On representations of star product algebras over cotangent spaces on Hermitian line bundles

For every formal power series $B=B_0+\lambda B_1+O({\lambda }^2)$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $\kappa\in [0,1]$weconstructaconcretestarproduct$\star^B_\kappa$ on the cotangent bundle $\pi :T^{\ast }Q\to Q$. The star product ${\star }_{\kappa }^B$ is associated to the formal symplectic form on $T^{\ast }Q$ given by the sum of the canonical symplectic form $\omega$ and the pull-back of $B$ to $T^{\ast }Q$. Deligne’s characteristic class of ${\star }_{\kappa }^B$ is calculated and shown to coincide with the formal de Rham cohomology class of ${\pi }^{\ast }B$ divided by $\text{im}\lambda$. Therefore, every star product on $T^{\ast }Q$ corresponding to the Poisson bracket induced by the symplectic form $\omega +{\pi }^{\ast }{B}_0$ is equivalent to some ${\star }_k^{B}appa$. It turns out that every ${\star }_k^{B}appa$ 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.