$c-a$ from the $N=1$ superconformal index

TL;DR

We present a universal prescription to extract the central-charge difference $c-a$ from the large-$N$ single-trace superconformal index. The method applies a differential operator to the suitably regularized index, yielding a finite $c-a$ term whose value is tested across holographic theories (including $ ext{N}=4$ SYM, toric quivers, SPP, and del Pezzo) and non-holographic large-$N models (e.g., U$(1)^N$ with matter, SQCD, and $A_k$ theories). The results agree with field-theory expectations and provide AdS/CFT matching for toric quivers, while also elucidating RG-flow relations between $ ext{N}=2$ UV theories and $ ext{N}=1$ IR theories and addressing finite-$N$ subtleties. This work demonstrates that protected operator data encoded in the superconformal index captures $c-a$, offering a powerful, field-theoretic route to access holographic and geometric information without explicit gravity computations.

Abstract

We present a prescription for obtaining the difference of the central charges, c-a, of a four dimensional superconformal quantum field theory from its single-trace index. The formula is derived from a one-loop holographic computation, but is expected to be valid independent of holography. We demonstrate the prescription with several holographic and non-holographic examples. As an application of our formula, we show the AdS/CFT matching of c-a for arbitrary toric quiver CFTs without adjoint matter that are dual to smooth Sasaki-Einstein 5-manifolds.