A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index

TL;DR

The paper establishes a precise holographic interpretation of the N=4 SYM superconformal index by matching a large-N Bethe Ansatz expansion with a discrete set of Euclidean gravity solutions. It identifies Hong–Liu BA solutions with Euclidean black holes and their Z_m orbifolds, capturing both the leading $\mathcal{O}(N^2)$ gravity action and non-perturbative D3-brane corrections, while proposing a stability criterion to exclude gravity saddles that would destabilize the correspondence. Through a detailed analysis of elliptic gamma functions, Jacobians, and non-perturbative D3-brane actions, the authors demonstrate a one-to-one mapping between field-theory BA contributions and gravitational saddles, including both perturbative $1/N$ terms and non-perturbative exponentials. The results reinforce the view that the supersymmetric partition function on $S^3\times S^1$ localizes onto a discrete set of gravity solutions, offering a concrete bridge between CFT BA solutions and the gravity dual, with potential implications for black hole microstate counting and generalizations to broader SCFTs and dimensions.

Abstract

The superconformal index of the $\mathcal{N}=4$ SU(N) supersymmetric Yang-Mills theory counts the 1/16-BPS states in this theory, and has been used via the AdS/CFT correspondence to count black hole microstates of 1/16-BPS black holes. On one hand, this index may be related to the Euclidean partition function of the theory on $S^3\times S^1$ with complex chemical potentials, which maps by the AdS/CFT correspondence to a sum over Euclidean gravity solutions. On the other hand, the index may be expressed as a sum over solutions to Bethe Ansatz Equations (BAEs). We show that the known solutions to the BAEs that have a good large $N$ limit, for the case of equal chemical potentials for the two angular momenta, have a one-to-one mapping to (complex) Euclidean black hole solutions on the gravity side. This mapping captures both the leading contribution from the classical gravity action (of order $N^2$), as well as non-perturbative corrections in $1/N$, which on the gravity side are related to wrapped D3-branes. Some of the BA solutions map to orbifolds of the standard Euclidean black hole solutions (that obey exactly the same boundary conditions as the other solutions). A priori there are many more gravitational solutions than Bethe Ansatz solutions, but we show that by considering the non-perturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides.

Figures (1)

• Figure 1: Fundamental strips for $[\Delta]_\tau$ and $[\Delta]'_\tau$. The function $[\Delta]_\tau$ is the restriction of $\Delta$ mod $1$ to the region $\mathop{\mathrm{\mathbb{I}m}}\nolimits(-1/\tau) > \mathop{\mathrm{\mathbb{I}m}}\nolimits(\Delta/\tau)>0$ (in yellow, on the left), while $[\Delta]'_\tau$ is the restriction of $\Delta$ mod $1$ to the region $0 > \mathop{\mathrm{\mathbb{I}m}}\nolimits(\Delta/\tau) > \mathop{\mathrm{\mathbb{I}m}}\nolimits(1/\tau)$ (in purple, on the right). We dubbed $\mathscr{L}_\tau$ the blue line through $0$ and $\tau$.