Wronskian指数等于$\mathbf{0}$或$\mathbf{2}$的三字符RCFTs的分类
Classifying three-character RCFTs with Wronskian Index equalling $\mathbf{0}$ or $\mathbf{2}$
摘要：
在对有理共形场论进行分类的模线性微分方程（MLDE）方法中，MLDE和RCFT都由一对非负整数$\textbf{[n,l]}$来标识。$\mathbf{n}$是RCFT的字符数以及字符求解的MLDE的阶数，并且$\mathbf{l}$（朗斯基索引）与字符的朗斯基的零的结构相关联。在本文中，我们研究了$\textbf{[3,0]}$和$\textbf{[3,2]}$ MLDE，以便对相应的CFTs进行分类。我们把这个问题归结为一个“有限”问题：要对具有中心电荷$ 0 < c \leq 96$的CFTs进行分类，我们需要对前者执行$6,720$计算，对后者执行$20,160$计算。每个计算涉及（i）首先找到一对丢番图方程的联立解，以及（ii）计算高阶傅立叶系数并检查正性。在$\textbf{[3,0]}$的情况下，对于$ 0 < c \leq 96$，我们得到了许多字符的解决方案：两个无限类和一个离散的$303$的集合。在考虑了各种已知的解决方案，包括Virasoro最小模型，WZW CFTs，Franc-Mason顶点算子代数和Gaberdiel-Hampapura-Mukhi新陪集CFTs之后，我们似乎有七个迄今未知的字符类解决方案，这些解决方案可能会产生新的CFTs。我们还对$ 0 < c \leq 96$的$\textbf{[3,2]}$ CFT进行了分类：在这种情况下，每个CFT都是通过将一个常数字符与$\textbf{[2,0]}$ CFT相邻而获得的，其分类是由Mathur-Mukhi-Sen在三十年前实现的。
英文摘要：
In the modular linear differential equation (MLDE) approach to classifying rational conformal field theories (RCFTs) both the MLDE and the RCFT are identified by a pair of non-negative integers $\textbf{[n,l]}$. $\mathbf{n}$ is the number of characters of the RCFT as well as the order of the MLDE that the characters solve and $\mathbf{l}$, the Wronskian index, is associated to the structure of the zeroes of the Wronskian of the characters. In this paper, we study $\textbf{[3,0]}$ and $\textbf{[3,2]}$ MLDEs in order to classify the corresponding CFTs. We reduce the problem to a "finite" problem: to classify CFTs with central charge $ 0 < c \leq 96$, we need to perform $6,720$ computations for the former and $20,160$ for the latter. Each computation involves (i) first finding a simultaneous solution to a pair of Diophantine equations and (ii) computing Fourier coefficients to a high order and checking for positivity. In the $\textbf{[3,0]}$ case, for $ 0 < c \leq 96$, we obtain many character-like solutions: two infinite classes and a discrete set of $303$. After accounting for various categories of known solutions, including Virasoro minimal models, WZW CFTs, Franc-Mason vertex operator algebras and Gaberdiel-Hampapura-Mukhi novel coset CFTs, we seem to have seven hitherto unknown character-like solutions which could potentially give new CFTs. We also classify $\textbf{[3,2]}$ CFTs for $ 0 < c \leq 96$: each CFT in this case is obtained by adjoining a constant character to a $\textbf{[2,0]}$ CFT, whose classification was achieved by Mathur-Mukhi-Sen three decades ago.
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