Magnetically charged AdS5 black holes from class $$ \mathcal{S} $$ theories on hyperbolic 3-manifolds

<jats:title>A<jats:sc>bstract</jats:sc>
</jats:title><jats:p>We study the twisted index of 4d <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>N</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> = 2 class <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{S} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>S</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> theories on a closed hyperbolic 3-manifold <jats:italic>M</jats:italic><jats:sub>3</jats:sub>. Via 6d picture, the index can be written in terms of topological invariants called analytic torsions twisted by irreducible flat connections on the 3-manifold. Using the topological expression, we determine the full perturbative 1/<jats:italic>N</jats:italic> expansion of the twisted index. The leading part nicely matches the Bekestein-Hawking entropy of a magnetically charged black hole in the holographic dual <jats:italic>AdS</jats:italic><jats:sub>5</jats:sub> with <jats:italic>AdS</jats:italic><jats:sub>2</jats:sub> × <jats:italic>M</jats:italic><jats:sub>3</jats:sub> near-horizon.</jats:p>