Quantum dilogarithms and partition q-series

TL;DR

This paper demonstrates that for specific mutation sequences in quiver theory, the partition q-series aligns with quantum dilogarithm products and offers a combinatorial perspective on Donaldson-Thomas invariants.

Contribution

It establishes a connection between partition q-series and quantum dilogarithms for reverse-ending mutation loops, advancing the understanding of their algebraic and combinatorial structures.

Findings

Partition q-series coincide with quantum dilogarithm products for certain mutation sequences.

Provides a state-sum description of Donaldson-Thomas invariants.

Extends previous work on mutation loops and their algebraic properties.

Abstract

In our previous work [arXiv:1403.6569], we introduced the partition q-series for mutation loop --- a loop in exchange quiver. In this paper, we show that for certain class of mutation sequences, called reverse-ending mutation loops, a 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 B. Keller.