Bloch torsion and modular combinations of Nahm sums

Abstract

We construct a positive-definite symmetric integral $12\times12$ matrix $A$ such that, for every solution $Q$ of Nahm's equations $1-Q=Q^A$ with $Q_i\ne0,1$ for all $i$, the associated element $\sum_i[Q_i]$ is torsion in the Bloch group of $\mathbb{Q}(Q_1,\ldots,Q_{12})$. However, the Nahm sum $F_{A,B,C}(τ)$ is nonmodular for every $B\in\mathbb{Q}^{12}$ and $C\in\mathbb{Q}$. Thus the torsion condition for all solutions in Nahm's conjecture is not sufficient for the existence of a modular Nahm sum.