Exceptional embeddings of $$N=2$$ minimal models

Abstract Vafa and Warner observed that the Landau–Ginzburg model associated to the potential $$E_6$$ E 6 (resp. $$E_8$$ E 8 ) is a product of two other models, associated to the potentials $$A_2$$ A 2 and $$ A_3$$ A 3 (resp. $$A_2 $$ A 2 and $$ A_4$$ A 4 ). We translate this along the Landau–Ginzburg/conformal field theory correspondence to a conjecture about the unitary minimal quotients $$M_d$$ M d of the $$N=2$$ N = 2 superconformal algebra of central charge $$c_d=3-\frac{6}{d}$$ c d = 3 - 6 d : there should be a conformal embedding $$M_{12}\hookrightarrow M_{3} \otimes M_4$$ M 12 ↪ M 3 ⊗ M 4 (resp. $$M_{30}\hookrightarrow M_{3} \otimes M_5$$ M 30 ↪ M 3 ⊗ M 5 ) that exhibits the product as Ostrik’s $$E_6$$ E 6 (resp. $$E_8$$ E 8 ) algebra in the $$\textbf{Rep}(su(2)_{10})$$ Rep ( s u ( 2 ) 10