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

Abstract

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 M 3 , 1 × K ⊂ ℝ 9 , 1 $$ {\mathcal{M}}^{3,1}\times \mathcal{K}\subset {\mathbb{R}}^{9,1} $$ , where M 3 , 1 $$ {\mathcal{M}}^{3,1} $$ is the noncommutative space-time brane, and K $$ \mathcal{K} $$ 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 K $$ \mathcal{K} $$ . 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 K $$ \mathcal{K} $$ 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.

Keywords

• M(atrix) Theories
• Models of Quantum Gravity
• Non-Commutative Geometry