A Bethe Ansatz type formula for the superconformal index
Francesco Benini, Elisa Milan
TL;DR
The paper tackles computing the 4d $\mathcal{N}=1$ superconformal index and introduces a Bethe Ansatz type finite-sum formula valid for rational fugacity ratios $\tau:a\omega$, $\sigma:b\omega$. It constructs BAEs from well-defined BA operators $Q_i(u;\xi,\nu_R,\omega)$ with a Jacobian $H$, and expresses the index as a sum over BA solutions $\hat{u}$ of $\mathcal{Z}_{tot}(\hat{u})$ divided by $H(\hat{u})$, up to a normalization constant $\kappa_G$. The continuation to generic fugacities is argued via holomorphy or continuity by reparametrizing flavor fugacities into $\Delta_a$ and $y_a$, ensuring a single-valued, analytic dependence. The method clarifies the relationship between Coulomb-branch localization (standard integral) and a Higgs-branch localization perspective, improves numerical tractability, and holds potential for applications to large-$N$ limits and black hole entropy in AdS$_5$.
Abstract
Inspired by recent work by Closset, Kim and Willett, we derive a new formula for the superconformal (or supersymmetric) index of 4d N=1 theories. Such a formula is a finite sum, over the solution set of certain transcendental equations that we dub Bethe Ansatz Equations, of a function evaluated at those solutions.