Abstract
In three-dimensional pseudo-Riemannian manifolds, the Cotton tensor arises as the variation of the gravitational Chern-Simons action with respect to the metric. It is Weyl-covariant, symmetric, traceless and covariantly conserved. Performing a reduction of the Cotton tensor with respect to Carrollian diffeomorphisms in a suitable frame, one discloses four sets of Cotton Carrollian relatives, which are conformal and obey Carrollian conservation equations. Each set of Carrollian Cotton tensors is alternatively obtained as the variation of a distinct Carroll-Chern-Simons action with respect to the degenerate metric and the clock form of a strong Carroll structure. The four Carroll-Chern-Simons actions emerge in the Carrollian reduction of the original Chern-Simons ascendant. They inherit its anomalous behaviour under diffeomorphisms and Weyl transformations. The extremums of these Carrollian actions are commented and illustrated.
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References
É. Cotton, Sur les variétés à trois dimensions (in French), Ann. Fac. Sci. Toulouse 1 (1899) 385.
L. Andrianopoli et al., N = 2 AdS4 supergravity, holography and Ward identities, JHEP 02 (2021) 141 [arXiv:2010.02119] [INSPIRE].
S. de Haro and A.C. Petkou, Holographic aspects of electric-magnetic dualities, J. Phys. Conf. Ser. 110 (2008) 102003 [arXiv:0710.0965] [INSPIRE].
G. Compère and D. Marolf, Setting the boundary free in AdS/CFT, Class. Quant. Grav. 25 (2008) 195014 [arXiv:0805.1902] [INSPIRE].
D.S. Mansi, A.C. Petkou and G. Tagliabue, Gravity in the 3 + 1-split formalism II: self-duality and the emergence of the gravitational Chern-Simons in the boundary, Class. Quant. Grav. 26 (2009) 045009 [arXiv:0808.1213] [INSPIRE].
D.S. Mansi, A.C. Petkou and G. Tagliabue, Gravity in the 3 + 1-split formalism II: self-duality and the emergence of the gravitational Chern-Simons in the boundary, Class. Quant. Grav. 26 (2009) 045009 [arXiv:0808.1213] [INSPIRE].
I. Bakas, Energy-momentum/Cotton tensor duality for AdS4 black holes, JHEP 01 (2009) 003 [arXiv:0809.4852] [INSPIRE].
O. Miskovic and R. Olea, Topological regularization and self-duality in four-dimensional anti-de Sitter gravity, Phys. Rev. D 79 (2009) 124020 [arXiv:0902.2082] [INSPIRE].
P.M. Petropoulos, Gravitational duality, topologically massive gravity and holographic fluids, Lect. Notes Phys. 892 (2015) 331 [arXiv:1406.2328] [INSPIRE].
M.P. Blencowe, A consistent interacting massless higher spin field theory in D = (2 + 1), Class. Quant. Grav. 6 (1989) 443 [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
M. Banados, R. Canto and S. Theisen, The action for higher spin black holes in three dimensions, JHEP 07 (2012) 147 [arXiv:1204.5105] [INSPIRE].
A. Mukhopadhyay et al., Holographic perfect fluidity, Cotton energy-momentum duality and transport properties, JHEP 04 (2014) 136 [arXiv:1309.2310] [INSPIRE].
J. Gath et al., Petrov classification and holographic reconstruction of spacetime, JHEP 09 (2015) 005 [arXiv:1506.04813] [INSPIRE].
P.M. Petropoulos and K. Siampos, Integrability, Einstein spaces and holographic fluids, in proceedings of the workshop About various kinds of interactions in honour of the 65th birthday of professor Philippe Spindel, N. Boulanger and S. Detournay eds., Mons, Belgium (2017) [arXiv:1510.06456] [INSPIRE].
M. Haack and A. Yarom, Nonlinear viscous hydrodynamics in various dimensions using AdS/CFT, JHEP 10 (2008) 063 [arXiv:0806.4602] [INSPIRE].
S. Bhattacharyya et al., Conformal nonlinear fluid dynamics from gravity in arbitrary dimensions, JHEP 12 (2008) 116 [arXiv:0809.4272] [INSPIRE].
G. Bernardi de Freitas and H.S. Reall, Algebraically special solutions in AdS/CFT, JHEP 06 (2014) 148 [arXiv:1403.3537] [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].
L. Ciambelli et al., Flat holography and Carrollian fluids, JHEP 07 (2018) 165 [arXiv:1802.06809] [INSPIRE].
N. Mittal, P.M. Petropoulos, D. Rivera-Betancour and M. Vilatte, Ehlers, Carroll, charges and dual charges, JHEP 07 (2023) 065 [arXiv:2212.14062] [INSPIRE].
L. Ravera, AdS Carroll Chern-Simons supergravity in 2 + 1 dimensions and its flat limit, Phys. Lett. B 795 (2019) 331 [arXiv:1905.00766] [INSPIRE].
E. Bergshoeff, J. Gomis and L. Parra, The symmetries of the Carroll superparticle, J. Phys. A 49 (2016) 185402 [arXiv:1503.06083] [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 [Erratum ibid. 38 (2021) 229501] [arXiv:1905.00971] [INSPIRE].
A. Campoleoni et al., Flat from anti de Sitter, JHEP 12 (2023) 078 [arXiv:2309.15182] [INSPIRE].
R. Akhoury, S. Choi and M.J. Perry, Holography from singular supertranslations on a black hole horizon, Phys. Rev. Lett. 129 (2022) 221603 [arXiv:2205.07923] [INSPIRE].
R. Akhoury, S. Choi and M.J. Perry, Singular supertranslations and Chern-Simons theory on the black hole horizon, Phys. Rev. D 107 (2023) 085019 [arXiv:2212.00170] [INSPIRE].
J.-M. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaré (in French), Ann. Inst. H. Poincaré III (1965) 1.
N.D. Sen Gupta, On an analogue of the Galilei group, Nuovo Cim. A 44 (1966) 512 [INSPIRE].
M. Henneaux, Geometry of zero signature space-times, Bull. Soc. Math. Belg. 31 (1979) 47 [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].
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].
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view, J. Math. Phys. 57 (2016) 022507 [arXiv:1412.8212] [INSPIRE].
X. Bekaert and K. Morand, Connections and dynamical trajectories in generalised Newton-Cartan gravity II. An ambient perspective, J. Math. Phys. 59 (2018) 072503 [arXiv:1505.03739] [INSPIRE].
K. Morand, Embedding Galilean and Carrollian geometries I. Gravitational waves, J. Math. Phys. 61 (2020) 082502 [arXiv:1811.12681] [INSPIRE].
L. Ciambelli, R.G. Leigh, C. Marteau and P.M. Petropoulos, Carroll structures, null geometry and conformal isometries, Phys. Rev. D 100 (2019) 046010 [arXiv:1905.02221] [INSPIRE].
Y. Herfray, Carrollian manifolds and null infinity: a view from Cartan geometry, Class. Quant. Grav. 39 (2022) 215005 [arXiv:2112.09048] [INSPIRE].
E. Bergshoeff, J. Figueroa-O’Farrill and J. Gomis, A non-lorentzian primer, SciPost Phys. Lect. Notes 69 (2023) 1 [arXiv:2206.12177] [INSPIRE].
J. Hartong, Gauging the Carroll algebra and ultra-relativistic gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].
L. Ciambelli et al., Covariant Galilean versus Carrollian hydrodynamics from relativistic fluids, Class. Quant. Grav. 35 (2018) 165001 [arXiv:1802.05286] [INSPIRE].
A.C. Petkou, P.M. Petropoulos, D.R. Betancour and K. Siampos, Relativistic fluids, hydrodynamic frames and their Galilean versus Carrollian avatars, JHEP 09 (2022) 162 [arXiv:2205.09142] [INSPIRE].
D. Hansen, N.A. Obers, G. Oling and B.T. Søgaard, Carroll expansion of general relativity, SciPost Phys. 13 (2022) 055 [arXiv:2112.12684] [INSPIRE].
A. Campoleoni et al., Magnetic Carrollian gravity from the Carroll algebra, JHEP 09 (2022) 127 [arXiv:2207.14167] [INSPIRE].
P. Kraus and F. Larsen, Holographic gravitational anomalies, JHEP 01 (2006) 022 [hep-th/0508218] [INSPIRE].
L. Ciambelli and C. Marteau, Carrollian conservation laws and Ricci-flat gravity, Class. Quant. Grav. 36 (2019) 085004 [arXiv:1810.11037] [INSPIRE].
V. Chandrasekaran, E.E. Flanagan, I. Shehzad and A.J. Speranza, Brown-York charges at null boundaries, JHEP 01 (2022) 029 [arXiv:2109.11567] [INSPIRE].
D. Rivera-Betancour and M. Vilatte, Revisiting the Carrollian scalar field, Phys. Rev. D 106 (2022) 085004 [arXiv:2207.01647] [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].
M. Henneaux and P. Salgado-Rebolledo, Carroll contractions of Lorentz-invariant theories, JHEP 11 (2021) 180 [arXiv:2109.06708] [INSPIRE].
A. Fiorucci, Carrollian physics: flat spacetime as the hologram of the wonderworld, lecture notes of the XIX Modave summer school in mathematical physics, Modave, Belgium (2023).
E. Bergshoeff, D. Grumiller, S. Prohazka and J. Rosseel, Three-dimensional spin-3 theories based on general kinematical algebras, JHEP 01 (2017) 114 [arXiv:1612.02277] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field theories with conformal Carrollian symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
A. Bagchi, R. Basu, A. Mehra and P. Nandi, Field theories on null manifolds, JHEP 02 (2020) 141 [arXiv:1912.09388] [INSPIRE].
N. Gupta and N.V. Suryanarayana, Constructing Carrollian CFTs, JHEP 03 (2021) 194 [arXiv:2001.03056] [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].
J. Figueroa-O’Farrill, E. Have, S. Prohazka and J. Salzer, The gauging procedure and Carrollian gravity, JHEP 09 (2022) 243 [arXiv:2206.14178] [INSPIRE].
E.T. Newman and R. Penrose, New conservation laws for zero rest-mass fields in asymptotically flat space-time, Proc. Roy. Soc. Lond. A 305 (1968) 175 [INSPIRE].
Acknowledgments
We would like to thank our colleagues Andrea Campoleoni, Sangmin Choi, Simon Pekar, Anastasios Petkou and Matthieu Vilatte for useful discussions. The work of David Rivera-Betancour was funded by Becas Chile (ANID) Scholarship No. 72200301. Marios Petropoulos thanks Olivera Mišković and Rodrigo Olea for financial support and hospitality in Instituto de Física, Pontificia Universidad Católica de Valparaíso and Departamento de Ciencias Físicas, Universidad Andrés Bello, Santiago. Olivera Mišković and Rodrigo Olea thank the Centre de Physique Théorique of the Ecole Polytechnique for hospitality during the completion of this work. This work has been funded in part by Anillo Grant ANID/ACT210100 Holography and its Applications to High Energy Physics, Quantum Gravity and Condensed Matter Systems and FONDECYT Regular Grants 1190533, 1230492 and 1231779. David Rivera-Betancour thanks the programme Erasmus+ of the Institut Polytechnique de Paris as well as the Aristotle University of Thessaloniki and the Kapodistrian University of Athens for hosting him with this fellowship. The Third Carroll Workshop held in the Aristotle University of Thessaloniki in October 2023 is also acknowledged for providing an utmost creative framework, where ideas related to the present work have been exchanged.
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Mišković, O., Olea, R., Petropoulos, P.M. et al. Chern-Simons action and the Carrollian Cotton tensors. J. High Energ. Phys. 2023, 130 (2023). https://doi.org/10.1007/JHEP12(2023)130
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DOI: https://doi.org/10.1007/JHEP12(2023)130