Vertex algebras, topological defects, and Moonshine

TL;DR

This paper explores topological defect lines in holomorphic vertex operator algebras, especially the Monster VOA and Conway module, revealing new formulas, properties, and conjectures related to Moonshine and lattice symmetries.

Contribution

It provides a general formula for defect McKay-Thompson series, describes properties of defect categories, and proposes a new perspective linking defects to Leech lattice endomorphisms.

Findings

Derived a formula for defect McKay-Thompson series.

Established a surjective ring homomorphism between defect Grothendieck ring and Leech lattice endomorphisms.

Proposed a generalization of Moonshine conjectures involving topological defect lines.

Abstract

We discuss topological defect lines in holomorphic vertex operators algebras and superalgebras, in particular Frenkel-Lepowsky-Meurman Monster VOA $V^\natural$ with central charge $c=24$, and Conway module SVOA $V^{f\natural}$ with $c=12$. First, we consider duality defects in $V^\natural$ for all non-anomalous Fricke elements of the Monster group, and provide a general formula for the corresponding defect McKay-Thompson series. Furthermore, we describe some general properties of the category of defect lines preserving the $N=1$ superVirasoro algebra in $V^{f\natural}$. We argue that, under some mild assumptions, every such defect in $V^{f\natural}$ is associated with a $\mathbb{Z}$-linear map form the Leech lattice to itself. This correspondence establishes a surjective (not injective) ring homomorphism between the Grothendieck ring of the category of topological defects and the ring of Leech lattice endomorphisms. Finally, we speculate about possible generalization of the Moonshine conjectures that include topological defect lines.