The Geometrical Principles of String Theory Constructions

  • Jan Govaerts
Part of the NATO ASI Series book series (NSSB, volume 223)


In their quest for a quantum consistent unified theory of all fundamental interactions among elementary particles, theoretical particle physicists in recent years have been entertaining the idea1 that such a theory may be constructed within the framework of superstring theories1. As opposed to the previous attempts in the past fifteen years, all formulated within the framework of ordinary quantum field theory, superstring theories seem to possess the very same desirable properties that stimulated previous attempts as well as the resolution of problems which led to the dismissal of these attempts2,3,4.


Open String Closed String Conformal Field Theory Bosonic String Ghost Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    For many references to original work, see “Superstrings, The First 15 Years of Superstring Theory”, ed. J.H. Schwarz (World Scientific, Singapore, 1985), 2 volumes.Google Scholar
  2. 2.
    P. Candeias, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985)46.ADSCrossRefGoogle Scholar
  3. 3.
    E. Witten, Phys.Lett. B155 (1985) 151;MathSciNetADSGoogle Scholar
  4. 3a.
    E. Witten, Nucl. Phys. B258 (1985) 75.MathSciNetADSCrossRefGoogle Scholar
  5. 3b.
    M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B156 (1985) 55.MathSciNetADSGoogle Scholar
  6. 3c.
    J.- P. Derendinger, L.E. Ibánez and H.P. Nilles, Nucl. Phys. B267 (1986) 365.ADSCrossRefGoogle Scholar
  7. 3d.
    N. Seiberg, in Proc. 3rd Jerusalem Winter School, 1985, eds. T. Piran and S. Weinberg (World Scientific, Singapore, 1987).Google Scholar
  8. 3e.
    B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Nucl.Phys. B278 (1986) 667;MathSciNetADSCrossRefGoogle Scholar
  9. 3f.
    B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Nucl. Phys. B292 (1987) 606;ADSCrossRefGoogle Scholar
  10. 3g.
    B.R. Greene, K.H. Kirklin, P.J. Miron and G.G. Ross, Phys. Lett. B180 (1986) 69.MathSciNetADSGoogle Scholar
  11. 4.
    For a recent discussion and references, see: G.G. Ross, Lectures at the Banff Summer Institute (CAP) on Particles and Fields, 1988, preprint CERN-TH.5109/88 (December 1988).Google Scholar
  12. 5.
    M.B. Green and J.H. Schwarz, Phys. Lett. B149 (1984) 117;MathSciNetADSGoogle Scholar
  13. 5a.
    M.B. Green and J.H. Schwarz, Nucl. Phys. B255 (1985) 93;MathSciNetADSCrossRefGoogle Scholar
  14. 5b.
    M.B. Green and J.H. Schwarz, Phys. Lett. B151 (1985) 21.MathSciNetADSGoogle Scholar
  15. 6.
    A. Font, L.E. Ibáñez, F. Quevedo and A. Sierra, preprint CERN-TH.5326/89, LAPP-TH.241/89 (March 1989), and references therein.Google Scholar
  16. 7.
    J.H. Schwarz, Int. J. Mod.Phys. A2 (1987) 593.ADSGoogle Scholar
  17. 8.
    See the Lectures by G. Altarelli, J.J Aubert , J.D. Bjorken, G. Feldman, F. Pauss, J. Steinberg and J. Stirling in these Proceedings.Google Scholar
  18. 9.
    See the Lectures by C. Itzykson in these Proceedings.Google Scholar
  19. 10.
    E. Witten, Comm. Math. Phys. 121 (1989) 351;MathSciNetADSMATHCrossRefGoogle Scholar
  20. 10a.
    E. Witten Nucl. Phys. B322 (1989) 629; and references therein.MathSciNetADSCrossRefGoogle Scholar
  21. 11.
    For a review and references , see: E. D’Hoker and D.H. Phong, Rev. Mod. Phys. 60 (1988) 917.MathSciNetADSCrossRefGoogle Scholar
  22. 12.
    L. Alvarez-Gaumé, P. Nelson, C. Gomez, G. Sierra and C. Vafa, Nucl.Phys. B311 (1988) 333; and references therein.ADSCrossRefGoogle Scholar
  23. 13.
    E. Witten, Nucl. Phys. B268 (1986) 253;MathSciNetADSCrossRefGoogle Scholar
  24. 13a.
    E. Witten, Nucl. Phys. B276 (1986) 291;MathSciNetADSCrossRefGoogle Scholar
  25. 13b.
    E. Witten, H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Lett. B172 (1986) 186, 195;ADSGoogle Scholar
  26. 13c.
    E. Witten, H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34 (1986) 2360;ADSGoogle Scholar
  27. 13d.
    E. Witten, H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Nucl. Phys. B283 (1987) 433;ADSGoogle Scholar
  28. 13e.
    A. Neveu and P. West, Nucl. Phys. B278 (1986) 601;MathSciNetADSCrossRefGoogle Scholar
  29. 13f.
    M. Kaku, Int. J. Mod. Phys. A2 (1987) 1; and references therein.MathSciNetADSGoogle Scholar
  30. 14.
    J. Scherk, Rev. Mod. Phys. 47 (1975) 123.MathSciNetADSCrossRefGoogle Scholar
  31. 14a.
    J. H. Schwarz, Physics Reports 89 (1982) 223.MathSciNetADSMATHCrossRefGoogle Scholar
  32. 15.
    M.B. Green, J.H. Schwarz and E. Witten, “Superstring Theory” (Cambridge, University, Cambridge, 1987) , 2 volumes.MATHGoogle Scholar
  33. 16.
    S. Shenker, in “Unified String Theories”, eds. M. Green and D. Gross (World Scientific , Singapore, 1986), p.141–161;Google Scholar
  34. 16a.
    D. Friedan, in “Unified String Theories”, eds. M. Green and D. Gross (World Scientific, Singapore, 1986), p. 162–213;Google Scholar
  35. 16b.
    D. Friedan, E. Martinec and S. Shenker, Nucl.Phys. B271 (1986)93;MathSciNetADSGoogle Scholar
  36. 16c.
    M. Peskin, Lectures at the 3rd Advanced Study Institute in Elementary Particle Physics, Santa-Cruz, June 23- July 19, 1986, preprint SLAC-PUB-4251 (March 1987)Google Scholar
  37. 17.
    J. Govaerts, in Proc. 2nd Mexican School of Particles and Fields, Cuernavaca-Morelos, 1986, eds. J.-L. Lucio and A. Zepeda (World Scientific, Singapore, 1987), p. 247–442.Google Scholar
  38. 18.
    J. Govaerts, Lectures at the 1st International Workshop on Mathematical Physics, Bujumbura, September 28-October 10, 1987, preprint CERN-TH.4953/88 (January 1988).Google Scholar
  39. 19.
    J. Govaerts, in Proc. X Autumn School “Physics Beyond the Standard Model”, Lisbon, October 10–14, 1988, eds. G.C. Branco and J.C. Romão (Nuclear Physics B- Proceedings Supplements Section), to appear; preprint CERN-TH.5306/89 (February 1989).Google Scholar
  40. 20.
    J. Govaerts, Int. J. Mod. Phys. A4 (1989) 173.MathSciNetADSGoogle Scholar
  41. 21.
    Y. Nambu, Lecture at the Copenhagen Symposium, 1970, unpublishedGoogle Scholar
  42. 21a.
    T. Goto, Prog. Theor. Phys. 46 (1971) 1560.MathSciNetADSMATHCrossRefGoogle Scholar
  43. 22.
    A.M. Polyakov, Phys. Lett. B103 (1981) 207, 211.MathSciNetADSGoogle Scholar
  44. 23.
    S. Deser and B. Zumino, Phys. Lett. B65 (1976) 369.MathSciNetADSGoogle Scholar
  45. 23a.
    L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B65 (1976) 471.ADSGoogle Scholar
  46. 24.
    J. Polchinski, Comm. Math. Phys. 104 (1986) 37.MathSciNetADSMATHCrossRefGoogle Scholar
  47. 25.
    For recent reviews and references , see: A.A. Tseythin, Int. J. Mod. Phys. A4 (1989) 1257ADSGoogle Scholar
  48. 25a.
    For recent reviews and references , see: A.A. Tseythin, Int. J. Mod. Phys. A4 (1989) 4279; Trieste preprint ICTP-IC/89/90.ADSGoogle Scholar
  49. 26.
    M. Kato and K. Ogawa, Nucl. Phys. B212 (1983) 443.ADSCrossRefGoogle Scholar
  50. 27.
    A. Bilal and J.L. Gervais, Nucl. Phys. B284 (1987) 397;MathSciNetADSCrossRefGoogle Scholar
  51. 27a.
    A. Bilal and J.L. Gervais, Phys. Lett. B187 (1987) 39;MathSciNetADSGoogle Scholar
  52. 27b.
    A. Bilal and J.L. Gervais, Nucl. Phys. B293 (1987) 1.MathSciNetADSCrossRefGoogle Scholar
  53. 28.
    A.M. Polyakov, Mod. Phys. Lett. A2 (1987) 893;MathSciNetADSGoogle Scholar
  54. 28a.
    V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988)819MathSciNetADSGoogle Scholar
  55. 28b.
    A. M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 1213;MathSciNetADSGoogle Scholar
  56. 28c.
    A.M. Chamseddine and M. Reuter, Nucl. Phys. B317 (1989) 757;MathSciNetADSCrossRefGoogle Scholar
  57. 28d.
    K. Fujikawa, N. Nakazawa and H. Suzuki, Phys. Lett. B221 (1989) 289;ADSGoogle Scholar
  58. 28e.
    Y. Matsuo, Phys. Lett. B227 (1989) 209;MathSciNetADSGoogle Scholar
  59. 28f.
    P. Forgács, A. Wipf, J. Balog, L. Fehér and L. O’Raifeartaigh, Phys. Lett. B227 (1989)214ADSGoogle Scholar
  60. 28g.
    J. Distler, Z. Hlousek and H. Kawai, Cornell preprints CLNS 88/854, CLNS 88/871; and references therein.Google Scholar
  61. 29.
    P. Goddard, A.J. Hanson and G. Ponzano, Nucl. Phys. B89 (1975) 76.ADSCrossRefGoogle Scholar
  62. 30.
    P. Goddard, J. Goldstone, C. Rebbi and C.B. Thorn, Nucl. Phys. B56 (1973) 109.ADSCrossRefGoogle Scholar
  63. 31.
    P.A.M. Dirac, “Lectures on Quantum Mechanics” (Yeshiva University, New York, 1964).Google Scholar
  64. 32.
    J. Govaerts, “Hamiltonian Quantization and Constrained Dynamics” (Leuven University Press), in preparation.Google Scholar
  65. 33.
    P.A.M. Dirac, Phys. Rev. 114 (1959) 924.MathSciNetADSMATHCrossRefGoogle Scholar
  66. 33a.
    L.D. Faddeev, Theor. Math. Phys. 1 (1969) 3.MathSciNetCrossRefGoogle Scholar
  67. 34.
    K. Sundermeyer, “Constrained Dynamics”, Lecture Notes in Physics 169 (Springer Verlag, Berlin, 1982).MATHGoogle Scholar
  68. 35.
    See for example P. Goddard and D. Olive, Int. J. Mod. Phys. A1 (1986) 303.MathSciNetADSGoogle Scholar
  69. 36.
    R.C. Brower, Phys. Rev. D6 (1972) 1655ADSGoogle Scholar
  70. 36a.
    P. Goddard and C.B. Thorn, Phys. Lett. B40 (1972) 235.ADSGoogle Scholar
  71. 37.
    C. Lovelace, Phys. Lett. B34 (1971) 500.ADSGoogle Scholar
  72. 38.
    J. Govaerts, Int. J. Mod. Phys. A4 (1989) 4353.MathSciNetADSGoogle Scholar
  73. 39.
    T. Yoneya, Progr. Theor. Phys. 51 (1974) 1907;ADSCrossRefGoogle Scholar
  74. 39a.
    J. Scherk and J.H. Schwarz, Nucl. Phys. B81 (1974) 118.ADSCrossRefGoogle Scholar
  75. 40.
    L. Brink and H.B. Nielsen, Phys. Lett. B45 (1973) 332.ADSGoogle Scholar
  76. 41.
    L.D. Faddeev and V.N. Popov, Phys. Lett. B25 (1967) 29; and references therein.ADSGoogle Scholar
  77. 42.
    M. Henneaux, Physics Reports 126 (1985) 1.MathSciNetADSCrossRefGoogle Scholar
  78. 43.
    I.A. Batalin and E.S. Fradkin, Riv. Nuovo Cim. 9 (1986) 1; and references therein.MathSciNetADSCrossRefGoogle Scholar
  79. 44.
    I.A. Batalin and G.A. Vilkovisky, Phys. Rev. D28 (1983) 2567; and references therein.MathSciNetADSGoogle Scholar
  80. 45.
    C. Becchi, A. Rouet and R. Stora, Phys. Lett. B52 (1974) 344;ADSGoogle Scholar
  81. 45a.
    C. Becchi, A. Rouet and R. Stora Ann. Phys. 98 (1976) 287MathSciNetADSCrossRefGoogle Scholar
  82. 45b.
    C. Becchi, A. Rouet and R. Stora I.V. Tyutin, Lebedev preprint FIAN No.39 (1975), unpublished.Google Scholar
  83. 46.
    J. Govaerts, Int. J. Mod. Phys. A4 (1989) 4487.MathSciNetADSGoogle Scholar
  84. 47.
    T.P. Killingback, Comm. Math. Phys. 100 (1985) 267;MathSciNetADSMATHCrossRefGoogle Scholar
  85. 47a.
    M.A. Solov’ev, JETP Lett. 44 (1986) 469;MathSciNetADSGoogle Scholar
  86. 47b.
    Z. Jaskolski, J. Math. Phys. 29 (1988) 1035.MathSciNetADSCrossRefGoogle Scholar
  87. 48.
    V.N. Gribov, Nucl. Phys. B139 (1978) 1.MathSciNetADSCrossRefGoogle Scholar
  88. 48a.
    I.M. Singer, Comm. Math. Phys. 60 (1978) 7.MathSciNetADSMATHCrossRefGoogle Scholar
  89. 49.
    S. Hwang, Phys. Rev. D28 (1983) 2614.ADSGoogle Scholar
  90. 50.
    M.D. Freeman and D.I. Olive, Phys. Lett. B175 (1986) 151;MathSciNetADSGoogle Scholar
  91. 50a.
    M. Henneaux, Phys. Lett. B177 (1986) 35.MathSciNetADSGoogle Scholar
  92. 51.
    P. Ramond, Phys. Rev. D3 (1971) 2415.MathSciNetADSGoogle Scholar
  93. 52.
    A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86.ADSCrossRefGoogle Scholar
  94. 53.
    M.B. Green and J.H. Schwarz, Phys. Lett. B109 (1982) 444;ADSGoogle Scholar
  95. 53a.
    M.B. Green and J.H. Schwarz, Phys. Lett. B136 (1984) 367;ADSGoogle Scholar
  96. 53b.
    M.B. Green and J.H. Schwarz, Nucl. Phys. B243 (1984) 285.MathSciNetADSCrossRefGoogle Scholar
  97. 54.
    F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys. B122 (1977) 253.ADSCrossRefGoogle Scholar
  98. 55.
    N. Seiberg and E. Witten, Nucl. Phys. B276 (1986) 272.MathSciNetADSCrossRefGoogle Scholar
  99. 56.
    R. Nepomechie, Phys. Lett. B178 (1986) 207;MathSciNetADSGoogle Scholar
  100. 56a.
    K. Arakawa and M. Maeno, Progr. Theor. Phys. 77 (1987) 145.MathSciNetADSCrossRefGoogle Scholar
  101. 57.
    R. Kallosh, Phys. Lett. B224 (1989) 273;MathSciNetADSGoogle Scholar
  102. 57a.
    R. Kallosh, Phys. Lett. B225 (1989) 49;MathSciNetADSGoogle Scholar
  103. 57b.
    S.J. Gates, M.T. Grisaru, M. Rocek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B225 (1989) 44;ADSGoogle Scholar
  104. 57c.
    M.B. Green and C.H. Hull, Phys. Lett. B225 (1989) 57.MathSciNetADSGoogle Scholar
  105. 58.
    A.H. Chamseddine and J.-P. Derendinger, Nucl. Phys. B301 (1988) 381;MathSciNetADSCrossRefGoogle Scholar
  106. 58a.
    A.H. Chamseddine, J.-P. Derendinger and M. Quirós, Nucl. Phys. B311 (1988) 140;ADSCrossRefGoogle Scholar
  107. 58b.
    A.H. Chamseddine, J.-P. Derendinger and M. Quirós, Phys. Lett. B227 (1989) 41.ADSGoogle Scholar
  108. 59.
    P. Goddard, C. Rebbi and C.B. Thorn, Nuovo Cim. 12 (1972) 425;MathSciNetADSCrossRefGoogle Scholar
  109. 59a.
    R.C. Brower and K.A. Friedman, Phys. Rev. D7 (1973) 535;ADSGoogle Scholar
  110. 59b.
    J.H. Schwarz, Nucl. Phys. B46 (1972) 61.ADSCrossRefGoogle Scholar
  111. 60.
    Y. Iwasaki and K. Kikkawa, Phys. Rev. D8 (1973) 440.ADSGoogle Scholar
  112. 61.
    M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502.ADSCrossRefGoogle Scholar
  113. 62.
    W. Lerche, D. Lust and A.N. Schellekens, Nucl. Phys. B287 (1987) 477.ADSCrossRefGoogle Scholar
  114. 63.
    For a review and references, see W. Lerche, A.N. Schellekens and N.P. Warner, Physics Reports 177 (1989) 1.MathSciNetADSCrossRefGoogle Scholar
  115. 64.
    N. Ohta , Phys. Lett. B179 (1986) 347;ADSGoogle Scholar
  116. 64a.
    N. Ohta , Phys. Rev. D33 (1986) 1681.ADSGoogle Scholar
  117. 65.
    M. Henneaux, Phys. Lett. B183 (1987) 59.MathSciNetADSGoogle Scholar
  118. 66.
    D. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B256 (1985) 253MathSciNetADSCrossRefGoogle Scholar
  119. 66a.
    D. Gross, J. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B267 (1986) 75.MathSciNetADSCrossRefGoogle Scholar
  120. 67.
    K. Narain, Phys. Lett. B169 (1986) 41MathSciNetADSGoogle Scholar
  121. 67a.
    K. Narain, M. Sarmadi and E. Witten, Nucl.Phys. B279 (1986) 369.MathSciNetADSGoogle Scholar
  122. 68.
    J.H. Schwarz , Int. J. Mod. Phys. A4 (1989) 2653.ADSGoogle Scholar
  123. 69.
    D. Friedan, Z. Qiu and S. Schenker, Phys. Rev. Lett. 52 (1984) 1575;ADSCrossRefGoogle Scholar
  124. 69a.
    D. Friedan, Z. Qiu and S. Schenker,Comm. Math. Phys. 107 (1986) 535.MathSciNetADSMATHCrossRefGoogle Scholar
  125. 70.
    M. Ademollo et al., Nucl.Phys. B111 (1976) 77;ADSCrossRefGoogle Scholar
  126. 70a.
    M. Ademollo et al., Nucl. Phys. B114 (1976) 297ADSCrossRefGoogle Scholar
  127. 70b.
    M. Ademollo et al., Phys. Lett. B62 (1976) 105ADSGoogle Scholar
  128. 70c.
    L. Brink and J.H. Schwarz, Nucl. Phys. B121 (1977) 285.MathSciNetADSCrossRefGoogle Scholar
  129. 71.
    J.E. Paton and H.M. Chan, Nucl. Phys. B10 (1969) 516.ADSCrossRefGoogle Scholar
  130. 72.
    J.H. Schwarz, Physics Reports 89 (1982) 223.MathSciNetADSMATHCrossRefGoogle Scholar
  131. 73.
    N. Marcus and A. Sagnotti, Phys. Lett. B119 (1982) 97.MathSciNetADSGoogle Scholar
  132. 74.
    G. Veneziano, Nuovo Cim. A57 (1968) 190.ADSCrossRefGoogle Scholar
  133. 75.
    M.A. Virasoro, Phys. Rev. 177 (1969) 2309;MathSciNetADSCrossRefGoogle Scholar
  134. 75a.
    J.A. Shapiro, Phys. Lett. B33 (1970) 361.ADSGoogle Scholar
  135. 76.
    M. Hayashi, N. Kawamoto, T. Kuramoto and K. Shigemoto, Nucl. Phys. B294 (1988) 459MathSciNetADSGoogle Scholar
  136. 76a.
    M. Hayashi, N. Kawamoto, T. Kuramoto and K. Shigemoto, Nucl. Phys. B296 (1988) 373.MathSciNetADSCrossRefGoogle Scholar
  137. 77.
    J. Govaerts, Phys. Lett. B220 (1989) 77;MathSciNetADSGoogle Scholar
  138. 77a.
    J. Govaerts, preprint CERN-TH.5149/88 (August 1988), to appear in the Proceedings of the XVIIth Intern.Colloquium on Group Theoretical Methods in Physics, June 27 — July 2, 1988, St. Adèle (Canada).Google Scholar
  139. 78.
    A. Neveu and P. West, Phys. Lett. B194 (1987) 200.MathSciNetADSGoogle Scholar
  140. 79.
    A. Sagnotti, in “Non-Perturbative Methods in Field Theory, Cargèse 1987” eds. G. Mack et al. (Plenum, New York, 1988);Google Scholar
  141. 79a.
    M. Bianchi and A. Sagnotti, Phys. Lett. B211 (1988) 407MathSciNetADSGoogle Scholar
  142. 79b.
    G. Pradisi and A. Sagnotti, Phys. Lett. B216 (1988) 59.MathSciNetADSGoogle Scholar
  143. 80.
    L. Clavelli, P.H. Cox and B. Harms, Phys. Rev. Lett. 61 (1988) 787;MathSciNetADSCrossRefGoogle Scholar
  144. 80a.
    L. Clavelli and S.T. Jones, Phys. Rev. D39 (1989) 3795ADSGoogle Scholar
  145. 80b.
    Z. Bern and D. Dunbar, Phys. Lett. B203 (1988) 109;MathSciNetADSGoogle Scholar
  146. 80c.
    D.C. Dunbar, Nucl. Phys. B319 (1989) 72:MathSciNetADSCrossRefGoogle Scholar
  147. 80d.
    Z. Bern and D.C. Dunbar, Nucl. Phys. B319 (1989) 104;MathSciNetADSCrossRefGoogle Scholar
  148. 80e.
    N. Ishibashi and T. Onogi, Nucl. Phys. B318 (1989) 239;MathSciNetADSCrossRefGoogle Scholar
  149. 80f.
    P. Hofava, Prague preprint PRA-HEP-89/1 (February 1989).Google Scholar
  150. 81.
    A. Parkes, Phys. Lett. B184 (1987) 19;MathSciNetADSGoogle Scholar
  151. 81a.
    A.N. Schellekens, Phys. Lett. B199 (1987) 427;MathSciNetADSGoogle Scholar
  152. 81b.
    R. Bluhm, L. Dolan and P. Goddard, Nucl. Phys. B309 (1988) 330; and references therein.MathSciNetADSCrossRefGoogle Scholar
  153. 82.
    L. Alvarez-Gaumé and P. Nelson, in “Supersymmetry, Supergravity and Super-strings ‘86”, eds. B. de Wit, P. Fayet and M. Grisaru (World Scientific, Singapore, 1987), p. 419–510; and references therein.Google Scholar
  154. 83.
    L. Alvarez-Gaumé, G. Moore and C. Vafa, Comm. Math. Phys. 106 (1986) 40.CrossRefGoogle Scholar
  155. 84.
    H. Kikkawa and M. Yamasaki, Phys. Lett. B149 (1984) 357;MathSciNetADSGoogle Scholar
  156. 84a.
    N. Sakai and I. Senda, Progr. Theor. Phys. 75 (1986) 692;MathSciNetADSCrossRefGoogle Scholar
  157. 84b.
    V.P. Nair, A. Shapere, A. Strominger and F. Wilczek, Nucl. Phys. B287 (1987) 402;MathSciNetADSCrossRefGoogle Scholar
  158. 84c.
    R. Dijkgraaf, E. Verlinde and H. Verlinde, Comm. Math. Phys. 115 (1988) 649;MathSciNetADSMATHCrossRefGoogle Scholar
  159. 84d.
    A. Giveon, E. Rabinovici and G. Veneziano, Nucl. Phys. B322(1989) 167;MathSciNetADSCrossRefGoogle Scholar
  160. 84f.
    M. Dine, P. Huet and N. Seiberg, Nucl. Phys. B322 (1989) 301;MathSciNetADSCrossRefGoogle Scholar
  161. 84g.
    J. Lauer , J. Mas and H. P. Nilles, Phys. Lett. B226 (1989) 251;MathSciNetADSGoogle Scholar
  162. 84f.
    J.M. Molera and B.A. Ovrut, Phys. Rev. D40 (1989) 1146.MathSciNetADSGoogle Scholar
  163. 85.
    L. Alvarez-Gaumé, P. Ginsparg, G. Moore and C. Vafa, Phys. Lett. B171 (1986) 155;ADSGoogle Scholar
  164. 85a.
    L.J. Dixon and J. A. Harvey, Nucl. Phys. B274 (1986) 93.MathSciNetADSCrossRefGoogle Scholar
  165. 86.
    P. Goddard, D.I. Olive and A. Schwimmer, Phys. Lett. B157 (1985) 393.MathSciNetADSGoogle Scholar
  166. 87.
    For a review , see: M.J. Duff, B.E.W. Nilsson and C.N. Pope, Physics Reports 130 (1986) 1.MathSciNetADSCrossRefGoogle Scholar
  167. 88.
    L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B261 (1985) 678MathSciNetADSCrossRefGoogle Scholar
  168. 88a.
    L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B274 (1986) 285;MathSciNetADSCrossRefGoogle Scholar
  169. 88b.
    M. Mueller and E. Witten, Phys. Lett. B182 (1986) 28;MathSciNetADSGoogle Scholar
  170. 88b.
    K. Narain ,M.H. Sarmadi and C. Vafa, Nucl. Phys. B288 (1987) 551;MathSciNetADSCrossRefGoogle Scholar
  171. 88c.
    L.J. Dixon, in “Superstrings, Unified Theories and Cosmology, 1987”, eds. G. Furlan et al. (World Scientific , Singapore, 1988), p. 67–126.Google Scholar
  172. 89.
    D. Nemenschansky and S. Yankielowicz, Phys. Rev. Lett. 54 (1985) 620;MathSciNetADSCrossRefGoogle Scholar
  173. 89a.
    D. Gepner and E. Witten, Nucl. Phys. B278 (1986) 493;MathSciNetADSCrossRefGoogle Scholar
  174. 89b.
    R. Bluhm, L. Dolan and P. Goddard, Nucl. Phys. B289 (1987) 364.MathSciNetADSCrossRefGoogle Scholar
  175. 90.
    H. Kawai, D.C. Lewellen and S.-H. H. Tye, Nucl. Phys. B288 (1987) 1;MathSciNetADSCrossRefGoogle Scholar
  176. 90a.
    I. Antoniadis, C. P. Bachas and C. Kounnas, Nucl. Phys. B289 (1987) 87;MathSciNetADSCrossRefGoogle Scholar
  177. 90b.
    I. Antoniadis and C. Bachas, Nucl. Phys. B298 (1988) 586.MathSciNetADSCrossRefGoogle Scholar
  178. 91.
    K. Bardacki, E. Rabinovici and B. Säring, Nucl. Phys. B299 (1988) 151.ADSCrossRefGoogle Scholar
  179. 92.
    D. Gepner, Phys. Lett. B199 (1987) 380MathSciNetADSGoogle Scholar
  180. 92a.
    D. Gepner, Nucl. Phys. B296 (1988) 757.MathSciNetADSCrossRefGoogle Scholar
  181. 93.
    P. Ginsparg, Phys. Rev. D35 (1987) 648;MathSciNetADSGoogle Scholar
  182. 93a.
    P. Ginsparg and C. Vafa, Nucl. Phys. B289 (1987) 414.MathSciNetADSCrossRefGoogle Scholar
  183. 94.
    L.J. Dixon, V.S. Kaplunovsky and C. Vafa, Nucl. Phys. B294 (1987) 43.MathSciNetADSCrossRefGoogle Scholar
  184. 95.
    T. Banks, L.J. Dixon, D. Friedan and E. Martinec, Nucl. Phys. B299 (1988) 613.MathSciNetADSCrossRefGoogle Scholar
  185. 96.
    D. Gepner, Lectures at the Spring School on Superstrings, Trieste, April 3–11, 1989, Princeton preprint PUPT-1121 (April 1989).Google Scholar
  186. 97.
    Y. Kazama and H. Suzuki, Phys. Lett. B216 (1989) 11MathSciNetGoogle Scholar
  187. 97a.
    Y. Kazama and H. Suzuki, Nucl. Phys. B321 (1989) 232;MathSciNetADSCrossRefGoogle Scholar
  188. 97b.
    Y. Kazama and H. Suzuki, Int. J. Mod. Phys. A4 (1989) 235.MathSciNetGoogle Scholar
  189. 98.
    D. Gepner, Nucl. Phys. B311 (1988) 191;MathSciNetADSCrossRefGoogle Scholar
  190. 98a.
    D. Gepner, Nucl. Phys. B322(1989) 65;MathSciNetADSCrossRefGoogle Scholar
  191. 98b.
    D. Gepner, P. Zoglin, Phys. Lett. B218 (1989) 444;ADSGoogle Scholar
  192. 98c.
    A. Font, L.E. Ibáñez and F. Quevedo, Phys. Lett. B224 (1989)79;ADSGoogle Scholar
  193. 98d.
    A. Font, L.E. Ibáñez , M. Mondragón, F. Quevedo and G.G. Ross, Phys. Lett. B227 (1989) 34;ADSGoogle Scholar
  194. 98f.
    K. Ito, Phys. Lett. B226 (1989) 264;ADSGoogle Scholar
  195. 98g.
    C.A. Lütken and G.G. Ross, Phys. Lett. B213 (1988) 152;ADSGoogle Scholar
  196. 98h.
    M. Lynker and R. Schimmrigk, Phys. Lett. B215 (1988) 681;MathSciNetADSGoogle Scholar
  197. 98i.
    T. Eguchi, H. Ooguri, A. Taormina and S.K. Yang, Nucl. Phys. B315 (1989) 193.MathSciNetADSCrossRefGoogle Scholar
  198. 99.
    A. Taormina, Lectures given at the Third Regional Conference in Mathematical Physics, Islamabad, February 18–25, 1989, preprint CERN-TH.5409/89 (May 1989);Google Scholar
  199. 99a.
    G. Mussardo, G. Sotkov and M. Stanishkov, Phys. Lett. B218 (1989) 191;MathSciNetADSGoogle Scholar
  200. 99b.
    A.N. Schellekens and S. Yankielowicz, Phys. Lett. B227 (1989) 387; preprint CERN-TH.5440/89 (June 1989);MathSciNetADSGoogle Scholar
  201. 99c.
    C. Vafa and N.P. Warner, Phys. Lett. B218 (1989) 51;MathSciNetADSGoogle Scholar
  202. 99d.
    C. Vafa, Mod. Phys. Lett. A4 (1989) 1169;MathSciNetADSGoogle Scholar
  203. 99e.
    W. Lerche, C. Vafa and N.P. Warner, Harvard preprint HUTP-88/A065; L.J. Dixon, M.E. Peskin and J. Lykken, preprint SLAC-PUB-4884 (March 1989).Google Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Jan Govaerts
    • 1
  1. 1.Instituut voor Theoretische FysicaKatholieke Universiteit LeuvenLeuvenBelgium

Personalised recommendations