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As an example, take the Virasoro algebra, i.e. $V$ is spanned by elements of the form $L_{-2}^{k_1} \cdots L_{-n}^{k_{n-1}} \Omega$ where $\Omega$ is the vacuum and $n \geq 2$. As I understand, we define

$$C_2(V) = \{\psi^i_{-h_i-1}v | v \in V\}$$

where $i$ labels all the operators strongly generating the VOA, and $h_i$ are the conformal dimensions. So, for the Virasoro algebra one has,

$$C_2(V) = \{L_{-3}v | v \in V\}.$$

Now it is claimed that $V/C_2(V) \cong \mathbb C[x]$, where the isomorphism maps $L_{-2}\Omega \mapsto x$. However, I do not see how this is the case. We are modding out all elements of the form $L_{-3}v$, i.e. all those that contain a factor of $L_{-3}$. Then say, $L_{-10}\Omega$ cannot be part of an equivalence class with a representative given by a power of $L_{-2}\Omega$, so I do not see how $V/C_2(V)$ is just generated by $L_{-2}\Omega$.

For reference, see page 14 of Arakawa's paper.

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You have underdefined the $C_2(V)$ subspace.

It is as you wrote but with $\psi^i$ being any element of $V$. You can intuitively think of this as being the subspace of iterated normally ordered products of vertex operators for which at least one is a derivative.

Thus in your Virasoro example, $L_{-10}\Omega\in C_2(V)$ by setting $\psi^i=\partial^7T$ where $T$ is the stress tensor.

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  • $\begingroup$ Thank you for your reply, and I would just add thank you for your chiral algebra construction for 4D N=2 SCFTs which I based my MSc thesis on last year :) $\endgroup$
    – JamalS
    Aug 12, 2020 at 10:07

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