The large-$N$ limit of 4d superconformal indices for general BPS charges

TL;DR

This paper analyzes the superconformal index of 4d $ ext{N}=1$ theories at large N for general BPS charges, especially unequal angular momenta, using Bethe Ansatz and elliptic methods, revealing their correspondence and extending previous results.

Contribution

It extends the Bethe Ansatz and elliptic extension methods to handle unequal angular momenta in superconformal indices, providing new insights and simplified derivations.

Findings

Good agreement between Bethe Ansatz and elliptic methods for $J_1 eq J_2$

Extended Bethe Ansatz to include a broader class of exponential terms

Established correspondence between saddle points in elliptic action and Bethe Ansatz terms

Abstract

We study the superconformal index of $\mathcal N=1$ quiver theories at large-$N$ for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic extension method. We are particularly interested in the case of unequal angular momenta, $J_1\ne J_2$, which has only been partially considered in the literature. We revisit the previous computation with the Bethe Ansatz formulation with generic angular momenta and extend it to encompass a large class of competing exponential terms. In the process, we also provide a simplified derivation of the original result. We consider the newly-developed elliptic extension method as well; we apply it to the $J_1\ne J_2$ case, finding a good match with the Bethe Ansatz results. We also investigate the relation between the two different approaches, finding in particular that for every saddle of the elliptic action there are corresponding terms in the Bethe Ansatz formula that match it at large-$N$.