Fedosov Deformation Quantization as a BRST Theory
M. A. Grigoriev, S. L. Lyakhovich
TL;DR
This work embeds a general symplectic manifold ${\mathcal M}$ as a second-class constraint surface in a modified cotangent bundle and then converts to a first-class system on an extended space. Through BFV-BRST quantization, it constructs a quantum BRST charge whose adjoint action reproduces the Abelian Fedosov connection and identifies BRST-invariant observables with Fedosov flat sections, yielding the fibrewise Weyl star-product. The authors prove existence of the quantum BRST charge and observables, and establish a precise BFV-Fedosov correspondence linking BRST cohomology to Fedosov deformation quantization on ${\mathcal M}$. This framework recasts deformation quantization as a cohomological problem within BRST theory and clarifies how Fedosov's geometric data arise from Abelian conversion and BRST structures.
Abstract
The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold $\mathcal M$ is presented as a second class constrained surface in the fibre bundle ${{\mathcal T}^*_ρ}{\mathcal M}$ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ${{\mathcal T}^*_ρ}{\mathcal M}$ and the tangent bundle $T {\mathcal M}$. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables.