On higher order geometric and renormalisation group flows

TL;DR

This paper investigates higher order geometric and renormalisation group flows in string theory, analyzing their effects on specific manifolds, and compares perturbative RG flows with purely geometric flows to understand corrections, singularities, and solitons.

Contribution

It provides a detailed analysis of higher order RG and geometric flows on various manifolds, highlighting the impact of higher order terms and their role in flow evolution and singularity formation.

Findings

Higher order corrections are small within perturbative regimes.

Separable solutions correspond to constant curvature AdS spacetimes.

Higher order terms modify the flow's scale factor and singularity behavior.

Abstract

Renormalisation group flows of the bosonic nonlinear \sigma-model are governed, perturbatively, at different orders of \alpha', by the perturbatively evaluated \beta--functions. In regions where \frac{\alpha'}{R_c^2} << 1 the flow equations at various orders in \alpha' can be thought of as \em approximating the full, non-perturbative RG flow. On the other hand, taking a different viewpoint, we may consider the abovementioned RG flow equations as viable {\em geometric} flows in their own right and without any reference to the RG aspect. Looked at as purely geometric flows where higher order terms appear, we no longer have the perturbative restrictions . In this paper, we perform our analysis from both these perspectives using specific target manifolds such as S^2, H^2, unwarped S^2 x H^2 and simple warped products. We analyze and solve the higher order RG flow equations within the appropriate perturbative domains and find the \em corrections arising due to the inclusion of higher order terms. Such corrections, within the perturbative regime, are shown to be small and they provide an estimate of the error which arises when higher orders are ignored. We also investigate the higher order geometric flows on the same manifolds and figure out generic features of geometric evolution, the appearance of singularities and solitons. The aim, in this context, is to demonstrate the role of the higher order terms in modifying the flow. One interesting aspect of our analysis is that, separable solutions of the higher order flow equations for simple warped spacetimes, correspond to constant curvature Anti-de Sitter (AdS) spacetime, modulo an overall flow--parameter dependent scale factor. The functional form of this scale factor (which we obtain) changes on the inclusion of successive higher order terms in the flow.