Abstract
This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to build examples of Ricci-flat connections. For a more extensive survey, see Bieliavsky et al. [Int. J. Geom. Methods Mod. Phys. 3, 375–420 2006]. This note also includes a moment map for the action of the group of symplectomorphisms on the space of symplectic connections, an algebraic construction of a large class of Ricci-flat symmetric symplectic spaces, and an example of global reduction in a non-symmetric case.
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References
Baguis P., Cahen M. (2001) A construction of symplectic connections through reduction. Lett. Math. Phys. 57, 149–160
Bayen F., Flato M., Fronsdal C., Lichnerowicz A., Sternheimer D. (1978) Deformation theory and quantization, part I. Ann. Phys. 111, 61–110
Bieliavsky, P.: Espaces Symétriques Symplectiques. PhD thesis, ULB (1995)
Bieliavsky, P., Cahen, M., Gutt, S.: Symmetric symplectic manifolds and deformation quantization. Modern group theoretical methods in physics (Paris, 1995). Math. Phys. Stud. 18, 63–73. Kluwer Academic, Dordrecht (1995)
Bieliavsky P., Cahen M., Gutt S., Rawnsley J., Schwachhöfer L. (2006) Symplectic connections. Int. J. Geom. Methods Mod. Phys. 3, 375–420
Bourgeois F., Cahen M. (1999) A variational principle for symplectic connections. J. Geome. Phys. 30, 233–265
Cahen, M., Gutt, S., Rawnsley, J.: Symmetric symplectic spaces with Ricci-type curvature. In: Dito, G., Sternheimer, D. (eds.) Conférence Moshé Flato 1999, vol. 2. Math. Phys. Stud. 22, 81–91 (2000)
Cahen M., Gutt S., Schwachhöfer L. (2004) Construction of Ricci-type connections by reduction and induction. In: Marsden J.E., Ratiu T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Birkhauser, Boston, pp. 41–57, Progress in Mathematics, vol. 232.
Cahen, M., Schwachhöfer, L.: Special symplectic connections, preprint DG0402221
Fedosov B.V. (1994) A simple geometrical construction of deformation quantization. J. Differ. Geom. 40, 213–238
Flato M., Lichnerowicz A., Sternheimer D. (1976) Crochet de Moyal–Vey et quantification. C. R. Acad. Sci. Paris I Math. 283, 19–24
Gutt S., Rawnsley J. (2003) Natural star products on symplectic manifolds and quantum moment maps. Lett. Math. Phys. 66, 123–139
Horowitz, J.: PhD Thesis, Université Libre de Bruxelles (2001)
Kostant B. (2004) Minimal coadjoint orbits and symplectic induction. In: Marsden J.E., Ratiu T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Birkhauser, Boston, pp. 391–422, Progress in Mathematics, vol. 232.
Lemlein V.G. (1957) On spaces with symmetric almost symplectic connection (in Russian). Dokl. Akad. nauk SSSR 115(4):655–658
Lichnerowicz A. (1982) Déformations d’algèbres associées à une variété symplectique (les \({*_\nu}\) -produits). Ann. Inst. Fourier Grenoble 32, 157–209
Tondeur P. (1961) Affine Zusammenhänge auf Mannigfaltigkeiten mit fast-symplektischer Struktur. Comment. Helv. Math. 36, 234–244
Vaisman I. (1985) Symplectic curvature tensors. Monats. Math. 100, 299–327
De Visher, M.: Mémoire de licence, Bruxelles (1999)
Vaisman I. (1998) Variations on the theme of twistor spaces. Balk. J. Geom. Appl. 3, 135–156
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Gutt, S. Remarks on Symplectic Connections. Lett Math Phys 78, 307–328 (2006). https://doi.org/10.1007/s11005-006-0126-y
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DOI: https://doi.org/10.1007/s11005-006-0126-y