which provide a local basis of orthonormal (1,0)-forms w.r.t. $J$. Then one can introduce forms $$\label{3.5}
\o^1:=\frac{1}{(1{+}\zeta{\bar{\zeta}})^\frac{1}{2}}(\th^1{-}\zeta\th^{\2})\ ,
\quad
\o^2:=\frac{1}{(1{+}\zeta{\bar{\zeta}})^\frac{1}{2}}(\th^2{+}\zeta\th^{\1})
\and
\o^3:=\frac{1}{1{+}\zeta{\bar{\zeta}}}(\diff\zeta{-}\Gamma_+^iL_i^\zeta)\ ,$$ which may serve as the definition of an almost complex structure ${{\cal J}}$ on ${{\cal T}}(M^4)$ such that $$\label{3.6}
{{\cal J}}\,\o^i =\im\,\o^i\quad{\rm for}\quad i=1,2,3\ .$$ Here $\Gamma_+=(\Gamma_+^i)$ is the self-dual part of the Levi-Civita connection on $M^4$ and $L_i^{\zeta}$ are holomorphic components of vector fields $L_i=L_i^{\zeta}\pa_{\zeta}{+}L_i^{{\bar{\zeta}}}\pa_{{\bar{\zeta}}}$ on fibres ${\mathbb C}P^1_x\hra {{\cal T}}(M^4)$ which give a realization of the generators of the group SU(2) acting on ${\mathbb C}P^1{=}$SU(2)/U(1). One can take e.g. $L_1-\im L_2=-2\im$, $L_1+\im L_2=-2\im\zeta^2$ and $L_3=-2\im\zeta$. Note that the forms (\[3.5\]) extend to a basis of globally defined forms on ${{\cal T}}(M^4)$ of type (1,0) w.r.t. ${{\cal J}}$. That is why our discussion does not depend on the choice of local coordinates, forms etc.

[**From ASDYM on $M^4$ to HYM on ${{\cal T}}(M^4)$**]{}. For the curvature $F=\diff A + A\wedge A$ of the vector bundle $E\to M^4$ we have $$\label{3.7}
F=\sfrac{1}{2}\,(F+*F) + \sfrac{1}{2}\,(F-*F)=:F^++F^-\ ,$$ where in the basis (\[3.4\]) of local forms $$\label{3.8}
F^+=\sfrac{1}{2}\,(F_{1\1}+F_{2\2})(\th^{1\1}+\th^{2\2})+F_{12}\th^{12}
+F_{\1\2}\th^{\1\2}\ ,$$ $$\label{3.9}
F^-=\sfrac{1}{2}\,(F_{1\1}-F_{2\2})(\th^{1\1}-\th^{2\2})+F_{1\2}\th^{1\2}
+F_{2\1}\th^{2\1}\ ,$$ with $\th^{1\1}:=\th^{1}\wedge\th^{\1},\ \th^{12}:=\th^{1}\wedge\th^{2}$ etc. Here $F^+$ and $F^-$ are self-dual and anti-self-dual parts of the curvature 