Modular symmetry at level 6 and a new route towards finite modular groups

Abstract

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as $\Gamma (N^{\prime })/\Gamma (N")$, and the modular group $SL(2,\mathbb{Z})$ is extended to a principal congruence subgroup $\Gamma (N^{\prime })$. The original modular invariant theory is reproduced when $N^{\prime }=1$. We perform a comprehensive study of ${\Gamma }_6^{\prime }$ modular symmetry corresponding to $N^{\prime }=1$ and $N"=6$, five types of models for lepton masses and mixing with ${\Gamma }_6^{\prime }$ modular symmetry are discussed and some example models are studied numerically. The case of $N^{\prime }=2$ and $N"=6$ is considered, the finite modular group is $\Gamma (2)/\Gamma (6)\cong T^{\prime }$, and a benchmark model is constructed.

Comments: 39 pages, 4 figures