Semi-infinite construction of one-dimensional lattice vertex superalgebras

Abstract

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 $\unicode{x2013}$ 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 $V_{\sqrt{2}\,\mathbb{Z}}$.

Figures (2)

• Figure 1: Weight diagram for $V_{\sqrt{2N}\,\mathbb{Z}}$ Notation from \ref{['DenoteModesEven']} Conformal vector is $\omega_0 = \frac{1}{2}\,a_{-1}^2$$\ch V_{\sqrt{2N}\,\mathbb{Z}} = \sum\limits_{m\in\mathbb{Z}}\,\frac{z^m\,q^{Nm^2}}{(q)_{\infty}}$
• Figure 2: Weight diagram for $V_{\sqrt{2N + 1}\,\mathbb{Z}}$ Notation from \ref{['DenoteModesOdd']} Conformal vector is $\omega_{\frac{\sqrt{2N + 1}}{2}} = \frac{1}{2}\,a_{-1}^2 + \frac{\sqrt{2N + 1}}{2}\,a_{-2}$$\ch V_{\sqrt{2N + 1}\,\mathbb{Z}} = \sum\limits_{m\in\mathbb{Z}}\,\frac{z^m\,q^{(2N + 1)\frac{m(m - 1)}{2}}}{(q)_{\infty}}$

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.1
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Lemma 3.1
• proof
• Remark 3.1
• Theorem 3.1
• proof