The projective modules over a following connes lott

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The projective modules over A , following Connes-Lott, are constructed in section 3 as M = P ( A ⊕ A ) where P = P a = 1 , P b = 1 2 ( 1 + n b · σ ) . In Connes’ work [1], the smooth manifold is four-dimensional so that, taking the four-sphere S 4 , the mappings n : S 4 S 2 are classified by π 4 ( S 2 ) = Z 2 . However, the local unitary transformations acting as P b U P b U have homotopy classes π 4 ( U (2)) = π 4 ( SU (2)) = π 4 ( S 3 ) = Z 2 . It follows that 2 all Bott projectors define isomorphic modules to the module considered by 2 We are indebted to prof.Balachandran of Syracuse University for discussions on this point. 2
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Connes, obtained by P b = 1 2 ( 1 + σ 3 ). For the two-sphere S 2 this does not happen since π 2 ( S 2 ) = Z and π 2 ( U (2)) = π 2 ( SU (2)) = π 2 ( S 3 ) = { 1 } . In section 4 we construct the full spectral triple obtained as the product of the Dirac-Pensov triple for the algebra A 1 with a discrete spectral triple for the algebra A 2 . Following again Connes’prescription ` a la lettre, we take the discrete Hilbert space as H dis = C N a C N b with chirality χ dis given as +1 on the a-sector and -1 on the b-sector. The discrete Dirac operator D dis is then the most general hermitian matrix, odd with respect to the grading defined by χ dis . ”Eliminating the junk” in the induced representation of the universal differential enveloppe Ω ( A ), yields bounded operators Ω D ( A ) in H = H ( s ) ⊗H dis . The standard use of the Dixmier trace and of Connes’trace theorem allows then to define a scalar product of operators in Ω D ( A ). This scalar product is used in section 4.1 to construct the Yang-Mills-Higgs action. The main new features, as compared with Connes’result, in this action are the appearance of an additional monopole potential of strength eg/ 4 π = ( n/ 2)¯ h , where n is the integer characterizing the homotopy class of P b , and the fact that the Higgs doublet is not globally defined on S 2 but transforms as a Pensov field of weight ± n/ 2. The particle sector is examined in section 4.2 and a covariant Dirac operator D acting on H p = M ⊗ A H is defined. Here, the new feature is that, while the ”a-doublet” continues as a doublet of Pensov spinors of weight s , the ”b- singlet” metamorphoses in a Pensov spinor of weight s + n/ 2. If one should insist on a comparison with the standard electroweak model on S 2 , this would mean that right-handed electrons see a different magnetic monopole than the left-handed and this is not really welcome. In section 5 we introduce a real Dirac-Pensov spectral triple by doubling the Hilbert space as H 1 = H ( s ) ⊕ H ( - s ) . It is seen that, with the same H dis as before, it is not possible to define a real structure. However, a more general discrete Hilbert space H 2 = C N aa C N ab C N ba C N bb as discussed in [10, 17], allows for the construction of a real structure on H new = H 1 ⊗ H 2 .
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