The Kastler-Kalau-Walze type theorem about Witten deformation for manifolds with boundary

In this paper, we compute the lower dimensional volume Vol6(1,3) for 6-dimensional spin manifolds with boundary and derive the gravity action on boundary by the noncommutative residue associated with Witten deformation. We also give a theoretical explanation of the gravitational action for 6 dimensional manifolds with boundary by the noncommutative residue associated with Witten deformation.