F-Yang–Mills equations

In differential geometry, the

F

{\displaystyle F}

-Yang–Mills equations (or

F

{\displaystyle F}

-YM equations) are a generalization of the Yang–Mills equations. Its solutions are called

F

{\displaystyle F}

-Yang–Mills connections (or

F

{\displaystyle F}

-YM connections). Simple important cases of

F

{\displaystyle F}

-Yang–Mills connections include exponential Yang–Mills connections using the exponential function for

F

{\displaystyle F}

and

p

{\displaystyle p}

-Yang–Mills connections using

p

{\displaystyle p}

as exponent of a potence of the norm of the curvature form similar to the

p

{\displaystyle p}

-norm. Also often considered are Yang–Mills–Born–Infeld connections (or YMBI connections) with positive or negative sign in a function

F

{\displaystyle 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

:

R

0

+

→

R

0

+

{\displaystyle F\colon \mathbb {R} _{0}^{+}\rightarrow \mathbb {R} _{0}^{+}}

be a strictly increasing

C

2

{\displaystyle C^{2}}

function (hence with

F

′

>

0

{\displaystyle F'>0}

) and

F

(

0

)

=

0

{\displaystyle F(0)=0}

. Let:

d

F

:=

sup

t

≥

0

t

F

′

(

t

)

F

(

t

)

.

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

Since

F

{\displaystyle F}

is a

C

2

{\displaystyle C^{2}}

function, one can also consider the following constant:

d

F

′

=

sup

t

≥

0

t

F

″

(

t

)

F

′

(

t

)

.

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

Let

G

{\displaystyle G}

be a compact Lie group with Lie algebra

g

{\displaystyle {\mathfrak {g}}}

and

E

↠

B

{\displaystyle E\twoheadrightarrow B}

be a principal

G

{\displaystyle G}

-bundle with an orientable Riemannian manifold

B

{\displaystyle B}

having a metric

g

{\displaystyle g}

and a volume form

vol

g

{\displaystyle \operatorname {vol} _{g}}

. Let

Ad

⁡

(

E

)

:=

E

×

G

g

↠

B

{\displaystyle \operatorname {Ad} (E):=E\times _{G}{\mathfrak {g}}\twoheadrightarrow B}

be its adjoint bundle.