the symplectic linear category with half-forms. The automorphism group of an object in this category is a concrete realization or the metaplectic group. It is the subject of section \[sec:metaplectic-group\], where the index of some of its element is defined. In section \[sec:gener-half-form\], we generalize the previous considerations, which is necessary for the application to the quantum representation of the mapping class group. In section \[sec:quant-kahl-manif\], we consider the quantization of K[ä]{}hler manifold with an auxiliary line bundle, introduce the operator quantizing the symplectomorphisms and estimate their trace. In section \[sec:quant-half-form\], we treat the particular case where the auxiliary bundle is a half-form bundle. In appendix \[sec:linear-quantization\], we introduce a functor from the category of symplectic vector spaces with polarization and half-form to the category of Hilbert spaces. Applying this functor to the automorphism group of a single object, we recover the well-known metaplectic representation. With this elementary construction in mind, one can view the quantization of symplectomorphisms studied in this article as a generalization of the metaplectic representation.

Half-forms {#sec:half-forms}
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Complex structures and canonical lines
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Let $S$ be a symplectic real vector space. A positive polarization $E$ of $S$ is a Lagrangian subspace of $S \otimes {{\mathbb{C}}}$ 