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Orbifolds of Lattice Vertex Algebras

Bojko Bakalov, Jason Elsinger, Victor G. Kac, Ivan Todorov

TL;DR

This work classifies and analyzes orbifolds $V_Q^\sigma$ of lattice vertex algebras $V_Q$ under a prime-order automorphism $\sigma$. It provides explicit constructions and classifications of irreducible $V_Q^\sigma$-modules, computes their modified characters, and derives their modular transformation properties, including explicit $S$ and $T$-matrix data. The authors express orbifold characters as ratios of theta-functions to determinant factors and determine asymptotic and quantum dimensions of all irreducibles, with detailed treatment of the cases $p=2,3$ and permutation orbifolds. The results advance understanding of regular orbifold VOAs and their modular tensor category structure, linking twisted lattice modules to orbifold representations via concrete group-theoretic and lattice-theoretic data.

Abstract

To a positive-definite even lattice $Q$, one can associate the lattice vertex algebra $V_Q$, and any automorphism $σ$ of $Q$ lifts to an automorphism of $V_Q$. In this paper, we investigate the orbifold vertex algebra $V_Q^σ$, which consists of the elements of $V_Q$ fixed under $σ$, in the case when $σ$ has prime order. We describe explicitly the irreducible $V_Q^σ$-modules, compute their characters, and determine the modular transformations of characters. As an application, we find the asymptotic and quantum dimensions of all irreducible $V_Q^σ$-modules. We consider in detail the cases when the order of $σ$ is $2$ or $3$, as well as the case of permutation orbifolds.

Orbifolds of Lattice Vertex Algebras

TL;DR

This work classifies and analyzes orbifolds of lattice vertex algebras under a prime-order automorphism . It provides explicit constructions and classifications of irreducible -modules, computes their modified characters, and derives their modular transformation properties, including explicit and -matrix data. The authors express orbifold characters as ratios of theta-functions to determinant factors and determine asymptotic and quantum dimensions of all irreducibles, with detailed treatment of the cases and permutation orbifolds. The results advance understanding of regular orbifold VOAs and their modular tensor category structure, linking twisted lattice modules to orbifold representations via concrete group-theoretic and lattice-theoretic data.

Abstract

To a positive-definite even lattice , one can associate the lattice vertex algebra , and any automorphism of lifts to an automorphism of . In this paper, we investigate the orbifold vertex algebra , which consists of the elements of fixed under , in the case when has prime order. We describe explicitly the irreducible -modules, compute their characters, and determine the modular transformations of characters. As an application, we find the asymptotic and quantum dimensions of all irreducible -modules. We consider in detail the cases when the order of is or , as well as the case of permutation orbifolds.
Paper Structure (36 sections, 74 theorems, 487 equations, 7 tables)

This paper contains 36 sections, 74 theorems, 487 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Vertex Algebras and Their Twisted Modules
  3. Conformal vertex algebras
  4. Twisted representations of vertex algebras
  5. Regular vertex algebras
  6. Theta functions and transformation laws
  7. Twisted Representations of Lattice Vertex Algebras
  8. Lattice vertex algebras
  9. Twisted Heisenberg algebra
  10. The groups $G_\sigma$ and $G_\sigma^\perp$
  11. The $G_\sigma$-modules $W(\mu,\zeta)$
  12. The $\sigma$-twisted $V_Q$-modules $M(\mu,\zeta)$
  13. $L_0^{\mathrm{tw}}$-action on the $V_Q$-module $M(\mu,\zeta)$
  14. The sublattice $\bar{Q}\subset Q$
  15. The subgroup $\Gamma_{\sigma,\mu,\zeta}$
  16. ...and 21 more sections

Key Result

Proposition 2.2

Let $V$ be a vertex algebra, $\sigma$ an automorphism of $V$, and $M$ a $\sigma$-twisted representation of $V$. Then for all $a,b\in V$, $v\in M$, $k\geq0$, and sufficiently large $N$. Conversely, locpr and locpr3 imply the Borcherds identity vert5.

Theorems & Definitions (155)

  • Definition 2.1
  • Proposition 2.2: BM
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5: MC
  • Theorem 2.6: Z, ABD
  • Theorem 2.7: H
  • Corollary 2.8
  • proof
  • Definition 2.9
  • ...and 145 more