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 $\operatorname {vol} _{g}$. Let $\operatorname {Ad} (E):=E\times _{G}{\mathfrak {g}}\twoheadrightarrow B$ be its adjoint bundle. $\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})\cong \Omega ^{1}(B,\operatorname {Ad} (E))$ is the space of connections, 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:

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

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:

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

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

$(\delta _{A}\mathrm {d} _{A}+{\mathcal {R}}_{A})(\delta _{A}F_{A})=0.$

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

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

for every smooth family $A\colon (-\varepsilon ,\varepsilon )\rightarrow \Omega ^{1}(B,\operatorname {Ad} (E))$ 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 $A\in \Omega ^{1}(B,\operatorname {Ad} (E))$, its curvature $F_{A}\in \Omega ^{2}(B,\operatorname {Ad} (E))$ 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). Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields. Birkhäuser. ISBN 978-3034805339.

External links

• Bi-Yang-Mills equation at the nLab