Abstract
Celestial holography expresses \( \mathcal{S} \)-matrix elements as correlators in a CFT living on the night sky. Poincaré invariance imposes additional selection rules on the allowed positions of operators. As a consequence, n-point correlators are only supported on certain patches of the celestial sphere, depending on the labeling of each operator as incoming/outgoing. Here we initiate a study of the celestial geometry, examining the kinematic support of celestial amplitudes for different crossing channels. We give simple geometric rules for determining this support. For n ≥ 5, we can view these channels as tiling together to form a covering of the celestial sphere. Our analysis serves as a stepping off point to better understand the analyticity of celestial correlators and illuminate the connection between the 4D kinematic and 2D CFT notions of crossing symmetry.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in 2022 Snowmass Summer Study, (2021) [arXiv:2111.11392] [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].
J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
J. D. Brown and J. W. York Jr., Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [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].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [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].
R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne, The analytic S-matrix, Cambridge University Press (1966).
N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06 (2007) 060 [hep-th/0601001] [INSPIRE].
N. Arkani-Hamed, T.-C. Huang and Y.-T. Huang, The EFT-Hedron, JHEP 05 (2021) 259 [arXiv:2012.15849] [INSPIRE].
N. Arkani-Hamed, Y.-t. Huang, J.-Y. Liu and G. N. Remmen, Causality, unitarity, and the weak gravity conjecture, JHEP 03 (2022) 083 [arXiv:2109.13937] [INSPIRE].
S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].
S. Pasterski, Soft Shadows, 978-0-9863685-4-7 (2017).
A. Atanasov, W. Melton, A.-M. Raclariu and A. Strominger, Conformal block expansion in celestial CFT, Phys. Rev. D 104 (2021) 126033 [arXiv:2104.13432] [INSPIRE].
A. Sharma, Ambidextrous light transforms for celestial amplitudes, JHEP 01 (2022) 031 [arXiv:2107.06250] [INSPIRE].
A. Ball, E. Himwich, S. A. Narayanan, S. Pasterski and A. Strominger, Uplifting AdS3/CFT2 to flat space holography, JHEP 08 (2019) 168 [arXiv:1905.09809] [INSPIRE].
S. Pasterski and H. Verlinde, Chaos in celestial CFT, JHEP 08 (2022) 106 [arXiv:2201.01630] [INSPIRE].
J. de Boer and S. N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [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].
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. Guevara, E. Himwich, M. Pate and A. Strominger, Holographic symmetry algebras for gauge theory and gravity, JHEP 11 (2021) 152 [arXiv:2103.03961] [INSPIRE].
A. Strominger, w(1+infinity) and the Celestial Sphere, arXiv:2105.14346 [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].
A. Atanasov, A. Ball, W. Melton, A.-M. Raclariu and A. Strominger, (2, 2) Scattering and the celestial torus, JHEP 07 (2021) 083 [arXiv:2101.09591] [INSPIRE].
E. Crawley, A. Guevara, N. Miller and A. Strominger, Black Holes in Klein Space, arXiv:2112.03954 [INSPIRE].
S. Mizera, Bounds on Crossing Symmetry, Phys. Rev. D 103 (2021) 081701 [arXiv:2101.08266] [INSPIRE].
S. Mizera, Crossing symmetry in the planar limit, Phys. Rev. D 104 (2021) 045003 [arXiv:2104.12776] [INSPIRE].
H. S. Hannesdottir and S. Mizera, What is the iε for the S-matrix?, to appear.
Y. T. A. Law and M. Zlotnikov, Poincaré constraints on celestial amplitudes, JHEP 03 (2020) 085 [Erratum ibid. 04 (2020) 202] [arXiv:1910.04356] [INSPIRE].
N. Arkani-Hamed, M. Pate, A.-M. Raclariu and A. Strominger, Celestial amplitudes from UV to IR, JHEP 08 (2021) 062 [arXiv:2012.04208] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Revisiting the conformally soft sector with celestial diamonds, JHEP 11 (2021) 143 [arXiv:2105.09792] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Goldilocks modes and the three scattering bases, JHEP 06 (2022) 124 [arXiv:2202.11127] [INSPIRE].
A. Schreiber, A. Volovich and M. Zlotnikov, Tree-level gluon amplitudes on the celestial sphere, Phys. Lett. B 781 (2018) 349 [arXiv:1711.08435] [INSPIRE].
J. Jackson, On the existence problem of linear programming, Pacific J. Math. 4 (1954) 29.
S. Pasterski, A Shorter Path to Celestial Currents, arXiv:2201.06805 [INSPIRE].
N. J. A. Sloane, Sequence A000125/M1100, The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A000125.
A. Brandhuber, G. R. Brown, J. Gowdy, B. Spence and G. Travaglini, Celestial superamplitudes, Phys. Rev. D 104 (2021) 045016 [arXiv:2105.10263] [INSPIRE].
C.-M. Chang, Y.-t. Huang, Z.-X. Huang and W. Li, Bulk locality from the celestial amplitude, SciPost Phys. 12 (2022) 176 [arXiv:2106.11948] [INSPIRE].
H. T. Lam and S.-H. Shao, Conformal Basis, Optical Theorem, and the Bulk Point Singularity, Phys. Rev. D 98 (2018) 025020 [arXiv:1711.06138] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Asymptotic Symmetries and Celestial CFT, JHEP 09 (2020) 176 [arXiv:2005.08990] [INSPIRE].
D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, Celestial Amplitudes: Conformal Partial Waves and Soft Limits, JHEP 10 (2019) 018 [arXiv:1904.10940] [INSPIRE].
W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, Conformal blocks from celestial gluon amplitudes, JHEP 05 (2021) 170 [arXiv:2103.04420] [INSPIRE].
W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, Conformal blocks from celestial gluon amplitudes. Part II. Single-valued correlators, JHEP 11 (2021) 179 [arXiv:2108.10337] [INSPIRE].
W. Fan, A. Fotopoulos, S. Stieberger, T. R. Taylor and B. Zhu, Elements of Celestial Conformal Field Theory, arXiv:2202.08288 [INSPIRE].
Y. Hu, L. Lippstreu, M. Spradlin, A. Y. Srikant and A. Volovich, Four-point correlators of light-ray operators in CCFT, JHEP 07 (2022) 104 [arXiv:2203.04255] [INSPIRE].
S. Stieberger and T. R. Taylor, Symmetries of Celestial Amplitudes, Phys. Lett. B 793 (2019) 141 [arXiv:1812.01080] [INSPIRE].
J. Bros, H. Epstein and V. Glaser, A proof of the crossing property for two-particle amplitudes in general quantum field theory, Commun. Math. Phys. 1 (1965) 240 [INSPIRE].
M. Gerstenhaber, Theory of convex polyhedral cones, in Chap. XVIII of Cowles Commission Monograph Activity analysis of production and allocation, T.C. Koopmans ed., no. 13, pp. 298–316 (1951).
H. Joos, On the Representation theory of inhomogeneous Lorentz groups as the foundation of quantum mechanical kinematics, Fortsch. Phys. 10 (1962) 65 [INSPIRE].
S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press (1995) [DOI].
I. Gel’fand, M. Graev and N. Vilenkin, Generalized Functions, Volume 5: Integral Geometry and Representation Theory, Elsevier Science (2014).
I. Gel’fand, R. Minlos and Z. Shapiro, Representations of the Rotation and Lorentz Groups and Their Applications, Dover Publications (2018).
I. M. Gel’fand and M. A. Naimark, Unitary representations of the Lorentz group, Izv. Rossiiskoi Akad. Nauk. Ser. Mat. 11 (1947) 411.
A. Chakrabarti, M. Levy-Nahas and R. Seneor, ‘Lorentz Basis’ of the Poincaré Group, J. Math. Phys. 9 (1968) 1274.
A. Chakrabarti, Lorentz basis of the Poincaré group. 2, J. Math. Phys. 12 (1971) 1822 [INSPIRE].
W. W. Macdowell and R. Roskies, Reduction of the Poincaré group with respect to the Lorentz group, J. Math. Phys. 13 (1972) 1585 [INSPIRE].
I. Shapiro, Expansion of the scattering amplitude in relativistic spherical functions, Phys. Lett. 1 (1962) 253.
I. Bars and F. Guersey, Operator treatment of the gel’fand-naimark basis for SL(2, C), J. Math. Phys. 13 (1972) 131 [INSPIRE].
W. Ruhl, The Lorentz Group and Harmonic Analysis, Mathematical physics monograph series, W. A. Benjamin (1970).
M. Carmeli, Group Theory and General Relativity: Representations of the Lorentz Group and Their Applications to the Gravitational Field, World Scientific (2000) [DOI].
M. Naimark and H. Farahat, Linear Representations of the Lorentz Group, Elsevier Science (2014).
S.-J. Chang and L. O’ Raifeartaigh, Unitary representations of SL(2, C) in an E2 basis, J. Math. Phys. 10 (1969) 21 [INSPIRE].
G. J. Iverson and G. Mack, E2-parametrization of SL(2, C ), J. Math. Phys. 11 (1970) 1581 [INSPIRE].
Y. V. Novozhilov and E. V. Prokhvatilov, Representations of the Poincaré group in E(2) bases, Theor. Math. Phys. 1 (1969) 78.
I. Bars and F. Guersey, Duality and the Lorentz group, Phys. Rev. D 4 (1971) 1769 [INSPIRE].
G. B. Smith, Matrix Element Expansion of a Spin Wave Function, J. Math. Phys. 19 (1978) 581 [INSPIRE].
J. S. Lomont and H. E. Moses, The Representations of the Inhomogeneous Lorentz Group in Terms of an Angular Momentum Basis, J. Math. Phys. 5 (1964) 294.
J. S. Zmuidzinas, Unitary Representations of the Lorentz Group on 4-Vector Manifolds, J. Math. Phys. 7 (1966) 764.
K.-C. Chou and L. G. Zastavenko, The Shapiro Integral Transformation, in Selected Papers of K C Chou, World Scientific (2009), pp. 33–38 [DOI].
V. Popov, On the theory of the relativistic transformations of the wave functions and density matrix of particles with spin, Sov. Phys. JETP 37 (1960).
K.-C. Chou and L. G. Zastavenko, Integral Transformations of the I.S. Shapiro Type for Particles of Zero Mass, in Selected Papers of K C Chou, World Scientific (2009), pp. 77–80 [DOI].
M. L. Paciello, A. Sciarrino and B. Taglienti, Projective invariance of dual-resonance models from spin analyticity and Lorentz invariance, Nuovo Cim. A 14 (1973) 591 [INSPIRE].
A. W. Weidemann, Quantum fields in a ‘Lorentz basis’, Nuovo Cim. A 57 (1980) 221 [INSPIRE].
N. Mukunda, Zero-Mass Representations of the Poincaré Group in an O(3, 1) Basis, J. Math. Phys. 9 (1968) 532.
W. Ruehl, The convolution of fourier transforms and its application to the decomposition of the momentum operator on the homogeneous Lorentz group, Nuovo Cim. A 63 (1969) 1131 [INSPIRE].
M. Daumens and M. Perroud, Internal Lorentz basis for two particle states, J. Math. Phys. 20 (1979) 2621 [INSPIRE].
M. Daumens, M. Perroud and P. Winternitz, Relativistic Energy Dependent Partial Wave Analysis for Particles With Spin, Phys. Rev. D 19 (1979) 3413 [INSPIRE].
A. D. Steiger, Poincaré-irreducible tensor operators for positive-mass one-particle states. I, J. Math. Phys. 12 (1971) 1178 [INSPIRE].
A. D. Steiger, Poincaré-irreducible tensor operators for positive-mass one-particle states. II, J. Math. Phys. 12 (1971) 1497 [INSPIRE].
B. Radhakrishnan and N. Mukunda, Spacelike representations of the inhomogeneous Lorentz group in a Lorentz basis, J. Math. Phys. 15 (1974) 477 [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: 2204.02505
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
Mizera, S., Pasterski, S. Celestial geometry. J. High Energ. Phys. 2022, 45 (2022). https://doi.org/10.1007/JHEP09(2022)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2022)045