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Supersymmetry, localization and quantum entropy function

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Abstract

AdS2/CFT1 correspondence leads to a prescription for computing the degeneracy of black hole states in terms of path integral over string fields living on the near horizon geometry of the black hole. In this paper we make use of the enhanced supersymmetries of the near horizon geometry and localization techniques to argue that the path integral receives contribution only from a special class of string field configurations which are invariant under a subgroup of the supersymmetry transformations. We identify saddle points which are invariant under this subgroup. We also use our analysis to show that the integration over infinite number of zero modes generated by the asymptotic symmetries of AdS2 generate a finite contribution to the path integral.

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References

  1. A. Sen, Quantum entropy function from AdS 2/CFT 1 correspondence, Int. J. Mod. Phys. A 24 (2009) 4225 [arXiv:0809.3304] [SPIRES].

    Google Scholar 

  2. N. Banerjee, D.P. Jatkar and A. Sen, Asymptotic expansion of the N = 4 dyon degeneracy, JHEP 05 (2009) 121 [arXiv:0810.3472] [SPIRES].

    Article  ADS  Google Scholar 

  3. A. Sen, Arithmetic of quantum entropy function, JHEP 08 (2009) 068 [arXiv:0903.1477] [SPIRES].

    Article  ADS  Google Scholar 

  4. D. Astefanesei, K. Goldstein, R.P. Jena, A. Sen and S.P. Trivedi, Rotating attractors, JHEP 10 (2006) 058 [hep-th/0606244] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  5. A. Sen, Entropy function and AdS 2/CFT 1 correspondence, JHEP 11 (2008) 075 [arXiv:0805.0095] [SPIRES].

    Article  ADS  Google Scholar 

  6. R.K. Gupta and A. Sen, AdS 3/CFT 2 to AdS 2/CFT 1, JHEP 04 (2009) 034 [arXiv:0806.0053] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  7. R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427 [gr-qc/9307038] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  8. T. Jacobson, G. Kang and R.C. Myers, On black hole entropy, Phys. Rev. D 49 (1994) 6587 [gr-qc/9312023] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  9. V. Iyer and R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (1994) 846 [gr-qc/9403028] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  10. T. Jacobson, G. Kang and R.C. Myers, Black hole entropy in higher curvature gravity, gr-qc/9502009 [SPIRES].

  11. A. Sen, Black hole entropy function and the attractor mechanism in higher derivative gravity, JHEP 09 (2005) 038 [hep-th/0506177] [SPIRES].

    Article  ADS  Google Scholar 

  12. A. Sen, Black hole entropy function, attractors and precision counting of microstates, Gen. Rel. Grav. 40 (2008) 2249 [arXiv:0708.1270] [SPIRES].

    Article  MATH  ADS  Google Scholar 

  13. J.J. Duistermaat and G.J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. E. Witten, The N matrix model and gauged WZW models, Nucl. Phys. B 371 (1992) 191 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  16. E. Witten, Mirror manifolds and topological field theory, hep-th/9112056 [SPIRES].

  17. E. Witten, Two-dimensional gauge theories revisited, J. Geom. Phys. 9 (1992) 303 [hep-th/9204083] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. A.S. Schwarz and O. Zaboronsky, Supersymmetry and localization, Commun. Math. Phys. 183 (1997) 463 [hep-th/9511112] [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  19. O.V. Zaboronsky, Dimensional reduction in supersymmetric field theories, J. Phys. A 35 (2002) 5511 [SPIRES].

    MathSciNet  Google Scholar 

  20. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].

    MathSciNet  Google Scholar 

  21. V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, arXiv:0712.2824 [SPIRES].

  22. M.F. Atiyah, Elliptic operators and compact groups, Springer-Verlag, Berlin Germany (1974).

    MATH  Google Scholar 

  23. P. Shanahan, The Atiyah-Singer index theorem: an introduction, Springer-Verlag, Berlin Germany (1976).

    Google Scholar 

  24. C. Beasley et al., Why Z BH = |Z top|2, hep-th/0608021 [SPIRES].

  25. J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in N = 4 super Yang-Mills, arXiv:0904.4486 [SPIRES].

  26. R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [SPIRES].

    Article  MATH  MathSciNet  ADS  Google Scholar 

  27. S. Murthy and B. Pioline, A Farey tale for N = 4 dyons, JHEP 09 (2009) 022 [arXiv:0904.4253] [SPIRES].

    Article  Google Scholar 

  28. S. Banerjee and A. Sen, Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on T 6, JHEP 03 (2008) 022 [arXiv:0712.0043] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  29. S. Banerjee and A. Sen, S-duality action on discrete T-duality invariants, JHEP 04 (2008) 012 [arXiv:0801.0149] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  30. A. Dabholkar, D. Gaiotto and S. Nampuri, Comments on the spectrum of CHL dyons, JHEP 01 (2008) 023 [hep-th/0702150] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  31. J.H. Schwarz and A. Sen, Type IIA dual of the six-dimensional CHL compactification, Phys. Lett. B 357 (1995) 323 [hep-th/9507027] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  32. R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Counting dyons in N = 4 string theory, Nucl. Phys. B 484 (1997) 543 [hep-th/9607026] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  33. G. Lopes Cardoso, B. de Wit, J. Kappeli and T. Mohaupt, Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy, JHEP 12 (2004) 075 [hep-th/0412287] [SPIRES].

    ADS  Google Scholar 

  34. D. Shih, A. Strominger and X. Yin, Recounting dyons in N = 4 string theory, JHEP 10 (2006) 087 [hep-th/0505094] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  35. D. Gaiotto, Re-recounting dyons in N = 4 string theory, hep-th/0506249 [SPIRES].

  36. J.R. David and A. Sen, CHL dyons and statistical entropy function from D1-D5 system, JHEP 11 (2006) 072 [hep-th/0605210] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  37. S. Banerjee, A. Sen and Y.K. Srivastava, Generalities of quarter BPS dyon partition function and dyons of torsion two, JHEP 05 (2008) 101 [arXiv:0802.0544] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  38. S. Banerjee, A. Sen and Y.K. Srivastava, Partition functions of torsion > 1 dyons in heterotic string theory on T 6, JHEP 05 (2008) 098 [arXiv:0802.1556] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  39. A. Dabholkar, J. Gomes and S. Murthy, Counting all dyons in N = 4 string theory, arXiv:0803.2692 [SPIRES].

  40. N. Berkovits, M. Bershadsky, T. Hauer, S. Zhukov and B. Zwiebach, Superstring theory on AdS 2 × S 2 as a coset supermanifold, Nucl. Phys. B 567 (2000) 61 [hep-th/9907200] [SPIRES].

    Article  MathSciNet  Google Scholar 

  41. A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  42. L.J. Romans, Selfduality for interacting fields: covariant field equations for six-dimensional chiral supergravities, Nucl. Phys. B 276 (1986) 71 [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

  43. F. Riccioni, Tensor multiplets in six-dimensional (2, 0) supergravity, Phys. Lett. B 422 (1998) 126 [hep-th/9712176] [SPIRES].

    MathSciNet  ADS  Google Scholar 

  44. S. Deger, A. Kaya, E. Sezgin and P. Sundell, Spectrum of D = 6, N = 4b supergravity on AdS 3 × S 3, Nucl. Phys. B 536 (1998) 110 [hep-th/9804166] [SPIRES].

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad, 211019, India

    Nabamita Banerjee, Shamik Banerjee, Rajesh Kumar Gupta, Ipsita Mandal & Ashoke Sen

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  1. Nabamita Banerjee
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  2. Shamik Banerjee
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  3. Rajesh Kumar Gupta
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  4. Ipsita Mandal
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  5. Ashoke Sen
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Correspondence to Nabamita Banerjee.

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ArXiv ePrint: 0905.2686

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Banerjee, N., Banerjee, S., Kumar Gupta, R. et al. Supersymmetry, localization and quantum entropy function. J. High Energ. Phys. 2010, 91 (2010). https://doi.org/10.1007/JHEP02(2010)091

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