Noncommutative Geometry

Noncommutative geometry

Gravity From Noncommutative Geometry

We introduce the linear connection in the noncommutative geometry model of the product of continuous manifold and the discrete space of two points. We discuss its metric properties, de ne the metric connection and calculate the curvature. We de ne also the Ricci tensor and the scalar curvature. We nd that the latter di ers from the standard scalar curvature of the manifold by a term, which migh...

Noncommutative Geometry for Pedestrians∗

A short historical review is made of some recent literature in the field of noncommutative geometry, especially the efforts to add a gravitational field to noncommutative models of space-time and to use it as an ultraviolet regulator. An extensive bibliography has been added containing reference to recent review articles as well as to part of the original literature. LMU-TPW 99-11 ∗Lecure given...

Duality in Noncommutative Geometry

The structure of spacetime duality and discrete worldsheet symmetries of compactified string theory is examined within the framework of noncommutative geometry. The full noncommutative string spacetime is constructed using the Fröhlich-Gawȩdzki spectral triple which incorporates the vertex operator algebra of the string theory. The duality group appears naturally as a subgroup of the automorphi...

Noncommutative Geometry and Quantization

The purpose of these lectures is to survey several aspects of noncommutative geometry, with emphasis on its applicability to particle physics and quantum field theory. By now, it is a commonplace statement that spacetime at short length —or high energy— scales is not the Cartesian continuum of macroscopic experience, and that older methods of working with functions on manifolds must be rethough...