A UNITARY INVARIANT IN RIEMANNIAN GEOMETRY

2008/10/12 by Alain Connes · 30 citations
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dirac operator #Geometry #Invariant (physics) #Mathematical physics #Mathematics #Noncommutative geometry #Physics #Pure mathematics #Riemannian geometry #Spectral triple #Symmetric space #Unitary state #hep-th #math.QA

paper · pdf · doi:10.1142/s0219887808003284

published in International Journal of Geometric Methods in Modern Physics 05(08), 1215-1242 (World Scientific) · 25 pages, 1 Figure

arxiv created 2008/10/12 · openalex publication_date 2008/12/01 · arxiv updated 2011/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi–Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.

Citations

• A finiteness result for commuting squares of matrix algebras
• Reconstruction of manifolds in noncommutative geometry
• Riemannian Coverings and Isospectral Manifolds
• Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren
• EIGENVALUES OF THE LAPLACE OPERATOR ON CERTAIN MANIFOLDS
• Embedding Riemannian manifolds by their heat kernel
• The Local Index Formula in Noncommutative Geometry
• The Spectral Action Principle
• On Rings of Operators. Reduction Theory

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