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 $F\colon\mathbb_0^+\rightarrow\mathbb_0^+$ be a strictly increasing $C^2$ function (hence with $F'>0$) and $F(0)=0$. Let:

: $d_F :=\sup_\frac.$

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

: $d_ =\sup_\frac.$

Let $G$ be a compact Lie group with Lie algebra $\mathfrak$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with an orientable Riemannian manifold $B$ having a metric $g$ and a volume form $\operatorname_g$. Let $\operatorname(E) :=E\times_G\mathfrak\twoheadrightarrow B$ be its adjoint bundle. $\Omega_^1(E,\mathfrak) \cong\Omega^1(B,\operatorname(E))$ is the space of connections, which are either under the adjoint representation $\operatorname$ 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 $\operatorname_g$, the second space is usually used.

The $F$-Yang–Mills action functional is given by:

: $\operatorname_F\colon \Omega^1(B,\operatorname(E))\rightarrow\mathbb, \operatorname_F(A) :=\int_BF\left(\frac\, F_A\, ^2\right)\mathrm\operatorname_g.$

For a flat connection $A\in\Omega^1(B,\operatorname(E))$ (with $F_A=0$), one has $\operatorname_F(A) =F(0)\operatorname(M)$. 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 $A\in\Omega^1(B,\operatorname(E))$ is called $F$''-Yang–Mills connection'', if it is a critical point of the $F$-Yang–Mills action functional, hence if:

: $\frac\operatorname_F(A(t))\vert_=0$

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

: $\mathrm_A\star\left( F'\left( \frac\, F_A\, ^2 \right)F_A \right) =0.$

For a $F$-Yang–Mills connection $A\in\Omega^1(B,\operatorname(E))$, its curvature $F_A\in\Omega^2(B,\operatorname(E))$ 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 $d_=\infty$. The exponential and normed exponential Yang–Mills action functional are denoted with $\operatorname_\mathrm$ and $\operatorname_\mathrm^0$ respectively.
* $F(t)=\frac(2t)^$ is called ''$p$-Yang–Mills connection/field''. In this case, one has $d_=\frac-1$. Usual Yang–Mills connections/fields are exactly the $2$-Yang–Mills connections/fields. The $p$-Yang–Mills action functional is denoted with $\operatorname_p$.
* $F(t)=\sqrt-1$ or $F(t)=\sqrt-1$ is called ''Yang–Mills–Born–Infeld connection/field'' (or ''YMBI connection/field'') with negative or positive sign respectively. In these cases, one has $d_=\infty$ and $d_=0$ respectively. The Yang–Mills–Born–Infeld action functionals with negative and positive sign are denoted with $\operatorname^-$ and $\operatorname^+$ respectively. The Yang–Mills–Born–Infeld equations with positive sign are related to the minimal surface equation:
*: $\mathrm_A\frac =0.$

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 $A\in\Omega^1(B,\operatorname(E))$ is called ''stable'' if:

: $\frac\operatorname_F(A(t))\vert_>0$

for every smooth family $A\colon (-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname(E))$ 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 $A\in\Omega^1(B,\operatorname(E))$, its curvature $F_A\in\Omega^2(B,\operatorname(E))$ 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:
*: $\, F_A\, \leq\sqrt$

: is unstable.

* Every non-flat Yang–Mills–Born–Infeld connection with negative sign over $S^n$ with $n\geq 5$ and:
*: $\, F_A\, \leq\sqrt$

: is unstable.

* All non-flat $F$-Yang–Mills connections over $S^n$ with $n>4(d_+1)$ 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.Chiang 2013, Theorem 3.1.9 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 $0\leq d_\leq\frac$, every non-flat $F$-Yang–Mills connection over the Cayley plane $F_4/\operatorname(9)$ is unstable.

Literature

*

See also

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

References

External links

* F-Yang-Mills equation at the ''n''Lab

{{DEFAULTSORT:F-Yang-Mills equations
Differential geometry
Mathematical physics
Partial differential equations