The application of star-products to noncommutative geometry and gauge theory

Due to the singularities arising in quantum field theory and the
difficulties in quantizing gravity it is often believed that the
description of spacetime by a smooth manifold should be given up at
small length scales or high energies. In this work we will replace
spacetime by noncommutative structures arising within the framework
of deformation quantization. The ordinary product between functions
will be replaced by a *-product, an associative product for the
space of functions on a manifold. We develop a formalism to realize
algebras defined by relations on function spaces. For this porpose
we construct the Weyl-ordered *-product and present a method how to
calculate *-products with the help of commuting vector fields.
Concepts developed in noncommutative differential geometry will be
applied to this type of algebras and we construct actions for
noncommutative field theories. In the classical limit these
noncommutative theories become field theories on manifolds with
nonvanishing curvature. It becomes clear that the application of
*-products is very fruitful to the solution of noncommutative
problems. In the semiclassical limit every *-product is related to
a Poisson structure, every derivation of the algebra to a vector
field on the manifold. Since in this limit many problems are
reduced to a couple of differential equations the *-product
representation makes it possible to construct noncommutative spaces
corresponding to interesting Riemannian manifolds. Derivations of
*-products makes it further possible to extend noncommutative gauge
theory in the Seiberg-Witten formalism with covariant derivatives.
The resulting noncommutative gauge fields may be interpreted as one
forms of a generalization of the exterior algebra of a manifold.
For the Formality *-product we prove the existence of the abelian
Seiberg-Witten map for derivations of these *-products. We
calculate the enveloping algebra valued non abelian Seiberg-Witten
map pertubatively up to second order for the Weyl-ordered
*-product. A general method to construct actions invariant under
noncommutative gauge transformations is developed. In the
commutative limit these theories are becoming gauge theories on
curved backgrounds. We study observables of noncommutative gauge
theories and extend the concept of so called open Wilson lines to
general noncommutative gauge theories. With help of this
construction we give a formula for the inverse abelian
Seiberg-Witten map on noncommutative spaces with nondegenerate
*-products.