Kappa deformed gauge theory and theta deformed gravity

Noncommutative (deformed, quantum) spaces are deformations of the
usual commutative space-time. They depend on parameters, such that
for certain values of parameters they become the usual space-time.
The symmetry acting on them is given in terms of a deformed quantum
group symmetry. In this work we discuss two special examples, the
$\theta$-deformed space and the $\kappa$-deformed space. In the
case of the $\theta$-deformed space we construct a deformed theory
of gravity. In the first step the deformed diffeomorphism symmetry
is introduced. It is given in terms of the Hopf algebra of deformed
diffeomorphisms. The algebra structure is unchanged (as compared to
the commutative diffeomorphism symmetry), but the comultiplication
changes. In the commutative limit we obtain the Hopf algebra of
undeformed diffeomorphisms. Based on this deformed symmetry a
covariant tensor calculus is constructed and concepts such as
metric, covariant derivative, curvature and torsion are defined. An
action that is invariant under the deformed diffeomorphisms is
constructed. In the zeroth order in the deformation parameter it
reduces to the commutative Einstein-Hilbert action while in higher
orders correction terms appear. They are given in terms of the
commutative fields (metric, vierbein) and the deformation parameter
enters as the coupling constant. One special example of this
deformed symmetry, the $\theta$-deformed global Poincar\' e
symmetry, is also discussed. In the case of the $\kappa$-deformed
space our aim is the construction of noncommutative gauge theories.
Starting from the algebraic definition of the $\kappa$-deformed
space, derivatives and the deformed Lorentz generators are
introduced. Choosing one particular set of derivatives, the
$\kappa$-Poincar\' e Hopf algebra is defined. The algebraic setting
is then mapped to the space of commuting coordinates. In the next
step, using the enveloping algebra approach and the Seiberg-Witten
map, a general nonabelian gauge theory on this deformed space is
constructed. As a consequence of the deformed Leibniz rules for the
derivatives used in the construction, the gauge field is
derivative-valued. As in the $\theta$-deformed case, in the zeroth
order of the deformation parameter the theory reduces to its
commutative analog and the higher order corrections are given in
terms of the usual (commutative) fields. In this way the field
content of the theory is unchanged, but new interactions appear.
The deformation parameter takes the role of the coupling constant.
For the special case of $U(1)$ gauge theory the action for the
gauge field coupled to fermionic matter is formulated and the
equations of motion and the conserved current(s) are calculated.
The ambiguities in the Seiberg-Witten map are discussed and
partially fixed, and an effective action (up to first order in the
deformation parameter) which is invariant under the usual Poincar\'
e symmetry is obtained.