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Journal of High Energy Physics

, 2019:51 | Cite as

Supersymmetric Liouville theory in AdS2 and AdS/CFT

  • Matteo BeccariaEmail author
  • Hongliang Jiang
  • Arkady A. Tseytlin
Open Access
Regular Article - Theoretical Physics
  • 21 Downloads

Abstract

In a series of recent papers, a special kind of AdS2/CFT1 duality was observed: the boundary correlators of elementary fields that appear in the Lagrangian of a 2d conformal theory in rigid AdS2 background are the same as the correlators of the corresponding primary operators in the chiral half of that 2d CFT in flat space restricted to the real line. The examples considered were: (i) the Liouville theory where the operator dual to the Liouville scalar in AdS2 is the stress tensor; (ii) the abelian Toda theory where the operators dual to the Toda scalars are the \( \mathcal{W} \) -algebra generators; (iii) the non-abelian Toda theory where the Liouville field is dual to the stress tensor while the extra gauged WZW theory scalars are dual to non-abelian parafermionic operators. By direct Witten diagram com- putations in AdS2 one can check that the structure of the boundary correlators is indeed consistent with the Virasoro (or higher) symmetry. Here we consider a supersymmetric generalization: the \( \mathcal{N} \) = 1 superconformal Liouville theory in AdS2. We start with the super Liouville theory coupled to 2d supergravity and show that a consistent restriction to rigid AdS2 background requires a non-zero value of the supergravity auxiliary field and thus a modification of the Liouville potential from its familiar flat-space form. We show that the Liouville scalar and its fermionic partner are dual to the chiral half of the stress tensor and the supercurrent of the super Liouville theory on the plane. We perform tests supporting the duality by explicitly computing AdS2 Witten diagrams with bosonic and fermionic loops.

Keywords

AdS-CFT Correspondence Supersymmetry and Duality 

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]
    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].ADSzbMATHCrossRefGoogle Scholar
  2. [2]
    M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in \( \mathcal{N} \) = 4 SYM and AdS 2/CFT 1 , JHEP05 (2019) 122 [arXiv:1903.04365] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  3. [3]
    H. Ouyang, Holographic four-point functions in Toda field theories in AdS 2, JHEP04 (2019) 159 [arXiv:1902.10536] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  4. [4]
    M. Beccaria and A.A. Tseytlin, On boundary correlators in Liouville theory on AdS 2, JHEP07 (2019) 008 [arXiv:1904.12753] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    M. Beccaria and G. Landolfi, Toda theory in AdS 2and \( \mathcal{W} \)An -algebra structure of boundary correlators, JHEP10 (2019) 003 [arXiv:1906.06485] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M. Beccaria, H. Jiang and A.A. Tseytlin, Non-abelian Toda theory on AdS 2and AdS 2/ \( CFT\frac{1/2}{2} \)duality, JHEP09 (2019) 036 [arXiv:1907.01357] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    E. D’Hoker and R. Jackiw, Space translation breaking and compactification in the Liouville theory, Phys. Rev. Lett.50 (1983) 1719 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    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
  9. [9]
    A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
  10. [10]
    A.M. Polyakov, Quantum Geometry of Bosonic Strings, Phys. Lett.B 103 (1981) 207 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    Y. Nakayama, Liouville field theory: A Decade after the revolution, Int. J. Mod. Phys.A 19 (2004) 2771 [hep-th/0402009] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    A.M. Polyakov, Quantum Geometry of Fermionic Strings, Phys. Lett.B 103 (1981) 211 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  13. [13]
    J. Distler, Z. Hlousek and H. Kawai, SuperLiouville Theory as a Two-Dimensional, Superconformal Supergravity Theory, Int. J. Mod. Phys.A 5 (1990) 391 [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    K. Higashijima, T. Uematsu and Y.-z. Yu, Dynamical Supersymmetry Breaking in Two-dimensional N = 1 Supergravity Theories, Phys. Lett.B 139 (1984) 161 [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    W.A. Bardeen and D.Z. Freedman, On the Energy Crisis in anti-de Sitter Supersymmetry, Nucl. Phys.B 253 (1985) 635 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    E.J. Martinec, Superspace Geometry of Fermionic Strings, Phys. Rev.D 28 (1983) 2604 [INSPIRE].ADSMathSciNetGoogle Scholar
  17. [17]
    E. D’Hoker, Classical and Quantal Supersymmetric Liouville Theory, Phys. Rev.D 28 (1983) 1346 [INSPIRE].ADSMathSciNetGoogle Scholar
  18. [18]
    T. Fukuda and K. Hosomichi, Super Liouville theory with boundary, Nucl. Phys.B 635 (2002) 215 [hep-th/0202032] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    S. Deser and B. Zumino, A Complete Action for the Spinning String, Phys. Lett.B 65 (1976) 369 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    L. Brink, P. Di Vecchia and P.S. Howe, A Locally Supersymmetric and Reparametrization Invariant Action for the Spinning String, Phys. Lett.B 65 (1976) 471 [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    P.S. Howe, Super Weyl Transformations in Two-Dimensions, J. Phys.A 12 (1979) 393 [INSPIRE].ADSMathSciNetGoogle Scholar
  22. [22]
    I.L. Buchbinder and S.M. Kuzenko, Ideas and methods of supersymmetry and supergravity: Or a walk through superspace, IOP Publishing, Bristol U.K. (1998), pg. 656.Google Scholar
  23. [23]
    G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP06 (2011) 114 [arXiv:1105.0689] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    S.M. Kuzenko, Supersymmetric Spacetimes from Curved Superspace, PoS(CORFU2014)140 (2015) [arXiv:1504.08114] [INSPIRE].
  25. [25]
    T. Uematsu, Structure of N = 1 Conformal and Poincaŕe Supergravity in (1 + 1)-dimensions and (2 + 1)-dimensions, Z. Phys.C 29 (1985) 143 [INSPIRE].ADSMathSciNetGoogle Scholar
  26. [26]
    T. Uematsu, Constraints and Actions in Two-dimensional and Three-dimensional N = 1 Conformal Supergravity, Z. Phys.C 32 (1986) 33 [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    N. Sakai and Y. Tanii, Effective Potential in Two-dimensional Anti-de Sitter Space, Nucl. Phys.B 255 (1985) 401 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    T. Inami and H. Ooguri, Dynamical breakdown of supersymmetry in two-dimensional anti-de Sitter space, Nucl. Phys.B 273 (1986) 487 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    E. D’Hoker and D.H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys.60 (1988) 917 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    E. Abdalla, M.C.B. Abdalla, D. Dalmazi and K. Harada, Correlation functions in superLiouville theory, Phys. Rev. Lett.68 (1992) 1641 [hep-th/9108025] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    C.G. Callan Jr., E.J. Martinec, M.J. Perry and D. Friedan, Strings in Background Fields, Nucl. Phys.B 262 (1985) 593 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    A.A. Tseytlin, σ-model approach to string theory, Int. J. Mod. Phys.A 4 (1989) 1257 [INSPIRE].
  33. [33]
    A.A. Tseytlin, On the Structure of the Renormalization Group β-functions in a Class of Two-dimensional Models, Phys. Lett.B 241 (1990) 233 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    E. D’Hoker and D.H. Phong, Vertex Operators for Closed Strings, Phys. Rev.D 35 (1987) 3890 [INSPIRE].ADSMathSciNetGoogle Scholar
  35. [35]
    L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
  36. [36]
    I.E. Cunha, N.L. Holanda and F. Toppan, From worldline to quantum superconformal mechanics with and without oscillatorial terms: D(2, 1; α) and sl(2|1) models, Phys. Rev.D 96 (2017) 065014 [arXiv:1610.07205] [INSPIRE].
  37. [37]
    A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].CrossRefGoogle Scholar
  38. [38]
    T. Kawano and K. Okuyama, Spinor exchange in AdS(d + 1), Nucl. Phys.B 565 (2000) 427 [hep-th/9905130] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  39. [39]
    D.W. Dusedau and D.Z. Freedman, Renormalization in Anti-de Sitter Supersymmetry, Phys. Rev.D 33 (1986) 395 [INSPIRE].ADSGoogle Scholar
  40. [40]
    Y.Y. Goldschmidt, On the renormalization of the supersymmetric Liouville action and the three-dimensional Ising model critical exponent, Phys. Lett.B 112 (1982) 359 [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    P. Menotti and E. Tonni, The Tetrahedron graph in Liouville theory on the pseudosphere, Phys. Lett.B 586 (2004) 425 [hep-th/0311234] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  42. [42]
    A. Denner, H. Eck, O. Hahn and J. Kublbeck, Compact Feynman rules for Majorana fermions, Phys. Lett.B 291 (1992) 278 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    A. Denner, H. Eck, O. Hahn and J. Kublbeck, Feynman rules for fermion number violating interactions, Nucl. Phys.B 387 (1992) 467 [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    T. Hahn, CUBA: A Library for multidimensional numerical integration, Comput. Phys. Commun.168 (2005) 78 [hep-ph/0404043] [INSPIRE].
  45. [45]
    F. Benini and S. Cremonesi, Partition Functions of \( \mathcal{N} \) = (2, 2) Gauge Theories on S 2and Vortices, Commun. Math. Phys.334 (2015) 1483 [arXiv:1206.2356] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  46. [46]
    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].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. [47]
    R. Camporesi, The Spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys.148 (1992) 283 [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  48. [48]
    W. Mueck, Spinor parallel propagator and Green’s function in maximally symmetric spaces, J. Phys.A 33 (2000) 3021 [hep-th/9912059] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  49. [49]
    A. Basu and L.I. Uruchurtu, Gravitino propagator in anti de Sitter space, Class. Quant. Grav.23 (2006) 6059 [hep-th/0603089] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar
  50. [50]
    E. D’Hoker, D.Z. Freedman, S.D. Mathur, A. Matusis and L. Rastelli, Graviton exchange and scomplete four point functions in the AdS/CFT correspondence, Nucl. Phys.B 562 (1999) 353 [hep-th/9903196] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  51. [51]
    F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].ADSMathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Fisica Ennio De GiorgiUniversità del Salento & INFNLecceItaly
  2. 2.Albert Einstein Center for Fundamental Physics, Institute for Theoretical PhysicsUniversity of BernBernSwitzerland
  3. 3.Blackett LaboratoryImperial CollegeLondonU.K.

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