Skip to main content
SpringerLink
Log in

First order parent formulation for generic gauge field theories

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We show how a generic gauge field theory described by a BRST differential can systematically be reformulated as a first order parent system whose spacetime part is determined by the de Rham differential. In the spirit of Vasiliev’s unfolded approach, this is done by extending the original space of fields so as to include their derivatives as new independent fields together with associated form fields. Through the inclusion of the antifield dependent part of the BRST differential, the parent formulation can be used both for on and off-shell formulations. For diffeomorphism invariant models, the parent formulation can be reformulated as an AKSZ-type sigma model. Several examples, such as the relativistic particle, parametrized theories, Yang-Mills theory, general relativity and the two dimensional sigma model are worked out in details.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Fierz and W. Pauli, On relativistic wave equations for particles of arbitrary spin in an electromagnetic field, Proc. Roy. Soc. Lond. A 173 (1939) 211 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  2. L.P.S. Singh and C.R. Hagen, Lagrangian formulation for arbitrary spin. 1. The boson case, Phys. Rev. D9 (1974) 898 [SPIRES].

    ADS  Google Scholar 

  3. C. Fronsdal, Massless fields with integer spin, Phys. Rev. D 18 (1978) 3624 [SPIRES].

    ADS  Google Scholar 

  4. M.A. Vasiliev, Equations of motion of interacting massless fields of all spins as a free differential algebra, Phys. Lett. B 209 (1988) 491 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  5. M.A. Vasiliev, Consistent equations for interacting massless fields of all spins in the first order in curvatures, Annals Phys. 190 (1989) 59 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. M.A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B 243 (1990) 378 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  7. O.V. Shaynkman and M.A. Vasiliev, Scalar field in any dimension from the higher spin gauge theory perspective, Theor. Math. Phys. 123 (2000) 683 [hep-th/0003123] [SPIRES].

    Article  MATH  Google Scholar 

  8. M.A. Vasiliev, Nonlinear equations for symmetric massless higher spin fields in (A)dS(d), Phys. Lett. B 567 (2003) 139 [hep-th/0304049] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. M.A. Vasiliev, Actions, charges and off-shell fields in the unfolded dynamics approach, Int. J. Geom. Meth. Mod. Phys. 3 (2006) 37 [hep-th/0504090] [SPIRES].

    Article  MathSciNet  Google Scholar 

  10. D. Sullivan, Infinitesimal computations in topology, Publications Mathématiques de L’IHÉS 47 (1977) 269.

    MATH  Google Scholar 

  11. R. D’Auria and P. Fre, Geometric supergravity in d = 11 and its hidden supergroup, Nucl. Phys. B 201 (1982) 101 [Erratum ibid. B 206 (1982) 496] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  12. P. Fre and P.A. Grassi, Free differential algebras, rheonomy, and pure spinors, arXiv:0801.3076 [SPIRES].

  13. G. Barnich, M. Grigoriev, A. Semikhatov, and I. Tipunin, Parent field theory and unfolding in BRST first-quantized terms, Commun. Math. Phys. 260 (2005) 147 [hep-th/0406192] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. G. Barnich and M. Grigoriev, Parent form for higher spin fields on anti-de Sitter space, JHEP 08 (2006) 013 [hep-th/0602166] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  15. B. Fedosov, Deformation quantization and index theory, Akademie-Verl., Berlin Germany (1996) [SPIRES].

    MATH  Google Scholar 

  16. X. Bekaert and M. Grigoriev, Manifestly conformal descriptions and higher symmetries of bosonic singletons, SIGMA 6 (2010) 038 [arXiv:0907.3195] [SPIRES].

    MathSciNet  Google Scholar 

  17. K.B. Alkalaev, M. Grigoriev and I.Y. Tipunin, Massless Poincaré modules and gauge invariant equations, Nucl. Phys. B 823 (2009) 509 [arXiv:0811.3999] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  18. K.B. Alkalaev and M. Grigoriev, Unified BRST description of AdS gauge fields, Nucl. Phys. B 835 (2010) 197 [arXiv:0910.2690] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  19. E.D. Skvortsov, Mixed-symmetry massless fields in Minkowski space unfolded, JHEP 07 (2008) 004 [arXiv:0801.2268] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  20. N. Boulanger, C. Iazeolla and P. Sundell, Unfolding mixed-symmetry fields in AdS and the BMV conjecture: I. General formalism, JHEP 07 (2009) 013 [arXiv:0812.3615] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  21. N. Boulanger, C. Iazeolla and P. Sundell, Unfolding mixed-symmetry fields in AdS and the BMV conjecture: II. Oscillator realization, JHEP 07 (2009) 014 [arXiv:0812.4438] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  22. E.D. Skvortsov, Gauge fields in (A)dS within the unfolded approach: algebraic aspects, JHEP 01 (2010) 106 [arXiv:0910.3334] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  23. I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981) 27 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  24. I.A. Batalin and G.A. Vilkovisky, Feynman rules for reducible gauge theories, Phys. Lett. B 120 (1983) 166 [SPIRES].

    ADS  Google Scholar 

  25. I.A. Batalin and G.A. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D 28 (1983) 2567 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  26. I.A. Batalin and G.A. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B 234 (1984) 106 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  27. I.A. Batalin and G.A. Vilkovisky, Existence theorem for gauge algebra, J. Math. Phys. 26 (1985) 172 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  28. M. Henneaux and C. Teitelboim, Quantization of gauge systems, Princeton University Press, Princeton U.S.A. (1992) [SPIRES].

    MATH  Google Scholar 

  29. J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept. 259 (1995) 1 [hep-th/9412228] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. F. Brandt, Gauge covariant algebras and local BRST cohomology, Contemp. Math. 219 (1999) 53 [hep-th/9711171] [SPIRES].

    Article  MathSciNet  Google Scholar 

  31. F. Brandt, Local BRST cohomology and covariance, Commun. Math. Phys. 190 (1997) 459 [hep-th/9604025] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  32. F. Brandt, Jet coordinates for local BRST cohomology, Lett. Math. Phys. 55 (2001) 149 [math-ph/0103006] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  33. R. Stora, Algebraic structure and topological origin of anomalies, seminar given at Cargese Summer Inst.: Progress in gauge field theory, Cargese, France, Sep 1–15, 1983, in Progress in gauge field theory, Plenum Press, New York U.S.A. (1984).

    Google Scholar 

  34. M. Alexandrov, M. Kontsevich, A. Schwartz and O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A 12 (1997) 1405 [hep-th/9502010] [SPIRES].

    ADS  Google Scholar 

  35. A.S. Cattaneo and G. Felder, A path integral approach to the Kontsevich quantization formula, Commun. Math. Phys. 212 (2000) 591 [math/9902090] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. M.A. Grigoriev and P.H. Damgaard, Superfield BRST charge and the master action, Phys. Lett. B 474 (2000) 323 [hep-th/9911092] [SPIRES].

    ADS  Google Scholar 

  37. I. Batalin and R. Marnelius, Superfield algorithms for topological field theories, in Multiple facets of quantization and supersymmetry, M. Olshanetsky and A. Vainshtein eds., World Scientific (2002), pg. 233–251 [hep-th/0110140] [SPIRES].

  38. I. Batalin and R. Marnelius, Generalized Poisson σ-models, Phys. Lett. B 512 (2001) 225 [hep-th/0105190] [SPIRES].

    ADS  Google Scholar 

  39. A.S. Cattaneo and G. Felder, On the AKSZ formulation of the Poisson σ-model, Lett. Math. Phys. 56 (2001) 163 [math/0102108] [SPIRES].

    Article  MATH  MathSciNet  Google Scholar 

  40. J.-S. Park, Topological open p-branes, hep-th/0012141 [SPIRES].

  41. D. Roytenberg, On the structure of graded symplectic supermanifolds and courant algebroids, math/0203110 [SPIRES].

  42. N. Ikeda, Deformation of Batalin-Vilkovisky structures, math/0604157 [SPIRES].

  43. F. Bonechi, P. Mnev and M. Zabzine, Finite dimensional AKSZ-BV theories, Lett. Math. Phys. 94 (2010) 197 [arXiv:0903.0995] [SPIRES].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  44. G. Barnich and M. Grigoriev, A Poincaré lemma for σ-models of AKSZ type, J. Geom. Phys. 61 (2011) 663 [arXiv:0905.0547] [SPIRES].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in gauge theories, Phys. Rept. 338 (2000) 439 [hep-th/0002245] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  46. G. Barnich and M. Henneaux, Consistent couplings between fields with a gauge freedom and deformations of the master equation, Phys. Lett. B 311 (1993) 123 [hep-th/9304057] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  47. P.J. Olver, Applications of Lie groups to differential equations, 2nd edition, Springer Verlag, New York U.S.A. (1993).

    Book  MATH  Google Scholar 

  48. I.M. Anderson, The variational bicomplex, technical report, Utah State University (1989).

  49. I.M. Anderson, Introduction to the variational bicomplex, Contemp. Math. 132 (1992) 51.

    Article  Google Scholar 

  50. L. Dickey, Soliton equations and Hamiltonian systems, Adv. Ser. Math. Phys. 12 (1991) 1 [SPIRES].

    MathSciNet  Google Scholar 

  51. A. Vinogradov, Translations of mathematical monographs. Vol. 204: Cohomological analysis of partial differential equations and secondary calculus, American Mathematical Society (2001).

  52. B.S. DeWitt, The global approach to quantum field theory. Vol. 1, 2, Int. Ser. Monogr. Phys. 114 (2003) 1 [SPIRES].

    MathSciNet  Google Scholar 

  53. G. Barnich and M. Grigoriev, BRST extension of the non-linear unfolded formalism, Physics AUC, 16 part II (2006) 46 [hep-th/0504119] [SPIRES].

    Google Scholar 

  54. G. Barnich and M. Grigoriev, Hamiltonian BRST and Batalin-Vilkovisky formalisms for second quantization of gauge theories, Commun. Math. Phys. 254 (2005) 581 [hep-th/0310083] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  55. A. Dresse, P. Grégoire and M. Henneaux, Path integral equivalence between the extended and nonextended Hamiltonian formalisms, Phys. Lett. B 245 (1990) 192 [SPIRES].

    ADS  Google Scholar 

  56. J.M.L. Fisch and M. Henneaux, Homological perturbation theory and the algebraic structure of the antifield-antibracket formalism for gauge theories, Commun. Math. Phys. 128 (1990) 627 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  57. M. Henneaux, Space-time locality of the BRST formalism, Commun. Math. Phys. 140 (1991) 1 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. G. Barnich, F. Brandt and M. Henneaux, Local BRST cohomology in the antifield formalism. 1. General theorems, Commun. Math. Phys. 174 (1995) 57 [hep-th/9405109] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  59. G. Barnich, M. Henneaux, T. Hurth and K. Skenderis, Cohomological analysis of gauge-fixed gauge theories, Phys. Lett. B 492 (2000) 376 [hep-th/9910201] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  60. M. Grigoriev, Off-shell gauge fields from BRST quantization, hep-th/0605089 [SPIRES].

  61. E.A. Ivanov and V.I. Ogievetsky, Gauge theories as theories of spontaneous breakdown, Lett. Math. Phys. 1 (1976) 309 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  62. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. B 77 (1978) 394 [SPIRES].

    ADS  Google Scholar 

  63. E.A. Ivanov, Yang-Mills theory in σ-model representation, JETP Lett. 30 (1979) 422 [SPIRES].

    ADS  Google Scholar 

  64. C.G. Torre, Natural symmetries of the Yang-Mills equations, J. Math. Phys. 36 (1995) 2113 [hep-th/9407129] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  65. A.B. Borisov and V.I. Ogievetsky, Theory of dynamical affine and conformal symmetries as gravity theory of the gravitational field, Theor. Math. Phys. 21 (1975) 1179 [Teor. Mat. Fiz. 21 (1974) 329] [SPIRES].

    Article  Google Scholar 

  66. A. Pashnev, Nonlinear realizations of the (super)diffeomorphism groups, geometrical objects and integral invariants in the superspace, hep-th/9704203 [SPIRES].

  67. N. Boulanger, A Weyl-covariant tensor calculus, J. Math. Phys. 46 (2005) 053508 [hep-th/0412314] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  68. R. Bonezzi, E. Latini and A. Waldron, Gravity, two times, tractors, Weyl invariance and six dimensional quantum mechanics, Phys. Rev. D 82 (2010) 064037 [arXiv:1007.1724] [SPIRES].

    ADS  Google Scholar 

  69. F. Brandt, W. Troost and A. Van Proeyen, The BRST-antibracket cohomology of 2d gravity, Nucl. Phys. B 464 (1996) 353 [hep-th/9509035] [SPIRES].

    Article  ADS  Google Scholar 

  70. G. Barnich, G. Bonelli and M. Grigoriev, From BRST to light-cone description of higher spin gauge fields, Physics AUC 15 part I (2005) 1 [hep-th/0502232] [SPIRES].

    Google Scholar 

  71. C.B. Thorn, Perturbation theory for quantized string fields, Nucl. Phys. B 287 (1987) 61 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  72. M. Bochicchio, Gauge fixing for the field theory of the bosonic string, Phys. Lett. B 193 (1987) 31 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  73. M. Bochicchio, String field theory in the Siegel gauge, Phys. Lett. B 188 (1987) 330 [SPIRES].

    MathSciNet  ADS  Google Scholar 

  74. C.B. Thorn, String field theory, Phys. Rept. 175 (1989) 1 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  75. C. Hull and B. Zwiebach, Double field theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  76. R.d.M. Koch, A. Jevicki, K. Jin and J.P. Rodrigues, AdS 4 /CFT 3 construction from collective fields, arXiv:1008.0633 [SPIRES].

  77. E. Sezgin and P. Sundell, Massless higher spins and holography, Nucl. Phys. B 644 (2002) 303 [hep-th/0205131] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  78. E. Sezgin and P. Sundell, 7D bosonic higher spin theory: symmetry algebra and linearized constraints, Nucl. Phys. B 634 (2002) 120 [hep-th/0112100] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  79. A. Sagnotti, E. Sezgin and P. Sundell, On higher spins with a strong Sp(2,R) condition, hep-th/0501156 [SPIRES].

  80. V.E. Didenko and M.A. Vasiliev, Static BPS black hole in 4d higher-spin gauge theory, Phys. Lett. B 682 (2009) 305 [arXiv:0906.3898] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  81. S. Giombi and X. Yin, Higher spin gauge theory and holography: the three-point functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  82. A. Sagnotti and M. Taronna, String lessons for higher-spin interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [SPIRES].

    Article  ADS  MathSciNet  Google Scholar 

  83. X. Bekaert, S. Cnockaert, C. Iazeolla and M.A. Vasiliev, Nonlinear higher spin theories in various dimensions, hep-th/0503128 [SPIRES].

Download references

Author information

Authors and Affiliations

  1. Physique Théorique et Mathématique, Université Libre de Bruxelles, and International Solvay Institutes, Campus Plaine C.P. 231, B-1050, Bruxelles, Belgium

    Glenn Barnich

  2. Tamm Theory Department, Lebedev Physics Institute, Leninsky prospect 53, 119991, Moscow, Russia

    Maxim Grigoriev

Authors
  1. Glenn Barnich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maxim Grigoriev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maxim Grigoriev.

Additional information

ArXiv ePrint: 1009.0190

Research Director of the Fund for Scientific Research-FNRS (Belgium) (Glenn Barnich)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barnich, G., Grigoriev, M. First order parent formulation for generic gauge field theories. J. High Energ. Phys. 2011, 122 (2011). https://doi.org/10.1007/JHEP01(2011)122

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2011)122

Keywords

Search

Navigation