Semiclassical Geometry of Quantum Line Bundles and Morita Equivalence of Star Products

Abstract

In this paper we show how deformation quantization of line bundles over a Poisson manifold $M$ produces a canonical action $\Phi$ of the Picard group $\Pic(M)\cong H^2(M,\mathbb Z)$ on the moduli space of equivalence classes of differential star products on $M$, $\Def(M)$. The orbits of $\Phi$ characterize Morita equivalent star products on $M$. We describe the semiclassical limit of $\Phi$ in terms of the characteristic classes of star products by studying the semiclassical geometry of deformed line bundles.

23 pages. Minor corrections made (Sec. 2.1), few typos fixed. Final version to appear at Internat. Math. Res. Notices