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

TL;DR

This work computes the large $N$ topologically twisted index for a wide class of ${\cal N}\ge 2$ 3d gauge theories, using localization to reduce to a matrix model and solving the Bethe Ansatz equations to obtain the topological free energy $\mathfrak{F}$. The analysis spans ${\cal N}=4,3,2$ quivers and includes models dual to CY$_4$ singularities and Sasaki–Einstein spaces such as $N^{0,1,0}$, $V^{5,2}$, and $Q^{1,1,1}$; in all cases $\mathfrak{F}$ scales as $N^{3/2}$ and often reduces to ABJM-type results multiplied by a geometry-dependent factor like $\sqrt{\frac{\text{(orbifold product)}}{k}}$. The paper provides explicit BAEs solutions, region-by-region analyses, and maps between different theories under dualities (mirror symmetry, SL$(2,\mathbb{Z})$), yielding nontrivial checks of holographic expectations and offering a framework to compare with black hole entropy in AdS$_4$ via the index theorem. Overall, the results establish a versatile, cross-checked program for computing and comparing topologically twisted indices across a broad landscape of 3d gauge theories with M-theory duals. The findings connect detailed field-theoretic indices to geometric data of CY$_4$ and Sasaki–Einstein manifolds, reinforcing the utility of the twisted index as a diagnostic for dualities and holographic entropy.

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.