555 in case 2b a similar result is obtained d ψ cov

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(5.55) In case 2.b) a similar result is obtained : (( D Ψ Cov ( ± ) aa )) = D 1( ± ) (( ψ ( ± ) aa )) - i γ r (( α a,r )) , (( ψ ( ± ) aa )) + γ 3 A + | η ab ψ ( ± ) ba | + γ 3 B + | ψ ( ± ) ab η ba | , |D Ψ Cov ( ± ) ab = D ( - n/ 2) 1( ± ) | ψ ( ± ) ab + γ 3 B (( ψ ( ± ) aa )) | η ab , - i γ r (( α a,r )) | ψ ( ± ) ab - | ψ ( ± ) ab α b,r , D Ψ Cov ( ± ) ba | = D (+ n/ 2) 1( ± ) ψ ( ± ) ba | + γ 3 A η ba | (( ψ ( ± ) aa )) , - i γ r α b ψ ( ± ) ba | - ψ ( ± ) ba | (( α a,r )) ( D Ψ Cov ) ( ± ) bb = D 1( ± ) ψ ( ± ) bb . (5.56) The difference is that here the Higgs field interact with the quadruplet (( ψ ( ± ) aa )), while in case 2.a) it interacts with the singlet ψ ( ± ) bb and it is this interaction that gives masses to the particles. The Dirac operators, D ( - n/ 2) 1( ± ) = D ( ± s - n/ 2) and D (+ n/ 2) 1( ± ) = D ( ± s + n/ 2) , acting on Pensov spinor fields of spin weight ± s + n/ 2 or ± s - n/ 2, arise from D 1( ± ) ( | ψ ( ± ) ab ν | ) | ν = D ( - n/ 2) 1( ± ) | ψ ( ± ) ab , ν | D 1( ± ) ( | ν ψ ( ± ) ba | ) = D (+ n/ 2) 1( ± ) ψ ( ± ) ba | , where (( P b )) = | ν ν | is the representative, chosen in (3.3), of the homotopy class [ n ] π 2 ( S 2 ). The induced contribution of the ”magnetic monopole” is 47
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hidden in this modification of the Dirac operator. The Higgs doublets of Pensov fields of weight n/ 2, | η ab and η ba | were defined in (4.24). A suitable action of the matter field would be S Mat ( Ψ Cov , D ) = Ψ Cov ; D Ψ Cov . (5.57) But, if we aim for a theory admitting only chiral matter, i.e. if we restrict the Hilbert space to those vectors obeying say Γ Ψ Cov = + Ψ Cov , (5.58) then the above action vanishes identically. Another choice for the action would be S Chiral ( Ψ Cov , D ) = J Cov Ψ Cov ; D Ψ Cov . (5.59) It is easy to show that this action does not vanish identically if = - 1 , (5.60) = +1 . (5.61) In two dimensions Connes’ sign table obeys the first but not the second condition 21 . It should however be stressed that Connes’ sign table, with its modulo eight periodicity, comes from representation theory of the real Clifford algebras and if we restrict our (generalized) spinors to Weyl spinors, we loose the Clifford algebra representation and the sign tables ceases to be mandatory. We could then go back to 5.3 and with J = J 1 J 2 require (5.32) to hold with 1 = - 1 1 or, with J = J 1 ( J 2 χ 2 ) require (5.34) to hold with 1 = +1 = 1 . The full story of this chirality problem, sometimes called the ”neutrino paradigm”, is postponed to forthcoming work on the four-sphere with C H . 21 In four dimensions it is the first condition that is not satisfied. 48
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6 Conclusions and Outlook The Connes-Lott model with C C , over the two-sphere, has been gener- alised so as to allow for nontrivial topological structure. In the complex case, i.e. without the real structure, a main feature lies in the fact that the Higgs fields are no longer functions over S 2 , but rather sections of nontrivial com- plex line bundles over S 2 . Furthermore, the covariantisation of the Hilbert space with the nontrivial module M , induces a ”spin” change in some of the matter fields. A real spectral triple has also been constructed even though C C is abelian. In contrast with the standard noncommutative geometry model of the standard model, here the continuum spectral triple has an S 0 - real structure while the discrete is not S 0 -real. Some physical plausability
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