F-Yang–Mills equations - Misplaced Pages

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 {\displaystyle F} -Yang–Mills connections over S n {\displaystyle S^{n}} with n > 4 ( d F ′ + 1 ) {\displaystyle 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.