# The Topologically Twisted Index in the 't Hooft Limit and the Dual AdS$_4$ Black Hole Entropy

## TL;DR

This paper computes the topologically twisted index of a supersymmetric Chern-Simons theory in the 't Hooft limit, showing it matches the entropy of dual AdS$_4$ black holes and exploring quantum corrections beyond leading order.

## Contribution

It provides an analytical expression for the index at large 't Hooft coupling and explores quantum corrections, connecting field theory results with supergravity predictions.

## Key findings

- Leading order index matches black hole entropy.
- Numerical analysis of genus expansion beyond leading order.
- Logarithmic term predicts one-loop supergravity correction.

## Abstract

We study the topologically twisted index of $\mathcal{N}=6$ supersymmetric Chern-Simons matter theory with $U(N)_k \times U(N)_{-k}$ gauge group in the 't Hooft limit, that is, for $N, k \, \to \infty$ with $\lambda=N/k$ fixed. In the regime where $\lambda$ is fixed and large we find an analytical expression for the leading order term of the index. The leading term of the index matches precisely the Bekenstein-Hawking entropy of the dual asymptotically AdS$_4$ magnetically charged black holes embedded in IIA supergrvity on AdS$_4\times \mathbb{CP}^3$, after a standard Legendre transformation. We numerically explore the genus expansion of the topologically twisteed index beyond the leading order, focusing on the genus one contribution, that is, $N^0$. We find qualitative agreement with the topological expansion of the free energy on $S^3$ at genus one. Our logarithmic in $\lambda$ term constitutes a prediction for the one-loop effective action on the IIA supergravity side.