2000/03/01 by Alain Connes
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Dirac equation #Dirac operator #Mathematical physics #Mathematics #Noncommutative algebraic geometry #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Noncommutative quantum field theory #Pure mathematics #Quantum differential calculus #Spectral triple #hep-th #math.QA
paper · pdf · doi:10.1063/1.533329
published as J.Math.Phys. 41 (2000) 3832-3866 · Invited lecture for JMP 2000, 45p
arxiv created 2000/03/01 · openalex publication_date 2000/06/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We give a survey of selected topics in noncommutative geometry, with some emphasis on those directly related to physics, including our recent work with Dirk Kreimer on renormalization and the Riemann–Hilbert problem. We discuss at length two issues. The first is the relevance of the paradigm of geometric space, based on spectral considerations, which is central in the theory. As a simple illustration of the spectral formulation of geometry in the ordinary commutative case, we give a polynomial equation for geometries on the four-sphere with fixed volume. The equation involves an idempotent e, playing the role of the instanton, and the Dirac operator D. It is of the form 〈(e−12)[D,e]4〉=γ5 and determines both the sphere and all its metrics with fixed volume form. The expectation 〈x〉 is the projection on the commutant of the algebra of 4 by 4 matrices. We also show, using the noncommutative analog of the Polyakov action, how to obtain the noncommutative metric (in spectral form) on the noncommutative tori from the formal naive metric. We conclude with some questions related to string theory.