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The round two-sphere has zero untwisted Spin Dirac kernel

Statement

Equip $S^{2} ≅ {C P}^{1}$ with its unit round metric and complex orientation. It has a Spin structure, its scalar curvature is $2$ , and every smooth untwisted harmonic spinor is zero.

Facts & Assumptions

Given: Full AC, the unit round sphere and the complex-projective identification with ${C P}^{1}$ .

[F1]

$c_{1} (T {C P}^{1}) = 2 x$ and $H^{2} ({C P}^{1}; Z) = Z x$ (The tangent bundle of complex projective space and its Pontryagin classes, Integral cohomology ring of complex projective space).

[F2]

For a complex tangent bundle, $w_{2} = ρ_{2} c_{1}$ ; an oriented bundle with $w_{2} = 0$ has a Spin lift (Mod-two reduction of Chern classes, Spin structures exist exactly when w2 vanishes and then form an H1 torsor).

[F3]

The unit round sphere has constant sectional curvature $1$ (The round sphere has positive constant sectional curvature); in dimension two the scalar curvature is the sum of the two ordered sectional-curvature terms (Riemann curvature four-tensor).

[F4]

On a compact Spin manifold with positive scalar curvature, the untwisted spinor Dirac kernel vanishes (Positive scalar curvature annihilates untwisted spinor harmonic sections).

[A1]

Full AC is inherited through the characteristic-class and Spin-lift suppliers (The Axiom of Choice).

Proof

Given: The oriented round sphere.

1.1A1F1F2algebra

By [F1]–[F2], $w_{2} (T S^{2}) = ρ_{2} (2 x) = 0$ ; the complex orientation orients its tangent bundle. The spin-structure criterion in [F2] supplies a Spin lift.

1.2F3givenalgebra

The sectional curvature is $1$ by [F3]. In an orthonormal frame on a surface, scalar curvature is $R_{1221} + R_{2112} = 2$ , with the sign convention of [F3]. It is strictly positive everywhere.

2.1A1F4step 1.1step 1.2∎

Apply [F4] to the Spin structure from step 1.1 and the positive scalar curvature from step 1.2. Every smooth untwisted harmonic spinor is zero. The assertion is for the untwisted operator; a line twist has its own curvature term and is treated separately.

Depends on

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Sources

Liviu I. Nicolaescu, Notes on the Atiyah–Singer Index Theorem, revision 15 November 2013 (standard reference, not scraped)
Konstantin Wernli, Lecture Notes on Spin Geometry (2019) (standard reference, not scraped)