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We study the large $N$ limit of the refined topologically twisted index ofthree-dimensional $\mathcal{N} = 2$ gauge theories with a holographic dual,which is relevant for counting the number of microstates of rotating,magnetically charged and twisted AdS$_4$ black holes. As a difference withprevious computations, we perform a direct saddle point analysis of therelevant matrix model. Among other things we explicitly identify thedistribution of gauge magnetic fluxes that contributes at large $N$. The indexexhibits a large $N$ factorized form in agreement with the supergravityexpectations based on gravitational blocks and correctly reproduces the entropyof magnetically charged rotating AdS$_4 \times S^7$ black holes.

The large $N$ limit of topologically twisted indices: a direct approach