AdS Taub-Nut Space and the O(N) Vector Model on a Squashed 3-Sphere

TL;DR

This paper explores the holographic relationship between a strongly coupled O(N) vector model on a squashed three-sphere and the dual AdS-Taub-NUT gravity space, providing analytical results and insights into higher spin contributions.

Contribution

It presents the first analytical solution for the boundary theory's action in the squashing limit and links it to the bulk AdS-Taub-NUT geometry, highlighting higher spin effects.

Findings

The boundary action scales as ln(1+α)/(1+α)^2 near the limit.

The bulk geometry is dual to the boundary theory in the squashing limit.

The results match previous numerical findings and suggest higher spin contributions.

Abstract

In this note, motivated by the Klebanov-Polyakov conjecture we investigate the strongly coupled O(N) vector model at large $N$ on a squashed three-sphere and its holographic relation to bulk gravity on asymptotically locally $AdS_4$ spaces. We present analytical results for the action of the field theory as the squashing parameter $\alpha\to-1$, when the boundary becomes effectively one dimensional. The dual bulk geometry is AdS-Taub-NUT space in the corresponding limit. In this limit we solve the theory exactly and show that the action of the strongly coupled boundary theory scales as $\ln(1+\alpha)/ (1+\alpha)^2$. This result is remarkably close to the $-1/(1+\alpha)^2$ scaling of the Einstein gravity action for AdS-Taub-NUT space. These results explain the numerical agreement presented in hep-th/0503238, and the soft logarithmic departure is interpreted as a prediction for the contribution due to higher spin fields in the bulk $AdS_4$ geometry.