论文标题
布朗循环汤的新食谱
New Recipes for Brownian Loop Soups
论文作者
论文摘要
我们在布朗尼循环集团中定义了一类新的保形主要运算符，以二维为“``棕色环汤'''，并在分析和封闭形式上计算其相关功能。循环汤是一种与中央电荷的共同不变的统计合奏，$ c =2λ$，其中$λ> 0 $是汤的强度。先前的工作将分层运算符的指数$ e^{iβn（z）} $作为主要操作员。每个Brownian Loop都被分配了$ \ pm 1 $，而$ n（z）$被定义为包围点$ z $的所有循环中这些数字的总和。然后，这些指数运算符具有共形尺寸$ {\fracλ{10}}（1- \cosβ）$。在这里，我们通过为每个循环分配一个更一般的随机值来概括此过程。运算符$ e^{iβn（z）} $保持主要尺寸$ \fracλ{10}（1 - ϕ（β））$，其中$ ϕ（β）$是用于将随机值分配给每个循环的概率分布的特征函数。使用最近的结果，我们以封闭形式计算上半平面中的确切两点函数，并在此非常通用的运算符类的整个平面中的四点函数。这些相关函数在分析上取决于参数$λ，β_i，z_i $以及特征函数$ ϕ（β）$。他们满足保形的病房身份，并越过对称。与以前的工作一样，四点函数的共形块扩展揭示了存在额外且尚未达到的保形的主要算子的存在。
We define a large new class of conformal primary operators in the ensemble of Brownian loops in two dimensions known as the ``Brownian loop soup,'' and compute their correlation functions analytically and in closed form. The loop soup is a conformally invariant statistical ensemble with central charge $c = 2 λ$, where $λ> 0$ is the intensity of the soup. Previous work identified exponentials of the layering operator $e^{i βN(z)}$ as primary operators. Each Brownian loop was assigned $\pm 1$ randomly, and $N(z)$ was defined to be the sum of these numbers over all loops that encircle the point $z$. These exponential operators then have conformal dimension ${\fracλ{10}}(1 - \cos β)$. Here we generalize this procedure by assigning a more general random value to each loop. The operator $e^{i βN(z)}$ remains primary with conformal dimension $\frac λ{10}(1 - ϕ(β))$, where $ϕ(β)$ is the characteristic function of the probability distribution used to assign random values to each loop. Using recent results we compute in closed form the exact two-point functions in the upper half-plane and four-point functions in the full plane of this very general class of operators. These correlation functions depend analytically on the parameters $λ, β_i, z_i$, and on the characteristic function $ϕ(β)$. They satisfy the conformal Ward identities and are crossing symmetric. As in previous work, the conformal block expansion of the four-point function reveals the existence of additional and as-yet uncharacterized conformal primary operators.
