  We study in detail certain brane solutions with compact extra dimensions M^4
x K in the IKKT matrix model, with K being a two-dimensional rotating torus
embedded in R^6. We focus on the compactification moduli and the fluctuations
of K \subset R^6 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.
