Localization and holography in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N}=2 $\end{document} gauge theories

We compare exact results from Pestun’s localization [1] of SU(N) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N}={2^{*}} $\end{document} gauge theory on S4 with available holographic models. While localization can explain the Coulomb branch vacuum of the holographic Pilch-Warner flow [2], it disagrees with the holographic Gauntlett et al. [3] vacuum of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N}=2 $\end{document} super Yang-Mills theory. We further compute the free energy of the Pilch-Warner flow on S4 and show that it disagrees with the localization result both for a finite S4 radius, and in the S4 decompactification limit. Thus, neither model represents holographic dual of supersymmetric S4 localization of [1].