Automorphic forms and fermion masses

Abstract

We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups Γ, 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 Γ. For a general choice of G, K, Γ and a generic matter content, we explicitly construct a minimal Kähler potential and a general superpotential, for both rigid and local $$ \mathcal{N} $$ N = 1 supersymmetric theories. We also specialize our construction to the case G = Sp(2g, ℝ), K = U(g) and Γ = 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.

CP symmetry and symplectic modular invariance

“…In the class of theories under consideration here, both the flavour symmetry and the fields responsible for SB have the same origin [2]. Scalars driving SB take values in a symmetric space of the type G/K, G being some noncompact continuous group and K a maximal compact subgroup of G. The flavour symmetry group is a discrete, modular subgroup Γ of G, acting on G/K.…”

“…It is convenient to base our construction on the invariant subspace τ 1 = τ 2 at genus g = 2 [2]. At level 2, the modular group N (H) 11 is S 4 × Z 2 , whose generators G 1 = T 1 T 2 , G 2 = T 3 and G 3 = S are shown in appendix A, in a convenient basis.…”

“…No exact flavour symmetry does the job, however, and realistic models should rely heavily on the properties of the symmetry breaking (SB) sector, comprising a set of scalar fields whose vacuum expectation values (VEV) are suitably oriented in flavour space. To reduce the vast arbitrariness associated with this construction, where both the flavour group and the SB sector are essentially unrestricted, we recently proposed a framework defined by a set of geometrical data [2]. Scalars responsible for SB span a moduli space, a symmetric space of the type G/K, G being a noncompact continuous group and K a maximal compact subgroup of G. A discrete, modular subgroup Γ of G, acting on G/K, plays the role of flavour symmetry.…”

“…In this way both the flavour symmetry group and the SB sector are closely linked and cannot be arbitrarily chosen. Modular invariant supersymmetric field theories [3,4] are a particular case of this general setting, related to the choice G = S L(2, ), K = SO (2) and Γ = S L (2, ).…”

SU(5) GUTs with A4 modular symmetry

“…The more fundamental theory such as string theory sometimes requires several compact space with more than one modulus parametrizing its shape. The modular invariance approach has been extended to incorporate several factorizable [40] and non-factorizable moduli [58].…”