Ricci flows and expansion in axion-dilaton cosmology

Abstract

We study renormalization-group flows by deforming a class of conformal sigmamodels. We consider overall scale factor perturbation of Einstein spaces as well as more general anisotropic deformations of three-spheres. At leading order in α , renormalizationgroup equations turn out to be Ricci flows. In the three-sphere background, the latter is the Halphen system, which is exactly solvable in terms of modular forms. We also analyze time-dependent deformations of these systems supplemented with an extra time coordinate and time-dependent dilaton. In some regimes time evolution is identified with renormalization-group flow and time coordinate can appear as Liouville field. The resulting space-time interpretation is that of a homogeneous isotropic Friedmann-Robertson-Walker universe in axion-dilaton cosmology. We find as general behaviour the superposition of a big-bang (polynomial) expansion with a finite number of oscillations at early times. Any initial anisotropy disappears during the evolution.

Geometric flows in Hořava-Lifshitz gravity

“…In physics they originally appeared in off-critical string theory via the renormalization-group equations of two-dimensional non-linear sigma models, where the evolution of the metric under the Ricci flow equations provides the running of the bulk coupling to lowest order in perturbation theory (see [22,23] for the original result). In this context, the renormalization-group time is provided by the logarithmic length scale of the world-sheet, but in some cases it can also assume the role of genuine time, describing realtime evolution in string theory in regimes where the friction due to the motion of the dilaton effectively reduces the second-order evolution equations to the first-order renormalizationgroup flow equations [24,25].…”

“…Although what we call space is somewhat arbitrary in Euclidean gravity, the latter statement can be made precise by assuming a foliation in three-dimensional leaves that are invariant under an isometry group of motions. For these particular vacuum solutions, it turns out that the Euclidean time evolution of the homogeneous leaves inside the gravitational instanton can be recast as Ricci flow equations for the corresponding geometry on the homogeneous model spaces [25][26][27][28][29].…”

Geometric flows and applications

“…As shown in [6], for finite positive initial conditions, the flow remains positive with universal late-time …”

Parafermions, brane distributions and FRW hierarchies

“…Time evolution usually emerges through a time-dependent dilaton as well as in a warping factor of the spatial metric. This genuine time evolution is often identified at late times, with RG evolution where the two-dimensional scale plays the role of time [22,23,24].…”