Holography for $\mathcal{N}=1^*$ on $S^4$ and Supergravity

We study the $SO(3)$ sector of the $\mathcal{N}=1^*$ mass deformation of $\mathcal{N}=4$ super Yang-Mills on $S^4$. The gravity dual of this sector is $\mathcal{N}=2$ supergravity coupled to two hypermultiplets. The scalar fields in the hypermultiplets span a quaternionic-Kahler manifold that is described by the coset $G_{2,2}/SU(2)\times SU(2)$. We use the $\mathcal{N}=2$ supergravity dual to study field configurations in the bulk that feature analytical solutions, and compute the corresponding $S^4$ free energy using the procedure of holographic renormalization. We find that the free energy of these configurations is quadratic in the mass and show that it is devoid of unphysical ambiguities, hence providing an analytical prediction for the $\mathcal{N}=1$ four-sphere partition function at large 't Hooft coupling in the planar limit.