Duality symmetries and noncommutative geometry of string spacetimes

Communications in Mathematical Physics

Volumn 197, Issue 3, 1998, Pages 667-712

  Lizzi, Fedele ^a   Szabo, Richard J ^a  

^a UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032475788     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050468     Document Type: Article

References (75)

1. Giveon, A., Porrati, M. and Rabinovici, E.: Target Space Duality in String Theory. Phys. Rep. 244, 77 (1994) (hep-th/9401139)
2. Alvarez-Gaumé, L. and Hassan, S.F.: Introduction to S-Duality in N = 2 Supersymmetric Gauge Theories: A Pedagogical Review of the work of Seiberg and Witten, Fortsch. Phys 45, 159 (1997) (hep-th/9701069)
3. Witten, E.: String Theory Dynamics in Various Dimensions. Nucl. Phys. B443, 85 (1995); Schwarz, J.H.: The Power of M Theory. Phys. Lett. B367, 97 (1996); Lectures on Superstring and M Theory Dualities. Nucl. Phys. B (Proc. Suppl.) 55B, 1 (1997); Duff, M.J.: M Theory (The Theory formerly known as Strings). Intern. J. Mod. Phys. A11, 5623 (1996)
4. Witten, E.: String Theory Dynamics in Various Dimensions. Nucl. Phys. B443, 85 (1995); Schwarz, J.H.: The Power of M Theory. Phys. Lett. B367, 97 (1996); Lectures on Superstring and M Theory Dualities. Nucl. Phys. B (Proc. Suppl.) 55B, 1 (1997); Duff, M.J.: M Theory (The Theory formerly known as Strings). Intern. J. Mod. Phys. A11, 5623 (1996)
5. Witten, E.: String Theory Dynamics in Various Dimensions. Nucl. Phys. B443, 85 (1995); Schwarz, J.H.: The Power of M Theory. Phys. Lett. B367, 97 (1996); Lectures on Superstring and M Theory Dualities. Nucl. Phys. B (Proc. Suppl.) 55B, 1 (1997); Duff, M.J.: M Theory (The Theory formerly known as Strings). Intern. J. Mod. Phys. A11, 5623 (1996)
6. Witten, E.: String Theory Dynamics in Various Dimensions. Nucl. Phys. B443, 85 (1995); Schwarz, J.H.: The Power of M Theory. Phys. Lett. B367, 97 (1996); Lectures on Superstring and M Theory Dualities. Nucl. Phys. B (Proc. Suppl.) 55B, 1 (1997); Duff, M.J.: M Theory (The Theory formerly known as Strings). Intern. J. Mod. Phys. A11, 5623 (1996)
7. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996) (hep-th/9510135); Shenker, S.H.: Another Length Scale in String Theory? hep-th/9509132; Danielsson, U.H., Ferretti, G. and Sundborg, B.: D-particle Dynamics and Bound States. Int. J. Mod. Phys. A11, 5463 (1996), (hep-th/9603081); Kabat, D. and Pouliot, P.: A Comment on Zero-brane Quantum Mechanics. Phys. Rev. Lett. 77, 1004 (1996), (hep-th/9603127); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85-127 (1997), (hep-th/9608024)
8. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996) (hep-th/9510135); Shenker, S.H.: Another Length Scale in String Theory? hep-th/9509132; Danielsson, U.H., Ferretti, G. and Sundborg, B.: D-particle Dynamics and Bound States. Int. J. Mod. Phys. A11, 5463 (1996), (hep-th/9603081); Kabat, D. and Pouliot, P.: A Comment on Zero-brane Quantum Mechanics. Phys. Rev. Lett. 77, 1004 (1996), (hep-th/9603127); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85-127 (1997), (hep-th/9608024)
9. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996) (hep-th/9510135); Shenker, S.H.: Another Length Scale in String Theory? hep-th/9509132; Danielsson, U.H., Ferretti, G. and Sundborg, B.: D-particle Dynamics and Bound States. Int. J. Mod. Phys. A11, 5463 (1996), (hep-th/9603081); Kabat, D. and Pouliot, P.: A Comment on Zero-brane Quantum Mechanics. Phys. Rev. Lett. 77, 1004 (1996), (hep-th/9603127); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85-127 (1997), (hep-th/9608024)
10. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996) (hep-th/9510135); Shenker, S.H.: Another Length Scale in String Theory? hep-th/9509132; Danielsson, U.H., Ferretti, G. and Sundborg, B.: D-particle Dynamics and Bound States. Int. J. Mod. Phys. A11, 5463 (1996), (hep-th/9603081); Kabat, D. and Pouliot, P.: A Comment on Zero-brane Quantum Mechanics. Phys. Rev. Lett. 77, 1004 (1996), (hep-th/9603127); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85-127 (1997), (hep-th/9608024)
11. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996) (hep-th/9510135); Shenker, S.H.: Another Length Scale in String Theory? hep-th/9509132; Danielsson, U.H., Ferretti, G. and Sundborg, B.: D-particle Dynamics and Bound States. Int. J. Mod. Phys. A11, 5463 (1996), (hep-th/9603081); Kabat, D. and Pouliot, P.: A Comment on Zero-brane Quantum Mechanics. Phys. Rev. Lett. 77, 1004 (1996), (hep-th/9603127); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85-127 (1997), (hep-th/9608024)
12. Banks, T., Fischler, W., Shenker, S.H. and Susskind, S.H.: M Theory as a Matrix Model: A Conjecture. Phys. Rev. D55, 5112-5128 (1997), (hep-th/9610043)
13. Connes, A.: Noncommutative Geometry. London-New York: Academic Press, 1994
14. Fröhlich, J. and Gawȩdzki, K.: Conformal Field Theory and Geometry of Strings. CRM Proc. Lecture Notes 7, 57-97 (1994), (hep-th/9310187)
15. Chamseddine, A.H. and Fröhlich, J.: Some Elements of Connes' Noncommutative Geometry and Space-time Geometry. In: Yang Festschrift, eds. C.S. Liu and S.-F. Yau, Boston: International Press, 1995, pp. 10-34
16. Fröhlich, J., Grandjean, O. and Recknagel, A.: Supersymmetric Quantum Theory, Noncommutative Geometry and Gravitation. In: Quantum Symmetries, Proc. LXIV Les Houches Session, eds. A. Connes and K. Gawȩdzki, to appear, (hep-th/9706132)
17. Chamseddine, A.H.: The Spectral Action Principle in Noncommutative Geometry and the Superstring. Phys. Lett. B400, 87 (1997), (hep-th/9701096); An Effective Superstring Spectral Action. Phys. Rev. D56, 3555, (1997), (hep-th/9705153)
18. Chamseddine, A.H.: The Spectral Action Principle in Noncommutative Geometry and the Superstring. Phys. Lett. B400, 87 (1997), (hep-th/9701096); An Effective Superstring Spectral Action. Phys. Rev. D56, 3555, (1997), (hep-th/9705153)
19. Chamseddine, A.H. and Connes, A.: Universal Formula for Noncommutative Geometry Actions: Unification of Gravity and the Standard Model. Phys. Rev. Lett. 77, 4868 (1996); The Spectral Action Principle. Commun. Math. Phys. 186, 731 (1997), (hep-th/9606001)
20. Chamseddine, A.H. and Connes, A.: Universal Formula for Noncommutative Geometry Actions: Unification of Gravity and the Standard Model. Phys. Rev. Lett. 77, 4868 (1996); The Spectral Action Principle. Commun. Math. Phys. 186, 731 (1997), (hep-th/9606001)
21. Frenkel, I.B., Lepowsky, J. and Meurman, A.: Vertex Operator Algebras and the Monster. Pure Appl. Math. 134, New York: Academic Press, 1988
22. Huang, Y.-Z.: Vertex Operator Algebras and Conformal Field Theory. Intern. J. Mod. Phys. A7, 2109-2151 (1992)
23. Gebert, R.W.: Introduction to Vertex Algebras, Borcherds Algebras and the Monster Lie Algebra. Intern. J. Mod. Phys. A8, 5441-5503 (1993), (hep-th/9308151)
24. Evans, M. and Giannakis, I.: T Duality in Arbitrary String Backgrounds. Nucl. Phys. B472, 139-162 (1996); Giannakis, I.: O(d, d; ℤ) Transformations as Automorphisms of the Operator Algebra. Phys. Lett. B388, 543-549 (1996)
25. Evans, M. and Giannakis, I.: T Duality in Arbitrary String Backgrounds. Nucl. Phys. B472, 139-162 (1996); Giannakis, I.: O(d, d; ℤ) Transformations as Automorphisms of the Operator Algebra. Phys. Lett. B388, 543-549 (1996)
26. Lizzi, F. and Szabo, R.J.: Target Space Duality in Noncommutative Geometry. Phys. Rev. Lett. 79, 3581 (1997), (hep-th/9706107)
27. Douglas, M.R.: Superstring Dualities, Dirichlet Branes and the Small Scale Structure of Space. In: Quantum Symmetries, Proc. LXIV Les Houches Session, eds. A. Connes and K. Gawȩdzki, to appear, (hep-th/9610041); Ho, P.-M. and Wu, Y.-S.: Noncommutative Geometry and D-branes. Phys. Lett. B398, 52-60 (1997), (hep-th/9611233); Kalkkinen, J.: Dimensionally Reduced Yang-Mills Theories in Noncommutative Geometry. Phys. Lett. B399, 243 (1997), (hep-th/9612027)
28. Douglas, M.R.: Superstring Dualities, Dirichlet Branes and the Small Scale Structure of Space. In: Quantum Symmetries, Proc. LXIV Les Houches Session, eds. A. Connes and K. Gawȩdzki, to appear, (hep-th/9610041); Ho, P.-M. and Wu, Y.-S.: Noncommutative Geometry and D-branes. Phys. Lett. B398, 52-60 (1997), (hep-th/9611233); Kalkkinen, J.: Dimensionally Reduced Yang-Mills Theories in Noncommutative Geometry. Phys. Lett. B399, 243 (1997), (hep-th/9612027)
29. Douglas, M.R.: Superstring Dualities, Dirichlet Branes and the Small Scale Structure of Space. In: Quantum Symmetries, Proc. LXIV Les Houches Session, eds. A. Connes and K. Gawȩdzki, to appear, (hep-th/9610041); Ho, P.-M. and Wu, Y.-S.: Noncommutative Geometry and D-branes. Phys. Lett. B398, 52-60 (1997), (hep-th/9611233); Kalkkinen, J.: Dimensionally Reduced Yang-Mills Theories in Noncommutative Geometry. Phys. Lett. B399, 243 (1997), (hep-th/9612027)
30. Várilly, J.C. and Gracia-Bondía, J.M.: Connes' Noncommutative Differential Geometry and the Standard Model. J. Geom. Phys. 12, 223 (1993)
31. Landi, G.: An Introduction to Noncommutative Spaces and their Geometries. Springer Lecture Notes in Physics 51, Berlin-Heidelberg: Springer-Verlag, 1997
32. Madore, J.: An Introduction to Noncommutative Geometry and its Physical Applications. LMS Lecture Notes 206, (1995)
33. Fell, J.M.G. and Doran, R.S.: Representations of *-Algebras, Locally Compact Groups and Banach *-Algebraic Bundles, London-New York: Academic Press, 1988
34. Dixmier, J.: Les C*-algèbres et leurs Représentations. Paris: Gauthier-Villars, 1964
35. Chamseddine, A.H., Felder, G. and Fröhlich, J.: Gravity in Noncommutative Geometry. Commun. Math. Phys. 155, 205-217 (1993)
36. Connes, A. and Lott, J.: Particle Models and Noncommutative Geometry. Nucl. Phys. B (Proc. Suppl.) 18B, 29 (1990); The Metric Aspect of Noncommutative Geometry. In: New Symmetry Principles in Quantum Field Theory, eds. J. Fröhlich, G. 't Hooft. A. Jaffe, G. Mack, P.K. Mitter and R. Stora, New York: Plenum Press, 1992, pp.53-94
37. Connes, A. and Lott, J.: Particle Models and Noncommutative Geometry. Nucl. Phys. B (Proc. Suppl.) 18B, 29 (1990); The Metric Aspect of Noncommutative Geometry. In: New Symmetry Principles in Quantum Field Theory, eds. J. Fröhlich, G. 't Hooft. A. Jaffe, G. Mack, P.K. Mitter and R. Stora, New York: Plenum Press, 1992, pp.53-94
38. Martín, C.P., Gracia-Bondía, J.M. and Várilly, J.C.: The Standard Model as a Noncommutative Geometry: The Low Energy Regime. Phys. Rep. 294, 363 (1998), (hep-th/9605001)
39. Iochum, B., Kastler, D. and Schücker, T.: On the Universal Chamseddine-Connes Action I. Details of the Action Computation. J. Math. Phys. 38, 4929 (1997), (hep-th/9607015)
40. Figueroa, H., Lizzi, F., Gracia-Bondía, J.M. and Várilly, J.C.: A Nonperturbative Form of the Spectral Action Principle in Noncommutative Geometry, hep-th/9701179, to appear in J. Geom. Phys.
41. Madore, J.: The Fuzzy Sphere. Class. Quant. Grav. 9, 69 (1992)
42. Balachandran, A.P., Bimonte, G., Ercolessi, I;., Landi, G., Lizzi, F., Sparano, G. and Teotonio-Sobrinho, P.: Noncommutative Lattices as Finite Approximations. J. Geom. Phys. 18, 163-194 P., (hep-th/9510217); Finite Quantum Physics and Noncommutative Geometry. Nucl. Phys. B (Proc. Suppl.) 37C, 20-45 (1995), (hep-th/9403067)
43. Balachandran, A.P., Bimonte, G., Ercolessi, I;., Landi, G., Lizzi, F., Sparano, G. and Teotonio-Sobrinho, P.: Noncommutative Lattices as Finite Approximations. J. Geom. Phys. 18, 163-194 P., (hep-th/9510217); Finite Quantum Physics and Noncommutative Geometry. Nucl. Phys. B (Proc. Suppl.) 37C, 20-45 (1995), (hep-th/9403067)
44. Doplicher, S., Fredenhagen, K. and Roberts, J.E.: Spacetime Quantization induced by Classical Gravity. Phys. Lett. B331, 39-44 (1994); The Quantum Structure of Spacetime at the Planck Scale and Quantum Fields. Commun. Math. Phys. 172, 187-220 (1995); Kempf, A. and Mangano, G.: Minimal Length Uncertainty Relation and Ultraviolet Regularization. Phys. Rev. D55, 7909-7920 (1997), (hep-th/9612084); Mangano, G.: Path Integral Approach to Noncommutative Spacetimes. gr-qc/9705040 to appear in J. Math. Phys.
45. Doplicher, S., Fredenhagen, K. and Roberts, J.E.: Spacetime Quantization induced by Classical Gravity. Phys. Lett. B331, 39-44 (1994); The Quantum Structure of Spacetime at the Planck Scale and Quantum Fields. Commun. Math. Phys. 172, 187-220 (1995); Kempf, A. and Mangano, G.: Minimal Length Uncertainty Relation and Ultraviolet Regularization. Phys. Rev. D55, 7909-7920 (1997), (hep-th/9612084); Mangano, G.: Path Integral Approach to Noncommutative Spacetimes. gr-qc/9705040 to appear in J. Math. Phys.
46. Doplicher, S., Fredenhagen, K. and Roberts, J.E.: Spacetime Quantization induced by Classical Gravity. Phys. Lett. B331, 39-44 (1994); The Quantum Structure of Spacetime at the Planck Scale and Quantum Fields. Commun. Math. Phys. 172, 187-220 (1995); Kempf, A. and Mangano, G.: Minimal Length Uncertainty Relation and Ultraviolet Regularization. Phys. Rev. D55, 7909-7920 (1997), (hep-th/9612084); Mangano, G.: Path Integral Approach to Noncommutative Spacetimes. gr-qc/9705040 to appear in J. Math. Phys.
47. Doplicher, S., Fredenhagen, K. and Roberts, J.E.: Spacetime Quantization induced by Classical Gravity. Phys. Lett. B331, 39-44 (1994); The Quantum Structure of Spacetime at the Planck Scale and Quantum Fields. Commun. Math. Phys. 172, 187-220 (1995); Kempf, A. and Mangano, G.: Minimal Length Uncertainty Relation and Ultraviolet Regularization. Phys. Rev. D55, 7909-7920 (1997), (hep-th/9612084); Mangano, G.: Path Integral Approach to Noncommutative Spacetimes. gr-qc/9705040 to appear in J. Math. Phys.
48. Majid, S.: Foundations of Quantum Group Theory. Cambridge: Cambridge University Press, 1995
49. Wess, J. and Zumino, B.: Covariant Differential Calculus on the Quantum Hyperplane. Nucl. Phys. B (Proc. Suppl.) 18B, 302 (1991)
50. Narain, K.S.: New Heterotic String Theories in Uncompactified Dimensions < 10. Phys. Lett. B169, 41 (1986); Narain, K.S., Sarmadi, M.H. and Witten, E.: A Note on Toroidal Compactification of Heterotic String Theory. Nucl. Phys. B279, 369 (1987)
51. Narain, K.S.: New Heterotic String Theories in Uncompactified Dimensions < 10. Phys. Lett. B169, 41 (1986); Narain, K.S., Sarmadi, M.H. and Witten, E.: A Note on Toroidal Compactification of Heterotic String Theory. Nucl. Phys. B279, 369 (1987)
52. Fröhlich, J., Grandjean, O. and Recknagel, A.: Supersymmetric Quantum Theory and (Noncommutative) Differential Geometry, hep-th/9612205, to appear in Commun. Math. Phys.
53. Witten, E.: Supersymmetry and Morse Theory. J. Diff. Geom. 17, 661 (1982); Constraints on Supersymmetry Breaking. Nucl. Phys. B202, 253-316 (1982)
54. Witten, E.: Supersymmetry and Morse Theory. J. Diff. Geom. 17, 661 (1982); Constraints on Supersymmetry Breaking. Nucl. Phys. B202, 253-316 (1982)
55. Witten, E.: Global Anomalies in String Theory. In: Anomalies, Geometry and Topology, eds. W. Bardeen and A. White, Sinagpore: World Scientific, 1985, pp. 61-99; Noncommutative Geometry and String Field Theory. Nucl. Phys. B268, 253-294 (1986)
56. Witten, E.: Global Anomalies in String Theory. In: Anomalies, Geometry and Topology, eds. W. Bardeen and A. White, Sinagpore: World Scientific, 1985, pp. 61-99; Noncommutative Geometry and String Field Theory. Nucl. Phys. B268, 253-294 (1986)
57. Goddard, P. and Olive, D.: Kac-Moody and Virasoro Algebras in relation to Quantum Physics. Intern. J. Mod. Phys. A1, 303-414 (1986)
58. Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-dimensional Quantum Field Theory. Nucl. Phys. B241, 333-380 (1984); Moore, G. and Seiberg, N.: Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177 (1989)
59. Belavin, A.A., Polyakov, A.M. and Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-dimensional Quantum Field Theory. Nucl. Phys. B241, 333-380 (1984); Moore, G. and Seiberg, N.: Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177 (1989)
60. Connes, A.: Gravity coupled with Matter and the Foundation of Noncommutative Geometry. Commun. Math. Phys. 182, 155 (1996)
61. Evans, M. and Ovrut, B.: Spontaneously Broken Intermass Level Symmetries in String Theory. Phys. Lett. B231, 80 (1989); Deformations of Conformal Field Theories and Symmetries of the String. Phys. Rev. D41, 3149 (1990); Evans, M. and Giannakis, I.: Gauge Covariant Deformations, Symmetries and Free Parameters of String Theory. Phys. Rev. D44, 2467-2479 (1991)
62. Evans, M. and Ovrut, B.: Spontaneously Broken Intermass Level Symmetries in String Theory. Phys. Lett. B231, 80 (1989); Deformations of Conformal Field Theories and Symmetries of the String. Phys. Rev. D41, 3149 (1990); Evans, M. and Giannakis, I.: Gauge Covariant Deformations, Symmetries and Free Parameters of String Theory. Phys. Rev. D44, 2467-2479 (1991)
63. Evans, M. and Ovrut, B.: Spontaneously Broken Intermass Level Symmetries in String Theory. Phys. Lett. B231, 80 (1989); Deformations of Conformal Field Theories and Symmetries of the String. Phys. Rev. D41, 3149 (1990); Evans, M. and Giannakis, I.: Gauge Covariant Deformations, Symmetries and Free Parameters of String Theory. Phys. Rev. D44, 2467-2479 (1991)
64. Ginsparg, P.: Applied Conformal Field Theory. In: Fields, Strings and Critical Phenomena, eds. E. Brézin and J. Zinn-Justin Amsterdam: North Holland, 1990, pp. 1-168
65. Hull, C.M.: Higher Spin Extended Conformal Algebras and W Gravities. Nucl. Phys. B353, 707-756 (1991); Witten, E. and Zwiebach, B.: Algebraic Structures and Differential Geometry in 2D String Theory. Nucl. Phys. B377, 55 (1992)
66. Hull, C.M.: Higher Spin Extended Conformal Algebras and W Gravities. Nucl. Phys. B353, 707-756 (1991); Witten, E. and Zwiebach, B.: Algebraic Structures and Differential Geometry in 2D String Theory. Nucl. Phys. B377, 55 (1992)
67. Evans, M., Giannakis, I. and Nanopoulos, D.V.: An Infinite-dimensional Symmetry Algebra in String Theory. Phys. Rev. D50, 4022-4031 (1994)
68. Moore, G.: Finite in all Directions, hep-th/9305139
69. Borcherds, R.E.: Generalized Kac-Moody Algebras. J. Algebra 115, 501 (1988); Monstrous Moonshine and Monstrous Lie Superalgebras. Invent. Math. 109, 405-444 (1992)
70. Borcherds, R.E.: Generalized Kac-Moody Algebras. J. Algebra 115, 501 (1988); Monstrous Moonshine and Monstrous Lie Superalgebras. Invent. Math. 109, 405-444 (1992)
71. Borcherds, R.E.: Proc. Nat. Acad. Sci USA 83, 3068 (1986); Dong, C.: Vertex Algebras associated with Even Lattices. J. Algebra 160, 245-265 (1993)
72. Borcherds, R.E.: Proc. Nat. Acad. Sci USA 83, 3068 (1986); Dong, C.: Vertex Algebras associated with Even Lattices. J. Algebra 160, 245-265 (1993)
73. Zhu, Y.: Modular Invariance of Characters of Vertex Operator Algebras. J. Am. Math. Soc. 9, 237-302 (1996) Dong, C., Li, H. and Mason, G.: Twisted representations of Vertex Operator Algebras. To appear in Mathematische Annalen, q-alg/9509005
74. Zhu, Y.: Modular Invariance of Characters of Vertex Operator Algebras. J. Am. Math. Soc. 9, 237-302 (1996) Dong, C., Li, H. and Mason, G.: Twisted representations of Vertex Operator Algebras. To appear in Mathematische Annalen, q-alg/9509005
75. Dong, C.: Li, H. and Mason, G.: Compact Automorphism Groups of Vertex Operator Algebras. q-alg/9608009