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Even-dimensional spinor bundles, chirality, and the spin connection

Definition

Assume full AC. Let $M$ be an oriented Riemannian manifold of even dimension $n = 2 m \geq 2$ with a specified Spin lift $P \to P_{S O} (T M)$ . Let $S_{0}$ be the complex irreducible Clifford module of dimension $2^{m}$ supplied by Complex spinor modules and complete reducibility for Euclidean Spin, with positive-square complex Clifford action $c_{+}$ and full-even Spin representation $σ$ . Put $c_{0} (v) = i c_{+} (v)$ for the negative-square cotangent action. This changes the Clifford square sign and preserves equivariance under $σ$ , since the scalar $i$ commutes with that action. Define the complex spinor bundle by the associated real bundle $P \times_{Spin (n)} (S_{0})_{R}$ using the realification of $σ$ , and descend multiplication by $i$ and the Clifford action through their commuting transition maps.

For an oriented orthonormal basis $(e_{1}, \dots, e_{2 m})$ , define the chirality operator on $S_{0}$ by $Γ_{S_{0}} = i^{m} c_{0} (e^{1}) \dots c_{0} (e^{2 m})$ . The calculation below proves that this is basis-independent, squares to $1$ , and anticommutes with each $c_{0} (ξ)$ . Its $\pm 1$ eigenspaces are $S_{0}^{\pm}$ and their associated bundles are $S^{\pm}$ . The Levi–Civita connection on $P_{S O} (T M)$ lifts through the differential of $Spin (n) \to S O (n)$ to the spin connection ${\tilde{\nabla}}^{S}$ on $S$ . This item defines these objects; Spinor and geometric Clifford bundles and connections are well defined ↗ supplies the global bundle, metric, connection, and transition verification.

Facts & Assumptions

Given: Full AC, the Clifford action and the selected oriented orthonormal basis in the Statement.

[F1]

The generators satisfy $c_{0} (e^{j})^{2} = - I$ and anticommute for distinct orthonormal covectors (Universal Clifford algebra of a quadratic space, Graded Hermitian Clifford modules and geometric Dirac operators).

[F2]

In even dimension the complex spinor module has dimension $2^{m}$ and its Spin restriction has two half-spin representations, each of dimension $2^{m - 1}$ (Complex spinor modules and complete reducibility for Euclidean Spin).

[A1]

The actual even-algebra identification Even subalgebras of real and complex Clifford algebras and full complex matrix classification Real and complex Clifford algebra classification with explicit signature give the spinor and two half-spin dimensions in [F2]. The negative-sign action here is obtained from the positive complex action by multiplication by $i$ , as checked below.

Proof

Given: $n = 2 m$ , the Clifford generators $c_{j} = c_{0} (e^{j})$ , and $Γ = i^{m} c_{1} \dots c_{n}$ .

1.1A1F1F2givenalgebra

The positive generators satisfy $c_{+} (v)^{2} = ∣ v ∣^{2}$ , so $c_{0} (v)^{2} = - ∣ v ∣^{2}$ and their polarized relation is the required negative one. Spin equivariance is preserved: $σ (h) c_{0} (v) σ (h)^{- 1} = i c_{+} (ρ (h) v) = c_{0} (ρ (h) v)$ . Its negative-sign volume chirality is $i^{m} \prod_{j} (i c_{+} (e^{j})) = (- 1)^{m} i^{m} \prod_{j} c_{+} (e^{j})$ , so the two half-spin labels are exchanged when $m$ is odd, with dimensions unchanged. Reversing the order of the $n$ pairwise anticommuting generators takes $n (n - 1) / 2 = m (2 m - 1)$ swaps, and their squares contribute $(- 1)^{n} = 1$ . Hence $(c_{1} \dots c_{n})^{2} = (- 1)^{m}$ . Since $(i^{m})^{2} = (- 1)^{m}$ , $Γ^{2} = 1$ . For any $k$ , moving $c_{k}$ through the other $n - 1$ factors changes the sign by $(- 1)^{n - 1} = - 1$ ; thus $Γ c_{k} = - c_{k} Γ$ , and linearity gives the same relation for every covector.

2.1A1F2step 1.1algebra∎

The product $c_{1} \dots c_{n}$ is fully alternating because distinct orthonormal generators anticommute. Under an oriented orthogonal basis change $A \in S O (n)$ , its top exterior coefficient is multiplied by $\det A = 1$ , so $Γ$ is independent of the chosen oriented orthonormal basis. Since $Γ^{2} = 1$ over $C$ , the projections $(I \pm Γ) / 2$ split $S_{0} = S_{0}^{+} \oplus S_{0}^{-}$ . Anticommutation shows each $c_{0} (ξ)$ exchanges the two eigenspaces. Their dimensions are equal to $2^{m - 1}$ by [F2]. The exact full and even matrix factors are those in [A1].

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Sources

Richard B. Melrose, Analysis, Loops and Spin (2024 lecture notes) (standard reference, not scraped)
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)