If the geometry of space-time is noncommutative, i.e. [x µ , x ν ] = iθ µν , then noncommutative CP violating effects may be manifest at low energies. For a noncommutative scale Λ ≡ θ −1/2 ≤ 2 T eV , CP violation from noncommutative geometry is comparable to that from the Standard Model (SM) alone: the noncommutative contributions to ǫ and ǫ ′ /ǫ in the K-system, and to sin 2β in the B-system, ...

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 f...

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 f...

In the past 15 years a study of “noncommutative projective geometry” has flourished. By using and generalizing techniques of commutative projective geometry, one can study certain noncommutative graded rings and obtain results for which no purely algebraic proof is known. For instance, noncommutative graded domains of quadratic growth, or “noncommutative curves,” have now been classified by geo...

We give a mostly self-contained review of some aspects of M(atrix) theory and noncommutative geometry. 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|Z)-duality, an elementary discussion of instantons an...

This paper is intended as an introduction to noncommutative geometry for readers with some knowledge of abstract algebra and differential geometry. We show how to extend the theory of differential forms to the “noncommutative spaces” studied in noncommutative geometry. We formulate and prove the Hochschild-Kostant-Rosenberg theorem and an extension of this result involving the Connes differential.

In this note I speculate on connections between the noncommutative geometry approach to the standard model on one side, and the internal space coming from strings on the other. The standard model in noncommutative geometry is described via the spectral action. I argue that an internal noncommutative manifold compactified at the renormalization scale, could give rise to the almost commutative ge...