Quantum dilogarithm identities for n-cycle quivers

TL;DR

This paper proves quantum dilogarithm identities for n-cycle quivers, linking them to maximal green sequences and Donaldson--Thomas invariants, and proposes a conjecture on their maximal lengths.

Contribution

It introduces new quantum dilogarithm identities for n-cycle quivers and connects them to maximal green sequences and Donaldson--Thomas invariants.

Findings

Identified quantum dilogarithm identities for n-cycle quivers

Connected identities to maximal green sequences of quiver mutations

Conjectured an upper bound on lengths of maximal green sequences

Abstract

We prove quantum dilogarithm identities for $n$-cycle quivers. By the combinatorial approach of Keller, each side of our identity determines a maximal green sequence of quiver mutations. Thus we interpret our identities as factorizations of the refined Donaldson--Thomas invariant for the quiver with potential. Finally, we conjecture an upper bound on the possible lengths of maximal green sequences.