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Spinors as automorphisms of the tangent bundle

Alexandru Scorpan
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Abstract

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field ϕ\in Γ(W^+) is the same as a bundle morphism ϕ: TM \to TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of ϕon tangent vectors, and that the squaring map σ: W^+ \to Λ^+ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.

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Alexandru Scorpan. 2003-04-11. Spinors as automorphisms of the tangent bundle. https://arxiv.org/abs/math/0210418

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