Orbifolds of Lattice Vertex Algebras

TL;DR

This paper studies orbifold vertex algebras derived from lattice vertex algebras under automorphisms of prime order, explicitly describing modules, characters, and modular properties, with applications to asymptotic and quantum dimensions.

Contribution

It provides explicit descriptions of irreducible modules and their characters for orbifold vertex algebras under prime order automorphisms, including special cases and permutation orbifolds.

Findings

Explicit irreducible module classification for $V_Q^\sigma$

Computed characters and modular transformations

Determined asymptotic and quantum dimensions

Abstract

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