Skip to main content
SpringerLink
Log in

Quantum Black Holes

  • Chapter
  • First Online:
Strings and Fundamental Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 851))

Abstract

In these notes we describe recent progress in understanding finite size corrections to the black hole entropy. Much of the earlier work concerning quantum black holes has been in the limit of large charges when the area of the even horizon is also large. In recent years there has been substantial progress in understanding the entropy of supersymmetric black holes within string theory going well beyond the large charge limit. It has now become possible to begin exploring finite size effects in perturbation theory in inverse size and even nonperturbatively, with highly nontrivial agreements between thermodynamics and statistical mechanics. Unlike the leading Bekenstein–Hawking entropy which follows from the two-derivative Einstein–Hilbert action, these finite size corrections depend sensitively on the ‘phase’ under consideration and contain a wealth of information about the details of compactification as well as the spectrum of nonperturbative states in the theory. Finite-size corrections are therefore very interesting as a valuable window into the microscopic degrees of freedom of the quantum theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This supersymmetry is a super Lie algebra containing \(ISO(1, 3) \times SU(4)\) as the bosonic subalgebra where \(ISO(1,3)\) is the Poincaré symmetry of the \({\mathbb{R}}^{1, 3}\) spacetime and SU(4) is an internal symmetry called R-symmetry in physics literature. The odd generators of the superalgebra are called supercharges. With \( {\fancyscript{N}}=4\) supersymmetry, there are eight complex supercharges which transform as a spinor of ISO(1,3) and a fundamental of SU(4).

  2. 2.

    The right-moving charges couple to the graviphoton vector fields associated with the right-moving chiral currents in the conformal field theory of the dual heterotic string.

  3. 3.

    There is an additional dependence on arithmetic T-duality invariants but the degeneracies for states with nontrivial values of these T-duality invariants can be obtained from the degeneracies discussed here by demanding S-duality invariance [26].

  4. 4.

    Such ‘phase transitions’ do occur and the degeneracies can jump upon crossing certain walls in the moduli space. This phenomenon called ‘wall-crossing’ occurs not because of Higgs mechanism but because at the walls, single particle states have the same mass as certain multi-particle states and can thus mix with the multi-particle continuum states. The wall-crossing phenomenon complicates the analytic continuation of the degeneracy from weak coupling from strong coupling since one may encounter various walls along the way. However, in many cases, the jumps across these walls can be taken into account systematically.

  5. 5.

    The physical degeneracies have an additional multiplicative factor of \((-1)^{\ell +1}\) which we omit here for simplicity of notation in later chapaters.

  6. 6.

    For an extensive description of this computation see [49].

  7. 7.

    F-type terms can be written as chiral integrals on superspace.

  8. 8.

    See [35, 59, 60] for a discussion of the connection with genus-two Riemann surfaces.

  9. 9.

    It is a ‘cusp’ form because it vanishes at ‘cusps’ which correspond to z = 0 and its images.

  10. 10.

    An SCFT with \((r, s) \) supersymmetries has r left-moving and s right-moving supersymmetries.

References

  1. Carroll, S.M.: Spacetime and Geometry: An Introduction to General Relativity, p. 513. Addison-Wesley, San Francisco (2004)

    MATH  Google Scholar 

  2. Wald, R.M.: General Relativity, pp. 491. University of Chicago Press, Chicago (1984)

    MATH  Google Scholar 

  3. Misner, C.W., Thorne, K.S., Wheeler, J.A., Gravitation, p. 1279. San Francisco (1973)

    Google Scholar 

  4. Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space, p. 340. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  5. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, vol. 1: Introduction, p. 469. Cambridge University Press, Cambridge (1987) (Cambridge Monographs On Mathematical Physics)

    Google Scholar 

  6. Green, M.B., Schwarz, J.H., Witten, E.: Superstring Theory, vol. 2: Loop Amplitudes, Anomalies and Phenomenology, p. 596. Cambridge University Press, Cambridge (1987) (Cambridge Monographs On Mathematical Physics).

    Google Scholar 

  7. Polchinski, J.: String Theory. vol. 1, Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  8. Polchinski, J.: String Theory. Vol. 2: Superstring Theory and Beyond, p. 531. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  9. Sen, A.: Black Hole Entropy Function, Attractors and Precision Counting of Microstates. Gen. Rel. Grav. 40, 2249–2431 (2008) ([arXiv:0708.1270 [hep-th]])

    Article  ADS  MATH  Google Scholar 

  10. Einstein, A.: PAW, p. 844 (1915)

    Google Scholar 

  11. Schwarzschild, K.: PAW, p. 189 (1916)

    Google Scholar 

  12. Bekenstein, J.D.: Black holes and entropy. Phys. Rev. D7, 2333–2346 (1973)

    MathSciNet  ADS  Google Scholar 

  13. Hawking, S.W.: Particle creation by black holes. Commun. Math. Phys. 43, 199–220 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  14. Wald, R.M.: Black hole entropy is the noether charge. Phys. Rev. D48, 3427–3431 (1993) ([gr-qc/9307038])

    MathSciNet  ADS  Google Scholar 

  15. Iyer, V., Wald, R.M.: Some properties of noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D50, 846–864 (1994) ([gr-qc/9403028])

    MathSciNet  ADS  Google Scholar 

  16. Jacobson, T., Kang, G., Myers, R.C.: Black hole entropy in higher curvature gravity. ([gr-qc/9502009])

    Google Scholar 

  17. Sen, A.: Entropy function and AdS(2)/CFT(1) correspondence. JHEP 0811, 075 (2008) ([arXiv:0805.0095 [hep-th]])

    Article  ADS  Google Scholar 

  18. Sen, A.: Quantum entropy function from AdS(2)/CFT(1) correspondence. ([arXiv:0809.3304 [hep-th]])

    Google Scholar 

  19. Hull, C.M., Townsend, P.K.: Unity of superstring dualities. Nucl. Phys. B438, 109–137 (1995) ([hep-th/9410167])

    Google Scholar 

  20. Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B443, 85–126 (1995) ([hep-th/9503124])

    Google Scholar 

  21. Sen, A.: Dyon-monopole bound states, selfdual harmonic forms on the multi-monopole moduli space, and sl(2, z) invariance in string theory. Phys. Lett. B329, 217–221 (1994) ([hep-th/9402032])

    Google Scholar 

  22. Sen, A.: Strong–weak coupling duality in four-dimensional string theory. Int. J. Mod. Phys. A9, 3707–3750 (1994) ([arXiv:hep-th/9402002])

    Google Scholar 

  23. Witten, E., Olive, D.I.: Supersymmetry algebras that include topological charges. Phys. Lett. B78, 97 (1978)

    ADS  Google Scholar 

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

    Google Scholar 

  25. Banerjee, S. Sen, A.: Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on \(T^6.\) ([arXiv:0712.0043 [hep-th]])

    Google Scholar 

  26. Banerjee, S., Sen, A.: S-duality action on discrete T-duality invariants. ([arXiv:0801.0149 [hep-th]])

    Google Scholar 

  27. Banerjee, S., Sen, A., Srivastava, Y.K.: Partition functions of torsion \(>1\) dyons in heterotic string theory on \(T^6.\) ([arXiv:0802.1556 [hep-th]])

    Google Scholar 

  28. Dabholkar, A., Gomes, J., Murthy, S.: Counting all dyons in N = 4 string theory. ([arXiv:0803.2692 [hep-th]])

    Google Scholar 

  29. Dabholkar, A., Harvey, J.A.: Nonrenormalization of the superstring tension. Phys. Rev. Lett. 63, 478 (1989)

    Article  ADS  Google Scholar 

  30. Dabholkar, A., Gibbons, G.W., Harvey, J.A., Ruiz Ruiz, F.: Superstrings and solitons. Nucl. Phys. B340, 33–55 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  31. Dijkgraaf, R., Moore, G.W., Verlinde, E.P., Verlinde, H.L.: Elliptic genera of symmetric products and second quantized strings. Commun. Math. Phys. 185, 197–209 (1997) ([hep-th/9608096])

    Google Scholar 

  32. Gaiotto, D., Strominger, A., Yin, X.: 5D black rings and 4D black holes. JHEP 02, 023 (2006) ([arXiv:hep-th/0504126])

    Google Scholar 

  33. Strominger, A., Vafa, C.: Microscopic origin of the Hekenstein–Hawking entropy. Phys. Lett. B379, 99–104 (1996) ([hep-th/9601029])

    Google Scholar 

  34. Breckenridge, J.C., Myers, R.C., Peet, A.W., Vafa, C.: D-branes and spinning black holes. Phys. Lett. B391, 93–98 (1997) ([hep-th/9602065])

    Google Scholar 

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

    Google Scholar 

  36. David, J.R., Sen, A.: CHL dyons and statistical entropy function from D1–D5 system. JHEP 0611, 072 (2006) ([hep-th/0605210])

    Google Scholar 

  37. Cheng, M.C.N., Verlinde, E.: Dying dyons don’t count. ([arXiv:0706.2363 [hep-th]])

    Google Scholar 

  38. Sen, A.: Walls of marginal stability and dyon spectrum in N = 4 supersymmetric string theories. JHEP 05, 039 (2007) ([hep-th/0702141])

    Google Scholar 

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

    Google Scholar 

  40. Sen, A.: Black hole solutions in heterotic string theory on a torus. Nucl. Phys. B440, 421–440 (1995) ([hep-th/9411187])

    Google Scholar 

  41. Cvetic, M., Youm, D.: Dyonic bps saturated black holes of heterotic string on a six torus. Phys. Rev. D53, 584–588 (1996) ([hep-th/9507090])

    Google Scholar 

  42. Ferrara, S., Kallosh, R., Strominger, A.: N = 2 extremal black holes. Phys. Rev. D52, 5412–5416 (1995) ([hep-th/9508072])

    Google Scholar 

  43. Ferrara, S., Kallosh, R.: Supersymmetry and attractors. Phys. Rev. D54, 1514–1524 (1996) ([hep-th/9602136])

    Google Scholar 

  44. Strominger, A.: Macroscopic entropy of n = 2 extremal black holes. Phys. Lett. B383 39–43 (1996) ([hep-th/9602111])

    Google Scholar 

  45. Lopes Cardoso, G., de Wit, B., Mohaupt, T.: Corrections to macroscopic supersymmetric black-hole entropy. Phys. Lett. B451, 309–316 (1999) ([hep-th/9812082])

    Google Scholar 

  46. Lopes Cardoso, G., de Wit, B., Mohaupt, T.: Deviations from the area law for supersymmetric black holes. Fortsch. Phys. 48, 49–64 (2000) ([hep-th/9904005])

    Google Scholar 

  47. Lopes Cardoso, G., de Wit, B., Mohaupt, T.: Area law corrections from state counting and supergravity. Class. Quant. Grav. 17, 1007–1015 (2000) ([hep-th/9910179])

    Google Scholar 

  48. Lopes Cardoso, G., de Wit, B., Kappeli, J., Mohaupt, T.: Stationary bps solutions in n = 2 supergravity with r**2 interactions. JHEP 12, 019 (2000) ([hep-th/0009234])

    Google Scholar 

  49. Sen, A.: Entropy function for heterotic black holes. ([hep-th/0508042])

    Google Scholar 

  50. Dabholkar, A., Denef, F., Moore, G.W. Pioline, B.: Exact and asymptotic degeneracies of small black holes. JHEP 08, 021 (2005) ([hep-th/0502157])

    Google Scholar 

  51. Dabholkar, A., Denef, F., Moore, G.W., Pioline, B.: Precision counting of small black holes. JHEP 10, 096 (2005) ([arXiv:hep-th/0507014])

    Google Scholar 

  52. Dabholkar, A.: Exact counting of black hole microstates. Phys. Rev. Lett. 94, 241301 (2005) ([hep-th/0409148])

    Google Scholar 

  53. Dabholkar, A., Kallosh, R., Maloney, A.: A stringy cloak for a classical singularity. JHEP 12, 059 (2004) ([hep-th/0410076])

    Google Scholar 

  54. Sen, A.: Extremal black holes and elementary string states Mod. Phys. Lett. A10, 2081–2094 (1995) ([hep-th/9504147])

    Google Scholar 

  55. Kraus, P., Larsen, F.: Microscopic black hole entropy in theories with higher derivatives. JHEP 09, 034 (2005) ([hep-th/0506176])

    Google Scholar 

  56. Kraus, P., Larsen, F.: Holographic gravitational anomalies. JHEP 01, 022 (2006) ([hep-th/0508218])

    Google Scholar 

  57. de Wit, B., Katmadas, S., van Zalk, M.: New supersymmetric higher-derivative couplings: Full N = 2 superspace does not count! ([arXiv:1010.2150 [hep-th]])

    Google Scholar 

  58. Kiritsis, E.: Introduction to non-perturbative string theory. ([hep-th/9708130])

    Google Scholar 

  59. Dabholkar, A., Nampuri, S.: Spectrum of dyons and black holes in CHL orbifolds using borcherds lift. JHEP 11, 077 (2007) ([arXiv:hep-th/0603066])

    Google Scholar 

  60. Banerjee, S., Sen, A., Srivastava, Y.K.: Genus two surface and quarter BPS dyons: the contour prescription. JHEP 03, 151 (2009) ([arXiv:0808.1746 [hep-th]])

    Google Scholar 

  61. Witten, E.: Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525 (1987)

    Google Scholar 

  62. Alvarez, O., Killingback, T.P., Mangano, M.L., Windey, P.: String theory and loop space index theorems. Commun. Math. Phys. 111, 1 (1987)

    Google Scholar 

  63. Ochanine, S.: Sur les genres multiplicatifs definis par des integrales elliptiques. Topology 26, 143 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  64. Ginsparg, P.H.: Applied conformal field theory. ([hep-th/9108028])

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique et Hautes Energies (LPTHE), Université Pierre et Marie Curie-Paris 6; CNRS UMR 7589, Tour 13-14, 5^éme étage, Boite 126, 4 Place Jussieu, 75252, Paris Cedex 05, France

    Atish Dabholkar

  2. Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333, München, Germany

    Suresh Nampuri

Authors
  1. Atish Dabholkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Suresh Nampuri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Atish Dabholkar .

Editor information

Editors and Affiliations

  1. Universität Hamburg, II. Institut für Theoretische Physik, Luruper Chaussee 149, Hamburg, 22761, Germany

    Marco Baumgartl

  2. , Department für Physik, Ludwig-Maximilians-Universität München, Theresienstr. 37, München, 80333, Germany

    Ilka Brunner

  3. , Department für Physik, Ludwig-Maximilians-Universität München, Theresienstr. 37, München, 80333, Germany

    Michael Haack

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dabholkar, A., Nampuri, S. (2012). Quantum Black Holes. In: Baumgartl, M., Brunner, I., Haack, M. (eds) Strings and Fundamental Physics. Lecture Notes in Physics, vol 851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25947-0_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-25947-0_5

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-25946-3

  • Online ISBN: 978-3-642-25947-0

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation