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Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

Ching Hung Lam, Hiroki Shimakura

Abstract

In this article, we study the automorphism group of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order $p$. We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over $\mathbb{Z}_p$ or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order $p$.

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

Abstract

In this article, we study the automorphism group of the cyclic orbifold of a vertex operator algebra associated with a rootless even lattice for a lift of a fixed-point free isometry of odd prime order . We prove that such a cyclic orbifold contains extra automorphisms, not induced from automorphisms of the lattice vertex operator algebra, if and only if the rootless even lattice can be constructed by Construction B from a code over or is isometric to the coinvariant lattice of the Leech lattice associated with a certain isometry of order .

Paper Structure

This paper contains 29 sections, 49 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. VOAs, modules and automorphisms
  5. Lattice VOAs and their automorphism groups
  6. Irreducible $V_L^{\hat{g}}$-modules and their conjugates
  7. Irreducible $V_L^{\hat{g}}$-modules of untwisted type
  8. Irreducible $V_L^{\hat{g}}$-modules of twisted type
  9. Irreducible $V_L^{\hat{g}}$-modules under $(1-g)L^*\subset L$
  10. Lattices associated with subgroups of $\bigoplus_{i=1}^t\mathbb{Z}_{k_i}$
  11. Root lattice of type $A_{k-1}$
  12. Constructions of lattices associated with subgroups of $\bigoplus_{i=1}^t\mathbb{Z}_{k_i}$
  13. Even lattices associated with codes over $\mathbb{Z}_p$
  14. Characterizations of even lattices associated with codes over $\mathbb{Z}_p$
  15. Coinvariant lattices of the Leech lattice and Construction B
  16. ...and 14 more sections

Key Result

Lemma 2.1

Let $L$ be a lattice and let $g$ be a fixed-point free isometry of $L$. Then we have $((1-g)L)^*=(1-g)^{-1}L^*$.

Theorems & Definitions (90)

  • Lemma 2.1
  • proof
  • Definition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Proposition 2.7
  • Definition 3.1
  • ...and 80 more