling limits of massive gravity, JHEP 09 (2018) 154 [arXiv:1807.08754] [INSPIRE].
C. de Rham, G. Gabadadze and A.J. Tolley, Helicity decomposition of ghost-free massive gravity, JHEP 11 (2011) 093 [arXiv:1108.4521] [INSPIRE].
S.F. Hassan, R.A. Rosen and A. Schmidt-May, Ghost-free massive gravity with a general reference metric, JHEP 02 (2012) 026 [arXiv:1109.3230] [INSPIRE].
D.G. Boulware and S. Deser, Can gravitation have a finite range?, Phys. Rev. D 6 (1972) 3368 [INSPIRE].
L. Brink, P. Di Vecchia and P.S. Howe, A locally supersymmetric and reparametrization invariant action for the spinning string, Phys. Lett. B 65 (1976) 471 [INSPIRE].
S. Deser and B. Zumino, A complete action for the spinning string, Phys. Lett. B 65 (1976) 369 [INSPIRE].
A. Hashimoto and D. Kutasov, \( T\overline{T},J\overline{T},T\overline{J} \)partition sums from string theory, JHEP 02 (2020) 080 [arXiv:1907.07221] [INSPIRE].
T.L. Curtright, D.B. Fairlie and H. Alshal, A Galileon primer, arXiv:1212.6972 [INSPIRE].
C. de Rham, M. Fasiello and A.J. Tolley, Galileon duality, Phys. Lett. B 733 (2014) 46 [arXiv:1308.2702] [INSPIRE].
C. De Rham, L. Keltner and A.J. Tolley, Generalized Galileon duality, Phys. Rev. D 90 (2014) 024050 [arXiv:1403.3690] [INSPIRE].
J. Polchinski and A. Strominger, Effective string theory, Phys. Rev. Lett. 67 (1991) 1681 [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Effective string theory revisited, JHEP 09 (2012) 044 [arXiv:1203.1054] [INSPIRE].
S. Hellerman, S. Maeda, J. Maltz and I. Swanson, Effective string theory simplified, JHEP 09 (2014) 183 [arXiv:1405.6197] [INSPIRE].
M. Dodelson, E. Silverstein and G. Torroba, Varying dilaton as a tracer of classical string interactions, Phys. Rev. D 96 (2017) 066011 [arXiv:1704.02625] [INSPIRE].
A.M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981) 207 [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
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 1911.06142
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Tolley, A.J. \( T\overline{T} \) deformations, massive gravity and non-critical strings. J. High Energ. Phys. 2020, 50 (2020). https://doi.org/10.1007/JHEP06(2020)050
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2020)050