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

Abstract

The superconformal index of the 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 × 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.

The exact Schur index in closed form

“…For N = 4 theories, such Bethe-Ansatz expansion can be interpreted holographically as accounting for the contributions of wrapped D3-branes [71]. While this approach does not seem to be naively applicable to the various limits of enhanced supersymmetry (Macdonald, Hall-Littlewood, Schur) of N = 2 superconformal indices, it could be of interest to compare our results with a carefully performed limit of the "BAE" final result.…”

Exact QFT duals of AdS black holes

“…Unlike the single-parallelogram ansatz, we find that after SL(3, Z) modular transformation there are extra poles included in the parallelogram as well as the zeros from the Haar measure, causing more complications. We feel that this is related to the gravitational stability issue of [24]. chemical potentials.…”

“…Our conditions imply their stability conditions Im(S D3 ) > 0, which makes our results consistent with the gravity analysis. Our conditions are stronger than their stability conditions, leading us to conjecture that there are more stability constraints from other instantons not discussed in [24]. Since our conditions (2.41), (2.45) come from the requirement that no branch points of the potential are included in the parallelogram, it would be interesting to establish a direct connection between the forces generated by the branch points and the gravitational instability.…”

“…We shall analyze in appendix A what kind of constraints are imposed by these conditions. These conditions are very reminiscent of the stability conditions of the Euclidean black hole solutions against D3-brane instantons wrapping S 3 ⊂ S 5 and S 1 ⊂ S 3 ⊂ AdS 5 [24]. Their contributions to the partition function is given by Z ← e iS D3 with S D3 = ±2πN δ I σ or ±2πN δ I τ in the two cases of (2.22).…”

“…To discuss these generalized solutions, we set the convention for δ I 's so that one of the two conditions (2.22) is met with σ, τ replaced by σ + r, τ + s. Then repeating the calculus of this section, one finds that the parallelogram ansatz with edges given by σ + r, τ + s solves the saddle point equation if (2.41) or (2.45) is met after replacing σ, τ by σ + r, τ + s. In appendix A, we show that (2.41) or (2.45) is always met by many choices of r, s, at least for typical choice of σ, τ which satisfies σ τ = real and Im(σ) Im(τ ) = rational. Following [24], we interpret our r, s as labelling multiple Euclidean solutions which map to the same Lorentzian solution once we compactify the temporal circle.…”