Bethe Ansatz for the superconformal index with unequal angular momenta

A few years ago it was shown that the superconformal index of the $\mathcal{N}=4$ supersymmetric $SU(N)$ Yang-Mills theory in the large $N$ limit matches with the entropy of $1/16$-supersymmetric black holes in type IIB string theory on ${\mathrm{AdS}}_5\times S^5$. In some cases, an even more detailed match between the two sides is possible. When the two angular momentum chemical potentials in the index are equal, the superconformal index can be written as a discrete sum of Bethe ansatz solutions, and it was shown that specific terms in this sum are in a one-to-one correspondence to stable black hole solutions, and that the matching can be extended to nonperturbative contributions from wrapped D3-branes. A Bethe ansatz approach to computing the superconformal index exists also when the ratio of the angular momentum chemical potentials is any rational number, but in those cases it involves a sum over a very large number of terms (growing exponentially with $N$). Benini et al. showed that a specific one of these terms matches with the black hole, but the role of the other terms is not clear. In this paper we analyze some of the additional contributions to the index in the Bethe ansatz approach, and we find that their matching to the gravity side is much more complicated than in the case of equal chemical potentials. In particular, we find some contributions that are larger than the one that was found to match the black holes, in which case they must cancel with other large contributions. We give some evidence that cancellations of this type are possible, but we leave a full understanding of how they work to the future.

Figure 1

The eigenvalue distribution for different choices of $M$ (which are divisors of $ab$) for the basic solution with $ab=6$, $N=30$, drawn on a torus with periodicities 1 and $ab\omega$. The $M=1$ case is the one analyzed in [10]. (a) $M=1$. (b) $M=2$. (c) $M=3$.

Figure 2

We plot ${\mathrm{\Theta }}_r^{u=\frac{\omega }{2}}$ for $a=9999$, $b=10000$, $\tau =-0.67+2.3i$, ${\mathrm{\Delta }}_1=0.3+1.1i$, and ${\mathrm{\Delta }}_2=0.11–0.6i$. The black vertical lines show two points in which $|{\mathrm{\Theta }}_r^u|=1$, that represent the beginning and end of a streak of $|{\mathrm{\Theta }}_r^u|$ values that are larger than 1. The blue horizontal lines mark the special values of 0 for $\log (|{\mathrm{\Theta }}_r^{\frac{\omega }{2}}|)$, and $-\pi ,0,\pi$ for $\arg ({\mathrm{\Theta }}_r^{\frac{\omega }{2}})$.

Figure 3

We plot $|Z_m^{\frac{\omega }{2}}|$ (blue) and $\log (|Z_m^{\frac{\omega }{2}}|)$ (green) as a function of $m$. The horizontal lines mark the beginning and end of $a$-point windows (here $a=30$, $b=31$, and the rest of the parameters are as in Fig. 2). The values of each window can be calculated from the values of the former window by multiplying its values by ${\mathrm{\Theta }}_r^{\frac{\omega }{2}}$ with the correct shift.