Dirac operators on noncommutative hypersurfaces

Abstract

This paper studies geometric structures on noncommutative hypersurfaces within a module-theoretic approach to noncommutative Riemannian (spin) geometry. A construction to induce differential, Riemannian and spinorial structures from a noncommutative embedding space to a noncommutative hypersurface is developed and applied to obtain noncommutative hypersurface Dirac operators. The general construction is illustrated by studying the sequence ${\mathbb{T}}_{\theta }^2\hookrightarrow {\mathbb{S}}_{\theta }^3\hookrightarrow {\mathbb{R}}_{\theta }^4$ of noncommutative hypersurface embeddings.

v2: 25 pages. Final version to appear in Journal of Geometry and Physics