gauge fields**]{}. For $M^4=S^4$ or $B_4$, let us consider a flat connection ${{\cal A}}$ on the trivial vector bundle $M^4\times{\mathbb C}^4\to M^4$ given by the one-form $$\label{4.8}
{{\cal A}}= Q^{-1}\diff Q =:
\begin{pmatrix}A^- &-\ve\p\\ \p^\+ & A^+\end{pmatrix} \ ,$$ where from (\[4.4\]) we obtain $$\label{4.9}
A^-= \sfrac{\ve}{f}\bar\h_{\m\n}x^\m\diff x^\n =:
\begin{pmatrix}\a_- &-\bar\b_-\\ \b_- & -\a_-\end{pmatrix}\in su(2) \ ,$$ $$\label{4.10}
A^+= \sfrac{\ve}{f}\h_{\m\n}x^\m\diff x^\n =:
\begin{pmatrix}\a_+ &-\bar\b_+\\ \b_+ & -\a_+\end{pmatrix}\in su(2) \ ,$$ $$\label{4.11}
\p = \frac{1}{f}\diff x = -\frac{\im}{f}
\begin{pmatrix}\diff x^3{+}\im\diff x^4&  \diff x^1{-}\im\diff x^2\\
\diff x^1{+}\im\diff x^2&  -(\diff x^3{-}\im\diff x^4)\end{pmatrix}
=-\frac{\im}{f}\begin{pmatrix}\diff z&\diff\bar y\\ \diff y&-\diff\bar z
\end{pmatrix}=:-\frac{\im}{2\Lambda}\begin{pmatrix}\th^2&\th^{\1}\\
\th^1&-\th^{\2}\end{pmatrix}\ ,$$ with $$\label{4.12}
\a_-= \sfrac{\ve}{2f}({\bar{y}}\,\diff y + {\bar{z}}\,\diff z - y\,\diff {\bar{y}}-
z\,\diff {\bar{z}})\ ,\quad \b_-= \sfrac{\ve}{f}(y\,\diff z -z \,\diff y)\ ,$$ $$\label{4.13}
\a_+= \sfrac{\ve}{2f}({\bar{y}}\,\diff y + z\,\diff {\bar{z}}- y\,\diff {\bar{y}}-
{\bar{z}}\,\diff z)\ ,\quad \b_+= \sfrac{\ve}{f}(y\,\diff {\bar{z}}-{\bar{z}}\,\diff y)\ ,$$ $$\label{4.14}
\th^1:=\frac{2\La\diff y}{1+\ve r^2}\ ,\quad
\th^2:=\frac{2\La\diff z}{1+\ve r^2}\und
\th^{\1}:=\frac{2\La\diff {\bar{y}}}{1+\ve r^2}\ ,\quad
\th^{\2}:=\frac{2\La\diff {\bar{z}}}{1+\ve r^2}\ .$$ Here, the bar denotes complex conjugation. Note that the real parameter $\La$ can be identified with the “radius” of $M^4=S^4$ or $B_4$.

The connection (\[4.8\]) is flat, i.e. $$\label{4.15}
{{\cal F}}= \diff{{\cal A}}+ {{\cal A}}\wedge{{\cal A}}= \begin{pmatrix}
F^--\ve\,\p\wedge\p^\+&-\ve (\diff\p + \p\wedge A^+ + A^-\wedge\p)\\
\diff\p^\+ + A^+\wedge\p^\+ + 