多矩阵模型的圈方程
Loop Equations for Multi-Matrix Models
AI总结：
本文提出一种按边界插入层级组织Schwinger-Dyson约束的圈方程方法，用于求解多矩阵模型的平面圆盘函数，并成功应用于三态Potts模型及多个未解模型，展示了其超越标准矩阵模型的实用性。
AI中文摘要：
我们重新审视了用于计算具有多项式势的多矩阵模型中平面圆盘振幅的组合圈方程方法。我们开发了一种程序，该程序根据边界插入的层级来组织Schwinger-Dyson约束，并且当所得系统闭合时，为平面圆盘函数产生一个代数方程。利用此方法，我们恢复了三态Potts模型中固定、混合和自由边界条件相关的圆盘配分函数，这些函数先前通过鞍点方法获得。作为新结果，我们推导了具有不等二次和三次耦合的三态Potts模型固定边界 resolvent 的代数方程。我们进一步展示了可以计算几个未解决的二矩阵和三矩阵模型的圆盘配分函数。这些例子表明，圈方程为识别和解决超出标准单矩阵和双矩阵情况的多矩阵模型扇区提供了一条实用途径。
英文摘要：
We revisit the combinatorial loop-equation method for computing planar disc amplitudes in multi-matrix models with polynomial potentials. We develop a procedure which organises the Schwinger-Dyson constraints by the level of their boundary insertions and, when the resulting system closes, produces an algebraic equation for the planar disc function. Using this method we recover the disc partition functions associated with fixed, mixed, and free boundary conditions in the three-state Potts model, previously obtained by saddle-point methods. As a novelty we derive an algebraic equation for the fixed-boundary resolvent of the three-state Potts model with unequal quadratic and cubic couplings. We further show one can compute the disc partition function for several unsolved two- and three-matrix models. These examples demonstrate that loop equations provide a practical route to identifying and solving sectors of multi-matrix models beyond the standard one- and two-matrix cases.
