Effective action ofβ-deformedN=4SYM theory and AdS/CFT

Abstract

We compute the one-loop effective action in $\mathcal{N}=1$ superconformal $SU(N)$ gauge theory which is an exactly marginal deformation of the $\mathcal{N}=4$ SYM. We consider an Abelian background of constant $\mathcal{N}=1$ gauge field and single chiral scalar. While for finite $N$ the effective action depends nontrivially on the deformation parameter $q={e}^{i\ensuremath{\pi}\ensuremath{\beta}}$, this dependence disappears in the large $N$ limit if the parameter $\ensuremath{\beta}$ is real. This conclusion matches the strong-coupling prediction coming from the form of a D3-brane probe action in the dual supergravity background: for the simplest choice of the D3-brane position the probe action happens to be the same as for a D3-brane in $Ad{S}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$ placed parallel to the boundary of $Ad{S}_{5}$. This suggests that in the real $\ensuremath{\beta}$ deformation case there exists a large $N$ nonrenormalization theorem for the ${F}^{4}$ term in the action.