零态偏微分方程组的解空间3
A solution space for a system of null-state partial differential equations 3
摘要：
这篇文章是四篇中的第三篇，完整地描述了共形场论（CFT）和多重Schramm-Lowner演化（SLE）中出现的$2N$变量的$2N+3$线性偏微分方程（PDE）的齐次系统的解空间$\mathcal{S}_N$。该系统包括$2N$零状态方程和三个共形Ward恒等式，它们控制$2N$单腿边界算子的CFT关联函数。在前两篇文章（第一部分和第二部分）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.扩展这些结果，我们证明在这篇文章中，$\dim\mathcal{S}_N=C_N$和$\mathcal{S}_N$完全由（实值）的解决方案构造的CFT库仑气体（轮廓积分）形式主义。为了证明这一点，我们证明了一定的一组$C_N$这样的解决方案是线性无关的。由于这些解的公式比较复杂，我们间接地证明了线性无关性。我们使用第一部分引理15的线性内射映射将上述集合的每个解发送到$\mathbb{R}^{C_N}$中的向量，其分量我们发现为Temperley-Lieb代数中元素的内积。我们将这些向量聚集在一起，作为对称$C_N$乘$C_N$矩阵的列，具有曲折矩阵的形式。如果该矩阵的行列式不为零，则$C_N$库仑气体解的集合是线性无关的。如果该行列式确实消失，那么我们构造另一组$C_N$库仑气体解，并遵循类似的程序来证明该组是线性独立的。后一种情况与CFT极小模型密切相关。
英文摘要：
This article is the third 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 that govern CFT correlation functions of $2N$ one-leg boundary operators. In the previous two articles (parts I and II), 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. Extending these results, we prove in this article that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of $C_N$ such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of lemma 15 in part I to send each solution of the mentioned set to a vector in $\mathbb{R}^{C_N}$, whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric $C_N$ by $C_N$ matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of $C_N$ Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of $C_N$ Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models.
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