Twisted Dirac Operators and the noncommutative residue for manifolds with boundary II

TL;DR

This paper proves new Kastler-Kalau-Walze type theorems for twisted Dirac and signature operators on six-dimensional manifolds with boundary, expanding the understanding of noncommutative residues in geometric analysis.

Contribution

It introduces two new Kastler-Kalau-Walze type theorems for twisted Dirac and signature operators on manifolds with boundary, considering non-unitary connections.

Findings

Established Kastler-Kalau-Walze type theorems for twisted operators

Extended residue formulas to six-dimensional manifolds with boundary

Analyzed effects of non-unitary connections on residues

Abstract

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Equations

$${\nabla }^F=\frac{{\tilde{\nabla }}^F+{\tilde{\nabla }}^{F,\ast }}{2},\Phi =\frac{{\tilde{\nabla }}^F-{\tilde{\nabla }}^{F,\ast }}{2},$$

$$c(a)=c(a)\otimes {\text{id}}_F,a\in TM,$$

$${\tilde{\nabla }}^{S(TM)\otimes F}={\nabla }^{S(TM)}\otimes {\text{id}}_F+{\text{id}}_{S(TM)}\otimes {\tilde{\nabla }}^F.$$

$${\nabla }^{S(TM)\otimes F}={\nabla }^{S(TM)}\otimes {\text{id}}_F+{\text{id}}_{S(TM)}\otimes {\nabla }^F,$$

$${\tilde{\nabla }}^{S(TM)\otimes F}={\nabla }^{S(TM)}\otimes {\text{id}}_F+{\text{id}}_{S(TM)}\otimes {\nabla }^F+{\text{id}}_{S(TM)}\otimes \Phi .$$

$$D_F=\sum _{i,j}{g}^{ij}c({\partial }_i){\nabla }_{{\partial }_j}^{S(TM)\otimes F}=\sum _j^{n}c(e_j){\nabla }_{e_j}^{S(TM)\otimes F},$$

$${\tilde{D}}_F(\psi \otimes \chi )=D_F(\psi \otimes \chi )+c(\Phi )(\psi \otimes \chi ),$$

$${\tilde{D}}_F^{\ast }(\psi \otimes \chi )=D_F(\psi \otimes \chi )-c({\Phi }^{\ast })(\psi \otimes \chi ),$$

$${\tilde{D}}_F=\sum _j^{n}c(e_j){\nabla }_{e_j}^{S(TM)\otimes F}+c(\Phi ),$$

$${\tilde{D}}_F^{\ast }=\sum _j^{n}c(e_j){\nabla }_{e_j}^{S(TM)\otimes F}-c({\Phi }^{\ast }).$$

$${\nabla }^{TM}(\widetilde{e_1},\cdots ,\widetilde{e_n})=(\widetilde{e_1},\cdots ,\widetilde{e_n})({\omega }_{s,t}).$$

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

$$\displaystyle\sigma_{-2}((\widetilde{D}^{*}_{F})^{-1})=\frac{c(\xi)\sigma_{0}(\widetilde{D}^{*})c(\xi)}{|\xi|^{4}}+\frac{c(\xi)}{|\xi|^{6}}\sum_{j}c(dx_{j})\Big{[}\partial_{x_{j}}[c(\xi)]|\xi|^{2}-c(\xi)\partial_{x_{j}}(|\xi|^{2})\Big{]};$$

$$\displaystyle\sigma_{-2}(\widetilde{D}^{-1}_{F})=\frac{c(\xi)\sigma_{0}(\widetilde{D})c(\xi)}{|\xi|^{4}}+\frac{c(\xi)}{|\xi|^{6}}\sum_{j}c(dx_{j})\Big{[}\partial_{x_{j}}[c(\xi)]|\xi|^{2}-c(\xi)\partial_{x_{j}}(|\xi|^{2})\Big{]},$$

$${\sigma }_0({\tilde{D}}^{\ast })$$

$${\sigma }_0(\tilde{D})$$

$${\sigma }_{-2}({\tilde{D}}_F^{-1})$$

$${\sigma }_{-2}(D^{-1})$$

$${\tilde{D}}_F{\tilde{D}}_F^{\ast }$$

$${\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast }$$

$${\sigma }_3({\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast })=\sqrt{-1}c(\xi )|\xi {|}^2;$$

$$\displaystyle\sigma_{2}(\widetilde{D}^{*}_{F}\widetilde{D}_{F}\widetilde{D}^{*}_{F})=c(dx_{n})h^{\prime}(0)|\xi|^{2}+c(\xi)(4\sigma^{k}-2\Gamma^{k})\xi_{k}+\sigma_{0}(D)|\xi|^{2}+\alpha|\xi|^{2}-2\Big{[}c(\xi)c(\Phi)c(\xi)+|\xi|^{2}c(\Phi^{*})\Big{]}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$$

$${\sigma }_2(D^3)=c(d{x}_n)h^{\prime }(0)|\xi {|}^2+c(\xi )(4{\sigma }^k-2{\Gamma }^k){\xi }_k+{\sigma }_0(D)|\xi {|}^2=c(\xi )(4{\sigma }^k-2{\Gamma }^k){\xi }_k-\frac{1}{4}|\xi {|}^2{h}^{\prime }(0)c(d{x}_n).$$

$$D_x^{\alpha }=(-\sqrt{-1}{)}^{|\alpha |}{\partial }_x^{\alpha };\sigma ({\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast })=p_3+p_2+p_1+p_0;\sigma (({\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast }{)}^{-1})=\sum _{j=3}^{\infty }{q}_{-j}.$$

$$1$$

$$q_{-3}=p_{3}^{-1};~{}q_{-4}=-p_{3}^{-1}\Big{[}p_{2}p_{3}^{-1}+\sum_{j}\partial_{\xi_{j}}p_{3}D_{x_{j}}(p_{3}^{-1})\Big{]}.$$

$${\sigma }_{-3}(({\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast }{)}^{-1})$$

$${\sigma }_{-4}(({\tilde{D}}_F^{\ast }{\tilde{D}}_F{\tilde{D}}_F^{\ast }{)}^{-1})$$

$$\sigma_{-4}(D^{-3})=\frac{c(\xi)\sigma_{2}(D^{3})c(\xi)}{|\xi|^{8}}+\frac{c(\xi)}{|\xi|^{10}}\sum_{j}\Big{[}c(dx_{j})|\xi|^{2}+2\xi_{j}c(\xi)\Big{]}\Big{[}\partial_{x_{j}}[c(\xi)]|\xi|^{2}-2c(\xi)\partial_{x_{j}}(|\xi|^{2})\Big{]}.$$

$$F:L^2({\mathbf{R}}_t)\to L^2({\mathbf{R}}_v);F(u)(v)=\int e^{-ivt}u(t)\text{d}t$$

$$r^{+}:C^{\infty }(\mathbf{R})\to C^{\infty }(\overline{{\mathbf{R}}^{+}});f\to f|\overline{{\mathbf{R}}^{+}};\overline{{\mathbf{R}}^{+}}=\{x\ge 0;x\in \mathbf{R}\}.$$

$$\frac{{\text{d}}^{l}h}{\text{d}{\xi }^l}(\xi )\sim \sum _{k=1}^{\infty }\frac{{\text{d}}^l}{\text{d}{\xi }^l}(\frac{c_k}{{\xi }^k})$$

$${\pi }^{+}h({\xi }_0)=\frac{1}{2\pi i}\underset{u\to 0^{-}}{\lim }{\int }_{{\Gamma }^{+}}\frac{h(\xi )}{{\xi }_0+iu-\xi }\text{d}\xi ,$$

$${\pi }^{\prime }h=\frac{1}{2\pi }{\int }_{{\Gamma }^{+}}h(\xi )\text{d}\xi .$$

Full text

††journal: J. Pseudo-Differ. Oper. Appl.

1 Introduction

The noncommutative residue found in [1, 2] plays a prominent role in noncommutative geometry. For one-dimensional manifolds, the noncommutative residue was discovered by Adler [3] in connection with geometric aspects of nonlinear partial differential equations. For arbitrary closed compact $n$-dimensional manifolds, the noncommutative residue was introduced by Wodzicki in [2] using the theory of zeta functions of elliptic pseudodifferential operators. In [4], Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy. Furthermore, Connes made a challenging observation that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action in [5]. In [6], Kastler gave a brute-force proof of this theorem. In [7], Kalau and Walze proved this theorem in the normal coordinates system simultaneously. And then, Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator ${\rm Wres}(D^{-2})$ in turn is essentially the second coefficient of the heat kernel expansion of $D^{2}$ in [8].

Recently, Ponge defined lower dimensional volumes of Riemannian manifolds by the Wodzicki residue [9]. Fedosov et al. defined a noncommutative residue on Boutet de Monvel’s algebra and proved that it was a unique continuous trace in [10]. In [11], Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary. In [12], Wang generalized the Kastler-Kalau-Walze type theorem to the cases of 3, 4-dimensional spin manifolds with boundary and proved a Kastler-Kalau-Walze type theorem. In [12, 13, 15, 16, 17], Y.Wang and his coauthors computed the lower dimensional volumes for 5,6,7-dimensional spin manifolds with boundary and also got some Kastler-Kalau-Walze type theorems. In [18], authors computed $\widetilde{{\rm Wres}}[(\pi^{+}D^{-2})\circ(\pi^{+}D^{-n+2})]$ for any-dimensional manifolds with boundary, and proved a general Kastler-Kalau-Walze type theorem.

In [19], J.Wang and Y.Wang proved the known Lichnerowicz formula for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection and got two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on four-dimensional manifolds with boundary.

The motivation of this paper is to establish two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators with non-unitary connections on six-dimensional manifolds with boundary.

This paper is organized as follows: In Section 2, we recall the definition of twisted Dirac operators and compute their symbols. In Section 3, we give a Kastler-Kalau-Walze type theorems for twisted Dirac operators on six-dimensional manifolds with boundary. In Section 4 and Section 5, we recall the definition of twisted signature operators and compute their symbols, and we give a Kastler-Kalau-Walze type theorems for twisted signature operators on six-dimensional manifolds with boundary.

2 Twisted Dirac operator and its symbol

In this section we consider a $n$-dimensional oriented Riemannian manifold $(M,g^{M})$ equipped with a fixed spin structure. We recall twisted Dirac operators. Let $S(TM)$ be the spinors bundle and $F$ be an additional smooth vector bundle equipped with a non-unitary connection $\widetilde{\nabla}^{F}$. Let $\widetilde{\nabla}^{F,\ast}$ be the dual connection on $F$, and define

[TABLE]

then $\nabla^{F}$ is a metric connection and $\Phi$ is an endomorphism of $F$ with a 1-form coefficient. We consider the tensor product vector bundle $S(TM)\otimes F$, which becomes a Clifford module via the definition:

[TABLE]

and which we equip with the compound connection:

[TABLE]

Let

[TABLE]

then the spinor connection $\widetilde{\nabla}$ induced by $\nabla^{S(TM)\otimes F}$ is locally given by

[TABLE]

Let $\{e_{i}\}(1\leq i,j\leq n)$ $(\{\partial_{i}\})$ be the orthonormal frames (natural frames respectively ) on $TM$,

[TABLE]

where $\nabla^{S(TM)\otimes F}_{\partial_{j}}=\partial_{j}+\sigma_{j}^{s}+\sigma_{j}^{F}$ and $\sigma_{j}^{s}=\frac{1}{4}\sum_{j,k}\langle\nabla^{TM}_{\partial_{i}}e_{j},e_{k}\rangle c(e_{j})c(e_{k})$, $\sigma_{j}^{F}$ is the connection matrix of $\nabla^{F}$, then the twisted Dirac operators $\widetilde{D}_{F}$, $\widetilde{D}^{*}_{F}$ associated to the connection $\widetilde{\nabla}$ as follows.

For $\psi\otimes\chi\in S(TM)\otimes F$, we have

[TABLE]

where $c(\Phi)=\sum_{i=1}^{n}c(e_{i})\otimes\Phi(e_{i})$ and $c(\Phi^{*})=\sum_{i=1}^{n}c(e_{i})\otimes\Phi^{*}(e_{i})$, $\Phi^{*}(e_{i})$ denotes the adjoint of $\Phi(e_{i})$.

Then, we have obtain

[TABLE]

Let $\nabla^{TM}$ denote the Levi-civita connection about $g^{M}$. In the local coordinates $\{x_{i};1\leq i\leq n\}$ and the fixed orthonormal frame $\{\widetilde{e_{1}},\cdots,\widetilde{e_{n}}\}$, the connection matrix $(\omega_{s,t})$ is defined by

[TABLE]

Let $c(\widetilde{e_{i}})$ denote the Clifford action, $g^{ij}=g(dx_{i},dx_{j})$,$\nabla^{TM}_{\partial_{i}}\partial_{j}=\sum_{k}\Gamma_{ij}^{k}\partial_{k};~{}\Gamma^{k}=g^{ij}\Gamma_{ij}^{k}$ and the cotangent vector $\xi=\sum\xi_{j}dx_{j}$ and $\xi^{j}=g^{ij}\xi_{i}$, by Lemma 1 in [13] and Lemma 2.1 in [12], for any fixed point $x_{0}\in\partial M$, we can choose the normal coordinates $U$ of $x_{0}$ in $\partial M$ (not in $M$), by the composition formula and (2.2.11) in [12], we obtain in [19],

Lemma 2.1**.**

Let $\widetilde{D}^{*}_{F},\widetilde{D}_{F}$ be the twisted Dirac operators on $\Gamma(S(TM)\otimes F)$, then

[TABLE]

where

[TABLE]

Let \alpha=\sum_{j=1}^{n}c(e_{j})\big{(}\sigma_{j}^{F}-\Phi^{*}(e_{j})\big{)},\beta=\sum_{j=1}^{n}c(e_{j})\big{(}\sigma_{j}^{F}+\Phi(e_{j})\big{)},\sigma_{0}(D)=-\frac{1}{4}\sum_{s,t}\omega_{s,t}(e_{l})c(e_{l})c(e_{s})c(e_{t}), $\partial^{j}=g^{ij}\partial_{i},\sigma^{i}=g^{ij}\sigma_{j}$, we note that

[TABLE]

where

[TABLE]

By (2.6), (2.9) and (2.10), we have

[TABLE]

Combining (2.10) and (2.20), we have

[TABLE]

By the above composition formulas, then we obtain:

Lemma 2.2**.**

Let $\widetilde{D}^{*}_{F},\widetilde{D}_{F}$ be the twisted Dirac operators on $\Gamma(S(TM)\otimes F)$,

[TABLE]

Write

[TABLE]

[TABLE]

By the composition formula of psudodifferential operators, we have

[TABLE]

Then

[TABLE]

By Lemma 2.1 in [12] and (2.21)-(2.27), we obtain

Lemma 2.3**.**

Let $\widetilde{D}^{*}_{F},\widetilde{D}_{F}$ be the twisted Dirac operators on $\Gamma(S(TM)\otimes F)$, then

[TABLE]

where

[TABLE]

3 A Kastler-Kalau-Walze type theorem for six-dimensional manifolds with boundary associated with twisted Dirac Operators

In this section, we shall prove a Kastler-Kalau-Walze type formula for six-dimensional compact manifolds with boundary. Some basic facts and formulae about Boutet de Monvel’s calculus are recalled as follows.

Let

[TABLE]

denote the Fourier transformation and $\varphi(\overline{{\bf R}^{+}})=r^{+}\varphi({\bf R})$ (similarly define $\varphi(\overline{{\bf R}^{-}}$)), where $\varphi({\bf R})$ denotes the Schwartz space and

[TABLE]

We define $H^{+}=F(\varphi(\overline{{\bf R}^{+}}));~{}H^{-}_{0}=F(\varphi(\overline{{\bf R}^{-}}))$ which are orthogonal to each other. We have the following property: $h\in H^{+}~{}(H^{-}_{0})$ iff $h\in C^{\infty}({\bf R})$ which has an analytic extension to the lower (upper) complex half-plane $\{{\rm Im}\xi<0\}~{}(\{{\rm Im}\xi>0\})$ such that for all nonnegative integer $l$,

[TABLE]

as $|\xi|\rightarrow+\infty,{\rm Im}\xi\leq 0~{}({\rm Im}\xi\geq 0)$.

Let $H^{\prime}$ be the space of all polynomials and $H^{-}=H^{-}_{0}\bigoplus H^{\prime};~{}H=H^{+}\bigoplus H^{-}.$ 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 $\tilde{H}$,

[TABLE]

where $\Gamma^{+}$ is a Jordan close curve included ${\rm Im}\xi>0$ surrounding all the singularities of $h$ in the upper half-plane and $\xi_{0}\in{\bf R}$. Similarly, define $\pi^{{}^{\prime}}$ on $\tilde{H}$,

[TABLE]

So, $\pi^{\prime}(H^{-})=0$. For $h\in H\bigcap L^{1}(R)$, $\pi^{\prime}h=\frac{1}{2\pi}\int_{R}h(v)\texttt{d}v$ and for $h\in H^{+}\bigcap L^{1}(R)$, $\pi^{\prime}h=0$. Denote by $\mathcal{B}$ Boutet de Monvel’s algebra (for details, see Section 2 of [14]).

An operator of order $m\in{\bf Z}$ and type $d$ is a matrix

[TABLE]

where $X$ is a manifold with boundary $\partial X$ and $E_{1},E_{2}~{}(F_{1},F_{2})$ are vector bundles over $X~{}(\partial X)$. Here, $P:C^{\infty}_{0}(\Omega,\overline{E_{1}})\rightarrow C^{\infty}(\Omega,\overline{E_{2}})$ is a classical pseudodifferential operator of order $m$ on $\Omega$, where $\Omega$ is an open neighborhood of $X$ and $\overline{E_{i}}|X=E_{i}~{}(i=1,2)$. $P$ has an extension: $~{}{\cal{E^{\prime}}}(\Omega,\overline{E_{1}})\rightarrow{\cal{D^{\prime}}}(\Omega,\overline{E_{2}})$, where ${\cal{E^{\prime}}}(\Omega,\overline{E_{1}})~{}({\cal{D^{\prime}}}(\Omega,\overline{E_{2}}))$ is the dual space of $C^{\infty}(\Omega,\overline{E_{1}})~{}(C^{\infty}_{0}(\Omega,\overline{E_{2}}))$. Let $e^{+}:C^{\infty}(X,{E_{1}})\rightarrow{\cal{E^{\prime}}}(\Omega,\overline{E_{1}})$ denote extension by zero from $X$ to $\Omega$ and $r^{+}:{\cal{D^{\prime}}}(\Omega,\overline{E_{2}})\rightarrow{\cal{D^{\prime}}}(\Omega,{E_{2}})$ denote the restriction from $\Omega$ to $X$, then define

[TABLE]

In addition, $P$ is supposed to have the transmission property; this means that, for all $j,k,\alpha$, the homogeneous component $p_{j}$ of order $j$ in the asymptotic expansion of the symbol $p$ of $P$ in local coordinates near the boundary satisfies:

[TABLE]

then $\pi^{+}P:C^{\infty}(X,{E_{1}})\rightarrow C^{\infty}(X,{E_{2}})$ by Section 2.1 of [14].

In the following, write \pi^{+}D^{-1}=\left(\begin{array}[]{lcr}\pi^{+}D^{-1}&0\\ 0&0\end{array}\right), we will compute

[TABLE]

Let $M$ be a compact manifold with boundary $\partial M$. We assume that the metric $g^{M}$ on $M$ has the following form near the boundary

[TABLE]

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

[TABLE]

such that $\hat{g}|_{M}=g$. We fix a metric $\hat{g}$ on the $\hat{M}$ such that $\hat{g}|_{M}=g$. Note $\widetilde{D}_{F}$ is the twisted Dirac operator on the spinor bundle $S(TM)\otimes F$ corresponding to the connection $\widetilde{\nabla}$.

Now we recall the main theorem in [10].

Theorem 3.4**.**

(Fedosov-Golse-Leichtnam-Schrohe)* Let $X$ and $\partial X$ be connected, ${\rm dim}X=n\geq 3$, 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:*

[TABLE]

Then

a) ${\rm\widetilde{Wres}}([A,B])=0$, for any $A,B\in\mathcal{B}$;

b) It is a unique continuous trace on $\mathcal{B}/\mathcal{B}^{-\infty}$.

Denote by $\sigma_{l}(A)$ the $l$-order symbol of an operator A. An application of (3.5) and (3.6) in [14] shows that

[TABLE]

where

[TABLE]

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

Locally we can use Theorem 2.4 in [19] to compute the interior term of (3.8), then

[TABLE]

So we only need to compute $\int_{\partial M}\Phi$.

From the formula (3.9) for the definition of $\Phi$, now we can compute $\Phi$. Since the sum is taken over $r+\ell-k-j-|\alpha|-1=-6,\ r\leq-1,\ell\leq-3$, then we have the $\int_{\partial_{M}}\Phi$ is the sum of the following five cases:  

 

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

By (3.9), we get

[TABLE]

By Lemma 2.2 in [12], for $i<n$, we have

[TABLE]

so case (a) (I) vanishes.

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

By (3.9), we have

[TABLE]

By (2.2.23) in [12], we have

[TABLE]

By (2.28) and direct calculations, we have

[TABLE]

Since $n=6$, ${\rm trace}_{S(TM)\otimes F}[-\texttt{id}]=-8{\rm dimF}$. By the relation of the Clifford action and ${\rm trace}AB={\rm trace}BA$, then

[TABLE]

By (3.14), (3.15) and (3.16), we get

[TABLE]

Then we obtain

[TABLE]

where ${\rm\Omega_{4}}$ is the canonical volume of $S_{4}.$

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

By (3.9), we have

[TABLE]

By (2.2.29) in [12], we have

[TABLE]

By (2.28) and direct calculations, we have

[TABLE]

Combining (3.16), (3.20) and (3.21), we have

[TABLE]

Then

[TABLE]

where ${\rm\Omega_{4}}$ is the canonical volume of $S_{4}.$

case (b) $r=-1,l=-4,|\alpha|=j=k=0$.

By (3.9), we have

[TABLE]

where

[TABLE]

By (2.2.44) in [12], we have

[TABLE]

In the normal coordinate, $g^{ij}(x_{0})=\delta^{j}_{i}$ and $\partial_{x_{j}}(g^{\alpha\beta})(x_{0})=0$, if $j<n$; $\partial_{x_{j}}(g^{\alpha\beta})(x_{0})=h^{\prime}(0)\delta^{\alpha}_{\beta}$, if $j=n$. So by Lemma A.2 in [12], we have $\Gamma^{n}(x_{0})=\frac{5}{2}h^{\prime}(0)$ and $\Gamma^{k}(x_{0})=0$ for $k<n$. By the definition of $\delta^{k}$ and Lemma 2.3 in [12], we have $\delta^{n}(x_{0})=0$ and $\delta^{k}=\frac{1}{4}h^{\prime}(0)c(\widetilde{e}_{k})c(\widetilde{e}_{n})$ for $k<n$. We obtain

[TABLE]

Then

[TABLE]

By (3.16),(3.29) and (3.31), we obtain

[TABLE]

By (3.25) and (3.32), we have

[TABLE]

Since

[TABLE]

then

[TABLE]

By the relation of the Clifford action and ${\rm trace}AB={\rm trace}BA$, then we have the equalities

[TABLE]

We note that $i<n,~{}\int_{|\xi^{\prime}|=1}\xi_{i}\sigma(\xi^{\prime})=0$, so ${\rm trace}[c(\xi^{\prime})\alpha]$ has no contribution for computing case (b).

By (3.26) and (3.35), then

[TABLE]

Since

[TABLE]

then

[TABLE]

By the relation of the Clifford action and ${\rm trace}AB={\rm trace}BA$, then we have the equalities

[TABLE]

We note that $i<n,~{}\int_{|\xi^{\prime}|=1}\xi_{i}\sigma(\xi^{\prime})=0$, so ${\rm trace}[c(\xi^{\prime})c(\Phi^{*})]$ has no contribution for computing case (b).

By (3.27) and (3.40), then

[TABLE]

Since

[TABLE]

then

[TABLE]

By the relation of the Clifford action and ${\rm trace}AB={\rm trace}BA$, then we have the equalities

[TABLE]

We note that $i<n,~{}\int_{|\xi^{\prime}|=1}\xi_{i}\sigma(\xi^{\prime})=0$, so ${\rm trace}[c(\xi^{\prime})c(\Phi)]$ has no contribution for computing case (b).

By (3.28) and (3.45), then

[TABLE]

By (3.24), then

[TABLE]

case (c) $r=-2,l=-3,|\alpha|=j=k=0$.

By (3.9), we have

[TABLE]

By (2.18), we have

[TABLE]

By (2.19), we have

[TABLE]

where

[TABLE]

On the other hand,

[TABLE]

By (2.28), we obtain

[TABLE]

By (3.53) and (3.56), then we have

[TABLE]

Similarly, we have

[TABLE]

By (3.57) and (3.58), we obtain

[TABLE]

By (3.55) and (3.56), we have

[TABLE]

By the relation of the Clifford action and ${\rm trace}AB={\rm trace}BA$, then we have the equalities

[TABLE]

We note that $i<n,~{}\int_{|\xi^{\prime}|=1}\xi_{i}\sigma(\xi^{\prime})=0$, so ${\rm trace}[c(\xi^{\prime})\beta]$ has no contribution for computing case (c).

Then, we obtain

[TABLE]

Then

[TABLE]

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

[TABLE]

By (4.2) in [12], we have

[TABLE]

and $K_{i,j}$ is the second fundamental form, or extrinsic curvature. For $n=6$, then

[TABLE]

Hence we conclude that

Theorem 3.5**.**

Let M be a 6-dimensional compact spin manifolds with the boundary $\partial M$. Then

[TABLE]

where $s$ is the scalar curvature.

4 Twisted signature operator and its symbol

Let us recall the definition of twisted signature operators. We consider a $n$-dimensional oriented Riemannian manifold $(M,g^{M})$. Let $F$ be a real vector bundle over $M$. let $g^{F}$ be an Euclidean metric on $F$. Let

[TABLE]

be the real exterior algebra bundle of $T^{\ast}M$. Let

[TABLE]

be the set of smooth sections of $\wedge^{\ast}(T^{\ast}M)\otimes F$. Let $\ast$ be the Hodge star operator of $g^{TM}$. It extends on $\wedge^{\ast}(T^{\ast}M)\otimes F$ by acting on $F$ as identity. Then $\Omega^{\ast}(M,F)$ inherits the following standardly induced inner product

[TABLE]

Let $\widehat{\nabla}^{F}$ be the non-Euclidean connection on $F$. Let $d^{F}$ be the obvious extension of $\nabla^{F}$ on $\Omega^{\ast}(M,F)$. Let $\delta^{F}=d^{F\ast}$ be the formal adjoint operator of $d^{F}$ with respect to the inner product. Let $\hat{D}^{F}$ be the differential operator acting on $\Omega^{\ast}(M,F)$ defined by

[TABLE]

Let

[TABLE]

Then $\nabla^{F,e}$ is an Euclidean connection on $(F,g^{F})$.

Let $\nabla^{\wedge^{\ast}(T^{\ast}M)}$ be the Euclidean connection on $\wedge^{\ast}(T^{\ast}M)$ induced canonically by the Levi-Civita connection $\nabla^{TM}$ of $g^{TM}$. Let $\nabla^{e}$ be the Euclidean connection on $\wedge^{\ast}(T^{\ast}M)\otimes F$ obtained from the tensor product of $\nabla^{\wedge^{\ast}(T^{\ast}M)}$ and $\nabla^{F,e}$. Let $\{e_{1},\cdots,e_{n}\}$ be an oriented (local) orthonormal basis of $TM$. The following result was proved by Proposition in [20].

The following identity holds

[TABLE]

Let $D_{F}^{e}=\sum_{j=1}^{n}c(e_{j})\nabla^{e}_{e_{j}}$ and $\omega(F,g^{F})$ be any element in $\Omega(M,EndF)$, then we define the generalized twisted signature operators $\hat{D}_{F}$, $\hat{D}^{*}_{F}$ as follows.

For sections $\psi\otimes\chi\in\wedge^{\ast}(T^{\ast}M)\otimes F$,

[TABLE]

Here $\omega^{*}(F,g^{F})(e_{i})$ denotes the adjoint of $\omega(F,g^{F})(e_{i})$.

In the local coordinates $\{x_{i};1\leq i\leq n\}$ and the fixed orthonormal frame $\{\widetilde{e_{1}},\cdots,\widetilde{e_{n}}\}$, the connection matrix $(\omega_{s,t})$ is defined by

[TABLE]

Let $M$ be a $6$-dimensional compact oriented Riemannian manifold with boundary $\partial M$. We define that $\hat{D}_{F}:~{}C^{\infty}(M,\wedge^{\ast}(T^{\ast}M)\otimes F)\rightarrow C^{\infty}(M,\wedge^{\ast}(T^{\ast}M)\otimes F)$ is the generalized twisted signature operator. Take the coordinates and the orthonormal frame as in Section 3. Let $\epsilon(\widetilde{e_{j}*}),~{}\iota(\widetilde{e_{j}*})$ be the exterior and interior multiplications respectively. Write

[TABLE]

We’ll compute ${\rm tr}_{\wedge^{*}(T^{*}M)\otimes F}$ in the frame $\{e^{\ast}_{i_{1}}\wedge\cdots\wedge e^{\ast}_{i_{k}}|~{}1\leq i_{1}<\cdots<i_{k}\leq 6\}.$ By (3.2) and (4.8) in [12], we have

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By the composition formula and (2.2.11) in [12], we obtain in [19],

Lemma 4.6**.**

Let $\hat{D}^{*}_{F},\hat{D}_{F}$ be the twisted signature operators on $\Gamma(\wedge^{*}(T^{*}M)\otimes F)$, then

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By the composition formula of pseudodifferential operators in Section 2.2.1 of [12], we have

Lemma 4.7**.**

The symbol of the twisted signature operators $\hat{D}^{*}_{F},\hat{D}_{F}$ as follows:

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Since $\Psi$ is a global form on $\partial M$, so for any fixed point $x_{0}\in\partial M$, we can choose the normal coordinates $U$ of $x_{0}$ in $\partial M$(not in $M$) and compute $\Psi(x_{0})$ in the coordinates $\widetilde{U}=U\times[0,1)$ and the metric $\frac{1}{h(x_{n})}g^{\partial M}+\texttt{d}x_{n}^{2}$. The dual metric of $g^{\partial M}$ on $\widetilde{U}$ is $\frac{1}{\tilde{h}(x_{n})}g^{\partial M}+\texttt{d}x_{n}^{2}.$ Write $g_{ij}^{M}=g^{M}(\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial x_{j}})$; $g^{ij}_{M}=g^{M}(dx_{i},dx_{j})$, then

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and

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Let $\{e_{1},\cdots,e_{n-1}\}$ be an orthonormal frame field in $U$ about $g^{\partial M}$ which is parallel along geodesics and $e_{i}=\frac{\partial}{\partial x_{i}}(x_{0})$, then $\{\widetilde{e_{1}}=\sqrt{h(x_{n})}e_{1},\cdots,\widetilde{e_{n-1}}=\sqrt{h(x_{n})}e_{n-1},\widetilde{e_{n}}=\texttt{d}x_{n}\}$ is the orthonormal frame field in $\widetilde{U}$ about $g^{M}.$ Locally $\wedge^{\ast}(T^{\ast}M)|\widetilde{U}\cong\widetilde{U}\times\wedge^{*}_{C}(\frac{n}{2}).$ Let $\{f_{1},\cdots,f_{n}\}$ be the orthonormal basis of $\wedge^{*}_{C}(\frac{n}{2})$. Take a spin frame field $\sigma:\widetilde{U}\rightarrow Spin(M)$ such that $\pi\sigma=\{\widetilde{e_{1}},\cdots,\widetilde{e_{n}}\}$ where $\pi:Spin(M)\rightarrow O(M)$ is a double covering, then $\{[\sigma,f_{i}],1\leq i\leq 4\}$ is an orthonormal frame of $\wedge^{\ast}(T^{\ast}M)|_{\widetilde{U}}.$ In the following, since the global form $\Psi$ is independent of the choice of the local frame, so we can compute $\texttt{tr}_{\wedge^{\ast}(T^{\ast}M)}$ in the frame $\{[\sigma,f_{i}],1\leq i\leq 4\}$. Let $\{E_{1},\cdots,E_{n}\}$ be the canonical basis of $R^{n}$ and $c(E_{i})\in cl_{C}(n)\cong\texttt{Hom}(\wedge^{*}_{C}(\frac{n}{2}),\wedge^{*}_{C}(\frac{n}{2}))$ be the Clifford action. By [12], then

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then we have $\frac{\partial}{\partial x_{i}}c(\widetilde{e_{i}})=0$ in the above frame. By Lemma 2.2 in [12], we have

Lemma 4.8**.**

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where $\xi=\xi^{\prime}+\xi_{n}\texttt{d}x_{n}$.

Then an application of Lemma 2.3 in [12] shows

Lemma 4.9**.**

The symbol of the twisted signature operators $\hat{D}^{*}_{F},\hat{D}_{F}$ as follows:

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We write

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Let $\hat{c}(\omega)=\sum_{i}c(e_{i})\omega(F,g^{F})(e_{i})$ and $\hat{c}(\omega^{*})=\sum_{i}c(e_{i})\omega^{*}(F,g^{F})(e_{i})$, then similar to (2.20), we have

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where $s$ is the scalar curvature, $R^{F,e}$ denotes the curvature-tensor on $F$.

Combining (4.9) and (4.10), we have

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By the above composition formulas, then we obtain:

Lemma 4.10**.**

Let $\hat{D}^{*}_{F},\hat{D}_{F}$ be the twisted signature operators on $\Gamma(\wedge^{*}(T^{*}M)\otimes F)$, then

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where,

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Write

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By the composition formula of psudodifferential operators, we have

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Then

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By Lemma 2.1 in [12] and (4.30)-(4.36), we obtain

Lemma 4.11**.**

Let $\hat{D}^{*}_{F},\hat{D}_{F}$ be the generalized twisted signature operators on $\Gamma(\wedge^{\ast}(T^{\ast}M)\otimes F)$, then

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where

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Hence we conclude that

Theorem 4.12**.**

[19]** For even $n$-dimensional oriented compact Riemainnian manifolds without boundary, the following equality holds:

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5 A Kastler-Kalau-Walze theorem for six-dimensional Riemannian manifolds with boundary associated to twisted Signature Operators

In this section, we shall prove a Kastler-Kalau-Walze type formula for $\hat{D}^{*}_{F}\hat{D}_{F}$. An application of (2.1.4) in [14] shows that

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where

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

Locally we can use Theorem 4.3 [19] to compute the interior term of (5.1), then

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So we only need to compute $\int_{\partial M}\Psi$. From the remark above, now we can compute $\Psi$ (see formula (3.6) for the definition of $\Psi$). Since the sum is taken over $r+\ell-k-j-|\alpha|-1=-6,\ r\leq-1,\ell\leq-3$, then we have the $\int_{\partial_{M}}\Psi$ is the sum of the following five cases:  

 

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

By (5.2), we get

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By Lemma 2.2 in [12], for $i<n$ we have

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so case a) I) vanishes.

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

By (5.2), we have

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By Lemma 2.2 in [12], we have

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By direct calculations we have

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By (5.7) and (5.8), we obtain

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Then we obtain

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where ${\rm\Omega_{4}}$ is the canonical volume of $S_{4}.$

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

By (5.2) and an integration by parts, we have

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By Lemma 2.2 in [12], we have

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By (4.37) and direct calculations, we have

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Combining (5.12) and (5.13), we have

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Then

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where ${\rm\Omega_{4}}$ is the canonical volume of $S_{4}.$

case b) $r=-2,l=-3,|\alpha|=j=k=0$.

By (5.2) and an integration by parts, we have

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Then an application of Lemma 4.3 shows

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Hence,

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where

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On the other hand,

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From (5.19) and (5.24), we have

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Similarly, we obtain

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For the signature operator case,

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and

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By Section 3 in [12], then

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where b_{6,m}=\left(\begin{array}[]{lcr}\ \ 4\\ \ m-2\end{array}\right)+\left(\begin{array}[]{lcr}\ \ 4\\ \ m\end{array}\right)-2\left(\begin{array}[]{lcr}\ \ 4\\ \ m-1\end{array}\right).

Then

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Hence in this case,

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We note that $\int_{|\xi^{\prime}|=1}\xi_{1}\cdots\xi_{2q+1}\sigma(\xi^{\prime})=0$, then ${\rm tr}_{\wedge^{*}(T^{*}M)}[c(\xi^{\prime})p(x_{0})]$ has no contribution for computing Case (b).

So, we obtain

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Then, we have

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

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Then $\texttt{tr}[\vartheta c(\xi^{\prime})]$ has no contribution for computing Case b.

Then, we have

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From (5.33), we obtain

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From (5.35), we obtain

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Then

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case c) $r=-1,l=-4,|\alpha|=j=k=0$.

By (5.2) and an integration by parts, we have

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By (3.12) in [19], we have

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In the normal coordinate, $g^{ij}(x_{0})=\delta^{j}_{i}$ and $\partial_{x_{j}}(g^{\alpha\beta})(x_{0})=0$, if $j<n$; $\partial_{x_{j}}(g^{\alpha\beta})(x_{0})=h^{\prime}(0)\delta^{\alpha}_{\beta}$, if $j=n$. So by Lemma A.2 in [12], we have $\Gamma^{n}(x_{0})=\frac{5}{2}h^{\prime}(0)$ and $\Gamma^{k}(x_{0})=0$ for $k<n$. By the definition of $\delta^{k}$ and Lemma 2.3 in [12], we have $\delta^{n}(x_{0})=0$ and $\delta^{k}=\frac{1}{4}h^{\prime}(0)c(\widetilde{e_{k}})c(\widetilde{e_{n}})$ for $k<n$. By ( 3.15) in [19], we obtain

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Then

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Combining (5.40) and (5.42), we obtain

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Then

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Since

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from (5.40) and (5.45), then we have

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We have (5.46) has no contribution for computing case b).

Similarly, we have

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Then

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Simiarly,

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Then

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

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By (4.2) in [12], we have

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and $K_{i,j}$ is the second fundamental form, or extrinsic curvature. For $n=6$, then

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Hence we conclude that

Theorem 5.13**.**

Let M be a 6-dimensional compact manifolds with the boundary $\partial M$. Then

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where $s$ is the scalar curvature.

Acknowledgements

This work was supported by NSFC. 11771070 . The authors thank the referee for his (or her) careful reading and helpful comments.