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Exceptional embeddings of $N=2$ minimal models

Ana Ros Camacho, Thomas A. Wasserman

TL;DR

The paper proves that certain unitary $N=2$ minimal quotients $M_d$ admit conformal embeddings into tensor products, specifically $M_{12} o M_3oxtimes M_4$ and $M_{30} o M_3oxtimes M_5$, which realize Ostrik-type algebras $E_6$ and $E_8$ inside $ ext{Rep}(M_{12})$ and $ ext{Rep}(M_{30})$. The authors build the embeddings by descending diagonal maps from universal $N=2$ VOAs and analyze the NS-sector modules, showing that the relevant algebras decompose as $M_3oxtimes M_4 \\cong C_1oxplus C_7$ and $M_3oxtimes M_5 \\cong C_1oxplus C_{11}oxplus C_{19}oxplus C_{29}$, with $E_6=C_1oxplus C_7$ in $ ext{Rep}(M_{12})$ and $E_8=C_1oxplus C_{11}oxplus C_{19}oxplus C_{29}$ in $ ext{Rep}(M_{30})$. These constructions provide a VOA-level realization of the LG/CFT prediction that products of ADE theories correspond to orbifold algebras in larger minimal-model categories, supporting the conjectured tensor-product structure in the LG/CFT correspondence. The work combines conformal embedding technology, analysis of Shapovalov radicals, and explicit module decompositions to substantiate the ADE-based embeddings and their algebra objects, with implications for the interplay between LG-models, ADE classifications, and VOA representation theory.

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.

Exceptional embeddings of $N=2$ minimal models

TL;DR

The paper proves that certain unitary minimal quotients admit conformal embeddings into tensor products, specifically and , which realize Ostrik-type algebras and inside and . The authors build the embeddings by descending diagonal maps from universal VOAs and analyze the NS-sector modules, showing that the relevant algebras decompose as and , with in and in . These constructions provide a VOA-level realization of the LG/CFT prediction that products of ADE theories correspond to orbifold algebras in larger minimal-model categories, supporting the conjectured tensor-product structure in the LG/CFT correspondence. The work combines conformal embedding technology, analysis of Shapovalov radicals, and explicit module decompositions to substantiate the ADE-based embeddings and their algebra objects, with implications for the interplay between LG-models, ADE classifications, and VOA representation theory.

Abstract

Vafa and Warner observed that the Landau-Ginzburg model associated to the potential (resp. ) is a product of two other models, associated to the potentials and (resp. and ). We translate this along the Landau-Ginzburg / Conformal Field Theory correspondence to a conjecture about the unitary minimal quotients of the superconformal algebra of central charge : there should be a conformal embedding (resp. ) that exhibits the product as Ostrik's (resp. ) algebra in the (resp. ) factor of the NS-sector of (resp. ). We motivate, formulate, and prove this conjecture.

Paper Structure

This paper contains 9 sections, 5 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. The LG/CFT correspondence
  3. Orbifolding
  4. ADE in the LG/CFT correspondence
  5. The Vafa-Warner prediction
  6. Proof of the main theorem
  7. $\mathbf{N=2}$ minimal quotients
  8. The conformal embedding
  9. The algebra objects

Key Result

Theorem 1

There are conformal embeddings between N=2 minimal quotients $M_d$ of central charge $3-\frac{6}{d}$, with $d \geq 2$, Furthermore, these embeddings exhibit $M_3\otimes M_4$ and $M_3 \otimes M_5$ as the algebras $E_6$ and $E_8$ in $\mathbf{Rep}(M_{12})$ and $\mathbf{Rep}(M_{30})$, respectively.

Theorems & Definitions (8)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof