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

Abstract

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 non-commutative (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 non-commutative \( \mathcal{N} = 4 \) 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 nontrivial loop computations in the gauge theory.