TITLE: Annihilation operators in a vertex algebra
QUESTION [9 upvotes]: Let $V=\bigoplus_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.
Recall that a vector $v\in V$ is called quasi-primary if $L_1v=0$.
For $v\in V(d)$, we write $Y(v,z)=\sum_{n\in\mathbb Z} z^{-n-d}v_{(n)}$.
With that convention, $v_{(n)}$ is an operator $V(k)\to V(k-n)$.

Let $d>n$.
  Is it true that for any quasi-primary $v\in V(d)$ and any vector $w\in V(n)$, we have $v_{(n)}w=0$?

If the above relation does not always hold, are there reasonable extra assumptions that one can impose on $V$ that imply it?
Do the above relations hold when $v$ is required to be primary instead of quasi-primary? (Add the assumption that $V$ is a VOA so that the notion of a primary vector make sense)

REPLY [4 votes]: The answer seems to be yes for quasi-primary $v$ if $V$ has a suitable invariant bilinear form. Then one can identify $v_{(n)} w$ with its pairing with the vacuum, and obtains it as the appropriate coefficient of
$$(\mathbf{1}, Y(v, x)w) = (-x^{-2})^d (Y(v, x^{-1})\mathbf{1}, w) = (-x^{-2})^d (e^{x^{-1} L(-1)} v, w) = 0$$
since the weight of $v$ is greater than that of $w$.
Of course, this argument does not work if $v$ is not quasi-primary since then $v$ must be replaced with $e^{x L(1)} v$ after the first equality in the calculation.
Note that it isn’t necessary to assume the bilinear form is nondegenerate, just nondegenerate on the one-dimensional vacuum space.