Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra

Robert L. Griess, Gerald Höhn

TL;DR

This paper fully determines the Virasoro frame stabilizers for the $E_8$ lattice VOA, showing there are exactly five Aut$(V)$-orbits on Virasoro frames and giving explicit stabilizer structures inside $E_8(\mathbb{C})$. It develops a lattice-code framework that expresses frame stabilizers as extensions of toral and Weyl components, with $G_D$ a $2B$-pure elementary abelian group and $G_C$ governed by a $\mathbb{Z}_4$-code $\Delta$ and a $D_X$-action, enabling precise classification by $k=\dim(\mathcal{D})$. The five frame types (k=1..5) are realized in $V_{E_8}$ with detailed group-theoretic descriptions, including extraspecial, wreath-product, and Alekseevski–Thompson structures, and the case $k=5$ is shown to be unique up to conjugacy. Appendices provide general results on lattice unimodularizations, VOAs automorphism lifts $\widetilde{W}$, and parabolic-orbit analysis, enriching the interplay between VOA theory, lattice theory, and finite group theory.

Abstract

The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine Kac-Moody VOA V. We analyze the frame stabilizers, both as abstract groups and as subgroups of the Lie group Aut(V) = E_8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V). In particular, we prove that there are exactly five orbits for the action of Aut(V) on the set of Virasoro frames, settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a result of general interest, on equivariant embeddings of lattices: embeddings of lattices into unimodular lattices which respect automorphism groups and definiteness. [DGH] C. Dong, R. Griess and G. Hoehn: "Framed vertex operator algebras, codes and the moonshine module", Comm. Math. Phys. 193 (1998), 407-448.

Virasoro frames and their Stabilizers for the E_8 lattice type Vertex Operator Algebra

TL;DR

This paper fully determines the Virasoro frame stabilizers for the lattice VOA, showing there are exactly five Aut-orbits on Virasoro frames and giving explicit stabilizer structures inside . It develops a lattice-code framework that expresses frame stabilizers as extensions of toral and Weyl components, with a -pure elementary abelian group and governed by a -code and a -action, enabling precise classification by . The five frame types (k=1..5) are realized in with detailed group-theoretic descriptions, including extraspecial, wreath-product, and Alekseevski–Thompson structures, and the case is shown to be unique up to conjugacy. Appendices provide general results on lattice unimodularizations, VOAs automorphism lifts , and parabolic-orbit analysis, enriching the interplay between VOA theory, lattice theory, and finite group theory.

Abstract

The concept of a framed vertex operator algebra was studied in [DGH] (q-alg/9707008). This article is an analysis of all Virasoro frame stabilizers of the lattice VOA V for the E_8 root lattice, which is isomorphic to the E_8-level 1 affine Kac-Moody VOA V. We analyze the frame stabilizers, both as abstract groups and as subgroups of the Lie group Aut(V) = E_8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V). In particular, we prove that there are exactly five orbits for the action of Aut(V) on the set of Virasoro frames, settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a result of general interest, on equivariant embeddings of lattices: embeddings of lattices into unimodular lattices which respect automorphism groups and definiteness. [DGH] C. Dong, R. Griess and G. Hoehn: "Framed vertex operator algebras, codes and the moonshine module", Comm. Math. Phys. 193 (1998), 407-448.

Paper Structure

This paper contains 19 sections, 49 theorems, 20 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Stabilizers for framed lattice VOAs
  3. General integral even lattices
  4. Lattices from marked binary codes
  5. General results about Virasoro frames in $V_{E_8}$
  6. The five classes of frames in $V_{E_8}$
  7. The case $k=1$
  8. The case $k=2$
  9. The case $k=3$
  10. The case $k=4$
  11. The case $k=5$
  12. Appendix
  13. Equivariant unimodularizations of even lattices
  14. Lifting ${\it Aut}(L)$ to the automorphism group of $V_L$
  15. The construction of $\widetilde{W}$ when ${\it det}(L)$ is odd
  16. ...and 4 more sections

Key Result

Proposition 1.1

Theorems & Definitions (77)

  • Proposition 1.1
  • Remark 1.2
  • Definition 2.1
  • Proposition 2.2: Cartan subalgebra
  • Corollary 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Definition 2.7: The code $\Delta$ and integers $k$, $\ell$
  • Theorem 2.8: The intersection $G\cap N$
  • ...and 67 more