C n 0 used here 7 acting on ψ s σ s 1 2 σ s 1 2

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C ( n, 0) used here. 7
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acting on ψ ( s ) = σ ( s - 1 / 2) σ ( s +1 / 2) , the Dirac operator can also be written as : D ( s ) ψ ( s ) = - i γ k LC k, ( s ) ψ ( s ) , (2.10) where the covariant derivative of the spinor ψ ( s ) is given by : LC k, ( s ) ψ ( s ) = q 2 ∂ψ ( s ) ∂ξ k + 1 2 Σ ( s ) ij Γ ij k ψ ( s ) . Here, Σ ( s ) 12 = i s 1 0 0 1 + - i / 2 0 0 + i / 2 reduces to 1 4 [ γ 1 2 ] for s = 0. In terms of the ”edth” operators, we may write LC ( s ) , + ψ ( s ) = 1 2 ˇ δ/ s - 1 / 2 0 0 ˇ δ/ s +1 / 2 ψ ( s ) , LC ( s ) , - ψ ( s ) = 1 2 ˇ δ / s - 1 / 2 0 0 ˇ δ / s +1 / 2 ψ ( s ) . With γ (+) = γ 1 + i γ 2 = 0 0 2 0 and γ ( - ) = γ 1 - i γ 2 = 0 2 0 0 , the Dirac operator of (2.8) is now written as D ( s ) ψ ( s ) = - i γ (+) LC ( s ) , + + γ ( - ) LC ( s ) , - ψ ( s ) . The transformation law for s -Pensov fields 6 under a local Zweibein rotation (2.2) is related to the Spin c structure of the Pensov spinors : σ ( s - 1 / 2) σ ( s +1 / 2) σ ( s - 1 / 2) σ ( s +1 / 2) = exp { α Σ ( s ) 12 } σ ( s - 1 / 2) σ ( s +1 / 2) , where exp { α Σ ( s ) 12 } = exp { i } exp( - i α/ 2) 0 0 exp(+ i α/ 2) . 6 A Pensov spinor of weight s can be interpreted as a usual Dirac spinor on S 2 , inter- acting with a Dirac monopole of strength s . Indeed, in the expression of the covariant derivative, the term i k Γ 12 k is the Clifford representative of the one-form (potential) μ s . = i k Γ 12 k = s 1+ | ζ | 2 ( ζ * - ζdζ * ), which, in { H B ; cos θ = +1 } , takes the usual form μ s | B = i s (1 - cosθ ) . 8
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The Clifford action of γ 3 = i ω yields a grading on the Pensov spinors γ 3 ψ ( s ) ψ ( s +1) = 1 0 0 - 1 ψ ( s - 1 / 2) ψ ( s +1 / 2) , ( γ 3 ) 2 = 1 , such that the Dirac operator (2.8) is odd D ( s ) γ 3 + γ 3 D ( s ) = 0 . According to (2.6), the scalar product of two Pensov spinors is defined as : Φ ( s ) | Ψ ( s ) = Σ ( s - 1 / 2) ,T ( s - 1 / 2) s - 1 / 2 + Σ ( s +1 / 2) ,T ( s +1 / 2) s +1 / 2 . (2.11) The adjointness (2.7) of - i ˇ δ/ s - 1 / 2 and - i ˇ δ/ s +1 / 2 implies that the Dirac operator is formally self-adjoint with respect to this scalar product. After completion, P ( s - 1 / 2) ⊕P ( s +1 / 2) becomes a bona fide Hilbert space H ( s ) on which D ( s ) acts as a self-adjoint (unbounded) operator. Its spectral resolution is completely solvable. Indeed, let X = ∂X + + * X - be a vector field on S 2 , then the Lie derivatives of the Zweibein along X are: L X θ = ∂X + ∂ζ - 1 q ( ∂q ∂ζ * X - + ∂q ∂ζ X + ) θ + ∂X + ∂ζ * θ * , L X θ * = ∂X - ∂ζ * - 1 q ( ∂q ∂ζ * X - + ∂q ∂ζ X + ) θ * + ∂X - ∂ζ θ. (2.12) A vector field X is said to be a conformal Killing vector field if L X g = μ g , where μ is a scalar function on S 2 . The expression of the Lie derivative (2.12) yields then the (anti-)holomorphic constraints : ∂X + ∂ζ * = 0 , ∂X - ∂ζ = 0 , and μ is given by : μ = q 2 ( X + /q 2 ) ∂ζ + ( X - /q 2 ) ∂ζ * . If X has to be globally defined, its Zweibein components (2 /q ) X + and (2 /q ) X - must be finite when | ζ |→ ∞ . For the standard metric q = 1+ | ζ | 2 9
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and this implies that X + , respectively X - , is a quadratic polynomial in ζ , respectively ζ * . There are thus six linearly independent conformal Killing vector fields, three of which are genuinely Killing,i.e. with μ = 0. They are chosen as 7 iL x = i 2 ( ζ 2 - 1) - ( ζ * 2 - 1) * iL y = 1 2 ( ζ 2 + 1) + ( ζ * 2 + 1) * , iL z = i ζ∂ - ζ * * .
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