Brane compactifications and 4-dimensional geometry in the IKKT model

We study in detail certain brane solutions with compact extra dimensions M4×K in the IKKT matrix model, with K being a two-dimensional rotating torus embedded in R6. We focus on the compactification moduli and the fluctuations of K⊂R6 and their physical significance. Mediated by the Poisson tensor, they contribute to the effective 4-dimensional metric on the brane, and thereby become gravitational degrees of freedom. We show that the zero modes corresponding to the global symmetries of the model lead to Ricci-flat 4-dimensional metric perturbations, wherever the energy-momentum tensor vanishes. Their coupling to the energy-momentum tensor depends on the extrinsic curvature of the brane.