Morita Equivalence and Spectral Triples on Noncommutative Orbifolds

TL;DR

This paper develops a Morita equivalence framework for spectral triples on noncommutative orbifolds, linking classical orbifold groupoids with noncommutative geometry and analyzing the impact of group action freeness.

Contribution

It introduces Morita equivalence for spectral triples on noncommutative orbifolds, extending classical orbifold groupoid concepts to noncommutative geometry.

Findings

Morita equivalence classes characterize noncommutative orbifolds.

Freeness of group action simplifies the spectral triple to the invariant function algebra.

The framework connects classical orbifold theory with noncommutative spectral triples.

Abstract

Let $G$ be a finite group. Noncommutative geometry of unital $G$-algebras is studied. A geometric structure is determined by a spectral triple on the crossed product algebra associated with the group action. This structure is to be viewed as a representative of a noncommutative orbifold. Based on a study of classical orbifold groupoids, a Morita equivalence for the crossed product spectral triples is developed. Noncommutative orbifolds are Morita equivalence classes of the crossed product spectral triples. As a special case of this Morita theory one can study freeness of the $G$-action on the noncommutative level. In the case of a free action, the crossed product formalism reduced to the usual spectral triple formalism on the algebra of $G$-invariant functions.