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

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.