Noncommutative Gauge Fields on Poisson Manifolds

TL;DR

This paper explores a framework for noncommutative quantum gauge theories on Poisson manifolds using deformation quantization, aiming to extend noncommutative gauge theories relevant to string theory beyond tori.

Contribution

It proposes a nonperturbative path integral approach to noncommutative gauge theories on Poisson manifolds, building on the Cattaneo-Felder formulation.

Findings

Discusses the nonperturbative path integral formulation for noncommutative gauge theories.

Analyzes classical and quantum aspects of noncommutative field theories.

Highlights potential applications in string theory and M-theory compactifications.

Abstract

It is shown by Connes, Douglas and Schwarz that gauge theory on noncommutative torus describes compactifications of M-theory to tori with constant background three-form field. This indicates that noncommutative gauge theories on more general manifolds also can be useful in string theory. We discuss a framework to noncommutative quantum gauge theory on Poisson manifolds by using the deformation quantization. The Kontsevich formula for the star product was given originally in terms of the perturbation expansion and it leads to a non-renormalizable quantum field theory. We discuss the nonperturbative path integral formulation of Cattaneo and Felder as a possible approach to construction of noncommutative quantum gauge theory on Poisson manifolds. Some other aspects of classical and quantum noncommutative field theory are also discussed.