Quantization and Morita equivalence for constant Dirac structures on tori
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1565-1580

We define a C * -algebraic quantization of constant Dirac structures on tori and prove that O(n,n|)-equivalent structures have Morita equivalent quantizations. This completes and extends from the Poisson case a theorem of Rieffel and Schwarz.

Nous définissons une quantification C * -algebrique des structures de Dirac constantes sur les tores, et nous démontrons que l’équivalence à O(n,n|) près des structures implique l’équivalence de Morita de leurs quantifications. Ce résultat complète et généralise un théorème de Rieffel et Schwarz, donné dans le cadre des structures de Poisson.

DOI : https://doi.org/10.5802/aif.2059
Classification:  46L65,  81S10
@article{AIF_2004__54_5_1565_0,
     author = {Tang, Xiang and Weinstein, Alan},
     title = {Quantization and Morita equivalence for constant Dirac structures on tori},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1565-1580},
     doi = {10.5802/aif.2059},
     zbl = {1068.46044},
     mrnumber = {2127858},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2004__54_5_1565_0}
}
Tang, Xiang; Weinstein, Alan. Quantization and Morita equivalence for constant Dirac structures on tori. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1565-1580. doi : 10.5802/aif.2059. http://www.numdam.org/item/AIF_2004__54_5_1565_0/

[1] J. Block & E. Getzler, Quantization of foliations, Vol. 1, 2, World Scientific, 1992, p. 471-487 | Zbl 0812.58028

[2] A. Connes, Noncommutative Geometry, Academic Press, 1994 | MR 1303779 | Zbl 0818.46076

[3] A. Connes, M.R. Douglas & A. Schwarz, Noncommutative Geometry and Matrix Theory: Compactification on Tori, J. High Energy Phys (1998) | MR 1613978 | Zbl 1018.81052

[4] T.J. Courant, Dirac manifolds, Trans. A.M.S 319 (1990) p. 631-661 | MR 998124 | Zbl 0850.70212

[5] G. A. Elliott, On the K-theory of the C * algebras generated by a projective representation of a torsion-free discrete abelian group, Pitman, 1984, p. 157-184 | Zbl 0542.46030

[6] G.A. Elliott & H. Li, Morita equivalence of smooth noncommutative tori, e-print, math.OA/0311502 | Zbl 1137.46030

[7] H. Kajiura, Kronecker foliation, D1-branes and Morita equivalence of noncommutative two-tori, J. High Energy Phys 8 (2002) no.50 | MR 1942142 | Zbl 1226.81097

[8] M. Kontsevich, Homological algebra of mirror symmetry., Vol. 1, 2, Birkhäuser, 1995, p. 120-139 | Zbl 0846.53021

[9] H. Li, Strong Morita equivalence of higher-dimensional noncommutative tori, e-print. To appear J. Reine Angew. Math., math.OA/0303123 | MR 2099203 | Zbl 1063.46057

[10] F. Lizzi & R. Szabo, Noncommutative Geometry and String Duality, J. High Energy Phys., 1999

[11] P.S. Muhly, J.N. Renault & D.P. Williams, Equivalence and isomorphism for groupoid C * -algebras, J. Operator Theory 17 (1987) p. 3-22 | MR 873460 | Zbl 0645.46040

[12] S. Mukai, Duality between D(X) and D(X ^) with its application to Picard sheaves, Nagoya Math. J. 81 (1981) p. 153-175 | MR 607081 | Zbl 0417.14036

[13] M.A. Rieffel, Morita equivalence for C * -algebras and W * -algebras, J. Pure Appl. Algebra 5 (1974) p. 51-96 | MR 367670 | Zbl 0295.46099

[14] M.A. Rieffel, C * -algebras associated with irrational rotations, Pacific. J. Math. 93 (1981) p. 415-429 | MR 623572 | Zbl 0499.46039

[15] M.A. Rieffel, Projective modules over higher-dimensional non-commutative noncommutative tori, Canadian J. Math 40 (1988) p. 257-338 | MR 941652 | Zbl 0663.46073

[16] M.A. Rieffel, Deformation quantization of Heisenberg manifolds, Commun. Math. Phys 122 (1989) p. 531-562 | MR 1002830 | Zbl 0679.46055

[17] M.A. Rieffel & A. Schwarz, Morita equivalence of multidimensional noncommutative tori, Int. J. Math 10 (1999) p. 289-299 | MR 1687145 | Zbl 0968.46060

[18] A. Schwarz, Morita equivalence and duality, Lett. Math. Phys 50 (1999) p. 309-321 | MR 1663471 | Zbl 0967.58004

[19] X. Tang, Deformation Quantization of Pseudo Symplectic (Poisson) Groupoids, e-print, math.QA/0405378 | Zbl 05051278

[20] A. Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, MSRI Series, Springer, 1991, p. 281-290 | Zbl 0731.58031

[21] P. Xu, Noncommutative Poisson algebras, Amer. J. Math 116 (1994) p. 101-125 | MR 1262428 | Zbl 0797.58012