Modular Linear Differential Equations for Hecke and Fricke Groups

Abstract

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 $\Gamma_{0}(N)$ and Fricke groups $\Gamma_{0}^{+}(N)$ 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 $\Gamma_{0}^{+}(p)$ where $p$ is a prime divisor of the Monster group $\mathbb{M}$ possess admissible single character solutions and we argue that the solutions for $\Gamma_{0}^{+}(11)$ are rendered inadmissible due to its Hauptmodul while those for $\Gamma_{0}^{+}(13)$ 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 $\Gamma_{0}(2)$, $\Gamma_{0}(7)$, and the subsequent group in its modular tower, $\Gamma_{0}(49)$. 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 $\Theta$-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 $\Theta$-series of level $p = 2$ has distinctive ties to the odd Leech lattice in $24$-dimensions.