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 $\mathfrak{𝔤}$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with a compact orientable Riemannian manifold $B$ having a metric $g$ and a volume form ${\mathrm{vol}}_g$. Let $\mathrm{Ad}(E):=E{\times }_G\mathfrak{𝔤}\twoheadrightarrow B$ be its adjoint bundle. ${\mathrm{\Omega }}_{\mathrm{Ad}}^1(E,\mathfrak{𝔤})\cong {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ is the space of connections, which are either under the adjoint representation $\mathrm{Ad}$ 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 ${\mathrm{vol}}_g$, the second space is usually used.

The Bi-Yang–Mills action functional is given by:

$\mathrm{BiYM}:{\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))\to \mathbb{ℝ},{\mathrm{BiYM}}_F(A):=\underset{B}{\int }\Vert {\delta }_A{F}_A{\Vert }^2\mathrm{d}{\mathrm{vol}}_g.$

Bi-Yang–Mills connections and equation

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

$\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{BiYM}(A(t)){|}_{t=0}=0$

for every smooth family $A:(-\epsilon ,\epsilon )\to {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ with $A(0)=A$. This is the case iff the Bi-Yang–Mills equations are fulfilled:

$({\delta }_A{\mathrm{d}}_A+{{ℛ}}_A)({\delta }_A{F}_A)=0.$

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

$\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}\mathrm{BiYM}(A(t)){|}_{t=0}>0$

for every smooth family $A:(-\epsilon ,\epsilon )\to {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ with $A(0)=A$. It is called weakly stable if only $\ge 0$ holds. A Bi-Yang–Mills connection, which is not weakly stable, is called unstable. For a (weakly) stable or unstable Bi-Yang–Mills connection $A\in {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$, its curvature $F_A\in {\mathrm{\Omega }}^2(B,\mathrm{Ad}(E))$ is furthermore called a (weakly) stable or unstable Bi-Yang–Mills field.

Properties

• Yang–Mills connections are weakly stable Bi-Yang–Mills connections.

See also

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

Literature

• Chiang, Yuan-Jen (2013-06-18) (in en). Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields. Frontiers in Mathematics. Birkhäuser. doi:10.1007/978-3-0348-0534-6. ISBN 978-3034805339. https://link.springer.com/book/10.1007/978-3-0348-0534-6. 

External links

• Bi-Yang-Mills equation at the nLab