Curved space resolution of the singularity of fractional D3-branes on a conifold

We construct a supergravity dual to the cascading $\mathrm{SU}\mathrm{}(N+M)\mathrm{}\times \mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ supersymmetric gauge theory (related to fractional D3-branes on a conifold according to Klebanov and co-workers) in the case when the three-space is compactified on ${\mathrm{S}}^3$ and in the phase with unbroken chiral symmetry. The size of ${\mathrm{S}}^3$ serves as an infrared cutoff on the gauge-theory dynamics. For a sufficiently large ${\mathrm{S}}^3$ the dual supergravity background is expected to be nonsingular. We demonstrate that this is indeed the case: we find a smooth type IIB supergravity solution using a perturbation theory that is valid when the radius of ${\mathrm{S}}^3$ is large. We consider also the case with the Euclidean world volume being ${\mathrm{S}}^4$ instead of $\mathrm{R}\times {\mathrm{S}}^3,$ where the supergravity solution is again found to be regular. This “curved space” resolution of the singularity of the fractional D3-branes on the conifold solution is analogous to the one in the nonextremal (finite temperature) case discussed in our previous work.