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Seeley-DeWitt coefficients in \( \mathcal{N} \) = 2 Einstein-Maxwell supergravity theory and logarithmic corrections to \( \mathcal{N} \) = 2 extremal black hole entropy

  • Sudip Karan
  • Gourav Banerjee
  • Binata PandaEmail author
Open Access
Regular Article - Theoretical Physics
  • 38 Downloads

Abstract

We investigate the heat kernel method for one-loop effective action following the Seeley-DeWitt expansion technique of heat kernel with Seeley-DeWitt coefficients. We also review a general approach of computing the Seeley-DeWitt coefficients in terms of background or geometric invariants. We, then consider the Einstein-Maxwell theory em-bedded in minimal \( \mathcal{N} \) = 2 supergravity in four dimensions and compute the first three Seeley-DeWitt coefficients of the kinetic operator of the bosonic and the fermionic fields in an arbitrary background field configuration. We find the applications of these results in the computation of logarithmic corrections to Bekenstein-Hawking entropy of the extremal Kerr-Newman, Kerr and Reissner-Nordström black holes in minimal \( \mathcal{N} \) = 2 Einstein-Maxwell supergravity theory following the quantum entropy function formalism.

Keywords

Black Holes Black Holes in String Theory Extended Supersymmetry 

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]
    I. Mandal and A. Sen, Black hole microstate counting and its macroscopic counterpart, Nucl. Phys. Proc. Suppl. 216 (2011) 147 [arXiv:1008.3801] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M.R. Setare, Logarithmic correction to the Cardy-Verlinde formula in topological Reissner-Nordström de Sitter space, Phys. Lett. B 573 (2003) 173 [hep-th/0308106] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  3. [3]
    S. Banerjee, R.K. Gupta and A. Sen, Logarithmic corrections to extremal black hole entropy from quantum entropy function, JHEP 03 (2011) 147 [arXiv:1005.3044] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    S. Banerjee, R.K. Gupta, I. Mandal and A. Sen, Logarithmic corrections to N = 4 and N = 8 black hole entropy: a one loop test of quantum gravity, JHEP 11(2011) 143 [arXiv:1106.0080] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  5. [5]
    A. Sen, Logarithmic corrections to N = 2 black hole entropy: an infrared window into the microstates, Gen. Rel. Grav. 44 (2012) 1207 [arXiv:1108.3842] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Sen, Logarithmic corrections to rotating extremal black hole entropy in four and five dimensions, Gen. Rel. Grav. 44 (2012) 1947 [arXiv:1109.3706] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Chowdhury, R.K. Gupta, S. Lal, M. Shyani and S. Thakur, Logarithmic corrections to twisted indices from the quantum entropy function, JHEP 11 (2014) 002 [arXiv:1404.6363] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    R.K. Gupta, S. Lal and S. Thakur, Logarithmic corrections to extremal black hole entropy in N =2, 4 and 8 supergravity, JHEP 11(2014) 072 [arXiv:1402.2441] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    S. Bhattacharyya, B. Panda and A. Sen, Heat kernel expansion and extremal Kerr-Newmann black hole entropy in Einstein-Maxwell theory, JHEP 08 (2012) 084 [arXiv:1204.4061] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    C. Keeler, F. Larsen and P. Lisbao, Logarithmic corrections to N ≥ 2 black hole entropy, Phys. Rev. D 90 (2014) 043011 [arXiv:1404.1379] [INSPIRE].ADSGoogle Scholar
  11. [11]
    A.M. Charles and F. Larsen, Universal corrections to non-extremal black hole entropy in N ≥ 2 supergravity, JHEP 06(2015) 200[arXiv:1505.01156] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F. Larsen and P. Lisbao, Quantum corrections to supergravity on AdS 2 × S 2, Phys. Rev. D 91 (2015) 084056 [arXiv:1411.7423] [INSPIRE].ADSMathSciNetGoogle Scholar
  13. [13]
    A. Castro, V. Godet, F. Larsen and Y. Zeng, Logarithmic corrections to black hole entropy: the non-BPS branch, JHEP 05 (2018) 079 [arXiv:1801.01926] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    A. Sen, Entropy function and AdS 2 /CFT 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [INSPIRE].ADSCrossRefGoogle Scholar
  15. [15]
    A. Sen, Quantum entropy function from AdS 2 /CFT 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    B.S. DeWitt, Dynamical theory of groups and fields, Gordon and Breach, New York, NY, U.S.A. (1965).Google Scholar
  18. [18]
    B.S. DeWitt, Quantum theory of gravity. 1. The canonical theory, Phys. Rev. 160 (1967) 1113 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  19. [19]
    B.S. DeWitt, Quantum theory of gravity. 2. The manifestly covariant theory, Phys. Rev. 162 (1967) 1195 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  20. [20]
    B.S. DeWitt, Quantum theory of gravity. 3. Applications of the covariant theory, Phys. Rev. 162 (1967) 1239 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  21. [21]
    R.T. Seeley, Singular integrals and boundary value problems, Amer. J. Math. 88 (1966) 781.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    R. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969) 889.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    M.J. Duff, Observations on conformal anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    S.M. Christensen and M.J. Duff, New gravitational index theorems and supertheorems, Nucl. Phys. B 154 (1979) 301 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  25. [25]
    S.M. Christensen and M.J. Duff, Quantizing gravity with a cosmological constant, Nucl. Phys. B 170 (1980) 480 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M.J. Duff and P. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys. Lett. B 94 (1980) 179 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    N.D. Birrel and P.C. W. Davis, Quantum fields in curved space, Cambridge University Press, New York, NY, U.S.A. (1982) [INSPIRE].CrossRefGoogle Scholar
  28. [28]
    P.B. Gilkey, Invariance theory, the heat equation and the Atiyah-Singer index theorem, Publish or Perish Inc., U.S.A. (1984) [INSPIRE].
  29. [29]
    M.J. Duff and S. Ferrara, Generalized mirror symmetry and trace anomalies, Class. Quant. Grav. 28 (2011) 065005 [arXiv:1009.4439] [INSPIRE].
  30. [30]
    D.V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    J.R. David, M.R. Gaberdiel and R. Gopakumar, The heat kernel on AdS 3 and its applications, JHEP 04 (2010) 125 [arXiv:0911.5085] [INSPIRE].
  32. [32]
    R. Gopakumar, R.K. Gupta and S. Lal, The heat kernel on AdS, JHEP 11 (2011) 010 [arXiv:1103.3627] [INSPIRE].
  33. [33]
    I. Lovrekovic, One loop partition function of six dimensional conformal gravity using heat kernel on AdS, JHEP 10 (2016) 064 [arXiv:1512.00858] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    F. Denef, S.A. Hartnoll and S. Sachdev, Black hole determinants and quasinormal modes, Class. Quant. Grav. 27 (2010) 125001 [arXiv:0908.2657] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].ADSGoogle Scholar
  36. [36]
    S.W. Hawking, Quantum gravity and path integrals, Phys. Rev. D 18 (1978) 1747 [INSPIRE].ADSGoogle Scholar
  37. [37]
    S.W. Hawking, Zeta function regularization of path integrals in curved space-time, Commun. Math. Phys. 55 (1977) 133 [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  38. [38]
    G. Denardo and E. Spallucci, Induced quantum gravity from heat kernel expansion, Nuovo Cim. A 69 (1982) 151 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    I.G. Avramidi, The heat kernel approach for calculating the effective action in quantum field theory and quantum gravity, hep-th/9509077 [INSPIRE].
  40. [40]
    G. De Berredo-Peixoto, A note on the heat kernel method applied to fermions, Mod. Phys. Lett. A 16 (2001) 2463 [hep-th/0108223] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    J.S. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev. 82 (1951) 664 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    B.S. DeWitt, Quantum field theory in curved space-time, Phys. Rept. 19 (1975) 295 [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    P.B. Gilkey, The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975) 601 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    S. Karan, S. Kumar and B. Panda, General heat kernel coefficients for massless free spin-3/2 Rarita-Schwinger field, Int. J. Mod. Phys. A 33 (2018) 1850063 [arXiv:1709.08063] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    A.M. Charles, F. Larsen and D.R. Mayerson, Non-renormalization for non-supersymmetric black holes, JHEP 08 (2017) 048 [arXiv:1702.08458] [INSPIRE].
  46. [46]
    H. Itoyama and K. Maruyoshi, U(N) gauged N = 2 supergravity and partial breaking of local N = 2 supersymmetry, Int. J. Mod. Phys. A 21(2006) 6191 [hep-th/0603180] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    R.C. Henry, Kretschmann scalar for a Kerr-Newman black hole, Astrophys. J. 535 (2000) 350 [astro-ph/9912320] [INSPIRE].
  48. [48]
    C. Cherubini, D. Bini, S. Capozziello and R. Ruffini, Second order scalar invariants of the Riemann tensor: applications to black hole space-times, Int. J. Mod. Phys. D 11 (2002) 827 [gr-qc/0302095] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia

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