In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills (YM) connection is a Yang–Mills connection around which the Yang–Mills action functional is positively or even strictly positively curved.
Yang–Mills connections are solutions of the Yang–Mills equations following from them being local extrema of the curvature, hence critical points of the Yang–Mills action functional, which are determined by a vanishing first derivative of a variation.
(Weakly) stable Yang–Mills connections 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
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
and the volume form
The Yang–Mills action functional is given by:[2][3] A Yang–Mills connection
, hence which fulfills the Yang–Mills equations, is called stable if:[4][5] for every smooth family
It is called weakly stable if only
A Yang–Mills connection, which is not weakly stable, is called instable.
For comparison, the condition to be a Yang–Mills connection is:[2] For a (weakly) stable or instable Yang–Mills connection
is called a (weakly) stable or instable Yang–Mills field.
A compact Riemannian manifold, for which no principal bundle over it (with a compact Lie group as structure group) has a stable Yang–Mills connection is called Yang–Mills-instable (or YM-instable).
because of the above result from James Simons.
A Yang–Mills-instable manifold always has a vanishing second Betti number.
[6] Central for the proof is that the infinite complex projective space
is the classifying space
(but even more generally every CW complex) are classified by its second cohomology (with integer coefficients):[11][13][12] On a non-trivial principal
, which exists for a non-trivial second cohomology, one could construct a stable Yang–Mills connection.
Open problems about Yang-Mills-instable manifolds include:[6]