Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary

Abstract

In this paper, we obtain two Lichnerowicz type formulas for the Dirac–Witten operators. And we give the proof of Kastler–Kalau–Walze type theorems for the Dirac–Witten operators on 4-dimensional and 6-dimensional compact manifolds with (resp. without) boundary.

1 Introduction

Until now, many geometers have studied noncommutative residues. In [5, 16], authors found noncommutative residues are of great importance to the study of noncommutative geometry. In [2], Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy. Connes showed us that the noncommutative residue on a compact manifold M coincided with the Dixmier’s trace on pseudodifferential operators of order-${{\mathrm{dim}}}M$ in [3]. And Connes claimed the noncommutative residue of the square of the inverse of the Dirac operator was proportioned to the Einstein–Hilbert action. Kastler [7] gave a brute-force proof of this theorem. Kalau and Walze proved this theorem in the normal coordinates system simultaneously in [6]. Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator ${\mathrm{Wres}}(D^{-2})$ in turn is essentially the second coefficient of the heat kernel expansion of $D^{2}$ in [1].

On the other hand, Wang generalized the Connes’ results to the case of manifolds with boundary in [10, 11], and proved the Kastler–Kalau–Walze type theorem for the Dirac operator and the signature operator on lower-dimensional manifolds with boundary [12]. In [12, 13], Wang computed $\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-1}\circ \pi ^+D^{-1}]$ and $\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-2}\circ \pi ^+D^{-2}]$, where two operators are symmetric. And in these cases, the boundary term vanished. But for $\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-1}\circ \pi ^+D^{-3}]$, Wang got a nonvanishing boundary term [14], and give a theoretical explanation for gravitational action on boundary. And then, Wang provides a kind of method to study the Kastler–Kalau–Walze type theorem for manifolds with boundary. In [8], López and his collaborators introduced an elliptic differential operator, which is called the Novikov operator. In [15], Wei and Wang proved Kastler–Kalau–Walze type theorem for modified Novikov operators on compact manifolds. In [17], in order to prove the nonsymmetric positive mass theorem, Zhang introduced the Dirac–Witten operator. The motivation of this paper is to prove the Kastler–Kalau–Walze type theorem for the Dirac–Witten operators.

The paper is organized in the following way. In Sect. 2, by using the definition of the Dirac–Witten operators, we compute the Lichnerowicz formulas for the Dirac–Witten operators. In Sects. 3 and  4, we prove the Kastler–Kalau–Walze type theorem for 4-dimensional and 6-dimensional manifolds with boundary for the Dirac–Witten operators respectively.

2 The Dirac–Witten Operators and Their Lichnerowicz Formulas

Firstly we introduce some notations about the Dirac–Witten operators. Let M be a n-dimensional ($n\ge 3$) oriented compact spin Riemannian manifold with a Riemannian metric $g^{M}$. And let $\nabla ^L$ be the Levi-Civita connection about $g^M$. In the local coordinates $\{x_i; 1\le i\le n\}$ and the fixed orthonormal frame $\{e_1,\cdots ,e_n\}$, the connection matrix $(\omega _{s,t})$ is defined by

$$\begin{aligned} \nabla ^L(e_1,\ldots ,e_n)= (e_1,\ldots ,e_n)(\omega _{s,t}). \end{aligned}$$

(2.1)

Let $c(e_j)$ be the Clifford action. Suppose that $\partial _{i}$ is a natural local frame on TM and $(g^{ij})_{1\le i,j\le n}$ is the inverse matrix associated to the metric matrix $(g_{ij})_{1\le i,j\le n}$ on M. By [12], we have the Dirac operator

$$\begin{aligned} D&=\sum ^n_{i=1}c(e_i)\bigg [e_i-\frac{1}{4}\sum _{s,t}\omega _{s,t} (e_i)c(e_s)c(e_t)\bigg ] \end{aligned}$$

(2.2)

Then the Dirac–Witten operators ${\widetilde{D}}$ and ${{\widetilde{D}}}^*$ are defined by

$$\begin{aligned} {\widetilde{D}}&=D+f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\nonumber \\&=\sum ^n_{i=1}c(e_i)\bigg [e_i-\frac{1}{4}\sum _{s,t}\omega _{s,t} (e_i)c(e_s)c(e_t)\bigg ]\nonumber \\&\quad +f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2,\nonumber \\ {{\widetilde{D}}}^*&=D-\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}\nonumber \\&=\sum ^n_{i=1}c(e_i)\bigg [e_i-\frac{1}{4}\sum _{s,t}\omega _{s,t} (e_i)c(e_s)c(e_t)\bigg ]\nonumber \\&\quad -\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}. \end{aligned}$$

(2.3)

where $f_1,f_2$ are complex numbers, p is a (0, 2)-tensor and $p_{uv}=p(e_u,e_v)$. Then when $f_1=\frac{\sqrt{-1}}{2}$, $f_2=-\frac{\sqrt{-1}}{2}\sum _{i}p_{ii}$, ${\widetilde{D}}$ is the Dirac–Witten operator defined by [17].

Then, we get the following Lichnerowicz formulas,

Theorem 2.1

The following equalities hold:

$$\begin{aligned} {{\widetilde{D}}}^*{\widetilde{D}}&=-\Big [g^{ij}(\nabla _{\partial _{i}}\nabla _{\partial _{j}}- \nabla _{\nabla ^{L}_{\partial _{i}}\partial _{j}})\Big ]+\frac{1}{4}s +(f_1\overline{f_2}-\overline{f_1}f_2)\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\nonumber \\&\quad +\frac{1}{4}\sum _{i}\left[ c(e_{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) c(e_i)\right] ^2\nonumber \\&\quad +\frac{1}{2}\sum _{j}[c(e_{j})e_j\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad \left. -e_j\left( -\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(e_j)\right] \nonumber \\&\quad -f_1\overline{f_1}\left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right] ^2+f_2\overline{f_2},\nonumber \\ {{\widetilde{D}}}^2&=-\Big [g^{ij}(\nabla _{\partial _{i}}\nabla _{\partial _{j}}- \nabla _{\nabla ^{L}_{\partial _{i}}\partial _{j}})\Big ]+\frac{1}{4}s +{f_1}^2\left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right] ^2\nonumber \\&\quad +\frac{1}{4}\sum _{i}\left[ c(e_{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(e_i)\right] ^2\nonumber \\&\quad -\frac{1}{2}\sum _{j}\left[ e_j\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) c(e_{j})\right. \nonumber \\&\quad \left. -c(e_{j})e_j\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right] \nonumber \\&\quad +2f_1f_2\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+{f_2}^2. \end{aligned}$$

(2.4)

where s is the scalar curvature.

Proof

Let M be a smooth compact oriented spin Riemannian n-dimensional manifolds without boundary and N be a vector bundle on M. If P is a differential operator of Laplace type, then it has locally the form

$$\begin{aligned} P=-(g^{ij}\partial _i\partial _j+A^i\partial _i+B), \end{aligned}$$

(2.5)

where $\partial _{i}$ is a natural local frame on TM and $(g^{ij})_{1\le i,\,j\le n}$ is the inverse matrix associated to the metric matrix $(g_{ij})_{1\le i,\,j\le n}$ on M, and $A^{i}$ and B are smooth sections of $\text {End}(N)$ on M (endomorphism). If a Laplace type operator P satisfies (2.5), then there is a unique connection $\nabla$ on N and a unique endomorphism E such that

$$\begin{aligned} P=-\Big [g^{ij}(\nabla _{\partial _{i}}\nabla _{\partial _{j}}- \nabla _{\nabla ^{L}_{\partial _{i}}\partial _{j}})+E\Big ], \end{aligned}$$

(2.6)

where $\nabla ^{L}$ is the Levi-Civita connection on M. Moreover (with local frames of $T^{*}M$ and N), $\nabla _{\partial _{i}}=\partial _{i}+\omega _{i}$ and E is related to $g^{ij}$, $A^{i}$ and B through

$$\begin{aligned} \omega _{i}= & {} \frac{1}{2}g_{ij}\big (A^{i}+g^{kl}\Gamma _{ kl}^{j} \texttt {id}\big ), \end{aligned}$$

(2.7)

$$\begin{aligned} E\,= & {} B-g^{ij}\big (\partial _{i}(\omega _{j})+\omega _{i}\omega _{j}-\omega _{k}\Gamma _{ ij}^{k} \big ), \end{aligned}$$

(2.8)

where $\Gamma _{ kl}^{j}$ are the Christoffel coefficients of $\nabla ^{L}$.

Let $g^{ij}=g(dx_{i},dx_{j})$, $\xi =\sum _{k}\xi _{j}dx_{j}$ and $\nabla ^L_{\partial _{i}}\partial _{j}=\sum _{k}\Gamma _{ij}^{k}\partial _{k}$, we denote that

$$\begin{aligned}&\sigma _{i}=-\frac{1}{4}\sum _{s,t}\omega _{s,t} (e_i)c(e_s)c(e_t);\nonumber \\&\xi ^{j}=g^{ij}\xi _{i};\quad \Gamma ^{k}=g^{ij}\Gamma _{ij}^{k}; \quad \sigma ^{j}=g^{ij}\sigma _{i}. \end{aligned}$$

(2.9)

Then the Dirac–Witten operators ${\widetilde{D}}$ and ${{\widetilde{D}}}^*$ can be written as

$$\begin{aligned} {\widetilde{D}}&=\sum ^n_{i=1}c(e_i)[e_i+\sigma _{i}] +f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2;\nonumber \\ {{\widetilde{D}}}^*&=\sum ^n_{i=1}c(e_i)[e_i+\sigma _{i}] -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}. \end{aligned}$$

(2.10)

By [7], we have

$$\begin{aligned} D^2&=-\Delta _0+\frac{1}{4}s\\&=-g^{ij}(\nabla _i^L\nabla _j^L-\Gamma _{ij}^k\nabla _k^L)+\frac{1}{4}s\nonumber \\&=-\sum _{ij}g^{ij}[\partial _i\partial _j+2\sigma _i\partial _j -\Gamma _{ij}^k\partial _k+\partial _i\sigma _j+\sigma _i\sigma _j -\Gamma _{ij}^k\sigma _k]+\frac{1}{4}s.\nonumber \end{aligned}$$

(2.11)

By (2.10), we have

$$\begin{aligned} {{\widetilde{D}}}^*{\widetilde{D}}&=D^2+D[f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2]\nonumber \\&\quad +\left[ -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2}\right] D\nonumber \\&\quad +\left[ f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right] \nonumber \\&\quad \left[ -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right] , \end{aligned}$$

(2.12)

$$\begin{aligned}&D\left[ f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right] +\left[ -\overline{f_1} \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right] D\nonumber \\&\quad =\sum _{i,j}g^{i,j}\left[ c(\partial _{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\qquad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(\partial _{i})\right] \partial _{j}\nonumber \\&\qquad -\sum _{i,j}g^{i,j}\left[ \left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)-\overline{f_2}\right) c(\partial _{i})\sigma _{j}\right. \nonumber \\&\qquad -c(\partial _{i}) \partial _{j}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\qquad \left. -c(\partial _{i})\sigma _j\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right] , \end{aligned}$$

(2.13)

then we obtain

$$\begin{aligned} {{\widetilde{D}}}^*{\widetilde{D}}&=-\sum _{i,j}g^{i,j}\left[ \partial _{i}\partial _{j} +2\sigma _{i}\partial _{j}-\Gamma _{i,j}^{k}\partial _{k}+\partial _{i}\sigma _{j} +\sigma _{i}\sigma _{j} -\Gamma _{i,j}^{k}\sigma _{k}\right] \nonumber \\&\quad +\frac{1}{4}s+\sum _{i,j}g^{i,j}\left[ c(\partial _{i}) \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(\partial _{i})\right] \partial _{j}\nonumber \\&\quad -\sum _{i,j}g^{i,j}\left[ \left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)-\overline{f_2}\right) c(\partial _i)\sigma _j\right. \nonumber \\&\quad -c(\partial _i)\partial _j\left( f_1 \sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad \left. -c(\partial _{i})\sigma _{j}\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \right] \nonumber \\&\quad -f_1\overline{f_1}\left[ \sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)\right] ^2\nonumber \\&\quad +\left( f_1\overline{f_2}-\overline{f_1}f_2\right) \sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\overline{f_2}. \end{aligned}$$

(2.14)

Similarly, we have

$$\begin{aligned} {{\widetilde{D}}}^2&=-\sum _{i,j}g^{i,j} \left[ \partial _{i}\partial _{j}+2\sigma _{i}\partial _{j} -\Gamma _{i,j}^{k}\partial _{k}+\partial _{i}\sigma _{j} +\sigma _{i}\sigma _{j}-\Gamma _{i,j}^{k}\sigma _{k}\right] \nonumber \\&\quad +\frac{1}{4}s+\sum _{i,j}g^{i,j}\left[ c(\partial _{i}) \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(\partial _{i})\right] \partial _{j}\nonumber \\&\quad +\sum _{i,j}g^{i,j}\left[ (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2)c(\partial _{i})\sigma _{j}\right. \nonumber \\&\quad +c(\partial _{i})\partial _{j} \left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad \left. +c(\partial _{i})\sigma _{j} \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right] \nonumber \\&\quad +f_1^2\left[ \sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)\right] ^2\nonumber \\&\quad +2f_1f_2\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2^2. \end{aligned}$$

(2.15)

By (2.6), (2.7), (2.8) and (2.14), we have

$$\begin{aligned} (\omega _{i})_{{{\widetilde{D}}}^*{{\widetilde{D}}}}&=\sigma _{i}-\frac{1}{2} \left[ c(\partial _{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(\partial _{i})\right] . \end{aligned}$$

(2.16)

$$\begin{aligned} E_{{{\widetilde{D}}}^*{\widetilde{D}}}&=-c(\partial _{i})\sigma ^{i}\left( f_1 \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad -\frac{1}{4}s+\left( \overline{f_1} \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)-\overline{f_2}\right) \nonumber \\&\quad c(\partial _i)\sigma ^i+c(\partial _i)\partial ^i\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad +\frac{1}{2}\partial ^j\left[ c(\partial _j) \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) c(\partial _j)\right] \nonumber \\&\quad -\frac{1}{2}\left[ c(\partial _j)\left( f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2} \right) c(\partial _j)\right] \sigma ^j\nonumber \\&\quad -\frac{g^{ij}}{4}\left[ c(\partial _{i})\left( f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) c(\partial ^i)\right] \nonumber \\&\quad \cdot \left[ c(\partial _{j})\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. -\left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)-\overline{f_2} \right) c(\partial _{j})\right] \nonumber \\&\quad -\frac{1}{2}\Gamma ^{k}\left[ c(\partial _{k})\left( f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(\partial _{k})\right] \nonumber \\&\quad -\frac{1}{2}\sigma ^{j}\left[ c(\partial _{j})\left( f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(\partial _{j})]\nonumber \\&\quad +f_1\overline{f_1}\left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right] ^2 -\left( f_1\overline{f_2}-\overline{f_1}f_2\right) \nonumber \\&\quad \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)-f_2\overline{f_2}. \end{aligned}$$

(2.17)

Since E is globally defined on M, taking normal coordinates at $x_0$, we have $\sigma ^{i}(x_0)=0$, $\partial ^{j}[c(\partial _{j})](x_0)=0$, $\Gamma ^k(x_0)=0$, $g^{ij}(x_0)=\delta ^j_i$, then

$$\begin{aligned} E_{{{\widetilde{D}}}^*{\widetilde{D}}}(x_0)&=-\frac{1}{4}s-\left( f_1\overline{f_2} -\overline{f_1}f_2\right) \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\nonumber \\&\quad +f_1\overline{f_1} \left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right] ^2\nonumber \\&\quad -f_2\overline{f_2}-\frac{1}{4}\sum _{i}\left[ c(e_{i})\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2}\right) c(e_i)\right] ^2\nonumber \\&\quad -\frac{1}{2}\left[ c(e_{j}) \nabla ^{\bigwedge ^*T^*M}_{e_j}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. -\nabla ^{\bigwedge ^*T^*M}_{e_j}\left( -\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) c(e_j)\right] . \end{aligned}$$

(2.18)

Similarly, we have

$$\begin{aligned} E_{{{\widetilde{D}}}^2}(x_0)&=-\frac{1}{4}s-f_1^2\left[ \sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)\right] ^2\nonumber \\&\quad -2f_1f_2\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) -f_2^2\nonumber \\&\quad -\frac{1}{4}\sum _{i}\left[ c(e_{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)+f_2\right) \right. \nonumber \\&\quad \left. +\left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) c(e_i)\right] ^2\nonumber \\&\quad +\frac{1}{2}\left[ \nabla ^{\bigwedge ^*T^*M}_{e_j} \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(e_j)\right. \nonumber \\&\quad \left. -c(e_{j})\nabla ^{\bigwedge ^*T^*M}_{e_j}\left( f_1\sum _{u<v}(p_{uv}-p_{vu} )c(e_u)c(e_v)+f_2\right) \right] , \end{aligned}$$

(2.19)

by (2.5), we get Theorem 2.1. $\square$

From [1], we know that the noncommutative residue of an appropriate power of a generalized laplacian ${\overline{\Delta }}$ is expressed as

$$\begin{aligned} (n-2)\Phi _{2}({\overline{\Delta }})=(4\pi )^{-\frac{n}{2}} \Gamma \left( \frac{n}{2}\right) Wres\left( {\overline{\Delta }}^{-\frac{n}{2}+1}\right) , \end{aligned}$$

(2.20)

where $\Phi _{2}({\overline{\Delta }})$ denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of ${\overline{\Delta }}$. Now let ${\overline{\Delta }}={{\widetilde{D}}}^*{{\widetilde{D}}}$ and ${{\widetilde{D}}}^*{{\widetilde{D}}}=\Delta -E$, then we have

$$\begin{aligned} {\mathrm{Wres}}({{{\widetilde{D}}}^{*}}{{\widetilde{D}}})^{-\frac{n-2}{2}}&=\frac{(n-2)(4\pi )^{\frac{n}{2}}}{\left( \frac{n}{2}-1\right) !}\int \limits _{M}{\mathrm{tr}} \left( \frac{1}{6}s+E_{{{\widetilde{D}}^{*}}{\widetilde{D}}}\right) d{\mathrm{Vol}_{\mathrm{M}} }, \end{aligned}$$

(2.21)

$$\begin{aligned} {\mathrm{Wres}}({{\widetilde{D}}}^2)^{-\frac{n-2}{2}}&=\frac{(n-2)(4\pi )^{\frac{n}{2}}}{(\frac{n}{2}-1)!}\int \limits _{M}{\mathrm{tr}} \left( \frac{1}{6}s+E_{{{\widetilde{D}}}^2}\right) d{\mathrm{Vol}_{\mathrm{M}} }, \end{aligned}$$

(2.22)

where ${\mathrm{Wres}}$ denotes the noncommutative residue. Then,

$$\begin{aligned} tr\left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right]&= \sum _{u<v}(p_{uv}-p_{vu})tr[c(e_u)c(e_v)]\nonumber \\&=\sum _{u<v}(p_{uv}-p_{vu})tr[c(e_v)c(e_u)]\nonumber \\&=-\sum _{u<v}(p_{uv}-p_{vu})tr[c(e_u)c(e_v)]=0. \end{aligned}$$

(2.23)

By

$$\begin{aligned}&tr\left[ \sum _{u<v}\sum _{s<t}c(e_u)c(e_v)c(e_s)c(e_t)\right] \nonumber \\&\quad =\left\{ \begin{array}{ll} -tr[id], &{}\quad {\mathrm{if }}~u=s,v=t, \\ 0,&{}\quad other~cases.\\ \end{array} \right. \end{aligned}$$

(2.24)

$$\begin{aligned}&tr\left\{ \left[ \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)c(e_u)c(e_v)\right] ^2\right\} \nonumber \\&\quad =tr\left[ \sum _{u<v}\sum _{s<t}(p_{uv}-p_{vu})(p_{st}-p_{ts})c(e_u)c(e_v)c(e_s)c(e_t)\right] \nonumber \\&\quad =-\sum _{u<v}(p_{uv}-p_{vu})^2tr[id]. \end{aligned}$$

(2.25)

Similarly, we have

$$\begin{aligned}&tr\left( \sum _{i}\left[ c(e_{i})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \right. \right. \left. \left. +\left( -{\overline{f}}_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +{\overline{f}}_2\right) c(e_i)\right] ^2\right) \nonumber \\&\quad =f_1^2(n-4)\sum _{u<v}(p_{uv}-p_{vu})^2tr[id]-nf_2^2tr[id] +{\overline{f}}_1^2(n-4)\sum _{u<v}(p_{uv}-p_{vu})^2tr[id]\nonumber \\&\qquad -n{\overline{f}}_2^2tr[id]-2nf_1{\overline{f}}_1\sum _{u<v} (p_{uv}-p_{vu})^2tr[id]-2nf_2{\overline{f}}_2\sum _{u<v}(p_{uv}-p_{vu})^2tr[id]\nonumber \\&\quad =\left[ f_1^2(n-4)\sum _{u<v}(p_{uv}-p_{vu})^2-nf_2^2\right] {\mathrm{tr}} [{{\texttt {id}}}];\nonumber \\&tr\left( \sum _{j}c(e_{j})\nabla ^{\bigwedge ^*T^*M}_{e_j} \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right) \nonumber \\&\quad =tr\left( \sum _{j}c(e_{j})\left[ \sum _{u<v}\nabla _{e_j}(f_1 (p_{uv}-p_{vu}))c(e_u)c(e_v)+f_1(p_{uv}-p_{vu})c(\nabla _{e_j}e_u)c(e_v)\right. \right. \nonumber \\&\qquad \left. \left. +f_1(p_{uv}-p_{vu})c(e_u)c(\nabla _{e_j}e_v)\right] +\sum _{j} c(e_j)\nabla _{e_j}(f_2)\right) \nonumber \\&\quad =0. \end{aligned}$$

(2.26)

Then by (2.23)–(2.26), we get

$$\begin{aligned} tr(E_{{{\widetilde{D}}}^*{\widetilde{D}}})&=\bigg [-\frac{s}{4} -\frac{1}{4}\{[(f_1^2+\overline{f_1}^2)(n-4)-2nf_1\overline{f_1}] \sum _{u<v}(p_{uv}-p_{vu})^2\nonumber \\&\quad +2nf_2\overline{f_2}-nf_2^2-n\overline{f_2}\}-f_2\overline{f_2} -f_1\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})^2\bigg ]{\mathrm{tr}}[{{\texttt {id}}}],\nonumber \\ tr(E_{{{\widetilde{D}}}^2})&=\left[ -\frac{s}{4}+(3-n)f_1^2 \sum _{u<v}(p_{uv}-p_{vu})^2+(n-1)f_2^2\right] {\mathrm{tr}}[{{\texttt {id}}}]. \end{aligned}$$

(2.27)

By (2.21), (2.22) and (2.27), we can get the following theorem,

Theorem 2.2

If M is a n-dimensional compact oriented spin manifolds without boundary, and n is even, then we get the following equalities:

$$\begin{aligned} {\mathrm{Wres}}({{\widetilde{D}}}^*{\widetilde{D}})^{-\frac{n-2}{2}}&=\frac{(n-2)(4\pi )^{\frac{n}{2}}}{(\frac{n}{2}-1)!}\int \limits _{M}2^\frac{n}{2}\bigg ( -\frac{1}{12}s-\frac{1}{4}\{[(f_1^2+\overline{f_1}^2)(n-4)-2nf_1\overline{f_1}]\nonumber \\&\sum _{u<v}(p_{uv}-p_{vu})^2+2nf_2\overline{f_2}-nf_2^2 -n\overline{f_2}^2\}-f_2\overline{f_2}-f_1\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})^2\bigg )d{\mathrm{Vol}_{\mathrm{M}}}. \end{aligned}$$

(2.28)

$$\begin{aligned} {\mathrm{Wres}}({{\widetilde{D}}}^2)^{-\frac{n-2}{2}}&=\frac{(n-2)(4\pi )^{\frac{n}{2}}}{(\frac{n}{2}-1)!}\int \limits _{M}2^\frac{n}{2}\bigg ( -\frac{1}{12}s+(3-n)f_1^2\sum _{u<v}(p_{uv}-p_{vu})^2 +(n-1)f_2^2\bigg )d{\mathrm{Vol}_{\mathrm{M}}}. \end{aligned}$$

(2.29)

where s is the scalar curvature.

3 A Kastler–Kalau–Walze Type Theorem for 4-Dimensional Manifolds with Boundary

We firstly recall some basic facts and formulas about Boutet de Monvel’s calculus and the definition of the noncommutative residue for manifolds with boundary which will be used in the following. For more details (see in Sect. 2 in [12]).

Let $U\subset M$ be a collar neighborhood of $\partial M$ which is diffeomorphic with $\partial M\times [0,1)$. By the definition of $h(x_n)\in C^{\infty }([0,1))$ and $h(x_n)>0$, there exists ${\widehat{h}}\in C^{\infty }((-\varepsilon ,1))$ such that ${\widehat{h}}|_{[0,1)}=h$ and ${\widehat{h}}>0$ for some sufficiently small $\varepsilon >0$. Then there exists a metric $g'$ on ${\widetilde{M}}=M\bigcup _{\partial M}\partial M\times (-\varepsilon ,0]$ which has the form on $U\bigcup _{\partial M}\partial M\times (-\varepsilon ,0 ]$

$$\begin{aligned} g'=\frac{1}{{\widehat{h}}(x_{n})}g^{\partial M}+dx _{n}^{2} , \end{aligned}$$

(3.1)

such that $g'|_{M}=g$. We fix a metric $g'$ on the ${\widetilde{M}}$ such that $g'|_{M}=g$.

Let Fourier transformation $F'$ be

$$\begin{aligned} F':L^2(\mathbf{R}_t)\rightarrow L^2(\mathbf{R}_v);~F'(u)(v)=\int e^{-ivt}u(t)dt \end{aligned}$$

(3.2)

and let

$$\begin{aligned} r^{+}:C^\infty (\mathbf{R})\rightarrow C^\infty (\widetilde{\mathbf{R}^+});~ f\rightarrow f|\widetilde{\mathbf{R}^+};~ \widetilde{\mathbf{R}^+}=\{x\ge 0;\,x\in \mathbf{R}\}. \end{aligned}$$

(3.3)

where $\Phi (\mathbf{R})$ denotes the Schwartz space and $\Phi (\widetilde{\mathbf{R}^+}) =r^+\Phi (\mathbf{R})$, $\Phi (\widetilde{\mathbf{R}^-}) =r^-\Phi (\mathbf{R})$.

We define $H^+=F'(\Phi (\widetilde{\mathbf{R}^+}));~ H^-_0=F'(\Phi (\widetilde{\mathbf{R}^-}))$ which satisfies $H^+\bot H^-_0$. We have the following property: $h\in H^+~(H^-_0)$ if and only if $h\in C^\infty (\mathbf{R})$ which has an analytic extension to the lower (upper) complex half-plane $\{{\mathrm{Im}}\xi <0\}~(\{{\mathrm{Im}}\xi >0\})$ such that for all nonnegative integer l,

$$\begin{aligned} \frac{d^{l}h}{d\xi ^l}(\xi )\sim \sum ^{\infty }_{k=1} \frac{d^l}{d\xi ^l}\left( \frac{c_k}{\xi ^k}\right) , \end{aligned}$$

(3.4)

as $|\xi |\rightarrow +\infty ,{\mathrm{Im}}\xi \le 0~({\mathrm{Im}}\xi \ge 0)$. Let $H'$ be the space of all polynomials and $H^-=H^-_0\bigoplus H';~H=H^+\bigoplus H^-.$ Denote by $\pi ^+~(\pi ^-)$ respectively the projection on $H^+~(H^-)$. For calculations, we take $H={{\widetilde{H}}}=\{$rational functions having no poles on the real axis$\}$ (${\tilde{H}}$ is a dense set in the topology of H). Then on ${\tilde{H}}$,

$$\begin{aligned} \pi ^+h(\xi _0)=\frac{1}{2\pi i}\lim _{u\rightarrow 0^{-}} \int \limits _{\Gamma ^+}\frac{h(\xi )}{\xi _0+iu-\xi }d\xi , \end{aligned}$$

(3.5)

where $\Gamma ^+$ is a Jordan close curve included ${\mathrm{Im}}(\xi )>0$ surrounding all the singularities of h in the upper half-plane and $\xi _0\in \mathbf{R}$. Similarly, define $\pi '$ on ${\tilde{H}}$,

$$\begin{aligned} \pi 'h=\frac{1}{2\pi }\int \limits _{\Gamma ^+}h(\xi )d\xi . \end{aligned}$$

(3.6)

So, $\pi '(H^-)=0$. For $h\in H\bigcap L^1(\mathbf{R})$, $\pi 'h=\frac{1}{2\pi }\int_{\mathbf{R}}h(v)dv$ and for $h\in H^+\bigcap L^1(\mathbf{R})$, $\pi 'h=0$.

Let M be a n-dimensional compact oriented spin manifold with boundary $\partial M$. Denote by ${\mathcal {B}}$ Boutet de Monvel’s algebra, we recall the main theorem in [4, 12].

Theorem 3.1

[4] (Fedosov–Golse–Leichtnam–Schrohe) Let X and $\partial X$ be connected, ${\mathrm{dim}}X=n\ge 3$, $A=\left( \begin{array}{lcr}\pi ^+P+G &{} K \\ T &{} S \end{array}\right)$ $\in {\mathcal {B}}$, and denote by p, b and s the local symbols of P, G and S respectively. Define:

$$\begin{aligned} {{{\widetilde{{{\mathrm{Wres}}}}}}}(A)&=\int \limits _X\int \limits _\mathbf{S}\mathrm{{tr}}_E \left[ p_{-n}(x,\xi )\right] \sigma (\xi )dx \nonumber \\&\quad +2\pi \int \limits _ {\partial X}\int \limits _\mathbf{S'}\left\{ {\mathrm{tr}}_E \left[ (\mathrm{{tr}}b_{-n})(x',\xi ')\right] +\mathrm{{tr}} _F\left[ s_{1-n}(x',\xi ')\right] \right\} \sigma (\xi ')dx', \end{aligned}$$

(3.7)

where ${{{\widetilde{{\mathrm{Wres}}}}}}$ denotes the noncommutative residue of an operator in the Boutet de Monvel’s algebra.

Then a) ${{{\widetilde{{\mathrm{Wres}}}}}}([A,B])=0$, for any $A,B\in {\mathcal {B}}$; b) It is a unique continuous trace on ${\mathcal {B}}/{\mathcal {B}}^{-\infty }$.

Definition 3.2

[12] Lower dimensional volumes of spin manifolds with boundary are defined by

$$\begin{aligned} {\mathrm{Vol}}^{(p_1,p_2)}_nM:= \widetilde{{\mathrm{Wres}}}[\pi ^+D^{-p_1}\circ \pi ^+D^{-p_2}], \end{aligned}$$

(3.8)

By [12], we get

$$\begin{aligned} \widetilde{{\mathrm{Wres}}}[\pi ^+D^{-p_1}\circ \pi ^+D^{-p_2}]= & {} \int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\sigma _{-n} (D^{-p_1-p_2})]\sigma (\xi )dx\nonumber \\&+\int \limits _{\partial M}\Phi , \end{aligned}$$

(3.9)

and

$$\begin{aligned} \Phi&=\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum \limits ^{\infty }_{j, k=0}\sum \frac{(-i)^{|\alpha |+j+k+1}}{\alpha !(j+k+1)!} \times {\mathrm{trace}}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\partial ^j_{x_n} \partial ^\alpha _{\xi '}\partial ^k_{\xi _n}\sigma ^+_{r}(D^{-p_1})(x',0,\xi ',\xi _n) \nonumber \\&\quad \times \partial ^\alpha _{x'}\partial ^{j+1}_{\xi _n}\partial ^k_{x_n} \sigma _{l}(D^{-p_2})(x',0,\xi ',\xi _n)]d\xi _n\sigma (\xi ')dx', \end{aligned}$$

(3.10)

where the sum is taken over $r+l-k-|\alpha |-j-1=-n,~~r\le -p_1,l\le -p_2$ and $\widetilde{{\mathrm{Wres}}}$ denotes the noncommutative residue for manifolds with boundary.

Since $[\sigma _{-n}(D^{-p_1-p_2})]|_M$ has the same expression as $\sigma _{-n}(D^{-p_1-p_2})$ in the case of manifolds without boundary, so locally we can compute the first term by [6, 7, 9, 12].

For any fixed point $x_0\in \partial M$, we choose the normal coordinates U of $x_0$ in $\partial M$ (not in M) and compute $\Phi (x_0)$ in the coordinates ${\widetilde{U}}=U\times [0,1)\subset M$ and the metric $\frac{1}{h(x_n)}g^{\partial M}+dx_n^2.$ The dual metric of $g^M$ on ${\widetilde{U}}$ is ${h(x_n)}g^{\partial M}+dx_n^2.$ Write $g^M_{ij}=g^M(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_j});~ g_M^{ij}=g^M(dx_i,dx_j)$, then

$$\begin{aligned} {[}g^M_{i,j}]= \left[ \begin{array}{lcr} \frac{1}{h(x_n)}[g_{i,j}^{\partial M}] &{} 0 \\ 0 &{} 1 \end{array}\right] ;~~~ [g_M^{i,j}]= \left[ \begin{array}{lcr} h(x_n)[g^{i,j}_{\partial M}] &{} 0 \\ 0 &{} 1 \end{array}\right] , \end{aligned}$$

(3.11)

and

$$\begin{aligned} \partial _{x_s}g_{ij}^{\partial M}(x_0)=0, 1\le i,j\le n-1; ~~~g_{ij}^M(x_0)=\delta _{ij}. \end{aligned}$$

(3.12)

From [12], we can get three lemmas.

Lemma 3.3

[12] With the metric $g^{M}$ on M near the boundary

$$\begin{aligned} \partial _{x_j}(|\xi |_{g^M}^2)(x_0)= & {} \left\{ \begin{array}{ll} 0, &{}\quad {\mathrm{if }}~j<n, \\ h'(0)|\xi '|^{2}_{g^{\partial M}}, &{}\quad {\mathrm{if }}~j=n; \end{array} \right. \end{aligned}$$

(3.13)

$$\begin{aligned} \partial _{x_j}[c(\xi )](x_0)= & {} \left\{ \begin{array}{ll} 0, &{}\quad {\mathrm{if }}~j<n,\\ \partial x_{n}(c(\xi '))(x_{0}), &{}\quad {\mathrm{if }}~j=n, \end{array} \right. \end{aligned}$$

(3.14)

where $\xi =\xi '+\xi _{n}dx_{n}$.

Lemma 3.4

[12] With the metric $g^{M}$ on M near the boundary

$$\begin{aligned} \omega _{s,t}(e_i)(x_0)&=\left\{ \begin{array}{ll} \omega _{n,i}(e_i)(x_0)=\frac{1}{2}h'(0), &{}\quad {\mathrm{if }}~s=n,t=i,i<n, \\ \omega _{i,n}(e_i)(x_0)=-\frac{1}{2}h'(0), &{}\quad {\mathrm{if }}~s=i,t=n,i<n,\\ \omega _{s,t}(e_i)(x_0)=0, &{}\quad other~cases,\\ \end{array} \right. \end{aligned}$$

(3.15)

where $(\omega _{s,t})$ denotes the connection matrix of Levi-Civita connection $\nabla ^L$.

Lemma 3.5

[12]

$$\begin{aligned} \Gamma _{st}^k(x_0)&=\left\{ \begin{array}{ll} \Gamma ^n_{ii}(x_0)=\frac{1}{2}h'(0) &{}\quad {\mathrm{if }}~s=t=i,k=n,i<n, \\ \Gamma ^i_{ni}(x_0)=-\frac{1}{2}h'(0),&{}\quad {\mathrm{if }}~s=n,t=i,k=i,i<n,\\ \Gamma ^i_{in}(x_0)=-\frac{1}{2}h'(0),&{}\quad {\mathrm{if }}~s=i,t=n,k=i,i<n,\\ \Gamma _{st}^i(x_0)=0,&{}\quad other~cases. \end{array} \right. \end{aligned}$$

(3.16)

By (3.6) and (3.7), we firstly compute

$$\begin{aligned} \widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+ ({{\widetilde{D}}}^*)^{-1}]= & {} \int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\sigma _{-4} (({{\widetilde{D}}}^*{{\widetilde{D}}})^{-1})]\sigma (\xi )dx\nonumber \\&+\int \limits _{\partial M}\Phi , \end{aligned}$$

(3.17)

where

$$\begin{aligned} \Phi&=\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum ^{\infty }_{j, k=0} \sum \frac{(-i)^{|\alpha |+j+k+1}}{\alpha !(j+k+1)!} \times {\mathrm{trace}}_{\wedge ^*T^*M \bigotimes {\mathbb {C}}} [\partial ^j_{x_n}\partial ^\alpha _{\xi '}\partial ^k_{\xi _n}\sigma ^+_{r} ({{\widetilde{D}}}^{-1}) (x',0,\xi ',\xi _n)\nonumber \\&\quad \times \partial ^\alpha _{x'}\partial ^{j+1}_{\xi _n} \partial ^k_{x_n}\sigma _{l}(({{\widetilde{D}}}^*)^{-1})(x',0,\xi ', \xi _n)]d\xi _n\sigma (\xi ')dx', \end{aligned}$$

(3.18)

the sum is taken over $r+l-k-j-|\alpha |=-3,~~r\le -1,l\le -1$ and $\widetilde{{\mathrm{Wres}}}$ denotes the noncommutative residue for manifolds with boundary.

By Theorem 2.2, we can compute the interior of $\widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+({{\widetilde{D}}}^*)^{-1}]$, so

$$\begin{aligned}&\int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M}[\sigma _{-4} (({{\widetilde{D}}}^*{\widetilde{D}})^{-1})]\sigma (\xi )dx\nonumber \\&\quad =32\pi ^2\int \limits _{M}\bigg \{-\frac{1}{3}s-12f_2\overline{f_2}+4f_2^2 +4\overline{f_2}^2+4f_1\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})^2\bigg \}d{\mathrm{Vol}_{\mathrm{M}}}. \end{aligned}$$

(3.19)

Now we need to compute $\int _{\partial M} \Phi$. Since, some operators have the following symbols.

Lemma 3.6

The following identities hold:

$$\begin{aligned} \sigma _1({{\widetilde{D}}})&=\sigma _1({{\widetilde{D}}}^*)=ic(\xi ); \nonumber \\ \sigma _0({{\widetilde{D}}})&= -\frac{1}{4}\sum _{i,s,t}\omega _{s,t}(e_i)c(e_i)c(e_s)c(e_t) +\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) ; \nonumber \\ \sigma _0({{\widetilde{D}}}^*)&= -\frac{1}{4}\sum _{i,s,t}\omega _{s,t}(e_i)c(e_i)c(e_s)c(e_t) +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) ; \end{aligned}$$

(3.20)

where $\xi =\Sigma _{i=1}^n\xi _id_{x_i}$ denotes the cotangent vector.

Write

$$\begin{aligned} D_x^{\alpha }&=(-i)^{|\alpha |}\partial _x^{\alpha }; ~\sigma ({\widetilde{D}})=p_1+p_0; ~\sigma ({\widetilde{D}})^{-1}=\sum \limits ^{\infty }_{j=1}q_{-j}. \end{aligned}$$

(3.21)

By the composition formula of pseudodifferential operators, we have

$$\begin{aligned} 1=\sigma ({\widetilde{D}}\circ {{\widetilde{D}}}^{-1})&=\sum _{\alpha }\frac{1}{\alpha !}\partial ^{\alpha }_{\xi }[\sigma ({{\widetilde{D}}})] {{\widetilde{D}}}_x^{\alpha }[\sigma ({{\widetilde{D}}}^{-1})]\nonumber \\&=(p_1+p_0)(q_{-1}+q_{-2}+q_{-3}+\cdots )\nonumber \\&\quad +\sum _j(\partial _{\xi _j}p_1+\partial _{\xi _j}p_0)( D_{x_j}q_{-1}+D_{x_j}q_{-2}+D_{x_j}q_{-3}+\cdots )\nonumber \\&=p_1q_{-1}+(p_1q_{-2}+p_0q_{-1}+\sum _j\partial _{\xi _j}p_1D_{x_j}q_{-1})+\cdots , \end{aligned}$$

(3.22)

so

$$\begin{aligned} q_{-1}=p_1^{-1};~q_{-2}=-p_1^{-1}\left[ p_0p_1^{-1}+\sum _j\partial _{\xi _j}p_1D_{x_j}(p_1^{-1})\right] . \end{aligned}$$

(3.23)

Lemma 3.7

The following identities hold:

$$\begin{aligned} \sigma _{-1}({{\widetilde{D}}}^{-1})&=\sigma _{-1}(({{\widetilde{D}}}^*)^{-1}) =\frac{ic(\xi )}{|\xi |^2};\nonumber \\ \sigma _{-2}({{\widetilde{D}}}^{-1})&=\frac{c(\xi )\sigma _{0} ({{\widetilde{D}}})c(\xi )}{|\xi |^4}+\frac{c(\xi )}{|\xi |^6}\sum _jc(dx_j) \Big [\partial _{x_j}(c(\xi ))|\xi |^2-c(\xi )\partial _{x_j}(|\xi |^2)\Big ] ;\nonumber \\ \sigma _{-2}(({{\widetilde{D}}}^*)^{-1})&=\frac{c(\xi )\sigma _{0} ({{\widetilde{D}}}^*)c(\xi )}{|\xi |^4}+\frac{c(\xi )}{|\xi |^6}\sum _jc(dx_j) \Big [\partial _{x_j}(c(\xi ))|\xi |^2-c(\xi )\partial _{x_j}(|\xi |^2)\Big ]. \end{aligned}$$

(3.24)

When $n=4$, then ${\mathrm{tr}}_{\wedge ^*T^*M}[{{ \texttt {id}}}]={\mathrm{dim}}(\wedge ^*(4))=4$, where ${\mathrm{tr}}$ as shorthand of ${\mathrm{trace}}$, the sum is taken over $r+l-k-j-|\alpha |=-3,~~r\le -1,l\le -1,$ then we have the following five cases:

Case (a) (I) $r=-1,~l=-1,~k=j=0,~|\alpha |=1$

By (3.18), we get

$$\begin{aligned} \Phi _1= & {} -\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum _{|\alpha |=1} {\mathrm{tr}}[\partial ^\alpha _{\xi '}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\nonumber \\&\times \partial ^\alpha _{x'}\partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^* )^{-1})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(3.25)

By Lemma 3.3, for $i<n$, then

$$\begin{aligned} \partial _{x_i}\left( \frac{ic(\xi )}{|\xi |^2}\right) (x_0)= \frac{i\partial _{x_i}[c(\xi )](x_0)}{|\xi |^2} -\frac{ic(\xi )\partial _{x_i}(|\xi |^2)(x_0)}{|\xi |^4}=0, \end{aligned}$$

(3.26)

so $\Phi _1=0$.

Case (a) (II) $r=-1,~l=-1,~k=|\alpha |=0,~j=1$

By (3.18), we get

$$\begin{aligned} \Phi _2= & {} -\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } \mathrm{trace} [\partial _{x_n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\nonumber \\&\times \partial _{\xi _n}^2\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(3.27)

By Lemma 3.7, we have

$$\begin{aligned} \partial ^2_{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})(x_0)= & {} i\left( -\frac{6\xi _nc(dx_n)+2c(\xi ')}{|\xi |^4}+\frac{8\xi _n^2c(\xi )}{|\xi |^6}\right) ; \end{aligned}$$

(3.28)

$$\begin{aligned} \partial _{x_n}\sigma _{-1}({{\widetilde{D}}}^{-1})(x_0)= & {} \frac{i\partial _{x_n}c(\xi ')(x_0)}{|\xi |^2}-\frac{ic(\xi )|\xi '|^2h'(0)}{|\xi |^4}. \end{aligned}$$

(3.29)

By (3.5), (3.6), we get

$$\begin{aligned} \pi ^+_{\xi _n}\left[ \frac{c(\xi )}{|\xi |^4}\right] (x_0)|_{|\xi '|=1}&=\pi ^+_{\xi _n}\left[ \frac{c(\xi ')+\xi _nc(dx_n)}{(1+\xi _n^2)^2}\right] \nonumber \\&=\frac{1}{2\pi i}{\mathrm{lim}}_{u\rightarrow 0^-}\int \limits _{\Gamma ^+}\frac{\frac{c(\xi ')+\eta _nc(dx_n)}{(\eta _n+i)^2(\xi _n+iu-\eta _n)}}{(\eta _n-i)^2}d\eta _n\nonumber \\&=-\frac{(i\xi _n+2)c(\xi ')+ic(dx_n)}{4(\xi _n-i)^2}. \end{aligned}$$

(3.30)

Similarly we have,

$$\begin{aligned} \pi ^+_{\xi _n}\left[ \frac{i\partial _{x_n}c(\xi ')}{|\xi |^2} \right] (x_0)|_{|\xi '|=1}=\frac{\partial _{x_n}[c(\xi ')](x_0)}{2(\xi _n-i)}. \end{aligned}$$

(3.31)

By (3.29), then

$$\begin{aligned} \pi ^+_{\xi _n}\partial _{x_n}\sigma _{-1}({{\widetilde{D}}}^{-1})|_{|\xi '|=1} =\frac{\partial _{x_n}[c(\xi ')](x_0)}{2(\xi _n-i)}+ih'(0) \left[ \frac{(i\xi _n+2)c(\xi ')+ic(dx_n)}{4(\xi _n-i)^2}\right] . \end{aligned}$$

(3.32)

By the relation of the Clifford action and ${\mathrm{tr}}{AB}={\mathrm{tr }}{BA}$, we have the equalities:

$$\begin{aligned}&{\mathrm{tr}}[c(\xi ')c(dx_n)]=0;~~{\mathrm{tr}}[c(dx_n)^2] =-4;~~{\mathrm{tr}}[c(\xi ')^2](x_0)|_{|\xi '|=1}=-4;\nonumber \\&{\mathrm{tr}}[\partial _{x_n}c(\xi ')c(dx_n)] =0;~~{\mathrm{tr}}[\partial _{x_n}c(\xi ')c(\xi ')](x_0)|_{|\xi '|=1}=-2h'(0). \end{aligned}$$

(3.33)

By (3.31), we have

$$\begin{aligned}&h'(0){\mathrm{tr}}\bigg [\frac{(i\xi _n+2)c(\xi ')+ic(dx_n)}{4(\xi _n-i)^2}\times \bigg (\frac{6\xi _nc(dx_n)+2c(\xi ')}{(1+\xi _n^2)^2} -\frac{8\xi _n^2[c(\xi ')+\xi _nc(dx_n)]}{(1+\xi _n^2)^3}\bigg ) \bigg ](x_0)|_{|\xi '|=1}\nonumber \\&\quad =-4h'(0)\frac{-2i\xi _n^2-\xi _n+i}{(\xi _n-i)^4(\xi _n+i)^3}. \end{aligned}$$

(3.34)

Similarly, we have

$$\begin{aligned}&-i\mathrm{tr}\bigg [\bigg (\frac{\partial _{x_n}[c(\xi ')](x_0)}{2(\xi _n-i)}\bigg ) \times \bigg (\frac{6\xi _nc(dx_n)+2c(\xi ')}{(1+\xi _n^2)^2} -\frac{8\xi _n^2[c(\xi ')+\xi _nc(dx_n)]}{(1+\xi _n^2)^3}\bigg )\bigg ](x_0)|_{|\xi '|=1}\nonumber \\&\quad =-2ih'(0)\frac{3\xi _n^2-1}{(\xi _n-i)^4(\xi _n+i)^3}. \end{aligned}$$

(3.35)

Then

$$\begin{aligned} \Phi _2&=-\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\frac{ih'(0)(\xi _n-i)^2}{(\xi _n-i)^4(\xi _n+i)^3}d\xi _n\sigma (\xi ')dx'\\&=-ih'(0)\Omega _3\int \limits _{\Gamma ^+}\frac{1}{(\xi _n-i)^2(\xi _n+i)^3}d\xi _ndx'\nonumber \\&=-ih'(0)\Omega _32\pi i\left[ \frac{1}{(\xi _n+i)^3}\right] ^{(1)}|_{\xi _n=i}dx'\nonumber \\&=-\frac{3}{8}\pi h'(0)\Omega _3dx',\nonumber \end{aligned}$$

(3.36)

where ${{\Omega _{3}}}$ is the canonical volume of $S^{3}$.

Case (a) (III) $r=-1,~l=-1,~j=|\alpha |=0,~k=1$

By (3.18), we get

$$\begin{aligned} \Phi _3= & {} -\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } {\mathrm{trace}}[\partial _{\xi _n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\nonumber \\&\times \partial _{\xi _n}\partial _{x_n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1}) ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(3.37)

By Lemma 3.7, we have

$$\begin{aligned} \partial _{\xi _n}\partial _{x_n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})(x_0)|_{|\xi '|=1}= & {} -ih'(0)\left[ \frac{c(dx_n)}{|\xi |^4}-4\xi _n\frac{c(\xi ') +\xi _nc(dx_n)}{|\xi |^6}\right] \nonumber \\&-\frac{2\xi _ni\partial _{x_n}c(\xi ')(x_0)}{|\xi |^4}; \end{aligned}$$

(3.38)

$$\begin{aligned} \partial _{\xi _n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1}) (x_0)|_{|\xi '|=1}= & {} -\frac{c(\xi ')+ic(dx_n)}{2(\xi _n-i)^2}. \end{aligned}$$

(3.39)

Similar to case a) II), we have

$$\begin{aligned}&{\mathrm{tr}}\left\{ \frac{c(\xi ')+ic(dx_n)}{2(\xi _n-i)^2}\times ih'(0)\left[ \frac{c(dx_n)}{|\xi |^4}-4\xi _n\frac{c(\xi ') +\xi _nc(dx_n)}{|\xi |^6}\right] \right\} \nonumber \\&\quad =2h'(0)\frac{i-3\xi _n}{(\xi _n-i)^4(\xi _n+i)^3} \end{aligned}$$

(3.40)

and

$$\begin{aligned} {\mathrm{tr}}\left[ \frac{c(\xi ')+ic(dx_n)}{2(\xi _n-i)^2}\times \frac{2\xi _ni\partial _{x_n}c(\xi ')(x_0)}{|\xi |^4}\right] =\frac{-2ih'(0)\xi _n}{(\xi _n-i)^4(\xi _n+i)^2}. \end{aligned}$$

(3.41)

So we have

$$\begin{aligned} \Phi _3&=-\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\frac{h'(0)(i-3\xi _n)}{(\xi _n-i)^4(\xi _n+i)^3}d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad -\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\frac{h'(0)i\xi _n}{(\xi _n-i)^4(\xi _n+i)^2}d\xi _n\sigma (\xi ')dx'\nonumber \\&=-h'(0)\Omega _3\frac{2\pi i}{3!}\left[ \frac{(i-3\xi _n)}{(\xi _n+i)^3} \right] ^{(3)}|_{\xi _n=i}dx'+h'(0)\Omega _3\frac{2\pi i}{3!}\left[ \frac{i\xi _n}{(\xi _n+i)^2}\right] ^{(3)}|_{\xi _n=i}dx'\nonumber \\&=\frac{3}{8}\pi h'(0)\Omega _3dx'. \end{aligned}$$

(3.42)

Case (b) $r=-2,~l=-1,~k=j=|\alpha |=0$

By (3.18), we get

$$\begin{aligned} \Phi _4&=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} [\pi ^+_{\xi _n}\sigma _{-2}({{\widetilde{D}}}^{-1})\times \partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(3.43)

By Lemma 3.7 we have

$$\begin{aligned} \sigma _{-2}({{\widetilde{D}}}^{-1})(x_0)=\frac{c(\xi ) \sigma _{0}({{\widetilde{D}}})(x_0)c(\xi )}{|\xi |^4}+\frac{c(\xi )}{|\xi |^6}c(dx_n) [\partial _{x_n}[c(\xi ')](x_0)|\xi |^2-c(\xi )h'(0)|\xi |^2_{\partial M}], \end{aligned}$$

(3.44)

where

$$\begin{aligned} \sigma _{0}({{\widetilde{D}}})(x_0)&=-\frac{1}{4}\sum _{s,t,i}\omega _{s,t}(e_i) (x_{0})c(e_i)c(e_s)c(e_t)\nonumber \\&\quad +f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2. \end{aligned}$$

(3.45)

We denote

$$\begin{aligned} Q(x_0)&=-\frac{1}{4}\sum _{s,t,i}\omega _{s,t}(e_i) (x_{0})c(e_i)c(e_s)c(e_t). \end{aligned}$$

(3.46)

Then

$$\begin{aligned}&\pi ^+_{\xi _n}\sigma _{-2}({{\widetilde{D}}}^{-1}(x_0))|_{|\xi '|=1}\nonumber \\&\quad =\pi ^+_{\xi _n} \Big [\frac{c(\xi )(f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2) (x_0)c(\xi )}{(1+\xi _n^2)^2}\Big ] \nonumber \\&\qquad +\pi ^+_{\xi _n}\Big [\frac{c(\xi )Q(x_0)c(\xi )+c(\xi )c(dx_n) \partial _{x_n}[c(\xi ')](x_0)}{(1+\xi _n^2)^2}-h'(0)\frac{c(\xi ) c(dx_n)c(\xi )}{(1+\xi _n^{2})^3}\Big ]. \end{aligned}$$

(3.47)

And

$$\begin{aligned}&{\mathrm{tr}}\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(dx_n)\right] \nonumber \\&\quad =f_1\sum _{u<v}(p_{uv}-p_{vu}){\mathrm{tr}}[c(e_u)c(e_v)c(dx_n)]\nonumber \\&\quad =0;\nonumber \\&{\mathrm{tr}}\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(\xi ')\right] \nonumber \\&\quad ={\mathrm{tr}}\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \sum _{j=1}^{n-1}\xi _jc(e_j)\right] \nonumber \\&\quad =0. \end{aligned}$$

(3.48)

Since

$$\begin{aligned} \partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})= \partial _{\xi _n}q_{-1}(x_0)|_{|\xi '|=1}=i\left[ \frac{c(dx_n)}{1+\xi _n^2} -\frac{2\xi _nc(\xi ')+2\xi _n^2c(dx_n)}{(1+\xi _n^2)^2}\right] . \end{aligned}$$

(3.49)

Then, we have

$$\begin{aligned}&\pi ^+_{\xi _n}\Big [\frac{c(\xi )Q(x_0)c(\xi )+c(\xi )c(dx_n)\partial _{x_n} [c(\xi ')](x_0)}{(1+\xi _n^2)^2}\Big ]-h'(0)\pi ^+_{\xi _n} \Big [\frac{c(\xi )c(dx_n)c(\xi )}{(1+\xi _n)^3}\Big ]\nonumber \\&\quad := C_1-C_2, \end{aligned}$$

(3.50)

where

$$\begin{aligned} C_1&=\frac{-1}{4(\xi _n-i)^2}[(2+i\xi _n)c(\xi ')Q(x_0)c(\xi ') +i\xi _nc(dx_n)Q(x_0)c(dx_n)\nonumber \\&\quad +(2+i\xi _n)c(\xi ')c(dx_n)\partial _{x_n}c(\xi ')+ic(dx_n)Q(x_0)c(\xi ') +ic(\xi ')Q(x_0)c(dx_n)-i\partial _{x_n}c(\xi ')] \end{aligned}$$

(3.51)

and

$$\begin{aligned} C_2&=\frac{h'(0)}{2}\left[ \frac{c(dx_n)}{4i(\xi _n-i)} +\frac{c(dx_n)-ic(\xi ')}{8(\xi _n-i)^2} +\frac{3\xi _n-7i}{8(\xi _n-i)^3}[ic(\xi ')-c(dx_n)]\right] . \end{aligned}$$

(3.52)

By (3.50) and (3.52), we have

$$\begin{aligned} {\mathrm{tr }}[C_2\times \partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})]|_{|\xi '|=1}= & {} \frac{i}{2}h'(0)\frac{-i\xi _n^2-\xi _n+4i}{4(\xi _n-i)^3 (\xi _n+i)^2}{\mathrm{tr}}[ \texttt {id}]\nonumber \\= & {} 2ih'(0)\frac{-i\xi _n^2-\xi _n+4i}{4(\xi _n-i)^3(\xi _n+i)^2}. \end{aligned}$$

(3.53)

By (3.50) and (3.51), we have

$$\begin{aligned} {\mathrm{tr }}[C_1\times \partial _{\xi _n}\sigma _{-1} (({{\widetilde{D}}}^*)^{-1})]|_{|\xi '|=1}= \frac{-2ic_0}{(1+\xi _n^2)^2}+h'(0)\frac{\xi _n^2 -i\xi _n-2}{2(\xi _n-i)(1+\xi _n^2)^2}, \end{aligned}$$

(3.54)

where $Q=c_0c(dx_n)$ and $c_0=-\frac{3}{4}h'(0)$.

By (3.53) and (3.54), we have

$$\begin{aligned}&-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} [(C_1-C_2)\times \partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})](x_0)d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad =-\Omega _3\int \limits _{\Gamma ^+}\frac{2c_0(\xi _n-i)+ih'(0)}{(\xi _n-i)^3 (\xi _n+i)^2}d\xi _ndx'\nonumber \\&\quad =\frac{9}{8}\pi h'(0)\Omega _3dx'. \end{aligned}$$

(3.55)

Then, we have

$$\begin{aligned}&-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \left[ \pi ^+_{\xi _n} \Big [\frac{c(\xi )(f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2)c(\xi )}{(1+\xi _n^2)^2}\Big ]\right. \nonumber \\&\quad \left. \times \partial _{\xi _n}\sigma _{-1}(({{\widetilde{D}}}^*)^{-1})\right] (x_0)d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad =\frac{\pi }{4}{\mathrm{tr}}\left[ c(dx_n)\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \right] \Omega _3dx'\nonumber \\&\quad =0. \end{aligned}$$

(3.56)

Then, we have

$$\begin{aligned} \Phi _4=\frac{9}{8}\pi h'(0)\Omega _3dx'. \end{aligned}$$

(3.57)

Case (c) $r=-1,~l=-2,~k=j=|\alpha |=0$

By (3.18), we get

$$\begin{aligned} \Phi _5=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} [\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \partial _{\xi _n}\sigma _{-2}(({{\widetilde{D}}}^*)^{-1})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(3.58)

By (3.5) and (3.6), Lemma 3.7, we have

$$\begin{aligned} \pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})|_{|\xi '|=1}= \frac{c(\xi ')+ic(dx_n)}{2(\xi _n-i)}. \end{aligned}$$

(3.59)

Since

$$\begin{aligned} \sigma _{-2}(({{\widetilde{D}}}^*)^{-1})(x_0)= & {} \frac{c(\xi ) \sigma _{0}({{\widetilde{D}}}^*)(x_0)c(\xi )}{|\xi |^4}\nonumber \\&+\frac{c(\xi )}{|\xi |^6}c(dx_n) \bigg [\partial _{x_n}[c(\xi ')](x_0)|\xi |^2-c(\xi )h'(0)|\xi |^2_{\partial _ M}\bigg ], \end{aligned}$$

(3.60)

where

$$\begin{aligned} \sigma _{0}({{\widetilde{D}}}^*)(x_0)&= -\frac{1}{4}\sum _{s,t,i}\omega _{s,t}(e_i)(x_{0})c(e_i)c(e_s)c(e_t)\nonumber \\&\quad +\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) (x_{0})\nonumber \\&=Q(x_0)+\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) (x_{0}), \end{aligned}$$

(3.61)

then

$$\begin{aligned}&\partial _{\xi _n}\sigma _{-2}(({{\widetilde{D}}}^*)^{-1})(x_0)|_{|\xi '|=1}\nonumber \\&\quad = \partial _{\xi _n}\bigg \{\frac{c(\xi )[Q(x_0) +(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0}) ]c(\xi )}{|\xi |^4}\nonumber \\&\qquad +\frac{c(\xi )}{|\xi |^6}c(dx_n)[\partial _{x_n}[c(\xi ')](x_0)|\xi |^2-c(\xi )h'(0) ]\bigg \}\nonumber \\&\quad =\partial _{\xi _n}\bigg \{\frac{c(\xi )}{|\xi |^6}c(dx_n)[\partial _{x_n}[c(\xi ') ](x_0)|\xi |^2-c(\xi )h'(0)]\bigg \}+\partial _{\xi _n}\frac{c(\xi )Q(x_0)c(\xi )}{|\xi |^4}\nonumber \\&\qquad +\partial _{\xi _n}\frac{c(\xi )(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2})(x_{0})c(\xi )}{|\xi |^4}. \end{aligned}$$

(3.62)

by $c(\xi )=c(\xi ')+\xi _nc(dx_n)$, $|\xi |^2=1+\xi _n^2$ and direct derivation, we get

$$\begin{aligned}&\partial _{\xi _n}\frac{c(\xi )(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0})c(\xi )}{|\xi |^4}\nonumber \\&\quad =\partial _{\xi _n}\frac{(c(\xi ')+\xi _nc(dx_n))(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0})(c(\xi ')+\xi _nc(dx_n))}{|\xi |^4}\nonumber \\&\quad =\frac{c(dx_n)(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0})c(\xi )}{|\xi |^4}\nonumber \\&\qquad +\frac{c(\xi )(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0})c(dx_n)}{|\xi |^4}\nonumber \\&\qquad -\frac{4\xi _n c(\xi )(-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2})(x_{0})c(\xi )}{|\xi |^4}.\nonumber \\ \end{aligned}$$

(3.63)

We denote

$$\begin{aligned} q_{-2}^{1}=\frac{c(\xi )Q(x_0)c(\xi )}{|\xi |^4}+\frac{c(\xi )}{|\xi |^6} c(dx_n)[\partial _{x_n}[c(\xi ')](x_0)|\xi |^2-c(\xi )h'(0)], \end{aligned}$$

then

$$\begin{aligned} \partial _{\xi _n}(q_{-2}^{1})&=\frac{1}{(1+\xi _n^2)^3} \bigg [(2\xi _n-2\xi _n^3)c(dx_n)Q(x_0)c(dx_n) +(1-3\xi _n^2)c(dx_n)Q(x_0)c(\xi ')\nonumber \\&\quad +(1-3\xi _n^2)c(\xi ')Q(x_0)c(dx_n) -4\xi _nc(\xi ')Q(x_0)c(\xi ') +(3\xi _n^2-1)\partial _{x_n}c(\xi ')\nonumber \\&\quad -4\xi _nc(\xi ')c(dx_n)\partial _{x_n}c(\xi ') +2h'(0)c(\xi ')+2h'(0)\xi _nc(dx_n)\bigg ]\nonumber \\&\quad +6\xi _nh'(0)\frac{c(\xi )c(dx_n)c(\xi )}{(1+\xi ^2_n)^4}. \end{aligned}$$

(3.64)

By (3.59) and (3.64), we have

$$\begin{aligned} {\mathrm{tr}}[\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \partial _{\xi _n}(q^1_{-2})](x_0)|_{|\xi '|=1}= & {} \frac{3h'(0)(i\xi ^2_n+\xi _n-2i)}{(\xi -i)^3(\xi +i)^3}\nonumber \\&+\frac{12h'(0)i\xi _n}{(\xi -i)^3(\xi +i)^4}, \end{aligned}$$

(3.65)

then

$$\begin{aligned}&-i\Omega _3\int \limits _{\Gamma _+}\left[ \frac{3h'(0)(i\xi _n^2+\xi _n-2i)}{(\xi _n-i)^3(\xi _n+i)^3}+\frac{12h'(0)i\xi _n}{(\xi _n-i)^3(\xi _n+i)^4}\right] d\xi _ndx'\nonumber \\&\quad = -\frac{9}{8}\pi h'(0)\Omega _3dx'. \end{aligned}$$

(3.66)

By $\int _{|\xi '|=1}\xi _{1}\cdot \cdot \cdot \xi _{2d+1}\sigma (\xi ')=0$, (3.59) and (3.62), we have

$$\begin{aligned}&-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{tr}}\left[ \pi ^+_{\xi _n} \sigma _{-1}({{\widetilde{D}}}^{-1})\times \partial _{\xi _n}\frac{c(\xi )(-\overline{f_1}\sum _{u<v}(p_{uv }-p_{vu})c(e_u)c(e_v)+\overline{f_2})c(\xi )}{|\xi |^4}\right] \nonumber \\&(x_0)d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad =-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\frac{i}{(\xi -i)(\xi +i)^3} {\mathrm{tr}}\left[ c(dx_n)\left( -\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) \right] \nonumber \\&(x_0)d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad =-\frac{\pi }{4}{\mathrm{tr}}\left[ c(dx_n)\left( -\overline{f_1}\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) \right] \Omega _3dx'\nonumber \\&\quad =0. \end{aligned}$$

(3.67)

Then,

$$\begin{aligned} \Phi _5=-\frac{9}{8}\pi h'(0)\Omega _3dx'. \end{aligned}$$

(3.68)

So $\Phi =\sum _{i=1}^5\Phi _i=0$.

By (3.17), (3.19) and (3.68), we can get

Theorem 3.8

Let M be a 4-dimensional compact oriented spin manifolds with the boundary $\partial M$ and the metric $g^M$ as above, ${{\widetilde{D}}}$ and ${{\widetilde{D}}}^*$ be the Dirac–Witten operators on ${\widetilde{M}}$ , then

$$\begin{aligned}&\widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+ ({{\widetilde{D}}}^*)^{-1}]\nonumber \\&\quad =32\pi ^2\int \limits _{M}\bigg \{-\frac{1}{3}s-12f_2\overline{f_2}+4f_2^2 +4\overline{f_2}^2+4f_1\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})^2 \bigg \}d{\mathrm{Vol}_{\mathrm{M}}}. \end{aligned}$$

(3.69)

where s is the scalar curvature.

4 A Kastler–Kalau–Walze type theorem for 6-dimensional manifolds with boundary

Firstly, we prove the Kastler–Kalau–Walze type theorems for 6-dimensional manifolds with boundary. From [14], we know that

$$\begin{aligned} \widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+ ({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1}]= & {} \int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\sigma _{-4}( ({{\widetilde{D}}}^*{{\widetilde{D}}})^{-2})]\sigma (\xi )dx\nonumber \\&+\int \limits _{{\partial _t}M}\Psi , \end{aligned}$$

(4.1)

where

$$\begin{aligned} \Psi&=\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum ^{\infty }_{j, k=0} \sum \frac{(-i)^{|\alpha |+j+k+1}}{\alpha !(j+k+1)!} \times {\mathrm{trace}}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\partial ^j_{x_n} \partial ^\alpha _{\xi '}{\partial _t}^k_{\xi _n}\sigma ^+_{r}({{\widetilde{D}}}^{-1})(x',0,\xi ',\xi _n) \nonumber \\&\quad \times \partial ^\alpha _{x'}\partial ^{j+1}_{\xi _n}\partial ^k_{x_n}\sigma _{l} (({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})(x',0,\xi ',\xi _n)]d\xi _n\sigma (\xi ')dx', \end{aligned}$$

(4.2)

and the sum is taken over $r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3$.

By Theorem 2.2, we compute the interior term of (4.1), then

$$\begin{aligned}&\int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}}[\sigma _{-4}(({{\widetilde{D} }}^*{{\widetilde{D}}})^{-2})]\sigma (\xi )dx\nonumber \\&\quad =128\pi ^3\int \limits _{M}\bigg ( -\frac{2}{3}s-4(f_1^2+\overline{f_1}^2-4f_1\overline{f_1}) \sum _{u<v}(p_{uv}-p_{vu})^2-32f_2\overline{f_2}+12f_2^2 +12\overline{f_2}^2\bigg )d{\mathrm{Vol}_{\mathrm{M}}}. \end{aligned}$$

(4.3)

Next, we compute $\int _{\partial M} \Psi$. By (2.12), we get

$$\begin{aligned} {{\widetilde{D}}}^*{\widetilde{D}}&=D^2+D\left[ -\overline{f_1}\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\right] \nonumber \\&\quad +\left[ f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right] D\nonumber \\&\quad +\left[ -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right] \nonumber \\&\quad \left[ f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right] , \end{aligned}$$

(4.4)

Then,

$$\begin{aligned} {{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*&=\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle (-g^{ij}\partial _{l}\partial _{i}\partial _{j})\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \bigg \{-(\partial _{l}g^{ij})\partial _{i}\partial _{j}-g^{ij}\bigg (4\sigma _{i} \partial _{j}-2\Gamma ^{k}_{ij}\partial _{k}\bigg )\partial _{l}\bigg \}\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \left\{ -2(\partial _{l}g^{ij})\sigma _{i}\partial _{j}+g^{ij} (\partial _{l}\Gamma ^{k}_{ij})\partial _{k}\right. \nonumber \\&\quad -2g^{ij}\partial _{l}\sigma _{i}\partial _{j} +(\partial _{l}g^{ij})\Gamma ^{k}_{ij}\partial _{k}\nonumber \\&\quad +\sum _{j,k} \left[ \partial _{l}\left( \left( f_1 \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(e_{j})\right. \right. \nonumber \\&\quad \left. \left. +c(e_{j})\left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) -\overline{f_2}\right) \right) \right] \langle e_{j},dx^{k}\rangle \partial _{k}\nonumber \\&\quad +\sum _{j,k}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) c(e_{j})\nonumber \\&\quad \left. \left. +c(e_{j})\left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)-\overline{f_2}\right) \right) \Big [\partial _{l}\langle e_{j},dx^{k}\rangle \Big ]\partial _{k} \right\} \nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \partial _{l} \left\{ -g^{ij}\Big [(\partial _{i}\sigma _{j})+\sigma _{i}\sigma _{j} -\Gamma _{i,j}^{k}\sigma _{k}\right. \nonumber \\&\quad +\sum _{i,j}g^{i,j}\left[ \Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\Bigg )c(\partial _{i})\sigma _{i}\right. \nonumber \\&\left. \quad +c(\partial _{i}) \partial _{i}\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) \right. \nonumber \\&\quad \left. +c(\partial _{i})\sigma _{i}\left( -\overline{f_1}\sum _{u<v} (p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2}\right) \right] \nonumber \\&\quad \left. +\frac{1}{4}s +\left( \overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) -\overline{f_2}\right) ^2\right\} \nonumber \\&\quad +\left[ \sigma _{i}-(\overline{f_1} \sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)-\overline{f_2})\right] (-g^{ij}\partial _{i}\partial _{j})\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \left\{ 2\sum _{j,k} \left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(e_{j})+c(e_{j})\right. \right. \nonumber \\&\quad \left. \left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) \right] \nonumber \\&\quad \left. \times \langle e_{i},dx_{k}\rangle \right\} _{l}\partial _{k} +\left[ \sigma _{i}+\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\right) \right] \nonumber \\&\quad \bigg \{-\sum _{i,j}g^{i,j}\Big [2\sigma _{i}\partial _{j} -\Gamma _{i,j}^{k}\partial _{k}+(\partial _{i}\sigma _{j})+\sigma _{i}\sigma _{j} -\Gamma _{i,j}^{k}\sigma _{k}\Big ]\nonumber \\&\quad -\sum _{i,j}g^{i,j}\Bigg [c(\partial _{i}) \Bigg (-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\nonumber \\&\quad +\overline{f_2}\Bigg )+\Big (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\Big )c(\partial _{i})\Bigg ]\partial _{j}\nonumber \\&\quad +\sum _{i,j}g^{i,j}\left[ \Bigg (f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\Bigg )c(\partial _{i})\right. \nonumber \\&\quad \sigma _{i}+c(\partial _{i})\partial _{i}\Bigg (-\overline{f_1} \sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\Bigg )\nonumber \\&\quad -c(\partial _{i}) \sigma _{i}\Bigg (-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +\overline{f_2}\Bigg )+c(\partial _{i})\nonumber \\&\quad \partial _{i}\Bigg (-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v )+\overline{f_2}\Bigg )c(\partial _{i})\sigma _{i}\nonumber \\&\quad \left. \left( -\overline{f_1}\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+\overline{f_2}\right) \right] \nonumber \\&\quad \left. +\frac{1}{4}s-\left( f_1\sum _{u<v} (p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) ^2\right\} . \end{aligned}$$

(4.5)

Then, by (4.5), we obtain

Lemma 4.1

The following identities hold:

$$\begin{aligned} \sigma _2({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)&=\sum _{i,j,l}c(dx_{l})\partial _{l}(g^{i,j})\xi _{i}\xi _{j} +c(\xi ) (4\sigma ^k-2\Gamma ^k)\xi _{k}\nonumber \\&\quad +2\left[ |\xi |^2\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2}\right) \right. \nonumber \\&\quad \left. -c(\xi )\left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) c(\xi )\right] \nonumber \\&\quad -\frac{1}{4}|\xi |^2\sum _{s,t,l}\omega _{s,t} c(e_l)c(e_s)c(e_t)\nonumber \\&\quad +|\xi |^2\left( -\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+\overline{f_2}\right) ^2;\nonumber \\ \sigma _{3}({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)&=ic(\xi )|\xi |^{2};\nonumber \\ \end{aligned}$$

(4.6)

where $\xi =\Sigma _{i=1}^n\xi _id_{x_i}$ denotes the cotangent vector.

Write

$$\begin{aligned} \sigma ({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)&=p_3+p_2+p_1+p_0; ~\sigma (({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^* )^{-1})=\sum ^{\infty }_{j=3}q_{-j}. \end{aligned}$$

(4.7)

By the composition formula of pseudodifferential operators, we have

$$\begin{aligned} 1=\sigma (({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*) \circ ({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})&= \sum _{\alpha }\frac{1}{\alpha !}\partial ^{\alpha }_{\xi } [\sigma ({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)]D^{\alpha }_{x} [({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1}] \nonumber \\&=(p_3+p_2+p_1+p_0)(q_{-3}+q_{-4}+q_{-5}+\cdots )\nonumber \\&\quad +\sum _j(\partial _{\xi _j}p_3+\partial _{\xi _j}p_2+\partial _{\xi _j}p_1 +\partial _{\xi _j}p_0)\nonumber \\ (D_{x_j}q_{-3}+D_{x_j}q_{-4}+D_{x_j}q_{-5}+\cdots )&=p_3q_{-3}+(p_3q_{-4}+p_2q_{-3}\nonumber \\&\quad +\sum _j\partial _{\xi _j}p_3D_{x_j}q_{-3})+\cdots , \end{aligned}$$

(4.8)

by (4.8), we have

$$\begin{aligned} q_{-3}=p_3^{-1};~q_{-4}=-p_3^{-1} \left[ p_2p_3^{-1}+\sum _j\partial _{\xi _j}p_3D_{x_j}(p_3^{-1})\right] . \end{aligned}$$

(4.9)

By Lemma 4.1, we have some symbols of operators.

Lemma 4.2

The following identities hold:

$$\begin{aligned} \sigma _{-3}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})&=\frac{ic(\xi )}{|\xi |^{4}};\nonumber \\ \sigma _{-4}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})&= \frac{c(\xi )\sigma _{2}({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*) c(\xi )}{|\xi |^8}\nonumber \\&\quad +\frac{ic(\xi )}{|\xi |^8}\Big (|\xi |^4c(dx_n)\partial _{x_n}c(\xi ') -2h'(0)c(dx_n)c(\xi )\nonumber \\&\quad +2\xi _{n}c(\xi )\partial _{x_n}c(\xi ')+4\xi _{n}h'(0)\Big ). \end{aligned}$$

(4.10)

When $n=6$, then ${\mathrm{tr}}_{\wedge ^*T^*M}[\texttt {id}]=8$, where ${\mathrm{tr}}$ as shorthand of ${\mathrm{trace}}$. Since the sum is taken over $r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3$, then we have the sum of the following five cases:

Case (a) (I) $r=-1, l=-3, j=k=0, |\alpha |=1$.

By (4.2), we get

$$\begin{aligned} \Psi _1= & {} -\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum _{|\alpha |=1}{\mathrm{trace}}\Big [\partial ^{\alpha }_{\xi '}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial ^{\alpha }_{x'}\partial _{\xi _{n}}\sigma _{-3} (({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})\Big ]\nonumber \\&(x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.11)

By Lemma 4.2, for $i<n$, we have

$$\begin{aligned} \partial _{x_{i}}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})(x_0)\,= \,& {} \partial _{x_{i}}\Big [\frac{ic(\xi )}{|\xi |^{4}}\Big ](x_{0})\nonumber \\=\, & {} i\partial _{x_{i}}[c(\xi )]|\xi |^{-4}(x_{0})-2ic(\xi )\partial _{x_{i}} [|\xi |^{2}]|\xi |^{-6}(x_{0})=0, \end{aligned}$$

(4.12)

so $\Psi _1=0$.

Case (a) (II) $r=-1, l=-3, |\alpha |=k=0, j=1$.

By (4.2), we have

$$\begin{aligned} \Psi _2=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } \mathrm{trace} \Big [\partial _{x_{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial ^{2}_{\xi _{n}}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})\Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.13)

By direct derivation, we have

$$\begin{aligned} \partial ^{2}_{\xi _{n}}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})&=\partial _{\xi _n}\left[ \frac{-4 i \xi _{n} c(\xi ')}{(1+\xi _{n}^{2})^{3}}+\frac{i(1- 3\xi _{n}^{2})c(\texttt {d}x_{n})}{(1+\xi _{n}^{2})^{3}}\right] \nonumber \\&=i\bigg [\frac{(20\xi ^{2}_{n}-4)c(\xi ')+ 12(\xi ^{3}_{n}-\xi _{n})c(dx_{n})}{(1+\xi _{n}^{2})^{4}}\bigg ]. \end{aligned}$$

(4.14)

Since $n=6$, ${\mathrm{tr}}[-\texttt {id}]=-8$. By the relation of the Clifford action and ${\mathrm{tr}}AB={\mathrm{tr}}BA$, then

$$\begin{aligned}&{\mathrm{tr}}[c(\xi ')c(dx_{n})]=0; \ {\mathrm{tr}}[c(dx_{n})^{2}]=-8;\ {\mathrm{tr}}[c(\xi ')^{2}](x_{0})|_{|\xi '|=1}=-8;\nonumber \\&{\mathrm{tr}}[\partial _{x_{n}}[c(\xi ')]c(\texttt {d}x_{n})]=0; \ {\mathrm{tr}}[\partial _{x_{n}}c(\xi ')c(\xi ')](x_{0})|_{|\xi '|=1}=-4h'(0). \end{aligned}$$

(4.15)

By (3.29), (4.13) and (4.15), we get

$$\begin{aligned}&{\mathrm{trace}} \Big [\partial _{x_{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial ^{2}_{\xi _{n}}\sigma _{-3}(({{\widetilde{D}}}^{*} {{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})\Big ](x_0)\nonumber \\&\quad =8 h'(0)\frac{-1-3\xi _{n}i+5\xi ^{2}_{n} +3i\xi ^{3}_{n}}{(\xi _{n}-i)^{6}(\xi _{n}+i)^{4}}. \end{aligned}$$

(4.16)

Then we obtain

$$\begin{aligned} \Psi _2&=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } h'(0) dimF\frac{-1-3\xi _{n}i+5\xi ^{2}_{n}+3i\xi ^{3}_{n}}{(\xi _{n}-i)^{6} (\xi _{n}+i)^{4}}d\xi _n\sigma (\xi ')dx'\nonumber \\&=h'(0)\Omega _{4}\int \limits _{\Gamma ^{+}}\frac{4+12\xi _{n}i-20\xi ^{2}_{n} -122i\xi ^{3}_{n}}{(\xi _{n}-i)^{6}(\xi _{n}+i)^{4}}d\xi _{n}dx'\nonumber \\&=h'(0)\Omega _{4}\frac{\pi i}{5!}\Big [\frac{1+3\xi _{n}i-5\xi ^{2}_{n} -3i\xi ^{3}_{n}}{(\xi _{n}+i)^{4}}\Big ]^{(5)}|_{\xi _{n}=i}dx'\nonumber \\&=-\frac{15}{16}\pi h'(0)\Omega _{4}dx', \end{aligned}$$

(4.17)

where ${{\Omega _{4}}}$ is the canonical volume of $S^{4}$.

Case (a) (III) $r=-1,l=-3,|\alpha |=j=0,k=1$.

By (4.2), we have

$$\begin{aligned} \Psi _3=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\partial _{\xi _{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\partial _{x_{n}}\sigma _{-3}(({{\widetilde{D}}}^{*} {{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})\Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.18)

By direct derivation, we have

$$\begin{aligned} \partial _{\xi _{n}}\partial _{x_{n}}\sigma _{-3}(({{\widetilde{D}}}^{*} {{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})&=\partial _{\xi _{n}} \frac{i\partial _{x_{n}}c(\xi ')(x_0)|\xi |^2-ic(\xi )|\xi '|^2h'(0)}{|\xi |^4}\nonumber \\&=-\frac{4 i\xi _{n}\partial _{x_{n}}c(\xi ')(x_{0})}{(1+\xi _{n}^{2})^{3}} + i\frac{12h'(0)\xi _{n}c(\xi ')}{(1+\xi _{n}^{2})^{4}} -i\frac{(2-10\xi ^{2}_{n})h'(0)c(dx_{n})}{(1+\xi _{n}^{2})^{4}}. \end{aligned}$$

(4.19)

Combining (3.36) and (4.19), we have

$$\begin{aligned}&{\mathrm{trace}} \Big [\partial _{\xi _{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\partial _{x_{n}}\sigma _{-3}(({{\widetilde{D}} }^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})\Big ](x_{0})|_{|\xi '|=1}\nonumber \\&\quad =h'(0)\frac{8i-32\xi _{n}-8i\xi ^{2}_{n}}{(\xi _{n}-i)^{5}(\xi +i)^{4}}. \end{aligned}$$

(4.20)

Then

$$\begin{aligned} \Psi _3&=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } h'(0) \frac{8i-32\xi _{n}-8i\xi ^{2}_{n}}{(\xi _{n}-i)^{5}(\xi +i)^{4}}d\xi _n\sigma (\xi ')dx'\nonumber \\&=-\frac{1}{2}h'(0)\Omega _{4}\int \limits _{\Gamma ^{+}}\frac{8i-32\xi _{n} -8i\xi ^{2}_{n}}{(\xi _{n}-i)^{5}(\xi +i)^{4}}d\xi _{n}dx'\nonumber \\&=-h'(0)\Omega _{4}\frac{\pi i}{4!}\Big [\frac{8i-32\xi _{n} -8i\xi ^{2}_{n}}{(\xi +i)^{4}}\Big ]^{(4)}|_{\xi _{n}=i}dx'\nonumber \\&=\frac{25}{16}\pi h'(0)\Omega _{4}dx'. \end{aligned}$$

(4.21)

Case (b) $r=-1,l=-4,|\alpha |=j=k=0$.

By (4.2), we have

$$\begin{aligned} \Psi _4&=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\sigma _{-4}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})\Big ](x_0)d\xi _n\sigma (\xi ')dx'\nonumber \\&=i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} [\partial _{\xi _n} \pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \sigma _{-4}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.22)

In the normal coordinate, $g^{ij}(x_{0})=\delta ^{j}_{i}$ and $\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=0$, if $j<n$; $\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=h'(0)\delta ^{\alpha }_{\beta }$, if $j=n$. So by [12], when $k<n$, we have $\Gamma ^{n}(x_{0})=\frac{5}{2}h'(0)$, $\Gamma ^{k}(x_{0})=0$, $\delta ^{n}(x_{0})=0$ and $\delta ^{k}=\frac{1}{4}h'(0)c(e_{k})c(e_{n})$. Then, we obtain

$$\begin{aligned}&\sigma _{-4}(({{\widetilde{D}}}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*} )^{-1})(x_{0})|_{|\xi '|=1}\nonumber \\&\quad = \frac{c(\xi )\sigma _{2}({{\widetilde{D}}}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*}) (x_{0})|_{|\xi '|=1}c(\xi )}{|\xi |^8} -\frac{c(\xi )}{|\xi |^4}\sum _j\partial _{\xi _j}\big (c(\xi )|\xi |^2\big ){D}_{x_j} \bigg (\frac{ic(\xi )}{|\xi |^4}\bigg )\nonumber \\&\quad =\frac{1}{|\xi |^8}c(\xi )\Bigg (\frac{1}{2}h'(0)c(\xi )\sum _{k<n}\xi _kc(e_k)c(e_n) -\frac{5}{2}h'(0)\xi _nc(\xi )-\frac{1}{4}h'(0)\nonumber \\&\qquad |\xi |^2c(dx_n) +2\Bigg [|\xi |^2\Bigg (-\overline{f_1}\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\Bigg ) -c(\xi )\nonumber \\&\qquad \Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\Bigg )c(\xi )\Bigg ]+|\xi |^2\Bigg (-\overline{f_1} \sum _{u<v}(p_{uv}-p_{vu})\nonumber \\&\qquad c(e_u)c(e_v)+\overline{f_2})\Bigg )c(\xi )+\frac{ic(\xi )}{|\xi |^8} \Big (|\xi |^4c(dx_n)\partial _{x_n}c(\xi ') -2h'(0)c(dx_n)c(\xi )\nonumber \\&\qquad +2\xi _{n}c(\xi )\partial _{x_n}c(\xi ')+4\xi _{n}h'(0)\Big ).\nonumber \\ \end{aligned}$$

(4.23)

By (3.32) and (4.23), we have

$$\begin{aligned}&{\mathrm{tr}} [\partial _{\xi _n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \sigma _{-4}({{\widetilde{D}}}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1}](x_0)|_{|\xi '|=1} \nonumber \\&\quad =\frac{1}{2(\xi _{n}-i)^{2}(1+\xi _{n}^{2})^{4}}\Bigg (\frac{3}{4}i+2+(3+4i)\xi _{n}+(-6+2i)\xi _{n}^{2}+3\xi _{n}^{3}+\frac{9i}{4}\xi _{n}^{4}\Bigg )h'(0){\mathrm{tr}} [id]\nonumber \\&\qquad +\frac{1}{2(\xi _{n}-i)^{2}(1+\xi _{n}^{2})^{4}}\big (-1-3i\xi _{n}-2\xi _{n}^{2}-4i\xi _{n}^{3}-\xi _{n}^{4}-i\xi _{n}^{5}\big ){\mathrm{tr}[c(\xi ')\partial _{x_n}c(\xi ')]}.\nonumber \\ \end{aligned}$$

(4.24)

Then by (4.24), we get

$$\begin{aligned} \Psi _4&= ih'(0)\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }8 \times \frac{\frac{3}{4}i+2+(3+4i)\xi _{n}+(-6+2i)\xi _{n}^{2} +3\xi _{n}^{3}+\frac{9i}{4}\xi _{n}^{4}}{2(\xi _n-i)^5(\xi _n+i )^4}d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad +ih'(0)\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }4\times \frac{1+3i\xi _{n} +2\xi _{n}^{2}+4i\xi _{n}^{3}+\xi _{n}^{4}+i\xi _{n}^{5}}{2(\xi _{n}-i)^{2} (1+\xi _{n}^{2})^{4}}d\xi _n\sigma (\xi ')dx'\nonumber \\&=\left( -\frac{41}{64}i-\frac{195}{64}\right) \pi h'(0)\Omega _4dx'.\nonumber \\ \end{aligned}$$

(4.25)

Case (c) $r=-2,l=-3,|\alpha |=j=k=0$.

By (4.2), we have

$$\begin{aligned} \Psi _5=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\pi ^{+}_{\xi _{n}}\sigma _{-2}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\sigma _{-3}(({{\widetilde{D}}}^{*}{ {\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})\Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.26)

By Lemma 4.1 and Lemma 4.2, we have

$$\begin{aligned} \sigma _{-2}({{\widetilde{D}}}^{-1})(x_0)&=\frac{c(\xi )\sigma _{0} ({{\widetilde{D}}})c(\xi )}{|\xi |^4}(x_0)+\frac{c(\xi )}{|\xi |^6}\sum _jc(dx_j) \Big [\partial _{x_j}(c(\xi ))|\xi |^2-c(\xi )\partial _{x_j}(|\xi |^2)\Big ](x_0), \end{aligned}$$

(4.27)

where

$$\begin{aligned} \sigma _0({{\widetilde{D}}})= -\frac{1}{4}\sum _{i,s,t}\omega _{s,t}(e_i)c(e_i)c(e_s)c(e_t) +\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) . \end{aligned}$$

(4.28)

On the other hand,

$$\begin{aligned} \partial _{\xi _{n}}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1})=\frac{-4 i \xi _{n}c(\xi ')}{(1+\xi _{n}^{2})^{3}} +\frac{i(1- 3\xi _{n}^{2})c(\texttt {d}x_{n})}{(1+\xi _{n}^{2})^{3}}. \end{aligned}$$

(4.29)

By (4.27), (3.5) and (3.6), we have

$$\begin{aligned} \pi ^{+}_{\xi _{n}}\Big (\sigma _{-2}({{\widetilde{D}}}^{-1})\Big )(x_{_{0}})|_{|\xi '|=1}&=\pi ^{+}_{\xi _{n}}\Big [\frac{c(\xi )\sigma _{0}({{\widetilde{D}}})(x_{0})c(\xi ) +c(\xi )c(dx_{n})\partial _{x_{n}}[c(\xi ')](x_{0})}{(1+\xi ^{2}_{n})^{2}}\Big ]\nonumber \\&\quad -h'(0)\pi ^{+}_{\xi _{n}}\Big [\frac{c(\xi )c(dx_{n})c(\xi )}{(1+\xi ^{2}_{n})^{3}}\Big ]. \end{aligned}$$

(4.30)

We denote

$$\begin{aligned} \sigma _{0}({{\widetilde{D}}})(x_0)|_{\xi _n=i}=Q(x_0) +\Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\Bigg ). \end{aligned}$$

(4.31)

Then, we obtain

$$\begin{aligned}&\pi ^{+}_{\xi _{n}}\Big (\sigma _{-2}({{\widetilde{D}}}^{-1})\Big )(x_{_{0}})|_{|\xi '|=1} \quad =\pi ^+_{\xi _n}\Big [\frac{c(\xi )Q(x_0)c(\xi )+c(\xi )c(dx_n) \partial _{x_n}[c(\xi ')](x_0)}{(1+\xi _n^2)^2}-h'(0)\frac{c(\xi )c(dx_n )c(\xi )}{(1+\xi _n^{2})^3}\Big ]\nonumber \\&\qquad +\pi ^+_{\xi _n}\Big [\frac{c(\xi )[(f_1\sum _{u<v}(p_{uv}-p_{vu} )c(e_u)c(e_v)+f_2)]c(\xi )(x_0)}{(1+\xi _n^2)^2}\Big ].\nonumber \\ \end{aligned}$$

(4.32)

Furthermore,

$$\begin{aligned}&\pi ^+_{\xi _n}\Big [\frac{c(\xi )[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(\xi )}{(1+\xi _n^2)^2}\Big ]\nonumber \\&\quad =\pi ^+_{\xi _n}\Big [\frac{c(\xi ')[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(\xi ')}{(1+\xi _n^2)^2}\Big ]\nonumber \\&\qquad +\pi ^+_{\xi _n}\Big [ \frac{\xi _nc(\xi ')[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2] (x_0)c(dx_{n})}{(1+\xi _n^2)^2}\Big ]\nonumber \\&\qquad +\pi ^+_{\xi _n}\Big [\frac{\xi _nc(dx_{n})[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(\xi ')}{(1+\xi _n^2)^2}\Big ]\nonumber \\&\qquad +\pi ^+_{\xi _n}\Big [\frac{\xi _n^{2}c(dx_{n})[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(dx_{n})}{(1+\xi _n^2)^2}\Big ]\nonumber \\&=-\frac{c(\xi ')[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(\xi ')(2+i\xi _{n})}{4(\xi _{n}-i)^{2}}\nonumber \\&\qquad +\frac{ic(\xi ')[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(dx_{n})}{4(\xi _{n}-i)^{2}}\nonumber \\&\qquad +\frac{ic(dx_{n})[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(\xi ')}{4(\xi _{n}-i)^{2}}\nonumber \\&\qquad +\frac{-i\xi _{n}c(dx_{n})[f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2](x_0)c(dx_{n})}{4(\xi _{n}-i)^{2}}.\nonumber \\ \end{aligned}$$

(4.33)

By $c(\xi )=c(\xi ')+\xi _nc(dx_n)$, we have

$$\begin{aligned}&\pi ^+_{\xi _n}\Big [\frac{c(\xi )Q(x_0)c(\xi )+c(\xi )c(dx_n)\partial _{x_n}(c(\xi '))(x_0)}{(1+\xi _n^2)^2}\Big ]-h'(0)\pi ^+_{\xi _n}\Big [\frac{c(\xi )c(dx_n)c(\xi )}{(1+\xi _n)^3}\Big ]\nonumber \\&\quad =\pi ^+_{\xi _n}\Big [\frac{(c(\xi ')+\xi _nc(dx_n))Q(x_0)(c(\xi ')+\xi _nc(dx_n))+(c(\xi ')+\xi _nc(dx_n))c(dx_n)\partial _{x_n}(c(\xi '))(x_0)}{(1+\xi _n^2)^2}\Big ]\nonumber \\&\qquad -h'(0)\pi ^+_{\xi _n}\Big [\frac{(c(\xi ')+\xi _nc(dx_n))c(dx_n)(c(\xi ')+\xi _nc(dx_n))}{(1+\xi _n)^3}\Big ]\nonumber \\&\quad =\frac{1}{2\pi i}\lim _{u\rightarrow 0^-}\int \limits _{\Gamma ^+}\frac{\frac{(c(\xi ')+\eta _nc(dx_n))Q(x_0)(c(\xi ')+\xi _nc(dx_n))+(c(\xi ')+\eta _nc(dx_n))c(dx_n)\partial _{x_n}(c(\xi '))(x_0)}{(\eta _n+i)^2(\xi _n+iu-\eta _n)}}{(\eta _n-i)^2}d\eta _n\nonumber \\&\qquad -\frac{1}{2\pi i}\lim _{u\rightarrow 0^-}\int \limits _{\Gamma ^+}h'(0)\frac{\frac{(c(\xi ')+\eta _nc(dx_n))c(dx_n)(c(\xi ')+\eta _nc(dx_n))}{(\eta _n+i)^3(\xi _n+iu-\eta _n)}}{(\eta _n-i)^3}d\eta _n\nonumber \\&\quad := C_1-C_2,\nonumber \\ \end{aligned}$$

(4.34)

where

$$\begin{aligned} C_1&=\frac{-1}{4(\xi _n-i)^2}\big [(2+i\xi _n)c(\xi ')Qc(\xi ')+i\xi _nc(dx_n)Qc(dx_n) \nonumber \\&\quad +(2+i\xi _n)c(\xi ')c(dx_n)\partial _{x_n}c(\xi ')+ic(dx_n)Q^2_0c(\xi ') +ic(\xi ')Qc(dx_n)-i\partial _{x_n}c(\xi ')\big ]\nonumber \\&=\frac{1}{4(\xi _n-i)^2}\Big [\frac{5}{2}h'(0)c(dx_n)-\frac{5i}{2}h'(0)c(\xi ') -(2+i\xi _n)c(\xi ')c(dx_n)\partial _{\xi _n}c(\xi ')+i\partial _{\xi _n}c(\xi ')\Big ]; \nonumber \\ C_2&=\frac{h'(0)}{2}\Big [\frac{c(dx_n)}{4i(\xi _n-i)}+\frac{c(dx_n)-ic(\xi ')}{8(\xi _n-i)^2} +\frac{3\xi _n-7i}{8(\xi _n-i)^3}\big (ic(\xi ')-c(dx_n)\big )\Big ].\nonumber \\ \end{aligned}$$

(4.35)

By (4.30) and (4.35), we have

$$\begin{aligned}&{\mathrm{tr }}[C_2\times \partial _{\xi _n}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})(x_0)]|_{|\xi '|=1}\nonumber \\&\quad ={\mathrm{tr }}\Big \{ \frac{h'(0)}{2}\Big [\frac{c(dx_n)}{4i(\xi _n-i)}+\frac{c(dx_n)-ic(\xi ')}{8(\xi _n-i)^2} +\frac{3\xi _n-7i}{8(\xi _n-i)^3}[ic(\xi ')-c(dx_n)]\Big ] \nonumber \\&\qquad \times \frac{-4i\xi _nc(\xi ')+(i-3i\xi _n^{2})c(dx_n)}{(1+\xi _n^{2})^3}\Big \} \nonumber \\&\quad =h'(0)\frac{4i-11\xi _n-6i\xi _n^{2}+3\xi _n^{3}}{(\xi _n-i)^5(\xi _n+i)^3}. \end{aligned}$$

(4.36)

Similarly, we have

$$\begin{aligned}&{\mathrm{tr }}[C_1\times \partial _{\xi _n}\sigma _{-3}(({{\widetilde{D}}}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1})(x_0)]|_{|\xi '|=1}\nonumber \\&\quad ={\mathrm{tr }}\Big \{ \frac{1}{4(\xi _n-i)^2}\Big [\frac{5}{2}h'(0)c(dx_n)-\frac{5i}{2}h'(0)c(\xi ')\nonumber \\&\qquad -(2+i\xi _n)c(\xi ')c(dx_n)\partial _{\xi _n}c(\xi ')+i\partial _{\xi _n}c(\xi ')\Big ]\nonumber \\&\qquad \times \frac{-4i\xi _nc(\xi ')+(i-3i\xi _n^{2})c(dx_n)}{(1+\xi _n^{2})^3}\Big \} \nonumber \\&\quad =h'(0)\frac{3+12i\xi _n+3\xi _n^{2}}{(\xi _n-i)^4(\xi _n+i)^3};\nonumber \\&{\mathrm{tr }}\bigg [\pi ^+_{\xi _n}\Big (\frac{c(\xi )[(f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2)] (x_0)c(\xi )}{(1+\xi _n^2)^2}\Big )\nonumber \\&\qquad \times \partial _{\xi _n}\sigma _{-3}(({D}^{*}{{\widetilde{D}}}{{\widetilde{D}}}^{*})^{-1}) (x_0)\bigg ]\bigg |_{|\xi '|=1}\nonumber \\&\quad =\frac{2-8i\xi _n-6\xi _n^2}{4(\xi _n-i)^{2}(1+\xi _n^2)^{3}}{\mathrm{tr }} \left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) (x_0)c(\xi ')\right] \nonumber \\&\quad =0.\nonumber \\ \end{aligned}$$

(4.37)

By $\int _{|\xi '|=1}\xi _{1}\ldots \xi _{2d+1}\sigma (\xi ')=0,$ we have

$$\begin{aligned} \Psi _5&= -i h'(0)\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } \times \frac{-7i+26\xi _n+15i\xi _n^{2}}{(\xi _n-i)^5(\xi _n+i)^3}d\xi _n\sigma (\xi ')dx' \nonumber \\&=-i h'(0)\times \frac{2 \pi i}{4!}\Big [\frac{-7i+26\xi _n+15i\xi _n^{2}}{(\xi _n+i)^3} \Big ]^{(5)}|_{\xi _n=i}\Omega _4dx'\nonumber \\&=\frac{55}{16}\pi h'(0)\Omega _4dx'. \end{aligned}$$

(4.38)

Now $\Psi$ is the sum of the cases (a), (b) and (c), then

$$\begin{aligned} \Psi =\sum _{i=1}^5\Psi _i=\left( \frac{65}{64}-\frac{41}{64}i\right) \pi h'(0)\Omega _4dx'. \end{aligned}$$

(4.39)

By (4.1), (4.3) and (4.39), we can get

Theorem 4.3

Let M be a 6-dimensional compact oriented spin manifold with the boundary $\partial M$ and the metric $g^M$ as above, ${{\widetilde{D}}}$ and ${{\widetilde{D}}}^*$ be the Dirac–Witten operators on ${\widetilde{M}}$ , then

$$\begin{aligned}&\widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+ ({{\widetilde{D}}}^{*}{{\widetilde{D}}} {{\widetilde{D}}}^{*})^{-1}]\nonumber \\&\quad =128\pi ^3\int \limits _{M}\bigg ( -\frac{2}{3}s-4(f_1^2+\overline{f_1}^2-4f_1\overline{f_1}) \sum _{u<v}(p_{uv}-p_{vu})^2-32f_2\overline{f_2}+12f_2^2 +12\overline{f_2}^2\bigg )d{\mathrm{Vol}_{\mathrm{M}}}\nonumber \\&\qquad +\int \limits _{\partial M}\left( \frac{65}{64}-\frac{41}{64}i\right) \pi h'(0)\,\Omega _4dx'. \end{aligned}$$

(4.40)

where s is the scalar curvature.

Next, we prove the Kastler–Kalau–Walze type theorem for 6-dimensional manifold with boundary associated to ${{\widetilde{D}}}^{3}$. From [14], we know that

$$\begin{aligned} \widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+{{\widetilde{D}}}^{-3} ]= & {} \int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M \bigotimes {\mathbb {C}}}[\sigma _{-4}({{\widetilde{D}}}^{-4})]\sigma (\xi )dx\nonumber \\&+\int \limits _{\partial M}{\overline{\Psi }}, \end{aligned}$$

(4.41)

where $\widetilde{{\mathrm{Wres}}}$ denote noncommutative residue on minifolds with boundary,

$$\begin{aligned} {\overline{\Psi }}&=\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum ^{\infty }_{j, k=0}\sum \frac{(-i)^{|\alpha |+j+k+1}}{\alpha !(j+k+1)!} \times {\mathrm{trace}}_{{\wedge ^*T^*M\bigotimes {\mathbb {C}}}}[\partial ^j_{x_n}\partial ^\alpha _{\xi '}\partial ^k_{\xi _n}\sigma ^+_{r}({{\widetilde{D}}}^{-1})(x',0,\xi ',\xi _n) \nonumber \\&\quad \times \partial ^\alpha _{x'}\partial ^{j+1}_{\xi _n}\partial ^k_{x_n}\sigma _{l} ({{\widetilde{D}}}^{-3})(x',0,\xi ',\xi _n)]d\xi _n\sigma (\xi ')dx',\nonumber \\ \end{aligned}$$

(4.42)

and the sum is taken over $r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3$.

By Theorem 2.2, we compute the interior term of (4.42), then

$$\begin{aligned}&\int \limits _M\int \limits _{|\xi |=1}\mathrm{trace}_{\wedge ^*T^*M\bigotimes {\mathbb {C}}} [\sigma _{-4}({{\widetilde{D}}}^{-4})]\sigma (\xi )dx\nonumber \\&\quad =128\pi ^3\int \limits _{M}\bigg [-\frac{2}{3}s -24f_1^2\sum _{u<v}(p_{uv}-p_{vu})^2+40f_2^2\bigg ]d{\mathrm{Vol}_{\mathrm{M}}}\nonumber \\. \end{aligned}$$

(4.43)

So we only need to compute $\int _{\partial M} {\overline{\Psi }}$. Let us now turn to compute the specification of ${{\widetilde{D}}}^3$.

$$\begin{aligned} {{\widetilde{D}}}^3&=\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle (-g^{ij}\partial _{l}\partial _{i}\partial _{j})\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \bigg \{-(\partial _{l}g^{ij})\partial _{i}\partial _{j}-g^{ij}\bigg (4\sigma _{i} \partial _{j}2\Gamma ^{k}_{ij}\partial _{k}\bigg ) \partial _{l}\bigg \}\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \left\{ -2(\partial _{l}g^{ij})\sigma _{i}\partial _{j}+g^{ij} (\partial _{l}\Gamma ^{k}_{ij})\partial _{k}-2g^{ij}[(\partial _{l}\sigma _{i}) +(\partial _{l}g^{ij})\Gamma ^{k}_{ij}\partial _{k}\right. \nonumber \\&\quad +\sum _{j,k}\left[ \partial _{l}\left( \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) c(e_{j})\right. \right. \nonumber \\&\quad \left. \left. +c(e_{j})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \right) \right] \langle e_{j},dx^{k}\rangle \partial _{k}\nonumber \\&\quad +\sum _{j,k}\left( \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(e_{j})\right. \nonumber \\&\quad \left. +c(e_{j})\left( f_1\sum _{u<v}(p_{uv} -p_{vu})c(e_u)c(e_v)+f_2\right) \right) \nonumber \\&\quad \left. \Big [\partial _{l}\langle e_{j},dx^{k}\rangle \Big ]\partial _{k} \right\} +\sum ^{n}_{i=1}c(e_{i})\langle e_{i},dx_{l}\rangle \partial _{l} \left\{ -g^{ij}\Big [(\partial _{i}\sigma _{j})+\sigma _{i}\sigma _{j}\right. \nonumber \\&\quad -\Gamma _{i,j}^{k}\sigma _{k}+\sum _{i,j}g^{i,j} \left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(\partial _{i})\sigma _{i}\right. \nonumber \\&\quad +c(\partial _{i})\partial _{i}\left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad \left. \left. +c(\partial _{i})\sigma _{i}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2\right) \right] +\frac{1}{4}s-\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu}) c(e_u)c(e_v)+f_2\right) \right] ^2\right\} \nonumber \\&\quad +\Big [\sigma _{i}+(f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v) +f_2)\Big ](-g^{ij}\partial _{i}\partial _{j})\nonumber \\&\quad +\sum ^{n}_{i=1}c(e_{i}) \langle e_{i},dx_{l}\rangle \left\{ 2\sum _{j,k}\left[ (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2)c(e_{j})\right. \right. \nonumber \\&\quad \left. +c(e_{j})\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)+f_2\right) \right] \nonumber \\&\quad \left. \times \langle e_{i},dx_{k}\rangle \right\} _{l}\partial _{k} +\Big [\sigma _{i}+\Big (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\Big )\Big ] \left\{ -\sum _{i,j}g^{i,j}\right. \nonumber \\&\quad \Big [2\sigma _{i}\partial _{j} -\Gamma _{i,j}^{k}\partial _{k}+(\partial _{i}\sigma _{j}) +\sigma _{i}\sigma _{j} -\Gamma _{i,j}^{k}\sigma _{k}\Big ]\nonumber \\&\quad +\sum _{i,j}g^{i,j}\Bigg [c(\partial _{i})\Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\nonumber \\&\quad +f_2\Bigg ) +\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) c(\partial _{i})\Bigg ] \partial _{j}\nonumber \\&\quad +\sum _{i,j}g^{i,j}\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)\right. \right. \nonumber \\&\quad +f_2)c(\partial _{i})\sigma _{i}+c(\partial _{i})\partial _{i} \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad +c(\partial _{i})\sigma _{i}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)+f_2\right) \nonumber \\&\quad +c(\partial _{i})\partial _{i}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \nonumber \\&\quad \left. +c(\partial _{i})\sigma _{i}\left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u) c(e_v)+f_2\right) \right] \nonumber \\&\quad \left. +\frac{1}{4}s-\left[ \left( f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\right) \right] ^2\right\} . \end{aligned}$$

(4.44)

Then, we obtain

Lemma 4.4

The following identities hold:

$$\begin{aligned} \sigma _2({{\widetilde{D}}}^3)&=\sum _{i,j,l}c(dx_{l})\partial _{l}(g^{i,j})\xi _{i}\xi _{j} +c(\xi )(4\sigma ^k-2\Gamma ^k)\xi _{k}-2[c(\xi )(f_1\sum _{u<v}(p_{uv}-p_{vu})\nonumber \\&\quad c(e_u)c(e_v)+f_2)c(\xi )-|\xi |^2(f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2)]\nonumber \\&\quad -\frac{1}{4}|\xi |^2\sum _{s,t,l}\omega _{s,t} (e_l)c(e_s)c(e_t)];\nonumber \\ \sigma _{3}({{\widetilde{D}}}^3)&=ic(\xi )|\xi |^{2}.\nonumber \\ \end{aligned}$$

(4.45)

Write

$$\begin{aligned} \sigma ({{\widetilde{D}}}^3)=p_3+p_2+p_1+p_0; ~\sigma ({{\widetilde{D}}}^{-3})=\sum ^{\infty }_{j=3}q_{-j}. \end{aligned}$$

(4.46)

By the composition formula of pseudodifferential operators, we have

$$\begin{aligned} 1=\sigma ({{\widetilde{D}}}^3\circ {{\widetilde{D}}}^{-3})&= \sum _{\alpha }\frac{1}{\alpha !}\partial ^{\alpha }_{\xi } [\sigma ({{\widetilde{D}}}^3)]{{\widetilde{D}}}^{\alpha }_{x} [\sigma ({{\widetilde{D}}}^{-3})] \nonumber \\&=(p_3+p_2+p_1+p_0)(q_{-3}+q_{-4}+q_{-5}+\cdots ) \nonumber \\&\quad +\sum _j(\partial _{\xi _j}p_3+\partial _{\xi _j}p_2+\partial _{\xi _j}p_1+\partial _{\xi _j}p_0)\nonumber \\&\quad (D_{x_j}q_{-3}+D_{x_j}q_{-4}+D_{x_j}q_{-5}+\cdots ) \nonumber \\&=p_3q_{-3}+(p_3q_{-4}+p_2q_{-3}+\sum _j\partial _{\xi _j}p_3D_{x_j}q_{-3})+\cdots , \end{aligned}$$

(4.47)

by (4.47), we have

$$\begin{aligned} q_{-3}=p_3^{-1};~q_{-4}=-p_3^{-1}\Bigg [p_2p_3^{-1}+\sum _j\partial _{\xi _j}p_3D_{x_j}(p_3^{-1})\Bigg ]. \end{aligned}$$

(4.48)

By (4.44)–(4.48), we have some symbols of operators.

Lemma 4.5

The following identities hold:

$$\begin{aligned} \sigma _{-3}({{\widetilde{D}}}^{-3})&=\frac{ic(\xi )}{|\xi |^{4}};\nonumber \\ \sigma _{-4}({{\widetilde{D}}}^{-3})&= \frac{c(\xi )\sigma _{2}({{\widetilde{D}}}^{3})c(\xi )}{|\xi |^8} +\frac{ic(\xi )}{|\xi |^8}\Big (|\xi |^4c(dx_n)\partial _{x_n}c(\xi ') -2h'(0)c(dx_n)c(\xi )\nonumber \\&\quad +2\xi _{n}c(\xi )\partial _{x_n}c(\xi ')+4\xi _{n}h'(0)\Big ). \end{aligned}$$

(4.49)

When $n=6$, then ${\mathrm{tr}}_{\wedge ^*T^*M}[\texttt {id}]=8$, where ${\mathrm{tr}}$ as shorthand of ${\mathrm{trace}}$. Since the sum is taken over $r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3$, then we have the following five cases:

Case (a) (I) $r=-1, l=-3, j=k=0, |\alpha |=1$.

By (4.42), we get

$$\begin{aligned} {\overline{\Psi }}_1=-\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }\sum _{|\alpha |=1}{\mathrm{trace}} \Big [\partial ^{\alpha }_{\xi '}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial ^{\alpha }_{x'}\partial _{\xi _{n}}\sigma _{-3}({{\widetilde{D}}}^{-3})\Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.50)

Case (a) (II) $r=-1, l=-3, |\alpha |=k=0, j=1$.

By (4.42), we have

$$\begin{aligned} {\overline{\Psi }}_2=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty } \mathrm{trace} \Big [\partial _{x_{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial ^{2}_{\xi _{n}}\sigma _{-3}({{\widetilde{D}}}^{-3}) \Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.51)

Case (a) (III) $r=-1,l=-3,|\alpha |=j=0,k=1$.

By (4.42), we have

$$\begin{aligned} {\overline{\Psi }}_3=-\frac{1}{2}\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\partial _{\xi _{n}}\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\partial _{x_{n}}\sigma _{-3}({{\widetilde{D}}}^{-3}) \Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.52)

By Lemma 4.2 and Lemma 4.5, we have $\sigma _{-3}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})=\sigma _{-3}({{\widetilde{D}}}^{-3})$, by (4.11)- (4.21), we obtain

$$\begin{aligned} {\overline{\Psi }}_1+{\overline{\Psi }}_2+{\overline{\Psi }}_3=\frac{5}{8}\pi h'(0)\Omega _{4}dx', \end{aligned}$$

where ${{\Omega _{4}}}$ is the canonical volume of $S^{4}.$

Case (b) $r=-1,l=-4,|\alpha |=j=k=0$.

By (4.42), we have

$$\begin{aligned} {\overline{\Psi }}_4&=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\pi ^{+}_{\xi _{n}}\sigma _{-1}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\sigma _{-4}({{\widetilde{D}}}^{-3})\Big ](x_0)d\xi _n \sigma (\xi ')dx'\nonumber \\&=i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} [\partial _{\xi _n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \sigma _{-4}({{\widetilde{D}}}^{-3})](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.53)

In the normal coordinate, $g^{ij}(x_{0})=\delta ^{j}_{i}$ and $\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=0$, if $j<n$; $\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=h'(0)\delta ^{\alpha }_{\beta }$, if $j=n$. So by [12], when $k<n$, we have $\Gamma ^{n}(x_{0})=\frac{5}{2}h'(0)$, $\Gamma ^{k}(x_{0})=0$, $\delta ^{n}(x_{0})=0$ and $\delta ^{k}=\frac{1}{4}h'(0)c(e_{k})c(e_{n})$. Then, we obtain

$$\begin{aligned}&\sigma _{-4}({{\widetilde{D}}}^{-3})(x_{0})|_{|\xi '|=1}\nonumber \\&\quad = \frac{c(\xi )\sigma _{2}({{\widetilde{D}}}^{3}) (x_{0})|_{|\xi '|=1}c(\xi )}{|\xi |^8} -\frac{c(\xi )}{|\xi |^4}\sum _j\partial _{\xi _j}\big (c(\xi )|\xi |^2\big ) {D}_{x_j}\bigg (\frac{ic(\xi )}{|\xi |^4}\bigg )\nonumber \\&\quad =\frac{1}{|\xi |^8}c(\xi )\Big (\frac{1}{2}h'(0)c(\xi )\sum _{k<n}\xi _k c(e_k)c(e_n)-\frac{5}{2}h'(0)\xi _nc(\xi )-\frac{1}{4}h'(0)|\xi |^2\nonumber \\&\qquad c(dx_n) -2\Bigg [c(\xi )\Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2\Bigg )c(\xi ) +|\xi |^2\Bigg (-\overline{f_1}\sum _{u<v}\nonumber \\&\qquad (p_{uv}-p_{vu})c(e_u)c(e_v)+\overline{f_2}\Bigg )\Bigg ] +|\xi |^2\Bigg (f_1\sum _{u<v}(p_{uv}-p_{vu})c(e_u)c(e_v)+f_2)\Bigg )\nonumber \\&\qquad c(\xi )+\frac{ic(\xi )}{|\xi |^8}\Bigg (|\xi |^4c(dx_n)\partial _{x_n}c(\xi ') -2h'(0)c(dx_n)c(\xi )+2\xi _{n}c(\xi )\partial _{x_n}c(\xi ')\nonumber \\&\qquad +4\xi _{n}h'(0)\Bigg ). \end{aligned}$$

(4.54)

By (3.29) and (4.54), we have

$$\begin{aligned}&{\mathrm{tr}} [\partial _{\xi _n}\pi ^+_{\xi _n}\sigma _{-1}({{\widetilde{D}}}^{-1})\times \sigma _{-4}({{\widetilde{D}}}^{-3}) ](x_0)|_{|\xi '|=1} \nonumber \\&\quad =\frac{1}{2(\xi _{n}-i)^{2}(1+\xi _{n}^{2})^{4}}\bigg (\frac{3}{4}i +2+(3+4i)\xi _{n}+(-6+2i)\xi _{n}^{2}+3\xi _{n}^{3}+\frac{9i}{4}\xi _{n}^{4}\bigg )h'(0){\mathrm{tr}} [id]\nonumber \\&\qquad +\frac{1}{2(\xi _{n}-i)^{2}(1+\xi _{n}^{2})^{4}}\big (-1-3i\xi _{n} -2\xi _{n}^{2}-4i\xi _{n}^{3}-\xi _{n}^{4}-i\xi _{n}^{5} \big ){\mathrm{tr}[c(\xi ')\partial _{x_n}c(\xi ')]}.\nonumber \\ \end{aligned}$$

(4.55)

By (4.55), we have

$$\begin{aligned} {\overline{\Psi }}_4&= ih'(0)\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }8\times \frac{\frac{3}{4}i +2+(3+4i)\xi _{n}+(-6+2i)\xi _{n}^{2}+3\xi _{n}^{3}+\frac{9i}{4} \xi _{n}^{4}}{2(\xi _n-i)^5(\xi _n+i)^4}d\xi _n\sigma (\xi ')dx'\nonumber \\&\quad +ih'(0)\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }4\times \frac{1+3i\xi _{n} +2\xi _{n}^{2}+4i\xi _{n}^{3}+\xi _{n}^{4}+i\xi _{n}^{5}}{2(\xi _{n}-i)^{2} (1+\xi _{n}^{2})^{4}}d\xi _n\sigma (\xi ')dx'\nonumber \\&=\left( -\frac{41}{64}i-\frac{195}{64}\right) \pi h'(0)\Omega _4dx'. \end{aligned}$$

(4.56)

Case (c) $r=-2,l=-3,|\alpha |=j=k=0$.

By (4.42), we have

$$\begin{aligned} {\overline{\Psi }}_5=-i\int \limits _{|\xi '|=1}\int \limits ^{+\infty }_{-\infty }{\mathrm{trace}} \Big [\pi ^{+}_{\xi _{n}}\sigma _{-2}({{\widetilde{D}}}^{-1}) \times \partial _{\xi _{n}}\sigma _{-3}({{\widetilde{D}}}^{-3})\Big ](x_0)d\xi _n\sigma (\xi ')dx'. \end{aligned}$$

(4.57)

By Lemma 4.2 and Lemma 4.5, we have $\sigma _{-3}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})=\sigma _{-3}({{\widetilde{D}}}^{-3})$, by (4.26)- (4.38), we obtain

$$\begin{aligned} {\overline{\Psi }}_5=\frac{55}{16}\pi h'(0)\Omega _4dx'. \end{aligned}$$

Now ${\overline{\Psi }}$ is the sum of the cases (a), (b) and (c), then

$$\begin{aligned} {\overline{\Psi }}=\sum _{i=1}^5{\overline{\Psi }}_i=\left( \frac{65}{64} -\frac{41}{64}i\right) \pi h'(0)\Omega _4dx'. \end{aligned}$$

(4.58)

By (4.41), (4.43) and (4.58), we can get

Theorem 4.6

Let M be a 6-dimensional compact oriented spin manifold with the boundary $\partial M$ and the metric $g^M$ as above, ${{\widetilde{D}}}$ be the Dirac-Witten operator on ${\widetilde{M}}$ , then

$$\begin{aligned}&\widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+( {{\widetilde{D}}}^{-3})]\nonumber \\&\quad =128\pi ^3\int \limits _{M}\bigg [-\frac{2}{3}s -24f_1^2\sum _{u<v}(p_{uv}-p_{vu})^2+40f_2^2\bigg ]d{\mathrm{Vol}_{\mathrm{M}}}\nonumber \\&\qquad +\int \limits _{\partial M}\left( \frac{65}{64}-\frac{41}{64}i\right) \pi h'(0)\Omega _4dx'. \end{aligned}$$

(4.59)

where s is the scalar curvature.