geometry.

[**(Anti-)self-dual gauge fields on ${\mathbb C}P^2$ and ${\mathbb C}B_2$**]{}. Let us introduce a flat connection on the trivial vector bundle $M^4{\times}{\mathbb C}^4$ given by the formula $$\label{5.8}
{{\cal A}}= V^{-1}\diff V =: \begin{pmatrix}2b&-\sfrac{\ve}{2\La}\, \th^{\+}\\
\sfrac{1}{2\La} \th&B\end{pmatrix}$$ with $b\in u(1)$ and $B\in u(2)$ on $M^4\cong {\mathbb C}P^2$ or ${\mathbb C}B_2$, where from (\[4.4\]) we obtain $$\label{5.9}
 b=\frac{\ve}{4\g^2}(T^{\+}\diff T - \diff T^{\+} T)\und
B=\frac{1}{\g^2}(W\diff W -T\diff T^{\+}- \frac{\ve}{2}\diff T^{\+}T -
\frac{\ve}{2}\,T^{\+}\diff T )\ ,$$ $$\label{5.10}
\th=\frac{2\La}{\g^2}\, W\,\diff T= \begin{pmatrix}\th^{\2}\\
\th^1\end{pmatrix}=\frac{2\La}{\g}\,
\begin{pmatrix}\diff{\bar{y}}^{\2}\\ \diff y^1\end{pmatrix}-
\frac{2\ve\La}{\g^2(\g +1)}\, \begin{pmatrix}{\bar{y}}^{\2}\\
y^1\end{pmatrix}({\bar{y}}^{\1}\diff y^{1} + y^2\diff{\bar{y}}^{\2})\ .$$ Here, $\th^1$ and $\th^2$ are local orthonormal basis of (1,0)-forms on ${\mathbb C}P^2$ for $\ve =1$ and ${\mathbb C}B_2$ for $\ve =-1$. The real parameter $\La$ characterizes “size” of these cosets.

The flatness condition, $\diff{{\cal A}}+ {{\cal A}}\wedge{{\cal A}}=0$, leads to the following component equations $$\label{5.11}
 f^-:=\diff b=\frac{\ve}{8\La^2}\th^{\+}\wedge\th = -\frac{\ve}{8\La^2}
(\th^{1}\wedge\th^{\1}-\th^{2}\wedge\th^{\2})\ ,$$ $$\label{5.12}
 B:= B^{+}  - b\cdot{\bf 1}_2=\begin{pmatrix}a_+&-\bar b_+\\
b_+&-a_+\end{pmatrix} - b\cdot{\bf 1}_2\ ,$$ $$\label{5.13}
 F=\diff B + B\wedge B=\frac{\ve}{4\La^2}\,\th\wedge\th^{\+}=
-\frac{\ve}{4\La^2}\,
\begin{pmatrix}\th^2\wedge\th^{\2}&\th^{\1}\wedge\th^{\2}\\
-\th^1\wedge\th^{2}&-\th^{1}\wedge\th^{\1}\end{pmatrix}=:
F^{+}-f^-\cdot{\bf 1}_2\ ,$$ where $$\label{5.14}
 F^{+}=\diff B^{+} + B^{+}\wedge B^{+}=-\frac{\ve}{8\La^2}\,
\begin{pmatrix}\th^{1}\wedge\th^{\1}+\th^2\wedge\th^{\2}&2\th^{\1}
\wedge\th^{\2}\\-2\th^1\wedge\th^{2}&-\th^{1}\wedge\th^{\1}
-\th^2\wedge\th^{\2}\end{pmatrix}\ .$$ From (\[5.11\]) and (\[5.14\]) it follows that $*f^-=-f^-$ and $*F^{+}= F^{+}$, i.e. $b$ is an anti-self-dual $u(1)$-connection on a complex line bundle over 