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We compute the quantum effective action induced by integrating out fermionsin Yang-Mills matrix models on a 4-dimensional background, expanded in powersof a gauge-invariant UV cutoff. The resulting action is recast into the form ofgeneralized matrix models, manifestly preserving the SO(D) symmetry of the bareaction. This provides noncommutative (NC) analogs of the Seeley-de Wittcoefficients for the emergent gravity which arises on NC branes, such ascurvature terms. From the gauge theory point of view, this provides strongevidence that the NC N=4 SYM has a hidden SO(10) symmetry even at the quantumlevel, which is spontaneously broken by the space-time background. Thegeometrical view proves to be very powerful, and allows to predict non-trivialloop computations in the gauge theory.

Heat kernel expansion and induced action for the matrix model Dirac operator