Quantum Dilogarithms and Partition q-Series

In our previous work (Kato and Terashima, Commun Math Phys. arXiv:1403.6569, 2014), we introduced the partition q-series for mutation loop γ—a loop in exchange quiver. In this paper, we show that for a certain class of mutation sequences, called reddening sequences, the graded version of partition q-series essentially coincides with the ordered product of quantum dilogarithm associated with each mutation; the partition q-series provides a state-sum description of combinatorial Donaldson–Thomas invariants introduced by Keller.