余切空间上的星积代数在Hermite线丛上的表示
On representations of star product algebras over cotangent spaces on Hermitian line bundles
摘要：
对于流形$Q$上的闭二形式的形式幂级数$B=B_0 + λB_1 + O(λ^2)$和序参数$κ\in [0,1]$的每一个值，我们构造了余切丛$π: T^*Q\to Q$上的具体星积$\star^B_κ$.星积$\star^B_κ$与$T^*Q$上的形式辛形式相关联，该形式辛形式由标准辛形式$ω$和$B$拉回到$T^*Q$的和给出。Deligne的$\star^B_κ$的特征类的计算结果表明，符合正式的德-拉姆上同调类的$π^*B$除以$\imλ$。因此，$T^*Q$上对应于由辛形式$ω+ π^*B_0$导出的泊松括号的每个星积都等价于某个$\star^B_kappa$。证明了每个$\star^B_kappa$都是强闭的。在本文中，我们还构造和分类明确的形式表示的变形代数，以及运营商表示给出了一定的整体符号演算的伪微分算子$Q$。此外，我们表明，后者的运营商表示诱导的形式表示由一定的泰勒展开。从而得到了WKB展开式的一个紧致公式。
英文摘要：
For every formal power series $B=B_0 + λB_1 + O(λ^2)$ of closed two-forms on a manifold $Q$ and every value of an ordering parameter $κ\in [0,1]$ we construct a concrete star product $\star^B_κ$ on the cotangent bundle $π: T^*Q\to Q$. The star product $\star^B_κ$ is associated to the formal symplectic form on $T^*Q$ given by the sum of the canonical symplectic form $ω$ and the pull-back of $B$ to $T^*Q$. Deligne's characteristic class of $\star^B_κ$ is calculated and shown to coincide with the formal de Rham cohomology class of $π^*B$ divided by $\imλ$. Therefore, every star product on $T^*Q$ corresponding to the Poisson bracket induced by the symplectic form $ω+ π^*B_0$ is equivalent to some $\star^B_kappa$. It turns out that every $\star^B_kappa$ 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.
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