The pseudo-differential perturbations of the Dirac operator and the Kastler-Kalau-Walze type theorems (Q6630802)

Basic references for this paper are the independent simultaneous contributions of \textit{D. Kastler} [Commun. Math. Phys. 166, No. 3, 633--643 (1995; Zbl 0823.58046)] and \textit{W. Kalau} and \textit{M. Walze} [J. Geom. Phys. 16, No. 4, 327--344 (1995; Zbl 0826.58008)], who proved a conjecture of A. Connes, stating that the Wodzicki residue of the square inverse of the Dirac operator is connected with the Einstein-Hilbert action on spin compact manifolds of dimension 4, or larger even dimension. Several extensions of this result were given subsequenly, see in particular [\textit{U. Battisti} and \textit{S. Coriasco}, J. Pseudo-Differ. Oper. Appl. 2, No. 3, 303--315 (2011; Zbl 1263.58009)] concerning non-compact manifolds, and a series of papers of the present authors, concerning manifolds with boundary. Here the authors consider the same issue for two types os pseudo-differential perturbations of the Dirac operator on 4-dimensional compact oriented spin manifolds. A precise expression of the Wodzicki residue is given in terms of integrals involving the scalar curvature. Some interesting examples are provided.