Semi-infinite construction of one-dimensional lattice vertex superalgebras

We construct the Feigin-Stoyanovsky (combinatorial) basis in case of one-dimensional lattice vertex superalgebras $V_{\sqrt{N}\mathbb{Z}}$. Our proof is based on invariance of semi-infinite monomials linear span under action of corresponding Heisenberg algebra. Semi-infinite monomials are parametrized by natural generalization of Maya diagrams $\text{unicode}x2013$ Fibonacci configurations on $\mathbb{Z}$, which allows us to construct a desired basis with character considerations. We also discuss some related questions such as functional realization of basic subspace’s dual and representational proof of Feigin-Stoyanovsky construction in case of $V_{\sqrt{2}\mathbb{Z}}$.