Non-holomorphic modular flavor symmetry

Understand

The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level $N$ polyharmonic Maa{\ss} forms satisfying the Laplacian condition.

• We find that the integer (even) weight polyharmonic Maa{\ss} forms of level $N$ can be decomposed into multiplets of the finite modular group $\Gamma'_N$ ($\Gamma_N$).
• The original modular invariance approach is extended by the presence of negative weight polyharmonic Maa{\ss} forms.
• The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry.