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The localization formula of Chern-Simons quiver gauge theory on $S^3$ nicelyreproduces the geometric data such as volume of Sasaki-Einstein manifolds inthe large-$N$ limit, at least for vector-like models. The validity ofchiral-like models is not established yet, due to technical problems in bothanalytic and numerical approaches. Recently Gulotta, Herzog and Pufu suggestedthat the counting of chiral operators can be used to find the eigenvaluedistribution of quiver matrix models. In this paper we apply this method tosome vector-like or chiral-like quiver theories, including the triangularquivers with generic Chern-Simons levels which are dual to in-homogeneousSasaki-Einstein manifolds $Y^{p,k}(\mathbb{CP}^2)$. The result is consistentwith AdS/CFT and the volume formula. We discuss the implication of ouranalysis.

Operator Counting for N=2 Chern-Simons Gauge Theories with Chiral-like Matter Fields