Non-commutative geometry and matrix models

TL;DR

This paper introduces noncommutative matrix geometries within matrix models, exploring their structure, effective geometry, and dynamical properties, especially in relation to 4-dimensional branes and supersymmetric models.

Contribution

It formulates a general notion of embedded noncommutative spaces (branes) and analyzes their effective Riemannian geometry and dynamical behavior in matrix models.

Findings

Effective Riemannian geometry for noncommutative branes

Preservation of geometric configurations under small deformations

Well-behaved one-loop effective action on 4D branes

Abstract

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.