We review the problem of BPS state counting described by the generalized quiver matrix model of ADHM type. In four dimensions the generating function of the counting gives the Nekrasov partition function and we obtain generalization in higher dimensions. By the localization theorem, the partition function is given by the sum of contributions from the fixed points of the torus action, which are labeled by partitions, plane partitions and solid partitions. The measure or the Boltzmann weight of the path integral can take the form of the plethystic exponential. Remarkably after integration the partition function or the vacuum expectation value is again expressed in plethystic form. We regard it as a characteristic property of the BPS state counting problem, which is closely related to the integrability.

中文翻译：

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ADHM类型的Quiver矩阵模型和不同维度的BPS状态计数

我们回顾了由 ADHM 类型的广义颤动矩阵模型描述的 BPS 状态计数问题。在四个维度上，计数的生成函数给出了 Nekrasov 分区函数，我们在更高维度上获得了泛化。根据定位定理，分区函数由环面动作的不动点的贡献之和给出，这些不动点用分区、平面分区和实体分区标记。路径积分的度量或玻尔兹曼权重可以采用体积指数的形式。值得注意的是，在积分后，分配函数或真空期望值再次以 plethystic 形式表示。我们将其视为 BPS 状态计数问题的一个特征属性，它与可积性密切相关。