Abstract
Non-abelian discrete symmetries are of particular importance in model building. They are mainly invoked to explain the various fermion mass hierarchies and forbid dangerous superpotential terms. In string models they are usually associated to the geometry of the compactification manifold and more particularly to the magnetised branes in toroidal compactifications. Motivated by these facts, in this note we propose a unified framework to construct representations of finite discrete family groups based on the automorphisms of the discrete and finite Heisenberg group. We focus in particular in the $PSL_2(p)$ groups which contain the phenomenologically interesting cases.
Highlights
Non-abelian discrete symmetries play a prominent rôle in model building
We have introduced an intriguing relation of the discrete flavour symmetries with the automorphisms of the magnetic translations of the finite and discrete Heisenberg Group
This relation is reminiscent of the discrete symmetries of the Quantum Hall effect, where in a toroidal two dimensional space the magnetic flux transforms the torus into a phase space and the Hilbert space of a charged particle becomes finite dimensional and the corresponding torus effectively discrete [32]
Summary
Non-abelian discrete symmetries play a prominent rôle in model building. Among other objectives, more than a decade ago, they have been widely used to interpret the neutrino data in various extensions of the Standard Mode [1,2,3]. For example, the GUT model is SU(5) which is the minimal symmetry accommodating the Standard Model, the commutant with respect to E8 is SU(5) (denoted usually as perpendicular, SU(5)⊥, to the GUT) The latter naturally incorporates phenomenologically viable non-abelian discrete groups [13], such as Sn, An where usually n ≤ 5 and more generally PSL2(p), where p ≤ 11. There exist cases in string theory [4] where nonabelian finite groups may emerge as well In this context a class of non-Abelian discrete symmetries may arise from discrete isometries of the torus geometry, on which the Heiseberg group has a natural action. The Weyl representation presented above, provides the interesting result that the unitary matrix corresponding to the SL2(p) element a = 0 −1 is – up to a phase – the Discrete Finite Fourier. This way, we get (2k + 1) and (2k − 1)-dimensional irreps of PSL2(p) correspondingly
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