Stable Yang–Mills–Higgs pair

In differential geometry and especially Yang–Mills theory, a (weakly) stable Yang–Mills–Higgs (YMH) pair is a Yang–Mills–Higgs pair around which the Yang–Mills–Higgs action functional is positively or even strictly positively curved. Yang–Mills–Higgs pairs are solutions of the Yang–Mills–Higgs equations following from them being local extrema of the curvature of both fields, hence critical points of the Yang–Mills-Higgs action functional, which are determined by a vanishing first derivative of a variation. (Weakly) stable Yang–Mills-Higgs pairs 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.

(Weakly) stable Yang–Mills–Higgs pairs are named after Yang Chen-Ning, Robert Mills and Peter Higgs.

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,^[1] 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–Higgs action functional is given by:^[2]

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

A Yang–Mills–Higgs pair ${\displaystyle A\in \Omega ^{1}(B,\operatorname {Ad} (E))}$ and ${\displaystyle \Phi \in \Gamma ^{\infty }(B,\operatorname {Ad} (E))}$, hence which fulfill the Yang–Mills–Higgs equations, is called stable if:^[3][4][5]

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

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

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

• Let ${\displaystyle (A,\Phi )}$ be a weakly stable Yang–Mills–Higgs pair on ${\displaystyle S^{n}}$, then the following claims hold:^[5]
  • If ${\displaystyle n=4}$, then ${\displaystyle A}$ is a Yang–Mills connection (${\displaystyle \mathrm {d} _{A}\star F_{A}=0}$) as well as ${\displaystyle \mathrm {d} _{A}\Phi =0}$ and ${\displaystyle \|\Phi \|=1}$.
  • If ${\displaystyle n\geq 5}$, then ${\displaystyle A}$ is flat (${\displaystyle F_{A}=0}$) as well as ${\displaystyle \mathrm {d} _{A}\Phi =0}$ and ${\displaystyle \|\Phi \|=1}$.

• Stable Yang–Mills connection

• stable Yang–Mills–Higgs pair at the nLab