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

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.