Bi-Yang–Mills equations

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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.

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]

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

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.

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.

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

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

• 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.

• Bi-Yang-Mills equation at the nLab