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Universal Logarithmic Behavior in Microstate Counting and the Dual One-loop Entropy of AdS$_4$ Black Holes

We numerically study the topologically twisted index of several three-dimensional supersymmetric field theories on a genus $g$ Riemann surface times a circle, $Σ_g\times S^1$. We show that for a large class of theories with leading term of the order $N^{3/2}$, where $N$ is generically the rank of the gauge group, there is a universal logarithmic correction of the form $\frac{g-1}{2} \log N$. We explain how this logarithmic subleading correction can be obtained as a one-loop effect on the dual supergravity theory for magnetically charged, asymptotically AdS$_4\times M^7$ black holes for a large class of Sasaki-Einstein manifolds, $M^7$. The matching of the logarithmic correction relies on a generic cohomological property of $M^7$ and it is independent of the black hole charges. We argue that our supergravity results apply also to rotating, electrically charged asymptotically AdS$_4\times M^7$ black holes. We present explicitly the quiver gauge theories and the gravity side corresponding to $M^7=N^{0,1,0}, V^{5,2}$ and $Q^{1,1,1}$.