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 $F:{\mathbb{ℝ}}_0^{+}\to {\mathbb{ℝ}}_0^{+}$ be a strictly increasing $C^2$ function (hence with $F^{\prime }>0$) and $F(0)=0$. Let:

$d_F:=\underset{t\ge 0}{\sup }\frac{t{F}^{\prime }(t)}{F(t)}.$

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

$d_{F^{\prime }}=\underset{t\ge 0}{\sup }\frac{t{F}^{″}(t)}{F^{\prime }(t)}.$

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 ${\mathrm{vol}}_g$. Let $\mathrm{Ad}(E):=E{\times }_G\mathfrak{𝔤}\twoheadrightarrow B$ be its adjoint bundle. ${\mathrm{\Omega }}_{\mathrm{Ad}}^1(E,\mathfrak{𝔤})\cong {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ is the space of connections, which are either under the adjoint representation $\mathrm{Ad}$ 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 ${\mathrm{vol}}_g$, the second space is usually used. The $F$-Yang–Mills action functional is given by:

${\mathrm{YM}}_F:{\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))\to \mathbb{ℝ},{\mathrm{YM}}_F(A):=\underset{B}{\int }F(\frac{1}{2}\Vert F_A{\Vert }^2)\mathrm{d}{\mathrm{vol}}_g.$

For a flat connection $A\in {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ (with $F_A=0$), one has ${\mathrm{YM}}_F(A)=F(0)\mathrm{vol}(M)$. 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 $F^{\prime }$ is of further importance.

F-Yang–Mills connections and equations

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

$\frac{\mathrm{d}}{\mathrm{d}t}{\mathrm{YM}}_F(A(t)){|}_{t=0}=0$

for every smooth family $A:(-\epsilon ,\epsilon )\to {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ with $A(0)=A$. This is the case iff the $F$-Yang–Mills equations are fulfilled:

${\mathrm{d}}_A\star \left(F^{\prime }(\frac{1}{2}\Vert F_A{\Vert }^2)F_A\right)=0.$

For a $F$-Yang–Mills connection $A\in {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$, its curvature $F_A\in {\mathrm{\Omega }}^2(B,\mathrm{Ad}(E))$ 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 $d_{F^{\prime }}=\mathrm{\infty }$. The exponential and normed exponential Yang–Mills action functional are denoted with ${\mathrm{YM}}_{\mathrm{e}}$ and ${\mathrm{YM}}_{\mathrm{e}}^0$ respectively.
• $F(t)=\frac{1}{p}(2t{)}^{\frac{p}{2}}$ is called $p$-Yang–Mills connection/field. In this case, one has $d_{F^{\prime }}=\frac{p}{2}-1$. Usual Yang–Mills connections/fields are exactly the $2$-Yang–Mills connections/fields. The $p$-Yang–Mills action functional is denoted with ${\mathrm{YM}}_p$.
• $F(t)=\sqrt{1-2t}-1$ 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 $d_{F^{\prime }}=\mathrm{\infty }$ and $d_{F^{\prime }}=0$ respectively. The Yang–Mills–Born–Infeld action functionals with negative and positive sign are denoted with ${\mathrm{YMBI}}^{-}$ and ${\mathrm{YMBI}}^{+}$ respectively. The Yang–Mills–Born–Infeld equations with positive sign are related to the minimal surface equation:

  ${\mathrm{d}}_A\frac{\star F_A}{\sqrt{1+\Vert F_A{\Vert }^2}}=0.$

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 $A\in {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ is called stable if:

$\frac{{\mathrm{d}}^2}{\mathrm{d}{t}^2}{\mathrm{YM}}_F(A(t)){|}_{t=0}>0$

for every smooth family $A:(-\epsilon ,\epsilon )\to {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$ with $A(0)=A$. It is called weakly stable if only $\ge 0$ holds. A $F$-Yang–Mills connection, which is not weakly stable, is called unstable. For a (weakly) stable or unstable $F$-Yang–Mills connection $A\in {\mathrm{\Omega }}^1(B,\mathrm{Ad}(E))$, its curvature $F_A\in {\mathrm{\Omega }}^2(B,\mathrm{Ad}(E))$ 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 $n\ge 5$ and:

  $\Vert F_A\Vert \le \sqrt{\frac{n-4}{2}}$

is unstable.

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

  $\Vert F_A\Vert \le \sqrt{\frac{n-4}{n-2}}$

is unstable.

• All non-flat $F$-Yang–Mills connections over $S^n$ with $n>4(d_{F^{\prime }}+1)$ 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 $0\le d_{F^{\prime }}\le \frac{1}{6}$, every non-flat $F$-Yang–Mills connection over the Cayley plane $F_4/\mathrm{Spin}(9)$ 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