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} = 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.
Highlights
The description of the flavour sector of particle physics requires up to 22 independent measurable parameters if neutrinos are Majorana particles
In this last section we present some concrete examples of models for fermion masses, separately in the leptonic and in the quark sectors. These are models invariant under rigid supersymmetry, where we restrict to Ω of eq (6.9) as moduli space and where the role of flavour symmetry is played by the discrete group N (H) generated by the elements given in eq (6.14)
An interesting possibility to reduce the ambiguity of a pure bottom-up approach comes from string theory where the background over which the string propagates provides a natural setup for the symmetry breaking sector
Summary
The description of the flavour sector of particle physics requires up to 22 independent measurable parameters if neutrinos are Majorana particles. Looking at the top-down approach offered by string theory, Yukawa couplings are not independent input parameters, but rather field dependent quantities. A discrete subgroup Γ of G, the duality group, realizes its natural action on the moduli space G/K [3] Such a framework is naturally equipped with a symmetry Γ and a symmetry breaking sector spanning G/K. Such a construction is based on the choice G = Sp(2g, R), K = U(g) and Γ = Sp(2g, Z). We present examples of Siegel modular invariant models for lepton and quark masses
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