String Corrections to the Holographic RG Flow of Supersymmetric SU(N) x SU(N+M) Gauge Theory

We study leading string corrections to the type IIB supergravity solution dual to the ${\cal N}=1$ supersymmetric $SU(N+M)\times SU(N)$ gauge theory coupled to bifundamental chiral superfields $A_i, B_j$, $i,j=1,2$. This solution was found in hep-th/0007191, and its asymptotic form describing logarithmic RG flow was constructed in hep-th/0002159. The leading tree-level string correction to the type IIB string effective action is represented by the invariant of the form $\alpha'^3 (R^4 + ...)$. Since the background contains 3-form field strengths, we need to know parts of this invariant that depend on them. By analyzing the 5-point superstring scattering amplitudes we show that certain $R^3 (H_3)^2$ and $R^3 (F_3)^2$ terms are absent from the effective action. This turns out to be crucial for finding agreement between $\alpha'$ corrected RG flow and the NSVZ beta functions in field theory. The agreement with field theory requires that the anomalous dimension of the operators $\Tr (A_i B_j)$ is corrected by a term of order $(M/N)^4 \lambda^{-1/2}$ from its value -1/2 found for M=0 ($\lambda$ is the appropriate 't Hooft coupling).