Abstract
We introduce a new gauge and solution space for three-dimensional gravity. As its name Bondi-Weyl suggests, it leads to non-trivial Weyl charges, and uses Bondi-like coordinates to allow for an arbitrary cosmological constant and therefore spacetimes which are asymptotically locally (A)dS or flat. We explain how integrability requires a choice of integrable slicing and also the introduction of a corner term. After discussing the holographic renormalization of the action and of the symplectic potential, we show that the charges are finite, symplectic and integrable, yet not conserved. We find four towers of charges forming an algebroid given by \( \mathfrak{vir}\oplus \mathfrak{vir}\oplus \) Heisenberg with three central extensions, where the base space is parametrized by the retarded time. These four charges generate diffeomorphisms of the boundary cylinder, Weyl rescalings of the boundary metric, and radial translations. We perform this study both in metric and triad variables, and use the triad to explain the covariant origin of the corner terms needed for renormalization and integrability.
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References
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].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A. Ashtekar, M. Campiglia and A. Laddha, Null infinity, the BMS group and infrared issues, Gen. Rel. Grav. 50 (2018) 140 [arXiv:1808.07093] [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].
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].
G. Compere and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
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].
C. Troessaert, Enhanced asymptotic symmetry algebra of AdS3, JHEP 08 (2013) 044 [arXiv:1303.3296] [INSPIRE].
G. Compère, L. Donnay, P.-H. Lambert and W. Schulgin, Liouville theory beyond the cosmological horizon, JHEP 03 (2015) 158 [arXiv:1411.7873] [INSPIRE].
L. Donnay, G. Giribet, H. A. Gonzalez and M. Pino, Supertranslations and Superrotations at the Black Hole Horizon, Phys. Rev. Lett. 116 (2016) 091101 [arXiv:1511.08687] [INSPIRE].
G. Compère, P. Mao, A. Seraj and M. M. Sheikh-Jabbari, Symplectic and Killing symmetries of AdS3 gravity: holographic vs boundary gravitons, JHEP 01 (2016) 080 [arXiv:1511.06079] [INSPIRE].
L. Donnay, G. Giribet, H. A. González and M. Pino, Extended Symmetries at the Black Hole Horizon, JHEP 09 (2016) 100 [arXiv:1607.05703] [INSPIRE].
H. Afshar et al., Soft Heisenberg hair on black holes in three dimensions, Phys. Rev. D 93 (2016) 101503 [arXiv:1603.04824] [INSPIRE].
A. Pérez, D. Tempo and R. Troncoso, Boundary conditions for General Relativity on AdS3 and the KdV hierarchy, JHEP 06 (2016) 103 [arXiv:1605.04490] [INSPIRE].
D. Grumiller and M. Riegler, Most general AdS3 boundary conditions, JHEP 10 (2016) 023 [arXiv:1608.01308] [INSPIRE].
B. Oblak, BMS Particles in Three Dimensions, Ph.D. Thesis, Université Libre de Bruxelles (2016) [arXiv:1610.08526] [DOI] [INSPIRE].
D. Grumiller, W. Merbis and M. Riegler, Most general flat space boundary conditions in three-dimensional Einstein gravity, Class. Quant. Grav. 34 (2017) 184001 [arXiv:1704.07419] [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].
R. Ruzziconi and C. Zwikel, Conservation and Integrability in Lower-Dimensional Gravity, JHEP 04 (2021) 034 [arXiv:2012.03961] [INSPIRE].
F. Alessio, G. Barnich, L. Ciambelli, P. Mao and R. Ruzziconi, Weyl charges in asymptotically locally AdS3 spacetimes, Phys. Rev. D 103 (2021) 046003 [arXiv:2010.15452] [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].
R. Penrose, Asymptotic properties of fields and space-times, Phys. Rev. Lett. 10 (1963) 66 [INSPIRE].
R. Penrose, Conformal treatment of infinity, Gen. Rel. Grav. 43 (2011) 901 [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. Sachs, Asymptotic symmetries in gravitational theory, Phys. Rev. 128 (1962) 2851 [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].
C. Fefferman and C. R. Graham, The ambient metric, Ann. Math. Stud. 178 (2011) 1 [arXiv:0710.0919] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
K. Bautier, Diffeomorphisms and Weyl transformations in AdS3 gravity, PoS tmr99 (1999) 006 [hep-th/9910134] [INSPIRE].
C. Imbimbo, A. Schwimmer, S. Theisen and S. Yankielowicz, Diffeomorphisms and holographic anomalies, Class. Quant. Grav. 17 (2000) 1129 [hep-th/9910267] [INSPIRE].
A. Schwimmer and S. Theisen, Diffeomorphisms, anomalies and the Fefferman-Graham ambiguity, JHEP 08 (2000) 032 [hep-th/0008082] [INSPIRE].
M. Rooman and P. Spindel, Holonomies, anomalies and the Fefferman-Graham ambiguity in AdS3 gravity, Nucl. Phys. B 594 (2001) 329 [hep-th/0008147] [INSPIRE].
J. Kalkkinen, D. Martelli and W. Mueck, Holographic renormalization and anomalies, JHEP 04 (2001) 036 [hep-th/0103111] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
L. Ciambelli and R. G. Leigh, Weyl Connections and their Role in Holography, Phys. Rev. D 101 (2020) 086020 [arXiv:1905.04339] [INSPIRE].
L. Ciambelli, C. Marteau, P. M. Petropoulos and R. Ruzziconi, Fefferman-Graham and Bondi Gauges in the Fluid/Gravity Correspondence, PoS CORFU2019 (2020) 154 [arXiv:2006.10083] [INSPIRE].
L. Ciambelli, C. Marteau, P. M. Petropoulos and R. Ruzziconi, Gauges in Three-Dimensional Gravity and Holographic Fluids, JHEP 11 (2020) 092 [arXiv:2006.10082] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4 charge algebra, JHEP 10 (2020) 205 [arXiv:2004.10769] [INSPIRE].
A. Poole, K. Skenderis and M. Taylor, (A)dS4 in Bondi gauge, Class. Quant. Grav. 36 (2019) 095005 [arXiv:1812.05369] [INSPIRE].
G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4 group of dS4 and new boundary conditions for AdS4, Class. Quant. Grav. 36 (2019) 195017 [arXiv:1905.00971] [INSPIRE].
R. Ruzziconi, On the Various Extensions of the BMS Group, Ph.D. Thesis (2020) [arXiv:2009.01926] [INSPIRE].
A. Fiorucci and R. Ruzziconi, Charge algebra in Al(A)dSn spacetimes, JHEP 05 (2021) 210 [arXiv:2011.02002] [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, Finite BMS transformations, JHEP 03 (2016) 167 [arXiv:1601.04090] [INSPIRE].
G. Barnich, P. Mao and R. Ruzziconi, BMS current algebra in the context of the Newman-Penrose formalism, Class. Quant. Grav. 37 (2020) 095010 [arXiv:1910.14588] [INSPIRE].
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, The Weyl BMS group and Einstein’s equations, JHEP 07 (2021) 170 [arXiv:2104.05793] [INSPIRE].
H. Adami, D. Grumiller, S. Sadeghian, M. M. Sheikh-Jabbari and C. Zwikel, T-Witts from the horizon, JHEP 04 (2020) 128 [arXiv:2002.08346] [INSPIRE].
H. Adami, D. Grumiller, M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, News and T-Witts From The Horizon, in preparation (2021).
L. Freidel, R. Oliveri, D. Pranzetti and S. Speziale, Extended corner symmetry, charge bracket and Einstein’s equations, arXiv:2104.12881 [INSPIRE].
H. Adami, M. M. Sheikh-Jabbari, V. Taghiloo, H. Yavartanoo and C. Zwikel, Chiral Massive News: Null Boundary Symmetries in Topologically Massive Gravity, JHEP 05 (2021) 261 [arXiv:2104.03992] [INSPIRE].
D. Grumiller, M. M. Sheikh-Jabbari and C. Zwikel, Horizons 2020, Int. J. Mod. Phys. D 29 (2020) 2043006 [arXiv:2005.06936] [INSPIRE].
E. De Paoli and S. Speziale, A gauge-invariant symplectic potential for tetrad general relativity, JHEP 07 (2018) 040 [arXiv:1804.09685] [INSPIRE].
R. Oliveri and S. Speziale, Boundary effects in General Relativity with tetrad variables, Gen. Rel. Grav. 52 (2020) 83 [arXiv:1912.01016] [INSPIRE].
R. Oliveri and S. Speziale, A note on dual gravitational charges, JHEP 12 (2020) 079 [arXiv:2010.01111] [INSPIRE].
L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, JHEP 11 (2020) 026 [arXiv:2006.12527] [INSPIRE].
L. Freidel, M. Geiller and D. Pranzetti, Edge modes of gravity. Part II. Corner metric and Lorentz charges, JHEP 11 (2020) 027 [arXiv:2007.03563] [INSPIRE].
L. Ciambelli and R. G. Leigh, Isolated surfaces and symmetries of gravity, Phys. Rev. D 104 (2021) 046005 [arXiv:2104.07643] [INSPIRE].
V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [INSPIRE].
D. Harlow and J.-Q. Wu, Covariant phase space with boundaries, JHEP 10 (2020) 146 [arXiv:1906.08616] [INSPIRE].
M. Geiller and P. Jai-akson, Extended actions, dynamics of edge modes, and entanglement entropy, JHEP 09 (2020) 134 [arXiv:1912.06025] [INSPIRE].
G. Barnich and F. Brandt, Covariant theory of asymptotic symmetries, conservation laws and central charges, Nucl. Phys. B 633 (2002) 3 [hep-th/0111246] [INSPIRE].
G. Barnich, Boundary charges in gauge theories: Using Stokes theorem in the bulk, Class. Quant. Grav. 20 (2003) 3685 [hep-th/0301039] [INSPIRE].
G. Barnich and G. Compere, Surface charge algebra in gauge theories and thermodynamic integrability, J. Math. Phys. 49 (2008) 042901 [arXiv:0708.2378] [INSPIRE].
G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12 (2011) 105 [arXiv:1106.0213] [INSPIRE].
M. Geiller, C. Goeller and N. Merino, Most general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more, JHEP 02 (2021) 120 [arXiv:2011.09873] [INSPIRE].
F. Hopfmüller and L. Freidel, Null Conservation Laws for Gravity, Phys. Rev. D 97 (2018) 124029 [arXiv:1802.06135] [INSPIRE].
V. Chandrasekaran and A. J. Speranza, Anomalies in gravitational charge algebras of null boundaries and black hole entropy, JHEP 01 (2021) 137 [arXiv:2009.10739] [INSPIRE].
T. Jacobson and A. Mohd, Black hole entropy and Lorentz-diffeomorphism Noether charge, Phys. Rev. D 92 (2015) 124010 [arXiv:1507.01054] [INSPIRE].
K. Prabhu, The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class. Quant. Grav. 34 (2017) 035011 [arXiv:1511.00388] [INSPIRE].
G. T. Horowitz, Exactly Soluble Diffeomorphism Invariant Theories, Commun. Math. Phys. 125 (1989) 417 [INSPIRE].
S. Carlip, Quantum gravity in 2 + 1 dimensions, Cambridge Monographs on Mathematical Physics, Cambridge University Press (2003) [DOI] [INSPIRE].
M. Geiller and C. Goeller, Dual diffeomorphisms and finite distance asymptotic symmetries in 3d gravity, arXiv:2012.05263 [INSPIRE].
W. Wieland, New boundary variables for classical and quantum gravity on a null surface, Class. Quant. Grav. 34 (2017) 215008 [arXiv:1704.07391] [INSPIRE].
M. Geiller, Lorentz-diffeomorphism edge modes in 3d gravity, JHEP 02 (2018) 029 [arXiv:1712.05269] [INSPIRE].
S. W. Hawking and C. J. Hunter, The Gravitational Hamiltonian in the presence of nonorthogonal boundaries, Class. Quant. Grav. 13 (1996) 2735 [gr-qc/9603050] [INSPIRE].
T. Takayanagi and K. Tamaoka, Gravity Edges Modes and Hayward Term, JHEP 02 (2020) 167 [arXiv:1912.01636] [INSPIRE].
J. Lee and R. M. Wald, Local symmetries and constraints, J. Math. Phys. 31 (1990) 725 [INSPIRE].
R. M. Wald and A. Zoupas, A General definition of ‘conserved quantities’ in general relativity and other theories of gravity, Phys. Rev. D 61 (2000) 084027 [gr-qc/9911095] [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. Gonzalez, Dual dynamics of three dimensional asymptotically flat Einstein gravity at null infinity, JHEP 05 (2013) 016 [arXiv:1303.1075] [INSPIRE].
M. Blagojevic and B. Cvetkovic, Canonical structure of 3-D gravity with torsion, gr-qc/0412134 [INSPIRE].
M. Blagojevic and B. Cvetkovic, Asymptotic charges in 3-D gravity with torsion, J. Phys. Conf. Ser. 33 (2006) 248 [gr-qc/0511162] [INSPIRE].
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Geiller, M., Goeller, C. & Zwikel, C. 3d gravity in Bondi-Weyl gauge: charges, corners, and integrability. J. High Energ. Phys. 2021, 29 (2021). https://doi.org/10.1007/JHEP09(2021)029
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DOI: https://doi.org/10.1007/JHEP09(2021)029