Abstract
The conformal symmetry algebra in 2D (Diff(S1)⊕Diff(S1)) is shown to be related to its ultra/non-relativistic version (BMS3≈GCA2) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT2, the BMS3 generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, T and \( \overline{T} \), closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS3 becomes a bona fide symmetry once the CFT2 is marginally deformed by the addition of a \( \sqrt{T\overline{T}} \) term to the Hamiltonian. The generic deformed theory is manifestly invariant under diffeomorphisms and local scalings, but it is no longer a CFT2 because its energy and momentum densities fulfill the BMS3 algebra. The deformation can also be described through the original CFT2 on a curved metric whose Beltrami differentials are determined by the variation of the deformed Hamiltonian with respect to T and \( \overline{T} \). BMS3 symmetries then arise from deformed conformal Killing equations, corresponding to diffeomorphisms that preserve the deformed metric and stress-energy tensor up to local scalings. As an example, we briefly address the deformation of N free bosons, which coincides with ultra-relativistic limits only for N = 1. Furthermore, Cardy formula and the S-modular transformation of the torus become mapped to their corresponding BMS3 (or flat) versions.
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References
A. Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic Conformal Field Theories, Phys. Rev. Lett. 105 (2010) 171601 [arXiv:1006.3354] [INSPIRE].
A. Ashtekar, J. Bicak and B.G. Schmidt, Asymptotic structure of symmetry reduced general relativity, Phys. Rev. D 55 (1997) 669 [gr-qc/9608042] [INSPIRE].
G. Barnich and G. Compere, Classical central extension for asymptotic symmetries at null infinity in three spacetime dimensions, Class. Quant. Grav. 24 (2007) F15 [gr-qc/0610130] [INSPIRE].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups and BMS symmetry, Class. Quant. Grav. 31 (2014) 092001 [arXiv:1402.5894] [INSPIRE].
F. Lizzi, B. Rai, G. Sparano and A. Srivastava, Quantization of the Null String and Absence of Critical Dimensions, Phys. Lett. B 182 (1986) 326 [INSPIRE].
J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Quantum null (super)strings, Phys. Lett. B 225 (1989) 335 [INSPIRE].
J. Isberg, U. Lindström, B. Sundborg and G. Theodoridis, Classical and quantized tensionless strings, Nucl. Phys. B 411 (1994) 122 [hep-th/9307108] [INSPIRE].
G. Barnich, A. Gomberoff and H.A. González, Three-dimensional Bondi-Metzner-Sachs invariant two-dimensional field theories as the flat limit of Liouville theory, Phys. Rev. D 87 (2013) 124032 [arXiv:1210.0731] [INSPIRE].
G. Barnich and H.A. González, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity, JHEP 05 (2013) 016 [arXiv:1303.1075] [INSPIRE].
R.F. Penna, BMS3 invariant fluid dynamics at null infinity, Class. Quant. Grav. 35 (2018) 044002 [arXiv:1708.08470] [INSPIRE].
O. Fuentealba et al., Integrable systems with BMS3 Poisson structure and the dynamics of locally flat spacetimes, JHEP 01 (2018) 148 [arXiv:1711.02646] [INSPIRE].
A. Campoleoni, L. Ciambelli, C. Marteau, P.M. Petropoulos and K. Siampos, Two-dimensional fluids and their holographic duals, Nucl. Phys. B 946 (2019) 114692 [arXiv:1812.04019] [INSPIRE].
A. Bagchi and R. Gopakumar, Galilean Conformal Algebras and AdS/CFT, JHEP 07 (2009) 037 [arXiv:0902.1385] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
S.W. Hawking, M.J. Perry and A. Strominger, Soft Hair on Black Holes, Phys. Rev. Lett. 116 (2016) 231301 [arXiv:1601.00921] [INSPIRE].
L. Donnay, G. Giribet, H.A. González and M. Pino, Supertranslations and Superrotations at the Black Hole Horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
S. Carlip, Black Hole Entropy from Bondi-Metzner-Sachs Symmetry at the Horizon, Phys. Rev. Lett. 120 (2018) 101301 [arXiv:1702.04439] [INSPIRE].
R.F. Penna, Near-horizon BMS symmetries as fluid symmetries, JHEP 10 (2017) 049 [arXiv:1703.07382] [INSPIRE].
D. Grumiller, A. Pérez, M.M. Sheikh-Jabbari, R. Troncoso and C. Zwikel, Spacetime structure near generic horizons and soft hair, Phys. Rev. Lett. 124 (2020) 041601 [arXiv:1908.09833] [INSPIRE].
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: I. Induced representations, JHEP 06 (2014) 129 [arXiv:1403.5803] [INSPIRE].
G. Barnich and B. Oblak, Notes on the BMS group in three dimensions: II. Coadjoint representation, JHEP 03 (2015) 033 [arXiv:1502.00010] [INSPIRE].
A. Campoleoni, H.A. González, B. Oblak and M. Riegler, Rotating Higher Spin Partition Functions and Extended BMS Symmetries, JHEP 04 (2016) 034 [arXiv:1512.03353] [INSPIRE].
C. Batlle, V. Campello and J. Gomis, Canonical realization of (2 + 1)-dimensional Bondi-Metzner-Sachs symmetry, Phys. Rev. D 96 (2017) 025004 [arXiv:1703.01833] [INSPIRE].
G. Barnich, L. Donnay, J. Matulich and R. Troncoso, Asymptotic symmetries and dynamics of three-dimensional flat supergravity, JHEP 08 (2014) 071 [arXiv:1407.4275] [INSPIRE].
N. Banerjee, D.P. Jatkar, I. Lodato, S. Mukhi and T. Neogi, Extended Supersymmetric BMS3 algebras and Their Free Field Realisations, JHEP 11 (2016) 059 [arXiv:1609.09210] [INSPIRE].
I. Lodato and W. Merbis, Super-BMS3 algebras from \( \mathcal{N} \) = 2 flat supergravities, JHEP 11 (2016) 150 [arXiv:1610.07506] [INSPIRE].
N. Banerjee, I. Lodato and T. Neogi, N = 4 Supersymmetric BMS3 algebras from asymptotic symmetry analysis, Phys. Rev. D 96 (2017) 066029 [arXiv:1706.02922] [INSPIRE].
R. Basu, S. Detournay and M. Riegler, Spectral Flow in 3D Flat Spacetimes, JHEP 12 (2017) 134 [arXiv:1706.07438] [INSPIRE].
O. Fuentealba, J. Matulich and R. Troncoso, Asymptotic structure of \( \mathcal{N} \) = 2 supergravity in 3D: extended super-BMS3 and nonlinear energy bounds, JHEP 09 (2017) 030 [arXiv:1706.07542] [INSPIRE].
H. Afshar, A. Bagchi, R. Fareghbal, D. Grumiller and J. Rosseel, Spin-3 Gravity in Three-Dimensional Flat Space, Phys. Rev. Lett. 111 (2013) 121603 [arXiv:1307.4768] [INSPIRE].
H.A. González, J. Matulich, M. Pino and R. Troncoso, Asymptotically flat spacetimes in three-dimensional higher spin gravity, JHEP 09 (2013) 016 [arXiv:1307.5651] [INSPIRE].
O. Fuentealba, J. Matulich and R. Troncoso, Extension of the Poincaré group with half-integer spin generators: hypergravity and beyond, JHEP 09 (2015) 003 [arXiv:1505.06173] [INSPIRE].
O. Fuentealba, J. Matulich and R. Troncoso, Asymptotically flat structure of hypergravity in three spacetime dimensions, JHEP 10 (2015) 009 [arXiv:1508.04663] [INSPIRE].
R. Caroca, P. Concha, E. Rodríguez and P. Salgado-ReboLledó, Generalizing the \( \mathfrak{bms} \)3 and 2D-conformal algebras by expanding the Virasoro algebra, Eur. Phys. J. C 78 (2018) 262 [arXiv:1707.07209] [INSPIRE].
R. Caroca, P. Concha, J. Matulich, E. Rodríguez and D. Tempo, Hypersymmetric extensions of Maxwell-Chern-Simons gravity in (2 + 1) dimensions, Phys. Rev. D 104 (2021) 064011 [arXiv:2105.12243] [INSPIRE].
H. Adami, M.M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, Symmetries at null boundaries: two and three dimensional gravity cases, JHEP 10 (2020) 107 [arXiv:2007.12759] [INSPIRE].
L. Donnay, G. Giribet and F. Rosso, Quantum BMS transformations in conformally flat space-times and holography, JHEP 12 (2020) 102 [arXiv:2008.05483] [INSPIRE].
C. Batlle, V. Campello and J. Gomis, A canonical realization of the Weyl BMS symmetry, Phys. Lett. B 811 (2020) 135920 [arXiv:2008.10290] [INSPIRE].
O. Fuentealba, H.A. González, A. Pérez, D. Tempo and R. Troncoso, Superconformal Bondi-Metzner-Sachs Algebra in Three Dimensions, Phys. Rev. Lett. 126 (2021) 091602 [arXiv:2011.08197] [INSPIRE].
G. Barnich, A. Gomberoff and H.A. González, The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [INSPIRE].
A. Bagchi, R. Gopakumar, I. Mandal and A. Miwa, GCA in 2d, JHEP 08 (2010) 004 [arXiv:0912.1090] [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].
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].
O. Aharony, S. Datta, A. Giveon, Y. Jiang and D. Kutasov, Modular invariance and uniqueness of T\( \overline{T} \) deformed CFT, JHEP 01 (2019) 086 [arXiv:1808.02492] [INSPIRE].
V. Gorbenko, E. Silverstein and G. Torroba, dS/dS and T\( \overline{T} \), JHEP 03 (2019) 085 [arXiv:1811.07965] [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].
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. Jorjadze and S. Theisen, Canonical maps and integrability in T\( \overline{T} \) deformed 2d CFTs, arXiv:2001.03563 [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [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].
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 G. Hernández-Chifflet, T\( \overline{T} \) partition function from topological gravity, JHEP 09 (2018) 158 [arXiv:1805.07386] [INSPIRE].
R. Conti, S. Negro and R. Tateo, The T\( \overline{\mathrm{T}} \) perturbation and its geometric interpretation, JHEP 02 (2019) 085 [arXiv:1809.09593] [INSPIRE].
U. Lindström, B. Sundborg and G. Theodoridis, The Zero tension limit of the superstring, Phys. Lett. B 253 (1991) 319 [INSPIRE].
J. Isberg, U. Lindström and B. Sundborg, Space-time symmetries of quantized tensionless strings, Phys. Lett. B 293 (1992) 321 [hep-th/9207005] [INSPIRE].
A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Strings from Worldsheet Symmetries, JHEP 01 (2016) 158 [arXiv:1507.04361] [INSPIRE].
A. Bagchi, S. Chakrabortty and P. Parekh, Tensionless Superstrings: View from the Worldsheet, JHEP 10 (2016) 113 [arXiv:1606.09628] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.M. Zhang, Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time, Class. Quant. Grav. 31 (2014) 085016 [arXiv:1402.0657] [INSPIRE].
E. Bergshoeff, J. Gomis, B. Rollier, J. Rosseel and T. ter Veldhuis, Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [INSPIRE].
G. Barnich, Entropy of three-dimensional asymptotically flat cosmological solutions, JHEP 10 (2012) 095 [arXiv:1208.4371] [INSPIRE].
A. Bagchi, S. Detournay, R. Fareghbal and J. Simón, Holography of 3D Flat Cosmological Horizons, Phys. Rev. Lett. 110 (2013) 141302 [arXiv:1208.4372] [INSPIRE].
A. Bagchi and R. Basu, 3D Flat Holography: Entropy and Logarithmic Corrections, JHEP 03 (2014) 020 [arXiv:1312.5748] [INSPIRE].
M. Riegler, Flat space limit of higher-spin Cardy formula, Phys. Rev. D 91 (2015) 024044 [arXiv:1408.6931] [INSPIRE].
G. Barnich, C. Troessaert, D. Tempo and R. Troncoso, Asymptotically locally flat spacetimes and dynamical nonspherically-symmetric black holes in three dimensions, Phys. Rev. D 93 (2016) 084001 [arXiv:1512.05410] [INSPIRE].
C. Troessaert, D. Tempo and R. Troncoso, Asymptotically flat black holes and gravitational waves in three-dimensional massive gravity, in 8th Aegean Summer School: Gravitational Waves: From Theory to Observations, (2015) [arXiv:1512.09046] [INSPIRE].
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Rodríguez, P., Tempo, D. & Troncoso, R. Mapping relativistic to ultra/non-relativistic conformal symmetries in 2D and finite \( \sqrt{T\overline{T}} \) deformations. J. High Energ. Phys. 2021, 133 (2021). https://doi.org/10.1007/JHEP11(2021)133
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DOI: https://doi.org/10.1007/JHEP11(2021)133