Skip to main content
SpringerLink
Log in

Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

  • Conference paper
  • First Online:
The Attractor Mechanism

Part of the book series: Springer Proceedings in Physics ((SPPHY,volume 134))

Abstract

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N = 2 Maxwell–Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N = 2 MESGTs are the quasiconformal groups QConf(J) associated with J that are isomorphic to their U-duality groups in three dimensions. We then review the work on spectrum generating symmetries of spherically symmetric stationary 4D BPS black holes, based on the equivalence of their attractor equations and the equations for geodesic motion of a fiducial particle on the target spaces of corresponding 3D supergravity theories obtained by timelike reduction. We also discuss the connection between harmonic superspace formulation of 4D, N = 2 sigma models coupled to supergravity and the minimal unitary representations of their isometry groups obtained by quantizing their quasiconformal realizations. We discuss the relevance of this connection to spectrum generating symmetries and conclude with a brief summary of more recent results.

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Nahm, Supersymmetries and their representations. Nucl. Phys. B 135, 149 (1978)

    Article  ADS  Google Scholar 

  2. E. Cremmer, B. Julia, J. Scherk, Supergravity theory in 11 dimensions. Phys. Lett. B 76, 409 (1978)

    Article  ADS  Google Scholar 

  3. C.M. Hull, P.K. Townsend, Unity of superstring dualities. Nucl. Phys. B 438, 109 (1995). http://www.arXiv.org/abs/hep-th/9410167hep-th/9410167

  4. N. Marcus, J.H. Schwarz, Three-dimensional supergravity theories. Nucl. Phys. B 228, 145 (1983)

    MathSciNet  Google Scholar 

  5. M. Günaydin, G. Sierra, P.K. Townsend, Exceptional supergravity theories and the magic square. Phys. Lett. B 133, 72 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  6. M. Günaydin, G. Sierra, P.K. Townsend, The geometry of \(\mathcal{N}\,=\,2\) Maxwell–Einstein supergravity and Jordan algebras. Nucl. Phys. B 242, 244 (1984)

    Article  ADS  Google Scholar 

  7. M. Günaydin, G. Sierra, P.K. Townsend, Gauging the d = 5 Maxwell–Einstein supergravity theories: more on Jordan algebras. Nucl. Phys. B 253, 573 (1985)

    Article  ADS  Google Scholar 

  8. M. Günaydin, G. Sierra, P.K. Townsend, More on d = 5 Maxwell–Einstein supergravity: symmetric spaces and kinks. Class. Quant. Grav. 3, 763 (1986)

    Article  MATH  ADS  Google Scholar 

  9. B. de Wit, F. Vanderseypen, A. Van Proeyen, Symmetry structure of special geometries. Nucl. Phys. B 400, 463 (1993). http://www.arXiv.org/abs/hep-th/9210068hep-th/9210068

  10. D. Kazhdan, A. Polishchuk, Minimal representations: spherical vectors and automorphic functionals, in Algebraic Groups and Arithmetic (Tata Institute of Fundamental Research, Mumbai, 2004), pp. 127–198

    Google Scholar 

  11. K. McCrimmon, A Taste of Jordan Algebras, Universitext (Springer, New York, 2004)

    Google Scholar 

  12. N. Jacobson, in Structure and Representations of Jordan Algebras. American Mathematical Society Colloquium Publications, Vol. XXXIX (American Mathematical Society, Providence, RI, 1968)

    Google Scholar 

  13. M. Günaydin, F. Gürsey, Quark structure and octonions. J. Math. Phys. 14, 1651 (1973)

    Article  MATH  Google Scholar 

  14. S. Ferrara, M. Günaydin, Orbits of exceptional groups, duality and BPS states in string theory. Int. J. Mod. Phys. A 13, 2075 (1998). http://www.arXiv.org/abs/hep-th/9708025hep-th/9708025

  15. M. Günaydin, K. Koepsell, H. Nicolai, Conformal and quasiconformal realizations of exceptional Lie groups. Commun. Math. Phys. 221, 57 (2001). http://www.arXiv.org/abs/hep-th/0008063hep-th/0008063

  16. M. Günaydin, Exceptional realizations of Lorentz group: supersymmetries and leptons. Nuovo Cim. A 29, 467 (1975)

    Article  ADS  Google Scholar 

  17. V.G. Kac, Lie superalgebras. Adv. Math. 26 (1977) 8

    Article  MATH  Google Scholar 

  18. M. Günaydin, Quadratic Jordan formulation of quantum mechanics and construction of Lie (super)algebras from Jordan (super)algebras. Ann. Israel Phys. Soc. 3, 279 (1980). Presented at 8th Int. Colloq. on Group Theoretical Methods in Physics, Kiriat Anavim, Israel, 25–29 March 1979

    Google Scholar 

  19. M. Günaydin, The exceptional superspace and the quadratic Jordan formulation of quantum mechanics, in Elementary Particles and the Universe: Essays in Honor of Murray Gell-Mann, Pasadena 1989, ed. by J. Schwarz (Cambridge University Press, Cambridge, 1989), pp. 99–119

    Google Scholar 

  20. M. Günaydin, Generalized conformal and superconformal group actions and Jordan algebras. Mod. Phys. Lett. A 8, 1407 (1993). http://www.arXiv.org/abs/hep-th/9301050hep-th/9301050

  21. G. Sierra, An application to the theories of Jordan algebras and Freudenthal triple systems to particles and strings. Class. Quant. Grav. 4, 227 (1987)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. M. Günaydin, O. Pavlyk, Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups. J. High Energy Phys. 08, 101 (2005). http://www.arXiv.org/abs/hep-th/0506010hep-th/0506010

  23. M. Günaydin, AdS/CFT dualities and the unitary representations of non-compact groups and supergroups: Wigner versus Dirac. http://www.arXiv.org/abs/hep-th/0005168hep-th/0005168

  24. G. Mack, M. de Riese, Simple space–time symmetries: generalizing conformal field theory. http://www.arXiv.org/abs/hep-th/0410277hep-th/0410277

  25. I.L. Kantor, Certain generalizations of Jordan algebras. Trudy Sem. Vektor. Tenzor. Anal. 16, 407 (1972)

    MathSciNet  Google Scholar 

  26. J. Tits, Une classe d’algèbres de Lie en relation avec les algèbres de Jordan. Nederl. Akad. Wetensch. Proc. Ser. A 65 = Indag. Math. 24, 530 (1962)

    Google Scholar 

  27. M. Koecher, Imbedding of Jordan algebras into Lie algebras. II. Am. J. Math. 90, 476 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  28. S. Ferrara, M. Gunaydin, Orbits and attractors for N = 2 Maxwell–Einstein supergravity theories in five dimensions. Nucl. Phys. B 759, 1 (2006). http://www.arXiv.org/abs/hep-th/0606108hep-th/0606108

  29. S. Ferrara, R. Kallosh, Universality of supersymmetric attractors. Phys. Rev. D 54, 1525 (1996). http://www.arXiv.org/abs/hep-th/9603090hep-th/9603090

  30. S. Ferrara, G.W. Gibbons, R. Kallosh, Black holes and critical points in moduli space. Nucl. Phys. B 500, 75 (1997). http://www.arXiv.org/abs/hep-th/9702103hep-th/9702103

  31. S. Ferrara, R. Kallosh, On N = 8 attractors. Phys. Rev. D 73, 125005 (2006). http://www.arXiv.org/abs/hep-th/0603247hep-th/ 0603247

  32. A.H. Chamseddine, S. Ferrara, G.W. Gibbons, R. Kallosh, Enhancement of supersymmetry near 5D black hole horizon. Phys. Rev. D 55, 3647 (1997). http://www.arXiv.org/abs/hep-th/9610155hep-th/9610155

  33. S. Ferrara, J.M. Maldacena, Branes, central charges and U-duality invariant BPS conditions. Class. Quant. Grav. 15, 749 (1998). http://www.arXiv.org/abs/hep-th/9706097hep-th/9706097

    Google Scholar 

  34. M. Günaydin, Unitary realizations of U-duality groups as conformal and quasiconformal groups and extremal black holes of supergravity theories. AIP Conf. Proc. 767, 268 (2005). http://www.arXiv.org/abs/hep-th/0502235hep-th/0502235

  35. M. Günaydin, Realizations of exceptional U-duality groups as conformal and quasiconformal groups and their minimal unitary representations. Comment. Phys. Math. Soc. Sci. Fenn. 166, 111 (2004). http://www.arXiv.org/abs/hep-th/0409263hep-th/0409263

  36. D. Gaiotto, A. Strominger, X. Yin, 5D black rings and 4D black holes. J. High Energy Phys. 02, 023 (2006). http://www.arXiv.org/abs/hep-th/0504126hep-th/0504126

  37. D. Gaiotto, A. Strominger, X. Yin, New connections between 4D and 5D black holes. J. High Energy Phys. 02, 024 (2006). http://www.arXiv.org/abs/hep-th/0503217hep-th/0503217

  38. H. Elvang, R. Emparan, D. Mateos, H.S. Reall, Supersymmetric 4D rotating black holes from 5D black rings. http://www.arXiv.org/abs/hep-th/0504125hep-th/0504125

  39. B. Pioline, BPS black hole degeneracies and minimal automorphic representations. J. High Energy Phys. 0508, 071 (2005). http://www.arXiv.org/abs/hep-th/0506228hep-th/0506228

  40. B. de Wit, A. Van Proeyen, Special geometry, cubic polynomials and homogeneous quaternionic spaces. Commun. Math. Phys. 149 (1992) 307, http://www.arXiv.org/abs/hep-th/9112027hep-th/9112027

  41. M. Günaydin, S. McReynolds, M. Zagermann, The R-map and the coupling of N = 2 tensor multiplets in 5 and 4 dimensions. J. High Energy Phys. 01, 168 (2006). http://www.arXiv.org/abs/hep-th/0511025hep-th/0511025

  42. M. Koecher, The Minnesota Notes on Jordan Algebras and Their Applications, ed. by A. Krieg, S. Walcher. Lecture Notes in Mathematics, vol. 1710 (Springer, Berlin, 1999)

    Google Scholar 

  43. H. Freudenthal, Lie groups in the foundations of geometry. Advances in Math. 1(2) 145 (1964)

    Google Scholar 

  44. H. Freudenthal, Beziehungen der E 7 und E 8 zur Oktavenebene. I. Nederl. Akad. Wetensch. Proc. Ser. A 57 = Indag. Math. 16, 218 (1954)

    Google Scholar 

  45. S. Bellucci, S. Ferrara, M. Gunaydin, A. Marrani, Charge orbits of symmetric special geometries and attractors. Int. J. Mod. Phys. A 21, 5043 (2006). http://www.arXiv.org/abs/hep-th/0606209hep-th/0606209

  46. S. Ferrara, S. Sabharwal, Quaternionic manifolds for type II superstring vacua of Calabi–Yau spaces. Nucl. Phys. B 332, 317 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  47. S. Bellucci, S. Ferrara, M. Gunaydin, A. Marrani, SAM lectures on extremal black holes in d = 4 extended supergravity. http://www.arXiv.org/abs/0905373909053739

  48. A. Ceresole, R. D’Auria, S. Ferrara, The symplectic structure of \(\mathcal{N}\,=\,2\) supergravity and its central extension. Nucl. Phys. Proc. Suppl. 46, 67 (1996). http://www.arXiv.org/abs/hep-th/9509160hep-th/9509160

  49. E. Cremmer, A. Van Proeyen, Classification of Kahler manifolds in \(\mathcal{N}\,=\,2\) vector multiplet supergravity couplings. Class. Quant. Grav. 2, 445 (1985)

    Article  MATH  ADS  Google Scholar 

  50. M. Günaydin, Realizations of exceptional U-duality groups as conformal and quasi-conformal groups and their minimal unitary representations. Prepared for 3rd International Symposium on Quantum Theory and Symmetries (QTS3), Cincinnati, OH, 10–14 Sept 2003

    Google Scholar 

  51. M. Günaydin, A. Neitzke, B. Pioline, A. Waldron, BPS black holes, quantum attractor flows and automorphic forms. Phys. Rev. D 73, 084019 (2006). http://www.arXiv.org/abs/hep-th/0512296hep-th/0512296

  52. M. Günaydin, A. Neitzke, B. Pioline, A. Waldron, Quantum attractor flows. J. High Energy Phys. 09, 056 (2007). http://www.arXiv.org/abs/0707026707070267

  53. I. Kantor, I. Skopets, Some results on Freudenthal triple systems. Sel. Math. Sov. 2, 293 (1982)

    MATH  Google Scholar 

  54. M. Günaydin, A. Neitzke, O. Pavlyk, B. Pioline, Quasi-conformal actions, quaternionic discrete series and twistors: SU(2, 1) and G 2(2). Commun. Math. Phys. 283, 169 (2008). http://www.arXiv.org/abs/0707166907071669

  55. B. Pioline, Lectures on Black Holes, Topological Strings and Quantum Attractors (2.0). Lecture Notes on Physics, vol. 755 (Springer, Berlin, 2008), pp. 1–91

    Google Scholar 

  56. S. Ferrara, R. Kallosh, A. Strominger, \(\mathcal{N}\,=\,2\) extremal black holes. Phys. Rev. D 52, 5412 (1995). http://www.arXiv.org/abs/hep-th/9508072hep-th/9508072

  57. P. Breitenlohner, G.W. Gibbons, D. Maison, Four-dimensional black holes from Kaluza–Klein theories. Commun. Math. Phys. 120, 295 (1988)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. M. Cvetic, D. Youm, All the static spherically symmetric black holes of heterotic string on a six torus. Nucl. Phys. B 472 (1996) 249, http://www.arXiv.org/abs/hep-th/9512127hep-th/9512127

  59. M. Cvetic, D. Youm, Dyonic BPS saturated black holes of heterotic string on a six torus. Phys. Rev. D 53, 584 (1996). http://www.arXiv.org/abs/hep-th/9507090hep-th/9507090

  60. M. Günaydin, A. Neitzke, B. Pioline, Topological wave functions and heat equations. J. High Energy Phys. 12, 070 (2006). http://www.arXiv.org/abs/hep-th/0607200hep-th/0607200

  61. J. Bagger, E. Witten, Matter couplings in \(\mathcal{N}\,=\,2\) supergravity. Nucl. Phys. B 222, 1 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  62. G.W. Moore, Arithmetic and attractors. http://www.arXiv.org/abs/hep-th/9807087hep-th/9807087

  63. F. Denef, Supergravity flows and D-brane stability. J. High Energy Phys. 08, 050 (2000). http://www.arXiv.org/abs/hep-th/0005049hep-th/0005049

  64. M. Gutperle, M. Spalinski, Supergravity instantons for \(\mathcal{N}\,=\,2\) hypermultiplets. Nucl. Phys. B 598, 509 (2001). http://www.arXiv.org/abs/hep-th/0010192hep-th/0010192

  65. A. Neitzke, B. Pioline, S. Vandoren, Twistors and black holes. J. High Energy Phys. 04, 038 (2007). http://www.arXiv.org/abs/hep-th/0701214hep-th/0701214

  66. M. Günaydin, Harmonic superspace, minimal unitary representations and quasiconformal groups. J. High Energy Phys. 05, 049 (2007). http://www.arXiv.org/abs/hep-th/0702046hep-th/0702046

  67. A. Galperin, E. Ivanov, S. Kalitsyn, V. Ogievetsky, E. Sokatchev, Unconstrained N = 2 Matter, Yang–Mills and supergravity theories in harmonic superspace. Class. Quant. Grav. 1, 469 (1984)

    Article  ADS  Google Scholar 

  68. J.A. Bagger, A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky, Gauging N = 2 sigma models in harmonic superspace. Nucl. Phys. B 303, 522 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  69. A. Galperin, E. Ivanov, O. Ogievetsky, Harmonic space and quaternionic manifolds. Ann. Phys. 230, 201 (1994). http://www.arXiv.org/abs/hep-th/9212155hep-th/9212155

    Google Scholar 

  70. A. Galperin, O. Ogievetsky, Harmonic potentials for quaternionic symmetric sigma models. Phys. Lett. B 301, 67 (1993). http://www.arXiv.org/abs/hep-th/9210153hep-th/9210153

  71. A.S. Galperin, E.A. Ivanov, V.I. Ogievetsky, E.S. Sokatchev, Harmonic Superspace (Cambridge University Press, Cambridge, 2001), 306 p

    Google Scholar 

  72. A. Galperin, V. Ogievetsky, N = 2 D = 4 supersymmetric sigma models and Hamiltonian mechanics. Class. Quant. Grav. 8, 1757 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  73. M. Günaydin, O. Pavlyk, A unified approach to the minimal unitary realizations of noncompact groups and supergroups. J. High Energy Phys. 09, 050 (2006). http://www.arXiv.org/abs/hep-th/0604077hep-th/0604077

  74. J.A. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033 (1965)

    MATH  MathSciNet  Google Scholar 

  75. M. Günaydin, K. Koepsell, H. Nicolai, The minimal unitary representation of E 8(8). Adv. Theor. Math. Phys. 5, 923 (2002). http://www.arXiv.org/abs/hep-th/0109005hep-th/0109005

  76. M. Günaydin, O. Pavlyk, Minimal unitary realizations of exceptional U-duality groups and their subgroups as quasiconformal groups. J. High Energy Phys. 01, 019 (2005). http://www.arXiv.org/abs/hep-th/0409272hep-th/0409272

  77. M. Günaydin, C. Saclioglu, Oscillator-like unitary representations of noncompact groups with a Jordan structure and the noncompact groups of supergravity. Commun. Math. Phys. 87, 159 (1982)

    Article  MATH  ADS  Google Scholar 

  78. M. Gunaydin, Exceptional supergravity theories, Jordan algebras and the magic square. Presented at 13th Int. Colloq. on Group Theoretical Methods in Physics, College Park, MD, 21–25 May 1984

    Google Scholar 

  79. P. Candelas, G.T. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings. Nucl. Phys. B 258, 46 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  80. A.C. Cadavid, A. Ceresole, R. D’Auria, S. Ferrara, Eleven-dimensional supergravity compactified on Calabi–Yau threefolds. Phys. Lett. B 357, 76 (1995). http://www.arXiv.org/abs/hep-th/9506144hep-th/9506144

  81. B.H. Gross, A remark on tube domains. Math. Res. Lett. 1(1), 1 (1994)

    MATH  ADS  Google Scholar 

  82. C. Vafa, E. Witten, Dual string pairs with N = 1 and N = 2 supersymmetry in four dimensions. Nucl. Phys. Proc. Suppl. 46 (1996) 225, http://www.arXiv.org/abs/hep-th/9507050hep-th/9507050

  83. A. Sen, C. Vafa, Dual pairs of type II string compactification. Nucl. Phys. B 455, 165 (1995). http://www.arXiv.org/abs/hep-th/9508064hep-th/9508064

  84. M. Günaydin. Talk at IAS, Princeton, Sept 2006

    Google Scholar 

  85. M. Günaydin, From d = 6, N = 1 to d = 4, N = 2, no-scale models and Jordan algebras. Talk at the Conference “30 Years of Supergravity”, Paris, October 2006. http://cft.igc.psu.edu/research/index.shtml\#gunaydin

  86. S. Ferrara, J.A. Harvey, A. Strominger, C. Vafa, Second quantized mirror symmetry. Phys. Lett. B 361, 59 (1995). http://www.arXiv.org/abs/hep-th/9505162hep-th/9505162

  87. A. Bouchareb et al., G 2 generating technique for minimal D=5 supergravity and black rings. Phys. Rev. D 76, 104032 (2007). http://www.arXiv.org/abs/0708236107082361

  88. D.V. Gal’tsov, N.G. Scherbluk, Generating technique for U(1)3, 5D supergravity. http://www.arXiv.org/abs/0805392408053924

  89. G. Compere, S. de Buyl, E. Jamsin, A. Virmani, G2 dualities in D = 5 supergravity and black strings. Class. Quant. Grav. 26, 125016 (2009). http://www.arXiv.org/abs/0903164509031645

    Google Scholar 

  90. M. Berkooz, B. Pioline, 5D black holes and non-linear sigma models. J. High Energy Phys. 05, 045 (2008). http://www.arXiv.org/abs/0802165908021659

  91. D. Gaiotto, W.W. Li, M. Padi, Non-supersymmetric attractor flow in symmetric spaces. J. High Energy Phys. 12, 093 (2007). http://www.arXiv.org/abs/0710163807101638

  92. W. Li, Non-Supersymmetric attractors in symmetric coset spaces. http://www.arXiv.org/abs/0801253608012536

  93. E. Bergshoeff, W. Chemissany, A. Ploegh, M. Trigiante, T. Van Riet, Generating geodesic flows and supergravity solutions. Nucl. Phys. B 812, 343 (2009). http://www.arXiv.org/abs/0806231008062310

  94. G. Bossard, H. Nicolai, K.S. Stelle, Universal BPS structure of stationary supergravity solutions. http://www.arXiv.org/abs/0902443809024438

  95. M. Gunaydin, O. Pavlyk, Spectrum generating conformal and quasiconformal U-duality groups, supergravity and spherical vectors. http://www.arXiv.org/abs/0901164609011646

  96. M. Gunaydin, O. Pavlyk, Quasiconformal realizations of E 6(6), E 7(7), E 8(8) and SO(n + 3, m + 3), N = 4 and N > 4 supergravity and spherical vectors. http://www.arXiv.org/abs/0904078409040784

  97. M. Gunaydin, P. van Nieuwenhuizen, N.P. Warner, General construction of the unitary representations of anti-de Sitter superalgebras and the spectrum of the S 4 compactification of eleven-dimensional supergravity. Nucl. Phys. B 255, 63 (1985)

    Article  ADS  Google Scholar 

  98. M. Gunaydin, N. Marcus, The spectrum of the S 5 compactification of the chiral N = 2, D = 10 supergravity and the unitary supermultiplets of U(2, 2 ∕ 4). Class. Quant. Grav. 2, L11 (1985)

    Article  MathSciNet  ADS  Google Scholar 

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

  100. M. Gunaydin, N.P. Warner, Unitary supermultiplets of OSp(8 ∕ 4, R) and the spectrum of the S 7 compactification of eleven-dimensional supergravity. Nucl. Phys. B 272, 99 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  101. B.H. Gross, N.R. Wallach, On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math. 481, 73 (1996)

    MATH  MathSciNet  Google Scholar 

  102. M. Günaydin. (In preparation)

    Google Scholar 

  103. Y. Dolivet, B. Julia, C. Kounnas, Magic N = 2 supergravities from hyper-free superstrings. J. High Energy Phys. 02, 097 (2008). http://www.arXiv.org/abs/0712286707122867

  104. M. Bianchi, S. Ferrara, Enriques and octonionic magic supergravity models. J. High Energy Phys. 02, 054 (2008). http://www.arXiv.org/abs/0712297607122976

  105. A.N. Todorov, CY manifolds with locally symmetric moduli spaces. http://www.arXiv.org/abs/arXiv:08064010arXiv:08064010

Download references

Acknowledgements

I would like to thank Stefano Bellucci for his kind invitation to deliver these lectures at SAM 2007 and the participants for numerous stimulating discussions. I wish to thank Stefano Bellucci, Sergio Ferrara, Kilian Koepsell, Alessio Marrani, Andy Neitzke, Hermann Nicolai, Oleksandr Pavlyk, Boris Pioline and Andrew Waldron for enjoyable collaborations and stimulating discussions on various topics covered in these lectures. Thanks are also due to the organizers of the “Fundamental Aspects of Superstring Theory 2009” Workshop at KITP, UC Santa Barbara and of the New Perspectives in String Theory 2009 Workshop at GGI, Florence where part of these lectures were written up. This work was supported in part by the National Science Foundation under grant number PHY-0555605. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Author information

Authors and Affiliations

  1. Institute for Gravitation and the Cosmos, Physics Department, Penn State University, University Park, PA, 16802, USA

    Murat Günaydin

Authors
  1. Murat Günaydin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Murat Günaydin .

Editor information

Editors and Affiliations

  1. Laboratori Nazionali di Frascati, Istituto Nazionale di Fisica Nucleare, via E. Fermi 40, Frascati, 00044, Italy

    Stefano Bellucci

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Günaydin, M. (2010). Lectures on Spectrum Generating Symmetries and U-Duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace. In: Bellucci, S. (eds) The Attractor Mechanism. Springer Proceedings in Physics, vol 134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10736-8_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-10736-8_2

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10735-1

  • Online ISBN: 978-3-642-10736-8

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

Publish with us

Policies and ethics

Search

Navigation