[Paper Review] Glueing vertex algebras

Thomas Creutzig, Shashank Kanade+1 more|arXiv (Cornell University)|May 31, 2019

Algebraic structures and combinatorial models54 references21 citations

TL;DR

This paper constructs a conformal vertex algebra $α$ by gluing two vertex operator algebras $Υ$ and $Υ$ along a map $\alpha$ from simple objects of $\mathcal{U}$ to objects of $\mathcal{V}$, forming $\mathsf{A} = \bigoplus \mathsf{X} \otimes \alpha(\mathsf{X})$. The key result is that $\mathsf{A}$ admits a simple conformal extension of $\mathsf{U} \otimes \mathsf{V}$ if and only if $\mathcal{U}$ and $\mathcal{V}$ are braided tensor categories with a braid-reversed equivalence mapping $\mathsf{X}$ to $\alpha(\mathsf{X})^*$. The construction provides a categorical criterion for extending vertex algebras via module category data.

ABSTRACT

Let $\mathsf{U}$ and $\mathsf{V}$ be vertex operator algebras with module (sub)categories $\mathcal{U}$ and $\mathcal{V}$, respectively, satisfying suitable assumptions which hold for example if $\mathcal{U}$ and $\mathcal{V}$ are semisimple rigid braided (vertex) tensor categories with countably many inequivalent simple objects. If $ au$ is a map from the set of inequivalent simple objects of $\mathcal{U}$ to the objects $\mathcal{V}$ with $ au(\mathsf{U})=\mathsf{V}$, then we glue $\mathsf{U}$ and $\mathsf{V}$ along $\mathcal{U}\boxtimes \mathcal{V}$ via $ au$ to obtain the object \[ \mathsf{A} = \bigoplus \mathsf{X} \otimes au(\mathsf{X}) \] where the sum is over all inequivalent simple objects of $\mathcal{U}$. Assuming $\mathsf{U}$ and $\mathsf{V}$ form a commuting pair in $\mathsf{A}$ in the sense that the multiplicity of $\mathsf{V}$ is $\mathsf{U}$, our main theorem is that there is a braid-reversed equivalence between $\mathcal{U}$ and $\mathcal{V}$ mapping $\mathsf{X}$ to $ au(\mathsf{X})^*$ if and only if $\mathsf{A}$ can be given the structure of a simple conformal vertex algebra that (conformally) extends $\mathsf{U}\otimes \mathsf{V}$.

Motivation & Objective

• To construct a new vertex algebra $\mathsf{A}$ by gluing two vertex operator algebras $\mathsf{U}$ and $\mathsf{V}$ using a map $\alpha$ between their simple objects.
• To identify categorical conditions under which the glued algebra $\mathsf{A}$ becomes a simple conformal vertex algebra extending $\mathsf{U} \otimes \mathsf{V}$.
• To establish a precise correspondence between the existence of such a conformal extension and the existence of a braid-reversed equivalence between the module categories $\mathcal{U}$ and $\mathcal{V}$.

Proposed method

• Define the object $\mathsf{A} = \bigoplus \mathsf{X} \otimes \alpha(\mathsf{X})$, where the sum runs over inequivalent simple objects of $\mathcal{U}$, using a map $\alpha$ from $\mathcal{U}$-objects to $\mathcal{V}$-objects with $\alpha(\mathsf{U}) = \mathsf{V}$.
• Assume $\mathsf{U}$ and $\mathsf{V}$ form a commuting pair in $\mathsf{A}$, ensuring compatibility of their actions.
• Use the structure of semisimple rigid braided tensor categories with countably many simple objects to ensure well-behaved module categories.
• Construct a vertex algebra structure on $\mathsf{A}$ by leveraging the tensor product and braided equivalence data.
• Apply the notion of conformal extension to determine when $\mathsf{A}$ becomes a simple conformal vertex algebra.
• Establish a braid-reversed equivalence between $\mathcal{U}$ and $\mathcal{V}$ mapping $\mathsf{X}$ to $\alpha(\mathsf{X})^*$ as a necessary and sufficient condition for the conformal extension.

Experimental results

Research questions

• RQ1 Under what categorical conditions can two vertex operator algebras $\mathsf{U}$ and $\mathsf{V}$ be glued into a new conformal vertex algebra $\mathsf{A}$?
• RQ2 What role does the map $\alpha$ from simple $\mathcal{U}$-objects to $\mathcal{V}$-objects play in constructing $\mathsf{A}$?
• RQ3 How is the existence of a conformal extension of $\mathsf{U} \otimes \mathsf{V}$ in $\mathsf{A}$ related to the braided equivalence between $\mathcal{U}$ and $\mathcal{V}$?
• RQ4 What is the significance of the braid-reversed equivalence mapping $\mathsf{X}$ to $\alpha(\mathsf{X})^*$ in the construction of $\mathsf{A}$?
• RQ5 Can the gluing procedure be characterized purely in terms of module category data without reference to explicit vertex operators?

Key findings

• The glued object $\mathsf{A} = \bigoplus \mathsf{X} \otimes \alpha(\mathsf{X})$ can be endowed with a simple conformal vertex algebra structure if $\mathsf{U}$ and $\mathsf{V}$ form a commuting pair in $\mathsf{A}$.
• The existence of a conformal extension of $\mathsf{U} \otimes \mathsf{V}$ in $\mathsf{A}$ is equivalent to the existence of a braid-reversed equivalence between the module categories $\mathcal{U}$ and $\mathcal{V}$.
• This braid-reversed equivalence maps each simple object $\mathsf{X}$ in $\mathcal{U}$ to the dual $\alpha(\mathsf{X})^*$ in $\mathcal{V}$, ensuring consistency with the braided tensor structure.
• The construction works under the assumption that $\mathcal{U}$ and $\mathcal{V}$ are semisimple rigid braided tensor categories with countably many simple objects.
• The result provides a categorical criterion for conformal extensions via module category data, generalizing known gluing constructions in vertex algebra theory.
• The condition that $\alpha(\mathsf{U}) = \mathsf{V}$ ensures compatibility of the gluing at the level of the vacuum modules.