Research Article | Open Access
Jian Wang, Yong Wang, "A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary", Abstract and Applied Analysis, vol. 2014, Article ID 465782, 18 pages, 2014. https://doi.org/10.1155/2014/465782
A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Manifolds with Boundary
We give a brute-force proof of the Kastler-Kalau-Walze type theorem for 7-dimensional manifolds with boundary.
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  in connection with geometric aspects of nonlinear partial differential equations. For arbitrary closed compact -dimensional manifolds, the noncommutative residue was introduced by Wodzicki in  using the theory of zeta functions of elliptic pseudodifferential operators. In , 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 . Let be the scalar curvature and let Wres denote the noncommutative residue. Then, the Kastler-Kalau-Walze theorem gives an operator-theoretic explanation of the gravitational action and says that, for a 4-dimensional closed spin manifold, there exists a constant , such that In , Kastler gave a brute-force proof of this theorem. In , Kalau and Walze proved this theorem in the normal coordinates system simultaneously. And then, Ackermann proved that the Wodzicki residue in turn is essentially the second coefficient of the heat kernel expansion of in .
On the other hand, Fedosov et al. defined a noncommutative residue on Boutet de Monvel’s algebra and proved that it was a unique continuous trace in . In , Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary. For an oriented spin manifold with boundary , by the composition formula in Boutet de Monvel’s algebra and the definition of , should be the sum of two terms from interior and boundary of , where is an element in Boutet de Monvel’s algebra . It is well known that the gravitational action for manifolds with boundary is also the sum of two terms from interior and boundary of . Considering the Kastler-Kalau-Walze theorem for manifolds without boundary, then the term from interior is proportional to gravitational action from interior, so it is natural to hope to get the gravitational action for manifolds with boundary by computing . Based on the motivation, Wang  proved a Kastler-Kalau-Walze type theorem for 4-dimensional spin manifolds with boundary where is the canonical volume of . Furthermore, Wang  found a Kastler-Kalau-Walze type theorem for higher dimensional manifolds with boundary and generalized the definition of lower-dimensional volumes in  to manifolds with boundary. For 5-dimensional spin manifolds with boundary , Wang got and for 6-dimensional spin manifolds with boundary,
In order to get the boundary term, we computed the lower-dimensional volume for 6-dimensional spin manifolds with boundary associated with and in  and obtained the volume with the boundary term where is the extrinsic curvature.
In , Wang proved a Kastler-Kalau-Walze type theorem for general form perturbations and the conformal perturbations of Dirac operators for compact manifolds with or without boundary. Let be 4-dimensional compact manifolds with the boundary and let be a general differential form on , from Theorem 10 in ; then
Recently, we computed for -dimensional spin manifolds with boundary in case of . In the present paper, we will restrict our attention to the case of . We compute for 7-dimensional manifolds with boundary. Our main result is as follows.
Main Theorem. The following identity for 7-dimensional manifolds with boundary holds: where and are, respectively, scalar curvatures on and . Compared with the previous results, up to the extrinsic curvature, the scalar curvature on and the scalar curvature on appear in the boundary term. This case essentially makes the whole calculations more difficult, and the boundary term is the sum of fifteen terms. As in computations of the boundary term, we will consider some new traces of multiplication of Clifford elements. And the inverse 4-order symbol of the Dirac operator and higher derivatives of -1-order and -3-order symbols of the Dirac operators will be extensively used.
This paper is organized as follows. In Section 2, we define lower-dimensional volumes of compact Riemannian manifolds with boundary. In Section 3, for 7-dimensional spin manifolds with boundary and the associated Dirac operators, we compute and get a Kastler-Kalau-Walze type theorem in this case.
2. Lower-Dimensional Volumes of Spin Manifolds with Boundary
In this section, we consider an -dimensional oriented Riemannian manifold with boundary equipped with a fixed spin structure. We assume that the metric on has the following form near the boundary: where is the metric on . Let be a collar neighborhood of which is diffeomorphic . By the definition of and , there exists such that and for some sufficiently small . Then, there exists a metric on which has the form on such that . We fix a metric on the such that .
Let us give the expression of Dirac operators near the boundary. Set and , where are orthonormal basis of . Let denote the Levi-Civita connection about . In the local coordinates and the fixed orthonormal frame , the connection matrix is defined by The Dirac operator is defined by By Lemma 6.1 in  and Propositions 2.2 and 2.4 in , we have the following lemma.
Lemma 1. Let and be a Riemannian manifold with the metric . For vector fields and in , then
Denote ; then we obtain the following lemma.
Lemma 2. The following identity holds: Others are zeros.
By Lemma 2, we have the following definition.
Definition 3. The following identity holds in the coordinates near the boundary:
To define the lower-dimensional volume, some basic facts and formulae about Boutet de Monvel’s calculus which can be found in Section 2 in  are needed.
Denote by the Fourier transformation and (similarly, define , where denotes the Schwartz space and We define and which are orthogonal to each other. We have the following property: iff which has an analytic extension to the lower (upper) complex half-plane such that, for all nonnegative integers , as , .
Let be the space of all polynomials and let ; . Denote by , respectively, the projection on . For calculations, we take rational functions having no poles on the real axis} ( is a dense set in the topology of ). Then, on , where is a Jordan close curve which included surrounding all the singularities of in the upper half-plane and . Similarly, define on : So, . For , and for , .
Let be an -dimensional compact oriented manifold with boundary . Denote by Boutet de Monvel’s algebra; we recall the main theorem in .
Theorem 4 (Fedosov-Golse-Leichtnam-Schrohe). Let and be connected, let , and let , and denote by , , and the local symbols of , and , respectively. Define Then, (a) , for any ; (b) it is a unique continuous trace on .
Let and be nonnegative integers and let . Then, by Section 2.1 of , we have the following definition.
Definition 5. Lower-dimensional volumes of spin manifolds with boundary are defined by
Denote by the -order symbol of an operator . An application of (2.1.4) in  shows that where and the sum is taken over , , .
3. A Kastler-Kalau-Walze Type Theorem for 7-Dimensional Spin Manifolds with Boundary
In this section, we compute the lower-dimensional volume for 7-dimensional compact manifolds with boundary and get a Kastler-Kalau-Walze type formula in this case. From now on, we always assume that carries a spin structure so that the spinor bundle and the Dirac operator are defined on .
The following proposition is the key of the computation of lower-dimensional volumes of spin manifolds with boundary.
Proposition 6 (see ). The following identity holds:
Nextly, for 7-dimensional spin manifolds with boundary, we compute . By Proposition 6, for 7-dimensional compact manifolds with boundary, we have
Recall the Dirac operator of Definition 3. Write By the composition formula of pseudodifferential operators, then we have Thus, we get
Lemma 7. Consider the symbol of the Dirac operator where
Since is a global form on , so for any fixed point , we can choose the normal coordinates of in (not in ) and compute in the coordinates and the metric . The dual metric of on is . Write and ; then,
Let be an orthonormal frame field in about which is parallel along geodesics and ; then, is the orthonormal frame field in about . Locally, . Let be the orthonormal basis of . Take a spin frame field such that , where is a double covering; then, is an orthonormal frame of . In the following, since the global form is independent of the choice of the local frame, we can compute in the frame . Let be the canonical basis of and let be the Clifford action. By , then then, we have in the above frame. By Lemma 2.2 in , we have the following.
Lemma 8. With the metric on near the boundary, where .
Then, the following lemma is introduced.
Lemma 9. The following identity holds:
Proof. From Lemma 5.7 in , we have Then, we obtain .
Lemma 10. Let be the metric on 7-dimensional spin manifolds near the boundary; then,
Proof. From Lemma 2.3 in , we have
Let , , and . Then,
When , ,
Similarly, when , , or , , . When , .
On the other hand, from definitions (10) and (11), then When , , Similarly, when , , . When , , . When ,
Lemma 11. When , When ,
Next, we can compute (see formula (23) for definition of ). Since the sum is taken over , , then we have that is the sum of the following fifteen cases.
Case 2. Consider , , , , and .
From (23), we have By Lemma 7, a simple computation shows By (18) and the Cauchy integral formula, then Similarly, we obtain From (51) and (53), we get Note that ; then, from (50), (54), and direct computations, we obtain Therefore, where is the canonical volume of .