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 in [1]. The mechanism requires a brane configuration with structure ${\cal M}^{3,1}\times {\cal K} \subset {\mathbb R}^{9,1}$, where ${\cal M}^{3,1}$ is the noncommutative space-time brane, and ${\cal K}$ 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 ${\cal K}$. 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 ${\cal K}$ 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.