Simplicities of VOAs associated to Jordan algebras of type B and character formulas for simple quotients

In this paper we study the VOA VJ,r constructed by Ashihara and Miyamoto in [2], which satisfies (VJ,r)0=C1,(VJ,r)1={0}, and the Griess algebra (VJ,r)2 is isomorphic to the type B simple Jordan algebra J. We construct simple quotients V¯J,r for r∈Z≠0 using dual-pair type constructions, and we reprove that VJ,r is simple if r∉Z. We also compute the character formula of the simple quotients V¯J,r for r=−2n,n∈Z>0.