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

The superconformal index of the $\mathcal{N}=4$ $SU(N)$ supersymmetric Yang-Mills theory counts the $1/16$-BPS (Bogomol’nyi-Prasad-Sommerfield) states in this theory, and has been used via the $\mathrm{AdS}/\mathrm{CFT}$ 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 $\mathrm{AdS}/\mathrm{CFT}$ 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 (BA) equations. We show that the solutions to the BA equations that are known to 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 nonperturbative 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 (which 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 nonperturbative effects, the extra solutions are ruled out, leading to a precise match between the solutions on both sides.

Figure 1

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