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A short survey of noncommutative geometry. (English) Zbl 0974.58008

Summary: 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 \(\langle(e - \frac 12)[D,e]^4\rangle = \gamma_5\) and determines both the sphere and all its metrics with fixed volume form. The expectation \(\langle x\rangle\) 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.

MSC:

58B34 Noncommutative geometry (à la Connes)
58J42 Noncommutative global analysis, noncommutative residues
46L99 Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)
81T18 Feynman diagrams
81T75 Noncommutative geometry methods in quantum field theory
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