536 the dirac operator d d d is given by in case 2a d

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(5.36) the Dirac operator : D = D (+) ⊕ D ( - ) is given by in case 2.a) : D a ( ± ) = D 1( ± ) 1 N 0 0 0 0 D 1( ± ) 1 N 0 γ 3 A + 0 0 D 1( ± ) 1 N γ 3 B + 0 γ 3 A γ 3 B D 1( ± ) 1 N , (5.37) 41
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in case 2.b) : D b ( ± ) = D 1( ± ) 1 N γ 3 B + γ 3 A + 0 γ 3 B D 1( ± ) 1 N 0 0 γ 3 A 0 D 1( ± ) 1 N 0 0 0 0 D 1( ± ) 1 N . (5.38) the chirality : χ = χ (+) χ ( - ) , is given by in case 2.a) : χ ( ± ) = χ γ 3 1 N 0 0 0 0 γ 3 1 N 0 0 0 0 γ 3 1 N 0 0 0 0 - γ 3 1 N , (5.39) in case 2.b) : χ ( ± ) = χ - γ 3 1 N 0 0 0 0 γ 3 1 N 0 0 0 0 γ 3 1 N 0 0 0 0 γ 3 1 N , (5.40) the real structure : J = 0 J ( - ) J (+) 0 , exchanges H (+) and H ( - ) and J ( ± ) = C 1 ⊗ C 2 χ 2 yields in case 2.a) : J a ( ± ) = χ C 1 1 N 0 0 0 0 0 C 1 1 N 0 0 C 1 1 N 0 0 0 0 0 -C 1 1 N K , (5.41) in case 2.b) : J b ( ± ) = χ -C 1 1 N 0 0 0 0 0 C 1 1 N 0 0 C 1 1 N 0 0 0 0 0 C 1 1 N K , (5.42) 42
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5.4 The ”Real” Yang-Mills-Higgs action The representations π ( ± ) of (5.35) and π o ( ± ) of (5.36), with the Dirac operators of (5.37) and (5.38), induce representations of Ω ( A ). Let F Ω (1) ( A ), then, using the same techniques which led to (4.10), we obtain in case 2.a) : π ( ± ) ( F ) = - ic ( σ (1) a ) 1 N 0 0 0 0 - ic ( σ (1) a ) 1 N 0 σ (0) ab γ 3 A + 0 0 - ic ( σ (1) b ) 1 N 0 0 σ (0) ba γ 3 A 0 - ic ( σ (1) b ) 1 N , and in case 2.b) : π ( ± ) ( F ) = - ic ( σ (1) a ) 1 N 0 σ (0) ab γ 3 A + 0 0 - ic ( σ (1) a ) 1 N 0 0 σ (0) ba γ 3 A 0 - ic ( σ (1) b ) 1 N 0 0 0 0 - ic ( σ (1) b ) 1 N , where the differential forms σ ( k ) ’s are given in (4.11). Introducing the 2 N × 2 N matrix in case 2.a) as M ( 2 . a ) = 0 0 0 A and in case 2.b) as M 2 . b ) = A 0 0 0 , we may also write: π ( ± ) ( F ) = - ic ( σ (1) a ) 1 2 N σ (0) ab γ 3 M + σ (0) ba γ 3 M - ic ( σ (1) b ) 1 2 N . (5.43) A universal two-form G Ω (2) ( A ) has representations π ( ± ) ( G ) given by a similar expression as in (4.12). The unwanted differential ideal J is removed using the orthonality condition analogous to (4.19) ρ (0) aaa - 1 2 N ρ (0) aba tr { M + M } = 0 ; ρ (0) bbb - 1 2 N ρ (0) bab tr { MM + } = 0 . (5.44) The representative of G in Ω (2) D ( A ), as in (4.20), is given by : π D ( ± ) ( G ) = π D ( ± ) ( G ) [ aa ] π D ( ± ) ( G ) [ ab ] π D ( ± ) ( G ) [ ba ] π D ( ± ) ( G ) [ bb ] , (5.45) 43
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with π D ( ± ) ( G ) [ aa ] = - c ( ρ (2) aaa ) 1 2 N + ρ (0) aba M + M NT , π D ( ± ) ( G ) [ ab ] = - ic ( ρ (1) ab ) γ 3( ± ) M + , π D ( ± ) ( G ) [ ba ] = - ic ( ρ (1) ba ) γ 3( ± ) M, π D ( ± ) ( G ) [ bb ] = - c ( ρ (2) bbb ) 1 2 N + ρ (0) bab MM + NT , where the differential forms ρ ( k ) are given in (4.13) and M + M NT = M + M - 1 2 N tr { M + M } , M + M NT = MM + - 1 2 N tr { MM + } . The scalar product in Ω (2) D ( A ) is the same for representatives in H (+) or H ( - ) and is given by a similar expression as (4.21) : π D ( ± ) ( G ); π D ( ± ) ( G ) 2 ,D = 1 2 π 2 N S 2 ρ (2) aaa * ρ (2) aaa + ρ (2) bbb * ρ (2) bbb + tr { A + A } S 2 ρ (1) ab * ρ (1) ab + ρ (1) ba * ρ (1) ba + tr { ( A + A ) 2 } - 1 2 N ( tr { A + A } ) 2 S 2 ρ (0) aba * ρ (0) aba + ρ (0) bab * ρ (0) bab . (5.46) From this expression of the scalar product, the Yang-Mills-Higgs action is essentially twice the action (4.28) obtained in section 4.1 : S Y MH ( D ) = λ π 2 N S 2 tr matrix (( F a )) + (( F a )) + F * b F b +2 tr { A + A } S 2 η ba | ∧ |∇ η ab + tr { ( A + A ) 2 } - 1 2 N ( tr { A + A } ) 2 S 2 2( η ba | η ab - 1) 2 + 1 .
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