A ${\varphi }_{1,3}$-Filtration of the Virasoro Minimal Series $M(p,p^{\prime })$ with $1<p^{\prime }/p<2$

Abstract

The filtration of the Virasoro minimal series representations $M_{r,s}^{(p,p^{\prime })}$ induced by the $(1,3)$-primary field ${\varphi }_{1,3}(z)$ is studied. For $1<p^{\prime }/p<2$, a conjectural basis of $M_{r,s}^{(p,p^{\prime })}$ compatible with the filtration is given by using monomial vectors in terms of the Fourier coeffcients of ${\varphi }_{1,3}(z)$. In support of this conjecture, we give two results. First, we establish the equality of the character of the conjectural basis vectors with the character of the whole representation space. Second, for the unitary series ($p^{\prime }=p+1$), we establish for each $m$ the equality between the character of the degree $m$ monomial basis and the character of the degree $m$ component in the associated graded module $\mathrm{gr}(M_{r,s}^{(p,p+1)})$ with respect to the filtration defined by ${\varphi }_{1,3}(z)$.