On supersymmetric geometric flows andR2inflation from scale invariant supergravity

Abstract

Models of geometric flows pertaining to R 2 scale invariant (super) gravity theories coupled to conformally invariant matter fields are investigated. Related to this work are supersymmetric scalar manifolds that are isomorphic to the Kählerian spaces M n = SU (1, 1 + k)/U (1) × SU (1 + k) as generalizations of the non-supersymmetric analogs with SO(1, 1 + k)/SO(1 + k) manifolds. For curved superspaces with geometric evolution of physical objects, a complete supersymmetric theory has to be elaborated on nonholonomic (super) manifolds and bundles determined by non-integrable superdistributions with additional constraints on (super) field dynamics and geometric evolution equations. We also consider generalizations of Perelman's functionals using such nonholonomic variables which result in the decoupling of geometric flow equations and Ricci soliton equations with supergravity modifications of the R 2 gravity theory. As such, it is possible to construct exact non-homogeneous and locally anisotropic cosmological solutions for various types of (super) gravity theories modelled as modified Ricci soliton configurations. Such solutions are defined by employing the general ansatz encompassing coefficients of generic off-diagonal metrics and generalized connections that depend generically on all spacetime coordinates. We consider nonholonomic constraints resulting in diagonal homogeneous configurations encoding contributions from possible nonlinear parametric geometric evolution scenarios, off-diagonal interactions and anisotropic polarization/ modification of physical constants. In particular, we analyze small parametric deformations when the underlying scale symmetry is preserved and the nontrivial anisotropic vacuum corresponds to generalized de Sitter spaces. Such configurations may mimic quantum effects whenever transitions to flat space are possible. Our approach allows us to generate solutions with scale violating terms induced by geometric flows, off-diagonal interactions and supersymmetric modifications of effective potentials. We study reconstructing the formalism for (super) geometric flows and modified gravity cosmological scenarios. We also analyze the conditions under which such modified mimetic type theories and solutions reproduce the Starobinsky inflationary models in the double scalar approach.

Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

“…In result, we can elaborate on advanced geometric methods for modeling relativistic geometric flows of classical and quantum mechanical systems, and modified commutative and noncommutative/ supersymmetric gravity theories etc. [36,37,35].…”

“…Introducing a respective thermodynamic generation function, all thermodynamic values can be defined and computed by integrating with corresponding measures defined by the metric structure and a corresponding normalizing function. Similar constructions can be elaborated for various relativistic, supersymmetric, commutative and noncommutative generalizations if the geometric flow evolution is modelled for corresponding nonholonomic fibered structures preserving causality and basic postulates for self-consistent stochastic, kinetic and thermodynamics models [33,34,35,36,37], see also [17,18,29,38,39] and references therein. Originally, such nonholonomic transforms of geometric objects and deformations of the (non) linear connection structures were considered in [40,41] where the theory of geometric flows was generalized for Finsler-Lagrange geometries.…”

“…Here we note that in this work we follow the notations on the so-called quantum geometric information flow, QGIF, theory (in brief, it is used GIF for classical models) introduced in [29]. Readers may study our previous works [33,34,35,36,37] and references therein, on nonholonomic (non) commutative / supersymmetric geometric flows and related kinetic and statistical thermodynamic models.…”

“…In this article, we study entanglement for quantum geometric flows of mechanical systems and do not concern issues on gravity, quantum field theory or condensed matter physics. On emergent gravity theories, modified Ricci flow theories and gravity, exact solutions and related classical and quantum mechanical entropic functionals from which generalized Einstein equations can be derived, see our recent results [33,34,35,36,37] and references therein.…”

“…For simplicity, we shall write, for instance, N(τ ) instead of N(τ, u) if that will not result in ambiguities. Relativistic nonholonomic phase spacetimes can be enabled with necessary types double nonholonomic (2+2)+(2+2) and (3+1)+(3+1) splitting[35,36,37,28,17]. Local (3+1)+(3+1) coordinates are labeled in the form u = { u α = u αs = (x i 1 , y a 2 ; pa 3 , pa 4 ) = (xì, u 4 = y 4 = t; pà, p8 = E)} for i1, j1, k1, ... = 1, 2; a1, b1, c1, ... = 3, 4; a2, b2, c2, ... = 5, 6; a3, b3, c3, ... = 7, 8.…”