Nonassociative geometry of nonholonomic phase spaces with star R-flux string deformations and (non) symmetric metrics

TL;DR

This paper develops a framework for nonassociative differential geometry on phase spaces with nonholonomic structures, star deformations, and (non)symmetric metrics, extending geometric methods to include R-flux string backgrounds.

Contribution

It introduces a nonassociative geometric approach using star products and nonholonomic frames, extending Einstein equations to R-flux deformed nonholonomic phase spaces.

Findings

Defined nonassociative torsion, curvature, and Ricci tensors.

Constructed R-flux deformations of Einstein equations.

Extended nonholonomic geometric methods to nonassociative settings.

Abstract

We elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non-integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi-Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R-flux background. Generalized Moyal-Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/ real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/ momentum variables like in (non) commutative Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N-connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N-adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N-adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R-flux deformations of vacuum Einstein equations for the case of N-adapted linear connections and symmetric and nonsymmetric metric structures.