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

Vafa and Warner observed that the Landau–Ginzburg model associated to the potential E₆ E 6 (resp. E₈ E 8 ) is a product of two other models, associated to the potentials A₂ A 2 and A₃ A 3 (resp. A₂ A 2 and A₄ A 4 ). We translate this along the Landau–Ginzburg/conformal field theory correspondence to a conjecture about the unitary minimal quotients Md M d of the $$N=2$$ N = 2 superconformal algebra of central charge cd=3-6/d c d = 3 - 6 d : there should be a conformal embedding M₁₂ M₃ ⊗ M₄ M 12 ↪ M 3 ⊗ M 4 (resp. M₃₀ M₃ ⊗ M₅ M 30 ↪ M 3 ⊗ M 5 ) that exhibits the product as Ostrik’s E₆ E 6 (resp. E₈ E 8 ) algebra in the Rep(su(2)₁₀) Rep ( s u ( 2 ) 10