On Noncommutative Geometry of Orbifolds

TL;DR

This paper explores the noncommutative geometric structure of orbifolds by constructing spectral triples over their associated algebras, providing a new framework for understanding orbifold geometry in noncommutative terms.

Contribution

It introduces a method to construct spectral triples for effective spin orbifolds over their convolution algebras, advancing noncommutative geometric analysis of orbifolds.

Findings

Spectral triples are associated with any compact spin orbifold.

Construction of spectral triples over convolution algebras for effective spin orbifolds.

Provides a new perspective on orbifold geometry using noncommutative tools.

Abstract

An orbifold is a Morita equivalence class of a proper {\' e}tale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the smooth convolution algebras of the representatives of the Morita equivalence class.