The noncommutative residue about Witten deformation
From the introduction: ``The motivation of this paper is to give two kinds of operator theoretic explanation of the gravitational action for manifolds with boundary by the noncommutative residue associated with the Witten deformation, and prove two Kastler-Kalau-Walze type theorems for manifolds with boundary. \par This paper is organized as follows: In Section 2, we recall the definition of Witten deformation $D_T$. In Section 3, we recall the definition of lower dimensional volumes of spin manifolds with boundary. In Section 4, we compute the lower dimensional volume $Vol_{4}^{(1,1)}$ and get a Kastler-Kalau-Walze type theorem in this case. We also give a theoretic explanation of the gravitational action for 4-dimensional manifolds with boundary by this noncommutative residue. In Section 5, we compute the lower dimensional volume $Vol_{6}^{(2,2)}$ and get a Kastler-Kalau-Walze type theorem in this case. We also give a theoretic explanation of the gravitational action for 6-dimensional manifolds with boundary by this noncommutative residue.'' \par The authors also explain in the introduction that the Kastler-Kalau-Walze theorem states the observation of of A. Connes, that the noncommutative residue of the inverse of the Dirac operator is proportional to the Hilbert-Einstein action.
scientific article; zbMATH DE number 7043622
scientific article