Abstract
We consider a one-parameter family of composite fields — bi-linear in the components of the stress-energy tensor — which generalise the \( \mathrm{T}\overline{\mathrm{T}} \) operator to arbitrary space-time dimension d ≥ 2. We show that they induce a deformation of the classical action which is equivalent — at the level of the dynamics — to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any d > 2, thus implying that the corresponding deformation can not be interpreted as a coordinate transformation. The central part of the paper is devoted to the development of a recursive algorithm to compute the coefficients of the power series expansion of the solution to the metric flow equation. We show that, under some quite restrictive assumptions on the stress-energy tensor, the power series yields an exact solution. Finally, we consider a class of theories in d = 4 whose stress-energy tensor fulfils the assumptions above mentioned, namely the family of abelian gauge theories in d = 4. For such theories, we obtain the exact expression of the deformed metric and the vierbein. In particular, the latter result implies that ModMax theory in a specific curved space is dynamically equivalent to its Born-Infeld-like extension in flat space. We also discuss a dimensional reduction of the latter theories from d = 4 to d = 2 in which an interesting marginal deformation of d = 2 field theories emerges.
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References
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].
M. Caselle, D. Fioravanti, F. Gliozzi and R. Tateo, Quantisation of the effective string with TBA, JHEP 07 (2013) 071 [arXiv:1305.1278] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity, JHEP 09 (2012) 133 [arXiv:1205.6805] [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].
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].
R. Conti, S. Negro and R. Tateo, The \( T\overline{T} \) perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [INSPIRE].
P. Caputa, S. Datta, Y. Jiang and P. Kraus, Geometrizing \( T\overline{T} \), JHEP 03 (2021) 140 [arXiv:2011.04664] [INSPIRE].
P. Ceschin, R. Conti and R. Tateo, \( T\overline{T} \)-deformed nonlinear Schrödinger, JHEP 04 (2021) 121 [arXiv:2012.12760] [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].
S. Frolov, \( T\overline{T} \) deformation and the light-cone gauge, Proc. Steklov Inst. Math. 309 (2020) 107 [arXiv:1905.07946] [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].
R. Conti, L. Iannella, S. Negro and R. Tateo, Generalised Born-Infeld models, Lax operators and the \( \mathrm{T}\overline{\mathrm{T}} \) perturbation, JHEP 11 (2018) 007 [arXiv:1806.11515] [INSPIRE].
J. Plebanski, Lectures on non-linear electrodynamics, unpublished (1970).
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].
C. Ferko, L. Smith and G. Tartaglino-Mazzucchelli, On current-squared flows and ModMax theories, SciPost Phys. 13 (2022) 012 [arXiv:2203.01085] [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].
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].
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].
L. Santilli, R.J. Szabo and M. Tierz, \( T\overline{T} \)-deformation of q-Yang-Mills theory, JHEP 11 (2020) 086 [arXiv:2009.00657] [INSPIRE].
L. Griguolo, R. Panerai, J. Papalini and D. Seminara, Exact TT deformation of two-dimensional Maxwell theory, Phys. Rev. Lett. 128 (2022) 221601 [arXiv:2203.09683] [INSPIRE].
H. Babaei-Aghbolagh, K.B. 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].
I. Mezo, The Lambert W function: its generalizations and applications, CRC Press (2022).
B.M. Barbashov and N.A. Chernikov, Scattering of two plane electromagnetic waves in the non-linear Born-Infeld electrodynamics, Commun. Math. Phys. 3 (1966) 313.
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, arXiv:2205.05094 [INSPIRE].
G. Bonelli, N. Doroud and M. Zhu, \( T\overline{T} \)-deformations in closed form, JHEP 06 (2018) 149 [arXiv:1804.10967] [INSPIRE].
M. Baggio, A. Sfondrini, G. Tartaglino-Mazzucchelli and H. Walsh, On \( T\overline{T} \) deformations and supersymmetry, JHEP 06 (2019) 063 [arXiv:1811.00533] [INSPIRE].
C.-K. Chang, C. Ferko and S. Sethi, Supersymmetry and \( T\overline{T} \) deformations, JHEP 04 (2019) 131 [arXiv:1811.01895] [INSPIRE].
C.-K. Chang, C. Ferko, S. Sethi, A. Sfondrini and G. Tartaglino-Mazzucchelli, \( T\overline{T} \) flows and (2, 2) supersymmetry, Phys. Rev. D 101 (2020) 026008 [arXiv:1906.00467] [INSPIRE].
N. Ondo and V. Shyam, The role of dRGT mass terms in cutoff holography and the Randall–Sundrum II scenario, arXiv:2206.04005 [INSPIRE].
C. Ferko, A. Sfondrini, L. Smith and G. Tartaglino-Mazzucchelli, Root-\( T\overline{T} \) deformations, arXiv:2206.10515 [INSPIRE].
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Conti, R., Romano, J. & Tateo, R. Metric approach to a \( \mathrm{T}\overline{\mathrm{T}} \)-like deformation in arbitrary dimensions. J. High Energ. Phys. 2022, 85 (2022). https://doi.org/10.1007/JHEP09(2022)085
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DOI: https://doi.org/10.1007/JHEP09(2022)085