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 stability 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.

中文翻译：

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热带盘上的唐纳森-托马斯不变量

我们证明，与温特尔电位颤动相关的量子DT不变量可以表示为热带盘的某些精确计数。这是基于Bridgeland在稳定性条件方面对簇散射图的描述的量子版本，以及在热带磁盘数方面对散射图的描述的新版本。计算热带圆盘的权重用某些颤抖标志品种的动力积分表示。通过显式的反例，我们还表明，霍尔代数的虚线不会产生一致的霍尔代数theta函数，即，它们违反了卡尔-彭佩拉-西伯特从经典环境引理的扩展。