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 $\cor$-algebras $\W[T^*X]$ of deformation quantization, where $\cor\eqdot \W[\rmptt]$ is a subfield of $\C[[\hbar,\opb{\hbar}]$. Here, we construct a new sheaf of $\cor$-algebras $\TW[T^*X]$ which contains $\W[T^*X]$ 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 $\W[T^*X]$, we show that $\exp(t\opb{\hbar} P)$ is well defined in $\TW[T^*X]$.