BRST quantization of quasi-symplectic manifolds and beyond

TL;DR

This paper extends BRST quantization and deformation techniques to a broad class of irregular Poisson structures, including those linked to symplectic Lie algebroids and n-algebroids, revealing new geometric and algebraic insights.

Contribution

It introduces a novel approach to quantize factorizable Poisson brackets using BRST theory and connects these structures to n-algebroids and generalized Yang-Baxter equations.

Findings

Extended Fedosov deformation quantization to irregular Poisson structures.

Established geometric framework using odd Poisson algebra bundles.

Linked factorizable brackets with n-algebroids and zero-curvature conditions.

Abstract

We consider a class of \textit{factorizable} Poisson brackets which includes almost all reasonable Poisson structures. A particular case of the factorizable brackets are those associated with symplectic Lie algebroids. The BRST theory is applied to describe the geometry underlying these brackets as well as to develop a deformation quantization procedure in this particular case. This can be viewed as an extension of the Fedosov deformation quantization to a wide class of \textit{irregular} Poisson structures. In a more general case, the factorizable Poisson brackets are shown to be closely connected with the notion of $n$-algebroid. A simple description is suggested for the geometry underlying the factorizable Poisson brackets basing on construction of an odd Poisson algebra bundle equipped with an abelian connection. It is shown that the zero-curvature condition for this connection generates all the structure relations for the $n$-algebroid as well as a generalization of the Yang-Baxter equation for the symplectic structure.