Simplicities of VOAs Associated to Jordan Algebras of Type $B$ and Character Formulas for Simple Quotients

In ths paper we study the VOA $V_{\mathcal{J},r}$ constructed by Ashihara and Miyamoto, We construct simple quotients of $V_{\mathcal{J},r},r\in {\mathbb{Z}}_{\mathrm{\ne }0}$ explicitly using dual-pair type constructions. We also compute the character formula of the simple quotients when $r=-2n,n\ge 1$.