Induced $C^*$-complexes in metaplectic geometry

TL;DR

This paper constructs an elliptic $C^*$-complex on symplectic manifolds with metaplectic structures, linking differential geometry, operator algebras, and PDEs, and revealing finitely generated projective modules in cohomology.

Contribution

It introduces a novel elliptic $C^*$-complex in metaplectic geometry, connecting differential operators with Hilbert $C^*$-modules and providing tools for PDE analysis on Hilbert bundles.

Findings

Construction of a Hilbert $C^*$-structure on symplectic spinor bundles.

Establishment of an elliptic $C^*$-complex with finitely generated projective cohomology.

Framework for analyzing differential complexes and PDEs on Hilbert bundles.

Abstract

For a symplectic manifold admitting a metaplectic structure and for a Kuiper map, we construct a complex of differential operators acting on exterior differential forms with values in the dual of the Kostant's symplectic spinor bundle. Defining a Hilbert $C^*$-structure on this bundle for a suitable $C^*$-algebra, we obtain an elliptic $C^*$-complex in the sense of Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^*$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles