Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems

We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang–Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero–Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero–Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero–Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^{\ast }$ deformation of $\mathcal{N}=4$ on ${\mathbb{R}}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{s}\mathfrak{u}(N)$: we exhibit an $\mathfrak{s}\mathfrak{o}(8)$ solution that flows to a pole of the Calogero–Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^{\ast }$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.