Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

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. Introduction, Motivations and GoalsWe initialize a research on nonassociative nonholonomic geometry induced by R-flux backgrounds in string gravity and classical and quantum models on phase spaces modelled as (co) tangent Lorentz bundles endowed with nonholonomic (non-integrable, equivalently, anholonomic) structures and nonsymmetric metrics. Such a nonholonomic geometric formalism was elaborated for various types of (non) commutative gravity and matter field theories, modified (relativistic) geometric flows, and classical and quantum information theories, see details in a series of our previous works [1][2][3][4][5][6][7][8][9][10][11] and references therein. In this article, nonassociative geometric and physical models are formulated in a form which is accessible to researchers in particle physics and modern cosmology. There are provided nonholonomic generalizations of two recent models of nonassociative geometry and vacuum gravity due to [12] and, with

Decoupling and integrability of nonassociative vacuum phase space gravitational equations with star and R-flux parametric deformations

“…In the first partner work [3], the nonassociative vacuum Einstein equations from [1,2] were generalized and written in canonical nonholonomic variables on a phase space modeled as a cotangent Lorentz bundle M = T * V enabled with nonlinear connection, N-connection structure N and a respectively defined N-adapted star product using general frame transforms. We use boldface indices for spaces and geometric objects enabled with (adapted to) N-connection structure as we introduce below.…”

“…In our first partner work on nonassociative nonholonomic geometry and gravity [3], we noted that because of generic nonlinearity and tensor coupling of (non) associative / commutative gravitational equations, even for vacuum configurations, the solutions of respective systems of PDEs can not be constructed in general off-diagonal exact or parametric forms depending on some phase and spacetime variables if we work only in holonomic variables determined with respect to local coordinate frames and for diagonalizable metrics. In such theories, R-flux star deformations extend the geometric approach to constructions on phase spaces modelled as (co) tangent bundles endowed with symmetric and nonsymmetric metrics and generalized connection structures depending both on spacetime and momentum like coordinates.…”

“…This work provides a nonholonomic dyadic formulation of nonassociative geometry and vacuum gravity when a decomposition on so-called "shell by shell" two dimensional variables determined by respective Nadapted frames (in brief, s-structures and/or s-decompositions) allows to prove some general decoupling and integrability properties of corresponding systems of nonlinear PDEs. We generalize the approaches toward to the theory of nonassociative gravity from proposed in [1,2] to a formalism when the geometric constructions are performed following the method of star nonholonomic deformations elaborated in [3]. Our main goal is to write the nonassociative vacuum gravitational equations in certain adapted nonholonomic variables with conventional 2+2+2+... splitting of dimensions and in terms of auxiliary distinguished connections 2 and, in result, to generate new classes of exact and parametric physically important solutions.…”

“…The stationary condition means that the metrics, connections, and related fundamental geometric objects are with Killing symmetry on a vector ∂ 4 = ∂ t for a time like spacetime coordinate x 4 = t. In our partner works, we elaborate on general methods for constructing exact and parametric stationary and cosmological type solutions (when the dependence on a time like variable is crucial) of nonassociative modified Einstein equations with nontrivial and/or effective matter sources and for generalized/ relativistic geometric flow evolution equations. Such a geometric techniques is called in our works as the anholonomic frame and connection deformation method, AFCDM, 3 of constructing exact solutions in (non) commutative/ supersymmetric/ string / massive and other type modified gravity and general relativity, GR, theories was developed in [19,20,21], see also references therein, on possible applications in modern cosmology and astrophysics, geometric thermodynamics and classical and quantum information flow theories [22,23]. Here we note that our nonholonomic dyadic 4 approach to generating exact solutions in (modified) gravity theories is more general and different from other type methods with dyadic decompositions (for instance, with the Newmann-Penrose formalism, see a review of geometric methods and main results in [24,25,26,27]).…”

“…Section 4 is devoted to elaborating a dyadic formalism with nonholonomic (2+2)+(2+2) splitting for nonassociative s-decomposition of vacuum Einstein equations encoding nontrivial and polarized cosomological constants and/or sources induced by R-fluxes and parametric star deformations of geometric objects. We follow the methods of nonassociative differential geometry from section 3 of [3] re-defined in s-adapted variables at the end of this paper in appendix A. There are considered s-variables for which the nonsymmetric components of Ricci s-tensors are related to parametric and s-decompositions of the nonsymmetric parts of s-metrics.…”