Abstract
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.
Highlights
Deformations of spacetime geometry through compactifications of string theory may help elucidate the precise mechanism by which closed strings provide a framework for a quantum theory of gravity
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
This has been the hope in some recent investigations surrounding non-geometric string theory, in which noncommutative and nonassociative deformations of target space geometry have been purported to be probed by closed strings in non-geometric flux compactifications [1, 9, 13, 15,16,17, 19, 25]
Summary
Deformations of spacetime geometry through compactifications of string theory may help elucidate the precise mechanism by which closed strings provide a framework for a quantum theory of gravity. We have chosen to develop the metric aspects of nonassociative gravity, because it represents the most direct way to explore the potential relevance of nonassociative gravity to string theory, in that in closed string theory the fundamental field is the metric tensor rather than the vielbein It is interesting in its own right to be able to implement general covariance under the quasi-Hopf algebra of deformed diffeomorphisms and to formulate Einstein equations in nonassociative space, in order to arrive at a theory that can be potentially considered as providing a low-energy effective action for closed strings in the presence of non-geometric fluxes, it is necessary to project nonassociative gravity from phase space to spacetime.
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