# Questions tagged [vertex-algebras]

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### Are umbral moonshine and umbral calculus connected?

In a 2013 article, Cheng, Duncan and Harvey introduce the concept of umbral moonshine as a generalization of monstrous moonshine. The terminology they use, starting with the title, is common in umbral ...
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### Alternative definition of physical states

Suppose that we have a vertex operator algebra $V$ with a conformal element $\omega$ and the associated conformal field $$Y(\omega,z) = \sum_{k\in \mathbb{Z}} L_kz^{-k-2}\,,$$ where $L_k$ satisfy ...
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### Vertex operator algebras and modular fusion categories

Let $\mathcal{V}$ be a vertex operator algebra (VOA), and let $\mathcal{C} = \text{Rep}(\mathcal{V})$ be the tensor category of $\mathcal{V}$-modules. It is a conjecture by Vaughan Jones whether every ...
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### q-series identity related to Jackson-Slater, proof required

The question: I have been trying to prove the following $q$-series identity for quite some time now:  \sum_{k \geq 0} \frac{q^{2k^2}}{(q)_{2k}} = \sum_{m,k \geq 0} \frac{q^{m^2 + 3k m + 4k^2}}{(q)...
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### Composition of operators in $w_{1+\infty}$ and $W_{1+\infty}$

The algebra $W_{1+\infty}$ can be defined as a central extension of the lie algebra $w_{1+\infty}$ (defined as being spanned by $\left(-\partial_z \right)^m z^{-k}$ ). See for example: Alexandrov, ...
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### Examples of simple vertex operator algebras (VOAs)

A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a ...
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### Vertex operator algebras and isomorphism of graded vector spaces

I have two vertex operator algebras and I would like to show that as graded vector spaces, they are isomorphic, rather than as algebras. The issue is I have not found anything in the literature that ...
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