Supersymmetric gauge theories on squashed five-spheres and their gravity duals

TL;DR

The paper constructs gravity duals for large N supersymmetric gauge theories on squashed five-spheres with SU(3) × U(1) symmetry by employing Euclidean Romans F(4) supergravity in six dimensions and uplifting to massive IIA. It derives 3/4 and 1/4 BPS bulk fillings, performs holographic renormalization, and shows exact agreement between holographic free energies and the large N gauge theory partition function obtained from a matrix model, as well as between holographic and matrix-model Wilson loops. A key outcome is the identification of a Killing vector K controlling both the free energy and Wilson loop observables, leading to conjectured universal formulas for these quantities on general supersymmetric five-sphere backgrounds. The results offer strong tests of gauge/gravity duality beyond conformally flat backgrounds and hint at a broader principle relating observables to Killing-vector data in supersymmetric backgrounds.

Abstract

We construct the gravity duals of large N supersymmetric gauge theories defined on squashed five-spheres with SU(3) x U(1) symmetry. These five-sphere backgrounds are continuously connected to the round sphere, and we find a one-parameter family of 3/4 BPS deformations and a two-parameter family of (generically) 1/4 BPS deformations. The gravity duals are constructed in Euclidean Romans F(4) gauged supergravity in six dimensions, and uplift to massive type IIA supergravity. We holographically renormalize the Romans theory, and use our general result to compute the renormalized on-shell actions for the solutions. The results agree perfectly with the large N limit of the dual gauge theory partition function, which we compute using large N matrix model techniques. In addition we compute BPS Wilson loops in these backgrounds, both in supergravity and in the large N matrix model, again finding precise agreement. Finally, we conjecture a general formula for the partition function on any five-sphere background, which for fixed gauge theory depends only on a certain supersymmetric Killing vector.