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 $${\mathcal{W}^t_{T*X}}$$ .