Additional sporadic Bethe solutions at special ranks

Determine whether Bethe Ansatz Equations for the four-dimensional N=2 SU(N)^K necklace theories possess additional solutions that arise only for particular fixed values of N or K, beyond the progenitor and descendant families constructed for generic N and K.

Background

The constructed progenitor and descendant solutions are defined uniformly for generic values of N and K. The authors explicitly caution that this classification may not be exhaustive because solutions could emerge sporadically at special ranks or quiver sizes. Such solutions would affect the counting of Bethe vacua and potentially the finite-rank superconformal index.

References

In analogy with the case of $\SU(N)$ $\mathcal{N}=4$ SYM we cannot exclude the existence of other solutions, for example solutions that can sporadically emerge for fixed values of $N$.

— New Bethe vacua for $\mathcal{N}=2$ elliptic models  (2609.20631 - Amariti et al., 17 Sep 2026) in Section 1, Introduction, footnote following the discussion of the solution classification