String geometry and the noncommutative torus

Communications in Mathematical Physics

Volumn 206, Issue 3, 1999, Pages 603-637

  Landi, Giovanni ^a,d,e   Lizzi, Fedele ^b   Szabo, Richard J ^c,f  

^a UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^b UNIVERSITY OF NAPLES FEDERICO II   (Italy)

^c UNIVERSITY OF OXFORD   (United Kingdom)

^d UNIVERSITY OF TRIESTE   (Italy)

^e INFN SEZIONE DI NAPOLI   (Italy)

^f UNIVERSITY OF COPENHAGEN   (Denmark)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033517394     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050839     Document Type: Article

References (67)

1. 1. Rieffel, M.A.: C*-algebras associated with Irrational Rotations. Pac. J. Math. 93, 415 (1981)
2. 2. Connes, A.: C*-algèbres et Géométries Différentielle. C. R. Acad. Sci. Paris A290, 599 (1980); Connes, A. and Rieffel, M.A.: Yang-Mills for Noncommutative Two-tori. Contemp. Math. 62, 237 (1987); Rieffel, M.A.: Projective Modules over Higher-dimensional Noncommutative Tori. Can. J. Math. 40, 257 (1988)
3. 2. Connes, A.: C*-algèbres et Géométries Différentielle. C. R. Acad. Sci. Paris A290, 599 (1980); Connes, A. and Rieffel, M.A.: Yang-Mills for Noncommutative Two-tori. Contemp. Math. 62, 237 (1987); Rieffel, M.A.: Projective Modules over Higher-dimensional Noncommutative Tori. Can. J. Math. 40, 257 (1988)
4. 2. Connes, A.: C*-algèbres et Géométries Différentielle. C. R. Acad. Sci. Paris A290, 599 (1980); Connes, A. and Rieffel, M.A.: Yang-Mills for Noncommutative Two-tori. Contemp. Math. 62, 237 (1987); Rieffel, M.A.: Projective Modules over Higher-dimensional Noncommutative Tori. Can. J. Math. 40, 257 (1988)
5. 3. Rieffel, M.A.: The Cancellation Theorem for Projective Modules over Irrational Rotation C*-algebras. Proc. London Math. Soc. 47, 285 (1983)
6. 4. Connes, A.: Noncommutative Geometry. London: Academic Press, 1994
7. 5. Landi, G.: An Introduction to Noncommutative Spaces and their Geometries. Berlin-Heidelberg-New York: Springer, 1997
8. 6. Connes, A., Douglas, M.R. and Schwarz, A.: Noncommutative Geometry and Matrix Theory: Compactification on Tori. J. High Energy Phys. 9802, 003 (1998)
9. 7. Douglas, M.R. and Hull, C.M.: D-branes and the Noncommutative Torus. J. High Energy Phys. 9802, 008 (1998); Li, M.: Comments on Supersymmetric Yang-Mills Theory on a Noncommutative Torus. hep-th/9802052; Berkooz, M.: Nonlocal Field Field Theories and the Noncommutative Torus. Phys. Lett. B430, 237 (1998); Cheung, Y.-K.E. and Krogh, M.: Noncommutative Geometry from 0-branes in a Background B-field. Nucl. Phys. B528, 185 (1998); Bigatti, D.: Noncommutative Geometry and Super Yang-Mills Theory. hep-th/9804120
10. 7. Douglas, M.R. and Hull, C.M.: D-branes and the Noncommutative Torus. J. High Energy Phys. 9802, 008 (1998); Li, M.: Comments on Supersymmetric Yang-Mills Theory on a Noncommutative Torus. hep-th/9802052; Berkooz, M.: Nonlocal Field Field Theories and the Noncommutative Torus. Phys. Lett. B430, 237 (1998); Cheung, Y.-K.E. and Krogh, M.: Noncommutative Geometry from 0-branes in a Background B-field. Nucl. Phys. B528, 185 (1998); Bigatti, D.: Noncommutative Geometry and Super Yang-Mills Theory. hep-th/9804120
11. 7. Douglas, M.R. and Hull, C.M.: D-branes and the Noncommutative Torus. J. High Energy Phys. 9802, 008 (1998); Li, M.: Comments on Supersymmetric Yang-Mills Theory on a Noncommutative Torus. hep-th/9802052; Berkooz, M.: Nonlocal Field Field Theories and the Noncommutative Torus. Phys. Lett. B430, 237 (1998); Cheung, Y.-K.E. and Krogh, M.: Noncommutative Geometry from 0-branes in a Background B-field. Nucl. Phys. B528, 185 (1998); Bigatti, D.: Noncommutative Geometry and Super Yang-Mills Theory. hep-th/9804120
12. 7. Douglas, M.R. and Hull, C.M.: D-branes and the Noncommutative Torus. J. High Energy Phys. 9802, 008 (1998); Li, M.: Comments on Supersymmetric Yang-Mills Theory on a Noncommutative Torus. hep-th/9802052; Berkooz, M.: Nonlocal Field Field Theories and the Noncommutative Torus. Phys. Lett. B430, 237 (1998); Cheung, Y.-K.E. and Krogh, M.: Noncommutative Geometry from 0-branes in a Background B-field. Nucl. Phys. B528, 185 (1998); Bigatti, D.: Noncommutative Geometry and Super Yang-Mills Theory. hep-th/9804120
13. 7. Douglas, M.R. and Hull, C.M.: D-branes and the Noncommutative Torus. J. High Energy Phys. 9802, 008 (1998); Li, M.: Comments on Supersymmetric Yang-Mills Theory on a Noncommutative Torus. hep-th/9802052; Berkooz, M.: Nonlocal Field Field Theories and the Noncommutative Torus. Phys. Lett. B430, 237 (1998); Cheung, Y.-K.E. and Krogh, M.: Noncommutative Geometry from 0-branes in a Background B-field. Nucl. Phys. B528, 185 (1998); Bigatti, D.: Noncommutative Geometry and Super Yang-Mills Theory. hep-th/9804120
14. 8. Ho, P.-M., Wu, Y.-Y. and Wu, Y.-S.: Towards a Noncommutative Geometric Approach to Matrix Compactification. Phys. Rev. D58, 026006 (1998); Kawano, T. and Okuyama, K.: Matrix Theory on Noncommutative Torus. Phys. Lett. B433, 29 (1998); Ardalan, F., Arfaei, H. and Sheikh-Jabbari, M.M.: Mixed Branes and Matrix Theory on Noncommutative Torus. hep-th/9803067; Ho, P.-M.: Twisted Bundle on Quantum Torus and BPS States in Matrix Theory. Phys. Lett. B434, 41 (1998)
15. 8. Ho, P.-M., Wu, Y.-Y. and Wu, Y.-S.: Towards a Noncommutative Geometric Approach to Matrix Compactification. Phys. Rev. D58, 026006 (1998); Kawano, T. and Okuyama, K.: Matrix Theory on Noncommutative Torus. Phys. Lett. B433, 29 (1998); Ardalan, F., Arfaei, H. and Sheikh-Jabbari, M.M.: Mixed Branes and Matrix Theory on Noncommutative Torus. hep-th/9803067; Ho, P.-M.: Twisted Bundle on Quantum Torus and BPS States in Matrix Theory. Phys. Lett. B434, 41 (1998)
16. 8. Ho, P.-M., Wu, Y.-Y. and Wu, Y.-S.: Towards a Noncommutative Geometric Approach to Matrix Compactification. Phys. Rev. D58, 026006 (1998); Kawano, T. and Okuyama, K.: Matrix Theory on Noncommutative Torus. Phys. Lett. B433, 29 (1998); Ardalan, F., Arfaei, H. and Sheikh-Jabbari, M.M.: Mixed Branes and Matrix Theory on Noncommutative Torus. hep-th/9803067; Ho, P.-M.: Twisted Bundle on Quantum Torus and BPS States in Matrix Theory. Phys. Lett. B434, 41 (1998)
17. 8. Ho, P.-M., Wu, Y.-Y. and Wu, Y.-S.: Towards a Noncommutative Geometric Approach to Matrix Compactification. Phys. Rev. D58, 026006 (1998); Kawano, T. and Okuyama, K.: Matrix Theory on Noncommutative Torus. Phys. Lett. B433, 29 (1998); Ardalan, F., Arfaei, H. and Sheikh-Jabbari, M.M.: Mixed Branes and Matrix Theory on Noncommutative Torus. hep-th/9803067; Ho, P.-M.: Twisted Bundle on Quantum Torus and BPS States in Matrix Theory. Phys. Lett. B434, 41 (1998)
18. 9. Ho, P.-M. and Wu, Y.-S.: Noncommutative Gauge Theories in Matrix Theory. Phys. Rev. D58, 066003 (1998): Casalbuoni, R.: Algebraic Treatment of Compactification on Noncommutative Tori. Phys. Lett. B431, 69 (1998)
19. 9. Ho, P.-M. and Wu, Y.-S.: Noncommutative Gauge Theories in Matrix Theory. Phys. Rev. D58, 066003 (1998): Casalbuoni, R.: Algebraic Treatment of Compactification on Noncommutative Tori. Phys. Lett. B431, 69 (1998)
20. 10. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85 (1997)
21. 10. Witten, E.: Bound States of Strings and p-branes. Nucl. Phys. B460, 335 (1996); Douglas, M.R., Kabat, D., Pouliot, P. and Shenker, S.H.: D-branes and Short Distances in String Theory. Nucl. Phys. B485, 85 (1997)
22. 11. Ho, P.-M. and Wu, Y.-S.: Noncommutative Geometry and D-branes. Phys. Lett. B398, 52 (1997)
23. 12. Lizzi, F., Mavromatos, N.E. and Szabo, R.J.: Matrix σ-models forMulti D-brane Dynamics. Mod. Phys. Lett. A13, 829 (1998)
24. 13. Banks, T., Fischler, W., Shenker, S.H. and Susskind, L.: M Theory as a Matrix Model: A Conjecture. Phys. Rev. D55, 5112 (1997)
25. 14. Fröhlich, J. and Gawȩdzki, K.: Conformal Field Theory and Geometry of Strings. CRM Proc. Lecture Notes 7, 57 (1994)
26. 15. Chamseddine, A.H.: The Spectral Action Principle in Noncommutative Geometry and the Superstring. Phys. Lett. B400, 87 (1997); An Effective Superstring Spectral Action. Phys. Rev. D56, 3555 (1997)
27. 15. Chamseddine, A.H.: The Spectral Action Principle in Noncommutative Geometry and the Superstring. Phys. Lett. B400, 87 (1997); An Effective Superstring Spectral Action. Phys. Rev. D56, 3555 (1997)
28. 16. Lizzi, F. and Szabo, R.J.: Target Space Duality in Noncommutative Geometry. Phys. Rev. Lett. 79, 3581 (1997)
29. 17. Lizzi, F. and Szabo, R.J.: Duality Symmetries and Noncommutative Geometry of String Spacetimes. Commun. Math. Phys. 197, 667 (1998)
30. 18. Rieffel, M.A. and Schwarz, A.: Morita Equivalence of Multidimensional Noncommutative Tori. math.QA/9803057, to appear in Int. J. of Math.
31. 19. Schwarz, A.: Morita Equivalence and Duality. Nucl. Phys. B534, 720 (1998)
32. 20. Lizzi, F. and Szabo, R.J.: Electric-magnetic Duality in Noncommutative Geometry. Phys. Lett. B417, 303 (1998)
33. 21. Lizzi, F. and Szabo, R.J.: Noncommutative Geometry and Spacetime Gauge Symmetries of String Theory. Chaos, Solitons and Fractals 10, 445 (1999)
34. 22. Faddeev, L.D.: Discrete Heisenberg-Weyl Group and Modular Group. Lett. Math. Phys. 34, 249 (1995)
35. 23. Connes, A. and Lott, J.: Particle Models and Noncommutative Geometry. Nucl. Phys. B (Proc. Suppl.) 18B, 29 (1990);
36. 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)
37. 24. Connes, A.: Noncommutative Geometry and Reality. J. Math. Phys. 36, 619 (1995)
38. 25. Connes, A.: Gravity Coupled with Matter and the Foundation of Noncommutative Geometry. Commun. Math. Phys. 182, 155 (1996)
39. 26. 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)
40. 26. 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)
41. 27. Zwanziger, D.: Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges. Phys. Rev. D3, 880 (1971); Deser, S. and Teitelboim, C.: Duality Transformations of Abelian and Nonabelian Gauge Fields. Phys. Rev. D13, 1592 (1976); Schwarz, J.H. and Sen, A.: Duality Symmetric Actions. Nucl. Phys. B411, 35 (1994); Deser, S., Gomberoff, A., Henneaux, M. and Teitelboim, C.: Duality, Selfduality, Sources and Charge Quantization in Abelian N-form Theories. Phys. Lett. B400, 80 (1997)
42. 27. Zwanziger, D.: Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges. Phys. Rev. D3, 880 (1971); Deser, S. and Teitelboim, C.: Duality Transformations of Abelian and Nonabelian Gauge Fields. Phys. Rev. D13, 1592 (1976); Schwarz, J.H. and Sen, A.: Duality Symmetric Actions. Nucl. Phys. B411, 35 (1994); Deser, S., Gomberoff, A., Henneaux, M. and Teitelboim, C.: Duality, Selfduality, Sources and Charge Quantization in Abelian N-form Theories. Phys. Lett. B400, 80 (1997)
43. 27. Zwanziger, D.: Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges. Phys. Rev. D3, 880 (1971); Deser, S. and Teitelboim, C.: Duality Transformations of Abelian and Nonabelian Gauge Fields. Phys. Rev. D13, 1592 (1976); Schwarz, J.H. and Sen, A.: Duality Symmetric Actions. Nucl. Phys. B411, 35 (1994); Deser, S., Gomberoff, A., Henneaux, M. and Teitelboim, C.: Duality, Selfduality, Sources and Charge Quantization in Abelian N-form Theories. Phys. Lett. B400, 80 (1997)
44. 27. Zwanziger, D.: Local-Lagrangian Quantum Field Theory of Electric and Magnetic Charges. Phys. Rev. D3, 880 (1971); Deser, S. and Teitelboim, C.: Duality Transformations of Abelian and Nonabelian Gauge Fields. Phys. Rev. D13, 1592 (1976); Schwarz, J.H. and Sen, A.: Duality Symmetric Actions. Nucl. Phys. B411, 35 (1994); Deser, S., Gomberoff, A., Henneaux, M. and Teitelboim, C.: Duality, Selfduality, Sources and Charge Quantization in Abelian N-form Theories. Phys. Lett. B400, 80 (1997)
45. 28. Pasti, P., Sorokin, D. and Tonin, M.: Duality Symmetric Actions with Manifest Spacetime Symmetries. Phys. Rev. D52, 4277 (1995); Berkovits, N.: Manifest Electromagnetic Duality in Closed Superstring Field Theory. Phys. Lett. B388, 743 (1996); Local Actions with Electric and Magnetic Sources. B395, 28 (1997); Super-Maxwell Actions with Manifest Duality. B398, 79 (1997); Medina, R. and Berkovits, N.: Pasti-Sorokin-Tonin Actions in the Presence of Sources. Phys. Rev. D56, 6388 (1997)
46. 28. Pasti, P., Sorokin, D. and Tonin, M.: Duality Symmetric Actions with Manifest Spacetime Symmetries. Phys. Rev. D52, 4277 (1995); Berkovits, N.: Manifest Electromagnetic Duality in Closed Superstring Field Theory. Phys. Lett. B388, 743 (1996); Local Actions with Electric and Magnetic Sources. B395, 28 (1997); Super-Maxwell Actions with Manifest Duality. B398, 79 (1997); Medina, R. and Berkovits, N.: Pasti-Sorokin-Tonin Actions in the Presence of Sources. Phys. Rev. D56, 6388 (1997)
47. 28. Pasti, P., Sorokin, D. and Tonin, M.: Duality Symmetric Actions with Manifest Spacetime Symmetries. Phys. Rev. D52, 4277 (1995); Berkovits, N.: Manifest Electromagnetic Duality in Closed Superstring Field Theory. Phys. Lett. B388, 743 (1996); Local Actions with Electric and Magnetic Sources. B395, 28 (1997); Super-Maxwell Actions with Manifest Duality. B398, 79 (1997); Medina, R. and Berkovits, N.: Pasti-Sorokin-Tonin Actions in the Presence of Sources. Phys. Rev. D56, 6388 (1997)
48. 28. Pasti, P., Sorokin, D. and Tonin, M.: Duality Symmetric Actions with Manifest Spacetime Symmetries. Phys. Rev. D52, 4277 (1995); Berkovits, N.: Manifest Electromagnetic Duality in Closed Superstring Field Theory. Phys. Lett. B388, 743 (1996); Local Actions with Electric and Magnetic Sources. B395, 28 (1997); Super-Maxwell Actions with Manifest Duality. B398, 79 (1997); Medina, R. and Berkovits, N.: Pasti-Sorokin-Tonin Actions in the Presence of Sources. Phys. Rev. D56, 6388 (1997)
49. 28. Pasti, P., Sorokin, D. and Tonin, M.: Duality Symmetric Actions with Manifest Spacetime Symmetries. Phys. Rev. D52, 4277 (1995); Berkovits, N.: Manifest Electromagnetic Duality in Closed Superstring Field Theory. Phys. Lett. B388, 743 (1996); Local Actions with Electric and Magnetic Sources. B395, 28 (1997); Super-Maxwell Actions with Manifest Duality. B398, 79 (1997); Medina, R. and Berkovits, N.: Pasti-Sorokin-Tonin Actions in the Presence of Sources. Phys. Rev. D56, 6388 (1997)
50. 29. Deser, S., Henneaux, M. and Teitelboim, C.: Electric-magnetic Black Hole Duality. Phys. Rev. D55, 826 (1997); Song, D.D. and Szabo, R.J.: Black String Entropy from Anomalous D-brane Couplings. hep-th/9805027
51. 29. Deser, S., Henneaux, M. and Teitelboim, C.: Electric-magnetic Black Hole Duality. Phys. Rev. D55, 826 (1997); Song, D.D. and Szabo, R.J.: Black String Entropy from Anomalous D-brane Couplings. hep-th/9805027
52. 30. Aganagic, M., Park, J., Popescu, C. and Schwarz, J.H.: Dual D-brane Actions. Nucl. Phys. B496, 215 (1997); Nurmagambetov, A.: Duality-symmetric Three-brane and its Coupling to Type IIB Supergravity. Phys. Lett. B436, 289 (1998)
53. 30. Aganagic, M., Park, J., Popescu, C. and Schwarz, J.H.: Dual D-brane Actions. Nucl. Phys. B496, 215 (1997); Nurmagambetov, A.: Duality-symmetric Three-brane and its Coupling to Type IIB Supergravity. Phys. Lett. B436, 289 (1998)
54. 31. Cheung, Y.-K.E. and Yin, Z.: Anomalies, Branes and Currents. Nucl. Phys. B517, 69 (1998)
55. 32. Frenkel, I.B., Lepowsky, J. and Meurman, A.: Vertex Operator Algebras and the Monster. Pure Appl. Math. 134 New York: Academic Press, 1988; Gebert, R.W.: Introduction to Vertex Algebras, Borcherds Algebras and the Monster Lie Algebra. Intern. J. Mod. Phys. A8, 5441
56. 32. Frenkel, I.B., Lepowsky, J. and Meurman, A.: Vertex Operator Algebras and the Monster. Pure Appl. Math. 134 New York: Academic Press, 1988; Gebert, R.W.: Introduction to Vertex Algebras, Borcherds Algebras and the Monster Lie Algebra. Intern. J. Mod. Phys. A8, 5441
57. 33. Green, M.B., Schwarz, J.H. and Witten, E.: Superstring Theory. Cambridge: Cambridge University Press, 1987
58. 34. Constantinescu, F. and Scharf, G.: Smeared and Unsmeared Chiral Vertex Operators. Commun. Math. Phys. 200, 275 (1999)
59. 35. De Bièvre, S.: Chaos, Quantization and the Classical Limit on the Torus Proc. XIV Workshop on Geometrical Methods in Physics, July 1995, Bialowieza, Poland, mp-arc 96-191; Várilly, J.C.: An Introduction to Noncommutative Geometry. Lectures at the EMS Summer School on Noncommutative Geometry and Applications, September 1997, Portugal, physics/9709045
60. 35. De Bièvre, S.: Chaos, Quantization and the Classical Limit on the Torus Proc. XIV Workshop on Geometrical Methods in Physics, July 1995, Bialowieza, Poland, mp-arc 96-191; Várilly, J.C.: An Introduction to Noncommutative Geometry. Lectures at the EMS Summer School on Noncommutative Geometry and Applications, September 1997, Portugal, physics/9709045
61. 36. Fairlie, D., Fletcher, P. and Zachos, C.: Trigonometric Structure Constants for New Infinite Dimensional Algebras. Phys. Lett. B218, 203 (1989); Infinite-Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras. J. Math. Phys. 31, 1088 (1990); Fairlie, D. and Zachos, C.: Infinite-Dimensional Algebras, Sine Brackets, and SU(∞). Phys. Lett. B224, 101 (1989)
62. 36. Fairlie, D., Fletcher, P. and Zachos, C.: Trigonometric Structure Constants for New Infinite Dimensional Algebras. Phys. Lett. B218, 203 (1989); Infinite-Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras. J. Math. Phys. 31, 1088 (1990); Fairlie, D. and Zachos, C.: Infinite-Dimensional Algebras, Sine Brackets, and SU(∞). Phys. Lett. B224, 101 (1989)
63. 36. Fairlie, D., Fletcher, P. and Zachos, C.: Trigonometric Structure Constants for New Infinite Dimensional Algebras. Phys. Lett. B218, 203 (1989); Infinite-Dimensional Algebras and a Trigonometric Basis for the Classical Lie Algebras. J. Math. Phys. 31, 1088 (1990); Fairlie, D. and Zachos, C.: Infinite-Dimensional Algebras, Sine Brackets, and SU(∞). Phys. Lett. B224, 101 (1989)
64. 37. Gracia-Bondía, J.M., Iochum, B. and Schücker, T.: The Standard Model in Noncommutative Geometry and Fermion Doubling. Phys. Lett. B416, 123 (1998)
65. 38. Rieffel, M.A.: Induced Representations of C*-algebras. Bull. Am. Math. Soc. 78, 606 (1972); Adv. Math. 13, 176 (1974); Morita Equivalence for Operator Algebras. In: Operator Algebras and Applications, Proc. Symp. Pure Math. 38, R.V. Kadison, ed. Providence, RI: American Mathematical Society, 1982, pp. 285-298
66. 38. Rieffel, M.A.: Induced Representations of C*-algebras. Bull. Am. Math. Soc. 78, 606 (1972); Adv. Math. 13, 176 (1974); Morita Equivalence for Operator Algebras. In: Operator Algebras and Applications, Proc. Symp. Pure Math. 38, R.V. Kadison, ed. Providence, RI: American Mathematical Society, 1982, pp. 285-298
67. 38. Rieffel, M.A.: Induced Representations of C*-algebras. Bull. Am. Math. Soc. 78, 606 (1972); Adv. Math. 13, 176 (1974); Morita Equivalence for Operator Algebras. In: Operator Algebras and Applications, Proc. Symp. Pure Math. 38, R.V. Kadison, ed. Providence, RI: American Mathematical Society, 1982, pp. 285-298