The Kastler–Kalau–Walze Type Theorems for the Perturbation of the de Rham Hodge Operator

Abstract

AbstractAs applications of the noncommutative residue, we prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds. To get these theorems, we give the Lichnerowicz type formulas for the perturbation of the de Rham Hodge operator on compact manifolds without boundary and the calculation formulas of the boundary term on compact mani-folds with boundary. Additionally, some concrete examples of the perturbation of the de Rham Hodge operator are provided for our main theorems. To the best of the authors' knowledge, the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator have not been considered in the literature. Thus, the current study hopes to serve such a need, in this paper, we investigate the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with or without boundary. We compute the Lichnerowicz formulas for the perturbation of the de Rham Hodge operator and the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on n-dimensional compact manifolds without boundary. And then we prove the Kastler–Kalau–Walze type theorems on 4-dimensional and 6-dimensional compact manifolds with boundary for the perturbation of the de Rham Hodge operator. Results: The motivation of this paper is to prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. Specically, we calculate $$\widetilde{\mathrm{Wres}}[\pi ^+D_{A}^{-1}\circ \pi ^+ {{D_A^*}^{-1}}]$$ Wres ~ [ π + D A - 1 ∘ π + D A ∗ - 1 ] , $$\widetilde{\mathrm{Wres}}[\pi ^+D_{A}^{-1}\circ \pi ^+ {{D_A}^{-1}}]$$ Wres ~ [ π + D A - 1 ∘ π + D A - 1 ] , $$\widetilde{\mathrm{Wres}}[\pi ^+D_{A}^{-1}\circ \pi ^+ {{D_A^*}D_{A}{D_A^*}^{-1}}]$$ Wres ~ [ π + D A - 1 ∘ π + D A ∗ D A D A ∗ - 1 ] . The main work of this paper is to prove the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on 4-dimensional and 6-dimensional compact manifolds with (resp.without) boundary. Theorem 2.2 shows that the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator on n-dimensional compact manifolds without boundary. On 4-dimensional compact manifolds with boundary, the main results of this paper are Theorem 3.8 and Theorem 3.18, which are the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. On the other hand, on 6-dimensional compact manifolds with 1 boundary, the main results of this paper are Theorem 4.3 and Theorem 4.15, which are the Kastler–Kalau–Walze type theorems for the perturbation of the de Rham Hodge operator. Comparing with the previous conclusion, we use new operator to study. Although the calculation process is more complex, the better conclusions we have. The results are useful for analyzing and interpreting certain physical theories.