We study the discrete flows generated by the symmetry group of the BPS quivers for Calabi–Yau geometries describing five-dimensional superconformal quantum field theories on a circle. These flows naturally describe the BPS particle spectrum of such theories and at the same time generate bilinear equations of q-difference type which, in the rank one case, are q-Painlevé equations. The solutions of these equations are shown to be given by grand canonical topological string partition functions which we identify with \(\tau \)-functions of the cluster algebra associated to the quiver. We exemplify our construction in the case corresponding to five-dimensional SU(2) pure super Yang–Mills and \(N_f=2\) on a circle.

我们研究了由卡拉奇-丘几何的BPS颤动对称群产生的离散流，这些几何描述了一个圆上的五维超共形量子场理论。这些流动自然地描述了这些理论的BPS粒子光谱，同时生成了q差类型的双线性方程，在一种情况下，它们是q-Painlevé方程。这些方程的解表明由大正则拓扑字符串划分函数给出，我们用与颤动相关的簇代数的\（\ tau \） -函数来标识。我们以圆上对应于五维SU（2）纯超杨格和\（N_f = 2 \）的情况为例来说明我们的构造。