A holography theory of Poisson sigma model and deformation quantization

TL;DR

This paper develops a topological quantum field theory based on Poisson structures, utilizing BV quantization and configuration space techniques, revealing invariants and algebraic structures related to Poisson and symplectic geometries.

Contribution

It introduces a novel Chern-Simons type theory for Poisson structures on Riemann surfaces with boundaries, employing BV quantization and identifying algebraic and topological invariants.

Findings

BV quantization is gauge-independent

Global observables serve as geometric invariants

Identifies Swiss-Cheese algebra structure in local observables

Abstract

We construct a Chern-Simons type of theory using the $l_\infty$ algebra encoded by a Poisson structure on arbitrary Riemann surfaces with boundaries. A deformation quantization within the Batalin-Vilkovisky framework is performed by constructing propagators with Dirichlet boundary condition on Fulton-MacPherson compactified configuration space. Our results show that the BV quantization is independent of several gauge choices in propagators, which leads to global observables that are candidates for geometric invariants of Poisson structure and topological invariants for the worldsheet structure. At the level of local observables, a Swiss-Cheese algebra structure has been identified. If the Poisson structure is symplectic, the two-dimensional theory is homotopic to a boundary theory. This is known in the classical case, and we confirm that the quantum homotopy exists as well.