Towards a Theory of Nonassociative Gravity

TL;DR

This paper develops a differential geometry framework for nonassociative phase-space structures arising from non-geometric fluxes in string theory, introducing star-generalized tensors, metrics, and connections.

Contribution

It provides a pedestrian derivation of nonassociative differential geometry on phase-space, complementing existing categorial approaches, and explores implications for gravity and string theory.

Findings

Defined star-tensors, covariant derivatives, torsion, and curvature in nonassociative geometry.

Introduced a star-metric and star-inverse with deviations from classical structures.

Discussed the star-Einstein-Hilbert action and connections to string theory.

Abstract

Violating the strong constraint of double field theory, non-geometric fluxes were argued to give rise to noncommutative/nonassociative structures. We derive in a rather pedestrian physicist way a differential geometry on the simplest nonassociative (phase-)space arising for a constant non-geometric R-flux. This provides a complementary presentation to the quasi-Hopf representation categorial one delivered by Barnes, Schenkel, Szabo in arXiv:1409.6331+1507.02792. As there, the notions of tensors, covariant derivative, torsion and curvature find a star-generalization. We continue the construction with the introduction of a star-metric and its star-inverse where, due to the nonassociativity, we encounter major deviations from the familiar structure. Comments on the Levi-Civita connection, a star-Einstein-Hilbert action and the relation to string theory are included, as well.