The $\mathcal{N}=4$ Bethe Ansatz beyond $\mathrm{SU}(N)$

Abstract

We solve the Bethe Ansatz equations (BAEs) to evaluate the superconformal index of four-dimensional $\mathcal{N}=4$ super-Yang--Mills with rank-two gauge algebra, at equal angular momentum fugacities $p=q$. The solutions for $A_2$ are known, and those for $D_2\cong A_1\oplus A_1$ follow readily from the (equally known) $A_1$ ones. The first genuinely new case is $B_2\cong C_2$, for which we obtain the complete solution set in closed form; for the exceptional case $G_2$ our results are numerical, with a single fully rational solution obtained in closed form. To the best of our knowledge, this is the first time any solution, analytic or numerical, has been found in non-$A$-type gauge algebra (for $\mathcal{N}=4$ or any other $\mathcal{N}=1$ theory). Along the way we uncover several phenomena absent in type $A$: isolated Weyl-fixed solutions contributing nontrivially to the index; isolated solutions in which at most half of the holonomies have rational coefficients (in contrast with the $A$-type fully rational Hong--Liu family); and solutions whose $|ω|\to0$ limit (with $p=q=:e^{2πiω}$) evades assumptions standardly made in the Cardy-like limit literature, landing on saddles that the usual analysis does not capture, for all types $BC\!D$. We also clarify the Bethe origin of the finite logarithmic correction to the Cardy expansion: it arises from the orbit size of a BAE solution under the gauge symmetries of the equations, rather than from the one-form center symmetry of the theory, to which the index is insensitive.