Non-commutative geometry and matrix models

Details Non-commutative geometry and matrix models Non-commutative geometry and matrix models

2011-09-26

arxiv.org/abs/1109.5521v2

Physics

High Energy Physics

High Energy Physics - Theory

26 pages. Lectures delivered at the 3rd Quantum Geometry and Quantum Gravity School, Zakopane 2011. V2: minor improvements

Scientific paper

These notes provide an introduction to the noncommutative matrix geometry which arises within matrix models of Yang-Mills type. Starting from basic examples of compact fuzzy spaces, a general notion of embedded noncommutative spaces (branes) is formulated, and their effective Riemannian geometry is elaborated. This class of configurations is preserved under small deformations, and is therefore appropriate for matrix models. A realization of generic 4-dimensional geometries is sketched, and the relation with spectral geometry and with noncommutative gauge theory is explained. In a second part, dynamical aspects of these matrix geometries are discussed. The one-loop effective action for the maximally supersymmetric IKKT or IIB matrix model is discussed, which is well-behaved on 4-dimensional branes.

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