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Fermions and noncommutative emergent gravity II: Curved branes in extra dimensions

We study fermions coupled to Yang-Mills matrix models from the point of view of emergent gravity. The matrix model Dirac operator provides an appropriate coupling for fermions to the effective gravitational metric for general branes with nontrivial embedding, albeit with a non-standard spin connection. This generalizes previous results for 4-dimensional matrix models. Integrating out the fermions in a nontrivial geometrical background induces indeed the Einstein-Hilbert action of the effective metric, as well as additional terms which couple the Poisson tensor to the Riemann tensor, and a dilaton-like term.