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The first Chern class as obstruction for asymptotic quantization. (La première classe de Chern comme obstruction à la quantification asymptotique.) (French) Zbl 0732.58020
Symplectic geometry, groupoids, and integrable systems, Sémin. Sud- Rhodan. Geom. VI, Berkeley/CA (USA) 1989, Math. Sci. Res. Inst. Publ. 20, 73-97 (1991).
The authors study the asymptotic quantization, considered by Karasëv, Maslov and Nazaĭkinskiĭ. By using the Maslov class the authors construct a cohomology class \([\varepsilon (E,\sigma)]\in H^ 2(M,{\mathbb Z})\) for every symplectic vector bundle \((E,\sigma\))\(\to M\). Then the fitting condition modulo \(O(h^ 2)\) for a certain presheaf \(\Pi (U_ i)\) is given by \((1/2\pi h)[\sigma]-(1/4)[\varepsilon]\in H^ 2(M;{\mathbb Z})\) (here \(h\) is the Planck constant appearing in the quantization operation). The authors show that \([\varepsilon (E,\sigma)]=2c_ 1(E,\sigma)\) where \(c_ 1\) is the first Chern class of \(E\). Thus it is obtained a purely symplectic definition of the first Chern class. In the case of a symplectic manifold \((M,\sigma)\) one gets the asymptotic quantization condition \((1/2\pi h)[\sigma]-(1/2)c_ 1(M,\sigma)\in H^ 2(M;{\mathbb Z})\). \(c_ 1(M,\sigma)\) is even if \((M,\sigma)\) has a metaplectic structure. Then the asymptotic quantization condition is reduced to the usual geometric quantization condition \((1/2\pi h)\in H^ 2(M;{\mathbb Z})\).
[For the entire collection see Zbl 0722.00026.]
Reviewer: V.Oproiu (Iaşi)

53D50 Geometric quantization