Automorphic forms and fermion masses

We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups Gamma, that allow the presence of several moduli and make connection with the theory of automorphic forms. Moduli span a coset space G/K, where G is a Lie group and K is a compact subgroup of G, modded out by Gamma. For a general choice of G, K, Gamma and a generic matter content, we explicitly construct a minimal Kahler potential and a general superpotential, for both rigid and local N = 1 supersymmetric theories. We also specialize our construction to the case G = Sp(2g, ), K = U(g) and Gamma = Sp(2g, Z), whose automorphic forms are Siegel modular forms. We show how our general theory can be consistently restricted to multi-dimensional regions of the moduli space enjoying residual symmetries. After choosing g = 2, we present several examples of models for lepton and quark masses where Yukawa couplings are Siegel modular forms of level 2.

Automated summary

This study extends the framework of modular invariant supersymmetric theories to encompass more general discrete groups Γ, involving multiple moduli and connection with the theory of automorphic forms. By constructing a minimal Kahler potential and a general superpotential, the theory can be applied to rigid and local N = 1 supersymmetric theories. Specializing the construction to the case G = Sp(2g, ), K = U(g) and Gamma = Sp(2g, Z), the study shows how the theory can be consistently restricted to multi-dimensional regions of the moduli space with residual symmetries.