One-loop effective action and emergent gravity on quantum spaces in the IKKT matrix model

Abstract

A detailed derivation of 3 + 1 dimensional induced or emergent gravity in the IKKT matrix model at one loop is given, as announced in [1]. The mechanism requires a brane configuration with structure {mathcal{M}}^{3,1}times mathcal{K}subset {mathbb{R}}^{9,1} , where {mathcal{M}}^{3,1} is the noncommutative space-time brane, and mathcal{K} are compact fuzzy extra dimensions embedded in target space. The 3+1-dimensional Einstein-Hilbert action then arises in the one loop effective action of the maximally supersymmetric IIB or IKKT matrix model, with effective Newton constant determined by the Kaluza-Klein scale of mathcal{K} . At weak coupling, all physical modes are confined to the brane, leading to 3 + 1-dimensional low-energy physics. The Einstein-Hilbert action can be interpreted as interaction of mathcal{K} with the space-time brane via IIB supergravity. The vacuum energy does not act as cosmological constant, but stabilizes the brane structure at one loop.