Conformal properties of harmonic spinors and lightlike geodesics in signature (1,1) (Q1871569)

Let \((M^{1+1},g)\) be a Lorentzian surface, that is a smooth and orientable two-dimensional manifold such that \(TM^{1+1}\) splits into the direct sum of a time-like bundle and a space-like bundle. The light-cone defined by \(g\) induces two one-dimensional light-like distributions which may be labelled unambiguously by \(\mathcal X\) and \(\mathcal Y\). Furthermore, a light-like vector field is a section either of the \(\mathcal X\) - or of the \(\mathcal Y\) - distribution. It makes therefore sense to speak of \(\mathcal X\) - or of \(\mathcal Y\) - conformal flatness depending on the divergence-free light-like vector field to be \(\mathcal X\) or \(\mathcal Y\). Let \(\delta_{\pm}=\dim \ker \left(D^{\pm}:\Gamma(S^{\pm})\to \Gamma(S^{\mp})\right)\) [resp. \(\tau_{\pm}=\dim \ker (P^{\pm}:\Gamma(S^{\pm})\to \Gamma(S^{\mp}))\)] be the dimensions of the spaces of positive/negative harmonic (resp. twistor) half-spinors. On Riemannian surfaces, harmonic spinors were studied by \textit{N. J. Hitchin} [Adv. Math. 14, 1-55 (1974; Zbl 0284.58016)] and by \textit{C. Bär} and \textit{P. Schmutz} [Ann. Glob. Anal. Geom. 10, 263-273 (1992; Zbl 0763.30017)]. The dimension of the space of harmonic spinors depends essentially-unlike the dimension of the space of harmonic forms - on the conformal class of the metric and the spin structure used to define the Dirac operator. Furthermore, the dimension is bounded by \([g+1/2]\), where \(g\) denotes the genus of the surface (see N. Hitchin, loc. cit.). If \((M^{1+1},g)\) is a compact \(\mathcal X\)-conformally flat Lorentzian surface, the author proves that \(\delta_{+}=\tau_{-}\) and the only possible dimensions for \(\delta_{+}\) are \(0,1\) (for the trivial spin structure) and \(+\infty\). Each of the these cases is geometrically characterized. Also he shows that in some sense \(\mathcal X\)-conformal flatness is forced by positive harmonic spinors which have non-zero ``mass''. The same conclusion holds for \(\mathcal Y\) and \(\delta_{-}\) instead of \(\mathcal X\) and \(\delta_{+}\), and an analogous assertion can be stated for twistor spinors.