Moduli spaces of conformal structures on Heisenberg vertex algebras

This paper is a continuation to understand Heisenberg vertex algebras in terms of moduli spaces of their conformal structures. We study the moduli space of the conformal structures on a Heisenberg vertex algebra that have the standard fixed conformal gradation. As we know in Proposition 3.1 in Sect.3, conformal vectors of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$ that have the standard fixed conformal gradation is parameterized by a complex vector $h$ of its weight-one subspace. First, we classify all such conformal structures of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$ by describing the automorphism group of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$ and then we describe moduli spaces of their conformal structures that have the standard fixed conformal gradation. Moreover, we study the moduli spaces of semi-conformal vertex operator subalgebras of each of such conformal structures of the Heisenberg vertex algebra $V_{\hat{\eta}}(1,0)$. In such cases, we describe their semi-conformal vectors as pairs consisting of regular subspaces and the projections of $h$ in these regular subspaces. Then by automorphism groups $G$ of Heisenberg vertex operator algebras, we get all $G$-orbits of varieties consisting of semi-conformal vectors of these vertex operator algebras. Finally, using properties of these varieties, we give two characterizations of Heisenberg vertex operator algebras.