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Massless hook field in AdS d+1 from the holographic perspective

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Abstract

We systematically consider the AdS/CFT correspondence for a simplest mixed- symmetry massless gauge field described by hook Young diagram. We introduce the radial gauge fixing and explicitly solve the Dirichlet problem for the hook field equations. Solution finding conveniently splits in two steps. We first define an incomplete solution characterized by a functional freedom and then impose the boundary conditions. The resulting complete solution is fixed unambiguously up to boundary values. Two-point correlation function of hook primary operators is found via the corresponding boundary effective action computed separately in even and odd boundary dimensions. In particular, the higher-derivative action for boundary conformal hook fields is identified with a singular part of the effective action in even dimensions. The bulk/boundary symmetry transmutation within the Dirichlet boundary problem is explicitly studied. It is shown that traces of boundary fields are Stueckelberg-like modes that can be algebraically gauged away so that boundary fields are traceless.

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References

  1. L. Brink, R. Metsaev and M.A. Vasiliev, How massless are massless fields in AdS(d), Nucl. Phys. B 586 (2000) 183 [hep-th/0005136] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  2. R. Metsaev, Light cone form of field dynamics in Anti-de Sitter space-time and AdS/CFT correspondence, Nucl. Phys. B 563 (1999) 295 [hep-th/9906217] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. A.A. Tseytlin, On limits of superstring in AdS 5 × S 5, Theor. Math. Phys. 133 (2002) 1376 [hep-th/0201112] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  4. M. Flato and C. Fronsdal, One Massless Particle Equals Two Dirac Singletons: Elementary Particles in a Curved Space. 6, Lett. Math. Phys. 2 (1978) 421 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  5. M. Vasiliev, Higher spin superalgebras in any dimension and their representations, JHEP 12 (2004) 046 [hep-th/0404124] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  6. K. Alkalaev, Mixed-symmetry tensor conserved currents and AdS/CFT correspondence, arXiv:1207.1079 [INSPIRE].

  7. J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  8. S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  9. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

    MathSciNet  ADS  MATH  Google Scholar 

  10. R. Metsaev, Massless arbitrary spin fields in AdS 5, Phys. Lett. B 531 (2002) 152 [hep-th/0201226] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  11. R. Metsaev, Light-cone formulation of conformal field theory adapted to AdS/CFT correspondence, Phys. Lett. B 636 (2006) 227 [hep-th/0512330] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  12. R. Metsaev, CFT adapted gauge invariant formulation of arbitrary spin fields in AdS and modified de Donder gauge, Phys. Lett. B 671 (2009) 128 [arXiv:0808.3945] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  13. R. Metsaev, CFT adapted gauge invariant formulation of massive arbitrary spin fields in AdS, Phys. Lett. B 682 (2010) 455 [arXiv:0907.2207] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  14. C.-M. Chang and X. Yin, Higher Spin Gravity with Matter in AdS 3 and Its CFT Dual, JHEP 10 (2012) 024 [arXiv:1106.2580] [INSPIRE].

    Article  ADS  Google Scholar 

  15. S. Giombi and X. Yin, Higher Spins in AdS and Twistorial Holography, JHEP 04 (2011) 086 [arXiv:1004.3736] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  16. M.A. Vasiliev, Holography, Unfolding and Higher-Spin Theory, arXiv:1203.5554 [INSPIRE].

  17. X. Bekaert and M. Grigoriev, Notes on the ambient approach to boundary values of AdS gauge fields, arXiv:1207.3439 [INSPIRE].

  18. V. Didenko and E. Skvortsov, Towards higher-spin holography in ambient space of any dimension, arXiv:1207.6786 [INSPIRE].

  19. X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, Nonlinear higher spin theories in various dimensions, in First Solvay Workshop on Higher Spin Gauge Theories, G. Barnich and G. Bonelli eds., International Solvay Institutes (2005). Lectures given by M.A. Vasiliev at the First Solvay Workshop on Higher Spin Gauge Theories, Brussels, Belgium, May 12-14 2004.

  20. H. Liu and A.A. Tseytlin, D = 4 super Yang-Mills, D = 5 gauged supergravity and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  21. G. Arutyunov and S. Frolov, On the origin of supergravity boundary terms in the AdS/CFT correspondence, Nucl. Phys. B 544 (1999) 576 [hep-th/9806216] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  22. W. Mueck and K. Viswanathan, Conformal field theory correlators from classical field theory on anti-de Sitter space. 2. Vector and spinor fields, Phys. Rev. D 58 (1998) 106006 [hep-th/9805145] [INSPIRE].

    ADS  Google Scholar 

  23. W. Mueck and K. Viswanathan, The graviton in the AdS-CFT correspondence: Solution via the Dirichlet boundary value problem, hep-th/9810151 [INSPIRE].

  24. A. Polishchuk, Massive symmetric tensor field on AdS, JHEP 07 (1999) 007 [hep-th/9905048] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  25. R. Metsaev, Massless mixed symmetry bosonic free fields in d-dimensional anti-de Sitter space-time, Phys. Lett. B 354 (1995) 78 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  26. K. Alkalaev, Two column higher spin massless fields in AdS(d), Theor. Math. Phys. 140 (2004) 1253 [hep-th/0311212] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Vasiliev, Bosonic conformal higher-spin fields of any symmetry, Nucl. Phys. B 829 (2010) 176 [arXiv:0909.5226] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  28. D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: A new method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  29. T. Curtright, Generalized gauge fields, Phys. Lett. B 165 (1985) 304 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  30. Y. Zinoviev, On massive mixed symmetry tensor fields in Minkowski space and (A)dS, hep-th/0211233 [INSPIRE].

  31. Y. Zinoviev, Towards frame-like gauge invariant formulation for massive mixed symmetry bosonic fields. II. General Young tableau with two rows, Nucl. Phys. B 826 (2010) 490 [arXiv:0907.2140] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  32. K. Alkalaev, O. Shaynkman and M. Vasiliev, On the frame - like formulation of mixed symmetry massless fields in (A)dS(d), Nucl. Phys. B 692 (2004) 363 [hep-th/0311164] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  33. K. Alkalaev, O. Shaynkman and M. Vasiliev, Frame-like formulation for free mixed-symmetry bosonic massless higher-spin fields in AdS(d), hep-th/0601225 [INSPIRE].

  34. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  35. 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] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  36. E. Skvortsov, Gauge fields in (A)dS(d) and Connections of its symmetry algebra, J. Phys. A 42 (2009) 385401 [arXiv:0904.2919] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  38. K. Alkalaev and M. Grigoriev, Unified BRST approach to (partially) massless and massive AdS fields of arbitrary symmetry type, Nucl. Phys. B 853 (2011) 663 [arXiv:1105.6111] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  39. C. Burdik and A. Reshetnyak, On representations of Higher Spin symmetry algebras for mixed-symmetry HS fields on AdS-spaces. Lagrangian formulation, J. Phys. Conf. Ser. 343 (2012) 012102 [arXiv:1111.5516] [INSPIRE].

    Article  ADS  Google Scholar 

  40. A. Campoleoni and D. Francia, Maxwell-like Lagrangians for higher spins, arXiv:1206.5877 [INSPIRE].

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

    Article  MathSciNet  ADS  Google Scholar 

  42. 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 

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

  44. M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  45. K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  46. R. Metsaev, Gauge invariant two-point vertices of shadow fields, AdS/CFT and conformal fields, Phys. Rev. D 81 (2010) 106002 [arXiv:0907.4678] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  47. I. Gelfand and G. Shilov, Generalized functions. Volume I (in Russian), (1959).

  48. E. Sezgin and P. Sundell, Towards massless higher spin extension of D = 5, N = 8 gauged supergravity, JHEP 09 (2001) 025 [hep-th/0107186] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  49. K. Alkalaev, Mixed-symmetry massless gauge fields in AdS 5, Theor. Math. Phys. 149 (2006) 1338 [hep-th/0501105] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  50. G. Arutyunov and S. Frolov, Antisymmetric tensor field on AdS 5, Phys. Lett. B 441 (1998) 173 [hep-th/9807046] [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  51. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (1972).

  52. W. Mueck and K. Viswanathan, A Regularization scheme for the AdS/CFT correspondence, JHEP 07 (1999) 018 [hep-th/9904039] [INSPIRE].

    Article  ADS  Google Scholar 

  53. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, (1963).

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  1. I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky ave. 53, 119991, Moscow, Russia

    Konstantin Alkalaev

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ArXiv ePrint: 1210.0217

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Alkalaev, K. Massless hook field in AdS d+1 from the holographic perspective. J. High Energ. Phys. 2013, 18 (2013). https://doi.org/10.1007/JHEP01(2013)018

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