零态偏微分方程组的解空间2
A solution space for a system of null-state partial differential equations 2
摘要：
这篇文章是第二个四个完整的特点的解决方案空间$\mathcal{S}_N$的一个齐次系统的$2N+3$线性偏微分方程（PDE）在2N个变量，出现在共形场论（CFT）和多重Schramm-Lowner演化（SLE）。该系统由$2N$零态方程和三个共形Ward恒等式组成，它们决定了$2N$单支边界算子的CFT关联函数.在第一篇文章（第一部分）中，我们用分析和线性代数的方法证明了$\dim\mathcal{S}_N\leq C_N$，其中$C_N$是$N$的第n个Catalan数.这篇文章的分析是完整的，除了它引用的一个引理的证明。这篇文章提供了证据。引理指出，如果$(x_2,x_3),$ $(x_3,x_4), ... ,(x_{2N-1},x_{2N})$之间的每个间隔都是$F\in\mathcal{S}_N$的两个分支间隔，则$F$为零。证明这个引理的矛盾，我们表明，这样一个非零函数的存在意味着存在一个非零的CFT两点函数涉及不同的共形权重的主要运营商，一个不可能的。这个证明（尽管我们偶尔提到CFT，但它是严格的）涉及两种不同类型的估计，一种是当一个区间的长度为零时给出F的渐近行为的估计，另一种是当两个区间的长度同时为零时给出F的渐近行为的估计。我们得到这些估计使用格林函数重写某些零状态偏微分方程的积分方程，结合其他零状态偏微分方程，以获得Schauder内部估计，然后反复集成这些估计的积分方程，直到我们获得最佳的界限。在两个相邻的区间长度消失的情况下，我们使用一个格林函数，包含雅可比热核作为其基本成分。
英文摘要：
This article is the second of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities which govern CFT correlation functions of $2N$ one-leg boundary operators. In the first article (part I), we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. This article provides the proof. The lemma states that if every interval among $(x_2,x_3),$ $(x_3,x_4), ... ,(x_{2N-1},x_{2N})$ is a two-leg interval of $F\in\mathcal{S}_N$, then $F$ vanishes. Proving this lemma by contradiction, we show that the existence of such a nonzero function implies the existence of a non-vanishing CFT two-point function involving primary operators with different conformal weights, an impossibility. This proof (which is rigorous in spite of our occasional reference to CFT) involves two different types of estimates, those that give the asymptotic behavior of F as the length of one interval vanishes, and those that give this behavior as the lengths of two intervals vanish simultaneously. We derive these estimates by using Green functions to rewrite certain null-state PDEs as integral equations, combining other null-state PDEs to obtain Schauder interior estimates, and then repeatedly integrating the integral equations with these estimates until we obtain optimal bounds. In the case where two adjacent interval lengths vanish, we use a Green function that contains the Jacobi heat kernel as its essential ingredient.
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