String unification, higher-level gauge symmetries, and exotic hypercharge normalizations

Nuclear Physics B

Volumn 467, Issue 1-2, 1996, Pages 44-99

  Dienes, Keith R ^a,c   Faraggi, Alon E ^a,b,c   March Russell, John ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b University of Florida ^*  (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030581990     PISSN: 05503213     EISSN: None     Source Type: Journal    
DOI: 10.1016/0550-3213(96)00085-5     Document Type: Article

References (83)

1. [1] For a review, see, M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory, Vols. 1 & 2 (Cambridge Univ. Press, Cambridge, 1987).
2. [2] P. Ginsparg, Phys. Lett. B 197 (1987) 139.
3. [3] M. Dine and N. Seiberg, Phys. Rev. Lett. 55 (1985) 366; V.S. Kaplunovsky, Phys. Rev. Lett. 55 (1985) 1036.
4. [3] M. Dine and N. Seiberg, Phys. Rev. Lett. 55 (1985) 366; V.S. Kaplunovsky, Phys. Rev. Lett. 55 (1985) 1036.
5. [4] V.S. Kaplunovsky, Nucl. Phys. B 307 (1988) 145; B 382 (1992) 436 (E).
6. [4] V.S. Kaplunovsky, Nucl. Phys. B 307 (1988) 145; B 382 (1992) 436 (E).
7. [5] K.R. Dienes and A.E. Faraggi, Phys. Rev. Lett. 75 (1995) 2646.
8. [6] K.R. Dienes and A.E. Faraggi, Nucl. Phys. B 457 (1995) 409.
9. [7] For a review, see, K.R. Dienes, preprint IASSNS-HEP-95/97, hep-th/9602045, to appear in Physics Reports.
10. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
11. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
12. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
13. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
14. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
15. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
16. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
17. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
18. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
19. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
20. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
21. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
22. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
23. [8] See, e.g., the following papers and references therein: L.J. Dixon, V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 355 (1991) 649; V.S. Kaplunovsky and J. Louis, Nucl. Phys. B 444 (1995) 191; L. Ibáñez, D. Lust and G. Ross, Phys. Lett. B 272 (1991) 251; L. Ibáñez and D. Lust, Nucl. Phys. B 382 (1992) 305; I. Antoniadis, K.S. Narain and T.R. Taylor, Phys. Lett. B 267 (1991) 37; I. Antoniadis, E. Gava and K.S. Narain, Phys. Lett. B 283 (1992) 209; J.P. Derendinger, S. Ferrara, C. Kounnas and F. Zwirner, Phys. Lett. B 271 (1991) 307; Nucl. Phys. B 372 (1992) 145; G. Lopes Cardoso and B.A. Ovrut, Nucl. Phys. B 369 (1992) 351; P. Mayr and S. Stieberger, Nucl. Phys. B 407 (1993) 725; Nucl. Phys. B 412 (1994) 502; D. Bailin and A. Love, Phys. Lett. B 292 (1992) 315; D. Bailin, A. Love, W.A. Sabra and S. Thomas, Phys. Lett. B 320 (1994) 21; Mod. Phys. Lett. A 10 (1995) 337.
24. [9] A. Font, L. Ibáñez, D. Lüst and F. Quevedo, Phys. Lett. B 245 (1990) 401; M. Cvetič et al., Nucl. Phys. B 361 (1991) 194; B. de Carlos, J.A. Casas and C. Munoz, Nucl. Phys. B 399 (1993) 623.
25. [9] A. Font, L. Ibáñez, D. Lüst and F. Quevedo, Phys. Lett. B 245 (1990) 401; M. Cvetič et al., Nucl. Phys. B 361 (1991) 194; B. de Carlos, J.A. Casas and C. Munoz, Nucl. Phys. B 399 (1993) 623.
26. [9] A. Font, L. Ibáñez, D. Lüst and F. Quevedo, Phys. Lett. B 245 (1990) 401; M. Cvetič et al., Nucl. Phys. B 361 (1991) 194; B. de Carlos, J.A. Casas and C. Munoz, Nucl. Phys. B 399 (1993) 623.
27. [10] See, e.g., the following papers and references therein: I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; P. Mayr, H.P. Nilles and S. Stieberger, Phys. Lett. B 317 (1993) 53; L. Dolan and J.T. Liu, Nucl. Phys. B 387 (1992) 86; D.M. Pierce, Phys. Rev. D 50 (1994) 6469; M. Chemtob, hep-th/9506018.
28. [10] See, e.g., the following papers and references therein: I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; P. Mayr, H.P. Nilles and S. Stieberger, Phys. Lett. B 317 (1993) 53; L. Dolan and J.T. Liu, Nucl. Phys. B 387 (1992) 86; D.M. Pierce, Phys. Rev. D 50 (1994) 6469; M. Chemtob, hep-th/9506018.
29. [10] See, e.g., the following papers and references therein: I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; P. Mayr, H.P. Nilles and S. Stieberger, Phys. Lett. B 317 (1993) 53; L. Dolan and J.T. Liu, Nucl. Phys. B 387 (1992) 86; D.M. Pierce, Phys. Rev. D 50 (1994) 6469; M. Chemtob, hep-th/9506018.
30. [10] See, e.g., the following papers and references therein: I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; P. Mayr, H.P. Nilles and S. Stieberger, Phys. Lett. B 317 (1993) 53; L. Dolan and J.T. Liu, Nucl. Phys. B 387 (1992) 86; D.M. Pierce, Phys. Rev. D 50 (1994) 6469; M. Chemtob, hep-th/9506018.
31. [10] See, e.g., the following papers and references therein: I. Antoniadis, J. Ellis, R. Lacaze and D.V. Nanopoulos, Phys. Lett. B 268 (1991) 188; P. Mayr, H.P. Nilles and S. Stieberger, Phys. Lett. B 317 (1993) 53; L. Dolan and J.T. Liu, Nucl. Phys. B 387 (1992) 86; D.M. Pierce, Phys. Rev. D 50 (1994) 6469; M. Chemtob, hep-th/9506018.
32. [11] P. Mayr and S. Stieberger, hep-th/9412196; Phys. Lett. B 355 (1995) 107.
33. [11] P. Mayr and S. Stieberger, hep-th/9412196; Phys. Lett. B 355 (1995) 107.
34. [12] H.P. Nilles and S. Stieberger, Phys. Lett. B 367 (1996) 126.
35. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
36. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
37. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
38. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
39. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
40. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
41. [13] D.C. Lewellen, Nucl. Phys. B 337 (1990) 61; J. Ellis, J.L. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 (1990) 375; A. Font, L.E. Ibáñez and F. Quevedo, Nucl. Phys. B 345 (1990) 389; S. Chaudhuri, S.-W. Chung, G. Hockney and J. Lykken, Nucl. Phys. B 456 ( 1995) 89; G. Aldazabal, A. Font, L.E. Ibáñez and A.M. Uranga, Nucl. Phys. B 452 (1995) 3; hep-th/9508033; G. Cleaver, hep-th/9506006; D. Finnell, hep-th/9508073.
42. [14] I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65; J. Lopez, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 399 (1993) 654.
43. [14] I. Antoniadis, J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. B 231 (1989) 65; J. Lopez, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 399 (1993) 654.
44. [15] S.M. Barr, Phys. Lett. B 112 (1982) 219; Phys. Rev. D 40 (1989) 2457; J.P. Derendinger, J.E. Kim and D.V Nanopoulos, Phys. Lett. B 139 (1984) 170.
45. [15] S.M. Barr, Phys. Lett. B 112 (1982) 219; Phys. Rev. D 40 (1989) 2457; J.P. Derendinger, J.E. Kim and D.V Nanopoulos, Phys. Lett. B 139 (1984) 170.
46. [15] S.M. Barr, Phys. Lett. B 112 (1982) 219; Phys. Rev. D 40 (1989) 2457; J.P. Derendinger, J.E. Kim and D.V Nanopoulos, Phys. Lett. B 139 (1984) 170.
47. [16] I. Antoniadis, G.K. Leontaris and J. Rizos, Phys. Lett. B 245 (1990) 161.
48. [17] J.C. Pati and A. Salam, Phys. Rev. D 8 (1973) 1240; Phys. Rev. Lett. 31 (1973) 661; Phys. Rev. D 10 (1974) 275.
49. [17] J.C. Pati and A. Salam, Phys. Rev. D 8 (1973) 1240; Phys. Rev. Lett. 31 (1973) 661; Phys. Rev. D 10 (1974) 275.
50. [17] J.C. Pati and A. Salam, Phys. Rev. D 8 (1973) 1240; Phys. Rev. Lett. 31 (1973) 661; Phys. Rev. D 10 (1974) 275.
51. [18] A.E. Faraggi, Phys. Lett. B 339 (1994) 223.
52. [19] I. Antoniadis, J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 272 (1991) 31; A.E. Faraggi, Phys. Lett. B 302 (1993) 202.
53. [19] I. Antoniadis, J. Ellis, S. Kelley and D.V. Nanopoulos, Phys. Lett. B 272 (1991) 31; A.E. Faraggi, Phys. Lett. B 302 (1993) 202.
54. [20] M.K. Gaillard and R. Xiu, Phys. Lett. B 296 (1992) 71; S.P. Martin and P. Ramond, Phys. Rev. D 51 (1995) 6515.
55. [20] M.K. Gaillard and R. Xiu, Phys. Lett. B 296 (1992) 71; S.P. Martin and P. Ramond, Phys. Rev. D 51 (1995) 6515.
56. [21] C. Bachas, C. Fabre and T. Yanagida, hep-th/9510094.
57. [22] J.A. Casas and C. Munoz, Phys. Lett. B 214 (1988) 543; L. Ibáñez, Phys. Lett. B 318 (1993) 73.
58. [22] J.A. Casas and C. Munoz, Phys. Lett. B 214 (1988) 543; L. Ibáñez, Phys. Lett. B 318 (1993) 73.
59. [23] See, e.g., H. Kawabe, T. Kobayashi and N. Ohtsubo, Nucl. Phys. B 434 (1995) 210, and references therein.
60. [24] A.N. Schellekens, Phys. Lett. B 237 (1990) 363.
61. [25] For a pedagogical introduction, see, e.g., P. Ginsparg, Applied Conformal Field Theory, published in Fields, Strings and Critical Phenomena: Proceedings of Les Houches, Session XLIX, 1988 (North-Holland, Amsterdam, 1989).
62. [26] See, e.g., Particle Data Group, L. Montanet et al., Phys. Rev. D 50 (1994) 1173; P. Langacker, hep-ph/9412361, and references therein; N. Polonsky, hep-ph/9411378, and references therein.
63. [26] See, e.g., Particle Data Group, L. Montanet et al., Phys. Rev. D 50 (1994) 1173; P. Langacker, hep-ph/9412361, and references therein; N. Polonsky, hep-ph/9411378, and references therein.
64. [26] See, e.g., Particle Data Group, L. Montanet et al., Phys. Rev. D 50 (1994) 1173; P. Langacker, hep-ph/9412361, and references therein; N. Polonsky, hep-ph/9411378, and references therein.
65. [27] A.E. Faraggi, Phys. Lett. B 278 (1992) 131; B 274 (1992) 47; Nucl. Phys. B 387 (1992) 239.
66. [27] A.E. Faraggi, Phys. Lett. B 278 (1992) 131; B 274 (1992) 47; Nucl. Phys. B 387 (1992) 239.
67. [27] A.E. Faraggi, Phys. Lett. B 278 (1992) 131; B 274 (1992) 47; Nucl. Phys. B 387 (1992) 239.
68. [28] D. Bailin and A. Love, Phys. Lett. B 203 (1988) 237.
69. [29] K.R. Dienes and J. March-Russell, Realizing Higher-Level Gauge Symmetries in String Theory, preprint IASSNS-HEP-95/56 (to appear).
70. [30] P.F. Smith, Ann. Rev. Nucl. Part. Sci. 39 (1989) 73.
71. [31] J. Ellis, J. Lopez and D. Nanopoulos, Phys. Lett. B 245 (1990) 375.
72. [32] I. Antoniadis and K. Benakli, Phys. Lett. B 295 (1992) 219.
73. [33] A.N. Schellekens and S. Yankielowicz, Nucl. Phys. B 327 (1989) 673.
74. [34] K. Intriligator, Nucl. Phys. B 332 (1990) 541.
75. [35] D. Verstegen, Nucl. Phys. B 346 ( 1990) 349.
76. [36] J. Fuchs, Affine Lie Algebras and Quantum Groups (Cambridge Univ. Press, Cambridge, 1992).
77. [37] I. Antoniadis, E. Gava and K.S. Narain, Nucl. Phys. B 383 (1992) 93; B. de Wit, V.S. Kaplunovsky, J. Louis and D. Lüst, Nucl. Phys. B 451 (1995) 53; I. Antoniadis et al., Nucl. Phys. B 447 (1995) 35.
78. [37] I. Antoniadis, E. Gava and K.S. Narain, Nucl. Phys. B 383 (1992) 93; B. de Wit, V.S. Kaplunovsky, J. Louis and D. Lüst, Nucl. Phys. B 451 (1995) 53; I. Antoniadis et al., Nucl. Phys. B 447 (1995) 35.
79. [37] I. Antoniadis, E. Gava and K.S. Narain, Nucl. Phys. B 383 (1992) 93; B. de Wit, V.S. Kaplunovsky, J. Louis and D. Lüst, Nucl. Phys. B 451 (1995) 53; I. Antoniadis et al., Nucl. Phys. B 447 (1995) 35.
80. [38] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B 289 (1987) 589; J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B 292 (1987) 109; S. Cecotti, S. Ferrara and M. Villasante, Int. J. Mod. Phys. A 2 (1987) 1839.
81. [38] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B 289 (1987) 589; J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B 292 (1987) 109; S. Cecotti, S. Ferrara and M. Villasante, Int. J. Mod. Phys. A 2 (1987) 1839.
82. [38] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B 289 (1987) 589; J.J. Atick, L.J. Dixon and A. Sen, Nucl. Phys. B 292 (1987) 109; S. Cecotti, S. Ferrara and M. Villasante, Int. J. Mod. Phys. A 2 (1987) 1839.
83. [39] S. Chaudhuri, G. Hockney and J. Lykken, hep-th/9510241.