Abstract
In this paper, we propose a novel way to construct off-shell actions of d-dimensional Carrollian field theories by considering the null-reduction of the Bargmann invariant actions in d +1 dimensions. This is based on the fact that d-dimensional Carrollian symmetry is the restriction of the (d + 1)-dimensional Bargmann symmetry to a null hypersurface. We focus on free scalar field theory and electromagnetic field theory, and show that the electric sectors and the magnetic sectors of these theories originate from different Bargmann invariant actions in one higher dimension. In the cases of massless free scalar field and d = 4 electromagnetic field, we verify the Carrollian conformal invariance of the resulting theories, and find that there appear naturally chain representations and staggered modules of Carrollian conformal algebra.
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J. Lévy-Leblond, Une nouvelle limite non-relativiste du groupe de Poincaré, Ann. Inst. H. Poincaré Phys. Théor. 3 (1965) 1.
N.D. Sen Gupta, On an Analogue of the Galileo Group, Nuovo Cim. 44 (1966) 512.
H. Bacry and J. Lévy-Leblond, Possible kinematics, J. Math. Phys. 9 (1968) 1605 [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].
E. Bergshoeff, J. Gomis and G. Longhi, Dynamics of Carroll Particles, Class. Quant. Grav. 31 (2014) 205009 [arXiv:1405.2264] [INSPIRE].
J.M. Souriau, Ondes et radiations gravitationnelles, Coll. Int. CNRS 220 (1973) 243.
C. Duval, G.W. Gibbons, P.A. Horvathy and P.-M. Zhang, Carroll symmetry of plane gravitational waves, Class. Quant. Grav. 34 (2017) 175003 [arXiv:1702.08284] [INSPIRE].
M. Henneaux, Geometry of Zero Signature Space-times, Bull. Soc. Math. Belg. 31 (1979) 47 [INSPIRE].
V. Belinski and M. Henneaux, The Cosmological Singularity, Cambridge University Press, Cambridge, U.K. (2017) [https://doi.org/10.1017/9781107239333] [INSPIRE].
R.F. Penna, Near-horizon Carroll symmetry and black hole Love numbers, arXiv:1812.05643 [INSPIRE].
L. Donnay and C. Marteau, Carrollian Physics at the Black Hole Horizon, Class. Quant. Grav. 36 (2019) 165002 [arXiv:1903.09654] [INSPIRE].
L. Freidel and P. Jai-akson, Carrollian hydrodynamics and symplectic structure on stretched horizons, arXiv:2211.06415 [INSPIRE].
J. Redondo-Yuste and L. Lehner, Non-linear black hole dynamics and Carrollian fluids, JHEP 02 (2023) 240 [arXiv:2212.06175] [INSPIRE].
J. de Boer et al., Carroll Symmetry, Dark Energy and Inflation, Front. in Phys. 10 (2022) 810405 [arXiv:2110.02319] [INSPIRE].
R. Casalbuoni, J. Gomis and D. Hidalgo, Worldline description of fractons, Phys. Rev. D 104 (2021) 125013 [arXiv:2107.09010] [INSPIRE].
F. Peña-Benitez, Fractons, symmetric gauge fields and geometry, Phys. Rev. Res. 5 (2023) 013101 [arXiv:2107.13884] [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].
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].
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].
J. Hartong, Holographic Reconstruction of 3D Flat Space-Time, JHEP 10 (2016) 104 [arXiv:1511.01387] [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].
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, Phys. Rev. D 107 (2023) 126027 [arXiv:2212.12553] [INSPIRE].
M. Henneaux and P. Salgado-Rebolledo, Carroll contractions of Lorentz-invariant theories, JHEP 11 (2021) 180 [arXiv:2109.06708] [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].
A. Bagchi, R. Basu, A. Mehra and P. Nandi, Field Theories on Null Manifolds, JHEP 02 (2020) 141 [arXiv:1912.09388] [INSPIRE].
A. Bagchi, A. Mehra and P. Nandi, Field Theories with Conformal Carrollian Symmetry, JHEP 05 (2019) 108 [arXiv:1901.10147] [INSPIRE].
E.A. Bergshoeff, J. Gomis and A. Kleinschmidt, Non-Lorentzian theories with and without constraints, JHEP 01 (2023) 167 [arXiv:2210.14848] [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].
D. Rivera-Betancour and M. Vilatte, Revisiting the Carrollian scalar field, Phys. Rev. D 106 (2022) 085004 [arXiv:2207.01647] [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].
B. Julia and H. Nicolai, Null Killing vector dimensional reduction and Galilean geometrodynamics, Nucl. Phys. B 439 (1995) 291 [hep-th/9412002] [INSPIRE].
A. Bagchi et al., Carroll covariant scalar fields in two dimensions, JHEP 01 (2023) 072 [arXiv:2203.13197] [INSPIRE].
A. Bagchi, A. Banerjee and H. Muraki, Boosting to BMS, JHEP 09 (2022) 251 [arXiv:2205.05094] [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].
A. Saha, Intrinsic approach to 1 + 1D Carrollian Conformal Field Theory, JHEP 12 (2022) 133 [arXiv:2207.11684] [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].
S. Dutta, Stress tensors of 3d Carroll CFTs, arXiv:2212.11002 [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].
C. Duval, G. Burdet, H.P. Kunzle and M. Perrin, Bargmann Structures and Newton-cartan Theory, Phys. Rev. D 31 (1985) 1841 [INSPIRE].
F. Rohsiepe, On reducible but indecomposable representations of the Virasoro algebra, hep-th/9611160 [INSPIRE].
M.R. Gaberdiel, An algebraic approach to logarithmic conformal field theory, Int. J. Mod. Phys. A 18 (2003) 4593 [hep-th/0111260] [INSPIRE].
K. Kytola and D. Ridout, On Staggered Indecomposable Virasoro Modules, J. Math. Phys. 50 (2009) 123503 [arXiv:0905.0108] [INSPIRE].
T. Creutzig and D. Ridout, Logarithmic Conformal Field Theory: Beyond an Introduction, J. Phys. A 46 (2013) 4006 [arXiv:1303.0847] [INSPIRE].
P.-X. Hao, W. Song, X. Xie and Y. Zhong, BMS-invariant free scalar model, Phys. Rev. D 105 (2022) 125005 [arXiv:2111.04701] [INSPIRE].
Z.-F. Yu and B. Chen, Free field realization of the BMS Ising model, JHEP 08 (2023) 116 [arXiv:2211.06926] [INSPIRE].
P.-X. Hao, W. Song, Z. Xiao and X. Xie, A BMS-invariant free fermion model, arXiv:2211.06927 [INSPIRE].
H.P. Jakobsen, Indecomposable finite-dimensional representations of a Lie algebras and Lie superalgebras, Lect. Notes Math. 2027 (2011) 125 [INSPIRE].
M. Hogervorst, M. Paulos and A. Vichi, The ABC (in any D) of Logarithmic CFT, JHEP 10 (2017) 201 [arXiv:1605.03959] [INSPIRE].
B. Chen, P.-X. Hao, R. Liu and Z.-F. Yu, On Galilean conformal bootstrap, JHEP 06 (2021) 112 [arXiv:2011.11092] [INSPIRE].
B. Chen and R. Liu, The shadow formalism of Galilean CFT2, JHEP 05 (2023) 224 [arXiv:2203.10490] [INSPIRE].
B. Chen, P.-X. Hao, R. Liu and Z.-F. Yu, On Galilean conformal bootstrap. Part II. ξ = 0 sector, JHEP 12 (2022) 019 [arXiv:2207.01474] [INSPIRE].
A. Banerjee, S. Dutta and S. Mondal, Carroll fermions in two dimensions, Phys. Rev. D 107 (2023) 125020 [arXiv:2211.11639] [INSPIRE].
I.M. Gel’fand and G.E. Shilov, Generalized Functions: Properties and Operations, American Mathematical Society (1964).
M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press (1992).
M. Islam, Carrollian Yang-Mills theory, JHEP 05 (2023) 238 [arXiv:2301.00953] [INSPIRE].
A. Bagchi et al., Magic fermions: Carroll and flat bands, JHEP 03 (2023) 227 [arXiv:2211.11640] [INSPIRE].
J. Hartong, Gauging the Carroll Algebra and Ultra-Relativistic Gravity, JHEP 08 (2015) 069 [arXiv:1505.05011] [INSPIRE].
E. Bergshoeff et al., Carroll versus Galilei Gravity, JHEP 03 (2017) 165 [arXiv:1701.06156] [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].
A. Campoleoni et al., Magnetic Carrollian gravity from the Carroll algebra, JHEP 09 (2022) 127 [arXiv:2207.14167] [INSPIRE].
J.E. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category \( \mathcal{O} \), American Mathematical Society (2008).
C.A. Weibel, An introduction to homological algebra, Cambridge University Press (1994) [https://doi.org/10.1017/CBO9781139644136].
S. Golkar and D.T. Son, Operator Product Expansion and Conservation Laws in Non-Relativistic Conformal Field Theories, JHEP 12 (2014) 063 [arXiv:1408.3629] [INSPIRE].
G. Hochschild and J.-P. Serre, Cohomology of group extensions, Trans. Am. Math. Soc. 74 (1953) 110.
G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Annals Math. 57 (1953) 591.
M.D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013) [https://doi.org/10.1017/9781139540940].
Acknowledgments
We are grateful to Zhe-fei Yu, Pengxiang Hao, Hongjie Chen, Yijun He, Yunsong Wei for valuable discussions. The work is partially supported by NSFC Grant No. 11735001, 12275004.
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Chen, B., Liu, R., Sun, H. et al. Constructing Carrollian field theories from null reduction. J. High Energ. Phys. 2023, 170 (2023). https://doi.org/10.1007/JHEP11(2023)170
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DOI: https://doi.org/10.1007/JHEP11(2023)170