A Kastler–Kalau–Walze type theorem for five-dimensional manifolds with boundary

Abstract

The Kastler-Kalau-Walze theorem, announced by Alain Connes, shows that the Wodzicki residue of the inverse square of the Dirac operator is proportional to the Einstein-Hilbert action of general relativity. In this paper, we prove a Kastler-Kalau-Walze type theorem for 5-dimensional manifolds with boundary.

Perturbations of Dirac Operators and A KKW Type Theorem for Five Dimensional Manifolds with Boundary

“…Since the sum is taken over −r − ℓ + k + j + |α| − 1 = 5, r, ℓ ≤ −1, then we have the ∂M Φ is the sum of the following fifteen cases. Such cases (1)-( 6) have been studied, similar to Case (1)-Case (6) in [15], we obtain Case (1):…”

“…Let us now consider the q −3 of the Dirac operators with one-form perturbations. From Lemma 3.7 in [15], we have Lemma 3.4. [15] For Dirac operators, the following identity holds:…”

“…Furthermore, using Dirac operators with perturbations,Wang defined a spectral triple and established an infinitesimal equivariant index formula in [13,14]. In [15], we proved a Kastler-Kalau-Walze type theorem for 5-dimensional manifolds with boundary. Motivated by [10][11][12][13][14][15], we study Dirac operators with one-form perturbations.…”

“…Now we discuss the cases ( 7)- (15). Case (7): r = −1, ℓ = −2, k = 0, j = 1, |α| = 0 From (2.9) and the Leibniz rule, we obtain…”

“…In [15], we proved a Kastler-Kalau-Walze type theorem for 5-dimensional manifolds with boundary. Motivated by [10][11][12][13][14][15], we study Dirac operators with one-form perturbations. In the present paper, we shall restrict our attention to the case of Wres[π + (D + c(X)) −1 • π + (D + c(X)) −1 ] for five dimensional manifolds with boundary.…”