Probing AdS4/CFT3 proposals beyond chiral rings

We calculate the superconformal Witten index for the Chern-Simons-matter theory which was proposed to describe multiple M2-branes on $C^2 X C^2/Z_k$. We consider a variant of this model, which exhibits explicit N=3 supersymmetry and has the advantage of not having an exotic branch of the moduli space. At $k=1$, we compare the index with that from the proposed gravity dual and find a disagreement.