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

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’