Exceptional embeddings of $N=2$ minimal models

Vafa and Warner observed that the Landau-Ginzburg model associated to the potential $E_6$ (resp. $E_8$) is a product of two other models, associated to the potentials $A_2$ and $A_3$ (resp. $A_2$ and $A_4$). We translate this along the Landau-Ginzburg / Conformal Field Theory correspondence to a conjecture about the unitary minimal quotients $M_d$ of the $N=2$ superconformal algebra of central charge $c_d=3-\frac{6}{d}$: there should be a conformal embedding $M_{12}\hookrightarrow M_{3} \otimes M_4$ (resp. $M_{30}\hookrightarrow M_{3} \otimes M_5$) that exhibits the product as Ostrik's $E_6$ (resp. $E_8$) algebra in the $\mathrm{Rep}(su(2)_{10})$ (resp. $\mathrm{Rep}(su(2)_{28})$) factor of the NS-sector of $\mathrm{Rep}(M_{12})$ (resp. $\mathrm{Rep}(M_{30})$). We motivate, formulate, and prove this conjecture.