Remarks on Nahm sums for symmetrizable matrices

Abstract

Nahm sums are specific $q$-hypergeometric series associated with symmetric positive definite matrices. In this paper we study Nahm sums associated with symmetrizable matrices. We show that one direction of Nahm's conjecture, which was proven by Calegari, Garoufalidis, and Zagier for the symmetric case, also holds for the symmetrizable case. This asserts that the modularity of a Nahm sum implies that a certain element in a Bloch group associated with the Nahm sum is a torsion element. During the proof, we investigate the radial asymptotics of Nahm sums. Finally, we provide lists of candidates of modular Nahm sums for symmetrizable matrices based on numerical experiments.