Algorithmic Construction of Vector-Valued Modular Forms and Application to an $A_4\times\mathbb{Z}_4$-invariant Model of Leptons

Abstract

We present an algorithmic procedure that constructs the Vector-Valued Modular Forms (VVMFs) of finite modular groups, as well as its implementation in a companion code. The usual method consists in building the first few VVMFs, whose tensor products then generate the others. We address the difficulty of systematically finding the minimal set of VVMFs that suffices to produce all higher-weight ones, and we propose a method to identify it that makes use of the Burnside-Brauer theorem. For any given group, whose basic group-theoretic and representation-theoretic description is provided by the user, the code determines the minimal set of VVMFs, explicitly builds it using general results of the theory of such functions, then performs the tensor products that generate any other VVMF. Importantly, it automatically takes into account all caveats that might arise during the study of new groups. Our approach reproduces the known results for the finite modular groups explored in the recent phenomenological literature on modular flavor symmetries, and works for additional groups which we tested. We illustrate our algorithm and the working of our code by studying a model of leptons based on the unexplored finite modular group $A_4\times\mathbb{Z}_4$.