The General Kastler-Kalau-Walze Type Theorems About Witten Deformation

A noncommutative residue under the framework of noncommutative geometry is a trace defined on the classical quasi-differential operator algebra of closed manifolds, but it is not an extension in the sense of the usual operator trace. Noncommutative residues on low-dimensional manifolds were discovered by Adler [1], and noncommutative residues on higher-dimensional manifolds were discovered simultaneously by Wodzicki and Guillemin [2, 3]. The study of noncommutative residues has profound theoretical and practical value. Connes [4–6] ’s research shows that non-commutative residues play an integral role in noncommutative geometry. The research work of Connes [4] and Ugalde [7] shows that non-commutative residues are closely related to conformal geometry. That is, noncommutative residues bridge conformal geometry and noncommutative geometry. Importantly, 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 [6]. This theorem was proved independently by Kastler, Kalau and Walze respectively, and is called Kastler-Kalau-Walze Theorem [8, 9]. This initiated the application of noncommutative residues in gravity action under the framework of non-commutative geometry, and gave an operator theory explanation for gravity action.


Introduction
A noncommutative residue under the framework of noncommutative geometry is a trace defined on the classical quasi-differential operator algebra of closed manifolds, but it is not an extension in the sense of the usual operator trace. Noncommutative residues on low-dimensional manifolds were discovered by Adler [1], and noncommutative residues on higher-dimensional manifolds were discovered simultaneously by Wodzicki and Guillemin [2,3]. The study of noncommutative residues has profound theoretical and practical value. Connes [4][5][6] 's research shows that non-commutative residues play an integral role in noncommutative geometry. The research work of Connes [4] and Ugalde [7] shows that non-commutative residues are closely related to conformal geometry. That is, noncommutative residues bridge conformal geometry and noncommutative geometry. Importantly, 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 [6]. This theorem was proved independently by Kastler, Kalau and Walze respectively, and is called Kastler-Kalau-Walze Theorem [8,9]. This initiated the application of noncommutative residues in gravity action under the framework of non-commutative geometry, and gave an operator theory explanation for gravity action.

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Wodzicki found that if the manifold M is not compact or manifold with boundary, the classical pseudodifferential operators algebra can not define trace. If one want to define noncommutative residue on these manifolds should have to consider another algebra structure. 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 Connes' results to the cases of 3, 4-dimensional spin manifolds with boundary and proved a Kastler-Kalau-Walze type theorem. In [13], Wang generalized the definition of lower dimensional volumes in [14] to manifolds with boundary, and found a Kastler-Kalau-Walze theorem for six-dimensional manifolds with boundary. Wang et al achieved many good conclusion in this fields. Recently, In [15], Wang et al computed W res[( + D −2 )•( + D −n+2 )] for any-dimensional manifolds with boundary, and proved a general Kastler-Kalau-Walze type theorem. Weiping Zhang introduced an elliptic differential operator-Witten deformation in [16]. In [17], we proved Kastler-Kalau-Walze type theorem for Witten deformation for 4, 6-dimensional manifolds with boundary [12]. Furthermore, we consider higher dimensional case. The motivation of this paper is to generalize the results of [15,18]

Witten Deformation
In this section, we will recall the definition of the Witten deformation D T [16].We also compute the local expression of operator D T , D 2 T and some symbols of oriented Riemannian manifold with boundary M equipped with a fixed spin structure. Let ∇ L denote the Levi-Civita connection about g M . For the fixed orthonormal frame {ẽ 1 , … ,ẽ n } , the connection matrix ( s,t ) is defined by Let (ẽ j * ), (ẽ j * ) be the exterior and interior multiplications respectively. Write The Witten deformation is defined by (2) c(� e j ) = (� e j * ) − (� e j * );c(� e j ) = (� e j * ) + (� e j * ).
By proposition 4.6 of [16], we have By [19], (d + ) 2 is expressed by Then D T can be written as By [19,20], we have then Denote l (A) the l-order symbol of an operator A. By (10) we have some symbols of operators: D T , D 2 T . (3)

Lemma 2
In the following, we will give some symbols of D −(2n−1) T , D −2n T . By formula (4.20) in [9], for n (even) we have We obtain by induction By the composition formula of psudodifferential operators, we have On the other hand So we have

Lower Dimensional Volumes of Spin Manifolds with Boundary
In order to define lower dimensional volumes of spin manifolds with boundary, we need some basic facts and formulae about Boutet de Monvel's calculus and the definition of the noncommutative residue for manifolds with boundary. such that ĝ | M = g M . To define the lower dimensional volume, some basic facts and formulae about Boutet de Monvel's calculus which can be found in Sect. 2 in [12] are needed. Let denote the Fourier transformation and Φ( + ) = r + Φ( ) (similarly define Φ( − )), where Φ( ) denotes the Schwartz space and which are orthogonal to each other. We have the following property: h ∈ H + (H − 0 ) iff h ∈ C ∞ ( ) which has an analytic extension to the lower (upper) complex half-plane {Im < 0} ({Im > 0}) such that for all nonnegative integer l, The Boutet de Monvel's algebra is denote by B , we recall the main theorem in [10]. Denote by l (A) the l-order symbol of an operator A. By (2.1.4)-(2.1.8) in [12], we get where the sum is taken over where denote the interior term and boundary term of

A Kastler-Kalau-Walze Type Theorem for Even Dimensional Manifolds with Boundary I
In this section, we will construct one kind of Kastler-Kalau-Walze type theorem for (2n + 2) dimensional manifolds with boundary. Firstly, we compute By [20], we have where W res denote noncommutative residue on minifolds with boundary, Wres denote noncommutative residue on minifolds without boundary.
for n = (2n + 2). (33) (34) We ues as shorthand of trace S(TM) . Since n = (2n + 2) , then tr[id] = dim(∧ * ( n 2 )) = 2 2n+2 2 = 2 n+1 . By the relation of the Clifford action and tr(AB) = tr (BA) , then we have following equalities: then we have So we only need to compute the boundary term W res and Let {E 1 , … , E n−1 } be an orthonormal frame field in U about g M which is parallel along geodesics and E i = In the following, we will give three instrumental Lemma. By Lemma 2.2 in [12], we have

Lemma 4 With the metric g M on M near the boundary
where = � + n x n , | � |= 1.

Denote
By the relation of the Clifford action and tr(AB) = tr(BA) , then we have following equalities.

By (29), we get
By Lemmas 3 and 4 , for i < n we have

By Lemmas 3 and 4, we have
By Lemma 5, we have

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where and Since Then By the relation of the Clifford action and tr(AB) = tr(BA) and Lemma 5, we have (57) (63) So B 3 has no contribution to case c),then

By Lemma 5, we have
Now Φ is the sum of the cases a), b) and c), then So we have where Φ is given by (72). Recall the Einstein-Hilbert action for manifolds with boundary (see [12,13]), (69) (73) where and K i,j is the second fundamental form, or extrinsic curvature. Then by Lemma A.2 in [12],

So we have
Now Φ � is the sum of the cases a), b) and c), then

So we have
where Φ � is given by (102).
Recall the Einstein-Hilbert action for manifolds with boundary (see [12,13]), where and K i,j is the second fundamental form, or extrinsic curvature. Take the metric in Sect. 2, then by Lemma A.2 in [12], K i,j (x 0 ) = −Γ n i,j (x 0 ) when i = j < 2n + 4 , otherwise is zero. Then Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.