Trace-normalized instanton number and self-dual sectors

On an overlap with $s_j=s_i{g}_{ij}$, [F3] gives $F_j={\mathrm{Ad}}_{g_{ij}^{-1}}{F}_i$. Since $B$ is invariant, $B(F_j\wedge F_j)=B(F_i\wedge F_i)$; these restrictions therefore define the global four-form ${\beta }_B$. The polynomial $P_B(Z)=B(Z,Z)\mathrm{/}(8{\pi }^2)$ is homogeneous, quadratic, and $G$-invariant, so its curvature evaluation [F4] is exactly ${\beta }_B$. It is closed because it has top degree on the four-manifold. To compare two supplied connections ${\omega }_0,{\omega }_1$, put ${\omega }_t=(1-t){\omega }_0+t{\omega }_1$ for $t\in [0,1]$. The reproduction and equivariance identities of [F2] are preserved, so each is a principal connection. In local sections put $a_i=A_{1,i}-A_{0,i}$ and $A_{t,i}=A_{0,i}+t{a}_i$; the inhomogeneous transition terms cancel in $a_i$, so $B(a_i\wedge F_{t,i})$ patches globally. Differentiating the curvature formula gives ${\mathrm{\partial }}_t{F}_{t,i}=D_{A_{t,i}}{a}_i$. Differentiating Ad-invariance gives $B([Z,X],Y)+B(X,[Z,Y])=0$; with the graded wedge rule this cancels the connection terms in $dB(a_i\wedge F_{t,i})=B(D_{A_{t,i}}{a}_i\wedge F_{t,i})-B(a_i\wedge D_{A_{t,i}}{F}_{t,i})$. The last term vanishes by Bianchi [F3]. Symmetry of $B$ and even degree of $F_t$ therefore give ${\mathrm{\partial }}_{t}B(F_t\wedge F_t)=2dB(a\wedge F_t)$. Integration in $t$ yields ${\beta }_{B,1}-{\beta }_{B,0}=d\zeta$, where $\zeta =(4{\pi }^2{)}^{-1}{\int }_0^{1}B(a\wedge F_t)dt$ is a global smooth three-form; its coefficients are polynomials in $t$. Thus the de Rham classes agree. Integration over compact boundaryless $X$ gives equal charges by [F10], whose choice hypothesis follows from [A1].

By step 1.1, $q_B$ is independent of the supplied connection on the fixed $G$-bundle. In the $SU(2)$ defining representation, the associated connection matrix is $A_i$ because ${\rho }_{\ast }$ is the identity inclusion, and its curvature matrix is ${\mathrm{\Omega }}_i=F_i$ by [F2, F3]. Put $C=i{\mathrm{\Omega }}_i\mathrm{/}(2\pi )=-{\mathrm{\Omega }}_i\mathrm{/}(2\pi i)$. Its entries have even form degree and commute, so the degree-four coefficient of $\det (I+C)$ is $\det C=\frac{1}{2}(({\mathrm{tr}}_{2}C{)}^2-{\mathrm{tr}}_2(C\wedge C))$. By [F5], ${\mathrm{tr}}_{2}C=0$, and $C\wedge C=-{\mathrm{\Omega }}_i\wedge {\mathrm{\Omega }}_i\mathrm{/}(4{\pi }^2)$; hence $c_2(\mathrm{\nabla })={\mathrm{tr}}_2({\mathrm{\Omega }}_i\wedge {\mathrm{\Omega }}_i)\mathrm{/}(8{\pi }^2)$ and ${\beta }_B=-c_2(\mathrm{\nabla })$ because $B=-{\mathrm{tr}}_2$. The associated connection is Hermitian by [F7], so [F6] sends this de Rham class to the negative realification of $c_2(E)\in H^4(X;\mathbb{Z})$. Evaluating by [F8] on the orientation class gives $q_B=-⟨c_2(E),[X]⟩\in \mathbb{Z}$. Full AC is used only through the stated characteristic-class comparison and integral-class construction.

By [F9], $F_i^{\pm }=(F_i\pm {\ast }_g{F}_i)\mathrm{/}2$ are the projections to the $\pm 1$ eigenspaces, so $F_i^{-}=0$ is equivalent to $F_i={\ast }_g{F}_i$, and $F_i^{+}=0$ is equivalent to $F_i=-{\ast }_g{F}_i$. The curvature transforms by the adjoint action on overlaps by [F3], while ${\ast }_g$ acts only on the form arguments, so each $F_i^{\pm }$ transforms by that same adjoint action. Thus $F^{\pm }$ are global adjoint-valued two-forms and the stated self-dual and ASD conditions do not depend on the local sections.