Equivariant quantization of orbifolds

TL;DR

This paper develops an equivariant quantization framework for orbifolds by desingularizing them into foliated smooth manifolds and establishing correspondences between their geometric objects, advancing the understanding of symmetries in quantum systems.

Contribution

It proves the existence of an equivariant quantization for orbifolds using a novel desingularization approach combined with foliated equivariant quantization techniques.

Findings

Established a method to desingularize orbifolds into foliated manifolds.

Defined geometric objects on orbifolds compatible with their desingularization.

Demonstrated the correspondence between singular orbifold objects and foliated objects.

Abstract

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces... In this work, we prove existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in \cite{PoRaWo}. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.