Modularity of principal tadpole Nahm sums for all ranks

Establish the modularity of the principal tadpole Nahm sum X_r(1,1,...,1; q) for every integer r ≥ 2 by determining a rational number a (depending on r) such that the q-series q^a X_r(1,1,...,1; q) is modular. Here X_r(x_1,...,x_r; q) denotes the generalized tadpole Nahm sum associated with the r×r tadpole Cartan matrix T_r, defined by X_r(x_1,...,x_r; q) = ∑_{(n_1,...,n_r)∈(Z_≥0)^r} q^{2 n^T T_r n} x_1^{n_1}...x_r^{n_r} / [(q; q)_{n_1}...(q; q)_{n_r}].

Background

The paper studies Nahm sums related to the tadpole Cartan matrix and their modularity. For r = 1 the modularity is classical, and prior works established the conjecture for r = 2 and r = 3. This paper proves the conjecture for the next cases r = 4 and r = 5 and develops rank reduction formulas to support these proofs.

Despite these advances, the general modularity statement across all ranks r ≥ 2 remains conjectural. Establishing this would complete the modularity classification for this family of Nahm sums.