A conjecture of Warnaar-Zudilin from deformations of lie superalgebras

We prove a collection of q-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k\left({\mathrm{osp}}_{1|2n}\right)$ into the principal subsuperspace of $L_k\left(s{l}_{1|2n+1}\right)$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers N, k the character of the principal subspace of type $A_N$ at level k can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.