Star products and local line bundles

TL;DR

This paper introduces local line bundles on manifolds, explores their role in twisting pseudodifferential operators, and connects these ideas to star products, index formulas, and residue traces in symplectic geometry.

Contribution

It establishes a link between local line bundles, twisted pseudodifferential operators, and star products, extending Fedosov's construction and residue trace concepts.

Findings

Twisted pseudodifferential operators have a real-valued index.

Star products are realized via twisted Toeplitz algebroids.

Trace on star algebra matches the residue trace of Wodzicki and Guillemin.

Abstract

The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([MR84g:17014]) see also Fedosov's construction in [MR92k:58267]. This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki and Guillemin