[Paper Review] Vertex operator algebras associated to modular invariant representations for $A_1 ^{(1)}$

Dražen Adamović, Antun Milas|ArXiv.org|Sep 22, 1995

Algebraic structures and combinatorial models3 references3 citations

TL;DR

This paper constructs and classifies vertex operator algebras L(k,0) associated with admissible-level representations of the affine Lie algebra $A_1^{(1)}$, proving rationality in the category $\mathcal{O}$ and determining all irreducible weight modules. The key contribution is a complete classification of irreducible representations within the category of weight modules for these VOAs at admissible rational levels.

ABSTRACT

We investigate vertex operator algebras $L(k,0)$ associated with modular-invariant representations for an affine Lie algebra $A_1 ^{(1)}$ , where k is 'admissible' rational number. We show that VOA $L(k,0)$ is rational in the category $\cal O$ and find all irreducible representations in the category of weight modules.

Motivation & Objective

• To study vertex operator algebras L(k,0) associated with modular-invariant representations of the affine Lie algebra $A_1^{(1)}$.
• To analyze the structure of these VOAs when k is an admissible rational number.
• To determine the full set of irreducible representations within the category of weight modules.
• To establish rationality of L(k,0) in the category $\mathcal{O}$ for admissible k.
• To provide a complete classification of irreducible modules in the weight module category for these VOAs.

Proposed method

• The authors use representation theory of affine Lie algebras, focusing on admissible representations of $A_1^{(1)}$ at rational levels.
• They construct the vertex operator algebra L(k,0) as the irreducible quotient of a Verma module at admissible level k.
• The analysis is conducted within the category $\mathcal{O}$, leveraging highest-weight module theory.
• The classification of irreducible weight modules is achieved through the structure theory of modules over affine Kac-Moody algebras.
• The proof of rationality relies on the finiteness of irreducible modules and the semisimplicity of the category of weight modules.
• The modular invariance of the character is used as a consistency condition to constrain the admissible levels.

Experimental results

Research questions

• RQ1 Which irreducible representations of $A_1^{(1)}$ at admissible rational levels give rise to well-defined vertex operator algebras L(k,0)?
• RQ2 How can the category of weight modules for L(k,0) be fully classified at admissible k?
• RQ3 What conditions ensure that L(k,0) is rational in the category $\mathcal{O}$?
• RQ4 How does modular invariance constrain the admissible levels k for $A_1^{(1)}$?
• RQ5 What is the structure of the irreducible modules in the weight module category for these VOAs?

Key findings

• The vertex operator algebra L(k,0) is rational in the category $\mathcal{O}$ for all admissible rational levels k.
• All irreducible representations of L(k,0) are classified and lie within the category of weight modules.
• The classification of irreducible modules is complete and finite for each admissible k.
• The VOAs L(k,0) at admissible rational levels exhibit modular invariance in their characters.
• The structure of the category of weight modules is semisimple, with finitely many irreducible objects.
• The results establish a complete framework for the representation theory of L(k,0) at admissible levels.