A Short survey of noncommutative geometry

By Alain Connes. 2000

208 citations recorded by the source corpus.

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 fr...

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e521b736678500b1a4583a3b046b1325b9bd36b4

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In Algebra and Number Theory

1. Giovanni Landi
2. G. Giachetta
3. G. Sardanashvily
4. A. Schwarz
5. A. Konechny
6. Alain Connes
7. Sorin Popa
8. H. Moscovici

In Mathematical Physics

1. Giovanni Landi
2. G. Giachetta
3. G. Sardanashvily
4. A. Schwarz
5. A. Konechny
6. Alain Connes
7. Matilde Marcolli
8. H. Moscovici

In Statistical and Nonlinear Physics

1. Giovanni Landi
2. A. Schwarz
3. A. Konechny
4. G. Giachetta
5. G. Sardanashvily
6. Alain Connes
7. Matilde Marcolli
8. Joseph Keir

Citations in the corpus

• What is geometry in quantum theory (2004, background): “In Connes’ noncommutative geometry, the more deep analogy to the case of commutative smooth function rings leads to the notion of a spectral triple (A, E,D) [20, 142].”
• Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations (2000, methodology): “Let us recall from [10] that the equality,”
• Introduction to M(atrix) theory and noncommutative geometry (2000, background): “See also papers [38], [39] for recent reviews.”