κ-deformed gauge theory and θ-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 θ-deformed space and the κ-deformed space. In the case of the θ-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 θ-deformed global Poincaré symmetry, is also discussed. In the case of the κ-deformed space our aim is the construction of noncommutative gauge theories. Starting from the algebraic definition of the κ-deformed space, derivatives and the deformed Lorentz generators are introduced. Choosing one particular set of derivatives, the κ-Poincaré 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 θ-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 Poincaré symmetry is obtained.