Journal of High Energy Physics

, 2013:11 | Cite as

Recurrence relations for finite-temperature correlators via AdS2/CFT1

  • Satoshi Ohya


This note is aimed at presenting a new algebraic approach to momentum-space correlators in conformal field theory. As an illustration we present a new Lie-algebraic method to compute frequency-space two-point functions for charged scalar operators of CFT1 dual to AdS2 black hole with constant background electric field. Our method is based on the real-time prescription of AdS/CFT correspondence, Euclideanization of AdS2 black hole and projective unitary representations of the Lie algebra \( sl \)(2, \( \mathbb{R} \)) ⊕ \( sl \)(2, \( \mathbb{R} \)). We derive novel recurrence relations for Euclidean CFT1 two-point functions, which are exactly solvable and completely determine the frequency- and charge-dependences of two-point functions. Wick-rotating back to Lorentzian signature, we obtain retarded and advanced CFT1 two-point functions that are consistent with the known results.


AdS-CFT Correspondence Conformal and W Symmetry Black Holes 


  1. [1]
    A.M. Polyakov, Conformal symmetry of critical fluctuations, JETP Lett. 12 (1970) 381 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 538] [INSPIRE].ADSGoogle Scholar
  2. [2]
    H. Osborn and A. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    C. Corianò, L. Delle Rose, E. Mottola and M. Serino, Solving the conformal constraints for scalar operators in momentum space and the evaluation of Feynmans master integrals, JHEP 07 (2013) 011 [arXiv:1304.6944] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Bzowski, P. McFadden and K. Skenderis, Implications of conformal invariance in momentum space, arXiv:1304.7760 [INSPIRE].
  5. [5]
    N. Iqbal and H. Liu, Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm, Phys. Rev. D 79 (2009) 025023 [arXiv:0809.3808] [INSPIRE].ADSGoogle Scholar
  6. [6]
    N. Iqbal and H. Liu, Real-time response in AdS/CFT with application to spinors, Fortsch. Phys. 57 (2009) 367 [arXiv:0903.2596] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Spradlin and A. Strominger, Vacuum states for AdS 2 black holes, JHEP 11 (1999) 021 [hep-th/9904143] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  8. [8]
    H.K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel. 16 (2013) 8 [arXiv:1306.2517] [INSPIRE].CrossRefzbMATHGoogle Scholar
  9. [9]
    T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].ADSGoogle Scholar
  10. [10]
    T. Faulkner, N. Iqbal, H. Liu, J. McGreevy and D. Vegh, Holographic non-Fermi liquid fixed points, Phil. Trans. Roy. Soc. A 369 (2011) 1640 [arXiv:1101.0597] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N. Iqbal, H. Liu and M. Mezei, Lectures on holographic non-Fermi liquids and quantum phase transitions, arXiv:1110.3814 [INSPIRE].
  12. [12]
    J. Derezinski and M. Wrochna, Exactly solvable Schrödinger operators, Annales Henri Poincaré 12 (2011) 397 [arXiv:1009.0541] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    B. Pioline and J. Troost, Schwinger pair production in AdS 2, JHEP 03 (2005) 043 [hep-th/0501169] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    G. Lindblad and B. Nagel, Continuous bases for unitary irreducible representations of SU(1, 1), Ann. Inst. Henri Poincaré 13 (1970) 27.MathSciNetzbMATHGoogle Scholar
  16. [16]
    A. Frank and K. Wolf, Lie algebras for potential scattering, Phys. Rev. Lett. 52 (1984) 1737 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    Y. Alhassid, J. Engel and J. Wu, Algebraic approach to the scattering matrix, Phys. Rev. Lett. 53 (1984) 17 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals Math. 48 (1947) 568 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L. Pukánszky, The Plancherel formula for the universal covering group of SL(R, 2), Math. Ann. 156 (1964) 96.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    K. Case, Singular potentials, Phys. Rev. 80 (1950) 797 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    V. Balasubramanian, I. García-Etxebarria, F. Larsen and J. Simón, Helical Luttinger liquids and three dimensional black holes, Phys. Rev. D 84 (2011) 126012 [arXiv:1012.4363] [INSPIRE].ADSGoogle Scholar
  23. [23]
    Y. Alhassid and J. Wu, An algebraic approach to the Morse potential scattering, Chem. Phys. Lett. 109 (1984) 81.ADSMathSciNetCrossRefGoogle Scholar
  24. [24]
    Y. Alhassid, F. Gürsey and F. Iachello, Group theory approach to scattering. II. The Euclidean connection, Annals Phys. 167 (1986) 181.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    A. Frank, Y. Alhassid and F. Iachello, Contractions and expansions of Lie groups and the algebraic approach to scattering, Phys. Rev. A 34 (1986) 677.ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    G. Kerimov, New algebraic approach to scattering problems, Phys. Rev. Lett. 80 (1998) 2976 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    G.A. Kerimov and M. Sezgin, On scattering systems related to the SO(2, 1) group, J. Phys. A 31 (1998) 7901.ADSMathSciNetzbMATHGoogle Scholar
  28. [28]
    G.A. Kerimov, Intertwining operators and S matrix, Phys. Atom. Nucl. 65 (2002) 1036.ADSMathSciNetCrossRefGoogle Scholar
  29. [29]
    K. Koller, The significance of conformal inversion in quantum field theory, Commun. Math. Phys. 40 (1975) 15.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    V. Dobrev, Intertwining operator realization of the AdS/CFT correspondence, Nucl. Phys. B 553 (1999) 559 [hep-th/9812194] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Department of Physics, Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PragueDěčínCzech Republic
  2. 2.Doppler Institute for Mathematical Physics and Applied MathematicsCzech Technical University in PraguePragueCzech Republic

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