Skip to main content
SpringerLink
Log in

A calculus for higher spin interactions

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

Abstract

Higher spin theories can be efficiently described in terms of auxiliary Stückelberg or projective space field multiplets. By considering how higher spin models couple to scale, these approaches can be unified in a conformal geometry/tractor calculus framework. We review these methods and apply them to higher spin vertices to obtain a generating function for massless, massive and partially massless three-point interactions.

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. A.K. Bengtsson, I. Bengtsson and L. Brink, Cubic interaction terms for arbitrary spin, Nucl. Phys. B 227 (1983) 31 [INSPIRE].

    Article  ADS  Google Scholar 

  3. R. Metsaev, Generating function for cubic interaction vertices of higher spin fields in any dimension, Mod. Phys. Lett. A 8 (1993) 2413 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  4. E. Fradkin and R. Metsaev, Cubic scattering amplitudes for all massless representations of the Poincaré group in any space-time dimension, Phys. Rev. D 52 (1995) 4660 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  5. R. Metsaev, Cubic interaction vertices of massive and massless higher spin fields, Nucl. Phys. B 759 (2006) 147 [hep-th/0512342] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. R. Metsaev, Cubic interaction vertices for fermionic and bosonic arbitrary spin fields, Nucl. Phys. B 859 (2012) 13 [arXiv:0712.3526] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  7. Y. Zinoviev, On spin 3 interacting with gravity, Class. Quant. Grav. 26 (2009) 035022 [arXiv:0805.2226] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  8. N. Boulanger, S. Leclercq and P. Sundell, On The Uniqueness of Minimal Coupling in Higher-Spin Gauge Theory, JHEP 08 (2008) 056 [arXiv:0805.2764] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  9. X. Bekaert, N. Boulanger and S. Leclercq, Strong obstruction of the Berends-Burgers-van Dam spin-3 vertex, J. Phys. A 43 (2010) 185401 [arXiv:1002.0289] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  10. R. Manvelyan, K. Mkrtchyan and W. Ruehl, Direct Construction of A Cubic Selfinteraction for Higher Spin gauge Fields, Nucl. Phys. B 844 (2011) 348 [arXiv:1002.1358] [INSPIRE].

    Article  ADS  Google Scholar 

  11. Y. Zinoviev, Spin 3 cubic vertices in a frame-like formalism, JHEP 08 (2010) 084 [arXiv:1007.0158] [INSPIRE].

    Article  ADS  Google Scholar 

  12. M. Henneaux, G. Lucena Gomez and R. Rahman, Higher-Spin Fermionic Gauge Fields and Their Electromagnetic Coupling, JHEP 08 (2012) 093 [arXiv:1206.1048] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  13. X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. 84 (2012) 987 [arXiv:1007.0435] [INSPIRE].

    Article  ADS  Google Scholar 

  14. R. Manvelyan, K. Mkrtchyan and W. Rühl, General trilinear interaction for arbitrary even higher spin gauge fields, Nucl. Phys. B 836 (2010) 204 [arXiv:1003.2877] [INSPIRE].

    Article  ADS  Google Scholar 

  15. M. Taronna, Higher Spins and String Interactions, arXiv:1005.3061 [INSPIRE].

  16. A. Sagnotti and M. Taronna, String Lessons for Higher-Spin Interactions, Nucl. Phys. B 842 (2011) 299 [arXiv:1006.5242] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  17. A. Fotopoulos and M. Tsulaia, On the Tensionless Limit of String theory, Off - Shell Higher Spin Interaction Vertices and BCFW Recursion Relations, JHEP 11 (2010) 086 [arXiv:1009.0727] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  18. E. Joung and M. Taronna, Cubic interactions of massless higher spins in (A)dS: metric-like approach, Nucl. Phys. B 861 (2012) 145 [arXiv:1110.5918] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  19. E. Joung, L. Lopez and M. Taronna, On the cubic interactions of massive and partially-massless higher spins in (A)dS, JHEP 07 (2012) 041 [arXiv:1203.6578] [INSPIRE].

    Article  ADS  Google Scholar 

  20. E. Joung, L. Lopez and M. Taronna, Solving the Noether procedure for cubic interactions of higher spins in (A)dS, J. Phys. A 46 (2013) 214020 [arXiv:1207.5520] [INSPIRE].

    ADS  Google Scholar 

  21. M. Taronna, Higher-Spin Interactions: three-point functions and beyond, arXiv:1209.5755 [INSPIRE].

  22. R. Manvelyan, R. Mkrtchyan and W. Ruehl, Radial Reduction and Cubic Interaction for Higher Spins in (A)dS space, Nucl. Phys. B 872 (2013) 265 [arXiv:1210.7227] [INSPIRE].

    Article  ADS  Google Scholar 

  23. E. Joung, L. Lopez and M. Taronna, Generating functions of (partially-)massless higher-spin cubic interactions, JHEP 01 (2013) 168 [arXiv:1211.5912] [INSPIRE].

    Article  ADS  Google Scholar 

  24. M. Vasiliev, Cubic Vertices for Symmetric Higher-Spin Gauge Fields in (A)dS d, Nucl. Phys. B 862 (2012) 341 [arXiv:1108.5921] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. N. Boulanger, D. Ponomarev and E. Skvortsov, Non-abelian cubic vertices for higher-spin fields in anti-de Sitter space, JHEP 05 (2013) 008 [arXiv:1211.6979] [INSPIRE].

    Article  ADS  Google Scholar 

  26. K. Alkalaev, FV-type action for AdS 5 mixed-symmetry fields, JHEP 03 (2011) 031 [arXiv:1011.6109] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  27. N. Boulanger, E. Skvortsov and Y. Zinoviev, Gravitational cubic interactions for a simple mixed-symmetry gauge field in AdS and flat backgrounds, J. Phys. A 44 (2011) 415403 [arXiv:1107.1872] [INSPIRE].

    Google Scholar 

  28. N. Boulanger and E. Skvortsov, Higher-spin algebras and cubic interactions for simple mixed-symmetry fields in AdS spacetime, JHEP 09 (2011) 063 [arXiv:1107.5028] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. L. Lopez, On cubic AdS interactions of mixed-symmetry higher spins, arXiv:1210.0554 [INSPIRE].

  30. R. Metsaev, Poincaré invariant dynamics of massless higher spins: fourth order analysis on mass shell, Mod. Phys. Lett. A 6 (1991) 359 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  31. R. Metsaev, S matrix approach to massless higher spins theory. 2: the Case of internal symmetry, Mod. Phys. Lett. A 6 (1991) 2411 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  32. M. Taronna, Higher-Spin Interactions: four-point functions and beyond, JHEP 04 (2012) 029 [arXiv:1107.5843] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. P. Dempster and M. Tsulaia, On the Structure of Quartic Vertices for Massless Higher Spin Fields on Minkowski Background, Nucl. Phys. B 865 (2012) 353 [arXiv:1203.5597] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  34. N. Boulanger, D. Ponomarev, E. Skvortsov and M. Taronna, On the uniqueness of higher-spin symmetries in AdS and CFT, arXiv:1305.5180 [INSPIRE].

  35. J. Scherk and J.H. Schwarz, How to Get Masses from Extra Dimensions, Nucl. Phys. B 153 (1979) 61 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. S. Rindani and M. Sivakumar, Gauge invariant description of massive higher spin particles by dimensional reduction, Phys. Rev. D 32 (1985) 3238 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  37. C. Aragone, S. Deser and Z. Yang, Massive higher spin from dimensional reduction of gauge fields, Annals Phys. 179 (1987) 76 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  38. T. Biswas and W. Siegel, Radial dimensional reduction: anti-de Sitter theories from flat, JHEP 07 (2002) 005 [hep-th/0203115] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. K. Hallowell and A. Waldron, Constant curvature algebras and higher spin action generating functions, Nucl. Phys. B 724 (2005) 453 [hep-th/0505255] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  40. P.A. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.

    Article  MathSciNet  Google Scholar 

  41. A. Gover, A. Shaukat and A. Waldron, Weyl Invariance and the Origins of Mass, Phys. Lett. B 675 (2009) 93 [arXiv:0812.3364] [INSPIRE].

    ADS  Google Scholar 

  42. A. Gover, A. Shaukat and A. Waldron, Tractors, Mass and Weyl Invariance, Nucl. Phys. B 812 (2009) 424 [arXiv:0810.2867] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  43. A. Shaukat and A. Waldron, Weyl’s Gauge Invariance: conformal Geometry, Spinors, Supersymmetry and Interactions, Nucl. Phys. B 829 (2010) 28 [arXiv:0911.2477] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  44. T.N. Bailey, M.G. Eastwood and A. Gover, Thomas’s structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994) 1191.

    Article  MathSciNet  MATH  Google Scholar 

  45. C. Fefferman and C.R. Graham, Conformal invariants, in Élie Cartan et les Mathématiques d’Aujourdui, Asterisque (1985), pg. 95–116.

  46. A. Cap and A.R. Gover, Standard tractors and the conformal ambient metric construction, Annals Global Anal. Geom. 24 (2003) 231 [math/0207016] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  47. A.R. Gover and L.J. Peterson, The Ambient obstruction tensor and the conformal deformation complex, Pacific J. Math. (2004) [math/0408229] [INSPIRE].

  48. A.R. Gover and A. Waldron, Boundary calculus for conformally compact manifolds, arXiv:1104.2991 [INSPIRE].

  49. A. Rod Gover, E. Latini and A. Waldron, Poincaré-Einstein Holography for Forms via Conformal Geometry in the Bulk, arXiv:1205.3489 [INSPIRE].

  50. A. Gover, Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature, J. Geom. Phys. 60 (2010) 182.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  51. A. Gover and A. Waldron, The so(d + 2, 2) Minimal Representation and Ambient Tractors: the Conformal Geometry of Momentum Space, Adv. Theor. Math. Phys. 13 (2009) [arXiv:0903.1394] [INSPIRE].

  52. P. Cherrier, Problèmes de Neumann non linéaires sur les variétés riemanniennes, J. Funct. Anal. 57 (1984) 154.

    Article  MathSciNet  MATH  Google Scholar 

  53. T. Damour and S. Deser, ’Geometry’ of spin 3 gauge theories, Annales Poincaré Phys. Theor. 47 (1987) 277 [INSPIRE].

    Google Scholar 

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

    MathSciNet  ADS  Google Scholar 

  55. K. Hallowell and A. Waldron, The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra, SIGMA 3 (2007) 089 [arXiv:0707.3164] [INSPIRE].

    MathSciNet  Google Scholar 

  56. M. Grigoriev and A. Waldron, Massive Higher Spins from BRST and Tractors, Nucl. Phys. B 853 (2011) 291 [arXiv:1104.4994] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  57. S. Ferrara, A. Grillo, G. Parisi and R. Gatto, The shadow operator formalism for conformal algebra. vacuum expectation values and operator products, Lett. Nuovo Cim. 4S2 (1972) 115.

    Google Scholar 

  58. F. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  59. M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].

    Article  ADS  Google Scholar 

  60. D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, arXiv:1204.3894 [INSPIRE].

  61. Y.S. Stanev, Correlation Functions of Conserved Currents in Four Dimensional Conformal Field Theory, Nucl. Phys. B 865 (2012) 200 [arXiv:1206.5639] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  62. A. Zhiboedov, A note on three-point functions of conserved currents, arXiv:1206.6370 [INSPIRE].

  63. X. Bekaert and M. Grigoriev, Notes on the ambient approach to boundary values of AdS gauge fields, J. Phys. A 46 (2013) 214008 [arXiv:1207.3439] [INSPIRE].

    ADS  Google Scholar 

  64. X. Bekaert and M. Grigoriev, Higher order singletons, partially massless fields and their boundary values in the ambient approach, arXiv:1305.0162 [INSPIRE].

  65. N. Colombo and P. Sundell, Twistor space observables and quasi-amplitudes in 4D higher spin gravity, JHEP 11 (2011) 042 [arXiv:1012.0813] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  66. N. Colombo and P. Sundell, Higher Spin Gravity Amplitudes From Zero-form Charges, arXiv:1208.3880 [INSPIRE].

  67. V. Didenko and E. Skvortsov, Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory, arXiv:1210.7963 [INSPIRE].

  68. O. Gelfond and M. Vasiliev, Operator algebra of free conformal currents via twistors, arXiv:1301.3123 [INSPIRE].

  69. V. Didenko, J. Mei and E. Skvortsov, Exact higher-spin symmetry in CFT: free fermion correlators from Vasiliev Theory, arXiv:1301.4166 [INSPIRE].

  70. I. Bars, The Standard Model of Particles and Forces in the Framework of 2T-physics, Phys. Rev. D 74 (2006) 085019 [hep-th/0606045] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  71. I. Bars, Gravity in 2T-Physics, Phys. Rev. D 77 (2008) 125027 [arXiv:0804.1585] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  72. I. Bars, Gauge Symmetry in Phase Space, Consequences for Physics and Spacetime, Int. J. Mod. Phys. A 25 (2010) 5235 [arXiv:1004.0688] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  73. 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] [INSPIRE].

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, 56126, Pisa, Italy

    Euihun Joung

  2. Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, D-14476, Golm, Germany

    Massimo Taronna

  3. Department of Mathematics, University of California Davis, 1 Shields Avenue, Davis, CA, 95616, USA

    Andrew Waldron

Authors
  1. Euihun Joung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Massimo Taronna
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andrew Waldron
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Massimo Taronna.

Additional information

ArXiv ePrint: 1305.5809

Rights and permissions

Reprints and permissions

About this article

Cite this article

Joung, E., Taronna, M. & Waldron, A. A calculus for higher spin interactions. J. High Energ. Phys. 2013, 186 (2013). https://doi.org/10.1007/JHEP07(2013)186

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2013)186

Keywords

Search

Navigation