Large N matrix models for 3d N = 2 theories: twisted index, free energy and black holes

We provide general formulae for the topologically twisted index of a general three-dimensional N ≥ 2 gauge theory with an M-theory or massive type IIA dual in the large N limit. The index is defined as the supersymmetric path integral of the theory on S2 × S1 in the presence of background magnetic fluxes for the R- and global symmetries and it is conjectured to reproduce the entropy of magnetically charged static BPS AdS4 black holes. For a class of theories with an M-theory dual, we show that the logarithm of the index scales indeed as N3/2 (and N5/3 in the massive type IIA case). We find an intriguing relation with the (apparently unrelated) large N limit of the partition function on S3. We also provide a universal formula for extracting the index from the large N partition function on S3 and its derivatives and point out its analogy with the attractor mechanism for AdS black holes.