Noncommutative Gauge Theory beyond the Canonical Case

Canonically deformed spacetime, where the commutator of two
coordinates is a constant, is the most commonly studied
noncommutative space. Noncommutative gauge theories that have
ordinary gauge theory as their commutative limit have been
constructed there. But these theories have their drawbacks: First
of all, constant noncommutativity can only be an approximation of a
realistic theory, and therefore it is necessary to study more
complicated space-dependent structures as well. Secondly, in the
canonical case, the noncommutativity didn't fulfill the initial
hope of curing the divergencies of quantum field theory. Therefore
it is very desirable to understand noncommutative spaces that
really admit finite QFTs. These two aspects of going beyond the
canonical case will be the main focus of this thesis. They will be
addressed within two different formalisms, each of which is
especially suited for the purpose. In the first part noncommutative
spaces created by star-products are studied. In the case of
nonconstant noncommutativity, the ordinary derivatives possess a
deformed Leibniz rule, i.e. d_i (f star g) \neq d_i f star g + f
star d_i g. Therefore we construct new objects that still have an
undeformed Leibniz rule. These derivations of the star-product
algebra can be gauged much in the same way as in the canonical case
and lead to function-valued gauge fields. By linking the
derivations to frames (vielbeins) of a curved manifold, it is
possible to formulate noncommutative gauge theories that admit
nonconstant noncommutativity and go to gauge theory on curved
spacetime in the commutative limit. We are also able to express the
dependence of the noncommutative quantities on their corresponding
commutative counterparts by using Seiberg-Witten maps. In the
second part we will study noncommutative gauge theory in the matrix
theory approach. There, the noncommutative space is the ground
state of a matrix action, the fluctuations around this ground state
creating the gauge theory. In the canonical case the matrices used
are infinite-dimensional (they are the Fock-space representation of
the Heisenberg algebra), leading to a number of problems,
especially with divergencies. Therefore we construct gauge theory
using finite dimensional matrices (fuzzy spaces). This gauge theory
is finite, goes to gauge theory on a 4-dimensional manifold in the
commutative limit and can also be used to regularize the
noncommutative gauge theory of the canonical case. In particular,
we are able to match parts of the known instanton sector of the
canonical case with the instantons of the finite theory.