Exact results on the Bethe Ansatz evaluation of the SCI

TL;DR

This paper evaluates the superconformal index of certain 4d supersymmetric gauge theories using the Bethe Ansatz, finding exact matches with direct calculations in specific cases without needing continuous solutions.

Contribution

It demonstrates that for some toric quiver gauge theories, the Bethe Ansatz approach with only discrete solutions suffices for exact superconformal index evaluation.

Findings

Exact match for T^{1,1} without continuous solutions

Discrete solutions are sufficient for certain theories

Continuous solutions are necessary for others like SU(N) with N≥3

Abstract

We evaluate the superconformal index using the Bethe Ansatz (BA) approach for 4d $\mathcal{N}=1$ toric quiver gauge theories with a small amount of gauge groups. We restrict to $\mathrm{SU}(2)$ gauge factors and compare the results with the ones obtained by a direct evaluation of the index. The answer obtained from the BA approach using only discrete solutions for the BA equations does not always reproduce the direct evaluation result. A similar problem affects the case of $\mathrm{SU}(N)$ $\mathcal{N}=4$ SYM for $N \geq 3$, and it requires to introduce continuous solutions to the BA equations. However, we find that for $T^{1,1}$ and partially for the suspended pinch point the matching is exact in absence of the contributions from continuous solutions.