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

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 $Ad{S}_4$ 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 $Ad{S}_4$ geometry.

11 pages, 3 figures. References added

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