Deformation quantization and homological reduction of a lattice gauge model

TL;DR

This paper constructs a Fedosov quantization of a lattice gauge model's symplectic manifold, refines homological reduction for star product construction, and demonstrates applicability for the SU(2) gauge group.

Contribution

It explicitly develops a Fedosov quantization and refines homological reduction methods for lattice gauge models, especially in singular cases.

Findings

Successfully quantized the symplectic manifold of the model.

Refined homological reduction for singular symplectic quotients.

Established applicability for SU(2) lattice gauge models.

Abstract

For a compact Lie group $G$ we consider a lattice gauge model given by the $G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$ with its canonical symplectic structure and standard moment map. We explicitly construct a Fedosov quantization of the underlying symplectic manifold using the Levi-Civita connection of the Killing metric on $G$. We then explain and refine quantized homological reduction for the construction of a star product on the symplectically reduced space in the singular case. Afterwards we show that for $G = \operatorname{SU} (2)$ the main hypotheses ensuring the method of quantized homological reduction to be applicable hold in the case of our lattice gauge model. For that case, this implies that the - in general singular - symplectically reduced phase space of the corresponding lattice gauge model carries a star product.