[Paper Review] Classification of some three-dimensional vertex operator algebras

Cameron Franc, Geoffrey Mason|arXiv (Cornell University)|May 17, 2019

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper classifies strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Using generalized hypergeometric series to analyze a family of vector-valued modular forms, the authors identify two infinite families of affine VOAs, known examples, and an exceptional U-series of eleven possible character vectors—only two of which are realized by known VOAs, suggesting no further realizations exist.

ABSTRACT

We discuss the classification of strongly regular vertex operator algebras (VOAs) with exactly three simple modules whose character vector satisfies a monic modular linear differential equation with irreducible monodromy. Our Main Theorem 1 provides a classification of all such VOAs in the form of two infinite families of affine VOAs, several other known examples, in addition to eleven possible exceptional character vectors and associated data that we call the U-series. Only two VOAs are known to realize any of the members of the U-series and we provide evidence that there are no more. The idea in the proof of our Main Theorem is to exploit properties of an algebraic family of vector-valued modular forms solving a family of modular linear differential equations in terms of generalized hypergeometric series.

Motivation & Objective

• To classify strongly regular vertex operator algebras (VOAs) with exactly three simple modules.
• To identify VOAs whose character vectors satisfy a monic modular linear differential equation with irreducible monodromy.
• To determine whether the exceptional U-series of eleven possible character vectors can be realized by actual VOAs.
• To establish the completeness of the classification by analyzing vector-valued modular forms via generalized hypergeometric series.

Proposed method

• Analyzing a family of vector-valued modular forms that solve modular linear differential equations.
• Employing generalized hypergeometric series to describe the structure of these modular forms.
• Using the monodromy properties of the differential equations to constrain possible character vectors.
• Applying algebraic techniques to study the monodromy group and its irreducibility.
• Deriving constraints on the character vectors from the modular invariance and modularity of VOAs.
• Comparing the resulting character data with known VOAs to identify realizations.

Experimental results

Research questions

• RQ1 Which strongly regular VOAs with exactly three simple modules satisfy a monic modular linear differential equation with irreducible monodromy?
• RQ2 Can the eleven exceptional character vectors in the U-series be realized by actual VOAs?
• RQ3 What is the role of generalized hypergeometric series in classifying modular vector-valued forms associated with VOAs?
• RQ4 How do the monodromy properties of the differential equations constrain the structure of the VOAs?
• RQ5 Are there any VOAs beyond the two infinite families and known examples that satisfy the classification criteria?

Key findings

• The classification includes two infinite families of affine VOAs and several known examples.
• Eleven exceptional character vectors are identified as part of the U-series, all with irreducible monodromy.
• Only two VOAs are currently known to realize members of the U-series, suggesting no further realizations exist.
• The structure of the modular linear differential equations is fully determined by the generalized hypergeometric series representation of the vector-valued modular forms.
• The monodromy group of the differential equations is irreducible, which is a key constraint in the classification.
• The analysis confirms the completeness of the classification under the given conditions, with no additional VOAs expected beyond the identified families and the U-series.