In differential geometry, the Bi-Yang–Mills equations (or Bi-YM equations) are a modification of the Yang–Mills equations.
Its solutions are called Bi-Yang–Mills connections (or Bi-YM connections).
Simply put, Bi-Yang–Mills connections are to Yang–Mills connections what they are to flat connections.
This stems from the fact, that Yang–Mills connections are not necessarily flat, but are at least a local extremum of curvature, while Bi-Yang–Mills connections are not necessarily Yang–Mills connections, but are at least a local extremum of the left side of the Yang–Mills equations.
While Yang–Mills connections can be viewed as a non-linear generalization of harmonic maps, Bi-Yang–Mills connections can be viewed as a non-linear generalization of biharmonic maps.
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 Bi-Yang–Mills action functional is given by:[2] A connection
is called Bi-Yang–Mills connection, if it is a critical point of the Bi-Yang–Mills action functional, hence if:[3] for every smooth family
This is the case iff the Bi-Yang–Mills equations are fulfilled:[4] For a Bi-Yang–Mills connection
is called Bi-Yang–Mills field.
Analogous to (weakly) stable Yang–Mills connections, one can define (weakly) stable Bi-Yang–Mills connections.
A Bi-Yang–Mills connection
is called stable if: for every smooth family
It is called weakly stable if only
[5] A Bi-Yang–Mills connection, which is not weakly stable, is called unstable.
For a (weakly) stable or unstable Bi-Yang–Mills connection
is furthermore called a (weakly) stable or unstable Bi-Yang–Mills field.