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

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 \({\mathcal{W}^t_{T*X}}\) .