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

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