On the Protected Spectrum of the Minimal Argyres-Douglas Theory

Abstract: Despite the power of supersymmetry, finding exact closed-form expressions for the protected operator spectra of interacting superconformal field theories (SCFTs) is difficult. In this paper, we take a step towards a solution for the "simplest" interacting 4D $\mathcal{N}=2$ SCFT: the minimal Argyres-Douglas (MAD) theory. We present two results that go beyond the well-understood Coulomb branch and Schur sectors. First, we find the exact closed-form spectrum of multiplets containing operators that are chiral with respect to any $\mathcal{N}=1\subset\mathcal{N}=2$ superconformal subalgebra. We argue that this "full" chiral sector (FCS) is as simple as allowed by unitarity for a theory with a Coulomb branch and that, up to a rescaling of $U(1)r$ quantum numbers and the vanishing of a finite number of states, the MAD FCS is isospectral to the FCS of the free $\mathcal{N}=2$ Abelian gauge theory. In the language of superconformal representation theory, this leaves only the spectrum of the poorly understood $\bar{\mathcal{C}}{R,r(j,\bar j)}$ multiplets to be determined. Our second result sheds light on these observables: we find an exact closed-form answer for the number of $\bar{\mathcal{C}}_{0,r(j,0)}$ multiplets, for any $r$ and $j$, in the MAD theory. We argue that this sub-sector is also as simple as allowed by unitarity for a theory with a Coulomb branch and that there is a natural map to the corresponding sector of the free $\mathcal{N}=2$ Abelian gauge theory. These results motivate a conjecture on the full local operator algebra of the MAD theory.