Abstract
In a 1 + 2D Carrollian conformal field theory, the Ward identities of the two local fields \( {S}_0^{+} \) and \( {S}_1^{+} \), entirely built out of the Carrollian conformal stress-tensor, contain respectively up to the leading and the subleading positive helicity soft graviton theorems in the 1 + 3D asymptotically flat space-time. This work investigates how the subsubleading soft graviton theorem can be encoded into the Ward identity of a Carrollian conformal field \( {S}_2^{+} \). The operator product expansion (OPE) \( {S}_2^{+}{S}_2^{+} \) is constructed using general Carrollian conformal symmetry principles and the OPE commutativity property, under the assumption that any time-independent, non-Identity field that is mutually local with \( {S}_0^{+} \), \( {S}_1^{+} \), \( {S}_2^{+} \) has positive Carrollian scaling dimension. It is found that, for this OPE to be consistent, another local field \( {S}_3^{+} \) must automatically exist in the theory. The presence of an infinite tower of local fields \( {S}_{k\ge 3}^{+} \) is then revealed iteratively as a consistency condition for the \( {S}_2^{+}{S}_{k-1}^{+} \) OPE. The general \( {S}_k^{+}{S}_l^{+} \) OPE is similarly obtained and the symmetry algebra manifest in this OPE is found to be the Kac-Moody algebra of the wedge sub-algebra of w1+∞. The Carrollian time-coordinate plays the central role in this purely holographic construction. The 2D Celestial conformally soft graviton primary \( {H}^k\left(z,\overline{z}\right) \) is realized to be contained in the Carrollian conformal primary \( {S}_{1-k}^{+}\left(t,z,\overline{z}\right) \). Finally, the existence of the infinite tower of fields \( {S}_k^{+} \) is shown to be directly related to an infinity of positive helicity soft graviton theorems.
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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].
S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2111.11392] [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, S.-H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [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].
L. Donnay, A. Fiorucci, Y. Herfray and R. Ruzziconi, Bridging Carrollian and celestial holography, Phys. Rev. D 107 (2023) 126027 [arXiv:2212.12553] [INSPIRE].
A. Saha, Carrollian approach to 1 + 3D flat holography, JHEP 06 (2023) 051 [arXiv:2304.02696] [INSPIRE].
J. Salzer, An embedding space approach to Carrollian CFT correlators for flat space holography, JHEP 10 (2023) 084 [arXiv:2304.08292] [INSPIRE].
K. Nguyen and P. West, Carrollian Conformal Fields and Flat Holography, Universe 9 (2023) 385 [arXiv:2305.02884] [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].
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].
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].
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].
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, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity \( \mathcal{S} \)-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [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].
H. Elvang, C.R.T. Jones and S.G. Naculich, Soft Photon and Graviton Theorems in Effective Field Theory, Phys. Rev. Lett. 118 (2017) 231601 [arXiv:1611.07534] [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].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys. 65 (1985) 1205 [INSPIRE].
A. Bagchi et al., Non-Lorentzian Kač-Moody algebras, JHEP 03 (2023) 041 [arXiv:2301.04686] [INSPIRE].
T. Klose et al., Double-Soft Limits of Gluons and Gravitons, JHEP 07 (2015) 135 [arXiv:1504.05558] [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].
A.B. Zamolodchikov, Physics reviews. Vol. 10, Pt. 4: Conformal field theory and critical phenomena in two-dimensional systems, [INSPIRE].
A. Saha, Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory, JHEP 12 (2022) 133 [arXiv:2207.11684] [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].
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].
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. Ball, S.A. Narayanan, J. Salzer and A. Strominger, Perturbatively exact w1+∞ asymptotic symmetry of quantum self-dual gravity, JHEP 01 (2022) 114 [arXiv:2111.10392] [INSPIRE].
B. Chen, R. Liu, H. Sun and Y.-F. Zheng, Constructing Carrollian field theories from null reduction, JHEP 11 (2023) 170 [arXiv:2301.06011] [INSPIRE].
C.N. Pope, Lectures on W algebras and W gravity, in the proceedings of the Summer School in High-energy Physics and Cosmology, Trieste, Italy, June 17 – August 09 (1991) [hep-th/9112076] [INSPIRE].
I. Bakas, The Large n Limit of Extended Conformal Symmetries, Phys. Lett. B 228 (1989) 57 [INSPIRE].
A. Strominger, w1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries, Phys. Rev. Lett. 127 (2021) 221601 [arXiv:2105.14346] [INSPIRE].
Y. Hamada and G. Shiu, Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities, Phys. Rev. Lett. 120 (2018) 201601 [arXiv:1801.05528] [INSPIRE].
Z.-Z. Li, H.-H. Lin and S.-Q. Zhang, Infinite Soft Theorems from Gauge Symmetry, Phys. Rev. D 98 (2018) 045004 [arXiv:1802.03148] [INSPIRE].
S. Banerjee, Symmetries of free massless particles and soft theorems, Gen. Rel. Grav. 51 (2019) 128 [arXiv:1804.06646] [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 and S. Pasterski, Revisiting the shadow stress tensor in celestial CFT, JHEP 04 (2023) 118 [arXiv:2212.00257] [INSPIRE].
C. Itzykson and J.B. Zuber, Quantum Field Theory, McGraw-Hill (1980), ISBN 978-0-486-44568-7.
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. 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. Fotopoulos and T.R. Taylor, Primary Fields in Celestial CFT, JHEP 10 (2019) 167 [arXiv:1906.10149] [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].
A.M. Polyakov, Quantum Gravity in Two-Dimensions, Mod. Phys. Lett. A 2 (1987) 893 [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, (Chiral) Virasoro invariance of the tree-level MHV graviton scattering amplitudes, JHEP 09 (2022) 236 [arXiv:2108.04262] [INSPIRE].
E. Himwich, M. Pate and K. Singh, Celestial operator product expansions and w1+∞ symmetry for all spins, JHEP 01 (2022) 080 [arXiv:2108.07763] [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].
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].
E. Conde and P. Mao, BMS Supertranslations and Not So Soft Gravitons, JHEP 05 (2017) 060 [arXiv:1612.08294] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, Higher spin dynamics in gravity and w1+∞ celestial symmetries, Phys. Rev. D 106 (2022) 086013 [arXiv:2112.15573] [INSPIRE].
S. Banerjee, H. Kulkarni and P. Paul, An infinite family of w1+∞ invariant theories on the celestial sphere, JHEP 05 (2023) 063 [arXiv:2301.13225] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic charges in massless QED revisited: A view from Spatial Infinity, JHEP 05 (2019) 207 [arXiv:1810.04619] [INSPIRE].
L.P. de Gioia and A.-M. Raclariu, Celestial Sector in CFT: Conformally Soft Symmetries, arXiv:2303.10037 [INSPIRE].
Acknowledgments
I would like to thank Shamik Banerjee for illuminating discussions on Celestial holography at the beginning stage of this work, for a clarification on the work [64] and also for his valuable comments on the manuscript. It is a pleasure to acknowledge the warm hospitality of the National Institute of Science Education and Research (NISER), Jatni where this work was started. I am also grateful to Raju Mandal for many helpful discussions on Celestial CFT during my stay at NISER. I am greatly indebted to Alok Laddha and Romain Ruzziconi for emphasizing to me the correspondence between the large Carrollian time limit on the boundary and the energetically soft limit in the bulk. Finally, I thank Arjun Bagchi for his continued support. This work is financially supported by the PMRF fellowship, MHRD, India.
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Saha, A. w1+∞ and Carrollian holography. J. High Energ. Phys. 2024, 145 (2024). https://doi.org/10.1007/JHEP05(2024)145
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DOI: https://doi.org/10.1007/JHEP05(2024)145