Bi-Yang–Mills equations

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.

Bi-Yang–Mills action functional Let be a compact Lie group with Lie algebra and be a principal -bundle with a compact orientable Riemannian manifold having a metric and a volume form . Let be its adjoint bundle. is the space of connections, 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 second space is usually used.

The Bi-Yang–Mills action functional is given by:Chiang 2013, Eq. (9)

Bi-Yang–Mills connections and equation A connection is called Bi-Yang–Mills connection, if it is a critical point of the Bi-Yang–Mills action functional, hence if:Chiang 2013, Eq. (5.1) and (6.1)

for every smooth family with . This is the case iff the Bi-Yang–Mills equations are fulfilled:Chiang 2013, Eq. (10), (5.2) and (6.3)

For a Bi-Yang–Mills connection , its curvature is called Bi-Yang–Mills field.

Stable Bi-Yang–Mills connections 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 with . It is called weakly stable if only holds.Chiang 2013, Definition 6.3.2 A Bi-Yang–Mills connection, which is not weakly stable, is called unstable. For a (weakly) stable or unstable Bi-Yang–Mills connection , its curvature is furthermore called a (weakly) stable or unstable Bi-Yang–Mills field.

Properties

Yang–Mills connections are weakly stable Bi-Yang–Mills connections.Chiang 2013, Proposition 6.3.3.

Bi-Yang–Mills equations

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