F-Yang–Mills equations

In differential geometry, the $F$-Yang–Mills equations (or $F$-YM equations) are a generalization of the Yang–Mills equations. Its solutions are called $F$-Yang–Mills connections (or $F$-YM connections). Simple important cases of $F$-Yang–Mills connections include exponential Yang–Mills connections using the exponential function for $F$ and $p$-Yang–Mills connections using $p$ as exponent of a potence of the norm of the curvature form similar to the $p$-norm. Also often considered are Yang–Mills–Born–Infeld connections (or YMBI connections) with positive or negative sign in a function $F$ involving the square root. This makes the Yang–Mills–Born–Infeld equation similar to the minimal surface equation.

F-Yang–Mills action functional

Let $F\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} _{0}^{+}$ be a strictly increasing $C^{2}$ function (hence with $F'>0$) and $F(0)=0$. Let:

$d_{F}:=\sup _{t\geq 0}{\frac {tF'(t)}{F(t)}}.$

Since $F$ is a $C^{2}$ function, one can also consider the following constant:

$d_{F'}=\sup _{t\geq 0}{\frac {tF''(t)}{F'(t)}}.$

Let $G$ be a compact Lie group with Lie algebra ${\mathfrak {g}}$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with an 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 $F$-Yang–Mills action functional is given by:

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

For a flat connection $A\in \Omega ^{1}(B,\operatorname {Ad} (E))$ (with $F_{A}=0$), one has $\operatorname {YM} _{F}(A)=F(0)\operatorname {vol} (M)$. Hence $F(0)=0$ is required to avert divergence for a non-compact manifold $B$, although this condition can also be left out as only the derivative $F'$ is of further importance.

F-Yang–Mills connections and equations

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

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

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 $F$-Yang–Mills equations are fulfilled:

$\mathrm {d} _{A}\star \left(F'\left({\frac {1}{2}}\|F_{A}\|^{2}\right)F_{A}\right)=0.$

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

A $F$-Yang–Mills connection/field with:

• $F(t)=t$ is just an ordinary Yang–Mills connection/field.
• $F(t)=\exp(t)$ (or $F(t)=\exp(t)-1$ for normalization) is called (normed) exponential Yang–Mills connection/field. In this case, one has $d_{F'}=\infty$. The exponential and normed exponential Yang–Mills action functional are denoted with $\operatorname {YM} _{\mathrm {e} }$ and $\operatorname {YM} _{\mathrm {e} }^{0}$ respectively.
• $F(t)={\frac {1}{p}}(2t)^{\frac {p}{2}}$ is called $p$-Yang–Mills connection/field. In this case, one has $d_{F'}={\frac {p}{2}}-1$. Usual Yang–Mills connections/fields are exactly the $2$-Yang–Mills connections/fields. The $p$-Yang–Mills action functional is denoted with $\operatorname {YM} _{p}$.
• $F(t)={\sqrt {1-2t}}-1$ or $F(t)={\sqrt {1+2t}}-1$ is called Yang–Mills–Born–Infeld connection/field (or YMBI connection/field) with negative or positive sign respectively. In these cases, one has $d_{F'}=\infty$ and $d_{F'}=0$ respectively. The Yang–Mills–Born–Infeld action functionals with negative and positive sign are denoted with $\operatorname {YMBI} ^{-}$ and $\operatorname {YMBI} ^{+}$ respectively. The Yang–Mills–Born–Infeld equations with positive sign are related to the minimal surface equation:

  $\mathrm {d} _{A}{\frac {\star F_{A}}{\sqrt {1+\|F_{A}\|^{2}}}}=0.$

Stable F-Yang–Mills connection

Analogous to (weakly) stable Yang–Mills connections, one can define (weakly) stable $F$-Yang–Mills connections. A $F$-Yang–Mills connection $A\in \Omega ^{1}(B,\operatorname {Ad} (E))$ is called stable if:

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

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 $F$-Yang–Mills connection, which is not weakly stable, is called unstable. For a (weakly) stable or unstable $F$-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 $F$-Yang–Mills field.

Properties

• For a Yang–Mills connection with constant curvature, its stability as Yang–Mills connection implies its stability as exponential Yang–Mills connection.
• Every non-flat exponential Yang–Mills connection over $S^{n}$ with $n\geq 5$ and:

  $\|F_{A}\|\leq {\sqrt {\frac {n-4}{2}}}$

is unstable.

• Every non-flat Yang–Mills–Born–Infeld connection with negative sign over $S^{n}$ with $n\geq 5$ and:

  $\|F_{A}\|\leq {\sqrt {\frac {n-4}{n-2}}}$

is unstable.

• All non-flat $F$-Yang–Mills connections over $S^{n}$ with $n>4(d_{F'}+1)$ are unstable. This result includes the following special cases:
  • All non-flat Yang–Mills connections with positive sign over $S^{n}$ with $n>4$ are unstable. James Simons presented this result without written publication during a symposium on "Minimal Submanifolds and Geodesics" in Tokyo in September 1977.
  • All non-flat $p$-Yang–Mills connections over $S^{n}$ with $n>2p$ are unstable.
  • All non-flat Yang–Mills–Born–Infeld connections with positive sign over $S^{n}$ with $n>4$ are unstable.
• For $0\leq d_{F'}\leq {\frac {1}{6}}$, every non-flat $F$-Yang–Mills connection over the Cayley plane $F_{4}/\operatorname {Spin} (9)$ is unstable.

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.

See also

• Bi-Yang–Mills equations, modification of the Yang–Mills equation

External links

• F-Yang-Mills equation at the nLab