Abstract
In this paper, we give two Lichnerowicz-type formulas for modified Novikov operators. We prove Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) a boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.
1. Introduction
As has been well known, the noncommutative residue plays a prominent role in noncommutative geometry which is found in [1, 2]. For this reason, it has been studied extensively by geometers. Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy in [3]. Connes showed us that the noncommutative residue on a compact manifold coincided with Dixmier’s trace on pseudodifferential operators of order in [4]. Connes has also observed that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action, which is called the Kastler-Kalau-Walze-type theorem now. Kastler [5] gave a brute-force proof of this theorem. Kalau and Walze proved this theorem in the normal coordinate system simultaneously in [6]. Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator in turn is essentially the second coefficient of the heat kernel expansion of in [7].
On the other hand, Wang generalized Connes’ results to the case of manifolds with a boundary in [8, 9] and proved the Kastler-Kalau-Walze-type theorem for the Dirac operator and the signature operator on lower-dimensional manifolds with a boundary [10]. In [10, 11], Wang computed and , where the two operators are symmetric; in these cases, the boundary term vanished. But for , Wang got a nonvanishing boundary term [12] and gave a theoretical explanation for gravitational action on the boundary. In others words, Wang provides a kind of method to study the Kastler-Kalau-Walze-type theorem for manifolds with a boundary. In [13], López and his collaborators introduced an elliptic differential operator which is called the Novikov operator. The motivation of this paper is to prove the Kastler-Kalau-Walze-type theorem for Novikov operators on manifolds with a boundary. In [14], Iochum and Levy computed heat kernel coefficients for Dirac operators with one-form perturbations and proved that there are no tadpoles for compact spin manifolds without a boundary. In [15], Sitarz and Zajac investigated the spectral action for scalar perturbations of Dirac operators. In [16], Hanisch and his collaborators derived a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional spin manifolds with totally antisymmetric torsion. In [17], Zhang introduced an elliptic differential operator which is called the Witten deformation. Motivated by [14–17], we will compute the spectral action for the Witten deformation on 4-dimensional compact manifolds in this paper.
The framework of this paper is organized as follows. Firstly, in Section 2, we give the definition of modified Novikov operators and the Lichnerowicz formulas associated with modified Novikov operators. We study the symbols of some operators associated with modified Novikov operators; by using symbols of operators associated with modified Novikov operators, we can prove the Kastler-Kalau-Walze-type theorem for manifolds with a boundary in Section 3 and in Section 4. In Section 5, we compute the spectral action for the Witten deformation on 4-dimensional compact manifolds.
2. Modified Novikov Operators and Their Lichnerowicz Formula
In this section, we firstly recall the definition of a Novikov operator (see details in [8]). Let be an -dimensional () oriented compact Riemannian manifold with a Riemannian metric . The de Rham derivative is an elliptic differential operator on . Then, we have the de Rham coderivative and the symmetric operators and (the Laplacian).
With more generality, we take any closed . For the sake of simplicity, we assume that is real. Then, we have the Novikov operators defined by , depending on in [8], where is the real part of , is the imaginary part of , , and . In this paper, we consider the modified Novikov operators; for θ, , we define that where and , where , .
Let be the Levi-Civita connection about . In the local coordinates and the fixed orthonormal frame , the connection matrix is defined by
Let and be the exterior and interior multiplications, respectively, and be the Clifford action. Suppose that is a natural local frame on and is the inverse matrix associated with the metric matrix on . Write
The modified Novikov operators and are defined by
We first establish the main theorem in this section. One has the following Lichnerowicz formulas.
Theorem 1. The following equalities hold: where is the scalar curvature.
In order to prove Theorem 1, we recall the basic notions of Laplace-type operators. Let be smooth compact-oriented Riemannian -dimensional manifolds without a boundary and be a vector bundle on . Any differential operator of the Laplace type has locally the form where is a natural local frame on , is the inverse matrix associated with the metric matrix on , and and are smooth sections of on (endomorphism). If is a Laplace-type operator with the form (7), then there is a unique connection on and a unique endomorphism such that where is the Levi-Civita connection on . Moreover (with local frames of and ), and are related to , , and through where is the Christoffel coefficient of .
By Proposition 4.6 of [17], we have
By [18], the local expression of is
Let , , and , we denote that
Then, the modified Novikov operators and can be written as
We note that then we obtain
Similarly, we have
By (8), (9), (10), and (17), we have
For a smooth vector field on , let denote the Clifford action. Since is globally defined on , taking normal coordinates at , we have , , , , and , so that
Similarly, we have which, together with (7), yields Theorem 1.
The noncommutative residue of a generalized Laplacian is expressed as by [7] where denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of . Now let . Since is a generalized Laplacian, we can suppose , then we have where denotes the noncommutative residue.
Similarly, we have where denotes the noncommutative residue.
Theorem 2. For even -dimensional compact-oriented manifolds without a boundary, the following equalities hold: where is the scalar curvature.
3. A Kastler-Kalau-Walze-Type Theorem for 4-Dimensional Manifolds with Boundary
In this section, we prove the Kastler-Kalau-Walze-type theorem for -dimensional compact-oriented manifold with a boundary. We firstly give some basic facts and formulas about Boutet de Monvel’s calculus and the definition of the noncommutative residue for manifolds with a boundary (see details in Section 2 in [10]).
Let be a 4-dimensional compact-oriented manifold with boundary . We assume that the metric on has the following form near the boundary, where is the metric on . for some and satisfies , where denotes the normal directional coordinate. Let be a collar neighborhood of which is diffeomorphic with . 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 denote 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: if and only if which has an analytic extension to the lower (upper) complex half-plane such that for all nonnegative integer , as , .
Let be the space of all polynomials and and . Denote by , respectively, the projection on . For calculations, we take ( is a dense set in the topology of ). Then, on , where is a Jordan close curve including surrounding all the singularities of in the upper half-plane and . Similarly, define on ,
So, . For , , and for , .
An operator of order and type is a matrix where is a manifold with boundary and are vector bundles over . Here, is a classical pseudodifferential operator of order on , where is an open neighborhood of and . has an extension: , where is the dual space of . Let denote extension by zero from to and denote the restriction from to , then define
In addition, is supposed to have the transmission property; this means that, for all , the homogeneous component of order in the asymptotic expansion of the symbol of in local coordinates near the boundary satisfies then by [19]. Let and be, respectively, the singular Green operator and the trace operator of order and type . is a potential operator and is a classical pseudodifferential operator of order along the boundary (for detailed definition, see [13]). Denote by the collection of all operators of order and type , and is the union over all and .
Recall is a Fréchet space. The composition of the above operator matrices yields a continuous map: Write
The composition is obtained by multiplication of the matrices (for more details, see [19]). For example, and are singular Green operators of type and
Here, is the usual composition of pseudodifferential operators, and called the leftover term is a singular Green operator of type . For our case, are classical pseudodifferential operators; in other words, and .
Let be an -dimensional compact-oriented manifold with boundary . Denote by Boutet de Monvel’s algebra, we recall the main theorem in [10, 20].
Theorem 3 ([20], Fedosov-Golse-Leichtnam-Schrohe). Let and be connected, , and denote by , , and the local symbols of , and , respectively. Define: Then, (a) , for any ; (b) it is a unique continuous trace on .
Formulas (2.1.4)–(2.1.8) from paper [10] still hold in the case when is an oriented (not necessarily spin) manifold, since these formulas come from a composition of pseudodifferential operators in Boutet de Monvel algebra (see p.23 in [20] and p.740 in [8]). These formulas hold for general pseudodifferential operators. Thus, these formulas hold for the modified Novikov operator.
By (2.1.4)–(2.1.8) in [10], we get where the sum is taken over , , . denotes the de Rham operator . In fact, for a general one-order elliptic differential operator, (41) and (42) are also correct.
Since has the same expression as in the case of manifolds without a boundary, locally, we can use the computations [5, 6, 10, 19] to compute the first term.
For any fixed point , we 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
For a general Clifford module, the conclusion of Section 2 and the Appendix in [10] is true. In our case, for , is the Clifford module, so we can use the conclusion of Section 2 and the Appendix in [10]. We will give the following three lemmas as computation tools.
Lemma 4 (see [10]). With the metric on near the boundary where .
Lemma 5 (see [10]). With the metric on near the boundary where denotes the connection matrix of Levi-Civita connection .
Lemma 6 (see [10]). When , then in other cases, .
By (41) and (42), we firstly compute where and the sum is taken over , , , denotes the modified Novikov operators.
Locally, we can use Theorem 2 (25) to compute the interior of ; we have
So we only need to compute . Let us now turn to compute the symbols of some operators. By (13)–(18), some operators have the following symbols.
Lemma 7. The following identities hold:
Write
By the composition formula of pseudodifferential operators, we have so
By Lemma 7, we have some symbols of operators.
Lemma 8. The following identities hold:
From the remark above, we can now compute (see formula (48) for the definition of ). We use as shorthand of . Since , then , since the sum is taken over , , , then we have the following five cases:
Case 1. (i) , , , and .
By (48), we get
By Lemma 4, for , then so Case 1 (i) vanishes.
Case 1. (ii) , , , and .
By (48), we get
By Lemma 8, we have
By (32), (33), and the Cauchy integral formula, we have
Similarly, we have,
By (59), then
By the relation of the Clifford action and , we have the equalities:
By (61) and a direct computation, we have
Similarly, we have
Then, where is the canonical volume of
Case 1. (iii) , , , and .
By (48), we get
By Lemma 8, we have
Similar to Case 1 (ii), we have
So we have
Case 2. , , and .
By (48), we get
By Lemma 8, we have where
We denote
Then,
By direct calculation, we have
Since then by the relation of the Clifford action and , we have the equalities:
Since
We note that , , so has no contribution for computing Case 2.
By direct calculation, we have where
By (80) and (83), we have where and .
Similar to (81), we have
Similar to (83), we have
Case 3. , , and .
By (48), we get
By (32) and (33) and Lemma 8, we have
Since where then
By direct calculation, we have
We denote then
By (79), we have
We note that , , so has no contribution for computing Case 3.
By (93) and (100), we have then
By and (81), we have
So we have
Since is the sum of Cases 1–3, so .
Theorem 9. Let be -dimensional compact-oriented manifolds with the boundary and the metric as above, and be modified Novikov operators on , then where is the scalar curvature.
On the other hand, we also prove the Kastler-Kalau-Walze-type theorem for -dimensional manifolds with a boundary associated to . By (41) and (42), we will compute where and the sum is taken over , , .
Locally, we can use Theorem 2 (26) to compute the interior of ; we have
So we only need to compute . From the remark above, now we can compute (see formula (110) for the definition of ). We use as shorthand of . Since , then , since the sum is taken over , , , then we have the following five cases:
Case 1. (i) , , , and .
By (110), we get
Case 1. (ii) , , , and .
By (110), we get
Case 1. (iii) , , , and .
By (110), we get
By Lemma 8, we have . By (55)-(71), so Case 1 vanishes.
Case 2. , , and .
By (110), we get
By Lemma 8, we have . By (72)–(91), we have where is the canonical volume of
Case 3. , , and .
By (110), we get
By (33) and (32) and Lemma 8, we have
Since where then
By direct calculation, we have
We denote then
By (79), we have
We note that , , so has no contribution for computing Case 3.
By (118) and (125), we have then
By and (79), we have
So we have
Since is the sum of Cases 1–3, so .
Theorem 10. Let be a -dimensional compact-oriented manifold with the boundary and the metric as above and be a modified Novikov operator on , then where is the scalar curvature.
4. A Kastler-Kalau-Walze-Type Theorem for 6-Dimensional Manifolds with Boundary
In this section, we prove the Kastler-Kalau-Walze-type theorems for 6-dimensional manifolds with a boundary. An application of (2.1.4) in [12] shows that where and the sum is taken over , , .
Locally, we can use Theorem 2 (25) to compute the interior term of (134); we have
So we only need to compute . Let us now turn to compute the specification of .
Then, we obtain
Lemma 11. The following identities hold:
Write
By the composition formula of pseudodifferential operators, we have by (140), we have
By Lemma 11, we have some symbols of operators.
Lemma 12. The following identities hold:
From the remark above, now we can compute (see formula (135) for the definition of ). We use as shorthand of . Since , then . Since the sum is taken over , , , then we have the as the sum of the following five cases:
Case 1. (i) , , , and .
By (135), we get
By Lemma 12, for , we have so Case 1 (i) vanishes.
Case 1. (ii) , , , and .
By (135), we have
By Lemma 12 and direct calculations, we have
Since , . By the relation of the Clifford action and , then
By (62), (146), and (147), we get
Then, we obtain where is the canonical volume of
Case 1. (iii) , , , and .
By (135), we have
By Lemma 12 and direct calculations, we have
Combining (69) and (152), we have
Then,
Case 2. , , and .
By (135), we have
In the normal coordinate, and , if ; , if . So by Lemma A.2 in [10], we have and for . By the definition of and Lemma 2.3 in [10], we have and for . By Lemma 12, we obtain
By direct calculation and the relation of the Clifford action and , we then have equalities:
Then,
So, we have
Case 3. , , and .
By (135), we have
By Lemmas 11 and 12, we have where
On the other hand,
By (163), (28), and (32), we have
We denote
Then, we obtain
Furthermore,
By the relation of the Clifford action and , we then have equalities:
Then, we have
By direct calculation, we have where
Similarly, we have
By we have
Now is the sum of Cases 1–3, then
Theorem 13. Let be a -dimensional compact-oriented manifold with the boundary and the metric as above and and be modified Novikov operators on , then where is the scalar curvature.
On the other hand, we prove the Kastler-Kalau-Walze-type theorem for a -dimensional manifold with a boundary associated with . An application of (2.1.4) in [12] shows that where denotes a noncommutative residue on manifolds with a boundary, and the sum is taken over , , .
Locally, we can use Theorem 2 (26) to compute the interior term of (181); we have
So we only need to compute . Let us now turn to compute the specification of .
Then, we obtain
Lemma 14. The following identities hold:
Write
By the composition formula of pseudodifferential operators, we have by (186), we have
By (183)–(187), we have some symbols of operators.
Lemma 15. The following identities hold:
From the remark above, we can now compute (see formula (181) for the definition of ). We use as shorthand of . Since , then . Since the sum is taken over , , , then we have the as the sum of the following five cases:
Case 1. (i) , , , and .
By (181), we get
Case 1. (ii) , , , and .
By (181), we have
Case 1. (iii) , , , and .
By (181), we have
By Lemmas 12 and 15, we have ; by (143)–(154), we obtain where is the canonical volume of
Case 2. , , .
By (181), we have
In the normal coordinate, and , if ; , if . So by Lemma A.2 in [10], we have and for . By the definition of and Lemma 2.3 in [10], we have and for . By Lemma 15, we obtain
Case 3. , , .
By (181), we have
By Lemmas 12 and 15, we have ; by (162)–(177), we obtain
Now is the sum of Cases 1–3, then
Theorem 16. Let be a -dimensional compact-oriented manifold with the boundary and the metric as above and be a modified Novikov operator on , then where is the scalar curvature.
5. The Spectral Action for Witten Deformation
In this section, we will compute the spectral action for the Witten deformation. Let be an -dimensional compact-oriented Riemannian manifold. Now we will recall the definition of the Witten deformation (see details in [17]).
Let denote the Levi-Civita connection about which is a Riemannian metric of . In the local coordinates and the fixed orthonormal frame , the connection matrix is defined by
Let and be the exterior and interior multiplications, respectively. The Witten deformation is defined by
By Proposition 4.6 of [17], we have
Let , , and , we denote
For a smooth vector field on , let denote the Clifford action. Since is globally defined on , we can perform computations of in normal coordinates. Taking normal coordinates about , then , , , , and , so that
For the Witten deformation , we will compute the spectral action for it on a 4-dimensional compact manifold. We will calculate the bosonic part of the spectral action for the Witten deformation. It is defined to be the number of eigenvalues of in the interval with . It is expressed as
Here, denotes the operator trace in the completion of and is a cut-off function with support in the interval which is constant near the origin. By Lemma 1.7.4 in [21], we have the heat trace asymptotics, for ,
One uses the Seeley-DeWitt coefficients and to obtain asymptotics for the spectral action when , with the first three moments of the cut-off function which are given by , , and .
We use Theorem 4.1.6 in [17] to obtain the first three coefficients of the heat trace asymptotics:
By the Clifford action and cyclicity of the trace, we have
So we obtain
And we have
And is globally defined, so we only compute it in normal coordinates about and the local orthonormal frame obtained by parallel transport along geodesics from . Then,
Then, we have
So we have
Proposition 17. The following equality holds: where is the scalar curvature.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by NSFC (11771070).