| Summary: | Abstract Holographic CFTs and holographic RG flows on space-time manifolds which are d-dimensional products of spheres are investigated. On the gravity side, this corresponds 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 × S 2, we show that the only regular five-dimensional bulk geometries have an IR endpoint where one of the sphere shrinks to zero size, while the other remains finite. In the Z 2-symmetric limit, where the two spheres have the same UV radii, we show the existence of a infinite discrete set of regular solutions, satisfying an Efimov-like discrete scaling. The Z 2-symmetric solution 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 identifying an unstable mode in the bulk around the would-be Z 2-symmetric solution. The space of theories have two branches that are connected by a conifold transition in the bulk, which is regular and correspond to a quantum first order transition. Our results also imply that AdS 5 does not admit a regular slicing by S 2 × S 2.
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