In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills–Higgs (YMH) pair is a Yang–Mills–Higgs pair around which the Yang–Mills–Higgs action functional is positively or even strictly positively curved.
Yang–Mills–Higgs pairs are solutions of the Yang–Mills–Higgs equations following from them being local extrema of the curvature of both fields, hence critical points of the Yang–Mills-Higgs action functional, which are determined by a vanishing first derivative of a variation.
(Weakly) stable Yang–Mills-Higgs pairs furthermore have a positive or even strictly positive curved neighborhood and hence are determined by a positive or even strictly positive second derivative of a variation.
be a compact Lie group with Lie algebra
-bundle with a compact orientable Riemannian manifold
and a volume form
be its adjoint bundle.
is the space of connections,[1] which are either under the adjoint representation
invariant Lie algebra–valued or vector bundle–valued differential forms.
Since the Hodge star operator
is defined on the base manifold
as it requires the metric
and the volume form
The Yang–Mills–Higgs action functional is given by:[2] A Yang–Mills–Higgs pair
, hence which fulfill the Yang–Mills–Higgs equations, is called stable if:[3][4][5] for every smooth family
α : ( − ε , ε ) →
It is called weakly stable if only
A Yang–Mills–Higgs pair, which is not weakly stable, is called instable.
For comparison, the condition to be a Yang–Mills–Higgs pair is: