[Paper Review] $C_2$-cofiniteness of orbifold models for finite groups

Masahiko Miyamoto|arXiv (Cornell University)|Dec 3, 2018

Algebraic structures and combinatorial models4 citations

TL;DR

This paper proves that if a $C_2$-cofinite, simple vertex operator algebra $V$ of CFT-type with a nonsingular invariant bilinear form admits a finite automorphism group $G$, then the corresponding orbifold construction $V^G$ is also $C_2$-cofinite. The result establishes a key finiteness property for orbifold models in vertex operator algebra theory, ensuring their algebraic and representation-theoretic well-behavedness under group actions.

ABSTRACT

We prove that if $V$ is a $C_2$-cofinite simple vertex operator algebra of CFT-type with a nonsingular invariant bilinear form and its an automorphism group $G$ is finite, then an orbifold model $V^G$ is also $C_2$-cofinite.

Motivation & Objective

• To establish the $C_2$-cofiniteness of orbifold vertex operator algebras constructed from finite group actions on $C_2$-cofinite VOAs of CFT-type.
• To address the open problem of whether orbifold constructions preserve the $C_2$-cofiniteness property under finite group automorphisms.
• To extend structural finiteness results in vertex operator algebra theory to the orbifold setting, particularly for VOAs with nonsingular invariant bilinear forms.

Proposed method

• Leveraging the $C_2$-cofiniteness and CFT-type conditions of the original vertex operator algebra $V$ to control the structure of the fixed-point subalgebra $V^G$.
• Using the existence of a nonsingular invariant bilinear form on $V$ to ensure duality and non-degeneracy properties in the orbifold construction.
• Applying representation-theoretic techniques to analyze the graded components of $V^G$ and their finite generation modulo $C_2$.
• Employing the finite group action to decompose $V$ into irreducible $G$-modules and study the fixed-point algebra structure via character theory and module theory.
• Analyzing the $C_2$-quotient space of $V^G$ to verify finite-dimensionality, a key criterion for $C_2$-cofiniteness.
• Utilizing the fact that $V$ is simple and of CFT-type to ensure the orbifold $V^G$ inherits sufficient algebraic control for finiteness.

Experimental results

Research questions

• RQ1 Does the orbifold construction $V^G$ preserve the $C_2$-cofiniteness property when $V$ is $C_2$-cofinite and $G$ is a finite automorphism group?
• RQ2 Under what conditions on $V$ and $G$ can one guarantee that $V^G$ is $C_2$-cofinite?
• RQ3 How does the presence of a nonsingular invariant bilinear form on $V$ contribute to the $C_2$-cofiniteness of $V^G$?
• RQ4 Is the $C_2$-cofiniteness of $V$ sufficient to ensure the same for its fixed-point subalgebra under a finite group action?

Key findings

• The orbifold model $V^G$ is $C_2$-cofinite whenever $V$ is a $C_2$-cofinite, simple vertex operator algebra of CFT-type with a nonsingular invariant bilinear form.
• The finiteness condition of $C_2$-cofiniteness is preserved under the orbifold construction for finite group actions.
• The existence of a nonsingular invariant bilinear form on $V$ plays a crucial role in ensuring the algebraic finiteness of $V^G$.
• The result confirms that $V^G$ satisfies the necessary structural conditions for well-behaved representation theory, including finite-dimensional $C_2$-quotients.
• The proof establishes that the $G$-fixed subalgebra $V^G$ inherits sufficient control from $V$ to ensure $C_2$-cofiniteness without additional assumptions.