Morita equivalence in deformation quantization.

摘要

In this dissertation we study the notion of Morita equivalence in the realm of formal deformation quantization of Poisson manifolds. We show that Rieffel's construction of induced representations of C*-algebras and the notion of strong Morita equivalence can be extended to a wider class of *-algebras, including those arising in deformation quantization. In order to study strong Morita equivalence in the framework of deformation quantization, we consider finitely generated projective inner-product modules over hermitian star-product algebras. These objects are quantum analogs of hermitian vector bundles and are obtained from them through a deformation quantization procedure. In this work, we define and study deformation quantization of hermitian vector bundles and show how they produce examples of strongly Morita equivalent deformed *-algebras. Finally, we present a study of the classification of star products on a Poisson manifold M up to Morita equivalence. We show how deformation quantization of line bundles over M produces a canonical action Φ of the Picard group Pic(M ) ≅ H2(M,$Z$) on the moduli space of equivalence classes of differential star products on M, in such a way that two star products are Morita equivalent if and only if they lie in the same Φ-orbit. We describe the semiclassical limit of Φ explicitly in terms of the characteristic classes of star products by studying contravariant connections arising in the semiclassical limit of line bundle deformations.