Toward Microstate Counting Beyond Large N in Localization and the Dual One-loop Quantum Supergravity

The topologically twisted index for ABJM theory with gauge group $U(N)k \times U(N){-k}$hasrecentlybeenshown,inthelarge-$N$ limit, to reproduce the Bekenstein-Hawking entropy of certain magnetically charged asymptotically AdS${}_4$ black holes. We numerically study the index beyond the large-$N$ limit and provide evidence that it contains a subleading logarithmic term of the form $-1\mathrm{/}2\log N$. On the holographic side, this term naturally arises from a one-loop computation. However, we find that the contribution coming from the near horizon states does not reproduce the field theory answer. We give some possible reasons for this apparent discrepancy.