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Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. (English) Zbl 1065.53063
The paper under review is divided into six sections. Section 1 is a short introduction where the authors fix the picture of the paper. Section 2 describes the role of coisotropic submanifolds in Poisson geometry. Section 3 deals with the role of branes in a Poisson sigma model. The quantization problem makes the object of section 4. Some examples are sketched in section 5 and finally Kontsevich’s formalism is discussed in section 6. In conclusion, a very nice paper.

MSC:
53D17 Poisson manifolds; Poisson groupoids and algebroids
53D20 Momentum maps; symplectic reduction
53D55 Deformation quantization, star products
81T70 Quantization in field theory; cohomological methods
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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[1] Alexandrov, M. Kontsevich, M. Schwarz A and Zaboronsky, O.: The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12(1997), 1405-1430. · Zbl 1073.81655
[2] Bursztyn H. and Weinstein, A.: Picard groups in Poisson geometry, math.SG/0304048. · Zbl 1068.53055
[3] Cattaneo, A. S.: On the integration of Poisson manifolds, Lie algebroids, and coisotropic submanifolds, math.SG/0308180. · Zbl 1059.53064
[4] Cattaneo, A. S. and Felder, G.: A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212, (3) (2000), 591-612. · Zbl 1038.53088
[5] Cattaneo, A. S. and Felder, G.: Poisson sigma models and symplectic groupoids, in: N. P. Landsman, M. Pflaum, and M. Schlichenmeier (eds), Quantization of Singular Symplectic Quotients, Progr. in Math. 198, Birkh”auser, Basel, 2001, pp. 61-93. · Zbl 1038.53074
[6] Cattaneo, A. S. and Felder, G.: On the AKSZ formulation of the Poisson sigma model, Lett. Math. Phys. 56(2001), 163-179. · Zbl 1058.81034
[7] Cattaneo, A. S. and Felder, G.: in preparation
[8] Cattaneo, A. S., Felder, G. and Tomassini, L.: From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115(2) (2002), 329-352. · Zbl 1037.53063
[9] Crainic, M and Fernandes, R. L.: Integrability of Lie brackets, Ann, Math. 157(2003),575-620. · Zbl 1037.22003
[10] Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory, Ann. Phys. 235(1994),435-464. · Zbl 0807.53070
[11] Kapustin, A and Orlov, D.: Remarks on A-branes, mirror symmetry and the Fukaya category, hep-th/0109098. · Zbl 1029.81058
[12] Karasev, M. V.: The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds. I, II, Selecta Math. Soviet. 8(1989), 212-234, 235-258. · Zbl 0704.58019
[13] Karasev, M. V.: Analogues of the objects of Lie group theory for nonlinear Poisson brackets (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 50(1986), 508-538, (English) Math. USSR-Izv. 28(1987), 497-527. Karasev M. V. and Maslov, V. P.: Nonlinear Poisson brackets, geometry and quantization, Transl. Math. Monogr. 119 (1993). · Zbl 0608.58023
[14] Kontsevich, M.: Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (3) (2003) 157-216. · Zbl 1058.53065
[15] Landsman, N. P.: Quantized reduction as a tensor product, In: N. P. Landsman, M. Pflaum, and M. Schlichenmeier (eds), Quantization of Singular Symplectic Quotients, Progr. in Math. 198, Birkh”auser, Basel, 2001, pp. 137-180. · Zbl 1026.53051
[16] Landsman, N. P.: Functorial quantization and the Guillemin?Sternberg conjecture, math-ph/0307059. · Zbl 1179.58012
[17] Oh, Y. G. and Park, J. S.: Deformations of coisotropic submanifolds and strongly homotopy Lie algebroid, math.SG/0305292.
[18] Schaller, P. and Strobl, T.: Poisson structure induced (topological) field theories, ModernPhys. Lett. A 9(33) (1994), 3129-3136. · Zbl 1015.81574
[19] Severa, P.: Some title containing the words ’homotopy’ and ’symplectic’, e.g. this one, math.SG/0105080. · Zbl 1102.58010
[20] Weinstein, A.: The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), 523-557. · Zbl 0524.58011
[21] Weinstein, A.: Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. 16(1987), 101-104. · Zbl 0618.58020
[22] Weinstein, A.: Coisotropic calculus and Poisson groupoids, J. Math, Soc. Japan 40(1988), 705-727. · Zbl 0642.58025
[23] Zakrzewski, S.: Quantum and classical pseudogroups, Part I: Union pseudogroups and their quantization, Comm. Math. Phys. 134(1990), 347-370; Quantum and classical pseudogroups, Part II: Differential and symplectic pseudogroups, Comm. Math. Phys. 134 (1990), 371-395. · Zbl 0708.58030
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