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We study the twisted index of 4d $\mathcal{N}$ = 2 class S theories on aclosed hyperbolic 3-manifold $M_3$. Via 6d picture, the index can be written interms of topological invariants called analytic torsions twisted by irreducibleflat connections on the 3-manifold. Using the topological expression, wedetermine the full perturbative 1/N expansion of the twisted index. The leadingpart nicely matches the Bekestein-Hawking entropy of a magnetically chargedblack hole in the holographic dual $AdS_5$ with $AdS_2\times M_3$ near-horizon.

Magnetically charged AdS5 black holes from class S theories on hyperbolic 3-manifolds