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Published July 2013 | Accepted Version
Journal Article Open

Real structures on almost-commutative spectral triples

Abstract

We refine the reconstruction theorem for almost-commutative spectral triples to a result for real almost-commutative spectral triples, clarifying in the process both concrete and abstract definitions of real commutative and almost-commutative spectral triples. In particular, we find that a real almost-commutative spectral triple algebraically encodes the commutative *-algebra of the base manifold in a canonical way, and that a compact oriented Riemannian manifold admits real (almost-)commutative spectral triples of arbitrary KO-dimension. Moreover, we define a notion of smooth family of real finite spectral triples and of the twisting of a concrete real commutative spectral triple by such a family, with interesting KK-theoretic and gauge-theoretic implications.

Additional Information

© 2013 Springer. Received: 21 September 2012 / Revised: 22 January 2013 / Accepted: 4 February 2013. Published online: 1 March 2013. The author would like to thank his advisor, Matilde Marcolli, and Susama Agarwala, Alan Lai, and Kevin Teh, for helpful comments and conversations, as well as the Erwin Schrödinger Institute for its hospitality and its financial and administrative support in the context of the thematic programme "K-Theory and Quantum Fields." The author was also partially supported by NSF grant DMS-1007207.

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