content_image/359436.jpg content_image/359433.jpg content_image/359435.jpg content_image/359434.jpg content_image/359432.jpg content_image/359431.jpg
where, β$_{ij}$ are the beta-functions , usually obtained order by order, perturbatively. In a field theory context, the parameter λ is related to the momentum cut-off scale Λ. Cut-off independence of the theory at length scales larger than inverse Λ gives rise to the above RG flow equation. 

As mentioned earlier, the β$_{ij}$ is usually obtained at various orders –i.e. as a series in the parameter α ′ (proportional to the inverse of the string tension). More specifically, we may write, 

where the various terms in the R. H. S. of the above equation, for the metric nonlinear σ -model, are given by: 

<img src='content_image/359436.jpg'>

<img src='content_image/359433.jpg'>

<img src='content_image/359435.jpg'>

<img src='content_image/359434.jpg'>

<img src='content_image/359432.jpg'>

with R$_{1}$ to R$_{13}$ representing various combinations of Riemann tensors(for explicit expressions see [15]). The above expressions are valid, in the RG flow context, only if α ′ R 2 c << 1 (where 1 R 2 c is the curvature scale). 

We rescale the flow parameter λ → λα ′ / 2 to bring the leading order term of Eq.(1) in the form of a Ricci flow. Thus we now have the modified flow equations as – 

<img src='content_image/359431.jpg'>

where ˇ Rc (2) is a symmetric 2 tensor defined as: ˇ Rc (2) = R$_{iklm}$R$_{jabc}$g $^{ka}$g$^{lb}$gmc , ˇ Rc (3) = 1 $_{2}$R$_{klmp}$ R$_{i}$ $^{mlr}$R$_{j}$kp r − 3 $_{8}$R$_{iklj}$ R $^{kspr}$Rl spr + ... + .. and ˇ Rc (4) = β (4) ij or, in component form, we 

