An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

Abstract

The cotangent bundle T ^* X to a complex manifold X is classically endowed with the sheaf of k-algebras ${\mathcal{W}_{T*X}}$ of deformation quantization, where k := ${\mathcal{W}_{\{pt\}}}$ is a subfield of ${\mathbb{C}[[\hbar, \hbar^{-1}]}$ . Here, we construct a new sheaf of k-algebras ${\mathcal{W}^t_{T*X}}$ which contains ${\mathcal{W}_{T*X}}$ as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of ${\mathcal{W}_{T*X}}$ , we show that ${{\rm exp}(t\hbar^{-1} P)}$ is well defined in ${\mathcal{W}^t_{T*X}}$ .