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An Introduction to Duality Symmetries in String Theory

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Unity from Duality: Gravity, Gauge Theory and Strings

Part of the book series: Les Houches - Ecole d’Ete de Physique Theorique ((LHSUMMER,volume 76))

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Abstract

In this review I discuss some basic aspects of non-perturbative string theory. The topics include test of duality symmetries based on the analysis of the low energy effective action and the spectrum of BPS states, relationship between different duality symmetries, and an introduction to M-theory.

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References

  1. A. Sen, Int. J. Mod. Phys. A 9 (1994) 3707 [hep-th/9402002].

    Article  MATH  ADS  Google Scholar 

  2. M. Duff, R. Khuri and J. Lu, Phys. Rep. 259 (1995) 213 [hep-th/9412184].

    Article  ADS  MathSciNet  Google Scholar 

  3. J. Schwarz, Nucl. Phys. Proc. Suppl. B 55(1) [hep-th/9607201].

    Google Scholar 

  4. S. Chaudhuri, C. Johnson and J. Polchinski [hep-th/9602052].

    Google Scholar 

  5. J. Polchinski, Rev. Mod. Phys. 68 (1996) 1245 [hep-th/9607050].

    Article  ADS  MathSciNet  Google Scholar 

  6. J. Polchinski [hep-th/9611050].

    Google Scholar 

  7. P. Townsend [hep-th/9612121; gr-qc/9707012; hep-th/9712004].

    Google Scholar 

  8. M. Duff [hep-th/9611203].

    Google Scholar 

  9. M. Douglas [hep-th/9610041].

    Google Scholar 

  10. W. Lerche [hep-th/9611190; hep-th/9710246].

    Google Scholar 

  11. S. Forste and J. Louis [hep-th/9612192].

    Google Scholar 

  12. C. Vafa [hep-th/9702201].

    Google Scholar 

  13. A. Klemm [hep-th/9705131].

    Google Scholar 

  14. E. Kiritsis [hep-th/9708130].

    Google Scholar 

  15. B. de Wit and J. Louis [hep-th/9801132].

    Google Scholar 

  16. M. Trigiante [hep-th/9801144].

    Google Scholar 

  17. S. Mukhi [hep-ph/9710470].

    Google Scholar 

  18. M. Green, J. Schwarz and E. Witten, Superstring Theory, Vols. 1 and 2 (Cambridge University Press, 1986); edited by D. Lust and S. Theisen, Lectures on String Theory (Springer, 1989); J. Polchinski [hep-th/9411028].

    Google Scholar 

  19. E. Witten, Nucl. Phys. B 443 (1995) 85 [hep-th/9503124].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. A. Dabholkar, Phys. Lett. B 357 (1995) 307 [hep-th/9506160].

    Article  ADS  MathSciNet  Google Scholar 

  21. C. Hull, Phys. Lett. B 357 (1995) 545 [hep-th/9506194].

    Article  ADS  MathSciNet  Google Scholar 

  22. J. Polchinski and E. Witten, Nucl. Phys. B 460 (1996) 525 [hep-th/9510169].

    Article  ADS  MathSciNet  Google Scholar 

  23. C. Hull and P. Townsend, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  24. M. Duff, Nucl. Phys. B 442 (1995) 47 [hep-th/9501030]. M. Duff and R. Khuri, Nucl. Phys. B 411 (1994) 473 [hep-th/9305142].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. A. Sen, Nucl. Phys. B 450 (1995) 103 [hep-th/9504027].

    Article  MATH  ADS  Google Scholar 

  26. J. Harvey and A. Strominger, Nucl. Phys. B 449 (1995) 535 [hep-th/9504047].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77 [hep-th/9401139].

    Article  ADS  MathSciNet  Google Scholar 

  28. P. Fre, Nucl. Phys. B [Proc. Sup.] 45B,C (1996) 59 [hep-th/9512043] and references therein.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. E. Witten and D. Olive, Phys. Lett. B 78 (1978) 97.

    Article  ADS  Google Scholar 

  30. M. de Roo, Nucl. Phys. B 255 (1985) 515.

    Article  ADS  Google Scholar 

  31. S. Ferrara, C. Kounnas and M. Porrati, Phys. Lett. B 181 (1986) 263.

    Article  ADS  MathSciNet  Google Scholar 

  32. M. Terentev, Sov. J. Nucl. Phys. 49 (1989) 713.

    MathSciNet  Google Scholar 

  33. S.F. Hassan and A. Sen, Nucl. Phys. B 375 (1992) 103 [hep-th/9109038].

    Article  ADS  MathSciNet  Google Scholar 

  34. J. Maharana and J. Schwarz, Nucl. Phys. B 390 (1993) 3 [hep-th/9207016].

    Article  ADS  MathSciNet  Google Scholar 

  35. A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A 6 (1991) 2677.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. A. Sen, Nucl. Phys. B 404 (1993) 109 [hep-th/9207053].

    Article  MATH  ADS  Google Scholar 

  37. A. Font, L. Ibanez, D. Lust and F. Quevedo, Phys. Lett. B 249 (1990) 35.

    Article  ADS  MathSciNet  Google Scholar 

  38. S.J. Rey, Phys. Rev. D 43 (1991) 526.

    Article  ADS  MathSciNet  Google Scholar 

  39. J. Schwarz [hep-th/9209125].

    Google Scholar 

  40. A. Sen, Phys. Lett. B 303 (1993) 22 [hep-th/9209016].

    Article  ADS  Google Scholar 

  41. A. Sen, Mod. Phys. Lett. A 8 (1993) 2023 [hep-th/9303057].

    Article  MATH  ADS  Google Scholar 

  42. J. Schwarz and A. Sen, Nucl. Phys. B 411 (1994) 35 [hep-th/9304154]; Phys. Lett. B 312 (1993) 105 [hep-th/9305185].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. N. Seiberg, Nucl. Phys. B 303 (1988) 286.

    Article  ADS  MathSciNet  Google Scholar 

  44. M. Green and J. Schwarz, Phys. Lett. B 122 (1983) 143; J. Schwarz and P. West, Phys. Lett. B 126 (1983) 301; J. Schwarz, Nucl. Phys. B 226 (1983) 269; P. Howe and P. West, Nucl. Phys. B 238 (1984) 181.

    Article  ADS  Google Scholar 

  45. E. Cremmer, in “Unification of Fundamental Particle Interactions” (Plenum, 1980); B. Julia, in “Superspace and Supergravity” (Cambridge Univ. Press, 1981).

    Google Scholar 

  46. S. Ferrara and C. Kounnas, Nucl. Phys. B 328 (1989) 406.

    Article  ADS  MathSciNet  Google Scholar 

  47. S. Chaudhuri, G. Hockney and J. Lykken, Phys. Rev. Lett. 75 (1995) 2264 [hep-th/9505054].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  48. S. Chaudhuri and J. Polchinski, Phys. Rev. D 52 (1995) 7168 [hep-th/9506048].

    Article  ADS  MathSciNet  Google Scholar 

  49. C. Montonen and D. Olive, Phys. Lett. B 72 (1977) 117.

    Article  ADS  Google Scholar 

  50. H. Osborn, Phys. Lett. B 83 (1979) 321.

    Article  ADS  Google Scholar 

  51. A. Sen, Phys. Lett. B 329 (1994) 217 [hep-th/9402032].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. M. Atiyah and N. Hitchin, Phys. Lett. A 107 (1985) 21; Phil. Trans. Roy. Soc. Lond. A 315 (1985) 459; Geometry and Dynamics of Magnetic Monopoles (Cambridge Univ. Press).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. G. Gibbons and N. Manton, Nucl. Phys. B 274 (1986) 183.

    Article  ADS  MathSciNet  Google Scholar 

  54. J. Gauntlett, Nucl. Phys. B 400 (1993) 103 [hep-th/9205008]; Nucl. Phys. B 411 (1994) 443 [hep-th/9305068].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  55. J. Blum, Phys. Lett. B 333 (1994) 92 [hep-th/9401133].

    Article  ADS  MathSciNet  Google Scholar 

  56. N. Manton and B. Schroers, Annals Phys. 225 (1993) 290.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  57. G. Gibbons and P. Ruback, Comm. Math. Phys. 115 (1988) 267.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  58. G. Segal and A. Selby, Comm. Math. Phys. 177 (1996) 775.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  59. M. Porrati, Phys. Lett. B 377 (1996) 67 [hep-th/9505187].

    Article  ADS  MathSciNet  Google Scholar 

  60. J. Gauntlett and J. Harvey [hep-th/9407111].

    Google Scholar 

  61. J. Schwarz, Phys. Lett. B 360 (1995) 13 [hep-th/9508143].

    Article  ADS  MathSciNet  Google Scholar 

  62. E. Witten, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  63. S. Das and S. Mathur, Phys. Lett. B 375 (1996) 103 [hep-th/9601152].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  64. A. Dabholkar and J. Harvey, Phys. Rev. Lett. 63 (1989) 478; A. Dabholkar, G. Gibbons, J. Harvey and F. Ruiz, Nucl. Phys. B 340 (1990) 33.

    Article  ADS  Google Scholar 

  65. J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 [hep-th/9510017].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  66. J. Dai, R. Leigh and J. Polchinski, Mod. Phys. Lett. A 4 (1989) 2073; R. Leigh, Mod. Phys. Lett. A 4 (1989) 2767; J. Polchinski, Phys. Rev. D 50 (1994) 6041 [hep-th/9407031].

    Article  ADS  MathSciNet  Google Scholar 

  67. M. Bershadsky, V. Sadov and C. Vafa, Nucl. Phys. B 463 (1996) 398 [hep-th/9510225].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  68. A. Sen, Phys. Rev. D 53 (1996) 2874 [hep-th/9711026].

    Article  ADS  MathSciNet  Google Scholar 

  69. C. Vafa, Nucl. Phys. B 469 (1996) 415 [hep-th/9511088]; Nucl. Phys. B 463 (1996) 435 [hep-th/9512078].

    Article  ADS  Google Scholar 

  70. S. Sethi and M. Stern, Phys. Lett. B 398 (1997) 47 [hep-th/9607145].

    Article  ADS  MathSciNet  Google Scholar 

  71. M. Bershadsky, V. Sadov and C. Vafa, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  72. E. Witten, Nucl. Phys. B 460 (1996) 541 [hep-th/9511030].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  73. M. Porrati, Phys. Lett. B 387 (1996) 492 [hep-th/9607082].

    Article  ADS  MathSciNet  Google Scholar 

  74. J. Blum, Nucl. Phys. B 506 (1997) 223 [hep-th/9705030].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  75. A. Sen [hep-th/9609176].

    Google Scholar 

  76. A. Sen, Nucl. Phys. B 434 (1995) 179 [hep-th/9408083]; Nucl. Phys. B 447 (1995) 62 [hep-th/9503057].

    Article  MATH  ADS  Google Scholar 

  77. C. Vafa and E. Witten [hep-th/9507050].

    Google Scholar 

  78. A. Sen, Nucl. Phys. B 474 (1996) 361 [hep-th/9604070].

    Article  MATH  ADS  Google Scholar 

  79. S. Kachru and C. Vafa, Nucl. Phys. B 450 ((1995) 69 [hep-th/9605105].

    Article  ADS  MathSciNet  Google Scholar 

  80. S. Ferrara, J. Harvey, A. Strominger and C. Vafa, Phys. Lett. B 361 (1995) 59 [hep-th/9505162].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  81. A. Klemm, W. Lerche and P. Mayr, Phys. Lett. B 357 (1995) 313 [hep-th/9506112].

    Article  ADS  MathSciNet  Google Scholar 

  82. G. Aldazabal, A. Font, L. Ibanez and F. Quevedo, Nucl. Phys. B 461 (1996) 537 [hep-th/9510093].

    Article  MathSciNet  Google Scholar 

  83. B. Hunt and R. Schimmrigk, Phys. Lett. B 381 (1996) 427 [hep-th/9512138]; B. Hunt, M. Lynker and R. Schimmtigk [hep-th/9609082].

    Article  ADS  MathSciNet  Google Scholar 

  84. B. de Wit, P. Lauwers and A. van Proeyen, Nucl. Phys. B 255 (1985) 569.

    Article  ADS  Google Scholar 

  85. V. Kaplunovsky, J. Louis and S. Theisen, Phys. Lett. B 357 (1995) 71 [hep-th/9506110].

    Article  ADS  MathSciNet  Google Scholar 

  86. I. Antoniadis, E. Gava, K. Narain and T. Taylor, Nucl. Phys. B 455 (1995) 109 [hep-th/9507115].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  87. S. Kachru, A. Klemm, W. Lerche, P. Mayr and C. Vafa, Nucl. Phys. B 459 (1996) 537 [hep-th/9508155]; A. Klemm, W. Lerche, P. Mayr, C. Vafa and N. Warner, Nucl. Phys. B 477 (1996) 746 [hep-th/9604034].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  88. N. Seiberg and E. Witten, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087]; Nucl. Phys. B 431 (1994) 484 [hep-th/9408099].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  89. P. Townsend, Phys. Lett. B 350 (1995) 184 [hep-th/9501068].

    Article  ADS  MathSciNet  Google Scholar 

  90. F. Giani and M. Pernici, Phys. Rev. D 30 (1984) 325; I. Campbell and P. West, Nucl. Phys. B 243 (1984) 112; M. Huq and M. Namazie, Class. Quant. Grav. 2 (1985) 293.

    Article  ADS  MathSciNet  Google Scholar 

  91. S. Sethi and M. Stern [hep-th/9705046].

    Google Scholar 

  92. A. Sen, Phys. Rev. D 54 (1996) 2964 [hep-th/9510229].

    Article  ADS  MathSciNet  Google Scholar 

  93. J. Schwarz, Phys. Lett. B 367 (1996) 97 [hep-th/9510086].

    Article  ADS  MathSciNet  Google Scholar 

  94. P. Aspinwall, Nucl. Phys. Proc. Suppl. 46 (1996) 30 [hep-th/9508154].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  95. P. Horava and E. Witten, Nucl. Phys. B 460 (1996) 506 [hep-th/9510209]; Nucl. Phys. B 475 (1996) 94 [hep-th/9603142].

    Article  ADS  MathSciNet  Google Scholar 

  96. K. Dasgupta and S. Mukhi, Nucl. Phys. B 465 (1996) 399 [hep-th/9512196].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  97. E. Witten, Nucl. Phys. B 463 (1996) 383 [hep-th/9512219].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  98. A. Sen, Mod. Phys. Lett. A 11 (1996) 1339 [hep-th/9603113].

    Article  MATH  ADS  Google Scholar 

  99. E. Caceres, V. Kaplunovsky and M. Mandelberg, Nucl. Phys. B 493 (1997) 73 [hep-th/9606036].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  100. A. Sagnotti, Open Strings and their Symmetry Groups (Talk at Cargese Summer Inst., 1987); G. Pradisi and A. Sagnotti, Phys. Lett. B 216 (1989) 59; M. Bianchi, G. Pradisi and A. Sagnotti, Nucl. Phys. B 376 (1992) 365; P. Horava, Nucl. Phys. B 327 (1989) 461; Phys. Lett. B 231 (1989) 251.

    Google Scholar 

  101. E. Gimon and J. Polchinski, Phys. Rev. D 54 (1996) 1667 [hep-th/9601038].

    Google Scholar 

  102. A. Chamseddine, Phys. Rev. D 24 (1981) 3065; E. Bergshoeff, M. de Roo, B. de Wit and P. van Niewenhuizen, Nucl. Phys. B 195 (1982) 97; E. Bergshoeff, M. de Roo and B. de Wit, Nucl. Phys. B 217 (1983) 143; G. Chapline and N. Manton, Phys. Lett. B 120 (1983) 105.

    Article  ADS  MathSciNet  Google Scholar 

  103. H. Nicolai, Phys. Lett. B 276 (1992) 333.

    Article  ADS  MathSciNet  Google Scholar 

  104. E. Weinberg, Phys. Rev. D 20 (1979) 936; E. Corrigan and P. Goddard, Comm. Math. Phys. 80 (1981) 575; C. Taubes, Comm. Math. Phys. 91 (1983) 235.

    Article  ADS  Google Scholar 

  105. N. Manton, Phys. Lett. B 110 (1982) 54.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  106. C. Callias, Comm. Math. Phys. 62 (1978) 213.

    Article  MATH  ADS  MathSciNet  Google Scholar 

  107. K. Narain, Phys. Lett. B 169 (1986) 41; K. Narain, H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369.

    Article  ADS  MathSciNet  Google Scholar 

  108. T. Banks, L. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B 299 (1988) 613; T. Banks and L. Dixon, Nucl. Phys. B 307 (1988) 93.

    Article  ADS  MathSciNet  Google Scholar 

  109. B. Greene [hep-th/9702155] and references therein.

    Google Scholar 

  110. B. de Wit and A. van Proeyen, Nucl. Phys. B 245 (1984) 89; E. Cremmer, C. Kounnas, A. van Proeyen, J. Derendinger, S. Ferrara, B. de Wit and L. Girardello, Nucl. Phys. B 250 (1985) 385; S. Ferrara and A. Strominger, in “Strings 89”, World Scientific (1989); A. Strominger, Comm. Math. Phys. 133 (1990) 163.

    Article  ADS  Google Scholar 

  111. J. Bagger and E. Witten, Nucl. Phys. B 222 (1983) 1; K. Galicki, Comm. Math. Phys. 108 (1987) 117.

    Article  ADS  MathSciNet  Google Scholar 

  112. E. Cremmer and B. Julia, Nucl. Phys. B 159 (1979) 141.

    Article  ADS  MathSciNet  Google Scholar 

  113. M. Gaillard and B. Zumino, Nucl. Phys. B 193 (1981) 221; S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B 121 (1977) 393; E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. B 74 (1978) 61; B. de Wit, Nucl. Phys. B 158 (1979) 189; B. de Wit and H. Nicolai, Nucl. Phys. B 208 (1982) 323; E. Cremmer and B. Julia, Phys. Lett. B 80 (1978) 48; B. de Wit and A. van Proeyen, Nucl. Phys. B 245 (1984) 89.

    Article  ADS  MathSciNet  Google Scholar 

  114. D. Gross, J. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B 256 (1985) 253; Nucl. Phys. B 267 (1986) 75.

    Article  ADS  MathSciNet  Google Scholar 

  115. M. Green and J. Schwarz, Phys. Lett. B 109 (1982) 444; Phys. Lett. B 149 (1984) 117; Phys. Lett. B 151 (1985) 21; Nucl. Phys. B 255 (1985) 93.

    Article  ADS  Google Scholar 

  116. C. Hull and P. Townsend, Nucl. Phys. B 451 (1995) 525 [hep-th/9505073].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  117. A. Strominger, Nucl. Phys. B 451 (1995) 96 [hep-th/9504090]; B. Greene, D. Morrison and A. Strominger, Nucl. Phys. B 451 (1995) 109 [hep-th/9504145].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  118. M. Douglas and G. Moore [hep-th/9603167]; J. Polchinski, Phys. Rev. B 55 (1997) 6423 [hep-th/9606165]; C. Johnson and R. Myers, Phys. Rev. D 55 (1997) 6382 [hep-th/9610140]; M. Douglas [hep-th/9612126] published in JHEP electronics journal; D. Diaconescu, M. Douglas and J. Gomis [hep-th/9712230].

    Google Scholar 

  119. E. Witten [hep-th/9507121].

    Google Scholar 

  120. M. Douglas [hep-th/9512077; hep-th/9604198].

    Google Scholar 

  121. L. Alvarez-Gaume, D. Freedman and S. Mukhi, Ann. Phys. (NY) 134 (1981) 85; L. Alvarez-Gaume and D. Freedman, Comm. Math. Phys. 80 (1981) 443.

    Article  ADS  Google Scholar 

  122. E. Bergshoeff, E. Sezgin and P. Townsend, Phys. Lett. B 189 (1987) 75; Ann. Phys. (NY) 185 (1988) 330; M. Duff, Class. Quant. Grav. 5 (1988) 189; B. de Wit, M. Luscher and H. Nicolai, Nucl. Phys. B 320 (1989) 135; B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B 305 (1988) 545.

    Article  ADS  MathSciNet  Google Scholar 

  123. P. Ramond, Phys. Rev. D 3 (1971) 2415.

    Article  ADS  MathSciNet  Google Scholar 

  124. A. Neveu and J. Schwarz, Nucl. Phys. B 31 (1971) 86; Phys. Rev. D 4 (1971) 1109.

    Article  ADS  Google Scholar 

  125. F. Gliozzi, J. Scherk and D. Olive, Phys. Lett. B 65 (1976) 282; Nucl. Phys. B 122 (1977) 253.

    Article  ADS  Google Scholar 

  126. P. Aspinwall and D. Morrison [hep-th/9404151]; P. Aspinwall [hep-th/9611137] and references therein.

    Google Scholar 

  127. M. Duff and R. Minasian, Nucl. Phys. B 436 (1995) 507 [hep-th/9406198]; C. Vafa and E. Witten, Nucl. Phys. B 447 (1995) 261 [hep-th/9505053]; M. Duff, J. Liu and R. Minasian, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126]; M. Duff, R. Minasian and E. Witten, Nucl. Phys. B 463 (1996) 435 [hep-th/9601036]; M. Berkooz, R. Leigh, J. Polchinski, J. Schwarz, N. Seiberg and E. Witten, Nucl. Phys. B 475 (1996) 115 [hep-th/9605184].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  128. P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46.

    Article  ADS  MathSciNet  Google Scholar 

  129. E. Gimonand, C. Johnson, Nucl. Phys. B 479 (1996) 285 [hep-th/9606176]; J. Blum and A. Zaffaroni, Phys. Lett. B 387 (1996) 71 [hep-th/9607019]; A. Dabholkar and J. Park, Phys. Lett. B 394 (1997) 302 [hep-th/9607041]; R. Gopakumar and S. Mukhi, Nucl. Phys. B 479 (1996) 260 [hep-th/9607057]; J. Park [hep-th/9611119]; A. Sen, Nucl. Phys. B 489 (1997) 139 [hep-th/9611186]; Nucl. Phys. B 498 (1997) 135 [hep-th/9702061]; Phys. Rev. D 55 (1997) 7345 [hep-th/9702165]; O. Aharony, J. Sonneschein, S. Yankielowicz and S. Theisen, Nucl. Phys. B 493 (1997) 177 [hep-th/9611222].

    Article  ADS  Google Scholar 

  130. M. Bershadsky, K. Intrilligator, S. Kachru, D. Morrison, V. Sadov and C. Vafa, Nucl. Phys. B 481 (1996) 215 [hep-th/9605200]; P. Aspinwall and M. Gross, Phys. Lett. B 387 (1996) 735 [hep-th/9605131].

    Article  ADS  Google Scholar 

  131. K. Dasgupta and S. Mukhi, Phys. Lett. B 385 (1996) 125 [hep-th/9606044].

    Article  ADS  MathSciNet  Google Scholar 

  132. A. Johansen, Phys. Lett. B 395 (1997) 36 [hep-th/9608186].

    Article  ADS  MathSciNet  Google Scholar 

  133. M. Gaberdiel and B. Zwiebach [hep-th/9709013]; M. Gaberdiel, T. Hauer and B. Zwiebach [hep-th/9801205].

    Google Scholar 

  134. E. Witten, Nucl. Phys. B 500 (1997) 3 [hep-th/9703166].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  135. A. Sen [hep-th/9707123] published in JHEP electronic journal.

    Google Scholar 

  136. E. Witten, Nucl. Phys. B 471 (1996) 135 [hep-th/9602070].

    Article  MATH  ADS  MathSciNet  Google Scholar 

  137. A. Cadavid, A. Ceresole, R. D’Auria and S. Ferrara [hep-th/9506144]; I. Antoniadis, S. Ferrara and T. Taylor, Nucl. Phys. B 460 (1996) 489; S. Ferrara, R. Khuri and R. Minasian [hep-th/9602102]; E. Witten, Nucl. Phys. B 471 (1996) 195 [hep-th/9603150].

    Google Scholar 

  138. M. Duff, B. Nilsson and C. Pope, Phys. Rep. 130 (1986) 1.

    Article  ADS  MathSciNet  Google Scholar 

  139. J. Cardy, Nucl. Phys. B 270 (1986) 186.

    Article  MATH  ADS  MathSciNet  Google Scholar 

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C. Bachas A. Bilal M. Douglas N. Nekrasov F. David

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Sen, A. (2002). An Introduction to Duality Symmetries in String Theory. In: Bachas, C., Bilal, A., Douglas, M., Nekrasov, N., David, F. (eds) Unity from Duality: Gravity, Gauge Theory and Strings. Les Houches - Ecole d’Ete de Physique Theorique, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36245-2_3

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