Donaldson-Thomas invariants from tropical disks

Abstract

We prove that the quantum DT invariants associated to quivers with genteel potential can be expressed in terms of certain refined counts of tropical disks. This is based on a quantum version of Bridgeland's description of cluster scattering diagrams in terms of stabilitiy conditions, plus a new version of the description of scattering diagrams in terms of tropical disk counts. The weights with which the tropical disks are counted are expressed in terms of motivic integrals of certain quiver flag varieties. We also show via explicit counterexample that Hall algebra broken lines do not result in consistent Hall algebra theta functions, i.e., they violate the extension of a lemma of Carl-Pumperla-Siebert from the classical setting.