PBW bases of irreducible Ising modules

To every $h+\mathbb{N}$-graded module $M$ over an $\mathbb{N}$-graded conformal vertex algebra $V$, we associate an increasing filtration $(G^{p}M{)}_{p\in \mathbb{Z}}$ which is compatible with the filtrations introduced by Haisheng Li. The associated graded vector space ${\mathrm{g}\mathrm{r}}^G(M)$ is naturally a module over the vertex Poisson algebra ${\mathrm{g}\mathrm{r}}^G(V)$. We study ${\mathrm{g}\mathrm{r}}^G(M)$ for the three irreducible modules of the Ising model ${\mathrm{V}\mathrm{i}\mathrm{r}}_{3,4}$, namely ${\mathrm{V}\mathrm{i}\mathrm{r}}_{3,4}=L(1\mathrm{/}2,0)$, $L(1\mathrm{/}2,1\mathrm{/}2)$ and $L(1\mathrm{/}2,1\mathrm{/}16)$. We obtain an explicit monomial basis of each of these modules and a formula for their refined characters which are related to Nahm sums for the matrix $\left(\begin{array}{cc}8& 3\\ 3& 2\end{array}\right)$.