Generators of relations for annihilating fields

For an untwisted affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}$, and a given positive integer level $k$, vertex operators $x(z)=\sum x(n)z^{-n-1}$, $x\in \mathfrak{g}$, generate a vertex operator algebra $V$. For the maximal root $\theta$ and a root vector $x_{\theta }$ of the corresponding finite-dimensional $\mathfrak{g}$, the field $x_{\theta }(z{)}^{k+1}$ generates all annihilating fields of level $k$ standard $\tilde{\mathfrak{g}}$-modules. In this paper we study the kernel of the normal order product map $r(z)\otimes Y(v,z)\mapsto :r(z) Y(v,z):$for$v\in V$and$r(z)$ in the space of annihilating fields generated by the action of $\frac{d}{dz}$ and $\mathfrak{g}$ on $x_{\theta }(z{)}^{k+1}$. We call the elements of this kernel the relations for annihilating fields, and the main result is that this kernel is generated, in certain sense, by the relation $x_\theta(z)\tfrac{d}{dz}(x_\theta(z)^{k+1})= (k+1)x_\theta(z)^{k+1}\tfrac{d}{dz}x_\theta(z)$. This study is motivated by Lepowsky-Wilson’s approach to combinatorial Rogers-Ramanujan type identities, and many ideas used here stem from a joint work with Arne Meurman.