Modular Nahm Sums for the Inverse Cartan Matrix of Type $D_r$

For $r\geq 3$ we denote by $\mathcal{C}(D_r)$ the Cartan matrix of type $D_r$.

Recently, Sun and Wang conjectured a Rogers--Ramanujan type identity for the Nahm sum associated with $\mathcal{C}(D_r)^{-1}$ and the zero vector.

They further conjecture that there exist $r-1$ companion modular Nahm sums associated with nonzero vectors.

We partially prove this conjecture by constructing $\lfloor (r+4)/2\rfloor$ modular Nahm sums for $\mathcal{C}(D_r)^{-1}$.

To prove their modularity, we utilize the method of Bailey pairs to establish various Rogers--Ramanujan type identities.

In particular, we confirm their conjectural identity.