Deformation Quantization of Hermitian Vector Bundles

Abstract

Motivated by deformation quantization, we consider in this paper $^*$-algebras $\mathcal A$ over rings $\ring C = \ring{R}(i)$, where $\ring R$ is an ordered ring and $i^2 = -1$, and study the deformation theory of projective modules over these algebras carrying the additional structure of a (positive) $\mathcal A$-valued inner product. For $A=C^\infty(M)$, M a manifold, these modules can be identified with Hermitian vector bundles E over M. We show that for a fixed Hermitian star-product on M, these modules can always be deformed in a unique way, up to (isometric) equivalence. We observe that there is a natural bijection between the sets of equivalence classes of local Hermitian deformations of $C^\infty(M)$ and $\Gamma^\infty(\End(E))$ and that the corresponding deformed algebras are formally Morita equivalent, an algebraic generalization of strong Morita equivalence of $C^*$-algebras. We also discuss the semi-classical geometry arising from these deformations.