Central charges from the $\mathcal{N} = 1$ superconformal index

TL;DR

This work shows how the four-dimensional $\mathcal{N}=1$ superconformal index encodes the central charges $a$ and $c$ of a SCFT. By applying differential operators to the single-trace index $\hat{I}$ and taking appropriate limits, the authors extract $a$ and $c$ from both holographic and non-holographic theories, including toric quivers without adjoint matter and certain finite-$N$ cases. They establish that the $t\to1$ expansion yields well-defined, scheme-independent pieces (notably the finite combination $3\hat{c}+\hat{a}$) and connect these to the supersymmetric Casimir energy, providing both AdS/CFT tests and a pathway to access central charges in non-holographic settings via plethystic expansions. The results reveal that the index carries not only $\,\mathcal{O}(1)$, but, in some large-$N$ non-holographic contexts, full $\mathcal{O}(N^2)$ information about $a$ and $c$, and they outline a finite-$N$ procedure to determine the central charges from the finite-$N$ single-trace index.

Abstract

We present prescriptions for obtaining the central charges, $a$ and $c$, of a four dimensional superconformal quantum field theory from the superconformal index. At infinite $N$, for holographic theories dual to Sasaki-Einstein 5-manifolds the prescriptions give the $\mathcal{O}(1)$ parts of the central charges. This allows us, among other things, to show the exact AdS/CFT matching of $a$ and $c$ for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds. In addition, we include evidence from non-holographic theories for the applicability of these results outside of a holographic setting and away from the large-$N$ limit.