Superconformal Chern-Simons theories from del Pezzo geometries

We present an explicit expression for the grand potential of the U(N )3 superconformal Chern-Simons theory with the Chern-Simons levels being (k, 0, −k). From the viewpoint of the Newton polygon, it is expected that the grand potential is given by the free energy of the topological string theory on the local D5 del Pezzo geometry, though the explicit identification was a puzzle for years. We show how the expectation is realized explicitly. As a bonus, we can also study the ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{Z}}_2 $$\end{document} orbifold of this theory and find the grand potential is now given in terms of the local E7 del Pezzo geometry.