# With the traceless matrices 14 m m nt m m 1 n a tr m

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with the traceless matrices 14 : M + M NT = M + M - 1 N a tr { M + M } , MM + NT = MM + - 1 N b tr { MM + } . The scalar product (4.15) in Ω 2 D ( A ) is calculated using, besides the trace theorems, the identities: 1 2 d/ 2 tr c ( c ( ρ ( k ) )) + c ( ρ ( k ) = k ! ( ρ ( k ) i 1 ...i k ) * ρ ( k ) j 1 ...j k δ i 1 j 1 ...δ i k j k = g - 1 ( ρ ( k ) * ; ρ ( k ) ) . In terms of the Hodge dual , defined by g - 1 ( ρ ( k ) * ; ρ ( k ) ) ω = ( ρ ( k ) ) * ρ ( k ) , the scalar product reads : π D ( G ); π D ( G ) 2 ,D = 1 2 π N a S 2 ρ (2) aaa * ρ (2) aaa + N b S 2 ρ (2) bbb * ρ (2) bbb + tr { MM + } S 2 ρ (1) ab * ρ (1) ab + tr { M + M } S 2 ρ (1) ba * ρ (1) ba + tr [ M + M ] 2 NT S 2 ρ (0) aba * ρ (0) aba + tr [ MM + ] 2 NT S 2 ρ (0) bab * ρ (0) bab . (4.21) 14 Note that when N a = N b = N and M is a scalar matrix, these traceless matrices M + M NT and MM + NT vanish and there is no ρ (0) α term in π D ( G ). Physically this implies that, in order to have a Higgs mechanism, a nontrivial mass spectrum is necessary! 27

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4.1 The Yang-Mills-Higgs action The universal connection in M , given by the matrices (( A αβ ( x,y ))) of (3.6), is represented in Ω (1) D ( A ) by an operator of the form (4.10) where the differential forms σ ( k ) ··· are matrix-valued. σ (1) a ( x ) = α a ( x ) e k,y (( ω aa ( x,y ))) | y = x θ k x , σ (1) b ( x ) = α b ( x )(( P b ( x ))) e k,y ω b ( x,y ) | y = x θ k x (( P b ( x ))) , σ (0) ab ( x ) = | Φ ab ( x,x ) ν ( x ) | , σ (0) ba ( x ) = | ν ( x ) Φ ba ( x,x ) | . (4.22) The monopole connection (3.13) also implements a differential one-form : μ b ( x ) = ( e k,y m b ( x,y )) | y = x θ k x = ν ( x ) | d | ν ( x ) , = 1 / 2 1 + | ν ( x ) | 2 ν ( x ) * d ν ( x ) - ν ( x )d ν ( x ) * . (4.23) It is also convenient to introduce the Higgs field doublets : | η ab ( x ) = | H ab ( x,x ) = | Φ ab ( x,x ) + | ν ( x ) , η ba ( x ) | = H ab ( x,x ) | = Φ ba ( x,x ) | + ν ( x ) | . (4.24) The hermiticity of the connection (3.9) yields : α a + = - α a , ( α b ) * = - α b , ( μ b ) * = - μ b , η ba | = | η ab + . (4.25) From (3.17) it follows that, under an active gauge transformation, the differ- ential forms (4.22) and (4.23) behave as : (( α U a )) = (( U a )) - 1 (( α a ))(( U a )) + (( U a )) - 1 d(( U a )) α U b = ( u b ) - 1 α b ( u b ) + ( u b ) - 1 d u b = α b + ( u b ) - 1 d u b μ U b = ( u b ) - 1 μ b ( u b ) = μ b | η U ab = (( U a )) - 1 | η ab u b , η U ba | = ( u b ) - 1 η ba | (( U a )) . On the other hand, under a passive gauge transformation H B H A , accord- ing to (3.10) and (3.14), they transform as : (( α A a )) = (( α B a )) , α A b = α B b , | η A ab = | η B ab ( c AB ) - n/ 2 , η A ba | = ( c AB ) + n/ 2 η B ba | , μ A b = μ B b + ( c AB ) + n/ 2 d( c AB ) - n/ 2 = μ B b - ( n/ 2)( c AB ) - 1 d c AB . 28
This means that the Higgs fields | η ab are actually Pensov fields of spin- weight n/ 2 and that the monopole potential cannot be represented by a globally defined one-form on the sphere, but adquires the inhomogeneous term - ( n/ 2)( c AB ) - 1 d c AB in H B H A . The canonical projection (4.14) induces a Ω (1) D ( A )-valued connection in M : D : M → M ⊗ A Ω (1) D ( A ) : X → ∇ D X, defined by D = 1 M π D ◦ ∇ . From (3.11) and (4.22) it follows that : | ( D X ) aa = - ic d | f a + (( α a )) | f a , | ( D X ) ab = | η ab f b - | f a γ 3 M + , | ( D X ) ba = | ν η ba | f a - f b γ 3 M, | ( D X ) bb = | ν - ic d f b + ( α b + μ b ) f b . The curvature (3.15) is represented in Ω (2) D ( A ) by π D ( R ) of the form (4.20), where the differential forms ρ ( k ) ...

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