On Supersymmetric Geometric Flows and $\mathcal{R}^2$ Inflation From Scale Invariant Supergravity

TL;DR

This paper explores supersymmetric geometric flows in scale-invariant supergravity, introducing new methods to derive exact cosmological solutions and connect them to inflationary models like Starobinsky's.

Contribution

It develops a supersymmetric framework for geometric flows in supergravity, utilizing nonholonomic variables and generalized Perelman's functionals to find exact solutions.

Findings

Constructed non-homogeneous, anisotropic cosmological solutions.

Demonstrated conditions for reproducing Starobinsky inflation.

Extended Ricci flow techniques to supersymmetric and supergravity contexts.

Abstract

Models of geometric flows pertaining to $\mathcal{R}^2$ 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 K\"{a}hlerian spaces $\mathcal{M}_n=SU(1,1+k)/U(1)\times SU(1+k)$ 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. We also analyze the conditions under which such modified mimetic type theories and solutions reproduce the Starobinsky inflationary models in the double scalar approach.