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Holographic CFTs and holographic RG flows on space-time manifolds which are$d$-dimensional products of spheres are investigated. On the gravity side, thiscorresponds to Einstein-dilaton gravity on an asymptotically $AdS_{d+1}$geometry, foliated by a product of spheres. We focus on holographic theories on$S^2\times S^2$, we show that the only regular five-dimensional bulk geometrieshave an IR endpoint where one of the sphere shrinks to zero size, while theother remains finite. In the $Z_2$-symmetric limit, where the two spheres havethe same UV radii, we show the existence of a infinite discrete set of regularsolutions, satisfying an Efimov-like discrete scaling. The $Z_2$-symmetricsolution in which both spheres shrink to zero at the endpoint is singular,whereas the solution with lowest free energy is regular and breaks $Z_2$symmetry spontaneously. We explain this phenomenon analytically by identifyingan unstable mode in the bulk around the would-be $Z_2$-symmetric solution. Thespace of theories have two branches that are connected by a conifold transitionin the bulk, which is regular and correspond to a quantum first ordertransition. Our results also imply that $AdS_5$ does not admit a regularslicing by $S^2\times S^2$.

Holographic QFTs on S$^2\times $S$^2$, spontaneous symmetry breaking and Efimov saddle points