Towards a Theory of Nonassociative Gravity

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.

38 pages, v2: ref. added

References in corpus (5)

• Double Field Theory
• A Gravity Theory on Noncommutative Spaces
• T-duality and closed string non-commutative (doubled) geometry
• Double Field Theory on Group Manifolds
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms

Cited by in corpus (17)

• Non-geometric backgrounds in string theory
• Double Field Theory and Membrane Sigma-Models
• Extended Riemannian Geometry I: Local Double Field Theory
• Nonassociative differential geometry and gravity with non-geometric fluxes
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Braided Symmetries in Noncommutative Field Theory
• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• $L_{\infty }$-Algebras of Einstein-Cartan-Palatini Gravity
• Magnetic monopoles and nonassociative deformations of quantum theory
• Nonassociative black holes in R-flux deformed phase spaces and relativistic models of G. Perelman thermodynamics
• Nonassociative gauge gravity theories with R-flux star products and Batalin-Vilkovisky quantization in algebraic quantum field theory
• Monopole star products are non-alternative
• The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories
• Noncommutativity and nonassociativity of closed bosonic string on T-dual toroidal backgrounds
• Nearly associative deformation quantization
• $L_{\infty }$-algebras and membrane sigma models