nLab stable Yang-Mills connection

Idea

A (weakly) stable Yang-Mills connection (or (weakly) stable YMH connection) is a Yang-Mills connection, around which the Yang-Mills action functional is positive or even strictly positively curved. Yang-Mills connections are critical points of the Yang-Mills action functional, where the first variational derivative vanishes. For (weakly) stable Yang-Mills connections, the second derivative is additionally required to be positive or even strictly positive.

Basics

Consider

• $G$ a Lie group and $\mathfrak{g}$ its Lie algebra,

• $X$ an orientable Riemannian manifold with Riemannian metric $g$ and volume form $vol_g$,

• $P\twoheadrightarrow X$ a principal $G$-bundle and $Ad(P)\coloneqq P\times_G\mathfrak{g}\twoheadrightarrow X$ its adjoint bundle,

• $A\in\Omega_{Ad}^1(P,\mathfrak{g})$ (affine space over $\Omega^1(X,Ad(P))$) a principal connection,

• $F_A\coloneqq\mathrm{d}A+\frac{1}{2}[A\wedge A]\in\Omega^2(X,Ad(P))$ its curvature. If $A$ is a Yang-Mills connection, $F_A$ is also called Yang-Mills field.

Definition

The Yang-Mills action functional (or YM action functional) is given by:

$$\operatorname{YM}\colon\Omega^1(X,\operatorname{Ad}(P))\rightarrow\mathbb{R}, \operatorname{YM}(A) \coloneqq\int_X\|F_A\|^2\mathrm{d}\operatorname{vol}_g.$$

$A$ is called a stable Yang-Mills connection (or stable YM connection) iff:

$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}\operatorname{YM}(\alpha(t))\vert_{t=0} \gt 0$$

for all smooth families $\alpha\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(X,\operatorname{Ad}(P))$ with $\alpha(0)=A$. It is called weakly stable if only $\geq 0$ holds. If it is not weakly stable, it is called unstable.

(Chiang 2013, Definition 3.1.7)

For comparison, the condition for a Yang-Mills connection (or YM connection) is:

$$\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{YM}(\alpha(t))\vert_{t=0} =0.$$

If $A$ is a (weakly) stable or unstable Yang-Mills connection, $F_A$ is also called (weakly) stable or unstable Yang-Mills field.

Theorem

(Formula for Yang–Mills stability) Let $\alpha\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(X,Ad(P))$ be a path with $A=\alpha(0)$, $B=\alpha'(0)$ and $C=\alpha''(0)$, then:

$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}YM(\alpha(t))\vert_{t=0} =\int_X\|\mathrm{d}_A B\|^2 +\langle F_A,[B\wedge B]\rangle +\langle\underbrace{\delta_A F_A}_{YM},C\rangle\mathrm{d}vol_g.$$

The Yang-Mills equations are $\delta_A F_A=0$ and therefore the formula simplifies for a Yang-Mills connection $A$. In this case it also becomes independent of $C$.

Proof

One has the following derivatives for the curvature:

$$\frac{\mathrm{d}}{\mathrm{d}t}F_{\alpha(t)} =\frac{\mathrm{d}}{\mathrm{d}t}\left(\mathrm{d}\alpha(t)+\frac{1}{2}[\alpha(t)\wedge\alpha(t)]\right) =\mathrm{d}\alpha'(t)+[\alpha(t)\wedge\alpha'(t)] =\mathrm{d}_{\alpha(t)}\alpha'(t);$$

$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}F_{\alpha(t)} =\mathrm{d}_{\alpha(t)}\alpha''(t) +[\alpha'(t)\wedge\alpha'(t)],$$

which combine together into the following second derivative for the Yang-Mills action functional:

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}t^2}YM(\alpha(t))\vert_{t=0} &=\int_X\left\|\frac{\mathrm{d}}{\mathrm{d}t}F_{\alpha(t)}\right\|^2 +\left\langle F_{\alpha(t)},\frac{\mathrm{d}^2}{\mathrm{d}t^2}F_{\alpha(t)}\right\rangle\mathrm{d}vol_g\vert_{t=0} \\ &=\int_X\left\|\mathrm{d}_{\alpha(t)}\alpha'(t)\right\|^2 +\left\langle F_{\alpha(t)},\mathrm{d}_{\alpha(t)}\alpha''(t) +[\alpha'(t)\wedge\alpha'(t)]\right\rangle\mathrm{d}vol_g\vert_{t=0} \\ &=\int_X\|\mathrm{d}_A B\|^2 +\langle F_A,\mathrm{d}_A C+[B\wedge B]\rangle. \end{aligned}$$

Since the first term is always positive, leaving it out directly implies:

Corollary

If $A\in\Omega^1(X,Ad(P))$ is a Yang-Mills connection with:

$$\int_X\langle F_A,[B\wedge B]\rangle\mathrm{d}\vol_g \gt 0\;\text{(or}\geq 0\text{)}$$

for all $B\in\Omega^1(X,Ad(P))$, then it is stable (or weakly stable).

Properties

Theorem

Every weakly stable Yang-Mills connection on $S^n$ for $n\geq 5$ is flat.

(Bourguignon & Lawson 81, Theorem A, Kobayashi, Ohnita & Takeuchi 86, Theorem 1.3., Kawai 86, Chiang 13, 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.

Theorem

If for a compact $n$-dimensional smooth submanifold of $\mathbb{R}^{n+1}$ a $\varepsilon\gt 0$ exists, so that:

$$\frac{2}{n-2}\varepsilon \lt\lambda_i \leq\varepsilon$$

at every point for all principal curvatures $\lambda_i$, then all weakly stable Yang-Mills connections on it are flat.

(Kawai 86)

This includes the previous theorem as a special case.

Theorem

Every weakly stable $SU(2)$-, $SU(3)$- or $U(2)$-Yang-Mills field on $S^4$ is either selfdual or anti-selfdual.

(Bourguignon & Lawson 81, Theorem B, Chiang 213, Theorem 3.1.10)

Theorem

All weakly stable Yang-Mills connections on a compact orientable homogeneous Riemannian $4$-manifold with structure group $SU(2)$ are either selfdual, antiselfdual or reduce to an abelian field.

(Bourguignon & Lawson 81, Theorem B’, Chiang 213, Theorem 3.1.11)

Theorem

For a connected compact Lie group with a simple Lie algebra, every weakly stable irreducible Yang-Mills field on $S^4$ is either selfdual or anti-selfdual.

(Ge & Xiao 26, Thrm. 1.3)

Yang-Mills-unstable manifolds

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-unstable (or YM-unstable). For example, the spheres $S^n$ are Yang–Mills-unstable for $n\geq 5$ because of the above result from James Simons. A Yang–Mills-unstable manifold always has a vanishing second Betti number.

(Kobayashi, Ohnita & Takeuchi 86, Theorem 2.17.)

Central for the proof is that the infinite complex projective space $\mathbb{C}P^\infty$ is the classifying space $\operatorname{BU}(1)$ as well as the Eilenberg-MacLane space $K(\mathbb {Z} ,2)$. Hence principal $\operatorname{U}(1)$-bundles over a Yang–Mills-unstable manifold $X$ (but even more generally every CW complex) are classified by its second cohomology (with integer coefficients):

$$\operatorname{Prin}_{\operatorname{U}(1)}(X) \cong[X,\operatorname{BU}(1)] =[X,K(\mathbb{Z},2)] =H^2(X,\mathbb{Z}).$$

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

Open problems about Yang-Mills-unstable manifolds include:

• Is a simply connected compact simple Lie group always Yang-Mills-unstable?

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

See also

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

References

• Jean-Pierre Bourguignon and H. Blaine Lawson Jr., Stability and Isolation Phenomena for Yang-Mills Fields (March 1981), Communications in Mathematical Physics 79, pp. 189–230, doi:10.1007/BF01942061
• S. Kobayashi, Y. Ohnita and M. Takeuchi, On instability of Yang-Mills connections (1986), Mathematische Zeitschrift 193, pp. 165–189, doi:10.1007/BF01174329
• Shigeo Kawai, A remark on the stability of Yang-Mills connections, Kodai Mathematical Journal 9, pp. 117–122, doi:10.2996/kmj/1138037154
• Yuan-Jen Chiang, Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields, ISBN 978-3034805339
• Jianquan Ge, Lixin Xiao, Weakly stable irreducible Yang-Mills fields over $S^4$ (2026) [arXiv:2603.17352]

See also:

• Wikipedia, stable Yang-Mills connection