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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 $N$ andfixed Chern-Simons levels. The topological free energy is defined as thelogarithm of the partition function of the theory on $S^2 \times S^1$ with atopological A-twist along $S^2$ and can be reduced to a matrix integral byexploiting the localization technique. The theories of our interest are dual toa variety of Calabi-Yau four-fold singularities, including a product of twoasymptotically locally Euclidean singularities and the cone over variouswell-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 bematched for theories which are related by dualities, including mirror symmetryand $\mathrm{SL}(2,\mathbb{Z})$ duality.

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