Twisted Reality and the Second-Order Condition

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 CℓD(A) into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence CℓD(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 CℓD(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 a “twist” and weaken one of the axioms of real spectral triples. We then investigate how the above mentioned conditions behave under products of spectral triples.