Twisted reality and the second-order condition

Abstract: An interesting feature of the finite-dimensional real spectral triple (A,H,D,J) of the Standard Model is that it satisfies a second-order'' condition: conjugation by J maps the Clifford algebra Cl_D(A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence Cl_D(A)-bimodule). This resembles a property of the canonical spectral triple of a closed oriented Riemannian manifold: there is a dense subspace of H which is a self-Morita equivalence Cl_D(A)-bimodule. In this paper we argue that on manifolds, in order for the self-Morita equivalence to be implemented by a reality operator J, one has to introduce atwist'' and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples.