Stable Yang–Mills connection

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. ${\displaystyle {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})}$, an affine vector space (not canonically) isomorphic to ${\displaystyle \Omega ^{1}(B,\operatorname {Ad} (E))}$, is the space of connections.^[1] These are under the adjoint representation ${\displaystyle \operatorname {Ad} }$ invariant ${\displaystyle {\mathfrak {g}}}$-valued (Lie algebra–valued) differential forms on ${\displaystyle E}$ and through pullback along smooth sections ${\displaystyle B\hookrightarrow E}$ differ by ${\displaystyle \operatorname {Ad} (E)}$-valued (vector bundle–valued) differential forms on ${\displaystyle B}$.

The Yang–Mills action functional is given by:^[2][3]

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

A Yang–Mills connection ${\displaystyle A\in {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})}$, hence which fulfills the Yang–Mills equations, is called stable if:^[4][5]

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

for every smooth family ${\displaystyle \alpha \colon (-\varepsilon ,\varepsilon )\rightarrow {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})}$ 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:^[2]

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

For a (weakly) stable or instable Yang–Mills connection ${\displaystyle A\in {\mathcal {A}}=\Omega _{\operatorname {Ad} }^{1}(E,{\mathfrak {g}})}$, its curvature ${\displaystyle F_{A}\in \Omega ^{2}(B,\operatorname {Ad} (E))}$ is called a (weakly) stable or instable Yang–Mills field.

• All weakly stable Yang–Mills connections on ${\displaystyle S^{n}}$ for ${\displaystyle n\geq 5}$ are flat.^[4][6][7][8] 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.^[7] 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.^[4][9]
• 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.^[4][10]

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.^[6] 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 ${\displaystyle K(\mathbb {Z} ,2)}$.^[11][12] 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):^[11][13][12]

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

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:^[6]

• 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.

• 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.

• Stable Yang–Mills–Higgs pair