Abstract

Within this thesis gauge-field theories and gravity on noncommutative spaces are studied. Starting with introducing the concepts underlying the making of field theories on noncommutative spaces. With a noncommutative space we mean a noncommutative algebra, which replaces the algebra of functions on ordinary space. We construct derivatives and deformed symmetries (“Quantum Group” symmetries) functioning on noncommutative spaces. Consistency requires us to alter the experience on the product of representations (“deformed coproducts”) this provides increase in particular to deformed Leibniz rules. We show the way a noncommutative space and also the generators of deformed symmetries functioning on it may be symbolized around the ordinary algebra of functions the commutative, point-wise method is substituted with a noncommutative one (“star-product”). One way possible to define gauge-field theories on noncommutative spaces would be to construct “Seiberg–Witten maps”. Within this approach you’ll be able to express all noncommutative quantities when it comes to their commutative counterparts. We illustrate this by two examples, the 2-dimensional q -deformed Euclidean plane and also the kappa -deformed Minkowski space-time. Additionally gauge-field theory on “fuzzy” S^times S^ is discussed like a multi-matrix model. We reveal that this model reduces within an appropriate limit to gauge-field theory on noncommutative mathbb^. We present a brand new method of deformed gauge theories, which is dependant on “twisted” gauge transformations. Within this setting new fields occur additionally towards the usual gauge fields. Consistent equations of movement and conserved currents are acquired. This is actually the very first time that conservation laws and regulations happen to be produced from a generalized, Quantum Group symmetry.

We discuss at length how to develop deformed infinitesimal diffeomorphisms by deformations via generic “twists”. Only then do we construct gravity like a theory, that is covariant regarding these diffeomorphisms. This can lead to a deformation of Einstein’s equations. For canonically deformed spaces, a deformed Einstein–Hilbert action could be even defined. It cuts down on towards the usual Einstein–Hilbert action within the commutative limit. All relevant quantities are expanded with regards to the usual, commutative fields as much as second order within the deformation parameter.

Thesis (Dissertation, LMU Munich)

Noncommutative Geometry, Gravity, Noncommutative Gauge Theories, Quantum Groups, Fuzzy Sphere

Faculty of Physics

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19. Jul 2016 16:20

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