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

We compute the quantum effective action induced by integrating out fermions in Yang-Mills matrix models on a 4-dimensional background, expanded in powers of a gauge-invariant UV cutoff. The resulting action is recast into the form of generalized matrix models, manifestly preserving the SO(D) symmetry of the bare action. This provides noncommutative (NC) analogs of the Seeley-de Witt coefficients for the emergent gravity which arises on NC branes, such as curvature terms. From the gauge theory point of view, this provides strong evidence that the NC N=4 SYM has a hidden SO(10) symmetry even at the quantum level, which is spontaneously broken by the space-time background. The geometrical view proves to be very powerful, and allows to predict non-trivial loop computations in the gauge theory.