论文标题
Hecke和Fricke组的模块化线性微分方程
Modular Linear Differential Equations for Hecke and Fricke Groups
论文作者
论文摘要
模块化线性微分方程（MLDE）在二维CFT的分类中起着重要作用，其中方程中的模块化形式属于$ \ text {sl}（2，\ mathbb {z}）$的空间。对hecke组的微分方程及其解决方案的系统研究$γ_{0}（n）$和弗里克组$γ_{0}^{+}（n）$可以更好地理解我们对CFT分类的理解，因为在MLDE分析中没有为$ \ text $ \ text text of text of text of pext {sl} $} $ {2，2 ,, zbbbb的MLDE分析进行重要工作。在本文中，我们以$ n \ leq 12 $的级别为Hecke和Fricke组设置并解决了MLDE，并报告了每个组中获得的类似于可接受的角色的解决方案。我们发现，只有前四个零属组$γ_{0}^{+}（p）$，其中$ p $是Monster $ \ Mathbb {M Mathbb {M} $ lotufloysible anplosissible的单个角色解决方案的主要分隔线$γ_{0}^{+}（13）$由于模块化形式空间的基本分解的性质而变得不可接受。我们在Hecke组的单个字符级别上提出了一个新的准字符解决方案$γ_{0}（2）$，$γ_{0}（7）$，以及其模块塔中的后续组，$γ_{0}（0}（49）$。我们还将Fricke组的单个字符解决方案的所有结果扩展到了$ \ Mathbb {M} $的所有Prime Divisor水平，并就可以在获得可允许的解决方案中发挥作用的每个组中的有利属性进行评论。最后，我们发现与$ p = 2,3,5,7 $相关的$θ$ - 系列以及每种情况下接吻数字和晶格半径的相应晶格数据。我们发现，$ p = 2 $的Fricke $θ$ - 列表与$ 24 $ dimensions的奇数水ech晶格有着独特的联系。
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.
