Spinors as automorphisms of the tangent bundle

Abstract:

We show that, on a $4$-manifold $M$ endowed with a $\operatorname {spin}^{\mathbb {C}}$-structure induced by an almost-complex structure, a self-dual (positive) spinor field $\phi \in \Gamma (W^+)$ is the same as a bundle morphism $\phi :T_M\to T_M$ acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of $\phi$ on tangent vectors, and that the squaring map $\sigma :\mathcal {W}^+\to \Lambda ^+$ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kähler and symplectic structures.