Non-commutative geometry and matrix models

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 considered. The one-loop effective action for the maximally supersymmetric IKKT or IIB matrix model is discussed, which is well-behaved on 4-dimensional branes.

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

“…It is easiest to use the x a description where the embedding is spherical, but the target space metric is g ab (3.10). The effective metric G µν in matrix models is determined by the kinetic term for a scalar field 6 as follows [9,10,30]…”

Closed star product on noncommutative ℝ 3 and scalar field dynamics

“…2 Note however that the matrix model interpretation of noncommutative gauge theories provided interesting results on (semi-)classical properties and/or one-loop computations. For a review on the related literature, see [46,47] (see also [48]- [51] and references therein). Recently, scalar field theories on R 3 λ , a deformation of R 3 introduced in [52] (see also [53]), have been studied in [54].…”

Noncommutative gauge theories on $ \mathbb{R}_{\theta}^2 $ as matrix models

“…To our knowledge this interpretation has not been explored so far for the action (1.1), although the matrix model formulation of NC gauge theory has been known since many years [61][62][63][64][65][66][67]. Keeping in mind that A µ is a natural variable of the Moyal geometry [46,47], a sensible question is to explore the properties of the action as a functional of the field A µ , in order to determine to what extent S Ω [A µ ] may give rise to a meaningful quantum theory.…”