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

In this paper, we calculate the topological free energy for a number of $$ mathcal{n} $$ ≥ 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 × 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 $$ mathrm{s}mathrm{l}left(2,mathbb{z}right) $$ duality.