Additional modular Nahm sums for (T1,Dr) with nonzero B-vectors

Identify, for each integer r≥3, r−1 nonzero vectors B∈ℚ^r (together with appropriate constants C∈ℚ) such that the Nahm sums f_{C(D_r)^{-1},B,C}(τ) are modular functions and admit expressions analogous to the B=0 case in terms of the Neveu–Schwarz and Ramond characters of the effective N=1 supersymmetric Virasoro minimal model SM_eff(8r+4,2).

Background

For the family (T1,Dr) with r≥3, the authors identify the B=0 Nahm sum as a specific linear combination of NS and R characters of SM_eff(8r+4,2), yielding generalized Rogers–Ramanujan-type product expansions.

They propose the existence of r−1 further modular Nahm sums associated with nonzero B-vectors that should have similar character decompositions, extending the uniform relation found for B=0.

References

We also conjecture that there exist r-1 non-zero vectors B such that the associated Nahm sums are modular and have similar expressions in terms of NS and R characters of \mathrm{SM}_{\rm eff}(8r+4,2).

— Dynkin diagrams, generalized Nahm sums and 2d CFTs  (2604.00847 - Sun et al., 1 Apr 2026) in Section 1 (Introduction), paragraph following equation (1.10) (displayed as \eqref{eq:T1Dr})