Stability conjecture for polarized chiral rings and SCFTs

Let $\mathcal{R}$ be a chiral ring equipped with a polarization $\zeta$, so that $({\mathcal{R}},\zeta)$ is a polarized chiral ring. A polarized chiral ring is stable if it has no destabilizing test chiral ring, where a test chiral ring is obtained as a flat limit under a one-parameter symmetry and is destabilizing when it gives no less central charge $a$ for the possible $U(1)_R$ symmetries $a\zeta+s\eta$ with $s\geq 0$. Stability conjecture. $({\mathcal{R}},\zeta)$ is the chiral ring of a superconformal field theory if and only if it is stable. This is proposed as a generalized $a$-maximization principle: stability is intended to characterize which chiral rings arise from four-dimensional $\mathcal{N}=1$ SCFTs, while the relation to Sasaki–Einstein and Ricci-flat conic metrics provides geometric motivation. The paper does not establish the conjecture.