Kernel(J) warns of false vacua

TL;DR

This paper explores the mathematical structure of false vacua in nonlinear sigma-models with spherical source and target spaces, linking homotopy theory to physical phenomena such as energy barriers.

Contribution

It introduces a novel connection between the kernel of the J-homomorphism and the existence of false vacua in certain sigma-models, with specific dimensional constraints.

Findings

False vacua exist in models with space dimension ≥8 and target dimension ≥5.

The kernel of the J-homomorphism informs the topology of these vacua.

Energy barriers separate false vacua from true vacua in the model.

Abstract

J.H.C. Whitehead defined a map $J_r:\pi_r(SO)\rightarrow \pi_r^s$ from the homotopy of the special orthogonal group to the stable homotopy of spheres. Within a toy model we show how the known computation for kernel$(J)$ leads to nonlinear $\sigma$-models with spherical source (space) and spherical target which admit false vacua separated from the true vacuum by an energy barrier. In this construction, the dimension of space must be at least 8 and the dimension of the $\sigma$-model target at least 5.