# 14 the connection is hermitian if d h xy h x free y h

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The connection is hermitian if d h ( X,Y ) = h ( X, free Y ) - h ( free X,Y ). For the product above, this yields (( ω )) i j = δ i k (( ω )) k δ j . The representation of ω by functions on M × M is given by : (( ω )) K αβ ( x,y ) S αβ ( x,y ) T αβ ( x,y ) L αβ ( x,y ) and the hermiticity condition reads : K αβ ( x,y ) = K βα ( y,x ) * , L αβ ( x,y ) = L βα ( y,x ) * , T αβ ( x,y ) = S βα ( y,x ) * . (3.2) The action of the connection (3.1) is represented by : ( free X ) i αβ ( x,y ) = f i β ( y ) - f i α ( x ) + (( ω )) i k,αβ ( x,y ) f k β ( y ) . A projective module is defined by an endomorphism P of A 2 which is idem- potent, P 2 = P , and hermitian, P = P , where the adjoint A of an endo- morphism A is defined by h ( X, A Y ) = h ( A X,Y ). In the basis { E i } , the projector is given by a 2 × 2 matrix ( P ) i j with entries in A and is represented by ( P ) i j,α ( x ). The projective module M is defined as the image of P : M = P X | X ∈ A 2 = X ∈ A 2 | P X = X . The hermiticity of the projector guarantees that h , restricted to M , defines a hermitian product in M . In the Connes-Lott model [4], the projectors are of the form (( P )) i j,a ( x ) = δ i j and (( P )) i j,b ( x ) = 1 2 1 + n ( x ) i j , where σ are the Pauli matrices and n ( x ) is a real unit vector mapping S 2 S 2 so that the projectors are classified by π 2 ( S 2 ) = Z . Furthermore, since π 2 ( U (2)) = π 2 ( SU (2)) = π 2 ( S 3 ) = { 1 } , projectors, belonging to different homoptopy classes, cannot be unitarily equivalent. The target sphere S 2 also has two coordinate charts H target B and H target A . 15
In these charts, the projector (( P b )) can be written as (( P B b )) = | ν B ν B | , respectively (( P A b )) = | ν A ν A | , where ν B , respectively ν A , is the complex coordinate of n in H target B , respectively H target A . We have used the Dirac ket- and bra-notation : | ν B = 1 1 + | ν B | 2 1 ν B , ν B | = 1 1 + | ν B | 2 1 ν * B , | ν A = 1 1 + | ν A | 2 - ν A 1 , ν A | = 1 1 + | ν A | 2 - ν * A 1 . An element X of A 2 is represented by the column matrix X | f a ( x ) | f b ( x ) , where | f a ( x ) = f 1 a ( x ) f 2 a ( x ) and | f b ( x ) = f 1 b ( x ) f 2 b ( x ) . It belongs to M if P X = X , which yields no restriction on | f a ( x ) but there is one on | f b ( x ) . In H target B it is expressed as | f b = | ν B f B b , where f B b = ν B | f b = 1 1 + | ν B | 2 ( f 1 b + ν * B f 2 b ) . In the same way, in H target A one has | f b = | ν A f A b , where f A b = ν A | f b = 1 1 + | ν A | 2 ( - ν * A f 1 b + f 2 b ) . As representatives of the homotopy class [ n ] π 2 ( S 2 ) Z , we choose a mapping n transforming H B , respectively H A of the range S 2 , into H target B , respectively H target A of the target S 2 . Such a choice is 11 ν B ( x ) = ζ B ζ * B n - 1 2 ζ B , ν A ( x ) = ζ A ζ * A n - 1 2 ζ A ; n Z , (3.3) where ζ B , ζ A are the complex coordinates of x S 2 . In the overlap H A H B , with transition function c AB given by (2.3), we have: f A b ( x ) = c AB ( x ) n/ 2 f B b ( x ) 11 Note that this choice is different from the one in previous work [14]. 16

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and this tells us that f b ( x ) is a Pensov field of spin-weight n/ 2.
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