关于复合微分算子的W-留数

The W-Residues of Complex Differential Operators

摘要: 本文首先综述近年来关于微分算子W-留数的一系列研究进展。之后研究一类复合微分算子$D+c\left(X\right)$ 的基本结构，并在法坐标系下导出了微分算子的主符号表示，最终结合Lichnerowicz公式给出了5维带边流形上复合微分算子的W-留数表示。

Abstract: In this paper, a series of research advances on W-residue of differential operators in recent years are reviewed. Then the basic structure of a class of complex differential operators $D+c\left(X\right)$ is studied, and the principal symbolic representation of differential operators is derived in normal coordinate system. Finally, the W-residue representation of complex differential operators on 5-dimensional manifolds with edges is given by combining Lichnerowicz formula.

1. 引言

近年来，非交换留数研究有了很多重要的结果，微分算子的Lichnerowicz公式及其带边流形上的留数研究获得了细致而深入的理解。本文在以往工作的基础上，探讨一类复合微分算子$D+c\left(X\right)$ 的基本结构，对该微分算子的Lichnerowicz公式和W-留数进行刻画。

20世纪90年代，Wodzicki对椭圆拟微分算子的zeta函数理论作了深入的研究，在非交换几何框架下给出了以他的名字命名的Wodzicki定理(参见文献[1])。Connes [2]应用非交换留数导出了四维情形下的微分算子的Polyakov作用。进一步，Connes [3]猜想Dirac算子逆的平方的非交换留数与Einstein-Hilbert作用成正比。Kastler [4]，Kalau和Walze [5]从不同的角度证明了这一结论，称之为Kastler-Kalau-Walze定理。Fedosov等人[6]结合紧致流形M上Wodzicki的相关理论将这种留数推广到Boutet代数上，并给出了带边流形W-留数的结构。结合FGLS定理[6]，带边流形上的W-留数$\widetilde{Wres}$ [7]，可以分为内部项和边界项两部分来刻画。Sitarz和Zajac [8]探讨了扰动Dirac算子的谱作用，Iochum和Levy [9]计算了具有单形式扰动的Dirac算子谱作用渐进展开中的热核系数。结合Dirac算子、Signature算子，Wang [10]证明了低维带边流形的Kastler-Kalau-Walze型定理。Wang [11]定义了具有边界的自旋流形的低维体积并计算了具有边界的5维和6维自旋流形的低维体积。进一步，相关文献[12]-[14]探讨了扭化Dirac算子的Lichnerowicz型公式，建立了高维带边流形上微分算子的Kastler-Kalau-Walze型定理。

本文主要考虑一类复合微分算子$D+c\left(X\right)$ ，结合Lichnerowicz公式计算出5维带边流形与复合微分Dirac算子相关的W-留数$\widetilde{Wres}\left[{\pi }^{+}{\left(D+c\left(X\right)\right)}^{-1}\circ {\pi }^{+}{\left(D+c\left(X\right)\right)}^{-1}\right]$ 。

2. 复合微分算子的Lichnerowicz公式

设$\left(M,g^M\right)$ 为n维带边流形，边界为$\partial M$ 。记${\nabla }^L$ 为Levi-Civita联络，在局部坐标$\left\{x_i;1\le i\le n\right\}$ 和标准正交坐标系$\left\{{\tilde{E}}_1,\cdots ,{\tilde{E}}_n\right\}$ 下，联络矩阵$\left({\omega }_{s,t}\right)$ 定义为

${\nabla }^L{\left({\tilde{E}}_1,\cdots ,{\tilde{E}}_n\right)}^t=\left(w_{s,t}\right){\left({\tilde{E}}_1,\cdots ,{\tilde{E}}_n\right)}^t$ (2.1)

定义2.1：记M为具有黎曼度量g的n维定向自旋流形，Dirac算子在切丛$TM$ 的正交框架$e_i$ ，$1\le i\le n$ 和自然框架${\partial }_i$ 下，

$D={\displaystyle \sum _{i,j}{g}^{i,j}c\left({\partial }_i\right){\nabla }_{{\partial }_i}^S}={\displaystyle \sum _{i}c\left(e_i\right){\nabla }_{e_i}^S},$ (2.2)

其中$c\left(e_i\right)$ 表示Clifford作用，满足关系式$c\left(e_i\right)c\left(e_j\right)+c\left(e_j\right)c\left(e_i\right)=-2{\delta }_i^j$ 。

定义2.2：记复合微分算子$\tilde{D}$ 为Dirac算子与微分形式的组合，

$\tilde{D}=D+c\left(X\right)={\displaystyle \sum _{i}c\left(e_i\right){\nabla }_{e_i}^S}+c\left(X\right).$ (2.3)

其中$c\left(X\right)$ 为M微分一形式，X为向量场。

结合复合微分算子$\tilde{D}$ 与${\tilde{D}}^2$ 的运算

${\tilde{D}}^2={\left(D+c\left(X\right)\right)}^2=D^2+Dc\left(X\right)+c\left(X\right)D+c^2\left(X\right)$ ,(2.4)

得到复合微分算子$\tilde{D}$ 的Lichnerowicz公式。

命题2.3：复合微分算子${\tilde{D}}^2$ 可表示为

$\begin{array}{c}{\tilde{D}}^2=-g^{ij}{\partial }_i{\partial }_j+\left[-2{\sigma }^j+{\Gamma }^j+c\left({\partial }^j\right)c\left(X\right)+c\left(X\right)c\left({\partial }^j\right)\right]{\partial }_j\\ +{\displaystyle \sum _{i,j}{g}^{ij}\left[-\left({\partial }_i{\sigma }_j\right)-{\sigma }_i{\sigma }_j+{\Gamma }_{ij}^k{\sigma }_k+c\left({\partial }_i\right){\partial }_j\left(c\left(X\right)\right)+c\left({\partial }_i\right){\sigma }_{j}c\left(X\right)+c\left(X\right)c\left({\partial }_i\right){\sigma }_j\right]}\\ +\frac{1}{4}s+c^2\left(X\right).\end{array}$(2.5)

结合命题2.3，得到如下结论。

定理2.4：对于偶数n维的紧致无边流形，复合微分算子$\tilde{D}$ 的非交换留数可表示为

$Wres\left({\tilde{D}}^{-n+2}\right)=\frac{\left(n-2\right){\pi }^{\frac{n}{2}}\dim \left(S\left(TM\right)\right)}{\left(\frac{n}{2}-1\right)!}{\displaystyle {\int }_M-\frac{5s}{12}\text{d}vo{l}_M}.$(2.6)

3. 带边流形上复合微分算子$\tilde{D}$ 的W-留数

定理3.1 [6]：对于带边流形x，边界为$\partial x$ ，$\dim V\ge 3$ ，$A=\left(\begin{array}{cc}{\pi }^{+}P+G& K\\ T& S\end{array}\right)\in B$ ，记$p,b$ 和s分别表示$P,B$ 和S的局部表示，微分算子的W-留数定义为

$\begin{array}{c}\widetilde{Wres}\left(A\right)={\displaystyle {\int }_X{\displaystyle {\int }_{S}t{r}_E\left[p_{-n}\left(x,\xi \right)\right]\sigma \left(\xi \right)\text{d}x}}\\ +2\pi {\displaystyle {\int }_X{\displaystyle {\int }_{S^{\prime }}\left\{t{r}_E\left[\left(tr{b}_{-n}\right)\left(x^{\prime },{\xi }^{\prime }\right)\right]+t{r}_F\left[s_{1-n}\left(x^{\prime },{\xi }^{\prime }\right)\right]\right\}\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}.\end{array}$ (3.1)

a) $\widetilde{Wres}\left(\left[A,B\right]\right)=0$ ，对于任意的$A,B\in {\rm B}$ ；b) 是${\rm B}/{{\rm B}}^{-\infty }$ 上唯一的连续迹。

记${\sigma }_l\left(A\right)$ 表示微分算子A的l阶符号，则带边流形微分算子A的W-留数为：

$\widetilde{Wres}\left[{\pi }^{+}{\tilde{D}}^{-p_1}\circ {\pi }^{+}{\tilde{D}}^{-p_2}\right]={\displaystyle {\int }_X{\displaystyle {\int }_{\left|\xi \right|=1}t{r}_{S\left(TM\right)}\left[{\sigma }_{-n}\left({\tilde{D}}^{-p_1-p_2}\right)\right]\sigma \left(\xi \right)\text{d}x}}+{\displaystyle {\int }_{\partial M}\Phi ,}$(3.2)

其中

$\begin{array}{c}\Phi ={\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{j,k=0}^{\infty }{\displaystyle \sum \frac{{\left(-i\right)}^{\left|a\right|+j+k+1}}{a!\left(j+k+1\right)}trac{e}_{S\left(TM\right)}[{\partial }_{x_n}^j{\partial }_{{\xi }^{\prime }}^a{\partial }_{{\xi }_n}^k{\sigma }_r^{+}\left({\tilde{D}}^{-p_1}\right)\left(x^{\prime },0,{\xi }^{\prime },{\xi }_n\right)}}}}\\ \times {\partial }_{x^{\prime }}^a{\partial }_{{\xi }_n}^{j+1}{\partial }_{x_n}^k{\sigma }_l\left({\tilde{D}}^{-p_2}\right)\left(x^{\prime },0,{\xi }^{\prime },{\xi }_n\right)]\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime },\end{array}$(3.3)

$r-k+\left|a\right|+l-j-1=-n,r\le -p_1,l\le -p_2.$

依据命题3.5 [11]可知，$p_1+p_2\equiv n\mathrm{mod}1$ ，$Vo{l}_n^{\left(p_1,p_2\right)}M={\displaystyle {\int }_{\partial M}\Phi }$ ，则5维带边流形情形下复合微分算子$\tilde{D}$ 的W-留数为$\widetilde{Wres}\left[{\pi }^{+}{\tilde{D}}^{-1}\circ {\pi }^{+}{\tilde{D}}^{-1}\right]={\displaystyle {\int }_{\partial M}\Phi }$ 。

引理3.2：复合微分算子$\tilde{D}$ 的主符号

${\sigma }_{-1}\left({\tilde{D}}^{-1}\right)=q_{-1}=\frac{\sqrt{-1}c\left(\xi \right)}{{\left|\xi \right|}^2};$

$\begin{array}{c}{\sigma }_{-2}\left({\tilde{D}}^{-1}\right)=q_{-2}=\frac{c\left(\xi \right)p_{0}c\left(\xi \right)}{{\left|\xi \right|}^4}+\frac{c\left(\xi \right)}{{\left|\xi \right|}^6}{\displaystyle \sum _{j}c\left(d_{x_j}\right)\left[{\partial }_{x_j}\left(c\left(\xi \right)\right){\left|\xi \right|}^2-c\left(\xi \right)\partial x_j\left({\left|\xi \right|}^2\right)\right]}\\ ={\sigma }_{-2}\left(D^{-1}\right)+\frac{c\left(X\right)}{{\left|\xi \right|}^2}-\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4};\end{array}$

${\sigma }_{-3}\left({\tilde{D}}^{-1}\right)=q_{-3}=-\frac{1}{p_1}\left[p_0{q}_{-2}+{\displaystyle \sum _{j=1}^{n-1}c\left(d{x}_j\right){\partial }_{x_j}{q}_{-2}+c\left(d{x}_n\right){\partial }_{x_n}{q}_{-2}}\right];$

其中，$p_0\left(x\right)=-h^{\prime }\left(0\right)c\left(d{x}_n\right)+c\left(X\right)$ 。

引理3.3：复合微分算子$\tilde{D}$ 的-3阶基本形式为

$\begin{array}{c}{{\sigma }_{-3}\left({\tilde{D}}^{-1}\right)\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}={{\sigma }_{-3}\left(D^{-1}\right)\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}+[-q_{1}c\left(X\right){\sigma }_{-2}\left(D^{-1}\right)-q_1\frac{c\left(X\right)}{{\left|\xi \right|}^2}+q_1\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}\\ -q_1{\displaystyle \sum _{j=1}^{n-1}c\left(d{x}_j\right){\partial }_{x_j}\left(\frac{c\left(X\right)}{{\left|\xi \right|}^2}\right)+q_1}{\displaystyle \sum _{j=1}^{n-1}c\left(d{x}_j\right){\partial }_{x_j}\left(\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}\right)}\\ -q_{1}c\left(d{x}_n\right){\partial }_{x_n}\left(\frac{c\left(X\right)}{{\left|\xi \right|}^2}\right)+{q_1{\partial }_{x_n}\left(\frac{2g\left(X,\xi \right)}{{\left|\xi \right|}^4}\right)]\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}\\ ={{\sigma }_{-3}\left(D^{-1}\right)\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}+{R_{-3}\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}.\end{array}$ (3.4)

对于5维带边流形，根据$-r-l+k+j-\left|a\right|-1=-5$ ，$r,l\le -1$ ，可得$\Phi ={\displaystyle \sum _{i=1}^{15}{\Phi }_i}$ 。

(1) $r=-1$ ，$l=-1$ ，$k=0$ ，$j=1$ ，$\left|a\right|=1$ ，

${\Phi }_1=\frac{i}{2}{\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{\left|\alpha \right|=1}trace\left[{\partial }_{x_n}{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{x_i}{\partial }_{{\xi }_n}^2{q}_{-1}\right]\left(x_0\right)\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }.}}}$

对于$j<n$ ，有

${\partial }_{x_i}{q}_{-1}\left(x_0\right)={\partial }_{x_i}\left(\frac{\sqrt{-1}c\left(\xi \right)}{{\left|\xi \right|}^2}\right)\left(x_0\right)=\frac{\sqrt{-1}{\partial }_{x_i}\left|c\left(\xi \right)\right|\left(x_0\right)}{{\left|\xi \right|}^2}-\frac{\sqrt{-1}c\left(\xi \right){\partial }_{x_i}\left({\left|\xi \right|}^2\right)\left(x_0\right)}{{\left|\xi \right|}^4}=0$ , ${\Phi }_1=0$ .

(2) $r=-1$ ，$l=-1$ ，$k=0$ ，$j=2$ ，$\left|a\right|=0$

${\Phi }_2=\frac{i}{6}{\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{j=2}trace\left[{\partial }_{x_n}^2{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{{\xi }_n}^3{q}_{-1}\right]\left(x_0\right)\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }.}}}$

直接计算得到

${{\partial }_{{\xi }_n}^3{q}_{-1}\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}=\frac{24{\xi }_n-24{\xi }_n^3}{{\left(1+{\xi }_n^2\right)}^4}\sqrt{-1}c\left({\xi }^{\prime }\right)+\frac{-6{\xi }_n^4+36{\xi }_n^2-6}{{\left(1+{\xi }_n^2\right)}^4}\sqrt{-1}c\left(d{x}_n\right)$ ,

$\begin{array}{c}{\partial x_n^2{\pi }_{{\xi }_n}^{+}{q}_{-1}\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}={{\pi }_{{\xi }_n}^{+}\partial x_n^2\frac{c\left(\xi \right)}{{\left|\xi \right|}^2}|}_{\left|{\xi }^{\prime }\right|=1}={\pi }_{{\xi }_n}^{+}\partial x_n\left[\frac{\partial x_n\left(c\left(\xi \right)\right)}{{\left|\xi \right|}^2}-\frac{c\left(\xi \right)\partial x_n\left({\left|\xi \right|}^2\right)}{{\left|\xi \right|}^4}\right]\\ ={\pi }_{{\xi }_n}^{+}\left(\frac{\partial x_n^2\left(c\left({\xi }^{\prime }\right)\right)}{{\left|\xi \right|}^2}-2\frac{h^{\prime }\left(0\right)\partial x_n\left(c\left({\xi }^{\prime }\right)\right)}{{\left|\xi \right|}^4}-\frac{h^{″}\left(0\right)c\left(\xi \right)}{{\left|\xi \right|}^4}+\frac{c\left(\xi \right)2{\left(h^{\prime }\left(0\right)\right)}^2}{{\left|\xi \right|}^6}\right)\\ ={\pi }_{{\xi }_n}^{+}\left(\frac{\partial x_n^2\left(c\left({\xi }^{\prime }\right)\right)}{{\left|\xi \right|}^2}\right)-2h^{\prime }\left(0\right)\partial x_n\left(c\left({\xi }^{\prime }\right)\right){\pi }_{{\xi }_n}^{+}\frac{1}{{\left|\xi \right|}^4}-h^{″}\left(0\right){\pi }_{{\xi }_n}^{+}\frac{c\left(\xi \right)}{{\left|\xi \right|}^4}+2{\left(h^{\prime }\left(0\right)\right)}^2{\pi }_{{\xi }_n}^{+}\frac{c\left(\xi \right)}{{\left|\xi \right|}^6}\\ =\left(\frac{3}{4}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{1}{2}{h}^{″}\left(0\right)\right)\frac{c\left({\xi }^{\prime }\right)}{2\left({\xi }_n-i\right)}-h^{\prime }\left(0\right)\frac{{\xi }_n-2i}{4{\left({\xi }_n-i\right)}^2}\partial x_n\left(c\left({\xi }^{\prime }\right)\right)\\ -h^{″}\left(0\right)\frac{\left({\xi }_n-2i\right)c\left({\xi }^{\prime }\right)+c\left(d{x}_n\right)}{4{\left({\xi }_n-i\right)}^2}+2i{\left(h^{\prime }\left(0\right)\right)}^2\frac{\left(-3i{\xi }_n^2-9{\xi }_n+8i\right)c\left({\xi }^{\prime }\right)-\left(i{\xi }_n+3\right)c\left(d{x}_n\right)}{16{\left({\xi }_n-i\right)}^3}.\end{array}$

结合Clifford运算及迹性质$trAB=trBA$ ，有

$\begin{array}{l}tr\left[c\left({\xi }^{\prime }\right)c\left(d{x}_n\right)\right]=0;tr\left[c{\left(d{x}_n\right)}^2\right]=-4;{tr\left[c{\left({\xi }^{\prime }\right)}^2\left(x_0\right)\right]\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}=-4;\\ tr\left[{\partial }_{x_n}c\left({\xi }^{\prime }\right)c\left(d{x}_n\right)\right]=0;{tr\left[{\partial }_{x_n}c\left({\xi }^{\prime }\right)\times c\left({\xi }^{\prime }\right)\right]\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}=-2h^{\prime }\left(0\right).\end{array}$

则

$\begin{array}{l}{trace\left[{\partial }_{x_n}^2{\pi }_{{\xi }_n}^2{q}_{-1}\times {\partial }_{{\xi }_n}^3{q}_{-1}\right]\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}\\ =i{\left(h^{\prime }\left(0\right)\right)}^2\frac{3\left(33{\xi }_n^5-75i{\xi }_n^4-94{\xi }_n^3+90i{\xi }_n^2+57{\xi }_n-3i\right)}{2{\left({\xi }_n-i\right)}^3{\left(1+{\xi }_n^2\right)}^4}\\ +i\left(h^{″}\left(0\right)\right)\frac{6\left(-9{\xi }_n^4+12i{\xi }_n^3+14{\xi }_n^2-12i{\xi }_n-1\right)}{2{\left({\xi }_n-i\right)}^2{\left(1+{\xi }_n^2\right)}^4}.\end{array}$

$\begin{array}{c}{\Phi }_2=-\frac{1}{6}{\left(h^{\prime }\left(0\right)\right)}^2{\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{3\left(33{\xi }_n^5-75i{\xi }_n^4-94{\xi }_n^3+90i{\xi }_n^2+57{\xi }_n-3i\right)}{2{\left({\xi }_n-i\right)}^3{\left(1+{\xi }_n^2\right)}^4}\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}\\ -\frac{1}{6}{h}^{″}\left(0\right){\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }\frac{6\left(-9{\xi }_n^4+12i{\xi }_n^3+14{\xi }_n^2-12i{\xi }_n-1\right)}{2{\left({\xi }_n-i\right)}^3{\left(1+{\xi }_n^2\right)}^4}\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}\\ =-\frac{1}{6}{\left(h^{\prime }\left(0\right)\right)}^2{\Omega }_3{\displaystyle {\int }_{{\Gamma }^{+}}\frac{3(33{\xi }_n^5-75i{\xi }_n^4-94{\xi }_n^3+90i{\xi }_n^2+57{\xi }_n-3i)}{2{\left({\xi }_n-i\right)}^3{\left(1+{\xi }_n^2\right)}^4}\text{d}{\xi }_n\text{d}{x}^{\prime }}\\ -\frac{1}{6}{h}^{″}\left(0\right){\Omega }_3{\displaystyle {\int }_{{\Gamma }^{+}}\frac{6\left(-9{\xi }_n^4+12i{\xi }_n^3+14{\xi }_n^2-12i{\xi }_n-1\right)}{2{\left({\xi }_n-i\right)}^3{\left(1+{\xi }_n^2\right)}^4}\text{d}{\xi }_n\text{d}{x}^{\prime }}\\ =\left(\frac{29}{64}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{3}{8}{h}^{″}\left(0\right)\right)\pi {\Omega }_3\text{d}{x}^{\prime },\end{array}$

其中${\Omega }_3$ 是$S^3$ 的标准体积。

(3) $r=-1$ ，$l=-2$ ，$k=0$ ，$j=0$ ，$\left|a\right|=1$

由(3.3)得：

$\begin{array}{c}{\Phi }_3=-{\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{\left|a\right|=1}trace\left[{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{x_i}{\partial }_{{\xi }_n}{q}_{-2}\right]\left(x_0\right)}\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}\\ ={\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{\left|a\right|=1}trace\left[{\partial }_{{\xi }_n}{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{x_i}{q}_{-2}\right]\left(x_0\right)}\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}\\ ={\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{\left|a\right|=1}trace\left[{\partial }_{{\xi }_n}{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{x_i}\left({\sigma }_{-2}\left(D^{-1}\right)+\frac{c\left(X\right)}{{\left|\xi \right|}^2}-\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}\right)\right]\left(x_0\right)}\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}.\end{array}$

在$x_0$ 点结合${\pi }_{{\xi }_n}^{+}$ 运算公式得到

${{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\left(x_0\right)|}_{\left|{\xi }^{\prime }\right|=1}=\frac{-1}{2{\left({\xi }_n-i\right)}^2}c\left(d{x}_i\right)-{\xi }_i\frac{3i-{\xi }_n}{2{\left({\xi }_n-i\right)}^3}c\left({\xi }^{\prime }\right)+{\xi }_i\frac{1}{{\left({\xi }_n-i\right)}^3}c\left(d{x}_n\right),$

${\partial }_{x_i}(\frac{c\left(X\right)}{{\left|\xi \right|}^2}-{\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}|}_{\left|{\xi }^{\prime }\right|=1}=\frac{1}{{\left|\xi \right|}^2}{\partial }_{x_i}\left[c\left(X\right)\right]-\frac{2c\left(\xi \right)}{{\left|\xi \right|}^4}{\partial }_{x_i}\left[g\left(X,\xi \right)\right].$

计算得

$\begin{array}{l}{\displaystyle \sum _{\left|a\right|=1}trace\left[{\partial }_{{\xi }_n}{\partial }_{{\xi }_i}{\pi }_{{\xi }_n}^{+}{q}_{-1}\times \partial x_i(\frac{c\left(X\right)}{{\left|\xi \right|}^2}-\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}\right]\left(x_0\right)}\\ =\frac{2}{{\left({\xi }_n-i\right)}^3\left({\xi }_n+i\right)}{\displaystyle \sum _{j-1}^{n-1}{\partial }_{x_j}\left[g\left(X,d{x}_j\right)\right]}+\frac{-2{\xi }_n^3+6i{\xi }_n^2-2{\xi }_n-2i}{{\left({\xi }_n-i\right)}^5{\left({\xi }_n+i\right)}^2}{\displaystyle \sum _{i,j-1}^{n-1}{\xi }_i{\xi }_j{\partial }_{x_j}}\left[g\left(X,d{x}_j\right)\right].\end{array}$

组合以上算式得

$\begin{array}{c}{\Phi }_3=\left(\frac{3}{16}-\frac{5}{32}i\right)s_{\partial M}\pi {\Omega }_3\text{d}{x}^{\prime }\\ +{\displaystyle {\int }_{\left|{\xi }^{\prime }\right|=1}{\displaystyle {\int }_{-\infty }^{+\infty }{\displaystyle \sum _{\left|a\right|=1}trace\left[{\partial }_{{\xi }_n}{\partial }_{{\xi }^{\prime }}^a{\pi }_{{\xi }_n}^{+}{q}_{-1}\times {\partial }_{x^{\prime }}^a(\frac{c\left(X\right)}{{\left|\xi \right|}^2}-\frac{2g\left(X,\xi \right)c\left(\xi \right)}{{\left|\xi \right|}^4}\right]\left(x_0\right)\text{d}{\xi }_n\sigma \left({\xi }^{\prime }\right)\text{d}{x}^{\prime }}}}\\ =\left(\frac{3}{16}-\frac{5}{32}i\right)s_{\partial M}\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{9}{16}\pi {\Omega }_3{\displaystyle \sum _{j=1}^{n-1}\partial x_j\left[g\left(X,\text{d}{x}_j\right)\right]\text{d}{x}^{\prime }}\\ =\left(\frac{3}{16}-\frac{5}{32}i\right)s_{\partial M}\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{9}{16}\pi {\Omega }_3{C}_1^1\left({\nabla }^{\partial M}{\left({X^{\prime }|}_{\partial M}\right)}^{*}\right)\text{d}{x}^{\prime }.\end{array}$

其中向量场${\left({X^{\prime }|}_{\partial M}\right)}^{*}$ 指代${X^{\prime }|}_{\partial M}$ 的度量对偶，${\nabla }^{\partial M}$ 是$\partial M$ 的Levi-Civita联络，$C_1^1$ 是(1,1)张量的缩并。

类似以上方法，计算得到其他情形。

${\Phi }_4=-\frac{5}{16}{\left(h^{\prime }\left(0\right)\right)}^2\pi {\Omega }_3\text{d}{x}^{\prime },$ ${\Phi }_5=0,$ ${\Phi }_6=\left(\frac{29}{64}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{3}{8}{h}^{″}\left(0\right)\right)\pi {\Omega }_3\text{d}{x}^{\prime },$

${\Phi }_7=\frac{39}{32}{\left(h^{\prime }\left(0\right)\right)}^2\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{5}{8}{a}_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime },$ ${\Phi }_8=-\frac{1}{4}{s}_{\partial M}{\pi }^3\text{d}{x}^{\prime },$

${\Phi }_9=\left(-\frac{367}{128}{h}^{\prime }{\left(0\right)}^2+\frac{103}{64}{h}^{″}\left(0\right)\right)\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{3}{8}\pi {\Omega }_3{\partial }_{x_n}\left(a_n\right)\text{d}{x}^{\prime }+\frac{15}{64}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime },$

${\Phi }_{10}=\left(-\frac{367}{128}{h}^{\prime }{\left(0\right)}^2+\frac{103}{64}{h}^{″}\left(0\right)\right)\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{3}{8}\pi {\Omega }_3{\partial }_{x_n}\left(a_n\right)\text{d}{x}^{\prime }+\frac{15}{64}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime },$

${\Phi }_{11}=0,$ ${\Phi }_{12}=\frac{39}{32}{\left(h^{\prime }\left(0\right)\right)}^2\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{5}{8}{a}_n{h}^{\prime }\left(0\right)\pi {\Omega }_3\text{d}{x}^{\prime },$

$\begin{array}{c}{\Phi }_{13}=-\frac{821}{256}{\left(h^{\prime }\left(0\right)\right)}^2\pi {\Omega }_3\text{d}{x}^{\prime }+\frac{15}{16}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime }+\frac{\pi }{2}{\left|X\right|}_g^2{}_{TM}{\Omega }_3\text{d}{x}^{\prime }\\ +\frac{35}{64}\pi {\left|X\right|}_g^2{}_{\partial M}{\Omega }_3\text{d}{x}^{\prime }-\pi a_n^2{\Omega }_3\text{d}{x}^{\prime },\end{array}$

$\begin{array}{c}{\Phi }_{14}=\left(\frac{239}{64}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{27}{16}{h}^{″}\left(0\right)-\frac{11}{192}{s}_{\partial M}\right)\pi {\Omega }_3\text{d}{x}^{\prime }-\frac{5}{8}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime }\\ +\frac{3}{4}\pi {\partial }_{x_n}\left(a_n\right){\Omega }_3\text{d}{x}^{\prime }+\frac{3}{4}\pi C_1^1\left(D{X}^{\ast }\right)\text{d}{x}^{\prime },\end{array}$

$\begin{array}{c}{\Phi }_{15}=\left(\frac{239}{64}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{27}{16}{h}^{″}\left(0\right)-\frac{11}{192}{s}_{\partial M}\right)\pi {\Omega }_3\text{d}{x}^{\prime }\\ -\frac{5}{8}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime }+\frac{3}{4}\pi {\partial }_{x_n}\left(a_n\right){\Omega }_3\text{d}{x}^{\prime }+\frac{3}{4}\pi {\Omega }_3{C}_1^1\left({\nabla }^{\partial M}{\left({X^{\prime }|}_{\partial M}\right)}^{\ast }\right)\text{d}{x}^{\prime },\end{array}$

组合上述结果得到

$\begin{array}{c}\Phi =\left(\frac{399}{256}{\left(h^{\prime }\left(0\right)\right)}^2-\frac{29}{32}{h}^{″}\left(0\right)-\left(\frac{17}{96}+\frac{5}{32}i\right)s_{\partial M}\right)\pi {\Omega }_3\text{d}{x}^{\prime }+\frac{25}{32}\pi a_n{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime }\\ -\frac{\pi }{2}{\left|X\right|}_g^2{}_{TM}{\Omega }_3\text{d}{x}^{\prime }+\frac{35\pi }{65}{\left|X\right|}_g^2{}_{TM}{h}^{\prime }\left(0\right){\Omega }_3\text{d}{x}^{\prime }-\pi a_n^2{\Omega }_3\text{d}{x}^{\prime }\\ -\frac{3}{4}\pi {\partial }_{x_n}\left(a_n\right){\Omega }_3\text{d}{x}^{\prime }+\frac{15}{16}\pi C_1^1\left({\nabla }^{\partial M}{\left({X^{\prime }|}_{\partial M}\right)}^{*}\right){\Omega }_3\text{d}{x}^{\prime }\end{array}$ (3.5)

4. 结论

依据带边流形的Einstein-Hilbert作用[10] [11]：

$I_{G_r}=\frac{1}{16\pi }{\displaystyle {\int }_{M}s\text{d}vo{l}_M}+2{\displaystyle {\int }_{\partial M}K\text{d}vo{l}_{\partial M}}:=I_{G_{r,i}}+I_{G_{r,b}},$

$K={\displaystyle \sum _{1\le i,j\le n-1}{K}_{i,j}{g}_{\partial M}^{i,j}}$ , $K_{i,j}\left(x_0\right)=-{\Gamma }_{i,j}^n\left(x_0\right)$ .

对于5维带边流形，$K\left(x_0\right)={\displaystyle \sum _{i,j}{K}_{i,j}\left(x_0\right)g_{\partial M}^{i,j}\left(x_0\right)}={\displaystyle \sum _{i,j}^4{K}_{i,j}\left(x_0\right)}=-2h^{\prime }\left(0\right)$ ，有

$s_M=3{\left(h^{\prime }\left(0\right)\right)}^2-4h^{″}\left(0\right)+s_{\partial M}\left(x_0\right).$ (4.1)

定理4.1：设M为5维带边流形，边界为$\partial M$ 。复合微分算子$\tilde{D}$ 的W-留数为

$\begin{array}{l}\widetilde{Wres}\left[{\pi }^{+}{\left(D+c\left(X\right)\right)}^{-1}\circ {\pi }^{+}{\left(D+c\left(X\right)\right)}^{-1}\right]\\ ={\displaystyle {\int }_{\partial M}[\frac{1}{16}\left(\frac{225}{32}{K}^2+\frac{29}{4}{s_M|}_{\partial M}-\left(\frac{155}{12}+5i\right)s_{\partial M}\right)}-\frac{25}{32}{a}_{n}K-{\left|X^{\prime }\right|}_g^2{}_{\partial M}\\ -\frac{35}{64}{\left|X^{\prime }\right|}_g^2{}_{\partial M}K-{3a_n^2|}_{\partial M}-\frac{3}{2}{{\partial }_{x_n}\left(a_n\right)|}_{\partial M}+\frac{15}{8}{C}_1^1\left({\nabla }^{\partial M}{\left({X^{\prime }|}_{\partial M}\right)}^{*}\right)]{\pi }^3\text{d}vo{l}_{\partial M}\end{array}$(4.2)

其中$s_M,s_{\partial M}$ 分别指代M和$\partial M$ 上的数量曲率，向量场${\left({X^{\prime }|}_{\partial M}\right)}^{*}$ 是${X^{\prime }|}_{\partial M}$ 的度量对偶，${\nabla }^{\partial M}$ 是$\partial M$ 的Levi-Civita联络，$C_1^1$ 是(1,1)张量的缩并。

NOTES

^*通讯作者。