Hecke群和Hecke群上的模线性微分方程
Modular Linear Differential Equations for Hecke and Fricke Groups
摘要：
模线性微分方程（MLDE）在二维CFTs的分类中起着重要的作用，其中方程的模形式属于$\text{SL}(2,\mathbb{Z})$空间。系统地研究Hecke群$Γ_{0}(N)$和Hecke群$Γ_{0}^{+}(N)$的微分方程及其解将有助于我们更好地理解CFT分类，因为$\text{SL}(2,\mathbb{Z})$的子群的MLDE分析还没有显著的工作。在本文中，我们建立和解决MLDE的Hecke和Hecke集团在$N\leq 12$水平和报告的容许字符型解决方案，在每个组。我们发现，只有前四个属零组$Γ_{0}^{+}(p)$，其中$p$是一个素因子的怪物组$\mathbb{M}$具有容许的单字符的解决方案，我们认为，解决方案$Γ_{0}^{+}(11)$是不可接受的，由于其Hauptmodul，而那些为$Γ_{0}^{+}(13)$是不可接受的，由于性质的基础分解的空间的模形式。本文给出了Hecke群$Γ_{0}(2)$、$Γ_{0}(7)$及其模塔中的后续群$Γ_{0}(49)$的单特征标水平上的一个新的拟特征标解.我们还扩展了所有的结果，单字符的解决方案的Sparkke群的所有素因子水平的$\mathbb{M}$和备注的有利性质，在每个组中，可以发挥作用，在获得容许的解决方案。最后，我们找到了与能级$p = 2,3,5,7$相关的$Θ$ -系列以及相应的Kissing数和晶格半径的晶格数据。我们发现，$p = 2$的$Θ$ -系列的水平$p = 2$有独特的联系，在$24$ -维的奇水蛭晶格。
英文摘要：
Modular linear differential equations (MLDE) play a significant role in the classification of two-dimensional CFTs, where the modular forms in the equations belonged to the space of $\text{SL}(2,\mathbb{Z})$. A systematic study of the differential equations and their solutions for the Hecke groups $Γ_{0}(N)$ and Fricke groups $Γ_{0}^{+}(N)$ would better our understanding of CFT classification as there has not been significant work on the MLDE analysis for subgroups of $\text{SL}(2,\mathbb{Z})$. In this paper, we set up and solve MLDEs for Hecke and Fricke groups at levels $N\leq 12$ and report on admissible character-like solutions obtained in each group. We find that only the first four genus zero groups $Γ_{0}^{+}(p)$ where $p$ is a prime divisor of the Monster group $\mathbb{M}$ possess admissible single character solutions and we argue that the solutions for $Γ_{0}^{+}(11)$ are rendered inadmissible due to its Hauptmodul while those for $Γ_{0}^{+}(13)$ are rendered inadmissible due to the nature of the basis decomposition of the space of modular forms. We present a new quasi-character solution at the single character level for the Hecke groups $Γ_{0}(2)$, $Γ_{0}(7)$, and the subsequent group in its modular tower, $Γ_{0}(49)$. We also extend all of the results for single character solutions of Fricke groups to all prime divisor levels of $\mathbb{M}$ and remark on favorable properties in each group that could play a role in obtaining admissible solutions. Finally, we find the $Θ$-series associated with levels $p = 2,3,5,7$ and the corresponding lattice data of Kissing numbers and lattice radii for each case. We find that the Fricke $Θ$-series of level $p = 2$ has distinctive ties to the odd Leech lattice in $24$-dimensions.
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