Modular Linear Differential Equations for Hecke and Fricke Groups

TL;DR

This paper systematically studies modular linear differential equations for Hecke and Fricke groups at levels up to 12, revealing new solutions and connections to lattice structures, enhancing understanding of conformal field theory classifications.

Contribution

It introduces new MLDE solutions for Hecke and Fricke groups, analyzes admissibility conditions, and links these to lattice structures and the Monster group.

Findings

Only the first four genus zero groups have admissible single character solutions.

New quasi-character solutions found for certain Hecke groups.

Fricke $ heta$-series at level 2 connects to the odd Leech lattice.

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.