On some vertex algebras related to $V_{-1}(\mathfrak{sl}(n))$ and their characters

Abstract

We consider several vertex operator (super)algebras closely related to $V_{-1}(\mathfrak{sl}(n))$, $n\ge 3$ : (a) the parafermionic subalgebra $K(\mathfrak{sl}(n),-1)$ for which we completely describe its inner structure, (b) the vacuum algebra $\Omega (V_{-1}(\mathfrak{sl}(n)))$, and (c) an infinite extension $\mathcal{U}$ of $V_{-1}(\mathfrak{sl}(n))$ constructed by combining certain irreducible ordinary modules with integral weights. It turns out that $\mathcal{U}$ is isomorphic to the coset vertex algebra $\mathfrak{psl}(n|n{)}_1/\mathfrak{sl}(n{)}_1$, $n\ge 3$. We show that $V_{-1}(\mathfrak{sl}(n))$ admits precisely $n$ ordinary irreducible modules, up to isomorphism. This leads to the conjecture that $\mathcal{U}$ is {\em quasi-lisse}. We present evidence in support of this conjecture: we prove that the (super)character of $\mathcal{U}$ is quasi-modular of weight one by virtue of being the constant term of a meromorphic Jacobi form of index zero. Explicit formulas and MLDE for characters and supercharacters are given for $\mathfrak{g}=\mathfrak{sl}(3)$ and outlined for general $n$. We present a conjectural family of 2nd order MLDEs for characters of vertex algebras $\mathfrak{psl}(n|n{)}_1$, $n\ge 2$. We finish with a theorem pertaining to characters of $\mathfrak{psl}(n|n{)}_1$ and $\mathcal{U}$-modules.

v2: 28 pages