Partition functions on squashed seven-spheres and holography

TL;DR

This paper investigates the free energies of gravity and field theories on squashed seven-spheres, revealing universal behaviors and confirming conjectured holographic relations without a Hawking-Page transition.

Contribution

It provides the first detailed analysis of renormalized free energies on squashed seven-spheres in both gravity and field theories, including universal results for free energy derivatives.

Findings

No Hawking-Page transition in Euclidean Einstein gravity on squashed spheres.

Universal behavior of free energies governed by conformal symmetry.

Agreement of results with holographic conjectures for U(1) bundle squashings.

Abstract

Our paper presents two main results. First, we study the renormalized free energies of Euclidean Einstein gravity in asymptotically AdS$_8$ and various field theories on a squashed seven sphere. In the gravity theory, we demonstrate the absence of the Hawking-Page transition, while in the field theory, we focus on the O($N$) vector model and the massless free fermion model. The conformal symmetry governs the universal behaviors of the free energies for small and large squashings, which we confirm numerically and analytically. Second, we evaluate the second-order derivative of CFT free energy with respect to the squashing parameter, finding universal results that hold for generic conformal field theories. We examine two different squashings, one with an SU(2) bundle, which is the primary focus of our paper, and another with a U(1) bundle, where our results align with the conjectured formula from the gravity side in the literature.