View source for F-Yang–Mills equations

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

== F-Yang–Mills action functional ==
Let <math>F\colon\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+</math> be a [[strictly increasing]] <math>C^2</math> function (hence with <math>F'>0</math>) and <math>F(0)=0</math>. Let:<ref name=":18">{{Cite arXiv |last=Wei |first=Shihshu Walter |date=2022-05-06 |title=On exponential Yang-Mills fields and p-Yang-Mills fields |class=math.DG |language=en |eprint=2205.03016}}</ref>

: <math>d_F
:=\sup_{t\geq 0}\frac{tF'(t)}{F(t)}.</math>

Since <math>F</math> is a <math>C^2</math> function, one can also consider the following constant:<ref name=":16">{{Cite arXiv |last1=Baba |first1=Kurando |last2=Shintani |first2=Kazuto |date=2023-01-11 |title=A Simons type condition for instability of F-Yang-Mills connections |class=math.DG |eprint=2301.04291 |language=en}}</ref>

: <math>d_{F'}
=\sup_{t\geq 0}\frac{tF''(t)}{F'(t)}.</math>

Let <math>G</math> be a [[Compact space|compact]] [[Lie group]] with [[Lie algebra]] <math>\mathfrak{g}</math> and <math>E\twoheadrightarrow B</math> be a [[Principal bundle|principal <math>G</math>-bundle]] with an [[Orientability|orientable]] [[Riemannian manifold]] <math>B</math> having a [[Metric (mathematics)|metric]] <math>g</math> and a [[volume form]] <math>\operatorname{vol}_g</math>. Let <math>\operatorname{Ad}(E)
:=E\times_G\mathfrak{g}\twoheadrightarrow B</math> be its [[adjoint bundle]]. <math>\Omega_{\operatorname{Ad}}^1(E,\mathfrak{g})
\cong\Omega^1(B,\operatorname{Ad}(E))</math> is the space of [[Connection (principal bundle)|connections]],<ref name=":62">{{Cite web |last=de los Ríos |first=Santiago Quintero |date=2020-12-16 |title=Connections on principal bundles |url=https://www.homotopico.com/assets/docs/notes/principal-connections.pdf |access-date=2024-11-09 |website=homotopico.com |at=Theorem 3.7 |language=en}}</ref> which are either under the [[adjoint representation]] <math>\operatorname{Ad}</math> invariant [[Lie algebra–valued differential form|Lie algebra–valued]] or [[vector bundle]]–valued [[Differential form|differential forms]]. Since the [[Hodge star operator]] <math>\star</math> is defined on the base manifold <math>B</math> as it requires the metric <math>g</math> and the volume form <math>\operatorname{vol}_g</math>, the second space is usually used.

The <math>F</math>-Yang–Mills action functional is given by:<ref name=":16" /><ref name=":6">{{Cite web |last=Baba |first=Kurando |date=2023-11-20 |title=On instability of F-Yang-Mills connections |url=https://www.rs.tus.ac.jp/kurando.baba/conference/record/2023/2023slide-baba.pdf |access-date=2024-11-02 |website=www.rs.tus.ac.jp |language=en}}</ref>

: <math>\operatorname{YM}_F\colon
\Omega^1(B,\operatorname{Ad}(E))\rightarrow\mathbb{R},
\operatorname{YM}_F(A)
:=\int_BF\left(\frac{1}{2}\|F_A\|^2\right)\mathrm{d}\operatorname{vol}_g.</math>

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

== F-Yang–Mills connections and equations ==
A connection <math>A\in\Omega^1(B,\operatorname{Ad}(E))</math> is called <math>F</math>''-Yang–Mills connection'', if it is a [[Critical point (mathematics)|critical point]] of the <math>F</math>-Yang–Mills action functional, hence if:

: <math>\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{YM}_F(A(t))\vert_{t=0}=0</math>

for every smooth family <math>A\colon
(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))</math> with <math>A(0)=A</math>. This is the case iff the <math>F</math>''-Yang–Mills equations'' are fulfilled:<ref name=":16" /><ref name=":6" />

: <math>\mathrm{d}_A\star\left(
F'\left(
\frac{1}{2}\|F_A\|^2
\right)F_A
\right)
=0.</math>

For a <math>F</math>-Yang–Mills connection <math>A\in\Omega^1(B,\operatorname{Ad}(E))</math>, its curvature <math>F_A\in\Omega^2(B,\operatorname{Ad}(E))</math> is called ''<math>F</math>-Yang–Mills field''.

A <math>F</math>-Yang–Mills connection/field with:<ref name=":18" /><ref name=":16" /><ref name=":6" />

* <math>F(t)=t</math> is just an ordinary Yang–Mills connection/field.
* <math>F(t)=\exp(t)</math> (or <math>F(t)=\exp(t)-1</math> for normalization) is called ''(normed) exponential Yang–Mills connection/field''. In this case, one has <math>d_{F'}=\infty</math>. The exponential and normed exponential Yang–Mills action functional are denoted with <math>\operatorname{YM}_\mathrm{e}</math> and <math>\operatorname{YM}_\mathrm{e}^0</math> respectively.<ref name=":17">{{Cite journal |last1=Matsura |first1=Fumiaki |last2=Urakawa |first2=Hajime |date=September 1995 |title=On exponential Yang-Mills connections |url=https://dx.doi.org/10.1016/0393-0440%2894%2900041-2 |journal=[[Journal of Geometry and Physics]] |language=en |volume=17 |issue=1 |pages=73–89|doi=10.1016/0393-0440(94)00041-2 }}</ref>
* <math>F(t)=\frac{1}{p}(2t)^{\frac{p}{2}}</math> is called ''<math>p</math>-Yang–Mills connection/field''. In this case, one has <math>d_{F'}=\frac{p}{2}-1</math>. Usual Yang–Mills connections/fields are exactly the <math>2</math>-Yang–Mills connections/fields. The <math>p</math>-Yang–Mills action functional is denoted with <math>\operatorname{YM}_p</math>.
* <math>F(t)=\sqrt{1-2t}-1</math> or <math>F(t)=\sqrt{1+2t}-1</math> is called ''Yang–Mills–Born–Infeld connection/field'' (or ''YMBI connection/field'') with negative or positive sign respectively. In these cases, one has <math>d_{F'}=\infty</math> and <math>d_{F'}=0</math> respectively. The Yang–Mills–Born–Infeld action functionals with negative and positive sign are denoted with <math>\operatorname{YMBI}^-</math> and <math>\operatorname{YMBI}^+</math> respectively. The Yang–Mills–Born–Infeld equations with positive sign are related to the [[minimal surface equation]]:
*: <math>\mathrm{d}_A\frac{\star F_A}{\sqrt{1+\|F_A\|^2}}
=0.</math>

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

: <math>\frac{\mathrm{d}^2}{\mathrm{d}t^2}\operatorname{YM}_F(A(t))\vert_{t=0}>0</math>

for every smooth family <math>A\colon
(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))</math> with <math>A(0)=A</math>. It is called ''weakly stable'' if only <math>\geq 0</math> holds. A <math>F</math>-Yang–Mills connection, which is not weakly stable, is called ''unstable''.<ref name=":6" /> For a (weakly) stable or unstable <math>F</math>-Yang–Mills connection <math>A\in\Omega^1(B,\operatorname{Ad}(E))</math>, its curvature <math>F_A\in\Omega^2(B,\operatorname{Ad}(E))</math> is furthermore called a ''(weakly) stable'' or ''unstable <math>F</math>-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.<ref name=":17" />
* Every non-flat exponential Yang–Mills connection over <math>S^n</math> with <math>n\geq 5</math> and:
*: <math>\|F_A\|
\leq\sqrt{\frac{n-4}{2}}</math>

: is unstable.<ref name=":16" /><ref name=":6" />

* Every non-flat Yang–Mills–Born–Infeld connection with negative sign over <math>S^n</math> with <math>n\geq 5</math> and:
*: <math>\|F_A\|
\leq\sqrt{\frac{n-4}{n-2}}</math>

: is unstable.<ref name=":16" />

* All non-flat <math>F</math>-Yang–Mills connections over <math>S^n</math> with <math>n>4(d_{F'}+1)</math> are unstable.<ref name=":16" /><ref name=":6" /> This result includes the following special cases:
** All non-flat Yang–Mills connections with positive sign over <math>S^n</math> with <math>n>4</math> are unstable.<ref name=":3">{{Cite journal |last1=Bourguignon |first1=Jean-Pierre |author-link=Jean-Pierre Bourguignon |last2=Lawson, Jr. |first2=H. Blaine |author-link2=H. Blaine Lawson, Jr. |date=March 1981 |title=Stability and Isolation Phenomena for Yang-Mills Fields |url=https://link.springer.com/article/10.1007/BF01942061 |journal=[[Communications in Mathematical Physics]] |language=en |volume=79 |issue=2 |pages=189–230 |doi=10.1007/BF01942061}}</ref><ref name=":0">{{Cite journal |last1=Kobayashi |first1=S. |last2=Ohnita |first2=Y. |last3=Takeuchi |first3=M. |year=1986 |title=On instability of Yang-Mills connections |url=https://www.digizeitschriften.de/download/pdf/266833020_0193/log19.pdf |journal=[[Mathematische Zeitschrift]] |publisher=[[Springer Science+Business Media|Springer]] |volume=193 |issue=2 |pages=165–189 |doi=10.1007/BF01174329}}</ref><ref>Chiang 2013, Theorem 3.1.9</ref> [[Jim Simons|James Simons]] presented this result without written publication during a symposium on "Minimal Submanifolds and Geodesics" in [[Tokyo]] in September 1977.
** All non-flat <math>p</math>-Yang–Mills connections over <math>S^n</math> with <math>n>2p</math> are unstable.
** All non-flat Yang–Mills–Born–Infeld connections with positive sign over <math>S^n</math> with <math>n>4</math> are unstable.
* For <math>0\leq d_{F'}\leq\frac{1}{6}</math>, every non-flat <math>F</math>-Yang–Mills connection over the [[Cayley plane]] <math>F_4/\operatorname{Spin}(9)</math> is unstable.<ref name=":6" />

== Literature ==

* {{cite book |last=Chiang |first=Yuan-Jen |url=https://link.springer.com/book/10.1007/978-3-0348-0534-6 |title=Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields |series=Frontiers in Mathematics |date=2013-06-18 |publisher=[[Birkhäuser]] |isbn=978-3034805339 |location= |doi=10.1007/978-3-0348-0534-6 |language=en}}

== See also ==

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

==References==
{{Reflist}}

== External links ==

* [[nlab:F-Yang-Mills+equation|F-Yang-Mills equation]] at the [[NLab|''n''Lab]]

{{DEFAULTSORT:F-Yang-Mills equations}}
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