Induced $C^{\ast }$-complexes in metaplectic geometry

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 $C^{\ast }$-structure on this bundle for a suitable $C^{\ast }$-algebra, we obtain an elliptic $C^{\ast }$-complex in the sense of Mishchenko--Fomenko. Its cohomology groups appear to be finitely generated projective Hilbert $C^{\ast }$-modules. The paper can serve as a guide for handling of differential complexes and PDEs on Hilbert bundles

37 pages, 3 figures, accepted in Communication in Mathematical Physics; an argumentation on the continuity of map T in the proof of Thm. 18 was corrected

References in corpus (5)

• Schwartz operators
• Contact Geometry and Quantum Mechanics
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Cauchy-Schwarz-type inequalities on Kähler manifolds-II
• Elliptic complexes over $C^{\ast }$-algebras of compact operators