Dirac operators with torsion and the noncommutative residue for manifolds with boundary

In this paper, we get the Kastler–Kalau–Walze theorem associated to Dirac operators with torsion on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of 4-dimensional compact manifolds with flat boundary. Furthermore, we get the Kastler–Kalau–Walze type theorem for four dimensional complex manifolds associated with nonminimal operators.