[Paper Review] Regularity of rational vertex operator algebras

Chongying Dong, Haisheng Li+1 more|ArXiv.org|Aug 24, 1995

Algebraic structures and combinatorial models5 references4 citations

TL;DR

This paper establishes sufficient conditions for rational vertex operator algebras to be regular, meaning all weak modules decompose into direct sums of irreducible ordinary modules. It proves regularity for key examples including the moonshine module $V^\natural$, affine $\widehat{\mathfrak{g}}$-modules $L(l,0)$, Virasoro discrete series $L(c_{p,q},0)$, and lattice vertex algebras $V_L$, extending complete reducibility to non-positive definite lattices.

ABSTRACT

A regular vertex operator algebra is a vertex operator algebra such that any weak module (without grading) is a direct sum of ordinary irreducible modules. In this paper we give several sufficient conditions under which a rational vertex operator algebra is regular. We prove that the moonshine module vertex operator algebra $V^{ atural},$ the vertex operator algebras $L(l,0)$ associated with the integrable representations of affine algebras of level $l,$ the vertex operator algebras $L(c_{p,q},0)$ associated with irreducible highest weight representations for the discrete series of the Virasoro algebra and the vertex operator algebras $V_L$ associated with positive definite even lattices $L$ are regular. Our result for $L(l,0)$ implies that any restricted integrable module of level $l$ for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules. The space $V_L$ in general is a vertex algebra if $L$ is not positive definite. In this case we establish the complete reducibility of any weak module.

Motivation & Objective

• To establish sufficient conditions under which rational vertex operator algebras are regular.
• To prove that weak modules over rational VOAs decompose into direct sums of irreducible ordinary modules.
• To extend the complete reducibility property to vertex algebras associated with non-positive definite even lattices.
• To demonstrate that restricted integrable modules for affine Lie algebras of level $l$ are direct sums of irreducible highest weight modules.

Proposed method

• Utilizes representation theory of vertex operator algebras and properties of weak modules.
• Applies the theory of integrable modules for affine Lie algebras to analyze $L(l,0)$ VOAs.
• Employs the structure of Virasoro algebra representations to study $L(c_{p,q},0)$ VOAs.
• Analyzes lattice vertex algebras $V_L$ for even lattices $L$, including non-positive definite cases.
• Uses the definition of regularity: every weak module is a direct sum of irreducible ordinary modules.
• Applies results from quantum algebra and vertex algebra theory to verify regularity across diverse VOAs.

Experimental results

Research questions

• RQ1 Under what conditions is a rational vertex operator algebra regular?
• RQ2 Can the complete reducibility of weak modules be established for $L(l,0)$ vertex operator algebras associated with affine Lie algebras?
• RQ3 Is the moonshine module $V^\natural$ regular, and does it satisfy the direct sum decomposition of weak modules?
• RQ4 Does the regularity property extend to vertex algebras $V_L$ when the lattice $L$ is not positive definite?
• RQ5 Are restricted integrable modules for affine Lie algebras of level $l$ completely reducible into irreducible highest weight modules?

Key findings

• The moonshine module vertex algebra $V^\natural$ is regular, meaning all weak modules decompose into direct sums of irreducible ordinary modules.
• The vertex operator algebra $L(l,0)$ associated with integrable representations of affine Lie algebras of level $l$ is regular.
• The Virasoro vertex algebra $L(c_{p,q},0)$ for discrete series of the Virasoro algebra is regular.
• The lattice vertex algebra $V_L$ is regular even when the lattice $L$ is not positive definite, ensuring complete reducibility of all weak modules.
• The regularity of $L(l,0)$ implies that any restricted integrable module for the corresponding affine Lie algebra is a direct sum of irreducible highest weight integrable modules.