Supersymmetry Breakings and Fermat's Last Theorem

TL;DR

This paper introduces a novel mechanism for supersymmetry breaking linked to Fermat's last theorem, showing that supersymmetry is exact at irrational points and broken at rational points in parameter space, with potential applications in superstring models.

Contribution

It presents the first explicit supersymmetry breaking mechanism sensitive to parameter irrationality, enabling small non-perturbative breakings in superstring model building.

Findings

Supersymmetry is exact at irrational parameter points.

Supersymmetry is broken at rational parameter points.

A superpotential for irrational parameters is constructed.

Abstract

A mechanism of supersymmetry breaking in two or four-dimensions is given, in which the breaking is related to the Fermat's last theorem. It is shown that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. Accordingly, supersymmetry is exact {\it almost everywhere}, as well as broken {\it almost everywhere} on the real axis in the parameter space at the same time. This is the first explicit mechanism of supersymmetry breaking with an arbitrarily small change of parameters around any exact supersymmetric model, which is possibly useful for realistically small non-perturbative supersymmetry breakings in superstring model building. As a byproduct, we also give a convenient superpotential for supersymmetry breaking only for irrational number parameters. Our superpotential can be added as a ``hidden'' sector to other useful supersymmetric models.