A First Approximation for Quantization of Singular Spaces

TL;DR

This paper proposes a novel approach to quantize singular spaces arising from symmetry reductions in physical models, using foliation theory as a desingularization technique to develop a compatible quantization procedure.

Contribution

It introduces a new method combining foliation theory and equivariant quantization to address the quantization of singular orbit spaces, advancing the mathematical framework for such models.

Findings

Develops a foliation-based desingularization technique

Constructs a quantization procedure compatible with symmetry reduction

Lays groundwork for quantization of orbit spaces and orbifolds

Abstract

Many mathematical models of physical phenomena that have been proposed in recent years require more general spaces than manifolds. When taking into account the symmetry group of the model, we get a reduced model on the (singular) orbit space of the symmetry group action. We investigate quantization of singular spaces obtained as leaf closure spaces of regular Riemannian foliations on compact manifolds. These contain the orbit spaces of compact group actions and orbifolds. Our method uses foliation theory as a desingularization technique for such singular spaces. A quantization procedure on the orbit space of the symmetry group - that commutes with reduction - can be obtained from constructions which combine different geometries associated with foliations and new techniques originated in Equivariant Quantization. The present paper contains the first of two steps needed to achieve these just detailed goals.