Stable Yang–Mills connection

Let $G$ be a compact Lie group with Lie algebra $\backslash mathfrak\{g\}$ and $E\backslash twoheadrightarrow\; B$ be a principal $G$-bundle with a compact orientable Riemannian manifold $B$ having a metric $g$ and a volume form $\backslash operatorname\{vol\}\_g$. Let $\backslash operatorname\{Ad\}(E)$

=E\times_G\mathfrak{g} be its adjoint bundle. $\backslash Omega\_\{\backslash operatorname\{Ad\}\}^1(E,\backslash mathfrak\{g\})$

\cong\Omega^1(B,\operatorname{Ad}(E)) is the space of connections,{{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}} which are either under the adjoint representation $\backslash operatorname\{Ad\}$ invariant Lie algebra–valued or vector bundle–valued differential forms. Since the Hodge star operator $\backslash star$ is defined on the base manifold $B$ as it requires the metric $g$ and the volume form $\backslash operatorname\{vol\}\_g$, the second space is usually used.

The Yang–Mills action functional is given by:{{cite web |title=Lecture 3: The Yang–Mills equations |url=https://empg.maths.ed.ac.uk/Activities/GT/Lect3.pdf |access-date=2024-11-24 |website=empg.maths.ed.ac.uk |language=en}}{{cite web |author=David Tong |title=Yang-Mills Theory |url=https://www.damtp.cam.ac.uk/user/tong/gaugetheory/2ym.pdf |access-date=2024-11-24 |website=www.damtp.cam.ac.uk |language=en}}

: $\backslash operatorname\{YM\}\backslash colon$

\Omega^1(B,\operatorname{Ad}(E))\rightarrow\mathbb{R},

\operatorname{YM}(A)

:=\int_B\|F_A\|^2\mathrm{d}\operatorname{vol}_g.

A Yang–Mills connection $A\backslash in\backslash Omega^1(B,\backslash operatorname\{Ad\}(E))$, hence which fulfills the Yang–Mills equations, is called stable if:{{Cite journal |last1=Bourguignon |first1=Jean-Pierre |author-link1=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|url-access=subscription }}Chiang 2013, Definition 3.1.7

: $\backslash frac\{\backslash mathrm\{d\}^2\}\{\backslash mathrm\{d\}t^2\}\backslash operatorname\{YM\}(\backslash alpha(t))\backslash vert\_\{t=0\}>0$

for every smooth family $\backslash alpha\backslash colon$

(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E)) with $\backslash alpha(0)=A$. It is called weakly stable if only $\backslash geq\; 0$ holds. A Yang–Mills connection, which is not weakly stable, is called instable. For comparison, the condition to be a Yang–Mills connection is:

: $\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\backslash operatorname\{YM\}(\backslash alpha(t))\backslash vert\_\{t=0\}=0.$

For a (weakly) stable or instable Yang–Mills connection $A\backslash in\backslash Omega^1(B,\backslash operatorname\{Ad\}(E))$, its curvature $F\_A\backslash in\backslash Omega^1(B,\backslash operatorname\{Ad\}(E))$ is called a (weakly) stable or instable Yang–Mills field.

• All weakly stable Yang–Mills connections on $S^n$ for $n\backslash geq\; 5$ are flat.{{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 |volume=193 |issue=2 |pages=165–189 |doi=10.1007/BF01174329}}{{Cite journal |last=Kawai |first=Shigeo |date=1986 |title=A remark on the stability of Yang-Mills connections |journal=Kodai Mathematical Journal |language=en |volume=9 |issue=1 |pages=117–122 |doi=10.2996/kmj/1138037154}}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.
• If for a compact $n$-dimensional smooth submanifold in $\backslash mathbb\{R\}^\{n+1\}$ an $\backslash varepsilon\backslash geq\; 0$ exists so that:
• : $\backslash frac\{2\}\{n-2\}\backslash varepsilon$

<\lambda_i

\leq\varepsilon

: for all principal curvatures $\backslash lambda\_1,\backslash ldots,\backslash lambda\_n$, then all weakly stable Yang–Mills connections on it are flat. As the inequality shows, the result can only be applied for $n\backslash geq\; 5$, for which it includes the previous result as a special case.

• Every weakly stable Yang–Mills field over $S^4$ with gauge group $\backslash operatorname\{SU\}(2)$, $\backslash operatorname\{SU\}(3)$, or $\backslash operatorname\{U\}(2)$ is either anti self-dual or self-dual.Chiang 2013, Theorem 3.1.10
• Every weakly stable Yang–Mills field over a compact orientable homogenous Riemannian $4$-manifold with gauge group $\backslash operatorname\{SU\}(2)$ is either anti self-dual, self-dual or reduces to an abelian field.Chiang 2013, Theorem 3.1.11

A compact Riemannian manifold, for which no principal bundle over it (with a compact Lie group as structure group) has a stable Yang–Mills connection is called Yang–Mills-instable (or YM-instable). For example, the spheres $S^n$ are Yang–Mills-instable for $n\backslash geq\; 5$ because of the above result from James Simons. A Yang–Mills-instable manifold always has a vanishing second Betti number. Central for the proof is that the infinite complex projective space $\backslash mathbb\{C\}P^\backslash infty$ is the classifying space $\backslash operatorname\{BU\}(1)$ as well as the Eilenberg–MacLane space $K(\backslash mathbb\{Z\},2)$.{{cite web |author=Ralph L. Cohen |date=January 1998 |title=The Topology of Fiber Bundles |url=https://math.stanford.edu/~ralph/fiber.pdf |access-date=2024-10-28 |publisher=Stanford University |pages=56-57, Example after Corollary 2.11, Theorem 2.12 and Theorem 2.13 |language=en |type=Lecture Notes}}{{Cite book |last=Hatcher |first=Allen |author-link=Allen Hatcher |url=https://pi.math.cornell.edu/~hatcher/AT/AT+.pdf |title=Algebraic Topology |publisher=Cambridge University Press |year=2002 |location=Cambridge |at=Example 4.50 and Theorem 4.57 |language=en}} Hence principal $\backslash operatorname\{U\}(1)$-bundles over a Yang–Mills-instable manifold $X$ (but even more generally every CW complex) are classified by its second cohomology (with integer coefficients):{{Cite web |last=Mitchell |first=Stephen A. |date=June 2011 |title=Notes on principal bundles and classifying spaces |url=https://sites.math.washington.edu/~mitchell/Notes/prin.pdf |access-date=2024-10-27 |at=Theorem 7.4 |language=en}}

: $\backslash operatorname\{Prin\}\_\{\backslash operatorname\{U\}(1)\}\backslash left(X\backslash right)$

=[X,\operatorname{BU}(1)]

=[X,K(\mathbb{Z},2)]

=H^2(X,\mathbb{Z}).

On a non-trivial principal $\backslash operatorname\{U\}(1)$-bundles over $X$, which exists for a non-trivial second cohomology, one could construct a stable Yang–Mills connection.

Open problems about Yang-Mills-instable manifolds include:

• Is a simply connected compact simple Lie group always Yang-Mills-instable?
• Is a Yang-Mills-instable simply connected compact Riemannian manifold always harmonically instable? Since $S^n\backslash times\; S^1$ for $n\backslash geq\; 5$ is Yang-Mills-instable, but not harmonically instable, the condition to be simply connected cannot be dropped.

• {{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 |publisher=Birkhäuser |date=2013-06-18 |isbn=978-3034805339 |location= |doi=10.1007/978-3-0348-0534-6 |language=en}}

• Stable Yang–Mills–Higgs pair

• stable Yang–Mills connection at the nLab

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Category:Differential geometry

Category:Gauge theories