E7(7) duality, BPS black-hole evolution and fixed scalars

Nuclear Physics B

Volumn 509, Issue 1-2, 1998, Pages 463-518

  Andrianopoli, Laura ^a,b   D'Auria, Riccardo ^b,c   Ferrara, Sergio ^d   Fré, Pietro ^b,e   Trigiante, Mario ^f,g  

^a UNIVERSITY OF GENOA   (Italy)

^b SEZIONE DI TORINO   (Italy)

^c POLITECNICO DI TORINO   (Italy)

^d CERN   (Switzerland)

^e UNIVERSITY OF TURIN   (Italy)

^f SISSA   (Italy)

^g SEZIONE DI TRIESTE   (Italy)

Author keywords

Black hole; Solvable algebra; Supergravity

Indexed keywords

EID: 0032484416     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0550-3213(97)00675-5     Document Type: Article

References (66)

1. [1] For a recent update, see J.M. Maldacena, Black holes in string theory, hep-th/9607235.
2. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
3. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
4. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
5. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
6. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
7. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
8. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
9. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
10. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
11. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
12. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
13. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
14. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
15. [2] References on SUSY black holes: G. Gibbons, in Unified theories of Elementary Particles. Critical Assessment and Prospects, Proc. Heisenberg Symp., München, West Germany, 1981, ed. P. Breitenlohner and H.P. Dürr, Lecture Notes in Physics Vol. 160 (Springer, Berlin, 1982); G.W. Gibbons and CM. Hull, Phys. Lett. B 109 (1982) 190; G.W. Gibbons, in Supersymmetry, Supergravity and Related Topics, Proc. 15th GIFT International Physics, Girona, Spain, 1984, ed. F. del Aguila, J. de Azcárraga and L. Ibáñez, (World Scientific, Singapore 1995) p. 147; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. Van Proeyen, Phys. Rev. D 46 (1992) 5278; R. Kallosh, T. Ortin and A. Peet, Phys. Rev. D 47 (1993) 5400; R. Kallosh, Phys. Lett. B 282 (1992) 80; R. Kallosh and A. Peet, Phys. Rev. D 46 (1992) 5223; A. Sen, Nucl. Phys. B 440 (1995) 421; Phys. Lett. B 303 (1993) 221; Mod. Phys. Lett. A 10 (1995) 2081; J. Schwarz and A. Sen, Phys. Lett. B 312 (1993) 105; M. Cvetic and D. Youm, Phys. Rev. D 53 (1996) 584; M. Cvetic and A.A. Tseytlin, Phys. Rev. D 53 (1996) 5619; M. Cvetic and C.M. Hull, Nucl. Phys. B 480 (1996) 296.
16. [3] K. Stelle, Lectures on supergravity p-branes, Lectures presented at 1996 ICTP Summer School, Trieste.
17. [4] M.J. Duff, R.R. Khuri and J.X. Lu, String solitons, Phys. Kept. 259 (1995) 213-326, hep-th/9412184.
18. [5] For recent reviews, see J. Schwarz, preprint CALTECH-68-2065, hep-th/9607201; M. Duff, preprint CPT-TAMU-33-96, hep-th/9608117; A. Sen, preprint MRI-PHY-96-28, hep-th/9609176.
19. [5] For recent reviews, see J. Schwarz, preprint CALTECH-68-2065, hep-th/9607201; M. Duff, preprint CPT-TAMU-33-96, hep-th/9608117; A. Sen, preprint MRI-PHY-96-28, hep-th/9609176.
20. [5] For recent reviews, see J. Schwarz, preprint CALTECH-68-2065, hep-th/9607201; M. Duff, preprint CPT-TAMU-33-96, hep-th/9608117; A. Sen, preprint MRI-PHY-96-28, hep-th/9609176.
21. [6] A. Sen and J. Schwarz, Phys. Lett. B 312 (1993) 105; Nucl. Phys. B 411 (1994) 35.
22. [6] A. Sen and J. Schwarz, Phys. Lett. B 312 (1993) 105; Nucl. Phys. B 411 (1994) 35.
23. [7] C.M. Hull and P.K. Townsend, Nucl. Phys. B 451 (1995) 525, hep-th/9505073; E. Witten, hep-th/9503124, Nucl. Phys. B 443 (1995) 85.
24. [7] C.M. Hull and P.K. Townsend, Nucl. Phys. B 451 (1995) 525, hep-th/9505073; E. Witten, hep-th/9503124, Nucl. Phys. B 443 (1995) 85.
25. [8] C.M. Hull and P.K. Townsend, hep-th/9410167, Nucl. Phys. B 438 (1995) 109.
26. [9] J.H. Schwarz, M-theory extension of T-duality, hep-th/9601077; C. Vafa, Evidence for F-theory, hep-th/9602022.
27. [9] J.H. Schwarz, M-theory extension of T-duality, hep-th/9601077; C. Vafa, Evidence for F-theory, hep-th/9602022.
28. [10] J. Polchinski, hep-th/9510017, Phys. Rev. Lett. 75 (1995) 4724.
29. [11] For a recent review, see J. Polchinski, TASI lectures on D-branes; J. Polchinski, S. Chaudhuri and C. Johnson, Notes on D-branes, hep-th/9602052.
30. [11] For a recent review, see J. Polchinski, TASI lectures on D-branes; J. Polchinski, S. Chaudhuri and C. Johnson, Notes on D-branes, hep-th/9602052.
31. [12] L. Andrianopoli, R. D'Auria, S. Ferrara, P. Fré and M. Trigiante, R-R scalars, U-duality and solvable Lie algebras, Nucl. Phys. B 496 (1997) 617, hep-th/9611014.
32. [13] L. Andrianopoli, R. D'Auria, S. Ferrara, P. Fré and M. Trigiante, Solvable Lie algebras in type IIA, type IIB and M-theories, Nucl. Phys. B 493 (1997) 249, hep-th/9612202.
33. [14] S. Helgason, Differential Geometry and Symmetric Spaces (Academic Press, New York, 1962).
34. [15] D.V. Alekseevski, Math. USSR Izv. 9 (1975) No. 2.
35. [16] E. Cremmer, in Supergravity '81, ed. S. Ferrara and J.G. Taylor, p. 313; B. Julia, in Superspace and Supergravity, ed. S.W. Hawking and M. Rocek (Cambridge Univ. Press, Cambridge, 1981) p. 331.
36. [16] E. Cremmer, in Supergravity '81, ed. S. Ferrara and J.G. Taylor, p. 313; B. Julia, in Superspace and Supergravity, ed. S.W. Hawking and M. Rocek (Cambridge Univ. Press, Cambridge, 1981) p. 331.
37. [17] S. Ferrara and R. Kallosh, Phys. Rev. D 54 (1996) 1525.
38. [18] K. Behrndt, G. Lopes Cardoso, B. de Wit, R. Kallosh, D. Lüst and T. Mohaupt, Classical and quantum N = 2 supersymmetric black holes, hep-th/9610105.
39. [19] K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
40. [20] M.J. Duff, J.T. Liu and J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125, hep-th/9508094; R. Kallosh, M. Shmakova and W.K. Wong, Freezing of moduli by N = 2 dyons, Phys. Rev. D 54 (1996) 6284, hep-th/9607077; K. Behrndt, R. Kallosh, J. Rahmfeld, M. Smachova and W.K. Wond, STU black holes and string triality, hep-th/9608059; G. Lopes Cardoso, D. Lust and T. Mohaupt, Modular symmetries of N = 2 black holes, Phys. Lett. B 388 (1996) 266, hep-th/9608099; K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
41. [20] M.J. Duff, J.T. Liu and J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125, hep-th/9508094; R. Kallosh, M. Shmakova and W.K. Wong, Freezing of moduli by N = 2 dyons, Phys. Rev. D 54 (1996) 6284, hep-th/9607077; K. Behrndt, R. Kallosh, J. Rahmfeld, M. Smachova and W.K. Wond, STU black holes and string triality, hep-th/9608059; G. Lopes Cardoso, D. Lust and T. Mohaupt, Modular symmetries of N = 2 black holes, Phys. Lett. B 388 (1996) 266, hep-th/9608099; K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
42. [20] M.J. Duff, J.T. Liu and J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125, hep-th/9508094; R. Kallosh, M. Shmakova and W.K. Wong, Freezing of moduli by N = 2 dyons, Phys. Rev. D 54 (1996) 6284, hep-th/9607077; K. Behrndt, R. Kallosh, J. Rahmfeld, M. Smachova and W.K. Wond, STU black holes and string triality, hep-th/9608059; G. Lopes Cardoso, D. Lust and T. Mohaupt, Modular symmetries of N = 2 black holes, Phys. Lett. B 388 (1996) 266, hep-th/9608099; K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
43. [20] M.J. Duff, J.T. Liu and J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125, hep-th/9508094; R. Kallosh, M. Shmakova and W.K. Wong, Freezing of moduli by N = 2 dyons, Phys. Rev. D 54 (1996) 6284, hep-th/9607077; K. Behrndt, R. Kallosh, J. Rahmfeld, M. Smachova and W.K. Wond, STU black holes and string triality, hep-th/9608059; G. Lopes Cardoso, D. Lust and T. Mohaupt, Modular symmetries of N = 2 black holes, Phys. Lett. B 388 (1996) 266, hep-th/9608099; K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
44. [20] M.J. Duff, J.T. Liu and J. Rahmfeld, Four dimensional string/string/string triality, Nucl. Phys. B 459 (1996) 125, hep-th/9508094; R. Kallosh, M. Shmakova and W.K. Wong, Freezing of moduli by N = 2 dyons, Phys. Rev. D 54 (1996) 6284, hep-th/9607077; K. Behrndt, R. Kallosh, J. Rahmfeld, M. Smachova and W.K. Wond, STU black holes and string triality, hep-th/9608059; G. Lopes Cardoso, D. Lust and T. Mohaupt, Modular symmetries of N = 2 black holes, Phys. Lett. B 388 (1996) 266, hep-th/9608099; K. Behrndt, D. Lust and W.A. Sabra, Stationary solutions of N = 2 supergravity, hep-th/9705169.
45. [21] S. Ferrara and J. Maldacena, Branes, central charges and U-duality invariant BPS conditions.
46. [22] L. Andrianopoli, M. Bertolini, A. Ceresole, R. D'Auria, S. Ferrara, P. Fré and T. Magri, N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance, gaugings and the momentum map, hep-th/9605032.
47. [23] P. Fré, Lectures on special Kahler geometry and electric-magnetic duality rotations, Nucl. Phys. B (Proc. Suppl.) 45BC (1996) 59.
48. [24] A. Ceresole, R. D'Auria, S. Ferrara and A. Van Proeyen, Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity, Nucl. Phys. B 444 (1995) 92.
49. [25] M.K. Gaillard and B. Zumino, Nucl. Phys. B 193 (1981) 221.
50. [26] B. de Wit and H. Nicolai, Phys. Lett. B 108 (1982) 285.
51. [27] P. Fré, Supersymmetry and first order equations for extremal states: monopoles, hyperinstantons, black holes and p-branes, Talk given at the Santa Margherita Workshop on Constrained Dynamics and Quantum Gravity, September 1996, hep-th/9701054.
52. [28] L. Andrianopoli, R. D'Auria and S. Ferrara, U-duality and central charges in various dimensions revisited, hep-th/9612105.
53. [29] L. Castellani, R. D'Auria and P. Fré, Supergravity and Superstrings: A Geometric Perspective (World Scientific, Singapore, 1991).
54. [30] P. Fré, U-duality, Solvable Lie algebras and extremal black Holes, Talk given at the B. Bertotti 65th birthday meeting, September 1996, hep-th/9702167.
55. [31] T. Ortin, Extremality versus supersymmetry in stringy black holes, hep-th/9612142.
56. [32] M. Cvetic and D. Youm, Singular BPS saturated states and enhanced symmetries of four-dimensional N = 4 supersymmetric string vacua, Phys. Lett. B 359 (1995) 87.
57. [33] M. Cvetic and A.A. Tseytlin, General class of BPS saturated dyonic black holes as exact superstring solutions, Phys. Lett. B 366 (1996) 95.
58. [34] J. Rahmfeld, Extremal black holes as bound states Phys. Lett. B 372 (1996) 198.
59. [35] S. Wolfram, MATHEMATICA (Addison-Wesley, New York, 1988).
60. [36] Humphrey, Introduction to Lie algebra and representation theory, GTM (Springer, Berlin).
61. [37] E. Witten, hep-th/9503124, Nucl. Phys. B 443 (1995) 85.
62. [38] S. Ferrara, M. Gunyadin and C.A. Savoy, unpublished.
63. [39] M. Cvetic and C.M. Hull, Black Holes and U-duality, Nucl. Phys. B 480 (1996) 296, hep-th/9606193; M. Cvetic and I. Gaida, Duality invariant non-extreme black holes in toroidally compactified string theory, hep-th/9703134; M. Cvetic and D. Youm, General static spherically symmetric black holes of heterotic string on a six torus, hep-th/9512127; M. Cvetic and A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619, hep-th/9512031.
64. [39] M. Cvetic and C.M. Hull, Black Holes and U-duality, Nucl. Phys. B 480 (1996) 296, hep-th/9606193; M. Cvetic and I. Gaida, Duality invariant non-extreme black holes in toroidally compactified string theory, hep-th/9703134; M. Cvetic and D. Youm, General static spherically symmetric black holes of heterotic string on a six torus, hep-th/9512127; M. Cvetic and A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619, hep-th/9512031.
65. [39] M. Cvetic and C.M. Hull, Black Holes and U-duality, Nucl. Phys. B 480 (1996) 296, hep-th/9606193; M. Cvetic and I. Gaida, Duality invariant non-extreme black holes in toroidally compactified string theory, hep-th/9703134; M. Cvetic and D. Youm, General static spherically symmetric black holes of heterotic string on a six torus, hep-th/9512127; M. Cvetic and A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619, hep-th/9512031.
66. [39] M. Cvetic and C.M. Hull, Black Holes and U-duality, Nucl. Phys. B 480 (1996) 296, hep-th/9606193; M. Cvetic and I. Gaida, Duality invariant non-extreme black holes in toroidally compactified string theory, hep-th/9703134; M. Cvetic and D. Youm, General static spherically symmetric black holes of heterotic string on a six torus, hep-th/9512127; M. Cvetic and A.A. Tseytlin, Solitonic strings and BPS saturated dyonic black holes, Phys. Rev. D 53 (1996) 5619, hep-th/9512031.