Symmetries and stabilisers in modular invariant flavour 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.

18 pages, 5 figures, 4 tables, accepted for publication in JHEP