Advertisement

Toda theory in AdS2 and 𝒲An-algebra structure of boundary correlators

  • Matteo BeccariaEmail author
  • Giulio Landolfi
Open Access
Regular Article - Theoretical Physics
First Online:

Abstract

We consider the conformal An Toda theory in AdS2. Due to the bulk full Virasoro symmetry, this system provides an instance of a non-gravitational AdS2/CFT1 correspondence where the 1d boundary theory enjoys enhanced \( ``\frac{1}{2}- Virasoro" \) symmetry. General boundary correlators are expected to be captured by the restriction of chiral correlators in a suitable WAn Virasoro extension. At next-to-leading order in weak coupling expansion they have been conjectured to match the subleading terms in the large central charge expansion of the dual 𝒲An correlators. We explicitly test this conjecture on the boundary four point functions of the Toda scalar fields dual to 𝒲An generators with next-to-minimal spin 3 and 4. Our analysis is valid in the generic rank case and extends previous results for specific rank-2 Toda theories. On the AdS side, the extension is straightforward and requires the computation of a finite set of tree Witten diagrams. This is due to simple rank dependence and selection rules of cubic and quartic couplings. On the boundary, we exploit crossing symmetry and specific meromorphic properties of the W-algebra correlators at large central charge. We present the required 4-point functions in closed form for any rank and verify the bulk-boundary correspondence in full details.

Keywords

AdS-CFT Correspondence Conformal Field Theory: 
Download to read the full article text

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited

References

  1. [1]
    C.G. Callan Jr. and F. Wilczek, Infrared behavior at negative curvature, Nucl. Phys.B 340 (1990) 366 [INSPIRE].ADSGoogle Scholar
  2. [2]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap. Part I: QFT in AdS, JHEP11 (2017) 133 [arXiv:1607.06109] [INSPIRE].
  3. [3]
    M.F. Paulos, J. Penedones, J. Toledo, B.C. van Rees and P. Vieira, The S-matrix bootstrap II: two dimensional amplitudes, JHEP11 (2017) 143 [arXiv:1607.06110] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  4. [4]
    D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP01 (2019) 200 [arXiv:1810.04185] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  5. [5]
    E. D’Hoker and R. Jackiw, Space translation breaking and compactification in the Liouville theory, Phys. Rev. Lett.50 (1983) 1719 [INSPIRE].ADSMathSciNetGoogle Scholar
  6. [6]
    E. D’Hoker, D.Z. Freedman and R. Jackiw, SO(2, 1) Invariant Quantization of the Liouville Theory, Phys. Rev.D 28 (1983) 2583 [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    T. Inami and H. Ooguri, Dynamical breakdown of supersymmetry in two-dimensional Anti-de Sitter space, Nucl. Phys.B 273 (1986) 487 [INSPIRE].ADSMathSciNetGoogle Scholar
  8. [8]
    A. Strominger, AdS 2quantum gravity and string theory, JHEP01 (1999) 007 [hep-th/9809027] [INSPIRE].ADSMathSciNetGoogle Scholar
  9. [9]
    N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS 5 × S 5: Semiclassical partition function, JHEP04 (2000) 021 [hep-th/0001204] [INSPIRE].ADSzbMATHGoogle Scholar
  10. [10]
    L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP11 (2007) 068 [arXiv:0710.1060] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  11. [11]
    A.M. Polyakov and V.S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys.B 581 (2000) 116 [hep-th/0002106] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  12. [12]
    J. Polchinski and J. Sully, Wilson Loop Renormalization Group Flows, JHEP10 (2011) 059 [arXiv:1104.5077] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  13. [13]
    N. Drukker and S. Kawamoto, Small deformations of supersymmetric Wilson loops and open spin-chains, JHEP07 (2006) 024 [hep-th/0604124] [INSPIRE].ADSMathSciNetGoogle Scholar
  14. [14]
    S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2/CFT 1, Nucl. Phys.B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].ADSzbMATHGoogle Scholar
  15. [15]
    M. Beccaria and A.A. Tseytlin, On non-supersymmetric generalizations of the Wilson-Maldacena loops in N = 4 SYM, Nucl. Phys.B 934 (2018) 466 [arXiv:1804.02179] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  16. [16]
    M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in N = 4 SYM and AdS 2/CFT 1, JHEP05 (2019) 122 [arXiv:1903.04365] [INSPIRE].ADSzbMATHGoogle Scholar
  17. [17]
    M. Hotta, Asymptotic isometry and two-dimensional anti-de Sitter gravity, gr-qc/9809035 [INSPIRE].
  18. [18]
    M. Cadoni and S. Mignemi, Asymptotic symmetries of AdS 2and conformal group in d = 1, Nucl. Phys.B 557 (1999) 165 [hep-th/9902040] [INSPIRE].ADSzbMATHGoogle Scholar
  19. [19]
    J. Navarro-Salas and P. Navarro, AdS 2/CFT 1correspondence and near extremal black hole entropy, Nucl. Phys.B 579 (2000) 250 [hep-th/9910076] [INSPIRE].ADSzbMATHGoogle Scholar
  20. [20]
    A. Almheiri and J. Polchinski, Models of AdS 2backreaction and holography, JHEP11 (2015) 014 [arXiv:1402.6334] [INSPIRE].ADSzbMATHGoogle Scholar
  21. [21]
    J. Maldacena, D. Stanford and Z. Yang, Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space, Prog. Theor. Exp. Phys.2016 (2016) 12C104 [arXiv:1606.01857] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  22. [22]
    J. EngelsĂśy, T.G. Mertens and H. Verlinde, An investigation of AdS 2backreaction and holography, JHEP07 (2016) 139 [arXiv:1606.03438] [INSPIRE].ADSzbMATHGoogle Scholar
  23. [23]
    M. Beccaria and A.A. Tseytlin, On boundary correlators in Liouville theory on AdS 2, JHEP07 (2019) 008 [arXiv:1904.12753] [INSPIRE].ADSGoogle Scholar
  24. [24]
    A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett.B 103 (1981) 207 [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    J. Teschner, Liouville theory revisited, Class. Quant. Grav.18 (2001) R153 [hep-th/0104158] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  26. [26]
    Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  27. [27]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
  28. [28]
    P. Menotti and E. Tonni, Standard and geometric approaches to quantum Liouville theory on the pseudosphere, Nucl. Phys.B 707 (2005) 321 [hep-th/0406014] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    H. Ouyang, Holographic four-point functions in Toda field theories in AdS 2, JHEP04 (2019) 159 [arXiv:1902.10536] [INSPIRE].ADSzbMATHGoogle Scholar
  30. [30]
    J.-L. Gervais and A. Neveu, New Quantum Treatment of Liouville Field Theory, Nucl. Phys.B 224 (1983) 329 [INSPIRE].ADSMathSciNetGoogle Scholar
  31. [31]
    P. Mansfield, Light Cone Quantization of the Liouville and Toda Field Theories, Nucl. Phys.B 222 (1983) 419 [INSPIRE].ADSGoogle Scholar
  32. [32]
    E. Braaten, T. Curtright, G. Ghandour and C.B. Thorn, A Class of Conformally Invariant Quantum Field Theories, Phys. Lett.B 125 (1983) 301 [INSPIRE].ADSGoogle Scholar
  33. [33]
    M. Beccaria, H. Jiang and A.A. Tseytlin, in preparation.Google Scholar
  34. [34]
    P. Bouwknegt and K. Schoutens, W symmetry in conformal field theory, Phys. Rept.223 (1993) 183 [hep-th/9210010] [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    V.A. Fateev and S.L. Lukyanov, The Models of Two-Dimensional Conformal Quantum Field Theory with Z nSymmetry, Int. J. Mod. Phys.A 3 (1988) 507 [INSPIRE].ADSGoogle Scholar
  36. [36]
    V.A. Fateev and A.V. Litvinov, Correlation functions in conformal Toda field theory. I, JHEP11 (2007) 002 [arXiv:0709.3806] [INSPIRE].
  37. [37]
    P. Christe and G. Mussardo, Integrable Systems Away from Criticality: The Toda Field Theory and S Matrix of the Tricritical Ising Model, Nucl. Phys.B 330 (1990) 465 [INSPIRE].ADSMathSciNetGoogle Scholar
  38. [38]
    P. Christe and G. Mussardo, Elastic S-Matrices in (1 + 1)-Dimensions and Toda Field Theories, Int. J. Mod. Phys.A 5 (1990) 4581 [INSPIRE].ADSMathSciNetGoogle Scholar
  39. [39]
    H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Affine Toda Field Theory and Exact S-Matrices, Nucl. Phys.B 338 (1990) 689 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  40. [40]
    H.W. Braden, E. Corrigan, P.E. Dorey and R. Sasaki, Extended Toda Field Theory and Exact S-Matrices, Phys. Lett.B 227 (1989) 411 [INSPIRE].ADSzbMATHGoogle Scholar
  41. [41]
    B. Gabai, D. Mazáč, A. Shieber, P. Vieira and Y. Zhou, No Particle Production in Two Dimensions: Recursion Relations and Multi-Regge Limit, JHEP02 (2019) 094 [arXiv:1803.03578] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  42. [42]
    F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  43. [43]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  44. [44]
    E. Perlmutter, Virasoro conformal blocks in closed form, JHEP08 (2015) 088 [arXiv:1502.07742] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  45. [45]
    A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].Google Scholar
  46. [46]
    P. Bowcock and G.M.T. Watts, On the classification of quantum W algebras, Nucl. Phys.B 379 (1992) 63 [hep-th/9111062] [INSPIRE].ADSMathSciNetGoogle Scholar
  47. [47]
    V.A. Fateev and A.B. Zamolodchikov, Conformal Quantum Field Theory Models in Two-Dimensions Having Z 3Symmetry, Nucl. Phys.B 280 (1987) 644 [INSPIRE].ADSGoogle Scholar
  48. [48]
    F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Extensions of the Virasoro Algebra Constructed from Kac-Moody Algebras Using Higher Order Casimir Invariants, Nucl. Phys.B 304 (1988) 348 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    F.A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Coset Construction for Extended Virasoro Algebras, Nucl. Phys.B 304 (1988) 371 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  50. [50]
    H.G. Kausch and G.M.T. Watts, A Study of W algebras using Jacobi identities, Nucl. Phys.B 354 (1991) 740 [INSPIRE].ADSMathSciNetGoogle Scholar
  51. [51]
    K. Thielemans, A Mathematica package for computing operator product expansions, Int. J. Mod. Phys.C 2 (1991) 787 [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  52. [52]
    A.L. Fitzpatrick, J. Kaplan, D. Li and J. Wang, Exact Virasoro Blocks from Wilson Lines and Background-Independent Operators, JHEP07 (2017) 092 [arXiv:1612.06385] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  53. [53]
    Y. Hikida and T. Uetoko, Correlators in higher-spin AdS 3holography from Wilson lines with loop correctionsCorrelators in higher-spin AdS 3holography from Wilson lines with loop corrections, Prog. Theor. Exp. Phys.2017 (2017) 113B03 [arXiv:1708.08657] [INSPIRE].
  54. [54]
    Y. Hikida and T. Uetoko, Conformal blocks from Wilson lines with loop corrections, Phys. Rev.D 97 (2018) 086014 [arXiv:1801.08549] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  55. [55]
    A. Bombini, S. Giusto and R. Russo, A note on the Virasoro blocks at order 1/c, Eur. Phys. J.C 79 (2019) 3 [arXiv:1807.07886] [INSPIRE].ADSGoogle Scholar
  56. [56]
    K. Hornfeck, The Minimal supersymmetric extension of WA n−1, Phys. Lett.B 275 (1992) 355 [INSPIRE].ADSGoogle Scholar
  57. [57]
    K. Hornfeck, Classification of structure constants for W-algebras from highest weights, Nucl. Phys.B 411 (1994) 307 [hep-th/9307170] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  58. [58]
    R. Blumenhagen, W. Eholzer, A. Honecker, K. Hornfeck and R. Hubel, Coset realization of unifying W-algebras, Int. J. Mod. Phys.A 10 (1995) 2367 [hep-th/9406203] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  59. [59]
    A.R. Linshaw, Universal two-parameter W ∞-algebra and vertex algebras of type W(2, 3, . . . , N ), arXiv:1710.02275 [INSPIRE].
  60. [60]
    M.R. Gaberdiel and R. Gopakumar, Triality in Minimal Model Holography, JHEP07 (2012) 127 [arXiv:1205.2472] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  61. [61]
    C. Candu, M.R. Gaberdiel, M. Kelm and C. Vollenweider, Even spin minimal model holography, JHEP01 (2013) 185 [arXiv:1211.3113] [INSPIRE].ADSGoogle Scholar
  62. [62]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and complete four point functions in the AdS/CFT correspondence, Nucl. Phys.B 562 (1999) 353 [hep-th/9903196] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  63. [63]
    G. Arutyunov, F.A. Dolan, H. Osborn and E. Sokatchev, Correlation functions and massive Kaluza-Klein modes in the AdS/CFT correspondence, Nucl. Phys.B 665 (2003) 273 [hep-th/0212116] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  64. [64]
    E. D’Hoker, D.Z. Freedman and L. Rastelli, AdS/CFT four point functions: How to succeed at z-integrals without really trying, Nucl. Phys.B 562 (1999) 395 [hep-th/9905049] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar

Copyright information

Š The Author(s) 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Fisica Ennio De GiorgiUniversitĂ  del Salento & INFNLecceItaly

Personalised recommendations