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

We provide general formulae for the topologically twisted index of a general three-dimensional $$ mathcal{n} $$ ≥ 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 s 2 × s 1 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 n 3/2 (and n 5/3 in the massive type iia case). we find an intriguing relation with the (apparently unrelated) large n limit of the partition function on s 3. we also provide a universal formula for extracting the index from the large n partition function on s 3 and its derivatives and point out its analogy with the attractor mechanism for ads black holes.