Appendix I A Gauss–Bonnet type identity

Proof. Since \(\pi : Y \to \Sigma\) admits smooth global sections, it follows that \(\pi^* : H^k(\Sigma) \to H^k(Y)\) is injective. Note that by construction the fibres of the bundle \(\pi : Y \to \Sigma\) are diffeomorphic to \(\left(\mathbb{R}_2\rtimes \mathrm{GL}^+(2,\mathbb{R})\right)/\mathrm{CO}(2)\) and hence diffeomorphic to \(\mathbb{R}_2\times D^2\). In particular, the fibre is contractible, thus we have \(H^2(Y)\simeq H^2(\Sigma)\simeq \mathbb{R}\) showing that \(\pi^* : H^2(\Sigma) \to H^2(Y)\) is an isomorphism. It follows that any two sections of \(Y \to \Sigma\) induce the same map on the second de Rham cohomology groups. It is therefore sufficient to construct a section \(s : \Sigma \to Y\) for which (I.1) holds. From the proof of the Lemma 4.3 we know that for every conformal structure \([g] : \Sigma \to Z\) there exists a lift \(\widetilde{[g]} : \Sigma \to Y\) so that on the pullback bundle \(P_{[g]}^{\prime}\) we have \[\zeta_2=2\overline{a}\,\overline{\zeta_1}, \qquad \zeta_3=k\zeta_1+2\overline{q}\,\overline{\zeta_1},\] Since \[\tau^*\Omega_\mathfrak{p}=-\frac{\mathrm{i}}{4}\left(\zeta_1\wedge\overline{\zeta_3}+\zeta_3\wedge\overline{\zeta_1}+\zeta_2\wedge\overline{\zeta_2}\right),\] computing as in Lemma 4.14 and using the above identities for \(\zeta_2,\zeta_3\) gives \[\begin{aligned} \int_{\Sigma} \widetilde{[g]}^*\Omega_\mathfrak{p}&=-\frac{\mathrm{i}}{4}\int_{\Sigma}\left(k+\overline{k}-4|a|^2\right)\zeta_1\wedge\overline{\zeta_1}=\frac{\mathrm{i}}{2}\int_{\Sigma}\mathrm{d}\varphi-\mathrm{d}\overline{\varphi}=2\pi\chi(\Sigma). \end{aligned}\]