Brane compactifications and 4-dimensional geometry in the IKKT model

We study in detail certain brane solutions with compact extra dimensions image in the IKKT matrix model, with image being a two-dimensional rotating torus embedded in image. We focus on the compactification moduli and the fluctuations of image 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.