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Twisting and localization in supergravity: equivariant cohomology of BPS black holes

  • Imtak JeonEmail author
  • Sameer Murthy
Open Access
Regular Article - Theoretical Physics
  • 42 Downloads

Abstract

We develop the formalism of supersymmetric localization in supergravity using the deformed BRST algebra defined in the presence of a supersymmetric background as recently formulated in [1]. The gravitational functional integral localizes onto the cohomology of a global supercharge Qeq, obeying Q eq 2  = H, where H is a global symmetry of the background. Our construction naturally produces a twisted version of supergravity whenever supersymmetry can be realized off-shell. We present the details of the twisted graviton multiplet and ghost fields for the superconformal formulation of four-dimensional \( \mathcal{N} \) = 2 supergravity. As an application of our formalism, we systematize the computation of the exact quantum entropy of supersymmetric black holes. In particular, we compute the one-loop determinant of the Qeq\( \mathcal{V} \) deformation operator for the off-shell fluctuations of the Weyl multiplet around the AdS2 × S2 saddle. This result, which is consistent with the corresponding large-charge on-shell analysis, is needed to complete the first-principles computation of the quantum entropy.

Keywords

Black Holes in String Theory BRST Quantization Gauge-gravity correspondence 

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]
    B. de Wit, S. Murthy and V. Reys, BRST quantization and equivariant cohomology: localization with asymptotic boundaries, JHEP 09 (2018) 084 [arXiv:1806.03690] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Dabholkar, J. Gomes and S. Murthy, Quantum black holes, localization and the topological string, JHEP 06 (2011) 019 [arXiv:1012.0265] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Dabholkar, J. Gomes and S. Murthy, Localization & Exact Holography, JHEP 04 (2013) 062 [arXiv:1111.1161] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    N. Berline and M. Vergne, Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante, C.R. Acad. Sci. Paris Sér. I Math. 295 (1982) 539.Google Scholar
  8. [8]
    M.F. Atiyah and R. Bott, The Moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003) 831 [hep-th/0206161] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. Baulieu and I.M. Singer, Topological Yang-Mills symmetry, Nucl. Phys. Proc. Suppl. B 5 (1988) 12.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    M.T. Grisaru and W. Siegel, Supergraphity. Part 1. Background field formalism, Nucl. Phys. B 187 (1981) 149 [INSPIRE].
  14. [14]
    M.T. Grisaru and D. Zanon, Quantum Superfield Supergravity With Off-shell Background Fields, Nucl. Phys. B 237 (1984) 32 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    M. de Roo, J.W. van Holten, B. de Wit and A. Van Proeyen, Chiral Superfields in \( \mathcal{N} \) = 2 Supergravity, Nucl. Phys. B 173 (1980) 175 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  16. [16]
    B. de Wit, J.W. van Holten and A. Van Proeyen, Structure of N = 2 Supergravity, Nucl. Phys. B 184 (1981) 77 [Erratum ibid. B 222 (1983) 516] [INSPIRE].
  17. [17]
    B. de Wit, P.G. Lauwers and A. Van Proeyen, Lagrangians of N = 2 Supergravity-Matter Systems, Nucl. Phys. B 255 (1985) 569 [INSPIRE].ADSCrossRefGoogle Scholar
  18. [18]
    N. Berkovits, A Ten-dimensional superYang-Mills action with off-shell supersymmetry, Phys. Lett. B 318 (1993) 104 [hep-th/9308128] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    L. Baulieu, N.J. Berkovits, G. Bossard and A. Martin, Ten-dimensional super-Yang-Mills with nine off-shell supersymmetries, Phys. Lett. B 658 (2008) 249 [arXiv:0705.2002] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    L. Baulieu, M. Bellon and V. Reys, Twisted N = 1, d = 4 supergravity and its symmetries, Nucl. Phys. B 867 (2013) 330 [arXiv:1207.4399] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    J. Bae, C. Imbimbo, S.-J. Rey and D. Rosa, New Supersymmetric Localizations from Topological Gravity, JHEP 03 (2016) 169 [arXiv:1510.00006] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  23. [23]
    K. Costello and S. Li, Twisted supergravity and its quantization, arXiv:1606.00365 [INSPIRE].
  24. [24]
    C. Imbimbo and D. Rosa, The topological structure of supergravity: an application to supersymmetric localization, JHEP 05 (2018) 112 [arXiv:1801.04940] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    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] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    R.K. Gupta, Y. Ito and I. Jeon, Supersymmetric Localization for BPS Black Hole Entropy: 1-loop Partition Function from Vector Multiplets, JHEP 11 (2015) 197 [arXiv:1504.01700] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    S. Murthy and V. Reys, Functional determinants, index theorems and exact quantum black hole entropy, JHEP 12 (2015) 028 [arXiv:1504.01400] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  29. [29]
    J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, Phys. Rept. 259 (1995) 1 [hep-th/9412228] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
  31. [31]
    D.Z. Freedman and A. Van Proeyen, Supergravity, Cambridge University Press, Cambridge, U.K. (2012) [INSPIRE].CrossRefzbMATHGoogle Scholar
  32. [32]
    T.W.B. Kibble, Lorentz invariance and the gravitational field, J. Math. Phys. 2 (1961) 212 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  34. [34]
    J.R. David, E. Gava, R.K. Gupta and K. Narain, Localization on AdS 2 × S 1, JHEP 03 (2017) 050 [arXiv:1609.07443] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    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
  36. [36]
    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
  37. [37]
    R.K. Gupta and S. Murthy, All solutions of the localization equations for N = 2 quantum black hole entropy, JHEP 02 (2013) 141 [arXiv:1208.6221] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    N. Banerjee, D.P. Jatkar and A. Sen, Asymptotic Expansion of the N = 4 Dyon Degeneracy, JHEP 05 (2009) 121 [arXiv:0810.3472] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    S. Murthy and B. Pioline, A Farey tale for N = 4 dyons, JHEP 09 (2009) 022 [arXiv:0904.4253] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    A. Dabholkar, J. Gomes and S. Murthy, Nonperturbative black hole entropy and Kloosterman sums, JHEP 03 (2015) 074 [arXiv:1404.0033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    S. Lee, Index, supersymmetry, and localization, lectures at The Pyeong-Chang Summer School, (2013) http://psi.kias.re.kr/2013/sub02/sub02_01.php.
  42. [42]
    K. Hosomichi, The localization principle in SUSY gauge theories, PTEP 2015 (2015) 11B101 [arXiv:1502.04543] [INSPIRE].
  43. [43]
    M.F. Atiyah, Elliptic operators and compact groups, Lect. Notes Math., Vol. 401, Springer Verlag (1974).Google Scholar
  44. [44]
    B. Assel, D. Martelli, S. Murthy and D. Yokoyama, Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds, JHEP 03 (2017) 095 [arXiv:1609.08071] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    J.R. David, E. Gava, R.K. Gupta and K. Narain, Boundary conditions and localization on AdS. Part I, JHEP 09 (2018) 063 [arXiv:1802.00427] [INSPIRE].
  46. [46]
    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
  47. [47]
    B. de Wit, S. Katmadas and M. van Zalk, New supersymmetric higher-derivative couplings: Full N = 2 superspace does not count!, JHEP 01 (2011) 007 [arXiv:1010.2150] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    D. Butter, B. de Wit and I. Lodato, Non-renormalization theorems and N = 2 supersymmetric backgrounds, JHEP 03 (2014) 131 [arXiv:1401.6591] [INSPIRE].ADSCrossRefGoogle Scholar
  49. [49]
    S. Murthy and V. Reys, Quantum black hole entropy and the holomorphic prepotential of N = 2 supergravity, JHEP 10 (2013) 099 [arXiv:1306.3796] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    A. Sen, Arithmetic of Quantum Entropy Function, JHEP 08 (2009) 068 [arXiv:0903.1477] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  51. [51]
    A. Dabholkar, J. Gomes, S. Murthy and A. Sen, Supersymmetric Index from Black Hole Entropy, JHEP 04 (2011) 034 [arXiv:1009.3226] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    K. Bringmann and S. Murthy, On the positivity of black hole degeneracies in string theory, Commun. Num. Theor Phys. 07 (2013) 15 [arXiv:1208.3476] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  53. [53]
    H. Ooguri, A. Strominger and C. Vafa, Black hole attractors and the topological string, Phys. Rev. D 70 (2004) 106007 [hep-th/0405146] [INSPIRE].ADSMathSciNetGoogle Scholar
  54. [54]
    G. Lopes Cardoso, B. de Wit and T. Mohaupt, Corrections to macroscopic supersymmetric black hole entropy, Phys. Lett. B 451 (1999) 309 [hep-th/9812082] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. [55]
    S. Murthy and V. Reys, Single-centered black hole microstate degeneracies from instantons in supergravity, JHEP 04 (2016) 052 [arXiv:1512.01553] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  56. [56]
    E. Witten, Topological Gravity, Phys. Lett. B 206 (1988) 601 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  57. [57]
    P. Benetti Genolini, P. Richmond and J. Sparks, Topological AdS/CFT, JHEP 12 (2017) 039 [arXiv:1707.08575] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    T.D. Brennan, F. Carta and C. Vafa, The String Landscape, the Swampland and the Missing Corner, PoS(TASI2017)015 (2017) [arXiv:1711.00864] [INSPIRE].
  59. [59]
    B. de Wit and V. Reys, Euclidean supergravity, JHEP 12 (2017) 011 [arXiv:1706.04973] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  60. [60]
    T. Mohaupt, Black hole entropy, special geometry and strings, Fortsch. Phys. 49 (2001) 3 [hep-th/0007195] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    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

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteAllahabadIndia
  2. 2.Department of MathematicsKing’s College LondonLondonU.K.

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