Generators of Relations for Annihilating Fields

For an untwisted affine Kac-Moody Lie algebra g̃, and a given positive integer level k, vertex operators x(z) = ∑ x(n)z−n−1, x ∈ g, generate a vertex operator algebra V . For the maximal root θ and a root vector xθ of the corresponding finite-dimensional g, the field xθ(z) k+1 generates all annihilating fields of level k standard g̃-modules. In this paper we study the kernel of the normal order product map r(z) ⊗ Y (v, z) 7→: r(z)Y (v, z) : for v ∈ V and r(z) in the space of annihilating fields generated by the action of d dz and g on xθ(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 xθ(z) d dz (xθ(z) k+1) = (k+ 1)xθ(z) k+1 d dz xθ(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.