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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)

##### MSC:
 53D50 Geometric quantization