  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.
