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

TL;DR

This paper explores holographic conformal field theories on S^2×S^2, revealing a discrete set of solutions with Efimov-like scaling, spontaneous symmetry breaking, and a conifold transition, advancing understanding of holographic RG flows on product spheres.

Contribution

It demonstrates the existence of Efimov-like solutions and spontaneous symmetry breaking in holographic theories on S^2×S^2, and analyzes the structure of the solution space including conifold transitions.

Findings

Regular solutions have one sphere shrinking to zero size at IR endpoint.

Existence of an infinite discrete set of Efimov-like solutions.

The lowest free energy solution spontaneously breaks Z_2 symmetry.

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 $S^2\times S^2$, 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 $S^2\times S^2$.