The heat kernel on AdS3 and its applications

  • Justin R. David
  • Matthias R. Gaberdiel
  • Rajesh Gopakumar
Article

Abstract

We derive the heat kernel for arbitrary tensor fields on S3 and (Euclidean) AdS3 using a group theoretic approach. We use these results to also obtain the heat kernel on certain quotients of these spaces. In particular, we give a simple, explicit expression for the one loop determinant for a field of arbitrary spin s in thermal AdS3. We apply this to the calculation of the one loop partition function of \( \mathcal{N} = 1 \) supergravity on AdS3. We find that the answer factorizes into left- and right-moving super Virasoro characters built on the \( {\text{SL}}\left( {2,\mathbb{C}} \right) \) invariant vacuum, as argued by Maloney and Witten on general grounds.

Keywords

AdS-CFT Correspondence Classical Theories of Gravity Supergravity Models 

References

  1. [1]
    A. Maloney and E. Witten, Quantum gravity partition functions in three dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [SPIRES].CrossRefGoogle Scholar
  2. [2]
    J.D. Brown and M. Henneaux, Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity, Commun. Math. Phys. 104 (1986) 207 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  3. [3]
    S. Giombi, A. Maloney and X. Yin, One-loop partition functions of 3D gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    A. Higuchi, Symmetric tensor spherical harmonics on the N sphere and their application to the de Sitter group SO(N, 1), J. Math. Phys. 28 (1987) 1553 [Erratum ibid. 43 (2002) 6385] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  5. [5]
    R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  6. [6]
    R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  7. [7]
    R. Camporesi and A. Higuchi, The Plancherel measure for p-forms in real hyperbolic spaces, J. Geom. Phys. 15 (1994) 57.MATHCrossRefMathSciNetADSGoogle Scholar
  8. [8]
    R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  9. [9]
    R. Camporesi and A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  10. [10]
    J.S. Dowker and Y.P. Dowker, Interactions of massless particles of arbitrary spin, Proc. Roy. Soc. London A 294 (1966) 175.CrossRefADSGoogle Scholar
  11. [11]
    J.S. Dowker, Arbitrary spin theory in the Einstein universe, Phys. Rev. D 28 (1983) 3013 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  12. [12]
    A. Chodos and E. Myers, Gravitational contribution to the Casimir energy in Kaluza-Klein theories, Ann. Phys. 156 (1984) 412 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  13. [13]
    M.A. Rubin and C.R. Ordonez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, J. Math. Phys. 25 (1984) 2888 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  14. [14]
    E. Elizalde, M. Lygren and D.V. Vassilevich, Antisymmetric tensor fields on spheres: functional determinants and non–local counterterms, J. Math. Phys. 37 (1996) 3105 [hep-th/9602113] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  15. [15]
    S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, U.S.A. (1978).MATHGoogle Scholar
  16. [16]
    I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, 6th ed., Academic Press, New York, U.S.A. (2000).MATHGoogle Scholar
  17. [17]
    T. Dray, The relationship between monopole harmonics and spin weighted spherical harmonics, J. Math. Phys. 26 (1985) 1030 [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  18. [18]
    A.W. Knapp, Representation theory of semisimple groups — an overview based on examples, Princeton University Press, Princeton, U.S.A. (1986).MATHGoogle Scholar
  19. [19]
    W. Rühl, Lorentz group and harmonic analysis, W.A. Benjamin, New York, U.S.A. (1970).MATHGoogle Scholar
  20. [20]
    M. Carmeli, Group theory and general relativity, World Scientific, Singapore (1977).MATHGoogle Scholar
  21. [21]
    A.A. Bytsenko, L. Vanzo and S. Zerbini, Ray-Singer torsion for a hyperbolic 3-manifold and asymptotics of Chern-Simons-Witten invariant, Nucl. Phys. B 505 (1997) 641 [hep-th/9704035] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    A.A. Bytsenko, G. Cognola, L. Vanzo and S. Zerbini, Quantum fields and extended objects in space-times with constant curvature spatial section, Phys. Rept. 266 (1996) 1 [hep-th/9505061] [SPIRES].CrossRefMathSciNetGoogle Scholar
  23. [23]
    A.A. Bytsenko and M.E.X. Guimaraes, Partition functions of three-dimensional quantum gravity and the black hole entropy, J. Phys. Conf. Ser. 161 (2009) 012023 [arXiv:0807.2222] [SPIRES].CrossRefADSGoogle Scholar
  24. [24]
    A.A. Bytsenko and M.E.X. Guimaraes, Truncated heat kernel and one-loop determinants for the BTZ geometry, Eur. Phys. J. C 58 (2008) 511 [arXiv:0809.1416] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  25. [25]
    G.W. Gibbons and M.J. Perry, Quantizing gravitational instantons, Nucl. Phys. B 146 (1978) 90 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  26. [26]
    S.M. Christensen and M.J. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B 170 (1980) 480 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    O. Yasuda, On the one loop effective potential in quantum gravity, Phys. Lett. B 137 (1984) 52 [SPIRES].ADSGoogle Scholar
  28. [28]
    W. Rarita and J. Schwinger, On a theory of particles with half integral spin, Phys. Rev. 60 (1941) 61 [SPIRES].MATHCrossRefADSGoogle Scholar
  29. [29]
    E.S. Fradkin and A.A. Tseytlin, On the new definition of off-shell effective action, Nucl. Phys. B 234 (1984) 509 [SPIRES].CrossRefADSGoogle Scholar
  30. [30]
    A. Maloney, W. Song and A. Strominger, Chiral gravity, log gravity and extremal CFT, Phys. Rev. D 81 (2010) 064007 [arXiv:0903.4573] [SPIRES].ADSGoogle Scholar
  31. [31]
    J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS 3 and the SL(2, R) WZW model. II: Euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  32. [32]
    A. Sen, Entropy function and AdS 2/CFT 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [SPIRES].CrossRefADSGoogle Scholar
  33. [33]
    A. Sen, Quantum entropy function from AdS 2/CFT 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [SPIRES].ADSGoogle Scholar
  34. [34]
    N. Banerjee, S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Supersymmetry, localization and quantum entropy function, JHEP 02 (2010) 091 [arXiv:0905.2686] [SPIRES].CrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  • Justin R. David
    • 1
  • Matthias R. Gaberdiel
    • 2
  • Rajesh Gopakumar
    • 3
  1. 1.Centre for High Energy PhysicsIndian Institute of ScienceBangaloreIndia
  2. 2.Institut für Theoretische PhysikETH ZurichZürichSwitzerland
  3. 3.Harish-Chandra Research InstituteJhusiIndia

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