Bethe Ansatz for the superconformal index with unequal angular momenta

A few years ago it was shown that the superconformal index of the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi mathvariant="script">N</a:mi><a:mo>=</a:mo><a:mn>4</a:mn></a:math> supersymmetric <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:mi>S</d:mi><d:mi>U</d:mi><d:mo stretchy="false">(</d:mo><d:mi>N</d:mi><d:mo stretchy="false">)</d:mo></d:math> Yang-Mills theory in the large <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>N</h:mi></h:math> limit matches with the entropy of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mn>1</j:mn><j:mo>/</j:mo><j:mn>16</j:mn></j:math>-supersymmetric black holes in type IIB string theory on <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>AdS</l:mi><l:mn>5</l:mn></l:msub><l:mo>×</l:mo><l:msup><l:mi>S</l:mi><l:mn>5</l:mn></l:msup></l:math>. 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:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>N</n:mi></n:math>). Benini 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.