Superconformal index of low-rank gauge theories via the Bethe Ansatz

We study the Bethe Ansatz formula for the superconformal index, in the case of 4d $\mathcal{N}=4$ super-Yang-Mills with gauge group $SU(N)$. We observe that not all solutions to the Bethe Ansatz Equations (BAEs) contribute to the index, and thus formulate "reduced BAEs" such that all and only their solutions contribute. We then propose, sharpening a conjecture of Arabi Ardehali et al. [arXiv:1912.04169], that there is a one-to-one correspondence between branches of solutions to the reduced BAEs and vacua of the 4d $\mathcal{N}=1^*$ theory. We test the proposal in the case of $SU(2)$ and $SU(3)$. In the case of $SU(3)$, we confirm that there is a continuous family of solutions, whose contribution to the index is non-vanishing.