Translation action with a unit Faddeev–Popov Jacobian

The set $SU(2)$ is closed and bounded in the finite-dimensional space $M_2(\mathbb{C})$, hence compact; the matrices $\mathrm{diag}(i,-i)$ and $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$ lie in $SU(2)$ and do not commute. The projection $P\to \{\star \}$ with right multiplication is a smooth principal bundle, $VP=TP$, and $H=0$ is a smooth right-invariant complement. For $Z\in \mathfrak{s}\mathfrak{u}(2)$, the defining one-parameter subgroup $t\mapsto {\exp }_G(tZ)$ has initial velocity $Z$, so differentiating $q{\exp }_G(tZ)$ gives the fundamental vector $Z_P(q)=qZ$ and ${\omega }_q(Z_P(q))=Z$; also $\ker \omega =0=H$ and ${\omega }_{qh}(Vh)=h^{-1}{\omega }_q(V)h$. Thus $\omega$ is a principal connection. The specified $s_0$ and $U$ satisfy the slice-data requirement. The regulator group is independent of the ambient full gauge group, as required by the Definition. Translation is a smooth right action because $((u,v)\cdot t)\cdot h=(u+t+h,v)=(u,v)\cdot (t+h)$. The zero set of $\chi$ is exactly $S$, and $d{\chi }_{(0,v)}(\delta u,\delta v)=\delta u$ has kernel $T_{(0,v)}S=\{(0,\delta v)\}$.

The orbit derivative sends $c\in T_0\mathbb{R}\cong \mathbb{R}$ to $(c,0)$; composing with $d\chi$ gives $M_s(c)=c$. Since $dt$ and $du$ have unit coefficients in these coordinates, ${\mathrm{\Delta }}_{\mathrm{F}\mathrm{P}}(s)=\mathrm{\mid }\det (1)\mathrm{\mid }=1$. Also $\mathrm{\Psi }(t,(0,v))=(t,v)$ preserves the product coordinate density, so $J_{\mathrm{\Psi }}=1$.

For $x=(a,b)$, $F_x(t)=a+t$, hence its unique zero is $t_0=-a$, and $x\cdot t_0=(0,b)\in S$. This is a transverse intersection because the operator there is the isomorphism $c\mapsto c$; no second group parameter produces the same point on this orbit.

Extend $\phi$ by zero outside $\mathrm{\Gamma }$; this is smooth because $\phi$ has compact support in $\mathrm{\Gamma }$. In the regularized expression, substitute $y=a+t$. Since $t_0\in \mathrm{\Gamma }$, we have $0=a+t_0\in a+\mathrm{\Gamma }$, so for all sufficiently small $ϵ$ the support of ${\eta }_{ϵ}$ lies in $a+\mathrm{\Gamma }$ and change of variables gives ${\int }_{\mathrm{\Gamma }}\phi (t){\eta }_{ϵ}(a+t)dt={\int }_{\mathbb{R}}\phi (y-a){\eta }_{ϵ}(y)dy$. The last integral differs from $\phi (-a)$ by at most ${\sup }_{\mathrm{\mid }y\mathrm{\mid }\le Cϵ}\mathrm{\mid }\phi (y-a)-\phi (-a)\mathrm{\mid }\int {\eta }_{ϵ}(y)dy$, where $C$ bounds the support of $\eta$. This tends to zero by continuity and normalization. Since ${\mathrm{\Delta }}_{\mathrm{F}\mathrm{P}}=1$, the required limit is $\phi (t_0)$, as asserted.