Spinors as automorphisms of the tangent bundle

Details Spinors as automorphisms of the tangent bundle Spinors as automorphisms of the tangent bundle

2002-10-27

arxiv.org/abs/math/0210418v2

Trans. Amer. Math. Soc., vol. 356 (2004), mo. 5, pp. 2049-2066

Mathematics

Differential Geometry

19 pages, 1 LaTeX figure. Minor revision, one figure added. To appear in Transaction of the AMS

Scientific paper

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 \phi \in \Gamma(W^+) is the same as a bundle morphism \phi: TM \to TM 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: W^+ \to \Lambda^+ 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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