Abstract
The isomorphism between the (extended) BMS4 algebra and the 1 + 2D Carrollian conformal algebra hints towards a co-dimension one formalism of flat holography with the field theory residing on the null-boundary of the asymptotically flat space-time enjoying a 1 + 2D Carrollian conformal symmetry. Motivated by this fact, we study the general symmetry properties of a source-less 1 + 2D Carrollian CFT, adopting a purely field-theoretic approach. After deriving the position-space Ward identities, we show how the 1 + 3D bulk super-translation and the super-rotation memory effects emerge from them, manifested by the presence of a temporal step-function factor in the same. Temporal-Fourier transforming these memory effect equations, we directly reach the bulk null-momentum-space leading and sub-leading soft graviton theorems. Along the way, we construct six Carrollian fields \( {S}_0^{\pm } \), \( {S}_1^{\pm } \), T and \( \overline{T} \) corresponding to these soft graviton fields and the Celestial stress-tensors, purely in terms of the Carrollian stress-tensor components. The 2D Celestial shadow-relations and the null-state conditions arise as two natural byproducts of these constructions. We then show that those six fields consist of the modes that implement the super-rotations and a subset of the super-translations on the quantum fields. The temporal step-function allows us to relate the operator product expansions (OPEs) with the operator commutation relations via a complex contour integral prescription. We deduce that not all of those six fields can be taken together to form consistent OPEs. So choosing \( {S}_0^{+} \), \( {S}_1^{+} \) and T as the local fields, we form their mutual OPEs using only the OPE-commutativity property, under two general assumptions. The symmetry algebra manifest in these holomorphic-sector OPEs is then shown to be Vir \( \overset{\wedge }{\ltimes \overline{\textrm{sl}\left(2,{\mathbb{R}}\right)}} \) with an abelian ideal.
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References
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].
S. Banerjee, Symmetries of free massless particles and soft theorems, Gen. Rel. Grav. 51 (2019) 128 [arXiv:1804.06646] [INSPIRE].
S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [arXiv:2108.04801] [INSPIRE].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
G. Travaglini et al., The SAGEX review on scattering amplitudes, J. Phys. A 55 (2022) 443001 [arXiv:2203.13011] [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity S-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].
T. He, D. Kapec, A.-M. Raclariu and A. Strominger, Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity, JHEP 08 (2017) 050 [arXiv:1701.00496] [INSPIRE].
L. Donnay, K. Nguyen and R. Ruzziconi, Loop-corrected subleading soft theorem and the celestial stress tensor, JHEP 09 (2022) 063 [arXiv:2205.11477] [INSPIRE].
H. Bondi, M.G.J. van der Burg and A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems, Proc. Roy. Soc. Lond. A 269 (1962) 21 [INSPIRE].
R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times, Proc. Roy. Soc. Lond. A 270 (1962) 103 [INSPIRE].
R. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [INSPIRE].
G. Barnich and C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited, Phys. Rev. Lett. 105 (2010) 111103 [arXiv:0909.2617] [INSPIRE].
G. Barnich and C. Troessaert, Supertranslations call for superrotations, PoS CNCFG2010 (2010) 010 [arXiv:1102.4632] [INSPIRE].
G. Barnich and C. Troessaert, Aspects of the BMS/CFT correspondence, JHEP 05 (2010) 062 [arXiv:1001.1541] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
A.B. Zamolodchikov, Physics reviews. Vol. 10, Pt. 4: Conformal field theory and critical phenomena in two-dimensional systems, (1989) [INSPIRE].
J. Distler, R. Flauger and B. Horn, Double-soft graviton amplitudes and the extended BMS charge algebra, JHEP 08 (2019) 021 [arXiv:1808.09965] [INSPIRE].
W. Fan, A. Fotopoulos and T.R. Taylor, Soft Limits of Yang-Mills Amplitudes and Conformal Correlators, JHEP 05 (2019) 121 [arXiv:1903.01676] [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Conformally Soft Theorem in Gauge Theory, Phys. Rev. D 100 (2019) 085017 [arXiv:1904.10831] [INSPIRE].
A. Guevara, Notes on Conformal Soft Theorems and Recursion Relations in Gravity, arXiv:1906.07810 [INSPIRE].
A. Fotopoulos and T.R. Taylor, Primary Fields in Celestial CFT, JHEP 10 (2019) 167 [arXiv:1906.10149] [INSPIRE].
M. Pate, A.-M. Raclariu, A. Strominger and E.Y. Yuan, Celestial operator products of gluons and gravitons, Rev. Math. Phys. 33 (2021) 2140003 [arXiv:1910.07424] [INSPIRE].
A. Fotopoulos, S. Stieberger, T.R. Taylor and B. Zhu, Extended BMS Algebra of Celestial CFT, JHEP 03 (2020) 130 [arXiv:1912.10973] [INSPIRE].
S. Banerjee, S. Ghosh and R. Gonzo, BMS symmetry of celestial OPE, JHEP 04 (2020) 130 [arXiv:2002.00975] [INSPIRE].
S. Banerjee, S. Ghosh and P. Paul, MHV graviton scattering amplitudes and current algebra on the celestial sphere, JHEP 02 (2021) 176 [arXiv:2008.04330] [INSPIRE].
S. Banerjee and S. Ghosh, MHV gluon scattering amplitudes from celestial current algebras, JHEP 10 (2021) 111 [arXiv:2011.00017] [INSPIRE].
A. Guevara, E. Himwich, M. Pate and A. Strominger, Holographic symmetry algebras for gauge theory and gravity, JHEP 11 (2021) 152 [arXiv:2103.03961] [INSPIRE].
S. Banerjee, S. Ghosh and S.S. Samal, Subsubleading soft graviton symmetry and MHV graviton scattering amplitudes, JHEP 08 (2021) 067 [arXiv:2104.02546] [INSPIRE].
A. Strominger, w1+∞ and the Celestial Sphere, arXiv:2105.14346 [INSPIRE].
S. Banerjee, S. Ghosh and P. Paul, (Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes, JHEP 09 (2022) 236 [arXiv:2108.04262] [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].
C. Duval, G.W. Gibbons and P.A. Horvathy, Conformal Carroll groups, J. Phys. A 47 (2014) 335204 [arXiv:1403.4213] [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].
J.M. Lévy-Leblond, Une nouvelle limite non-relativiste du group de Poincaré, Ann. Inst. H. Poincaré 3 (1965) 1.
N.D. Sen Gupta, On an analogue of the Galilei group, Nuovo Cim. A 44 (1966) 512.
G. Barnich, A. Gomberoff and H.A. Gonzalez, The Flat limit of three dimensional asymptotically anti-de Sitter spacetimes, Phys. Rev. D 86 (2012) 024020 [arXiv:1204.3288] [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].
G. Barnich and H.A. Gonzalez, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity, JHEP 05 (2013) 016 [arXiv:1303.1075] [INSPIRE].
A. Bagchi, R. Basu, D. Grumiller and M. Riegler, Entanglement entropy in Galilean conformal field theories and flat holography, Phys. Rev. Lett. 114 (2015) 111602 [arXiv:1410.4089] [INSPIRE].
H. Jiang, W. Song and Q. Wen, Entanglement Entropy in Flat Holography, JHEP 07 (2017) 142 [arXiv:1706.07552] [INSPIRE].
E. Hijano and C. Rabideau, Holographic entanglement and Poincaré blocks in three-dimensional flat space, JHEP 05 (2018) 068 [arXiv:1712.07131] [INSPIRE].
A. Bagchi, D. Grumiller and W. Merbis, Stress tensor correlators in three-dimensional gravity, Phys. Rev. D 93 (2016) 061502 [arXiv:1507.05620] [INSPIRE].
J. Hartong, Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [arXiv:1511.01387] [INSPIRE].
A. Bagchi, R. Basu, A. Kakkar and A. Mehra, Flat Holography: Aspects of the dual field theory, JHEP 12 (2016) 147 [arXiv:1609.06203] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Carrollian Perspective on Celestial Holography, Phys. Rev. Lett. 129 (2022) 071602 [arXiv:2202.04702] [INSPIRE].
A. Bagchi, S. Banerjee, R. Basu and S. Dutta, Scattering Amplitudes: Celestial and Carrollian, Phys. Rev. Lett. 128 (2022) 241601 [arXiv:2202.08438] [INSPIRE].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and Celestial Holography, arXiv:2212.12553 [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, Sub-subleading soft graviton theorem from asymptotic Einstein’s equations, JHEP 05 (2022) 186 [arXiv:2111.15607] [INSPIRE].
L.P. de Gioia and A.-M. Raclariu, Celestial Sector in CFT: Conformally Soft Symmetries, arXiv:2303.10037 [INSPIRE].
S. Banerjee, Null Infinity and Unitary Representation of The Poincaré Group, JHEP 01 (2019) 205 [arXiv:1801.10171] [INSPIRE].
A. Saha, Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory, JHEP 12 (2022) 133 [arXiv:2207.11684] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
A.M. Polyakov, Quantum Gravity in Two-Dimensions, Mod. Phys. Lett. A 2 (1987) 893 [INSPIRE].
S. Banerjee and S. Pasterski, Revisiting the shadow stress tensor in celestial CFT, JHEP 04 (2023) 118 [arXiv:2212.00257] [INSPIRE].
T. Klose et al., Double-Soft Limits of Gluons and Gravitons, JHEP 07 (2015) 135 [arXiv:1504.05558] [INSPIRE].
A. Bagchi et al., Non-Lorentzian Kač-Moody algebras, JHEP 03 (2023) 041 [arXiv:2301.04686] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries and subleading soft graviton theorem, Phys. Rev. D 90 (2014) 124028 [arXiv:1408.2228] [INSPIRE].
M. Campiglia and A. Laddha, New symmetries for the Gravitational S-matrix, JHEP 04 (2015) 076 [arXiv:1502.02318] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra, JHEP 11 (2018) 200 [Erratum ibid. 04 (2020) 172] [arXiv:1810.00377] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Asymptotic Symmetries and Celestial CFT, JHEP 09 (2020) 176 [arXiv:2005.08990] [INSPIRE].
M. Campiglia and J. Peraza, Generalized BMS charge algebra, Phys. Rev. D 101 (2020) 104039 [arXiv:2002.06691] [INSPIRE].
B. Oblak, From the Lorentz Group to the Celestial Sphere, arXiv:1508.00920 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: with applications to String theory, Springer (2009) [https://doi.org/10.1007/978-3-642-00450-6] [INSPIRE].
B. Chen, R. Liu and Y.-F. Zheng, On Higher-dimensional Carrollian and Galilean Conformal Field Theories, SciPost Phys. 14 (2023) 088 [arXiv:2112.10514] [INSPIRE].
A. Bagchi, R. Basu, A. Mehra and P. Nandi, Field Theories on Null Manifolds, JHEP 02 (2020) 141 [arXiv:1912.09388] [INSPIRE].
S. Baiguera, G. Oling, W. Sybesma and B.T. Søgaard, Conformal Carroll scalars with boosts, SciPost Phys. 14 (2023) 086 [arXiv:2207.03468] [INSPIRE].
C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980) [INSPIRE].
B. Chen, R. Liu, H. Sun and Y.-F. Zheng, Constructing Carrollian Field Theories from Null Reduction, arXiv:2301.06011 [INSPIRE].
M.J. Ablowitz and A.S. Fokas, Complex Variables: Introduction and Applications, Second edition, Cambridge University Press (2003) [https://doi.org/10.1017/CBO9780511791246].
S. Banerjee, P. Pandey and P. Paul, Conformal properties of soft operators: Use of null states, Phys. Rev. D 101 (2020) 106014 [arXiv:1902.02309] [INSPIRE].
S. Banerjee and P. Pandey, Conformal properties of soft-operators. Part II. Use of null-states, JHEP 02 (2020) 067 [arXiv:1906.01650] [INSPIRE].
L. Donnay, A. Puhm and A. Strominger, Conformally Soft Photons and Gravitons, JHEP 01 (2019) 184 [arXiv:1810.05219] [INSPIRE].
A. Puhm, Conformally Soft Theorem in Gravity, JHEP 09 (2020) 130 [arXiv:1905.09799] [INSPIRE].
X. Bekaert and B. Oblak, Massless scalars and higher-spin BMS in any dimension, JHEP 11 (2022) 022 [arXiv:2209.02253] [INSPIRE].
W.-B. Liu and J. Long, Symmetry group at future null infinity: Scalar theory, Phys. Rev. D 107 (2023) 126002 [arXiv:2210.00516] [INSPIRE].
G. Barnich, Centrally extended BMS4 Lie algebroid, JHEP 06 (2017) 007 [arXiv:1703.08704] [INSPIRE].
L. Donnay and R. Ruzziconi, BMS flux algebra in celestial holography, JHEP 11 (2021) 040 [arXiv:2108.11969] [INSPIRE].
J.H. Schwarz, Diffeomorphism Symmetry in Two Dimensions and Celestial Holography, arXiv:2208.13304 [INSPIRE].
Acknowledgments
The author would like to thank Arjun Bagchi for helpful discussions at various stages of this work and also for his valuable comments on the initial version of the draft. A.S. is financially supported by the PMRF fellowship, MHRD, India.
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Saha, A. Carrollian approach to 1 + 3D flat holography. J. High Energ. Phys. 2023, 51 (2023). https://doi.org/10.1007/JHEP06(2023)051
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DOI: https://doi.org/10.1007/JHEP06(2023)051