Triproducts, nonassociative star products and geometry of R-flux string compactifications

TL;DR

This paper explores the nonassociative geometric structures arising in non-geometric string theory, deriving triproducts from phase space star products and extending to differential geometry, with implications for nonassociative gravity theories.

Contribution

It establishes a precise connection between phase space star products and configuration space triproducts, extending to differential forms and tensors, and develops a framework for nonassociative gravity in string compactifications.

Findings

Nonassociativity vanishes on-shell in all cases.

Derived families of Moyal-Weyl type deformations of triproducts.

Proposed a method to deform configuration space geometry from phase space nonassociativity.

Abstract

We elucidate relations between different approaches to describing the nonassociative deformations of geometry that arise in non-geometric string theory. We demonstrate how to derive configuration space triproducts exactly from nonassociative phase space star products and extend the relationship in various directions. By foliating phase space with leaves of constant momentum we obtain families of Moyal-Weyl type deformations of triproducts, and we generalize them to new triproducts of differential forms and of tensor fields. We prove that nonassociativity disappears on-shell in all instances. We also extend our considerations to the differential geometry of nonassociative phase space, and study the induced deformations of configuration space diffeomorphisms. We further develop general prescriptions for deforming configuration space geometry from the nonassociative geometry of phase space, thus paving the way to a nonassociative theory of gravity in non-geometric flux compactifications of string theory.