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 $G$ be a compact Lie group with Lie algebra $\mathfrak$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with a compact orientable Riemannian manifold $B$ having a metric $g$ and a volume form $\operatorname_g$. Let $\operatorname(E) :=E\times_G\mathfrak\twoheadrightarrow B$ be its adjoint bundle. $\Omega_^1(E,\mathfrak) \cong\Omega^1(B,\operatorname(E))$ is the space of connections, which are either under the adjoint representation $\operatorname$ invariant Lie algebra–valued or vector bundle–valued differential forms. Since the Hodge star operator $\star$ is defined on the base manifold $B$ as it requires the metric $g$ and the volume form $\operatorname_g$, the second space is usually used.

The Bi-Yang–Mills action functional is given by:
: $\operatorname\colon \Omega^1(B,\operatorname(E))\rightarrow\mathbb, \operatorname_F(A) :=\int_B\, \delta_AF_A\, ^2\mathrm\operatorname_g.$

Bi-Yang–Mills connections and equation

A connection $A\in\Omega^1(B,\operatorname(E))$ is called ''Bi-Yang–Mills connection'', if it is a critical point of the Bi-Yang–Mills action functional, hence if:

: $\frac\operatorname(A(t))\vert_=0$

for every smooth family $A\colon (-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname(E))$ with $A(0)=A$. This is the case iff the ''Bi-Yang–Mills equations'' are fulfilled:

: $(\delta_A\mathrm_A+\mathcal_A)(\delta_AF_A) =0.$

For a Bi-Yang–Mills connection $A\in\Omega^1(B,\operatorname(E))$, its curvature $F_A\in\Omega^2(B,\operatorname(E))$ 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 $A\in\Omega^1(B,\operatorname(E))$ is called ''stable'' if:

: $\frac\operatorname(A(t))\vert_>0$

for every smooth family $A\colon (-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname(E))$ with $A(0)=A$. It is called ''weakly stable'' if only $\geq 0$ holds. A Bi-Yang–Mills connection, which is not weakly stable, is called ''unstable''. For a (weakly) stable or unstable Bi-Yang–Mills connection $A\in\Omega^1(B,\operatorname(E))$, its curvature $F_A\in\Omega^2(B,\operatorname(E))$ 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.

See also

* F-Yang–Mills equations, generalization of the Yang–Mills equation

Literature

*

References

External links

* Bi-Yang-Mills equation at the ''n''Lab

{{DEFAULTSORT:Bi-Yang-Mills equations
Differential geometry
Mathematical physics
Partial differential equations