回忆, $\alpha$-杨米尔斯–希格斯场的方程为方程组
$$
\Delta a_j^\beta=2\left( -a_i^\gamma \partial_ia_j^\delta g_{\gamma\delta}^\beta+\Upsilon h_{ab}(u)u_{|j}^a\lambda_\beta^b(u)+F_{ij}^\gamma a_i^\delta g_{\beta\delta}^\gamma \right),
$$
与
$$
\mathrm{div}(\Upsilon h_{ab}u_{|i}^a)=\Upsilon h_{ad}(u)u_{|i}^a\left( \Gamma_{bc}^d(u)\partial_iu^c+A_{bi}^d(u) \right)+\mu(u)\cdot\left[ \nabla_{\partial_{f^b}}\mu \right](u),
$$
其中$\Upsilon=\alpha(1+\lvert \nabla_Au \rvert^2)^{\alpha-1}=\alpha\left(1+h_{ab}(u)(\partial_iu^a+a_i^\beta\lambda_\beta^a(u))(\partial_iu^b+a_i^\gamma\lambda_\gamma^b(u))\right)^{\alpha-1}$, $u_{|i}^a=\partial_iu^a+a_i^\beta \lambda_\beta^a(u)$, $F_{ij}^\gamma=(\partial_ia_j^\gamma-\partial_ja_i^\gamma)+2a_i^\beta a_j^\delta g_{\beta\delta}^\gamma$, $A_{bi}^d(u)=a_i^\beta\left(\partial_{f^b}\lambda_\beta^d(u)+\lambda_\beta^c(u)\Gamma_{bc}^d(u)\right)$, 而$h$, $\Gamma$, $\lambda$, $\mu$ 是 $u$的光滑函数. 按照定义 $[v_\beta,v_\gamma]=g_{\beta\gamma}^\delta v_\delta$, $\left\{ g_{\beta\gamma}^\delta \right\}$ 称为李代数的结构常数.
现在, 注意到
$$
\mathrm{div}(\Upsilon h_{ab}u_{|i}^a)=\Upsilon h_{ab}(u)\partial_i u_{|i}^a+\Upsilon\partial_ch_{ab}u_{|i}^a\partial_iu^c+\partial_i\Upsilon h_{ab}(u)u_{|i}^a
\mathpunct{:}=\o{1}+\o{2}+\o{3}.
$$
为方便起见, 又令
$$
\begin{aligned}
\o{4}&\mathpunct{:}=\Upsilon h_{ad}u_{|i}^a\left( \Gamma_{bc}^d\partial_iu^c+A_{bi}^d \right),\\
\o{5}&\mathpunct{:}=\mu\cdot\nabla_{\partial_{f^b}}\mu.
\end{aligned}
$$
则直接计算得到
$$
\begin{aligned}
\o{1}&=\Upsilon h_{ab}\Delta u^a+\Upsilon h_{ab}\partial_i(a_i^\beta\lambda_\beta^a),\\
\o{2}&=\Upsilon \partial_ch_{ab}\nabla u^a\cdot\nabla u^c+\Upsilon\partial_ch_{ab}\partial_iu^ca_i^\beta\lambda_\beta^a,\\
\o{3}&=\frac{2(\alpha-1)\Upsilon}{1+\lvert \nabla_Au \rvert^2}h_{ab}\left( \partial_ju^a+a_j^\beta\lambda_\beta^a \right)
h_{cd}\left( \partial_iu^c+a_i^\beta\lambda_\beta^c \right)
\left( \partial_{ij}u^d+\partial_i(a_j^\beta \lambda_\beta^d(u)) \right)\\
&\qquad+\frac{(\alpha-1)\Upsilon}{1+\lvert \nabla_Au \rvert^2}h_{ab}u_{|j}^a\partial_eh_{cd}\partial_ju^eu_{|i}^cu_{|i}^d,\\
\o{4}&=\Upsilon h_{ad}\Gamma_{bc}^d\partial_iu^a\partial_iu^c+\Upsilon h_{ad}a_i^{\beta}\lambda_\beta^a\Gamma_{bc}^d\partial_iu^c
+\Upsilon h_{ad}(\partial_iu^a+a_i^\beta\lambda_\beta^a)A_{bi}^d.
\end{aligned}
$$
注意到$h$,$\Gamma$, $\lambda$, $\mu$, 作为$u$的函数它们本身及其导数都是有界的(因为靶流形是紧致的), 如下, 我们将用$\#$来表示系数为有界函数的乘积\footnote{其上界依赖于$\lVert \lambda \rVert_{1,\infty}$, $\lVert h \rVert_{1,\infty}$, $\alpha-1$, $\lVert \Gamma \rVert_{\infty}$.}. 则
$$
\begin{aligned}
\o{1}&=\Upsilon h_{ab}\Delta u^a+\Upsilon h_{ab}\left( \nabla a+a\#\nabla u \right),\\
\o{2}&=\Upsilon \partial_ch_{ab}\nabla u^a\cdot \nabla u^c+\Upsilon a\#\nabla u,\\
\o{3}&=2(\alpha-1)\Upsilon h_{ab} \frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^cu_{|i}^d}\left( \partial_{ij}u^d+\nabla a+a\#\nabla u \right)\\
&\quad+(\alpha-1)\Upsilon h_{ab}\frac{\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\lvert \nabla_Au \rvert^2}u_{|j}^a\partial_ju^e\\
&=2(\alpha-1)\Upsilon h_{ab} \frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^cu_{|i}^d}\partial_{ij}u^d+(\alpha-1)\Upsilon h_{ab}\frac{\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\lvert \nabla_Au \rvert^2}u_{|j}^a\partial_ju^e+\Upsilon h_{ab}\left( \nabla a+a\#\nabla u +\nabla u\#\nabla u\right)\\
\o{4}&=\Upsilon h_{ad}\Gamma_{bc}^d\nabla u^a\cdot\nabla u^c+\Upsilon h_{ad}(a\#\nabla u+(\nabla u+a)\# a),\\
\o{5}&=\mu\cdot\nabla\mu
\end{aligned}
$$
关于$u$的方程可以化为
$$
\Delta u^a+2(\alpha-1) \frac{h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^c u_{|i}^d}\partial_{ij}u^d
=(\alpha-1)\frac{\partial_eh_{cd}u_{|i}^cu_{|i}^d}{1+\lvert \nabla_Au \rvert^2}u_{|j}^a\partial_ju^e-\Gamma_{bc}^a\nabla u^b\cdot \nabla u^c+\nabla a+a\#\nabla u+a\#a+\mu\cdot\nabla\mu.
$$
即
$$
\Delta u^a+(\alpha-1) \frac{2h_{cd}u_{|i}^au_{|j}^c}{1+h_{cd}u_{|i}^c u_{|i}^d}\partial_{ij}u^d=\nabla u\#\nabla u+\nabla a+a\#\nabla u+a\#a+\mu\cdot\nabla\mu.
$$
完全类似地, 我们可以计算关于联络$a$的方程.
$$
\Delta a=a\#\nabla a+\Upsilon\#(\nabla u+a)+(\nabla a+a\#a)\#a.
$$
即
$$
\Delta a=a\#\nabla a+\Upsilon\#(\nabla u+a)+a\#a\#a
$$
这里, $\#$还依赖于李代数的结构常数的$L^\infty$模.