Non-abelian field cohomology, its relation with spontaneous symmetry breaking and Morse's Theorem
TLDR
Spontaneous symmetry breaking alters SU(N) gauge field cohomology, creating matter-like fields that solve the Gribov problem on-shell.
Key contributions
- Demonstrates how spontaneous symmetry breaking modifies SU(N) gauge field cohomology.
- Shows this process generates new fields with characteristic matter-like cohomological properties.
- Proposes a renormalizable unitary gauge fixing that automatically solves the Gribov condition on-shell.
- Explains this solution by identifying specific field combinations as matter-like, thus avoiding the Gribov problem.
Why it matters
This paper offers a novel perspective on gauge field theory by linking spontaneous symmetry breaking to changes in field cohomology. It provides a mechanism to resolve the long-standing Gribov problem, simplifying gauge fixing in quantum field theories. This could lead to more robust and consistent theoretical frameworks.
Original 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.
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