Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

CFTs with Large Gap from Barnes-Wall Lattice Orbifolds

Christoph A. Keller, Ashley Winter Roberts, Jeremy Roberts

Abstract

We investigate orbifolds of lattice conformal field theories with the goal of constructing theories with large gap. We consider Barnes-Wall lattices, which are a family of lattices with no short vectors, and orbifold by an extraspecial 2-group of lattice automorphisms. To construct the orbifold CFT, we investigate the orbifold vertex operator algebra and its twisted modules. To obtain a holomorphic CFT, a certain anomaly 3-cocycle $ω$ needs to vanish; based on evidence we provide, we conjecture that it indeed does. Granting this conjecture, we construct a holomorphic CFT of central charge 128 with gap 4.

CFTs with Large Gap from Barnes-Wall Lattice Orbifolds

Abstract

We investigate orbifolds of lattice conformal field theories with the goal of constructing theories with large gap. We consider Barnes-Wall lattices, which are a family of lattices with no short vectors, and orbifold by an extraspecial 2-group of lattice automorphisms. To construct the orbifold CFT, we investigate the orbifold vertex operator algebra and its twisted modules. To obtain a holomorphic CFT, a certain anomaly 3-cocycle needs to vanish; based on evidence we provide, we conjecture that it indeed does. Granting this conjecture, we construct a holomorphic CFT of central charge 128 with gap 4.

Paper Structure

This paper contains 38 sections, 13 theorems, 127 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. The Monster VOA
  4. The steps in more detail
  5. Orbifolds of lattice VOAs
  6. Orbifold theory
  7. Lattice VOAs and twisted group algebras
  8. Twisted modules and the defect representation
  9. Barnes-Wall lattices and the extraspecial group
  10. Barnes-Wall lattices and their automorphism groups
  11. Some properties of $E(m)$
  12. Lifting $E(m)$
  13. Defect representation and intertwiners
  14. Intertwining representation and Schur averaging
  15. Coordinate notation
  16. ...and 23 more sections

Key Result

Proposition 3.1

For $m\geq2$, under conjugation by elements of $BRW(m)$, $E(m)$ has 4 conjugacy classes with representatives $\mathbb{1},-\mathbb{1}, \sigma_2\otimes\mathbb{1}_2\otimes\cdots \otimes \mathbb{1}_2$ and $\sigma_1\sigma_2\otimes\mathbb{1}_2\otimes\cdots\otimes\mathbb{1}_2$. Their eigenvalues $\lambda$ Their centralizers in $E(m)$ are isomorphic to $E(m),E(m), \mathop{\mathrm{\mathbb{Z}}}\nolimits_2\

Theorems & Definitions (27)

  • Definition 3.1
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • Proposition 3.4
  • Theorem 3.5
  • proof
  • ...and 17 more