Bi-Yang–Mills equations explained

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 ak{g} 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{vol} _{g} . Let \operatorname{Ad}(E) : = E x _{Gak{g}\twoheadrightarrowB} be its adjoint bundle. \Omega _{\operatorname{Ad} }^1(E,\mathfrak)\cong\Omega^1(B,\operatorname(E)) is the space of connections,^[1] which are either under the adjoint representation \operatorname{Ad} 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{vol} _{g} , the second space is usually used.

The Bi-Yang–Mills action functional is given by:^[2]

\operatorname{BiYM}\colon \Omega ^{1 (B,\operatorname{Ad}(E)) → R , \operatorname{BiYM}} _{F(A) : = \int} _{B\|\delta} _{AF} ^{2 d \operatorname{vol}} ^{g.}

Bi-Yang–Mills connections and equation

A connection

A\in\Omega ^{1(B,\operatorname{Ad}(E))} is called Bi-Yang–Mills connection, if it is a critical point of the Bi-Yang–Mills action functional, hence if:^[3] (d)/(d t) \operatorname{BiYM}(A(t))\vert _{t=0} =0

for every smooth family

A\colon ( - \varepsilon,\varepsilon) → \Omega ^{1(B,\operatorname{Ad}(E))} with A( 0 ) = A . This is the case iff the Bi-Yang–Mills equations are fulfilled:^[4] (\delta _{Ad} _{A+l{R}} _{A)(\delta} _{AF} _{A) =0.}

For a Bi-Yang–Mills connection

A\in\Omega ^{1(B,\operatorname{Ad}(E))} , its curvature F ^{2 (B,\operatorname{Ad}(E))} _{A\in\Omega} 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{Ad}(E))} is called stable if:(d ^{2})/(d t ^{2}) \operatorname{BiYM}(A(t))\vert _{t=0} > 0

for every smooth family

A\colon ( - \varepsilon,\varepsilon) → \Omega ^{1(B,\operatorname{Ad}(E))} with A( 0 ) = A . It is called weakly stable if only \geq 0 holds.^[5] 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{Ad}(E))} , its curvature F ^{2 (B,\operatorname{Ad}(E))} _{A\in\Omega} is furthermore called a (weakly) stable or unstable Bi-Yang–Mills field.

Properties

• Yang–Mills connections are weakly stable Bi-Yang–Mills connections.^[6]

See also

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

Literature

• Book: Chiang, Yuan-Jen . Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields . Frontiers in Mathematics . . 2013-06-18 . 978-3034805339 . 10.1007/978-3-0348-0534-6 . en.

External links

• Bi-Yang-Mills equation at the nLab

Notes and References

1. Web site: de los Ríos . Santiago Quintero . 2020-12-16 . Connections on principal bundles . 2024-11-09 . homotopico.com . Theorem 3.7 . en.
2. Chiang 2013, Eq. (9)
3. Chiang 2013, Eq. (5.1) and (6.1)
4. Chiang 2013, Eq. (10), (5.2) and (6.3)
5. Chiang 2013, Definition 6.3.2
6. Chiang 2013, Proposition 6.3.3.