Non-abelian field cohomology, its relation with spontaneous symmetry breaking and Morse's Theorem

TL;DR

This paper explores how spontaneous symmetry breaking in non-abelian gauge theories alters field cohomology, leading to matter-like properties and resolving the Gribov problem on-shell.

Contribution

It demonstrates that symmetry breaking modifies cohomology, creating matter-like fields and automatically solving the Gribov problem in gauge fixing.

Findings

Spontaneous symmetry breaking changes gauge field cohomology.

A matter-like cohomological property emerges in broken symmetry.

The Gribov problem is automatically resolved on-shell due to cohomological properties.

Abstract

We show that, for an $SU(2)$ gauge field (the reasoning extends trivially to $SU(N)$), spontaneous symmetry breaking changes the field cohomology. This defines a new field with cohomological properties characteristic of matter fields. Consequently, the construction of a renormalizable unitary gauge fixing, following Morse's problem of functional extremization, leads to the Gribov condition being automatically solved on-shell. This result occurs because a specific combination of fields is cohomologically matter-like and therefore free of the Gribov problem.