Characterization of the non-CM BAE sector

Characterize the nontrivial Bethe Ansatz equation solutions of four-dimensional N=4 super-Yang–Mills theory that flow to poles of the untwisted elliptic Calogero–Moser potential under the limit Δ_a→0, determine whether such solutions form complete PSL(2,Z) orbits at higher rank, and establish whether they can be distinguished by a property of the superconformal index rather than by the BAE themselves.

Background

Some nontrivial BAE solutions do not map to CM extrema: instead, a root pairing reaches the period lattice and the limiting configuration is a pole of the CM potential. The paper exhibits this phenomenon for B_2 and G_2; in B_2, the three such solutions form a complete PSL(2,Z) orbit.

No general criterion is known for identifying these solutions before taking the limit. The authors also do not know whether the orbit structure observed for B_2 persists for higher-rank gauge algebras or whether the non-CM sector has an interpretation detectable directly from the index.

References

The HL solutions are the only known discrete solutions with non-vanishing contributions to the SCI so far. However, there is no proof of the fact that they are the only discrete solutions that contribute to the SCI. Moreover, as discussed above, the explicit form of the continuous solutions is known only for $N=3$. The exact evaluation of the SCI at finite rank remains then an open question, because the contribution of solutions beyond the HL family is required to reproduce the full index.

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

We have no characterization of the solutions that flow to poles beyond the observation that, for $B_2$, they form a complete $\mathrm{PSL}(2,\mathbb{Z})$ orbit, so that the CM sector is itself a union of orbits; whether this persists at higher rank, and whether these solutions are distinguished by some property of the index rather than of the equations, we do not know.

— Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems  (2608.19324 - Fazzi et al., 19 Aug 2026) in Section 3.1, ‘Interpretation of the non-CM sector’; Section 4, ‘Conclusions’