Topological Classification of Symmetry Breaking and Vacuum Degeneracy

TL;DR

This paper introduces a mathematical framework using groupoid bundles and singular foliations to classify symmetry breaking patterns and vacuum degeneracy in gauge theories.

Contribution

It provides a novel geometric approach linking principal groupoid bundles and singular foliations to classify vacuum degeneracy patterns.

Findings

Classifies vacuum degeneracy using singular foliations.

Connects symmetry breaking to Lie groupoid structures.

Provides a mathematical classification scheme for vacuum patterns.

Abstract

We argue that a general system of scalar fields and gauge fields manifesting vacuum degeneracy induces a principal groupoid bundle over spacetime and that the pattern of spontaneous symmetry breaking and the Higgs mechanism are encoded by the singular foliation canonically induced on the moduli space of scalar vacuum expectation values by the Lie groupoid structure. Recent mathematical results in the classification of singular foliations then provide a qualitative classification of the possible patterns of vacuum degeneracy.