Minimal Potentials with Very Many Minima

TL;DR

This paper constructs simple renormalizable matrix potentials with symmetric groups that can have an exponentially large number of deep local minima, challenging previous assumptions about the landscape of such potentials.

Contribution

It introduces a novel class of potentials with S_N symmetry that exhibit exponentially many minima, expanding understanding of potential landscapes in theoretical physics.

Findings

Potential landscapes with S_N symmetry can have exponentially many minima.

Constructed explicit examples of such potentials.

Implications for symmetry breaking and vacuum structure in field theories.

Abstract

We demonstrate, by construction, that simple renormalizable matrix potentials with S_N, as opposed to O(N), symmetry can exhibit an exponentially large number of inequivalent deep local minima.