The covariant dirac operator on m a h new a m is

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The covariant Dirac operator on M ⊗ A H new A M * is defined and it is seen that, with the use of a non trivial projective module, the abelian gauge fields are not slain, as they do when M = A [21]. Clearly this model building led us far from a toy electroweak model. The main purpose however was not to reproduce such a model on the two-sphere, but rather to examine some of the topologically nontrivial structures in model building with the simplest manifold allowing for such possibilities. 3
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2 The Hilbert space of Pensov spinors on S 2 The standard atlas of the two-sphere S 2 = { ( x,y,z ) R 3 | x 2 + y 2 + z 2 = 1 } consists of two charts, the boreal, H B = { ( x,y,z ) S 2 | - 1 <z +1 } , and austral chart, H A = { ( x,y,z ) S 2 | - 1 z< +1 } , with coordinates : ζ B = ξ 1 B + i ξ 2 B = + x + i y 1 + z in H B , ζ A = ξ 1 A + i ξ 2 A = - x - i y 1 - z in H A . In the overlap H B H A , they are related by ζ A ζ B = - 1 and the usual spherical coordinates ( θ,ϕ ), given by ζ B = - 1 A = tan θ/ 2 exp i ϕ , are nonsingular. In each chart, dual coordinate bases of the complexified tangent and cotangent spaces are : = ∂ζ = 1 2 ∂ξ 1 - i ∂ξ 2 ,∂ * = ∂ζ * = 1 2 ∂ξ 1 + i ∂ξ 2 = 1 + i 2 ,dζ * = 1 - i 2 . In H B H A they are related by A * A = B * B ζ B 2 0 0 ζ * B 2 , B * B = ζ B 2 0 0 ζ * B 2 A * A . The euclidean metric in R 3 induces a metric on the sphere : g = 4 q 2 δ ij i j = 2 q 2 * + * , where q = 1 + | ζ | 2 . Real and complex Zweibein fields are given by: θ i = 2 q i ; i = 1 , 2 and θ = 2 q dζ,θ * = 2 q * , (2.1) with duals e i = q 2 ∂ξ i ; i = 1 , 2 and e = q 2 ∂ζ , e * = q 2 ∂ζ * . 4
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A rotation of the real Zweibein by an angle α : θ 1 θ 2 ˜ θ 1 ˜ θ 2 = cos α sin α - sin α cos α θ 1 θ 2 , becomes diagonal for the complex Zweibein : θ θ * ˜ θ ˜ θ * = exp( - i α ) 0 0 exp( i α ) θ θ * . (2.2) This means that the complexified cotangent bundle ( T * S 2 ) C splits, in an SO (2) invariant way, into the direct sum of two line bundles ( T * S 2 ) and ( T * S 2 ) with one-dimensional local bases of sections given by { θ } and { θ * } . In the overlap H B H A , the Zweibein in H A and in H B are related by : θ A = ( c AB ) - 1 θ B , θ * A = c AB θ * B , (2.3) whith the transition function c AB = ζ B * B = ζ * A A = exp(2 i ϕ ), ϕ being the azimuthal angle, well defined (modulo 2 π ) in H B H A . A section of ( T * S 2 ) , respectively ( T * S 2 ) is written as Σ = σ (+1) θ , respec- tively Σ = σ ( - 1) θ * and, in H B H A , σ ( ± 1) | A = ( c AB ) ± 1 σ ( ± 1) | B . Following Staruszkiewicz [18], who refers to Pensov, we call such a field a Pensov field of spin-weight ( ± 1). The question is now adressed to define Pensov fields of spin-weight s on S 2 . In general this would require a cocycle condition on transition functions in triple overlaps. However, since the sphere is covered by only two charts, it is enough that the overlap equation σ ( s ) | A = ( c AB ) s σ ( s ) | B be well defined. Now, ( c AB ) s = exp(2 i ) is well defined when 2 s takes integer values 3 .
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