Bijection between the CM sector of BAE solutions and untwisted CM extrema

Prove that the map from Bethe Ansatz equation solutions of four-dimensional N=4 super-Yang–Mills theory with semisimple gauge algebra g to extrema or poles of the untwisted elliptic Calogero–Moser potential is a bijection when restricted to the BAE solutions whose limits under vanishing flavor chemical potentials are CM extrema.

Background

The paper constructs a map by continuously tracking a BAE solution as the flavor chemical potentials Δ_a tend to zero. The limiting holonomy either satisfies the extremum equations of the untwisted elliptic Calogero–Moser system or reaches a pole of its potential. The authors define the CM sector as the preimage of the extrema and conjecture that every CM extremum arises from exactly one BAE solution in this sector.

The conjecture is verified for all rank-two semisimple gauge algebras considered in the paper, but it is not proved in general. Establishing it would provide a structural correspondence between the BAE solutions relevant to the N=4 superconformal index and the extrema of an untwisted integrable system.

References

Our main conjecture is that the map restricted to this sector is a bijection.

— Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems  (2608.19324 - Fazzi et al., 19 Aug 2026) in Section 1, ‘Introduction and motivation’; Section 3, ‘The bijectivity conjecture’

A second question is the extension to continuous families.

— Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems  (2608.19324 - Fazzi et al., 19 Aug 2026) in Section 3, ‘The bijectivity conjecture’; Section 4, ‘Conclusions’

While we do not prove here that these lists are complete, a semi-analytic proof will be presented in using techniques from computational commutative algebra.

— Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems  (2608.19324 - Fazzi et al., 19 Aug 2026) in Section 3.2, ‘The exceptional G2 algebra’

Finally, one may ask whether the twisted system has a Bethe-like counterpart of its own. Our construction shows that the $\mathcal{N}=4$ BAEs relevant to compute the index do not produce it. If the correspondence with $\mathcal{N}=1\ast$ is to be restored outside type $A$ (or any simply-laced type), some other set of equations must do so, and identifying them would presumably require an object that distinguishes long from short roots in a way the index does not. We leave this to future work.

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