Arguments for this to happen were given in section 51

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arguments for this to happen, were given in section 5.1 . It was also shown that the covariantisation of the real spectral triple with the nontrivial M allows the abelian gauge fields to survive, while they are slain if covaraianti- sation is done with a trivial module. Finally a possibility of solution to the chirality problem 22 has been indicated. 22 By this we mean the non vanishing of the chiral matter action. 49
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A On modules and connections Let M be a right module over the -algebra A . The set of endomorphisms B = END A ( M ) has an obvious algebra structure and he left action of A ∈ B on X ∈ M commutes with the right action of a ∈ A : ( A X ) a = A ( Xa ) so that M acquires canonicall a B - A bimodule structure. The dual module M * = HOM A ( M , A ) is a left A -module and A ∈ B acts on the right on ξ ∈ M * by ( ξ A ,X ) = ( ξ, A X ) so that M * is a A - B bimodule. When M is endowed with a sesquilinear, hermitian and non-degenerate form h : M×M → A : ( X,Y ) h ( X,Y ), there is a canonical bijective mapping h : M → M * : X h ( X ) h ( X ) ,Y = h ( X,Y ) . (A.1) Since h ( Xa ) = a * h ( X ), h is an anti-isomorphism with inverse ( h ) - 1 = h : ξ h ( ξ ) , and h ( ) = h ( ξ ) a * . The inverse form h - 1 is defined by : h - 1 : M * × M * → A : ( ξ,η ) h - 1 ( ξ,η ) = h h ( ξ ) , h ( η ) . The hermitian conjugate of A ∈ B is defined by h ( X, A + Y ) = h ( A X,Y ). Let Ω ( A ) = k Z Ω ( k ) ( A ) be a graded differential * -envelope of A , then h can be extended to M = M ⊗ A Ω ( A ) by : h ( X A F,Y A G ) = F + h ( X,Y ) G, (A.2) where X,Y ∈ M and F,G Ω ( A ). Defining M *• . = Ω ( A ) A M * , h can be extended as a mapping M → M *• by ( X A F ) = F + A X , where h ( ) is written as ( ) . A connection in M is an additive map : M → M ⊗ A Ω (1) ( A ) : X → ∇ X, (A.3) which is additive and obeys the Leibniz rule ( Xa ) = ( X ) a + X A d a. It defines an associate dual connection * in M * by : ( * ξ,X ) + ( ξ, X ) = d ( ξ,X ) . (A.4) 50
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It obeys * ( ) = d a A ξ + a ( * ξ ). The extension of to M by ( X A F ) = ( X ) F + X A d F , allows to define the curvature as 2 : M → M and it is seen that 2 is in fact an endomorphism of M considered as a right Ω -module : 2 ( X A F ) G = 2 ( X A F ) G. (A.5) Similarly, * is extended to M *• and * 2 ( G A ξ ) ,X A F = G A ξ, 2 ( X A F ) . When M has an hermitian structure, the connection is said to be compatible with this hermitian structure, if 23 d ( h ( X,Y )) = - h ( X,Y ) + h ( X, Y ) (A.6) or equivalently if d h - 1 ( ξ,η ) = h - 1 ( * ξ,η ) - h - 1 ( ξ, * η ) . The mapping h relates both connections through * h ( X ) = - h ( X ) or * X = - ( X ) . (A.7) The curvature of a compatible connection is hermitian : h ( 2 X,Y ) = h ( X, 2 Y ) . (A.8) The algebras A and B = END A ( M ) are said to be Morita equivalent in the sense that there exists a B -A bimodule M and a A-B bimodule M * such that M ⊗ A M * B , with the identification ( X A ξ ) Y = X ( ξ,Y ) and η ( X A ξ ) = ( η,X ) ξ , and M * B M A with Y ( ξ B X ) = Y ( ξ,X ) and ( ξ B X ) η = ( ξ,X ) η .
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