2000/12/31 by Anatoly Konechny, A. Konechny, Albert Schwarz +1 · 258 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Black Holes and Theoretical Physics #Geometry #Mathematical physics #Noncommutative and Quantum Gravity Theories #Noncommutative geometry #Physics #Theoretical physics #hep-th
paper · pdf · doi:10.1016/s0370-1573(01)00096-5
published in Physics Reports 360(5-6), 353-465 (Elsevier BV) · v. 3: section 8.2 expanded, typos corrected, two references added. The second part of this review is now available as a preprint hep-th/0107251. Both parts will be published together supplemented by the English translation of A. Connes' 1980 seminal paper "C*algebras and differential geometry", Compt. Rend. Acad. Sci. (Ser.I Math.) A290 (1980) 599-604. This translation is also available as hep-th/0101093
arxiv created 2001/07/29 · openalex publication_date 2002/04/01 · arxiv updated 2015/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Noncommutative geometry is based on an idea that an associative algebra can be regarded as "an algebra of functions on a noncommutative space". The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang-Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang-Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics. In this paper we give a mostly self-contained review of some aspects of M(atrix) theory, of Connes' noncommutative geometry and of applications of noncommutative geometry to M(atrix) theory. The topics include introduction to BFSS and IKKT matrix models, compactifications on noncommutative tori, a review of basic notions of noncommutative geometry with a detailed discussion of noncommutative tori, Morita equivalence and SO(d,d|\mathbb Z)-duality, an elementary discussion of instantons and noncommutative orbifolds. The review is primarily intended for physicists who would like to learn some basic techniques of noncommutative geometry and how they can be applied in string theory and to mathematicians who would like to learn about some new problems arising in theoretical physics.