2012/07/31 by Koen van den Dungen, Mario Paschke, Adam Rennie · 53 citations
Mathematics · Medicine · Physics and Astronomy · #Advanced Neuroimaging Techniques and Applications #Advanced Operator Algebra Research #Atiyah–Singer index theorem #Class (philosophy) #Computer science #Context (archaeology) #Geometric Analysis and Curvature Flows #Harmonic oscillator #Mathematical analysis #Mathematics #Noncommutative geometry #Noncommutative quantum field theory #Physics #Pure mathematics #Quantum mechanics #Riemannian geometry #Spectral triple #math-ph #math.KT #math.MP #math.OA
paper · pdf · doi:10.1016/j.geomphys.2013.04.011
published in Journal of Geometry and Physics 73, 37-55 (Elsevier BV)
arxiv created 2013/04/17 · openalex publication_date 2013/05/09 · arxiv updated 2015/03/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We define pseudo-Riemannian spectral triples, an analytic context broad enough to encompass a spectral description of a wide class of pseudo-Riemannian manifolds, as well as their noncommutative generalisations. Our main theorem shows that to each pseudo-Riemannian spectral triple we can associate a genuine spectral triple, and so a K-homology class. With some additional assumptions we can then apply the local index theorem. We give a range of examples and some applications. The example of the harmonic oscillator in particular shows that our main theorem applies to much more than just classical pseudo-Riemannian manifolds.