A deformation of the algebra of diffeomorphisms is constructed for canonically deformed spaces with constant deformation parameter θ. The algebraic relations remain the same, whereas the comultiplication rule (Leibniz rule) is different from the undeformed one. Based on this deformed algebra a covariant tensor calculus is constructed and all the concepts like metric, covariant derivatives, curvature and torsion can be defined on the deformed space as well. The construction of these geometric quantities is presented in detail. This leads to an action invariant under the deformed diffeomorphism algebra and can be interpreted as a θ-deformed Einstein-Hilbert action. The metric or the vierbein field will be the dynamical variable as they are in the undeformed theory. The action and all relevant quantities are expanded up to second order in θ.

We study a deformation of infinitesimal diffeomorphisms of a smooth manifold. The deformation is based on a general twist. This leads to a differential geometry on a noncommutative algebra of functions whose product is a star-product. The class of noncommutative spaces studied is very rich. Non-anticommutative superspaces are also briefly considered.The differential geometry developed is covariant under deformed diffeomorphisms and it is coordinate independent. The main target of this work is the construction of Einstein's equations for gravity on noncommutative manifolds.

A general formalism is developed that allows the construction of a field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime (Poincaré group) is replaced by a quantum group. This formalism is demonstrated for the κ-deformed Poincaré algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable ⋆-product. Fields are elements of this function algebra. The Dirac and Klein-Gordon equation are defined and an action is found from which they can be derived.

Gauge theories on a space-time that is deformed by the Moyal-Weyl product are constructed by twisting the coproduct for gauge transformations. This way a deformed Leibniz rule is obtained, which is used to construct gauge invariant quantities. The connection will be enveloping algebra valued in a particular representation of the Lie algebra. This gives rise to additional fields, which couple only weakly via the deformation parameter and reduce in the commutative limit to free fields. Consistent field equations that lead to conservation laws are derived and some properties of such theories are discussed.Comment: 11 pages, V2: details and appendix adde

This study of gauge field theories on κ-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the SeibergWitten formalism. Derivative-valued gauge fields lead to field strength tensors as the sum of curvature-and torsion-like terms. We construct the Lagrangians explicitly to first order in the deformation parameter. This is the first example of a gauge theory that possesses a deformed Lorentz covariance.

The model of κ-deformed space is an interesting example of a noncommutative space, since it allows a deformed symmetry.In this paper we present new results concerning different sets of derivatives on the coordinate algebra of κ-deformed Euclidean space. We introduce a differential calculus with two interesting sets of one-forms and higher-order forms. The transformation law of vector fields is constructed in accordance with the transformation behaviour of derivatives. The crucial property of the different derivatives, forms and vector fields is that in an n-dimensional spacetime there are always n of them. This is the key difference with respect to conventional approaches, in which the differential calculus is (n + 1)-dimensional.This work shows that derivative-valued quantities such as derivative-valued vector fields appear in a generic way on noncommutative spaces.

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A ⋆-product is defined via a set of commuting vector fields X a = e µ a (x)∂ µ , and used in a φ ⋆4 theory coupled to the e µ a (x) fields. The ⋆-product is dynamical, and the vacuum solution φ = 0, e µ a = δ µ a reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy-momentum and angular momentum tensors are explicitly derived.leonardo.castellani@mfn.unipmn.it aschieri, dimitrij@to.infn.it 1 The realization (1.2) of the ⋆-product f ⋆ g holds for a limited class of functions (e.g. polynomials, or analytic and rapidly decreasing functions). For a richer class of functions, e.g. smooth and rapidly decreasing (Schwarz test functions), an integral representation of the ⋆-product is needed. One such representation is f ⋆ g (x) = (2π) −2D f (x + 1 2 θu)g(x + s)e ius d D u d D s [22] and explicitly encodes the nonlocality of the ⋆-product.