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

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

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

Bi-Yang–Mills connections and equation

A connection ${\displaystyle A\in \Omega ^{1}(B,\operatorname {Ad} (E))}$ is called Bi-Yang–Mills connection, if it is a critical point of the Bi-Yang–Mills action functional, hence if:^[3]

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

for every smooth family ${\displaystyle A\colon (-\varepsilon ,\varepsilon )\rightarrow \Omega ^{1}(B,\operatorname {Ad} (E))}$ with ${\displaystyle A(0)=A}$. This is the case iff the Bi-Yang–Mills equations are fulfilled:^[4]

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

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

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

for every smooth family ${\displaystyle A\colon (-\varepsilon ,\varepsilon )\rightarrow \Omega ^{1}(B,\operatorname {Ad} (E))}$ with ${\displaystyle A(0)=A}$. It is called weakly stable if only ${\displaystyle \geq 0}$ 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 ${\displaystyle A\in \Omega ^{1}(B,\operatorname {Ad} (E))}$, its curvature ${\displaystyle F_{A}\in \Omega ^{2}(B,\operatorname {Ad} (E))}$ is furthermore called a (weakly) stable or unstable Bi-Yang–Mills field.

Properties

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

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. Frontiers in Mathematics. Birkhäuser. doi:10.1007/978-3-0348-0534-6. ISBN 978-3034805339.

References

1. de los Ríos, Santiago Quintero (2020-12-16). "Connections on principal bundles" (PDF). homotopico.com. Theorem 3.7. Retrieved 2024-11-09.
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.

External links

• Bi-Yang-Mills equation at the nLab