# S are now 2 2 matrix valued the diagonal elements of

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’s are now 2 × 2-matrix valued. The diagonal elements of π D ( R ) are given by : ρ (2) aaa = (( F a )) = d(( α a )) + (( α a )) (( α a )) , ρ (2) bbb = F b (( P b )) = (d α b + d μ b )(( P b )) , ρ (0) aba = | η ab η ba | - (( Id )) , ρ (0) bab = ( η ba | η ab - 1) (( P b )) . The off-diagonal elements are given in terms of the covariant differentials of the Higgs fields (4.24): |∇ η ab = d | η ab + (( α a )) | η ab - | η ab ( α b + μ b ) , η ba | = d η ba | - η ba | (( α a )) + ( α b + μ b ) η ba | . (4.26) They read ρ (1) ab = |∇ η ab ν | , ρ (1) ba = | ν η ba | . 29

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The Yang-Mills-Higgs action is constructed as : S Y MH ( D ) = λ tr matrix π D ( R ); π D ( R ) 2 ,D = λ tr matrix Tr Dix π D ( R ) π D ( R ) | D | - 2 , (4.27) where λ is a coupling constant and tr matrix is the trace of the 2 × 2 matrices, product of matrices ρ ( k ) + with ρ ( k ) . Since the curvature transforms as R R U = U - 1 R U , the gauge invariance of the action follows from the obvious extension of the representation (4.16) of the unitaries in H (2) D . With the scalar product given by (4.21), the action (4.26) reads : S Y MH ( D ) = λ 2 π N a S 2 tr matrix (( F a )) + (( F a )) + N b S 2 ( F b ) * F b +2 tr { MM + } S 2 η ba | ∧ |∇ η ab + tr M + M NT 2 S 2 ( η ba | η ab - 1) 2 + 1 + tr MM + NT 2 S 2 ( η ba | η ab - 1) 2 . (4.28) 4.2 The Hilbert space of particle states and the covari- ant Dirac operator The tensor product over A of the right A -module M with the (left-module) Hilbert space H is itself a Hilbert space H p = M ⊗ A H , with scalar product induced by the scalar product (4.6) in H and the hermitian structure h in the module M : ( X A Ψ ; Y A Φ ) = ( Ψ ; π ( h ( X,Y )) Φ ) . A generic element of H p can be written as Ψ p = E i A Ψ i , where Ψ i ∈ H obeys π ( P i j ) Ψ j = Ψ i . In the model considered here, H = H ( s ) C N a C N b and the projective module is M = P A 2 , with P defined by the homotopy class [ g ] in (3.3). A state Ψ p describing particles, is thus represented by : 30
1. A pair of Pensov spinors of H ( s ) , given by : | ψ a ( x ) = ψ 1 a ( x ) ψ 2 a ( x ) , each with N a values of the generation index. 2. A single Pensov spinor ψ b ( x ) of H ( s + n/ 2) , with a N b -valued generation index, such that | ψ b ( x ) = ψ 1 b ( x ) ψ 2 b ( x ) = | ν ( x ) ψ b ( x ) in H B . The -representation π of Ω ( A ) in H induces a mapping π 1 : M ⊗ A Ω ( A ) → B ( H , H p ) : X A F π 1 ( X A F ) , (4.29) where B ( H , H p ) are the bounded linear operators from H to H p . It is defined by π 1 ( X A F ) Ψ = X A π ( F ) Ψ . Furthermore, there is a mapping π 2 : HOM A ( M , M ⊗ A Ω ) → B ( H p ) : T π 2 ( T ) , (4.30) defined by π 2 ( T ) X A Ψ = π 1 ( T X ) Ψ . The covariant Dirac operator in H p is defined, using (4.29), as D X A Ψ = X A + π 1 ( X ) Ψ . (4.31) It is easy to check that D Xf A Ψ = D X A π ( f ) Ψ so that D is well defined in H p . A grading in H p is defined by Γ p : X A Ψ X A Γ Ψ (4.32) and the covariant Dirac operator is odd with respect to this grading : D Γ p + Γ p D = 0 (4.33) With Ψ p as above, D is calculated as follows.

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