arXiv:1002.3464v3 [hep-th] 30 Apr 2010 On higher order geometric and renormalisation group flows Kartik Prabhu, Sanjit Das and Sayan Kar ∗ Department of Physics and Meteorology and Centre for Theoretical Studies Indian Institute of Technology, Kharagpur, 721302, India 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 con- text, 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 (of the kind used in the bulk-brane models with a single extra dimension), cor- respond 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. Renormalisation group (RG) flows of the bosonic nonlinear σ -model are governed, perturbatively, at different orders of α ′ , by the perturbatively evaluated β –functions. In regions where α ′ R 2 c << 1 ( 1 R 2 c represents the curvature scale) the flow equations at various orders in α ′ can be thought of as 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 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 (small curvatures). In this paper, we perform our analysis from both these perspectives using specific target manifolds such as S $^{2}$, H $^{2}$, unwarped S 2 × H 2 and simple warped products. We analyze and solve the higher order RG flow equations within the appropriate perturbative domains and find the 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. Abstract 1 $^{∗}$Electronic address: kartikprabhu.iitkgp@gmail.com, sanjit@phy.iitkgp.ernet.in, sayan@iitkgp.ac.in