[en] We propose a systematic generating procedure to construct free Lagrangians for massive, massless and partially massless, totally-symmetric tensor fields on AdSd+1 starting from the Becchi–Rouet–Stora–Tyutin (BRST) Lagrangian description of massless fields in the flat ambient space Rd,2. A novelty is that the Lagrangian is described by a d + 1 form on Rd,2 whose pullback to AdSd+1 gives the genuine Lagrangian defined on anti de Sitter spacetime. Our derivation uses the triplet formulation originating from the first-quantized BRST approach, where the action principle is determined by the BRST operator and the inner product of a first-quantised system. In this way we build, in a manifestly so(2, d)-covariant manner, a unifying action principle for the three types of fields mentioned above. In particular, our derivation justifies the form of some actions proposed earlier for massive and massless fields on (anti)-de Sitter. We also give a general setup for ambient Lagrangians in terms of the respective jet-bundles and variational bi-complexes. In particular we introduce a suitable ambient-space Euler–Lagrange differential which allows one to derive the equations of motion ambiently, i.e., without the need to explicitly derive the respective spacetime Lagrangian.
Research center :
AGIF - Algèbre, Géométrie et Interactions fondamentales
Disciplines :
Physics Mathematics
Author, co-author :
Bekaert, Xavier; Institut Denis Poisson, Unité Mixte de Recherche 7013, CNRS, Université de Tours, Université d’Orléans 1 , Parc de Grandmont, 37200 Tours, France
BOULANGER, Nicolas ; Université de Mons - UMONS > Faculté des Sciences > Service de Physique de l'Univers, Champs et Gravitation
Goncharov, Yegor ; Université de Mons - UMONS > Faculté des Sciences > Service de Physique de l'Univers, Champs et Gravitation ; Institut Denis Poisson, Unité Mixte de Recherche 7013, CNRS, Université de Tours, Université d’Orléans 1 , Parc de Grandmont, 37200 Tours, France
Grigoriev, Maxim ; Université de Mons - UMONS > Faculté des Sciences > Service de Physique de l'Univers, Champs et Gravitation ; I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute 3 , Leninsky Ave. 53, 119991 Moscow, Russia
Language :
English
Title :
Ambient-space variational calculus for gauge fields on constant-curvature spacetimes
Fierz, M. and Pauli, W., “ On relativistic wave equations for particles of arbitrary spin in an electromagnetic field,” Proc. R. Soc. London, Ser. A 173, 211- 232 ( 1939). 10.1098/rspa.1939.0140
Singh, L. P. S. and Hagen, C. R., “ Lagrangian formulation for arbitrary spin. I. The boson case,” Phys. Rev. D 9, 898- 909 ( 1974). 10.1103/physrevd.9.898
Singh, L. P. S. and Hagen, C. R., “ Lagrangian formulation for arbitrary spin. II. The fermion case,” Phys. Rev. D 9, 910- 920 ( 1974). 10.1103/physrevd.9.910
Fronsdal, C., “ Massless fields with integer spin,” Phys. Rev. D 18, 3624 ( 1978). 10.1103/physrevd.18.3624
Fang, J. and Fronsdal, C., “ Massless fields with half-integral spin,” Phys. Rev. D 18, 3630 ( 1978). 10.1103/physrevd.18.3630
Vasiliev, M. A., “‘ Gauge’ form of description of massless fields with arbitrary spin,” Yad. Fiz. 32, 855- 861 ( 1980).
Lopatin, V. E. and Vasiliev, M. A., “ Free massless bosonic fields of arbitrary spin in d-dimensional de Sitter space,” Mod. Phys. Lett. A 3, 257 ( 1988). 10.1142/s0217732388000313
Fronsdal, C., “ Singletons and massless, integral-spin fields on de Sitter space,” Phys. Rev. D 20, 848- 856 ( 1979). 10.1103/physrevd.20.848
Hallowell, K. and Waldron, A., “ Constant curvature algebras and higher spin action generating functions,” Nucl. Phys. B 724, 453- 486 ( 2005); arXiv:hep-th/0505255. 10.1016/j.nuclphysb.2005.06.021
Zinoviev, Y., “ On massive high spin particles in (A)dS,” arXiv:hep-th/0108192 ( 2001).
Metsaev, R. R., “ Massive totally symmetric fields in AdSd,” Phys. Lett. B 590, 95- 104 ( 2004); arXiv:hep-th/0312297. 10.1016/s0370-2693(04)00512-x
Skvortsov, E. D. and Vasiliev, M. A., “ Geometric formulation for partially massless fields,” Nucl. Phys. B 756, 117- 147 ( 2006); arXiv:hep-th/0601095. 10.1016/j.nuclphysb.2006.06.019
Zinoviev, Y. M., “ Frame-like gauge invariant formulation for massive high spin particles,” Nucl. Phys. B 808, 185- 204 ( 2009); arXiv:0808.1778 [hep-th]. 10.1016/j.nuclphysb.2008.09.020
Ponomarev, D. S. and Vasiliev, M. A., “ Frame-like action and unfolded formulation for massive higher-spin fields,” Nucl. Phys. B 839, 466- 498 ( 2010); arXiv:1001.0062 [hep-th]. 10.1016/j.nuclphysb.2010.06.007
Dirac, P. A. M., “ The electron wave equation in de-Sitter space,” Ann. Math. 36, 657- 669 ( 1935). 10.2307/1968649
Biswas, T. and Siegel, W., “ Radial dimensional reduction: (Anti) de Sitter theories from flat,” J. High Energy Phys. 2002( 07), 005; arXiv:hep-th/0203115. 10.1088/1126-6708/2002/07/005
Metsaev, R. R., “ Massless mixed-symmetry bosonic free fields in d-dimensional anti-de Sitter space-time,” Phys. Lett. B 354, 78- 84 ( 1995). 10.1016/0370-2693(95)00563-z
Metsaev, R. R., “ Arbitrary spin massless bosonic fields in d-dimensional anti-de Sitter space,” Lect. Notes Phys. 524, 331- 340 ( 1999); arXiv:hep-th/9810231. 10.1007/BFb0104614
Boulanger, N., Iazeolla, C., and Sundell, P., “ Unfolding mixed-symmetry fields in AdS and the BMV conjecture: II. Oscillator realization,” J. High Energy Phys. 2009( 07), 014; arXiv:0812.4438 [hep-th]. 10.1088/1126-6708/2009/07/014
Fefferman, C. and Graham, C. R., “ The ambient metric,” Ann. Math. Stud. 178, 1- 128 ( 2011); arXiv:0710.0919 [math.DG]. 10.23943/princeton/9780691153131.001.0001
Scherk, J. and Schwarz, J. H., “ How to get masses from extra dimensions,” Nucl. Phys. B 153, 61- 88 ( 1979). 10.1016/0550-3213(79)90462-0
Barnich, G. and Grigoriev, M., “ Parent form for higher spin fields on anti-de Sitter space,” J. High Energy Phys. 2006( 08), 013; arXiv:hep-th/0602166. 10.1088/1126-6708/2006/08/013
Alkalaev, K. B. and Grigoriev, M., “ Unified BRST description of AdS gauge fields,” Nucl. Phys. B 835, 197- 220 ( 2010); arXiv:0910.2690 [hep-th]. 10.1016/j.nuclphysb.2010.04.004
Alkalaev, K. and Grigoriev, M., “ Unified BRST approach to (partially) massless and massive AdS fields of arbitrary symmetry type,” Nucl. Phys. B 853, 663- 687 ( 2011); arXiv:1105.6111 [hep-th]. 10.1016/j.nuclphysb.2011.08.005
Fradkin, E. S. and Vilkovisky, G. A., “ Quantization of relativistic systems with constraints: Equivalence of canonical and covariant formalisms in quantum theory of gravitational field,” Report No. CERN-TH-2332, 6, ( 1977).
Batalin, I. A. and Vilkovisky, G. A., “ Relativistic S-matrix of dynamical systems with boson and fermion constraints,” Phys. Lett. B 69, 309- 312 ( 1977). 10.1016/0370-2693(77)90553-6
Fradkin, E. S. and Fradkina, T. E., “ Quantization of relativistic systems with boson and fermion first- and second-class constraints,” Phys. Lett. B 72, 343- 348 ( 1978). 10.1016/0370-2693(78)90135-1
Batalin, I. A. and Vilkovisky, G. A., “ Gauge algebra and quantization,” Phys. Lett. B 102, 27- 31 ( 1981). 10.1016/0370-2693(81)90205-7
Batalin, I. A. and Vilkovisky, G. A., “ Quantization of gauge theories with linearly dependent generators,” Phys. Rev. D 28, 2567- 2582 ( 1983); 10.1103/physrevd.28.2567
Erratum 30, 508 ( 1984). 10.1103/PhysRevD.30.508
Becchi, C., Rouet, A., and Stora, R., “ Renormalization of the abelian Higgs-Kibble model,” Commun. Math. Phys. 42, 127- 162 ( 1975). 10.1007/bf01614158
Becchi, C., Rouet, A., and Stora, R., “ Renormalization of gauge theories,” Ann. Phys. 98, 287- 321 ( 1976). 10.1016/0003-4916(76)90156-1
Tyutin, I. V., “ Gauge invariance in field theory and statistical physics in operator formalism,” Preprint of P. N. Labedev Physical Institute, No. 39, 1975; arXiv:0812.0580 [hep-th] ( 1975).
Neveu, A. and West, P. C., “ Gauge covariant local formulation of bosonic strings,” Nucl. Phys. B 268, 125- 150 ( 1986). 10.1016/0550-3213(86)90204-x
Neveu, A., Nicolai, H., and West, P. C., “ Gauge covariant local formulation of free strings and superstrings,” Nucl. Phys. B 264, 573- 587 ( 1986). 10.1016/0550-3213(86)90499-2
Banks, T. and Peskin, M. E., “ Gauge invariance of string fields,” Nucl. Phys. B 264, 513- 547 ( 1986). 10.1016/0550-3213(86)90496-7
Siegel, W. and Zwiebach, B., “ Gauge string fields,” Nucl. Phys. B 263, 105- 128 ( 1986). 10.1016/0550-3213(86)90030-1
Itoh, K., Kugo, T., Kunitomo, H., and Ooguri, H., “ Gauge invariant local action of string field from BRS formalism,” Prog. Theor. Phys. 75, 162 ( 1986). 10.1143/ptp.75.162
Bengtsson, A. K. H., “ A unified action for higher spin gauge bosons from covariant string theory,” Phys. Lett. B 182, 321- 325 ( 1986). 10.1016/0370-2693(86)90100-0
Ouvry, S. and Stern, J., “ Gauge fields of any spin and symmetry,” Phys. Lett. B 177, 335- 340 ( 1986). 10.1016/0370-2693(86)90763-x
Henneaux, M. and Teitelboim, C., “ First and second quantized point particles of any spin,” in 2nd Meeting on Quantum Mechanics of Fundamental Systems (CECS), 1987.
Bengtsson, A. K. H., “ BRST quantization in anti-de Sitter space and gauge fields,” Nucl. Phys. B 333, 407- 418 ( 1990). 10.1016/0550-3213(90)90044-e
Buchbinder, I. L., Pashnev, A., and Tsulaia, M., “ Lagrangian formulation of the massless higher integer spin fields in the AdS background,” Phys. Lett. B 523, 338- 346 ( 2001); arXiv:hep-th/0109067. 10.1016/s0370-2693(01)01268-0
Bonelli, G., “ On the covariant quantization of tensionless bosonic strings in AdS spacetime,” J. High Energy Phys. 2003( 11), 028; arXiv:hep-th/0309222. 10.1088/1126-6708/2003/11/028
Sagnotti, A. and Tsulaia, M., “ On higher spins and the tensionless limit of string theory,” Nucl. Phys. B 682, 83- 116 ( 2004); arXiv:hep-th/0311257. 10.1016/j.nuclphysb.2004.01.024
Buchbinder, I., Krykhtin, V., and Lavrov, P., “ Gauge invariant Lagrangian formulation of higher spin massive bosonic field theory in AdS space,” Nucl. Phys. B 762, 344- 376 ( 2007); arXiv:hep-th/0608005. 10.1016/j.nuclphysb.2006.11.021
Reshetnyak, A. and Moshin, P., “ Gauge invariant Lagrangian formulations for mixed symmetry higher spin bosonic fields in AdS spaces,” Universe 9( 12), 495 ( 2023); arXiv:2305.00142 [hep-th] ( 2023). 10.3390/universe9120495
Fotopoulos, A., Panigrahi, K. L., and Tsulaia, M., “ Lagrangian formulation of higher spin theories on AdS space,” Phys. Rev. D 74, 085029 ( 2006); arXiv:hep-th/0607248. 10.1103/physrevd.74.085029
Fotopoulos, A. and Tsulaia, M., “ Gauge invariant Lagrangians for free and interacting higher spin fields. A review of the BRST formulation,” Int. J. Mod. Phys. A 24, 1- 60 ( 2009); arXiv:0805.1346 [hep-th]. 10.1142/s0217751x09043134
Buchbinder, I. L. and Reshetnyak, A. A., “ General cubic interacting vertex for massless integer higher spin fields,” Phys. Lett. B 820, 136470 ( 2021); arXiv:2105.12030 [hep-th]. 10.1016/j.physletb.2021.136470
Buchbinder, I. L. and Reshetnyak, A. A., “ Covariant cubic interacting vertices for massless and massive integer higher spin fields,” Symmetry 15( 12), 2124 ( 2023); arXiv:2212.07097 [hep-th] ( 2022). 10.3390/sym15122124
Barnich, G. and Grigoriev, M., “ Hamiltonian BRST and Batalin-Vilkovisky formalisms for second quantization of gauge theories,” Commun. Math. Phys. 254, 581- 601 ( 2005); arXiv:hep-th/0310083. 10.1007/s00220-004-1275-4
Barnich, G., Grigoriev, M., Semikhatov, A., and Tipunin, I., “ Parent field theory and unfolding in BRST first-quantized terms,” Commun. Math. Phys. 260, 147- 181 ( 2005); arXiv:hep-th/0406192. 10.1007/s00220-005-1408-4
Grigoriev, M., “ Parent formulation at the Lagrangian level,” J. High Energy Phys. 2011( 07), 061; arXiv:1012.1903 [hep-th]. 10.1007/jhep07(2011)061
Thorn, C. B., “ String field theory,” Phys. Rep. 175, 1- 101 ( 1989). 10.1016/0370-1573(89)90015-x
Bochicchio, M., “ Gauge fixing for the field theory of the bosonic string,” Phys. Lett. B 193, 31 ( 1987). 10.1016/0370-2693(87)90451-5
Joung, E. and Taronna, M., “ Cubic interactions of massless higher spins in (A)dS: Metric-like approach,” Nucl. Phys. B 861, 145- 174 ( 2012); arXiv:1110.5918 [hep-th]. 10.1016/j.nuclphysb.2012.03.013
Joung, E., Lopez, L., and Taronna, M., “ On the cubic interactions of massive and partially-massless higher spins in (A)dS,” J. High Energy Phys. 2012( 07), 041; arXiv:1203.6578 [hep-th]. 10.1007/jhep07(2012)041
Joung, E., Lopez, L., and Taronna, M., “ Solving the Noether procedure for cubic interactions of higher spins in (A)dS,” J. Phys. A: Math. Theor. 46, 214020 ( 2013); arXiv:1207.5520 [hep-th]. 10.1088/1751-8113/46/21/214020
Bekaert, X. and Oblak, B., “ Massless scalars and higher-spin BMS in any dimension,” J. High Energy Phys. 2022( 11), 022; arXiv:2209.02253 [hep-th]. 10.1007/jhep11(2022)022
Chekmenev, A., “ Lagrangian BRST formulation of massive higher spin fields of generic symmetry type,” Theor. Math. Phys. 2, 1599- 1619 ( 2021); arXiv:1912.12079 [hep-th] ( 2019). 10.1134/s0040577921110076
Metsaev, R. R., “ BRST-BV approach to cubic interaction vertices for massive and massless higher-spin fields,” Phys. Lett. B 720, 237- 243 ( 2013); arXiv:1205.3131 [hep-th]. 10.1016/j.physletb.2013.02.009
Metsaev, R. R., “ Cubic interaction vertices for continuous-spin fields and arbitrary spin massive fields,” J. High Energy Phys. 2017( 11), 197; arXiv:1709.08596 [hep-th]. 10.1007/jhep11(2017)197
Francia, D. and Sagnotti, A., “ On the geometry of higher-spin gauge fields,” Classical Quantum Gravity 20, S473- S485 ( 2003); arXiv:hep-th/0212185. 10.1088/0264-9381/20/12/313
Alkalaev, K. B., Grigoriev, M., and Tipunin, I. Y., “ Massless Poincare modules and gauge invariant equations,” Nucl. Phys. B 823, 509- 545 ( 2009); arXiv:0811.3999 [hep-th]. 10.1016/j.nuclphysb.2009.08.007
Bekaert, X., Buchbinder, I. L., Pashnev, A., and Tsulaia, M., “ On higher spin theory: Strings, BRST, dimensional reductions,” Classical Quantum Gravity 21, S1457- S1463 ( 2004); arXiv:hep-th/0312252. 10.1088/0264-9381/21/10/018
Deser, S. and Waldron, A., “ Partial masslessness of higher spins in (A)dS,” Nucl. Phys. B 607, 577- 604 ( 2001); arXiv:hep-th/0103198. 10.1016/s0550-3213(01)00212-7
Anderson, I. M., “ Introduction to the variational bicomplex,” Contemp. Math. 132, 51- 73 ( 1992). 10.1090/conm/132/1188434
Bekaert, X. and Meunier, E., “ Higher spin interactions with scalar matter on constant curvature spacetimes: Conserved current and cubic coupling generating functions,” J. High Energy Phys. 2010( 11), 116; arXiv:1007.4384 [hep-th]. 10.1007/jhep11(2010)116
