Symmetries and stabilisers in modular invariant flavour models

TL;DR

This paper develops an algorithm to identify stabilisers in finite modular groups, aiding the construction of flavor models with specific residual symmetries for fermionic mixing.

Contribution

It introduces a novel algorithm for finding stabilisers in finite modular groups and applies it to groups with N=2 to 5, facilitating flavor model building.

Findings

Identified all stabilisers for groups with N=2 to 5.

Each stabiliser preserves a cyclic subgroup of the modular group.

Provides a systematic method for residual symmetry analysis in flavor models.

Abstract

The idea of modular invariance provides a novel explanation of flavour mixing. Within the context of finite modular symmetries $\Gamma_N$ and for a given element $\gamma \in \Gamma_N$, we present an algorithm for finding stabilisers (specific values for moduli fields $\tau_\gamma$ which remain unchanged under the action associated to $\gamma$). We then employ this algorithm to find all stabilisers for each element of finite modular groups for $N=2$ to $5$, namely, $\Gamma_2\simeq S_3$, $\Gamma_3\simeq A_4$, $\Gamma_4\simeq S_4$ and $\Gamma_5\simeq A_5$. These stabilisers then leave preserved a specific cyclic subgroup of $\Gamma_N$. This is of interest to build models of fermionic mixing where each fermionic sector preserves a separate residual symmetry.