Relational Symplectic Groupoid Quantization for Constant Poisson Structures

TL;DR

This paper applies the BV-BFV formalism to quantize relational symplectic groupoids for constant Poisson structures, demonstrating how it induces Kontsevich's deformation quantization and extending the framework to manifolds with boundary.

Contribution

It introduces a novel BV-BFV-based quantization method for constant Poisson structures, linking it explicitly to deformation quantization and boundary conditions.

Findings

Quantization induces Kontsevich's Moyal product.

Extension of BV-BFV formalism to manifolds with boundary.

Provides a practical approach to learning BV-BFV technology.

Abstract

As a detailed application of the BV-BFV formalism for the quantization of field theories on manifolds with boundary, this note describes a quantization of the relational symplectic groupoid for a constant Poisson structure. The presence of mixed boundary conditions and the globalization of results is also addressed. In particular, the paper includes an extension to space-times with boundary of some formal geometry considerations in the BV-BFV formalism, and specifically introduces into the BV-BFV framework a "differential" version of the classical and quantum master equations. The quantization constructed in this paper induces Kontsevich's deformation quantization on the underlying Poisson manifold, i.e., the Moyal product, which is known in full details. This allows focussing on the BV-BFV technology and testing it. For the unexperienced reader, this is also a practical and reasonably simple way to learn it.