Exceptional embeddings of \(N=2\) minimal models

Abstract

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 \(\textbf{Rep}(su(2)_{10})\) (resp. \(\textbf{Rep}(su(2)_{28})\)) factor of the NS sector of \(\textbf{Rep}(M_{12})\) (resp. \(\textbf{Rep}(M_{30})\)). We motivate, formulate, and prove this conjecture.