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 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 S²×S¹ with a topological A-twist along S² 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 SL(2,ℤ) duality.