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# Its elements are of the form ξ ξ aa ξ ab ξ ba ξ

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Its elements are of the form ( ξ ) = ξ aa ξ ab ξ ba ξ bb , 17 For a general discussion on real discrete spectral triples, we refer to ([10],[17]) 36

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where each ξ αβ is a column vector with N αβ rows. An element λ = ( λ a b ) of A 1 acts on the left on H 2 by: π 2 ( λ ) = λ a 1 N aa 0 0 0 0 λ a 1 N ab 0 0 0 0 λ b 1 N ba 0 0 0 0 λ b 1 N bb , (5.16) and on the right by: π o 2 ( λ ) = λ a 1 N aa 0 0 0 0 λ b 1 N ab 0 0 0 0 λ a 1 N ba 0 0 0 0 λ b 1 N bb . (5.17) Although A 2 is an abelian algebra and, as such, isomorphic to its opposite algebra , it is not a simple algebra so that, in general, π 2 ( λ ) = π o 2 ( λ ). The dis- crete real structure, given by J 2 = C 2 K , relates both by π o ( λ ) = J 2 π ( λ ) + J - 1 2 so that C 2 is an intertwining operator for the two representations: π o 2 ( λ ) = C 2 π 2 ( λ ) C - 1 2 . (5.18) This requires that N ab = N ba . = N and, since we require J 2 to be anti-unitary, the basis in H 2 may be chosen such that : C 2 = 1 N aa 0 0 0 0 0 1 N 0 0 1 N 0 0 0 0 0 1 N bb . (5.19) This implies that : J 2 2 = 2 1 2 , with 2 = +1 . (5.20) The chirality, χ 2 , defining the orientation of the spectral triple is the image of a Hochschild 0-cycle, i.e. an element of A 2 ⊗ A o 2 . This implies that χ 2 is diagonal and χ αβ = ± 1 on each subspace C N αβ . Furthermore, demanding that 18 J 2 χ 2 = 2 χ 2 J 2 , with 2 = +1 , (5.21) 18 If we should require that 2 = - 1, then N aa = N bb = 0 and χ ab = - χ ba and the corresponding odd Dirac operator would not satisfy the first order condition. 37
requires χ ab = χ ba = χ so that the chirality in H 2 reads: χ 2 = χ aa 1 N aa 0 0 0 0 χ 1 N 0 0 0 0 χ 1 N 0 0 0 0 χ bb 1 N bb . (5.22) We consider the following three possibilities leading to a non trivial hermitian Dirac operator, odd with respect to this chirality : 2.a + χ aa = + χ = - χ bb = ± 1 2.b - χ aa = + χ = + χ bb = ± 1 2.c - χ aa = + χ = - χ bb = ± 1 The corresponding Dirac operators have the form 2.a,2.b D 2 .a = 0 0 0 K + 0 0 0 A + 0 0 0 B + K A B 0 ; D 2 .b = 0 B + A + K + B 0 0 0 A 0 0 0 K 0 0 0 2.c D 2 .c = 0 B + A + 0 B 0 0 A + A 0 0 B + 0 A B 0 . The first-order condition [[ D 2 2 ( λ )] o 2 ( μ )] = 0, satisfied in case 2.c) , im- plies that in case 2.a) and 2.b) K must vanish. If we asssume J 2 D 2 = 2 D 2 J 2 , with 2 = +1 , (5.23) then B = A * ; B = A * . (5.24) It should be stressed that, in order to have a non trivial Dirac operator, necessarily N = 0. This confirms that the discrete Hilbert space H dis used 38

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in 4 does not allow for a real structure in the above sense. At last, it can be shown[10, 17] that noncommutative Poincar´ e duality, in the discrete case, amounts to the non degeneracy of the intersection matrix with elements αβ = χ αβ N αβ . This non degeneracy condition in case 2.a) and 2.b) reads N aa N bb + N 2 = 0 and is always satisfied. In case 2.c) it is required that N aa N bb - N 2 = 0 and if all N ’s should be equal, this would not be satisfied.
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