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 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.

Twisted Reality and the Second-Order Condition / Dabrowski, L.; D'Andrea, F.; Magee, A. M.. - In: MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY. - ISSN 1385-0172. - 24:2(2021), pp. 1-27. [10.1007/s11040-021-09384-4]

Twisted Reality and the Second-Order Condition

Dabrowski, L.;Magee, A. M.
2021-01-01

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 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.
2021
24
2
1
27
13
https://arxiv.org/abs/1912.13364
Dabrowski, L.; D'Andrea, F.; Magee, A. M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135233
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