On K-K-W Type Theorems for Conformal Perturbations of Twisted Dirac Operators

Abstract

In this paper, we prove two Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of twisted signature operators on four-dimensional manifolds with (resp. without) boundary.

Conformal Perturbations of Twisted Dirac Operators and Noncommutative residue

“…ij and the cotangent vector ξ = ξ j dx j and ξ j = g ij ξ i , by Lemma 1 in [13] and Lemma 2.1 in [12], for any fixed point x 0 ∈ ∂M , we can choose the normal coordinates U of x 0 in ∂M (not in M ), by the composition formula and (2.2.11) in [12], we obtain in [19],…”

“…For convenience, let ĉ(ω) = i ĉ(e i )ω(F, g F )(e i ) and ĉ(ω * ) = i ĉ(e i )ω * (F, g F )(e i ), by the composition formula and (2.2.11) in [12], we obtain in [19],…”

“…In the normal coordinate, [12], we have Γ n (x 0 ) = 5 2 h ′ (0) and Γ k (x 0 ) = 0 for k < n. By the definition of δ k and Lemma 2.3 in [12], we have δ n (x 0 ) = 0 and δ k = 1 4 h ′ (0)c( e k )c( e n ) for k < n. By (3.15) in [19], we obtain…”

“…In [19], J. Wang and Y. Wang proved two kinds of Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of signature operators by a vector bundle with a non-unitary connection on four-dimensional manifolds with (respectively without)boundary.…”

Twisted dirac operators and Kastler-Kalau-Walze theorems for six-dimensional manifolds with boundary

“…Theorem 2.2. [18] For even m-dimensional compact spin manifolds without boundary, the following equality holds:…”