Poisson sigma models and deformation quantization

TL;DR

This paper reviews how Poisson sigma models relate to deformation quantization, highlighting their topological nature, phase space structures, and connections to Kontsevich's star product, bridging classical and quantum perspectives.

Contribution

It provides a comprehensive comparison of Hamiltonian and Lagrangian approaches to Poisson sigma models and their role in deformation quantization.

Findings

Classical phase space described as a symplectic groupoid

Connection established between string boundary non-commutativity and Poisson structures

Perturbative approach linked to Kontsevich's star product

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.