Large $N$ topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces

TL;DR

This paper computes the large N topological free energy for certain supersymmetric gauge theories on S^2 x S^1, revealing dualities and geometric correspondences with Calabi-Yau and Sasaki-Einstein spaces.

Contribution

It provides explicit calculations of the topological free energy for a class of theories and demonstrates duality relations, including mirror symmetry and SL(2,Z) duality.

Findings

Matching of free energies for dual theories

Connection between gauge theories and Calabi-Yau geometries

Validation of dualities through free energy comparisons

Abstract

In this paper, we calculate the topological free energy for a number of ${\mathcal N} \geq 2$ Yang-Mills-Chern-Simons-matter theories at large $N$ and fixed Chern-Simons levels. The topological free energy is defined as the logarithm of the partition function of the theory on $S^2 \times S^1$ with a topological A-twist along $S^2$ and can be reduced to a matrix integral by exploiting the localization technique. The theories of our interest are dual to a variety of Calabi-Yau four-fold singularities, including a product of two asymptotically locally Euclidean singularities and the cone over various well-known homogeneous Sasaki-Einstein seven-manifolds, $N^{0,1,0}$, $V^{5,2}$, and $Q^{1,1,1}$. We check that the large $N$ topological free energy can be matched for theories which are related by dualities, including mirror symmetry and $\mathrm{SL}(2,\mathbb{Z})$ duality.