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: 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 – 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 4 ∂g ∂λ = − 2 Rc − α ′ ˇ Rc (2) − 2 α ′ 2 ˇ Rc (3) − 2 α ′ 3 ˇ Rc $^{(4)}$. . . (7) β (4) ij = − 1 16 R$_{1}$ + 1 48 R$_{2}$ − 1 16 [ 1 2 + ζ (3)] R$_{3}$ + 1 4 [1 + ζ (3)] R$_{4}$ + 1 16 [ 13 3 − 3 ζ (3)] R$_{5}$ + 1 8 [ 2 3 − ζ (3)] R$_{6}$ + 1 4 [ 8 3 + ζ (3)] R$_{7}$ + 1 4 [ − 1 3 + ζ (3)] R$_{8}$ + 1 12 R$_{9}$ + 1 12 R$_{10}$ − 1 6 R$_{11}$ + 1 16 [ 4 3 + ζ (3)] R$_{12}$ − 1 4 [ 4 3 + ζ (3)] R$_{13}$ + higher derivatives (6) β (3) ij = 1 8 ∇ $_{p}$R$_{iklm}$ ∇ $^{p}$R$_{j}$ klm − 1 16 ∇ $_{i}$R$_{klmp}$ ∇ $_{j}$R klmp + 1 2 R$_{klmp}$R$_{i}$ $^{mlr}$R$_{j}$kp r − 3 8 R$_{iklj}$R $^{kspr}$Rl spr (5) β (2) ij = 1 2 R$_{iklm}$R$_{j}$ klm (4) β (1) ij = R$_{ij}$ (3) β$_{ij}$ = α ′ β (1) ij + α ′ $^{2}$β (2) ij + α ′ $^{3}$β (3) ij + α ′ $^{4}$β (4) ij + O ( α ′ 5 ) (2)