Mathematics

According to A. Connes the topology and metric structure of a spin manifold can be encoded in terms of the canonical spectral triple, of which the crucial ingredient is the Dirac operator on Dirac spinors. Another natural spectral triple on a manifold incorporates the Hodge-de Rham operator acting on differential forms. Both of them can be characterized in terms of certain Morita equivalence, which involves the Clifford algebra. These notions admit a generalization to noncommutative algebras, and in particular we can determine of which type is the internal finite spectral triple in the almost commutative formulation of the Standard Model of fundamental particles and their interactions.