We systematically develop the metric aspects of nonassociative differential geometry tailored to the parabolic phase space model of constant locally non-geometric closed string vacua, and use it to construct preliminary steps towards a nonassociative theory of gravity on spacetime. We obtain explicit expressions for the torsion, curvature, Ricci tensor and Levi-Civita connection in nonassociative Riemannian geometry on phase space, and write down Einstein field equations. We apply this formalism to construct R-flux corrections to the Ricci tensor on spacetime, and comment on the potential implications of these structures in non-geometric string theory and double field theory.

Aiming to search for a signal of space-time noncommutativity, we study a quasinormal mode spectrum of the Reissner-Nordström black hole in the presence of a deformed space-time structure. In this context we study a noncommutative (NC) deformation of a scalar field, minimally coupled to a classical (commutative) Reissner-Nordström background. The deformation is performed via a particularly chosen Killing twist to ensure that the geometry remains undeformed (commutative). An action describing a noncommutative scalar field minimally coupled to the RN geometry is manifestly invariant under the deformed U(1) gauge symmetry group. We find the quasinormal mode solutions of the equations of motion governing the matter content of the model in some particular range of system parameters which corresponds to a near extremal limit. In addition, we obtain a well defined analytical condition which allows for a detailed numerical analysis. Moreover, there exists a parameter range, rather restrictive though, which allows for obtaining a QNMs spectrum in a closed analytic form. We also argue within a semiclassical approach that NC deformation does not affect the Hawking temperature of thermal radiation.

Noncommutative (NC) gravity is constructed on the canonical noncommutative (Moyal-Weyl) space-time as a noncommutative SO(2, 3) ⋆ gauge theory. The NC gravity action consists of three different terms: the first term is of Mac-Dowell Mansouri type, while the other two are generalizations of the Einstein-Hilbert action and the cosmological constant term. The expanded NC gravity action is then calculated using the Seiberg-Witten (SW) map and the expansion is done up second order in the deformation parameter. We analyze in details the low energy sector of the full model. We calculate the equations of motion, discuss their general properties and present one solution: the NC correction to Minkowski spacetime. Using this solution, we explain breaking of the diffeomorphism symmetry as a consequence of working in a particular coordinate system given by the Fermi normal coordinates.

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We give a detailed account of the cyclic L ∞ -algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We develop a local formulation as well as a global covariant framework, and derive an explicit isomorphism between the two L ∞ -algebras in the case of parallelizable spacetimes. We show that our L ∞ -algebras describe the complete BV-BRST formulation of Einstein-Cartan-Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of L ∞ -algebras. We use this to extend the on-shell equivalence between gravity and Chern-Simons theory in three dimensions to an explicit L ∞ -quasi-isomorphism between differential graded Lie algebras which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the L ∞ -algebra underlying gravity and the differential graded Lie algebra governing BF theory in four dimensions.