In the rest of this paper we shall omit the a and b

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In the rest of this paper we shall omit the A and B labels except when relating quantities in H A with those in H B in the overlap H A H B . An element of M is thus represented by : X | f a | ν f b and its scalar product with Y | g a | ν g b is given by : h P ( X,Y ) a ( x ) = f 1 a ( x ) * g 1 a ( x ) + f 2 a ( x ) * g 2 a ( x ) , h P ( X,Y ) b ( x ) = f b ( x ) * g b ( x ) . An active gauge transformation in the free module A 2 is given by a unitary 2 × 2 matrix U with values in A . It retricts to a gauge transformation in M when it commutes with P : PU = UP . An element X ∈ M transforms as X U X given by : | U X a ( x ) = (( U a ( x ))) | f a ( x ) , | U X b ( x ) = | ν ( x ) u b ( x ) f b ( x ) , (3.4) where (( U a ( x ))) U (2) and u b ( x ) U (1). The connection in the free module (3.1) induces a connection in the projective module M given by X = P free X where X ∈ M . It is represented by | ( X ) αβ ( x,y ) = (( P α ( x )))( | f β ( y ) - | f α ( x ) ) + (( A αβ ( x,y ))) | f β ( y ) . (3.5) The 2 × 2 matrices (( A αβ ( x,y ))) are given by : (( A aa ( x,y ))) = (( ω aa ( x,y ))) , (( A ab ( x,y ))) = | Φ ab ( x,y ) ν ( y ) | , (( A ba ( x,y ))) = | ν ( x ) Φ ba ( x,y ) | , (( A bb ( x,y ))) = | ν ( x ) ω b ( x,y ) ν ( y ) | , (3.6) where we have introduced the Ω (1) ( A )-valued ket- and bra- vectors : | Φ ab ( x,y ) = (( ω ab ( x,y ))) | ν ( y ) , Φ ba ( x,y ) | = ν ( x ) | (( ω ba ( x,y ) )) , (3.7) 17
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and the universal one-form : ω b ( x,y ) = ν ( x ) | (( ω bb ( x,y ))) | ν ( y ) . (3.8) The hermiticity condition (3.2) yields ; (( ω aa ( x,y ))) + = (( ω aa ( y,x ))) , ( ω b ( x,y )) * = ω b ( y,x ) , | Φ ab ( x,y ) + = Φ ba ( y,x ) | . (3.9) In H B H A : | Φ A ab ( x,y ) = | Φ B ab ( x,y ) c AB ( y ) - n/ 2 , Φ A ba ( x,y ) | = c AB ( x ) n/ 2 Φ B ba ( x,y ) | , ω A b ( x,y ) = c AB ( x ) n/ 2 ω B b ( x,y ) c AB ( y ) - n/ 2 . (3.10) The action of on X ∈ M is obtained using (3.6), (3.7) and (3.8) : | ( X ) aa ( x,y ) = | f a ( y ) - | f a ( x ) + (( ω aa ( x,y ))) | f a ( y ) , | ( X ) ab ( x,y ) = | H ab ( x,y ) f b ( y ) - | f a ( x ) , | ( X ) ba ( x,y ) = | ν ( x ) H ba ( x,y ) | f a ( y ) - f b ( x ) , | ( X ) bb ( x,y ) = | ν ( x ) f b ( y ) - f b ( x ) +( ω b ( x,y ) + m b ( x,y )) f b ( y ) , (3.11) where | H ab ( x,y ) = | Φ ab ( x,y ) + | ν ( y ) , H ba ( x,y ) | = Φ ba ( x,y ) | + ν ( x ) | . (3.12) and the ”monopole” connection m b ( x,y ) appears as : m b ( x,y ) = ν ( x ) | ν ( y ) - 1 . (3.13) As seen from (3.10) and (3.12), the off-diagonal connections | H ab ( x,y ) , H ba ( x,y ) | and also ω b ( x,y ) transform homogeneously from H B to H A but m b ( x,y ) transforms with the expected inhomogeneous term : m A b ( x,y ) = ( c AB ( x )) n/ 2 m B b ( x,y )( c AB ( y )) - n/ 2 +( c AB ( x )) n/ 2 [( c AB ( y )) - n/ 2 - ( c AB ( x )) - n/ 2 ] . (3.14) 18
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In terms of abstract universal differential one forms, (3.13) and (3.14) read : m b = 1 1 + | ν | 2 ν * d ν - 1 + | ν | 2 d 1 + | ν | 2 1 1 + | ν | 2 , m A b = ( c AB ) n/ 2 m B b ( c AB ) - n/ 2 + ( c AB ) n/ 2 d ( c AB ) - n/ 2 . The curvature of the connection is defined by : |∇ 2 X = (( R )) | X . It is a right-module homomorphism M → M⊗ A Ω (2) ( A ) given in the basis { E i } by the 2 × 2 matrix with values in Ω (2) ( A ) : (( R )) = (( P )) d (( A )) (( P )) + (( A )) 2 + (( P )) d (( P )) d (( P )) (( P )) , or, within the used realisation, by (( R αβγ ( x,y,z ))) = (( P α ( x ))) (( A βγ ( y,z ))) - (( A αγ ( x,z ))) + (( A αβ ( x,y ))) (( P β ( z ))) +(( A αβ ( x,y ))) (( A βγ ( y,z ))) +(( P α ( x ))) (( P β ( y ))) - (( P α ( x ))) (( P γ ( z ))) - (( P β ( y ))) (( P γ ( z ))) . (3.15) A connection compatible with the hermitian structure in M implies in a self-adjoint curvature : R i j = δ i ¯ ( R k ) + δ ¯ kj . (3.16) Let the connection be extended to M ⊗ A Ω ( A ) by X A F = ( X ) F + X A d F , then 2 becomes an endomorphism of the right Ω ( A )-module M⊗ A Ω ( A ).
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