\] The $0$-forms $\omega^{+,+}$, $\omega^{+,-}$, $\omega^{-,+}$ and $\omega^{-,-}$ determine smooth, closed $0$-forms on the compactified configuration space $\mathcal C_{0,0,0}^+$, which enjoy the following properties:  -   $$\omega^{+,+}\vert_\alpha=\mathrm d\varphi