Noncommutative Gauge Fields on Poisson Manifolds ∗

It is shown by Connes, Douglas and Schwarz that gauge theory on noncom-mutative 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.