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Twisted logarithmic modules of lattice vertex algebras

Bojko Bakalov, McKay Sullivan

Abstract

Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called twisted logarithmic module, involves the logarithm of the formal variable and is related to logarithmic conformal field theory. We investigate twisted logarithmic modules of lattice vertex algebras, reducing their classification to the classification of modules over a certain group. This group is a semidirect product of a discrete Heisenberg group and a central extension of the additive group of the lattice.

Twisted logarithmic modules of lattice vertex algebras

Abstract

Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called twisted logarithmic module, involves the logarithm of the formal variable and is related to logarithmic conformal field theory. We investigate twisted logarithmic modules of lattice vertex algebras, reducing their classification to the classification of modules over a certain group. This group is a semidirect product of a discrete Heisenberg group and a central extension of the additive group of the lattice.

Paper Structure

This paper contains 22 sections, 14 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Vertex Algebras
  4. Lattice Vertex Algebras
  5. Twisted Logarithmic Modules
  6. Twisted Logarithmic Modules of Free Bosons
  7. The $\varphi$-twisted Vertex Operators
  8. Action of the Heisenberg Subalgebra
  9. The Exponentials $E_\lambda(z)$
  10. The Operators $U_\lambda(z)$
  11. Twisted Vertex Operators
  12. Products of Twisted Vertex Operators
  13. Special Functions
  14. Products of Exponentials
  15. Products of Operators $U_\lambda$
  16. ...and 7 more sections

Key Result

Proposition 2.7

Let $\mathfrak{h}$ be a finite-dimensional vector space endowed with a nondegenerate symmetric bilinear form $(\cdot|\cdot)$ and with commuting linear operators $\sigma$, $\mathcal{N}$ satisfying inv, such that $\sigma$ is invertible and semisimple and $\mathcal{N}$ is nilpotent. Then $\mathfrak{h}$

Theorems & Definitions (38)

  • Definition 2.1: Bak
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Bak
  • Example 2.5: $d = 2\ell$
  • Example 2.6: $d=2\ell-1$ and $\lambda=\pm 1$
  • Proposition 2.7: Bak
  • Proposition 2.8: BakBS
  • Remark 3.1
  • Proposition 3.2
  • ...and 28 more