Superconformal Chern-Simons Theories from del Pezzo Geometries

TL;DR

This paper explicitly derives the grand potential for a specific superconformal Chern-Simons theory and connects it to topological string theory on del Pezzo geometries, resolving a long-standing puzzle.

Contribution

It provides an explicit expression for the grand potential of the U(N)^3 superconformal Chern-Simons theory and links it to topological string theory on local del Pezzo geometries.

Findings

Grand potential expressed explicitly for the theory.

Connection established with topological string theory on D_5 del Pezzo.

Extended analysis for Z_2 orbifold relating to E_7 del Pezzo geometry.

Abstract

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 D_5 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 Z_2 orbifold of this theory and find the grand potential is now given in terms of the local E_7 del Pezzo geometry.