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Black Holes in String Theory

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Black Objects in Supergravity

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Abstract

These lectures notes provide a fast-track introduction to modern developments in black hole physics within string theory, including microscopic computations of the black hole entropy as well as construction and quantization of microstates using supergravity. These notes are largely self-contained and should be accessible to students at an early PhD or Masters level. Topics covered include the black holes in supergravity, D-branes, Strominger-Vafa’s computation of the black hole entropy via D-branes, AdS-CFT and its applications to black hole phyisics, multicenter solutions, and the geometric quantization of the latter.

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Notes

  1. 1.

    The more general time-independent solution, a stationary black hole, is fully determined by its mass ánd angular momentum. When GR is coupled to an electromagnetic field, a black hole can have an electric and a magnetic charge as well. However, there is no additional memory of what formed the black hole: there are no higher multipole moments etc.

  2. 2.

    For comparison, the famous cosmological constant problem is the large ratio \(\varLambda _{QFT}/\varLambda _{obs} \sim 10^{120}\) between the “expected” value \(\varLambda _{QFT}\) and the observed value \(\varLambda _{obs}\). This number is peanuts compared to the required number of black hole microstates!

  3. 3.

    Often people refer to the entire framework as “M-theory”. We like to view this eleven-dimensional theory as one of the corners of the string web instead.

  4. 4.

    We choose to write the time directions as \(x^0\) and space time directions \(x^1,x^2,\ldots \). However, we choose the ‘eleventh’ dimensions to be \(x^{11}\) and skip \(x^{10}\).

  5. 5.

    Type IIA supergravity is one of the two possible ten-dimensional supergravity theories invariant under \(\mathcal{N}=2\) supersymmetry, namely the one for which the two supersymmetry generators (spinors) have opposite chirality. The other \(\mathcal{N}=2\) supergravity in ten dimensions is type IIB supergravity, the low-energy limit of IIB superstring theory, which has two supersymmetry generators with the same chirality.

  6. 6.

    Different boundary conditions for the fermionic fields living on the world-volume of the type II string give different possible fields in the string spectrum. In the massless spectrum we observe that Neveu-Schwarz-boundary conditions (anti-periodic) give the NS-fields: metric \(g_{\mu \nu }\), B-field \(B_{\mu \nu }\), and dilaton \(\phi \). Ramond boundary conditions (periodic) give RR fields \(C^{(0)},C^{(2)},C^{(4)}\).

  7. 7.

    We adopt common notation \(C\) for the Ramon-Ramond gauge field in ten dimensions that couple to D-branes, and \(A\) for the gauge field in elven dimensions that couples to M-branes.

  8. 8.

    In the near-horizon geometry of a D3 brane, which is \(AdS_5 \times S^5\) as we will see below, S-duality becomes the strong-weak coupling duality of \(N=4\) super Yang-mills, the theory dual to the \(AdS_5 \times S^5\) background through the AdS/CFT duality.

  9. 9.

    Although there is a naked singularity in the supergravity solution, as a solution to string theory, a D-brane is well-defined. As \(r\rightarrow 0\), the dilaton \(\phi \) blows up. Since it sets the length of the eleven-dimensional compactification circle of M-theory, the eleventh dimension decompactifies near \(r\rightarrow 0\). We hence get the near-M2-brane solution of eleven-dimensional M-theory, which is well-defined in all of space-time.

  10. 10.

    For the aficionados: this is the same mechanism that forces the extremal Reissner-Nordstrom black hole to have a near-horizon region of the form \(AdS_2 \times S^2\).

  11. 11.

    If we did not smear the individual D-branes making up the black hole solution, then the metric would depend on some of the internal coordinates as well. We only want dependence on four-dimensionsional space-time. In addition, if we T-dualize one D-brane, then the result becomes smeared along the dualization direction. To get a four-dimensional black hole that looks the same in all duality frames, we need to work in a duality frame where the branes are smeared on orthogonal compact directions.

  12. 12.

    At the position of the horizon, we have a degenerate coordinate system, but there is no physical singularity at \(r = 0\).

  13. 13.

    The \(T^6\) radii \(L_i\) are defined by identifying the \(x_i\) periodically as \(x_i = x_i + 2 \pi L_i\).

  14. 14.

    Remember the analogy with electromagnetism, for a magnetic monopole we find the quantized monopole charge \(N\) is \(N \propto \int _{S^2} F_2\).

  15. 15.

    This does not mean that it is a realistic astrophysical black hole. In nature, black holes will shed (almost) all their charge and be charge neutral. Supersymmetric black holes are extremal; they have the maximum amount of charge allowed for their mass and are hence not the black holes we observe in the sky.

  16. 16.

    These are not the lowest-energy modes of the string. Those are tachyonic (negative energy) modes, that can be consistently projected out of the spectrum of string theory.

  17. 17.

    Extrapolating from toy models is many a string theoriest’s idea of a mathematical proof of complicated string theory effects.

  18. 18.

    Note that for “2 D-branes” on a compact circle, we have either 2 distinct D-branes or a D-brane wrapping the circle 2 times.

  19. 19.

    There are also contributions from 1-1 and 5-5 strings, but these have momentum quantized in units of \(p \sim 1/N_1\) and \(-\sim 1/N_5\) and are hence subleading.

  20. 20.

    In natural units \(\hbar = c = 1\), energy is measured in dimensions of inverse length \([E] = L^{-1}\) .

  21. 21.

    Remember that the metric of the D1-D5-P system looks like

    $$\begin{aligned} ds^2 = -dt^2 + dx_5^2 + Z_p dx_-^2, \end{aligned}$$
    (2.192)

    with \(dx_- = dt - dx_5\). This fixes a particular chirality of the plane wave.

  22. 22.

    The information paradox leads to a breakdown of unitarity in quantum theory and hence a breakdown of quantum mechanics itself. If we want to save quantum mechanics, we need to make sure there is no information loss.

  23. 23.

    It is also important for confinement in supersymmetric holographically dual gauge theories through the AdS/CFT correspondence, but that is another matter. See [59, 60].

  24. 24.

    Stated without reference to a set of coordinates, ‘static’ means that the metric admits a global, nowhere zero, time-like hypersurface orthogonal Killing vector field. A generalization are the ‘stationary’ space-times, which admit a global, nowhere zero time-like Killing vector field.

  25. 25.

    In fact, the requirement of supersymmetry only requires the base space to be hyperkähler. The additional constraint of a Taub-NUT of Gibbons-Hawking metric makes it possible to solve for the metric explicitly. For more information, see [41] and reference therein.

  26. 26.

    In fact, there are certain conditions the harmonic functions \(H\) have to obey such that the multi-center geometry is also smooth and horizonless at each center. We will not dwell on that, see [41] for more information.

  27. 27.

    By ‘infinite throat’, people mean that the spatial metric distance \(\int ds\) to the horizon from any point outside the horizon blows up.

  28. 28.

    Note: only a subset of this multi-center solutions are actual fuzzballs. We need some more information to discuss them, we will leave it at this for the moment.

  29. 29.

    Only the relative positions are of importance, hence the degrees of freedom of one of the centers do not count and we get \(3N-3\) coordinates that specify a physical solution with \(N\) centers.

References

  1. N. Deruelle, Black Holes in General Relativity, Course de Physique Théorique at IPhT, CEA Saclay,ipht.cea.fr/Docspht//articles/t09/346/public/Cours2009.pdfipht.cea.fr/Docspht//articles/t09/346/public/Cours2009.pdf

    Google Scholar 

  2. J.D. Bekenstein, Black holes and the second law. Nuovo Cim. Lett. 4, 737–740 (1972)

    Article  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Google Scholar 

  5. M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory, Loop qamplitudes, anomalies and phenomenology, Cambridge Monographs On Mathematical Physics, vol 2 (Cambridge University Press, UK, 1987), p. 596

    Google Scholar 

  6. J. Polchinski, String Theory, An Introduction to the Bosonic String, vol 1 (Cambridge University Press, UK, 1998), p. 402

    Google Scholar 

  7. J. Polchinski, String Theory, Superstring Theory and Beyond, vol 2 (Cambridge University Press, UK, 1998), p. 531

    Google Scholar 

  8. K. Becker, M. Becker, J.H. Schwarz, String Theory and M-theory: A Modern Introduction (Cambridge University Press, UK, 2007)

    Google Scholar 

  9. E. Kiritsis, String Theory in a Nutshell (Princeton University Press, USA 2007)

    Google Scholar 

  10. D.Z. Freedman, A. Van Proeyen, Supergravity (Cambridge University Press, Cambridge, UK, 2012), p. 607

    Google Scholar 

  11. E. Cremmer, B. Julia, J. Scherk, Supergravity theory in eleven-dimensions. Phys. Lett. B76, 409–412 (1978)

    ADS  Google Scholar 

  12. G.W. Gibbons, M.B. Green, M.J. Perry, Instantons and seven-branes in type IIB superstring theory. Phys. Lett. B370, 37–44 (1996). http://www.arXiv.org/abs/hep-th/9511080 hep-th/9511080

    Google Scholar 

  13. F. Denef, Les Houches Lectures on Constructing String Vacua, http://www.arXiv.org/abs/0803.1194 0803.1194

  14. E.A. Bergshoeff, J. Hartong, T. Ortin, D. Roest, Seven-branes and supersymmetry. JHEP 0702, 003 (2007). http://www.arXiv.org/abs/hep-th/0612072 hep-th/0612072

    Google Scholar 

  15. E. Bergshoeff, J. Hartong, D. Sorokin, Q7-branes and their coupling to IIB supergravity. JHEP 0712, 079 (2007). http://www.arXiv.org/abs/0708.2287 0708.2287

    Google Scholar 

  16. J. Raeymaekers, D. Van den Bleeken, B. Vercnocke, Chronology protection and the stringy exclusion principle. JHEP 1104, 037 (2011). http://www.arXiv.org/abs/1011.5693 1011.5693

  17. E. Bergshoeff, p-branes, D-branes and M-branes, http://www.arXiv.org/abs/hep-th/9611099 hep-th/9611099

  18. K.S. Stelle, BPS Branes in Supergravity, http://www.arXiv.org/abs/hep-th/9803116 hep-th/9803116

  19. R. Argurio, Brane Physics in M-theory, http://www.arXiv.org/abs/hep-th/9807171 hep-th/9807171

  20. A.W. Peet, TASI Lectures on Black Holes in String Theory, http://www.arXiv.org/abs/hep-th/0008241 hep-th/0008241

  21. C.W. Misner, K. Thorne, J. Wheeler, Gravitation

    Google Scholar 

  22. A.A. Tseytlin, Harmonic superpositions of M-branes. Nucl. Phys. B475, 149–163 (1996). http://www.arXiv.org/abs/hep-th/9604035 hep-th/9604035

  23. A. Strominger, C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy. Phys. Lett. B379, 99–104 (1996). http://www.arXiv.org/abs/hep-th/9601029 hep-th/9601029

    Google Scholar 

  24. J.M. Maldacena, The Large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231–252 (1998). http://www.arXiv.org/abs/hep-th/9711200 hep-th/9711200

    Google Scholar 

  25. E. Witten, Anti-de Sitter space and holography. Adv. Theor. Math. Phys. 2, 253–291, (1998). http://www.arXiv.org/abs/hep-th/9802150 hep-th/9802150

    Google Scholar 

  26. S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from noncritical string theory. Phys. Lett. B428, 105–114 (1998). http://www.arXiv.org/abs/hep-th/9802109 hep-th/9802109

    Google Scholar 

  27. G. ’t Hooft, Large N, http://www.arXiv.org/abs/hep-th/0204069 hep-th/0204069

  28. I. Bena, M. Berkooz, J. de Boer, S. El-Showk, D. Van den Bleeken, Scaling BPS solutions and pure-Higgs states. JHEP 1211, 171 (2012). http://www.arXiv.org/abs/1205.5023 1205.5023

  29. I.R. Klebanov, TASI lectures: Introduction to the AdS/CFT Correspondence, http://www.arXiv.org/abs/hep-th/0009139 hep-th/0009139

  30. J.-P. Blaizot, E. Iancu, U. Kraemmer and A. Rebhan, Hard thermal loops and the entropy of supersymmetric Yang-Mills theories, JHEP 0706, 035 (2007). http://www.arXiv.org/abs/hep-ph/0611393 hep-ph/0611393

    Google Scholar 

  31. A. Bernamonti and R. Peschanski, Time-dependent AdS/CFT correspondence and the Quark-Gluon plasma. Nucl. Phys. Proc. Suppl. 216, 94–120 (2011). http://www.arXiv.org/abs/1102.0725 1102.0725

    Google Scholar 

  32. M. Petrini, The AdS/CFT correspondence, Course de Physique Théorique at IPhT, CEA Saclay, http://ipht.cea.fr/Docspht/search/article.php?id=t10/312http://ipht.cea.fr/Docspht/search/article.php?id=t10/312

  33. J.R. Peschanski Robi, The dynamics of Quark-Gluon Plasma and AdS/CFT correspondence, Course de Physique Théorique at IPhT, CEA Saclay, http://ipht.cea.fr/Docspht/search/article.php?id=t10/310http://ipht.cea.fr/Docspht/search/article.php?id=t10/310

  34. L. Susskind, Some Speculations About Black Hole Entropy in String Theory, http://www.arXiv.org/abs/hep-th/9309145 hep-th/9309145

  35. E. Halyo, A. Rajaraman, L. Susskind, Braneless black holes. Phys. Lett. B392, 319–322 (1997). http://www.arXiv.org/abs/hep-th/9605112 hep-th/9605112

    Google Scholar 

  36. E. Halyo, B. Kol, A. Rajaraman, L. Susskind, Counting Schwarzschild and charged black holes. Phys. Lett. B401, 15–20 (1997). http://www.arXiv.org/abs/hep-th/9609075 hep-th/9609075

    Google Scholar 

  37. G.T. Horowitz, J. Polchinski, A Correspondence principle for black holes and strings. Phys. Rev. D55, 6189–6197 (1997). http://www.arXiv.org/abs/hep-th/9612146 hep-th/9612146

    Google Scholar 

  38. T. Damour, G. Veneziano, Selfgravitating fundamental strings and black holes. Nucl. Phys. B568, 93–119 (2000). http://www.arXiv.org/abs/hep-th/9907030 hep-th/9907030

    Google Scholar 

  39. S.D. Mathur, The Information paradox: A Pedagogical introduction. Class. Quant. Grav. 26, 224001 (2009). http://www.arXiv.org/abs/0909.1038 0909.1038

    Google Scholar 

  40. S.D. Mathur, The fuzzball proposal for black holes: An elementary review. Fortsch. Phys. 53, 793–827 (2005). http://www.arXiv.org/abs/hep-th/0502050 hep-th/0502050

    Google Scholar 

  41. I. Bena, N.P. Warner, Black holes, black rings and their microstates. Lect. Notes Phys. 755, 1–92 (2008). http://www.arXiv.org/abs/hep-th/0701216 hep-th/0701216

  42. S.D. Mathur, Fuzzballs and the Information Paradox: A Summary and Conjectures, http://www.arXiv.org/abs/0810.4525 0810.4525

  43. V. Balasubramanian, J. de Boer, S. El-Showk and I. Messamah, Black holes as effective geometries. Class. Quant. Grav. 25, 214004 (2008). http://www.arXiv.org/abs/0811.0263 0811.0263

    Google Scholar 

  44. K. Skenderis, M. Taylor, The fuzzball proposal for black holes. Phys. Rept. 467, 117–171 (2008). http://www.arXiv.org/abs/0804.0552 0804.0552

    Google Scholar 

  45. B.D. Chowdhury, A. Virmani, Modave Lectures on Fuzzballs and Emission from the D1–D5 System, http://www.arXiv.org/abs/1001.1444 1001.1444

  46. V. Jejjala, O. Madden, S.F. Ross, G. Titchener, Non-supersymmetric smooth geometries and D1–D5-P bound states. Phys. Rev. D71, 124030 (2005). http://www.arXiv.org/abs/hep-th/0504181 hep-th/0504181

  47. S. Giusto, S.F. Ross, A. Saxena, Non-supersymmetric microstates of the D1–D5-KK system. JHEP 12, 065 (2007). http://www.arXiv.org/abs/0708.3845 0708.3845

    Google Scholar 

  48. V. Cardoso, O.J. Dias, J.L. Hovdebo, R.C. Myers, Instability of non-supersymmetric smooth geometries. Phys. Rev. D73, 064031 (2006). http://www.arXiv.org/abs/hep-th/0512277 hep-th/0512277

  49. B.D. Chowdhury, S.D. Mathur, Radiation from the non-extremal fuzzball. Class. Quant. Grav. 25, 135005 (2008). http://www.arXiv.org/abs/0711.4817 0711.4817

    Google Scholar 

  50. B.D. Chowdhury, S.D. Mathur, Non-extremal fuzzballs and ergoregion emission. Class. Quant. Grav. 26, 035006 (2009). http://www.arXiv.org/abs/0810.2951 0810.2951

    Google Scholar 

  51. B.D. Chowdhury, S.D. Mathur, Pair creation in non-extremal fuzzball geometries. Class. Quant. Grav. 25, 225021 (2008). http://www.arXiv.org/abs/0806.2309 0806.2309

    Google Scholar 

  52. I. Bena, J. de Boer, M. Shigemori, N.P. Warner, Double, Double Supertube Bubble. JHEP 1110, 116 (2011). http://www.arXiv.org/abs/1107.2650 1107.2650

  53. I. Bena, S. Giusto, M. Shigemori, N.P. Warner, Supersymmetric solutions in six dimensions: a linear structure. JHEP 1203, 084 (2012). http://www.arXiv.org/abs/1110.2781 1110.2781

  54. S. Giusto, R. Russo, D. Turton, New D1–D5-P geometries from string amplitudes. JHEP 1111, 062 (2011). http://www.arXiv.org/abs/1108.6331 1108.6331

  55. I. Bena, S. Giusto, C. Ruef, N.P. Warner, Supergravity solutions from floating branes. JHEP 1003, 047 (2010). http://www.arXiv.org/abs/0910.1860 0910.1860

  56. G. Dall’Agata, S. Giusto, C. Ruef, U-duality and non-BPS solutions. JHEP 1102, 074 (2011). http://www.arXiv.org/abs/1012.4803 1012.4803

  57. G. Bossard, C. Ruef, Interacting non-BPS black holes. Gen. Rel. Grav. 44, 21–66 (2012). http://www.arXiv.org/abs/1106.5806 1106.5806

    Google Scholar 

  58. G. Bossard, S. Katmadas, Duality covariant non-BPS first order systems. JHEP 1209, 100 (2012). http://www.arXiv.org/abs/1205.5461 1205.5461

  59. I.R. Klebanov, A.A. Tseytlin, Gravity duals of supersymmetric SU(N) x SU(N+M) gauge theories. Nucl. Phys. B578, 123–138 (2000). http://www.arXiv.org/abs/hep-th/0002159 hep-th/0002159

  60. I.R. Klebanov, M.J. Strassler, Supergravity and a confining gauge theory: Duality cascades and chi SB resolution of naked singularities. JHEP 0008, 052 (2000). http://www.arXiv.org/abs/hep-th/0007191 hep-th/0007191

    Google Scholar 

  61. M. Grana, Flux compactifications in string theory: a comprehensive review. Phys. Rept. 423, 91–158 (2006). http://www.arXiv.org/abs/hep-th/0509003 hep-th/0509003

    Google Scholar 

  62. J. Breckenridge, R.C. Myers, A. Peet, C. Vafa, D-branes and spinning black holes. Phys. Lett. B391, 93–98 (1997). http://www.arXiv.org/abs/hep-th/9602065 hep-th/9602065

    Google Scholar 

  63. J.B. Gutowski, H.S. Reall, General supersymmetric AdS(5) black holes. JHEP 0404, 048 (2004). http://www.arXiv.org/abs/hep-th/0401129 hep-th/0401129

  64. J.P. Gauntlett, J.B. Gutowski, Concentric black rings. Phys. Rev. D71, 025013 (2005). http://www.arXiv.org/abs/hep-th/0408010 hep-th/0408010

  65. I. Bena, P. Kraus, N.P. Warner, Black rings in Taub-NUT. Phys. Rev. D72, 084019 (2005). http://www.arXiv.org/abs/hep-th/0504142 hep-th/0504142

  66. F. Denef, Supergravity flows and D-brane stability. JHEP 0008, 050 (2000). http://www.arXiv.org/abs/hep-th/0005049 hep-th/0005049

    Google Scholar 

  67. B. Bates, F. Denef, Exact solutions for supersymmetric stationary black hole composites. JHEP 1111, 127 (2011). http://www.arXiv.org/abs/hep-th/0304094 hep-th/0304094

  68. I. Bena, C.-W. Wang, N.P. Warner, Plumbing the Abyss: Black ring microstates. JHEP 0807, 019 (2008). http://www.arXiv.org/abs/0706.3786 0706.3786

    Google Scholar 

  69. J. de Boer, S. El-Showk, I. Messamah, D. Van den Bleeken, Quantizing N=2 multicenter solutions. JHEP 0905, 002 (2009). http://www.arXiv.org/abs/0807.4556 0807.4556

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    Iosif Bena, Sheer El-Showk & Bert Vercnocke

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Bena, I., El-Showk, S., Vercnocke, B. (2013). Black Holes in String Theory . In: Bellucci, S. (eds) Black Objects in Supergravity. Springer Proceedings in Physics, vol 144. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00215-6_2

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