A Kastler-Kalau-Walze type theorem and the spectral action for   perturbations of Dirac operators on manifolds with boundary
  In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds. 
Yong Wang
math.DG math-ph math.MP