Modified Novikov Operators and the Kastler-Kalau-Walze type theorem for manifolds with boundary

In this paper, we give two Lichnerowicz type formulas for modified Novikov operators. We prove KastlerKalau-Walze type theorems for modified Novikov operators on compact manifolds with (resp.without) boundary. We also compute the spectral action for Witten deformation on 4-dimensional compact manifolds.


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 M coincided with Dixmier's trace on pseudodifferential operators of order −dim M 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 bruteforce 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 WresðD −2 Þ in turn is essentially the second coefficient of the heat kernel expansion of D 2 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 g Wres½π + D −1 ∘ π + D −1 and g Wres½π + D −2 ∘ π + D −2 , where the two operators are symmetric; in these cases, the boundary term vanished. But for g Wres½π + D −1 ∘ π + D −3 , 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 oneform 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][15][16][17], we will compute the spectral action for the Witten deformation on 4dimensional compact manifolds in this paper.
Let ∇ L be the Levi-Civita connection about g M . In the local coordinates fx i ; 1 ≤ i ≤ ng and the fixed orthonormal frame f e e 1 ,⋯, e e n g, the connection matrix ðω s,t Þ is defined by ∇ L e e 1 ,⋯, e e n ð Þ= e e 1 ,⋯, e e n ð Þω s,t ð Þ: ð3Þ Let ϵðg e j * Þ and ιðg e j * Þ be the exterior and interior multiplications, respectively, and cð e e j Þ be the Clifford action. Suppose that ∂ i is a natural local frame on TM and ðg ij Þ 1≤i,j≤n is the inverse matrix associated with the metric matrix ðg ij Þ 1≤i,j≤n on M. Write c e e j À Á = ε g e j * À Á − ι g e j * À Á , c e e j À Á = ε g e j * À Á + ι g e j * À Á : ð4Þ The modified Novikov operatorsD andD * are defined We first establish the main theorem in this section. One has the following Lichnerowicz formulas. Theorem 1. The following equalities hold: − g e e j , ∇ TM e j θ′ , where s is the scalar curvature.
In order to prove Theorem 1, we recall the basic notions of Laplace-type operators. Let M be smooth compactoriented Riemannian n-dimensional manifolds without a 2 Advances in Mathematical Physics boundary and V ′ be a vector bundle on M. Any differential operator P of the Laplace type has locally the form where ∂ i is a natural local frame on TM, ðg ij Þ 1≤i,j≤n is the inverse matrix associated with the metric matrix ðg ij Þ 1≤i,j≤n on M, and A i and B are smooth sections of EndðV ′ Þ on M (endomorphism). If P is a Laplace-type operator with the form (7), then there is a unique connection ∇ on V ′ and a unique endomorphism E such that where ∇ L is the Levi-Civita connection on M. Moreover (with local frames of T * M and V′), where Γ j kl is the Christoffel coefficient of ∇ L . By Proposition 4.6 of [17], we have By [18], the local expression of ðd + δÞ 2 is c e e k ð Þc e e l ð Þ + 1 4 s: ð12Þ Then, the modified Novikov operatorsD andD * can be written aŝ By [7,18], we have We note that then we obtain
The noncommutative residue of a generalized Laplaciañ Δ is expressed as by [7] where Φ 2 ðΔÞ denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion ofΔ. Now letΔ =D * D . SinceD * D is a generalized Laplacian, we can supposeD * D = Δ − E, then we have where Wres denotes the noncommutative residue.

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Similarly, we have where Wres denotes the noncommutative residue.

Theorem 2.
For even n-dimensional compact-oriented manifolds without a boundary, the following equalities hold: where s is the scalar curvature.

A Kastler-Kalau-Walze-Type Theorem for 4-Dimensional Manifolds with Boundary
In this section, we prove the Kastler-Kalau-Walze-type theorem for 4-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 M be a 4-dimensional compact-oriented manifold with boundary ∂M. We assume that the metric g M on M has the following form near the boundary, where g ∂M is the metric on ∂M. hðx n Þ ∈ C ∞ ð½0, 1ÞÞ = fhj ½0,1Þ j h ∈ C ∞ ðð−ε, 1ÞÞg for some ε > 0 and satisfies hðx n Þ > 0, hð0Þ = 1 where x n denotes the normal directional coordinate. Let U ⊂ M be a collar neighborhood of ∂M which is diffeomorphic with ∂M × ½0, 1Þ. By the definition of hðx n Þ ∈ C ∞ ð½0, 1ÞÞ and hðx n Þ > 0, there existsh ∈ C ∞ ðð−ε, 1ÞÞ such that h| ½0,1Þ = h andh > 0 for some sufficiently small ε > 0. Then, there exists a metricĝ onM = M S ∂M ∂M × ð−ε, 0 which has the form on U S ∂M ∂M × ð−ε, 0 such thatĝj M = g. We fix a metricĝ on theM such that gj M = g. Let denote the Fourier transformation and ΦðR + Þ = r + ΦðRÞ (similarly define ΦðR À Þ), where ΦðRÞ denotes the Schwartz space and We define H + = FðΦðR + ÞÞ and H − 0 = FðΦðR À ÞÞ which are orthogonal to each other. We have the following property: h ∈ H + ðH − 0 Þ if and only if h ∈ C ∞ ðRÞ which has an analytic extension to the lower (upper) complex halfplane fIm ξ < 0gðfIm ξ > 0gÞ such that for all nonnegative integer l, as |ξ| ⟶ +∞, Im ξ ≤ 0ðImξ ≥ 0Þ.
An operator of order m ∈ Z and type d is a matrix where X is a manifold with boundary ∂X and E 1 , E 2 ðF 1 , F 2 Þ are vector bundles over Xð∂XÞ. Here, P : In addition, P is supposed to have the transmission property; this means that, for all j, k, α, 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 [19]. Let G and T be, respectively, the singular Green operator and the trace operator of order m and type d. K is a potential operator and S is a classical pseudodifferential operator of order m along the boundary (for detailed definition, see [13]). Denote by B m,d the collection of all operators of order m and type d, and B is the union over all m and d.
Recall B m,d is a Fréchet space. The composition of the above operator matrices yields a continuous map: The composition AA′ is obtained by multiplication of the matrices (for more details, see [19]). For example, π + P ∘ G′ and G ∘ G ′ are singular Green operators of type d ′ and Here, PP ′ is the usual composition of pseudodifferential operators, and LðP, P′Þ called the leftover term is a singular Green operator of type m′ + d. For our case, P, P′ are classical pseudodifferential operators; in other words, π + P ∈ B ∞ and π + P′ ∈ B ∞ .
Let M be an n-dimensional compact-oriented manifold with boundary ∂M. Denote by B Boutet de Monvel's algebra, we recall the main theorem in [10,20].

Theorem 3 ([20]
, Fedosov-Golse-Leichtnam-Schrohe). Let X and ∂X be connected, dim X = n ≥ 3, and denote by p, b, and s the local symbols of P, G, and S, respectively. Define: Then, (a) g Wresð½A, Formulas (2.1.4)-(2.1.8) from paper [10] still hold in the case when M 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.
where the sum is taken over In fact, for a general one-order elliptic differential operator, (41) and (42) are also correct.
Since ½σ −n ðD −p 1 −p 2 Þj M has the same expression as σ −n ðD −p 1 −p 2 Þ 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 x 0 ∈ ∂M, we choose the normal coor-

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For a general Clifford module, the conclusion of Section 2 and the Appendix in [10] is true. In our case, for ∧ * T * M, cðdx j Þ = dx j ∧−ðdx j ∧ Þ * 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 g M on M near the boundary
Lemma 5 (see [10]). With the metric g M on M near the boundary where ðω s,t Þ denotes the connection matrix of Levi-Civita connection ∇ L .
By (41) and (42), we firstly compute and the sum is taken over Locally, we can use Theorem 2 (25) to compute the interior of g So we only need to compute Ð ∂M Φ. 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 7 Advances in Mathematical Physics so By Lemma 7, we have some symbols of operators.

Lemma 8.
The following identities hold: From the remark above, we can now compute Φ (see formula (48) for the definition of Φ). We use tr as shorthand of trace. Since n = 4, then tr ∧ * T * M ½id = dim ð∧ * ð4ÞÞ = 16, since the sum is taken over then we have the following five cases: By (48), we get By Lemma 4, for i < n, then so Case 1 (i) vanishes.
By (48), we get By Lemma 8, we have By (32), (33), and the Cauchy integral formula, we have Similarly, we have, 8 Advances in Mathematical Physics By (59), then By the relation of the Clifford action and trAB = trBA, we have the equalities: By (61) and a direct computation, we have Similarly, we have Then, where Ω 3 is the canonical volume of S 3 : By (48), we get By Lemma 8, we have Similar to Case 1 (ii), we have

Theorem 9.
Let M be 4-dimensional compact-oriented manifolds with the boundary ∂M and the metric g M as above,D andD * be modified Novikov operators onM, then where s is the scalar curvature.
On the other hand, we also prove the Kastler-Kalau-Walze-type theorem for 4-dimensional manifolds with a boundary associated toD 2 . By (41) and (42), we will computẽ where and the sum is taken over Locally, we can use Theorem 2 (26) to compute the interior of g So we only need to compute Ð ∂M b Φ. From the remark above, now we can compute b Φ (see formula (110) for the def-inition of b Φ). We use tr as shorthand of trace. Since n = 4, then tr ∧ * T * M ½id = dim ð∧ * ð4ÞÞ = 16, since the sum is taken over r + l − k − j − |α| = −3, r ≤ −1, l ≤ −1, then we have the following five cases: Case 1. (i) r = −1, l = −1, k = j = 0, and |α| = 1.

Theorem 10.
Let M be a 4-dimensional compact-oriented manifold with the boundary ∂M and the metric g M as above andD be a modified Novikov operator onM, then where s is the scalar curvature.

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 ð So we only need to compute Ð ∂M Ψ. Let us now turn to compute the specification ofD * DD * .

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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.

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Lemma 12. The following identities hold: From the remark above, now we can compute Ψ (see formula (135) for the definition of Ψ). We use tr as shorthand of trace. Since n = 6, then tr ∧ * T * M ½id = 64. Since the sum is taken over r Ψ as the sum of the following five cases: By (135), we get By Lemma 12, for i < n, we have By (135), we have By Lemma 12 and direct calculations, we have Since n = 6, tr½−id = −64. By the relation of the Clifford action and trAB = trBA, then tr By (62), (146), and (147), we get Then, we obtain where Ω 4 is the canonical volume of S 4 :
By (135), we have By Lemmas 11 and 12, we have

The Spectral Action for Witten Deformation
In this section, we will compute the spectral action for the Witten deformation. Let ðM, g M Þ be an n-dimensional compact-oriented Riemannian manifold. Now we will recall the definition of the Witten deformation D θ (see details in [17]). Let ∇ L denote the Levi-Civita connection about g M which is a Riemannian metric of M. In the local coordinates fx i , 1 ≤ i ≤ ng and the fixed orthonormal frame f e e 1 ,⋯, e e n g, the connection matrix ðω s,t Þ is defined by ∇ L e e 1 ,⋯, e e n ð Þ= e e 1 ,⋯, e e n ð Þω s,t ð Þ: ð201Þ Let εðg e j * Þ and ιðg e j * Þ be the exterior and interior multiplications, respectively. The Witten deformation is defined by Let g ij = gðdx i , dx j Þ, ξ = ∑ k ξ j dx j , and ∇ L ∂ i ∂ j = ∑ k Γ k ij ∂ k , we denote For a smooth vector field X on M, let cðXÞ denote the Clifford action. Since E is globally defined on M, we can perform computations of E in normal coordinates. Taking normal coordinates about x 0 , then σ i ðx 0 Þ = 0, a i ðx 0 Þ = 0, ∂ j ½cð∂ j Þðx 0 Þ = 0, Γ k ðx 0 Þ = 0, and g ij ðx 0 Þ = δ j i , so that For the Witten deformation D θ , 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 D θ in the interval ½−∧, ∧ with ∧∈ℝ + . It is expressed as Here, tr denotes the operator trace in the L 2 completion of ΓðM, SðTMÞÞ andF : ℝ + ⟶ ℝ + is a cut-off function with support in the interval ½0, 1 which is constant near the origin. By Lemma 1.7.4 in [21], we have the heat trace asymptotics, for t ⟶ 0, One uses the Seeley-DeWitt coefficients a 2m ðD 2 θ Þ and t = ∧ −2 to obtain asymptotics for the spectral action when dim M = 4, with the first three moments of the cut-off function which are given by F 4 = Ð ∞ 0 sFðsÞds, F 2 =