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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}\ell _{D}(A)$\end{document} into its commutant, which in fact is isomorphic to the Clifford algebra itself (H is a self-Morita equivalence CℓD(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}\ell _{D}(A)$\end{document}-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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {C}\ell _{D}(A)$\end{document}-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.