The corresponding line bundle 4 will be denoted by p

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The corresponding line bundle 4 will be denoted by P ( s ) . 3 The integer 2 s can be identified with the integer representing an element of the second ˇ Cech cohomology group of S 2 with integer values, ˇ H 2 ( S 2 , Z ) = Z , classifying the line bundles over the sphere S 2 . 4 In the sequel, abusing the notation, we shall denote bundles and their spaces of sections by the same symbol. 5
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A local basis of its sections in H B is denoted by Θ ( s ) | B , and in H A by Θ ( s ) | A . They are related in H B H A by the generalisation of (2.3): Θ ( s ) | A = ( c AB ) - s Θ ( s ) | B . A local section is thus given, in H A say, by : Σ ( s ) = σ ( s ) | A Θ ( s ) | A and σ ( s ) | A = ( c AB ) s σ ( s ) | B . On S 2 with metric g = δ ij θ i θ j , the Levi-Civita reads LC θ i = - Γ i j θ j = - Γ i k j θ k θ j , where Γ i k j = - 1 2 { δ i k ∂q ∂ξ j - δ i ∂q ∂ξ δ kj } . In terms of the complexified Zweibein (2.1), we have : LC θ = - Γ θ, LC θ * = - Γ * θ * , (2.4) where Γ = - Γ * = - 1 2 { ∂q ∂ζ θ - ∂q ∂ζ * θ * } = - 1 2 { ζ * θ - ζθ * } . It is easy to see that LC Θ ( s ) = - s Γ Θ ( s ) defines a connection in the module of s -Pensov fields generalising (2.4) above. This connection maps P ( s ) in ( T * ( S 2 )) C ⊗ P ( s ) . Now ( T * ( S 2 )) C , the space of complex-valued one- forms, is isomorphic to P (+1) ⊕ P ( - 1) , so that LC is a mapping : LC : P ( s ) → P ( s +1) ⊕ P ( s - 1) : Σ ( s ) → ∇ LC Σ ( s ) LC Σ ( s ) = d σ ( s ) - s Γ σ ( s ) Θ ( s ) = 1 2 ˇ δ / s σ ( s ) Θ ( s +1) + ˇ δ/ s σ ( s ) Θ ( s - 1) . Here we have introduced the ”edth” operators of Newman and Penrose [15], projecting LC Ψ ( s ) on each component of P ( s +1) ⊕ P ( s - 1) : ˇ δ/ s σ ( s ) = q - s +1 ∂ζ ( q s σ ( s ) ) = q ∂σ ( s ) ∂ζ + s ∂q ∂ζ σ ( s ) , ˇ δ/ s σ ( s ) = q s +1 ∂ζ * ( q - s σ ( s ) ) = q ∂σ ( s ) ∂ζ * - s ∂q ∂ζ * σ ( s ) . (2.5) 6
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With respect to the scalar product of Pensov fields : Σ ( s ) ,T ( s ) s = S 2 σ ( s ) * τ ( s ) ω, (2.6) where ω = θ 1 θ 2 = i 2 θ θ * is the invariant volume element on S 2 , the operators ˇ δ / s and ˇ δ/ s are formally anti-adjoint : σ ( s +1) , ˇ δ/ s τ ( s ) s +1 = - ˇ δ/ s +1 σ ( s +1) ( s ) s . (2.7) In a previous paper [14], the Dirac operator on K¨ ahler spinors was defined as the restriction of - i ( d - δ ) to the left ideals of the Clifford algebra bundle. Now, these ideals are identified with I E + = P (0) ⊕P (+1) , with basis { 1 + i ω,θ } , and I E - = P ( - 1) ⊕ P ( ¯ 0) , with basis { θ * , 1 - i ω } . In these bases, the local expressions of the Dirac operators were given as : D E + σ (0) σ (+1) = - i 0 ˇ δ/ +1 ˇ δ / 0 0 σ (0) σ (+1) , D E - σ ( - 1) σ ( ¯ 0) = - i 0 ˇ δ / 0 ˇ δ / - 1 0 σ ( - 1) σ ( ¯ 0) . This suggests to define a Pensov spinor field of weight s as a section Ψ ( s ) = Σ ( s - 1 / 2) Σ ( s +1 / 2) of the Whitney sum P ( s - 1 / 2) ⊕ P ( s +1 / 2) , with a Dirac operator locally ex- pressed as : D ( s ) σ ( s - 1 / 2) σ ( s +1 / 2) = - i 0 ˇ δ / s +1 / 2 ˇ δ/ s - 1 / 2 0 σ ( s - 1 / 2) σ ( s +1 / 2) . (2.8) The usual Dirac spinors on S 2 are recovered when s = 0. With the complex representation of the real Clifford algebra 5 C (2 , 0) γ 1 0 1 1 0 , γ 2 0 i - i 0 , (2.9) 5 The real Clifford algebra C ( p, q ) is defined by γ k γ + γ γ k = 2 η k , where the flat metric tensor η k is diagonal with p times +1 and q times - 1. This entails some differences with other work using the Clifford algebra C (0 , n ) for Riemannian manifolds instead of
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