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The differential geometry of Fedosov’s quantization. (English) Zbl 0846.58031

Brylinski, Jean-Luc (ed.) et al., Lie theory and geometry: in honor of Bertram Kostant on the occasion of his 65th birthday. Invited papers, some originated at a symposium held at MIT, Cambridge, MA, USA in May 1993. Boston, MA: Birkhäuser. Prog. Math. 123, 217-239 (1994).
Some years ago in a remarkable paper, B. V. Fedosov [Sov. Phys., Dokl. 34, No. 4, 319-321 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 4, 835-838 (1989)] has presented a simple and very natural construction of a deformation quantization for any symplectic manifold. The construction begins with a linear symplectic connection on the tangent bundle of a manifold and proceeds by iteration to produce a flat connection on the associated bundle of formal Weyl algebras.
In the paper under review a classical analog of Fedosov’s operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version of the above construction is also pointed out. Finally, some remarks are made on the implications for deformation quantization of Fedosov’s index theorem on general symplectic manifolds.
For the entire collection see [Zbl 0807.00014].

MSC:

53D50 Geometric quantization
81S10 Geometry and quantization, symplectic methods
58J20 Index theory and related fixed-point theorems on manifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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