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Uniqueness of vertex operator algebras arising from GKO-construction

Gu Yuhan, Zheng Wen

TL;DR

The paper addresses the uniqueness of VOA structures arising from GKO-based constructions $\mathcal{U}_k$ by leveraging fusion rules and braiding matrices of unitary Virasoro VOAs. The authors build $\mathcal{U}_k$ as commutants within a GKO framework, prove nonvanishing of the key scalars $\lambda_{i,j}^k$ in all admissible fusion channels, and use mirror-extension and braid-compatibility techniques to show any two VOA structures with those properties are isomorphic. They further prove $\mathcal{U}_k$ is generated by the Griess algebra, connecting the weight-two subspace to full VOA generation. The results extend known cases (3A and 6A algebras) and provide a framework to approach Conjecture A.14-type questions for a whole family of GKO-constructed VOAs. Open problems highlight the need for a uniform determination of irreducible modules and fusion rules across all $k$ and the role of Ising-vector generation in these algebras.

Abstract

A series of vertex operator algebras are constructed by GKO-construction, which is a generalization of 3A-algebra and 6A-algebra. It is proved their vertex operator algebra structures are unique under nonzero assumptions on some elements of braiding matrices. Furthermore, we show each of them is generated by weight two subspace, i.e. the Griess algebra.

Uniqueness of vertex operator algebras arising from GKO-construction

TL;DR

The paper addresses the uniqueness of VOA structures arising from GKO-based constructions by leveraging fusion rules and braiding matrices of unitary Virasoro VOAs. The authors build as commutants within a GKO framework, prove nonvanishing of the key scalars in all admissible fusion channels, and use mirror-extension and braid-compatibility techniques to show any two VOA structures with those properties are isomorphic. They further prove is generated by the Griess algebra, connecting the weight-two subspace to full VOA generation. The results extend known cases (3A and 6A algebras) and provide a framework to approach Conjecture A.14-type questions for a whole family of GKO-constructed VOAs. Open problems highlight the need for a uniform determination of irreducible modules and fusion rules across all and the role of Ising-vector generation in these algebras.

Abstract

A series of vertex operator algebras are constructed by GKO-construction, which is a generalization of 3A-algebra and 6A-algebra. It is proved their vertex operator algebra structures are unique under nonzero assumptions on some elements of braiding matrices. Furthermore, we show each of them is generated by weight two subspace, i.e. the Griess algebra.
Paper Structure (17 sections, 42 theorems, 113 equations)

This paper contains 17 sections, 42 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Basics
  4. Quantum Galois theory
  5. The unitary series of the Virasoro VOAs
  6. Braiding matrices
  7. GKO construction of the unitary Virasoro VOA
  8. Structure of 6A-algebra
  9. Mirror extensions for rational vertex operator algebras
  10. Construction of $U_k^i$ and $\{\mathcal{U}_k\}$
  11. $\lambda_{i,j}^k\neq 0$
  12. Fundamental Results
  13. When $m$ is odd
  14. When $m$ is even
  15. The uniqueness of $(\mathcal{U},Y)$
  16. ...and 2 more sections

Key Result

Proposition 2.10

If $N_{M^{1},M^{2}}^{M^{3}}<\infty$ or $N_{N^{1},N^{2}}^{N^{3}}<\infty,$ then

Theorems & Definitions (93)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 83 more