We study the Yang–Mills measure on the sphere with unitary structure group. In the limit where the structure group has high dimension, we show that the traces of loop holonomies converge in probability to a deterministic limit, which is known as the master field on the sphere. The values of the master field on simple loops are expressed in terms of the solution of a variational problem. We show that, given its values on simple loops, the master field is characterized on all loops of finite length by a system of differential equations, known as the Makeenko–Migdal equations. We obtain a number of further properties of the master field. On specializing to families of simple loops, our results identify the high-dimensional limit, in non-commutative distribution, of the Brownian bridge in the group of unitary matrices starting and ending at the identity.

中文翻译：

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Yang-Mills 测量和球面上的主场

我们研究了具有单一结构群的球面上的杨-米尔斯测度。在结构群具有高维数的极限中，我们表明循环完整的轨迹在概率上收敛到一个确定性极限，这被称为球体上的主场。简单循环上主域的值用变分问题的解来表示。我们表明，给定它在简单环上的值，主场在所有有限长度环上都通过一个微分方程系统来表征，称为 Makeenko-Migdal 方程。我们获得了主字段的许多其他属性。在专门研究简单循环族时，我们的结果确定了非交换分布中的高维极限，