AI 中文总结
介绍圆上非阿贝尔乘法混沌测度推广，其重整化过程以特定表示和测度为输入输出随机测度，在对应\(L^2\)相的\(\kappa\)值范围有效，证明依赖相关公式和方程。
AI 中文摘要
在本笔记中，我们引入了一种乘法混沌测度的推广，它既非高斯的也非阿贝尔的。重整化过程以一个紧致连通李群的不可约酉表示以及某个层级\(\kappa\)处的卡茨 - 穆迪酉化测度作为输入，并在圆上输出一个取值于表示空间的正定埃尔米特自同态空间的随机测度。到目前为止，我们的构造对于对应于\(L^2\)相的\(\kappa\)值范围是有效的。证明遵循乘法混沌理论中的常规路线，依赖于单点函数的精确公式和碰撞点处两点函数的界。这些表达式是利用该理论丰富的代数结构推导出来的：即，我们建立了克尼兹尼克 - 扎莫洛季科夫方程的一维版本。
英文摘要
In this note, we introduce a generalisation of multiplicative chaos measures which is both non-Gaussian and non-Abelian. The renormalisation procedure takes as inputs an irreducible unitary representation of a compact connected Lie group, together with a Kac-Moody unitarising measure at some level $κ$, and outputs a random measure on the circle with values in the space of positive definite Hermitian endomorphisms of the representation space. So far, our construction is valid for the range of $κ$-values corresponding to the $L^2$-phase. The proof follows the usual route in the theory of multiplicative chaos, relying on an exact formula for the one-point function and a bound on the two-point function at colliding points. These expressions are derived using the rich algebraic structure of the theory: namely, we establish a one-dimensional version of the Knizhnik-Zamolodchikov equations.
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