Poisson sigma models and deformation quantization

TL;DR

The paper investigates how Poisson sigma models realize deformation quantization of Poisson manifolds by analyzing both a classical Hamiltonian reduction to a symplectic groupoid and a perturbative Lagrangian quantization that yields Kontsevich's star product. It demonstrates endpoint noncommutativity and a groupoid (often symplectic) structure arising from composing boundary data, and it connects these classical structures to a formal symplectic groupoid with a generating function F that leads to a semiclassical star product. The perturbative approach on the disk reproduces Kontsevich's formula via boundary observables and BRST/BV/AKSZ methods, while considerations of associativity motivate higher-order corrections and potential lattice BV reformulations. Overall, the work links geometric groupoid ideas with field-theoretic constructions to illuminate deformation quantization from both a Hamiltonian and a Lagrangian perspective, and discusses globalization and structural compatibility with Weinstein's program.

Abstract

This is a review aimed at a physics audience on the relation between Poisson sigma models on surfaces with boundary and deformation quantization. These models are topological open string theories. In the classical Hamiltonian approach, we describe the reduced phase space and its structures (symplectic groupoid), explaining in particular the classical origin of the non-commutativity of the string end-point coordinates. We also review the perturbative Lagrangian approach and its connection with Kontsevich's star product. Finally we comment on the relation between the two approaches.