Abstract
We show that method of characteristics provides a powerful new point of view on \( T\overline{T} \)-and related deformations. Previously, the method of characteristics has been applied to \( T\overline{T} \)-deformation mainly to solve Burgers’ equation, which governs the deformation of the quantum spectrum. In the current work, we study classical deformed quantities using this method and show that \( T\overline{T} \) flow can be seen as a characteristic flow. Exploiting this point of view, we re-derive a number of important known results and obtain interesting new ones. We prove the equivalence between dynamical change of coordinates and the generalized light-cone gauge approaches to \( T\overline{T} \)-deformation. We find the deformed Lagrangians for a class of \( T\overline{T} \)-like deformations in higher dimensions and the (\( T\overline{T} \))α-deformation in 2d with generic α, generalizing recent results in [1] and [2].
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
R. Conti, J. Romano and R. Tateo, Metric approach to a \( \textrm{T}\overline{\textrm{T}} \)-like deformation in arbitrary dimensions, JHEP 09 (2022) 085 [arXiv:2206.03415] [INSPIRE].
C. Ferko, A. Sfondrini, L. Smith and G. Tartaglino-Mazzucchelli, Root-\( T\overline{T} \) Deformations in Two-Dimensional Quantum Field Theories, Phys. Rev. Lett. 129 (2022) 201604 [arXiv:2206.10515] [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].
Y. Jiang, A pedagogical review on solvable irrelevant deformations of 2D quantum field theory, Commun. Theor. Phys. 73 (2021) 057201 [arXiv:1904.13376] [INSPIRE].
J. Cardy, \( T\overline{T} \) deformations of non-Lorentz invariant field theories, arXiv:1809.07849 [INSPIRE].
Y. Jiang, \( \textrm{T}\overline{\textrm{T}} \)-deformed 1d Bose gas, SciPost Phys. 12 (2022) 191 [arXiv:2011.00637] [INSPIRE].
B. Chen, J. Hou and J. Tian, Note on the nonrelativistic TT¯ deformation, Phys. Rev. D 104 (2021) 025004 [arXiv:2012.14091] [INSPIRE].
B. Pozsgay, Y. Jiang and G. Takács, \( T\overline{T} \)-deformation and long range spin chains, JHEP 03 (2020) 092 [arXiv:1911.11118] [INSPIRE].
E. Marchetto, A. Sfondrini and Z. Yang, \( T\overline{T} \) Deformations and Integrable Spin Chains, Phys. Rev. Lett. 124 (2020) 100601 [arXiv:1911.12315] [INSPIRE].
Y. Jiang, F. Loebbert and D.-L. Zhong, Irrelevant deformations with boundaries and defects, J. Stat. Mech. 2204 (2022) 043102 [arXiv:2109.13180] [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].
P. Kraus, J. Liu and D. Marolf, Cutoff AdS3 versus the \( T\overline{T} \) deformation, JHEP 07 (2018) 027 [arXiv:1801.02714] [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. Guica and R. Monten, \( T\overline{T} \) and the mirage of a bulk cutoff, SciPost Phys. 10 (2021) 024 [arXiv:1906.11251] [INSPIRE].
G. Jafari, A. Naseh and H. Zolfi, Path Integral Optimization for \( T\overline{T} \) Deformation, Phys. Rev. D 101 (2020) 026007 [arXiv:1909.02357] [INSPIRE].
S. Khoeini-Moghaddam, F. Omidi and C. Paul, Aspects of Hyperscaling Violating Geometries at Finite Cutoff, JHEP 02 (2021) 121 [arXiv:2011.00305] [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. Frolov, \( T\overline{T} \) Deformation and the Light-Cone Gauge, Proc. Steklov Inst. Math. 309 (2020) 107 [arXiv:1905.07946] [INSPIRE].
C. Esper and S. Frolov, \( T\overline{T} \) deformations of non-relativistic models, JHEP 06 (2021) 101 [arXiv:2102.12435] [INSPIRE].
A. Sfondrini and S.J. van Tongeren, \( T\overline{T} \) deformations as TsT transformations, Phys. Rev. D 101 (2020) 066022 [arXiv:1908.09299] [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].
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].
J. Cardy, The \( T\overline{T} \) deformation of quantum field theory as random geometry, JHEP 10 (2018) 186 [arXiv:1801.06895] [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].
A.J. Tolley, \( T\overline{T} \) deformations, massive gravity and non-critical strings, JHEP 06 (2020) 050 [arXiv:1911.06142] [INSPIRE].
R. Conti, S. Negro and R. Tateo, The \( \textrm{T}\overline{\textrm{T}} \) perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [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].
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].
P. Ceschin, R. Conti and R. Tateo, \( \textrm{T}\overline{\textrm{T}} \)-deformed nonlinear Schrödinger, JHEP 04 (2021) 121 [arXiv:2012.12760] [INSPIRE].
J. Cardy and B. Doyon, \( T\overline{T} \) deformations and the width of fundamental particles, JHEP 04 (2022) 136 [arXiv:2010.15733] [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
S. Ebert, H.-Y. Sun and Z. Sun, \( T\overline{T} \) deformation in SCFTs and integrable supersymmetric theories, JHEP 09 (2021) 082 [arXiv:2011.07618] [INSPIRE].
H. Babaei-Aghbolagh, K.B. Velni, D.M. Yekta and H. Mohammadzadeh, Emergence of non-linear electrodynamic theories from TT−-like deformations, Phys. Lett. B 829 (2022) 137079 [arXiv:2202.11156] [INSPIRE].
A. Banerjee, A. Bhattacharyya and S. Chakraborty, Entanglement Entropy for TT deformed CFT in general dimensions, Nucl. Phys. B 948 (2019) 114775 [arXiv:1904.00716] [INSPIRE].
M. Taylor, TT deformations in general dimensions, arXiv:1805.10287 [INSPIRE].
J. Caetano, W. Peelaers and L. Rastelli, Maximally supersymmetric RG flows in 4D and integrability, JHEP 12 (2021) 119 [arXiv:2006.04792] [INSPIRE].
H. Babaei-Aghbolagh, K. Babaei Velni, D.M. Yekta and H. Mohammadzadeh, \( T\overline{T} \)-like flows in non-linear electrodynamic theories and S-duality, JHEP 04 (2021) 187 [arXiv:2012.13636] [INSPIRE].
C. Ferko, L. Smith and G. Tartaglino-Mazzucchelli, On Current-Squared Flows and ModMax Theories, SciPost Phys. 13 (2022) 012 [arXiv:2203.01085] [INSPIRE].
H. Babaei-Aghbolagh, K. Babaei Velni, D. Mahdavian Yekta and H. Mohammadzadeh, Marginal TT−-like deformation and modified Maxwell theories in two dimensions, Phys. Rev. D 106 (2022) 086022 [arXiv:2206.12677] [INSPIRE].
I. Bandos, K. Lechner, D. Sorokin and P.K. Townsend, A non-linear duality-invariant conformal extension of Maxwell’s equations, Phys. Rev. D 102 (2020) 121703 [arXiv:2007.09092] [INSPIRE].
B.P. Kosyakov, Nonlinear electrodynamics with the maximum allowable symmetries, Phys. Lett. B 810 (2020) 135840 [arXiv:2007.13878] [INSPIRE].
I. Bandos, K. Lechner, D. Sorokin and P.K. Townsend, On p-form gauge theories and their conformal limits, JHEP 03 (2021) 022 [arXiv:2012.09286] [INSPIRE].
P. Rodríguez, D. Tempo and R. Troncoso, Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite \( \sqrt{T\overline{T}} \) deformations, JHEP 11 (2021) 133 [arXiv:2106.09750] [INSPIRE].
A. Bagchi, A. Banerjee and H. Muraki, Boosting to BMS, JHEP 09 (2022) 251 [arXiv:2205.05094] [INSPIRE].
J. Levandosky, First-Order Equations: Method of Characteristics, (2002), https://web.stanford.edu/class/math220a/handouts/firstorder.pdf.
D. Pavshinkin, \( T\overline{T} \) deformation of Calogero-Sutherland model via dimensional reduction, arXiv:2111.12080 [INSPIRE].
W. Donnelly and V. Shyam, Entanglement entropy and \( T\overline{T} \) deformation, Phys. Rev. Lett. 121 (2018) 131602 [arXiv:1806.07444] [INSPIRE].
P. Caputa, S. Datta and V. Shyam, Sphere partition functions \& cut-off AdS, JHEP 05 (2019) 112 [arXiv:1902.10893] [INSPIRE].
B. Chen, J. Hou and J. Tian, Lax connections in \( T\overline{T} \)-deformed integrable field theories, Chin. Phys. C 45 (2021) 093112 [arXiv:2102.01470] [INSPIRE].
G. Giribet, \( T\overline{T} \)-deformations, AdS/CFT and correlation functions, JHEP 02 (2018) 114 [arXiv:1711.02716] [INSPIRE].
S. He and H. Shu, Correlation functions, entanglement and chaos in the \( T\overline{T}/J\overline{T} \)-deformed CFTs, JHEP 02 (2020) 088 [arXiv:1907.12603] [INSPIRE].
S. He and Y. Sun, Correlation functions of CFTs on a torus with a \( T\overline{T} \) deformation, Phys. Rev. D 102 (2020) 026023 [arXiv:2004.07486] [INSPIRE].
S. He, Note on higher-point correlation functions of the \( T\overline{T} \) or \( J\overline{T} \) deformed CFTs, Sci. China Phys. Mech. Astron. 64 (2021) 291011 [arXiv:2012.06202] [INSPIRE].
C. Ferko and S. Sethi, Sequential Flows by Irrelevant Operators, arXiv:2206.04787 [INSPIRE].
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: 2208.05391
Rights and permissions
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.
About this article
Cite this article
Hou, J. \( T\overline{T} \) flow as characteristic flows. J. High Energ. Phys. 2023, 243 (2023). https://doi.org/10.1007/JHEP03(2023)243
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2023)243