The Bethe-Ansatz approach to the $\mathcal N=4$ superconformal index at finite rank

TL;DR

This paper examines the Bethe-Ansatz method for calculating the superconformal index of ${ m SU}(N)$ ${ m N}=4$ SYM at finite rank, revealing the importance of both standard and non-standard solutions for accurate index reconstruction.

Contribution

It classifies Bethe-Ansatz solutions into standard and non-standard types and analyzes their roles in computing the superconformal index at finite $N$, especially for $N=2,3$.

Findings

Standard solutions fully recover the index for N=2.

For N≥3, standard solutions are insufficient, requiring non-standard solutions.

Continuous families of non-standard solutions significantly influence the index calculation.

Abstract

We investigate the Bethe-Ansatz (BA) approach to the superconformal index of ${\cal N}=4$ supersymmetric Yang-Mills with SU($N$) gauge group in the context of finite rank, $N$. We explicitly explore the role of the various types of solutions to the Bethe-Ansatz Equations (BAE) in recovering the exact index for $N=2,3$. We classify the BAE solutions as standard (corresponding to a freely acting orbifold $T^2/\mathbb{Z}_m \times \mathbb{Z}_n$) and non-standard. For $N=2$, we find that the index is fully recovered by standard solutions and displays an interesting pattern of cancellations. However, for $N\ge 3$, the standard solutions alone do not suffice to reconstruct the index. We present quantitative arguments in various regimes of fugacities that highlight the challenging role played by the continuous families of non-standard solutions.