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Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

Hsian-Yang Chen, Ching Hung Lam

Abstract

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry $g\in O(L)$ such that $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$. We show that when $L_2=\emptyset$ and $g^i$ is fixed point free on $L$ for any $1\leq i\leq |g|-1$, $V_L^{\hat{g}}$ has extra automorphisms implies either (1) the order of $g$ is a prime or (2) $L$ is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.

Completely fixed point free isometry and cyclic orbifold of lattice vertex operator algebras

Abstract

We continue our study of cyclic orbifolds of lattice vertex operator algebras and their full automorphism groups. We consider some special isometry such that is fixed point free on for any . We show that when and is fixed point free on for any , has extra automorphisms implies either (1) the order of is a prime or (2) is isometric to the Leech lattice or some coinvariant sublattices of the Leech lattice.
Paper Structure (16 sections, 15 theorems, 46 equations)

This paper contains 16 sections, 15 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices
  4. Lattice VOAs and their automorphism groups
  5. Irreducible modules for cyclic orbifolds of lattice VOAs
  6. Orbifold VOAs having Extra automorphisms
  7. Completely fixed point free isometries
  8. Characterization of $L$ and $g$
  9. Case (I): $\sigma$-conjugation of $V_L(1)$ is of untwisted type
  10. Case (II): $\sigma$-conjugation of $V_L(1)$ is of twisted type
  11. Coinvariant lattices of the Leech lattice
  12. Coinvariant lattices associated with prime order elements
  13. Elements of non-prime orders
  14. Some explicit examples
  15. $h=g^s\in 2A$ for some positive integer $s$
  16. ...and 1 more sections

Key Result

Theorem 2.1

Let $L$ be a positive definite even lattice. Then Moreover, the intersection $N(V_L)\cap O(\hat{L})$ contains a subgroup $\mathrm{Hom} (L,\mathbb{Z}/2\mathbb{Z})$ and the quotient $\mathrm{Aut}\, (V_L)/N(V_L)$ is isomorphic to a quotient group of $O(L)$.

Theorems & Definitions (24)

  • Theorem 2.1: DN
  • Remark 2.2
  • Theorem 2.3
  • Definition 3.1
  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Lemma 4.5
  • Lemma 4.6
  • ...and 14 more