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On representations of star product algebras over cotangent spaces on Hermitian line bundles. (English) Zbl 1038.53087
From the authors’ abstract: For every formal power series $$B$$ of closed two-forms on a manifold $$Q$$ and every value of an ordering parameter from $$[0,1]$$ we construct a concrete star product $$B$$ on the cotangent bundle $$T^{*}Q$$. The star product $$B$$ is associated to the symplectic form on $$T^{*}Q$$ given by the sum of the canonical symplectic form and the pull back of $$B$$ to $$T^{*}Q$$. Deligne’s characteristic class of $$B$$ is calculated and shown to coincide with the formal de Rham cohomology class of $$^{*}B$$ divided by $$i$$. Therefore, every star product on $$T^{*}Q$$ corresponding to the canonical Poisson bracket is equivalent to some $$B$$. It turns out that every $$B$$ is strongly closed.
In this paper, we also construct and classify explicitly formal representations of the deformed algebra as well as operator representations given by a certain global symbol calculus for pseudodifferential operators on $$Q$$. Moreover, we show that the latter operator representations induce the formal representations by a certain Taylor expansion. We thereby obtain a compact formula for the WKB expansion.

##### MSC:
 53D55 Deformation quantization, star products 81S10 Geometry and quantization, symplectic methods
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##### References:
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