On the Product of Real Spectral Triples

The product of two real spectral triples {A1,H1,D1,J1, γ1} and {A2,H2,D2,J2(, γ2)}, the first of which is necessarily even, was defined by A.Connes [3] as {A,H,D,J (, γ)} given by A = A1 ⊗ A2, H = H1 ⊗ H2, D = D1 ⊗ Id2 + γ1 ⊗ D2, J = J1 ⊗ J2 and, in the even-even case, by γ = γ1⊗γ2. Generically it is assumed that the real structure J obeys the relations J 2 = ǫId, JD = ǫ ′DJ , J γ = ǫ ′′γJ , where the ǫ-sign table depends on the dimension n modulo 8 of the spectral triple. If both spectral triples obey Connes’ ǫ-sign table, it is seen that their product , defined in the straightforward way above, does not necessarily obey this ǫ-sign table. In this note, we propose an alternative definition of the product real structure such that the ǫ-sign table is also satisfied by the product. PACS numbers : 11.15.-q, 02.40.-k ;