2022/10/12 by Umasankar, Naveen Balaji
#FOS: Physical sciences #High Energy Physics - Theory (hep-th)
paper · doi:10.48550/arxiv.2210.07186
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 SL(2,ℤ). A systematic study of the differential equations and their solutions for the Hecke groups Γ0(N) and Fricke groups Γ0+(N) would better our understanding of CFT classification as there has not been significant work on the MLDE analysis for subgroups of SL(2,ℤ). In this paper, we set up and solve MLDEs for Hecke and Fricke groups at levels N≤ 12 and report on admissible character-like solutions obtained in each group. We find that only the first four genus zero groups Γ0+(p) where p is a prime divisor of the Monster group \mathbbM possess admissible single character solutions and we argue that the solutions for Γ0+(11) are rendered inadmissible due to its Hauptmodul while those for Γ0+(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 Γ0(2), Γ0(7), and the subsequent group in its modular tower, Γ0(49). We also extend all of the results for single character solutions of Fricke groups to all prime divisor levels of \mathbbM 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.