共形场论中一种精确关联函数分解的通解
General solution of an exact correlation function factorization in conformal field theory
摘要：
我们讨论了一个关联函数分解，它将一个三点函数与三个两点函数的平方根联系起来。这种因式分解已知在二维渗流点和其他一些情况下对某些标度算子成立。关联函数在上半平面（或任何共形等价区域）中用实轴上的两个任意点和实轴上或内部的第三个任意点处的算子进行评估。这种类型的结果是感兴趣的，因为它是准确的和普遍的，涉及高阶关联函数低阶的，并有一个简单的解释在几个统计模型的集群或循环概率。这促使我们使用共形场论的技巧来确定其有效性的一般条件。在这里，我们发现了一个关联函数，以这种方式分解任何中心荷c，推广以前的结果。特别是，因式分解适用于FK（Fortuin-Kasteleyn）或Q状态Potts模型中的自旋团簇;它也适用于O（n）循环模型的稠密或稀相。此外，只有另一个非平凡的最高权重算子集（在不可约Verma模中）以这种方式分解。在这种情况下，算子具有负维数（c < 1），似乎没有物理实现。
英文摘要：
We discuss a correlation function factorization, which relates a three-point function to the square root of three two-point functions. This factorization is known to hold for certain scaling operators at the two-dimensional percolation point and in a few other cases. The correlation functions are evaluated in the upper half-plane (or any conformally equivalent region) with operators at two arbitrary points on the real axis, and a third arbitrary point on either the real axis or in the interior. This type of result is of interest because it is both exact and universal, relates higher-order correlation functions to lower-order ones, and has a simple interpretation in terms of cluster or loop probabilities in several statistical models. This motivated us to use the techniques of conformal field theory to determine the general conditions for its validity. Here, we discover a correlation function which factorizes in this way for any central charge c, generalizing previous results. In particular, the factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the Q-state Potts models; it also applies to either the dense or dilute phases of the O(n) loop models. Further, only one other non-trivial set of highest-weight operators (in an irreducible Verma module) factorizes in this way. In this case the operators have negative dimension (for c < 1) and do not seem to have a physical realization.
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