Exceptional embeddings of $N=2$ minimal models

TL;DR

This paper proves a conjecture relating to the embedding of certain minimal models in N=2 superconformal algebra, revealing exceptional algebraic structures connected to Landau-Ginzburg models and conformal field theory.

Contribution

It formulates and proves a conjecture about conformal embeddings of N=2 minimal models, linking Landau-Ginzburg potentials to algebraic structures in conformal field theory.

Findings

Confirmed the conformal embedding $M_{12} o M_3 imes M_4$

Confirmed the conformal embedding $M_{30} o M_3 imes M_5$

Identified the algebraic structure as Ostrik's $E_6$ and $E_8$ in specific representation categories

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 $\mathrm{Rep}(su(2)_{10})$ (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.