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A Symplectic Approach to Yang Mills Theory for Non Commutative Tori

Published online by Cambridge University Press:  20 November 2018

Mauro Spera*
Affiliation:
Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate Università di PadovaVia Belzoni 7 35131 Padova, Italy
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Abstract

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In this note we give a symplectic approach to Yang Mills theory for non commutative n-tori, inspired by the classical theory of Atiyah and Bott.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Atiyah, M.F. and Bott, R., The Yang Mills Equations over Riemann Surfaces, Phil. Trans. R. Soc. London A 308(1982), 523615.Google Scholar
2. Abraham, R. and Marsden, J.E., Foundations of Mechanics (2nd edition), Benjamin London, Amsterdam, 1978.Google Scholar
3. Connes, A., C* algebres et géométrie différentielle, C.R. Acad. Sci. Paris, Série I, 290( 1980), 599604.Google Scholar
4. Connes, A., Non Commutative Differential Geometry, Publ. Math. IHES 62(1986), 41144.Google Scholar
5. Connes, A., A Survey ofC* algebras of Foliations, Proc. Symp. Pure Math. 38, 1(1982), 521628.Google Scholar
6. Connes, A. and Rieffel, M.A., Yang Mills for Non Commutative Two Tori, Proceedings of the Conference on Operator Algebras and Mathematical Physics, Univ. of Iowa, 1985 Contemporary Mathematics 62(1987), 237266.Google Scholar
7. Dixmier, J., Sur la Relation i(PQ - QP)= /, Compos. Math. 13(1956), 263269.Google Scholar
8. Donaldson, S.K., A New Proof of a Theorem ofNarasimhan and Seshadri, J. Diff. Geom. 18(1983), 269- 277.Google Scholar
9. Donaldson, S.K., Anti Self-dual Yang Mills Connections over Complex Algebraic Surfaces, Proc. London Math. Soc. 50(1985), 126.Google Scholar
10. Freed, D. and Uhlenbeck, K., Istantons on Four-Manifolds, Springer-Verlag Berlin, Heidelberg 1984.Google Scholar
11. Guillemin, V. and Sternberg, S., Geometric Quantization and Multiplicity of Group Representations, Inv. Math. 67(1982), 515538.Google Scholar
12. Howe, R., On the role of the Heisenberg group in Harmonic Analysis, Bull. Am. Math. Soc. 3(1980), 822- 843.Google Scholar
13. Kirillov, A., Eléments de la Théorie des Représentations, MIR Publishers Moscow 1974.Google Scholar
14. Lawson, H.B., The theory of gauge fields in four dimensions, Reg. Conf. Ser. in Math 58 Providence, Rhode Island 1985.Google Scholar
15. Reed, M. and Simon, B., Methods of Modern Mathematical Physics, (Volumes I and II), Academic Press New York 1972–75.Google Scholar
16. Rieffel, M.A., Vector Bundles over Higher Dimensional Non Commutative Tori, Lecture Notes in Mathematics 1132, 456467.Springer-Verlag, Berlin, Heidelberg, New York 1985.Google Scholar
17. Rieffel, M.A., Projective modules over higher dimensional non commutative tori, Canad. J. Math., (2) XL( 1988), 257338.Google Scholar
18. Rieffel, M.A., Critical Points of Yang Mills for Non Commutative Two Tori, J. Diff. Geom. 31(1990), 535546.Google Scholar
19. Spera, M., Quantum Mechanical Commutation Relations and Differential Geometry (Classical and Non Commutative), in Proceedings of the Open University Conference on Statistical Mechanics (Solomon, A. Ed.) World Scientific Press (1988), 74100.Google Scholar
20. Spera, M., Yang Mills Equations and Holomorphic Structures on C*-dynamical Systems, Preprint (1988) (unpublished).Google Scholar
21. Spera, M., Yang Mills theory in non commutative differential geometry, Rend. Sem. Fac. Scienze Univ. Cagliari, Suppl. 58(1988), 409421.Google Scholar
22. Spera, M., A Non Commutative Geometric Re interpretation of the Canonical Commutation Relations, Bollettino dell'Unione Matematica Italiana, 58(1991), 5363.Google Scholar
23. Spera, M., Sobolev theory for non commutative tori, Rend. Sem. Mat. Univ. Padova (1991) (to appear).Google Scholar
24. Spera, M., A Note on Yang Mills Minima on Rieffel Modules over Higher Dimensional Non Commutative Tori, Preprint (1990).Google Scholar
25. von Neumann, J., Die Eindeutigkeit der Schrodingerschen Operatoren, Math. Ann. 104(1931), 570578.Google Scholar