Discrete Flavour Symmetries from the Heisenberg Group

TL;DR

This paper introduces a unified framework for constructing representations of finite discrete family groups using automorphisms of the Heisenberg group, with a focus on the phenomenologically relevant $PSL_2(p)$ groups.

Contribution

It proposes a novel method to derive discrete flavor symmetries from the automorphisms of the Heisenberg group, connecting string theory and particle physics model building.

Findings

Framework successfully constructs representations of finite discrete groups.

Highlights the relevance of $PSL_2(p)$ groups in flavor symmetry models.

Provides a unified approach linking geometry and group theory in string compactifications.

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.