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In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills (YM) connection is a Yang–Mills connection around which the Yang–Mills action functional is positively or even strictly positively curved. Yang–Mills connections are solutions of the Yang–Mills equations following from them being local extrema of the curvature, hence critical points of the Yang–Mills action functional, which are determined by a vanishing first derivative of a variation. (Weakly) stable Yang–Mills connections furthermore have a positive or even strictly positive curved neighborhood and hence are determined by a positive or even strictly positive second derivative of a variation.

Definition

Let ${\displaystyle G}$ be a compact Lie group with Lie algebra ${\displaystyle {\mathfrak {g}}}$ and ${\displaystyle E\twoheadrightarrow B}$ be a principal ${\displaystyle G}$-bundle with a compact orientable Riemannian manifold ${\displaystyle B}$ having a metric ${\displaystyle g}$ and a volume form ${\displaystyle \operatorname {vol} _{g}}$. Let ${\displaystyle \operatorname {Ad} (E)=E\times _{G}{\mathfrak {g}}}$ be its adjoint bundle. ${\displaystyle \Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})\cong \Omega ^{1}(B,\operatorname {Ad} (E))}$ is the space of connections, which are either under the adjoint representation ${\displaystyle \operatorname {Ad} }$ invariant Lie algebra–valued or vector bundle–valued differential forms. Since the Hodge star operator ${\displaystyle \star }$ is defined on the base manifold ${\displaystyle B}$ as it requires the metric ${\displaystyle g}$ and the volume form ${\displaystyle \operatorname {vol} _{g}}$, the second space is usually used.

The Yang–Mills action functional is given by:

${\displaystyle \operatorname {YM} \colon \Omega ^{1}(B,\operatorname {Ad} (E))\rightarrow \mathbb {R} ,\operatorname {YM} (A):=\int _{B}\|F_{A}\|^{2}\mathrm {d} \operatorname {vol} _{g}.}$

A Yang–Mills connection ${\displaystyle A\in \Omega ^{1}(B,\operatorname {Ad} (E))}$, hence which fulfills the Yang–Mills equations, is called stable if:

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

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

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

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

Properties

• All weakly stable Yang–Mills connections on ${\displaystyle S^{n}}$ for ${\displaystyle n\geq 5}$ are flat. 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 ${\displaystyle n}$-dimensional smooth submanifold in ${\displaystyle \mathbb {R} ^{n+1}}$ an ${\displaystyle \varepsilon \geq 0}$ exists so that:

  ${\displaystyle {\frac {2}{n-2}}\varepsilon <\lambda _{i}\leq \varepsilon }$

for all principal curvatures ${\displaystyle \lambda _{1},\ldots ,\lambda _{n}}$, then all weakly stable Yang–Mills connections on it are flat. As the inequality shows, the result can only be applied for ${\displaystyle n\geq 5}$, for which it includes the previous result as a special case.

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

Yang–Mills-instable 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-instable (or YM-instable). For example, the spheres ${\displaystyle S^{n}}$ are Yang–Mills-instable for ${\displaystyle n\geq 5}$ 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 ${\displaystyle \mathbb {C} P^{\infty }}$ is the classifying space ${\displaystyle \operatorname {BU} (1)}$ as well as the Eilenberg–MacLane space ${\displaystyle K(\mathbb {Z} ,2)}$. Hence principal ${\displaystyle \operatorname {U} (1)}$-bundles over a Yang–Mills-instable manifold ${\displaystyle X}$ (but even more generally every CW complex) are classified by its second cohomology (with integer coefficients):

${\displaystyle \operatorname {Prin} _{\operatorname {U} (1)}\left(X\right)===H^{2}(X,\mathbb {Z} ).}$

On a non-trivial principal ${\displaystyle \operatorname {U} (1)}$-bundles over ${\displaystyle X}$, 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 ${\displaystyle S^{n}\times S^{1}}$ for ${\displaystyle n\geq 5}$ is Yang-Mills-instable, but not harmonically instable, the condition to be simply connected cannot be dropped.

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

• Stable Yang–Mills–Higgs pair

References

1. de los Ríos, Santiago Quintero (2020-12-16). "Connections on principal bundles" (PDF). homotopico.com. Theorem 3.7. Retrieved 2024-11-09.
2. ^ "Lecture 3: The Yang–Mills equations" (PDF). empg.maths.ed.ac.uk. Retrieved 2024-11-24.
3. David Tong. "Yang-Mills Theory" (PDF). www.damtp.cam.ac.uk. Retrieved 2024-11-24.
4. ^ Bourguignon, Jean-Pierre; Lawson, Jr., H. Blaine (March 1981). "Stability and Isolation Phenomena for Yang-Mills Fields". Communications in Mathematical Physics. 79 (2): 189–230. doi:10.1007/BF01942061.
5. Chiang 2013, Definition 3.1.7
6. ^ Kobayashi, S.; Ohnita, Y.; Takeuchi, M. (1986). "On instability of Yang-Mills connections" (PDF). Mathematische Zeitschrift. 193 (2). Springer: 165–189. doi:10.1007/BF01174329.
7. ^ Kawai, Shigeo (1986). "A remark on the stability of Yang-Mills connections". Kodai Mathematical Journal. 9 (1): 117–122. doi:10.2996/kmj/1138037154.
8. Chiang 2013, Theorem 3.1.9
9. Chiang 2013, Theorem 3.1.10
10. Chiang 2013, Theorem 3.1.11
11. ^ Ralph L. Cohen (January 1998). "The Topology of Fiber Bundles" (PDF) (Lecture Notes). Stanford University. pp. 56–57, Example after Corollary 2.11, Theorem 2.12 and Theorem 2.13. Retrieved 2024-10-28.
12. ^ Hatcher, Allen (2002). Algebraic Topology (PDF). Cambridge: Cambridge University Press. Example 4.50 and Theorem 4.57.
13. Mitchell, Stephen A. (June 2011). "Notes on principal bundles and classifying spaces" (PDF). Theorem 7.4. Retrieved 2024-10-27.

External links

• stable Yang–Mills connection at the nLab

• Differential geometry
• Gauge theories