Dynkin diagrams, generalized Nahm sums and 2d CFTs

A folklore conjecture states that the Nahm sum associated with a pair of Dynkin diagrams of type $ADET$ is a modular function. In this paper, we extend this conjecture to Dynkin diagrams of type $ABCDEFGT$ in the context of generalized Nahm sums. The modular Nahm sums are closely related to the characters of 2d rational conformal field theories. In this work, we identify many specific generalized Nahm sums with characters of some well-studied 2d CFTs. For example, we find that the generalized Nahm sums associated with $(T_1,C_r)$ and $(T_1,D_r)$ correspond to the supersymmetric Virasoro minimal models $\mathrm{S}\mathrm{M}(4r+6,4)$ and $\mathrm{S}\mathrm{M}(8r+4,2)$, respectively.