# Modified Novikov Operators and the Kastler-Kalau-Walze-Type Theorem for Manifolds with Boundary

Authors: Sining Wei; Yong Wang
Journal: Advances in Mathematical Physics
 (2020)

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## 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.


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## Body

## 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 M coincided with Dixmier’s trace on pseudodifferential operators of order −dimM 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 WresD−2 in turn is essentially the second coefficient of the heat kernel expansion of D2 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 Wres~π+D−1∘π+D−1 and Wres~π+D−2∘π+D−2, where the two operators are symmetric; in these cases, the boundary term vanished. But for Wres~π+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 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 Section2, 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 M be an n-dimensional (n≥3) oriented compact Riemannian manifold with a Riemannian metric gM. The de Rham derivative d is an elliptic differential operator on C∞M;∧∗T∗M. Then, we have the de Rham coderivative δ=d∗ and the symmetric operators D=d+δ and Δ=D2=dδ+δd (the Laplacian).With more generality, we take any closedθ∈C∞M;T∗M. For the sake of simplicity, we assume that θ is real. Then, we have the Novikov operators defined by θ, depending on z∈ℂ in [8],
(1)dz=d+zθ∧,δz=dz∗=δ+z¯θ∧∗,Dz=dz+δz=d+δ+zθ∧+z¯θ∧∗=d+δ+Rezθ∧+Rezθ∧∗+iImzθ∧−Imzθ∧∗=d+δ+Rezθ∧+θ∧∗+iImzθ∧−θ∧∗=d+δ+Rezc¯θ+iImzcθ,where Rez is the real part of z, Imz is the imaginary part of z, c¯θ=θ∗∧+θ∧∗, and cθ=θ∗∧−θ∧∗. In this paper, we consider the modified Novikov operators; for θ, θ′∈ΓTM, we define that
(2)D^=d+δ+c¯θ+cθ′,D^∗=d+δ+c¯θ−cθ′,where c¯θ=θ∗∧+θ∧∗ and cθ′=θ′∗∧−θ′∧∗, where θ∗=gθ,·, θ′∗=gθ′,·.Let∇L be the Levi-Civita connection about gM. In the local coordinates xi;1≤i≤n and the fixed orthonormal frame e1~,⋯,en~, the connection matrix ωs,t is defined by
(3)∇Le1~,⋯,en~=e1~,⋯,en~ωs,t.Letϵej∗~ and ιej∗~ be the exterior and interior multiplications, respectively, and cej~ be the Clifford action. Suppose that ∂i is a natural local frame on TM and gij1≤i,j≤n is the inverse matrix associated with the metric matrix gij1≤i,j≤n on M. Write
(4)cej~=εej∗~−ιej∗~,c¯ej~=εej∗~+ιej∗~.The modified Novikov operatorsD^ and D^∗ are defined by
(5)D^=d+δ+c¯θ+cθ′=∑i=1ncei~ei~+14∑s,tωs,tei~c¯es~c¯et~−ces~cet~+c¯θ+cθ′,D^∗=d+δ+c¯θ−cθ′=∑i=1ncei~ei~+14∑s,tωs,tei~c¯es~c¯et~−ces~cet~+c¯θ−cθ′.We first establish the main theorem in this section. One has the following Lichnerowicz formulas.Theorem 1.
The following equalities hold:(6)D^∗D^=−gij∇∂i∇∂j−∇∇∂iL∂j−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TMθ+14s−cθ′c¯θ+c¯θcθ′+θ2+θ′2+14∑icei~cθ′−cθ′cei~2−gej~,∇ej~TMθ′,D^2=−gij∇∂i∇∂j−∇∇∂iL∂j−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TMθ+14s+cθ′c¯θ+c¯θcθ′+θ2−θ′2+14∑icei~cθ′+cθ′cei~2−12c∇ej~TMθ′cej~−cej~c∇ej~TMθ′,where s is the scalar curvature.In order to prove Theorem1, we recall the basic notions of Laplace-type operators. Let M be smooth compact-oriented Riemannian n-dimensional manifolds without a boundary and V′ be a vector bundle on M. Any differential operator P of the Laplace type has locally the form
(7)P=−gij∂i∂j+Ai∂i+B,where ∂i is a natural local frame on TM, gij1≤i,j≤n is the inverse matrix associated with the metric matrix gij1≤i,j≤n on M, and Ai and B are smooth sections of EndV′ 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
(8)P=−gij∇∂i∇∂j−∇∇∂iL∂j+E,where ∇L is the Levi-Civita connection on M. Moreover (with local frames of T∗M and V′), ∇∂i=∂i+ωi and E are related to gij, Ai, and B through
(9)ωi=12gijAi+gklΓkljid,(10)E=B−gij∂iωj+ωiωj−ωkΓijk,where Γklj is the Christoffel coefficient of ∇L.By Proposition 4.6 of [17], we have
(11)d+δ+c¯θ2=d+δ2+∑icei~c¯∇ei~TMθ+θ2.By [18], the local expression of d+δ2 is
(12)d+δ2=−△0−18∑ijklRijklc¯ei~c¯ej~cek~cel~+14s.Letgij=gdxi,dxj, ξ=∑kξjdxj, and ∇∂iL∂j=∑kΓijk∂k, we denote that
(13)σi=−14∑s,tωs,tei~ces~cet~,ai=14∑s,tωs,tei~c¯es~c¯et~,ξj=gijξi,Γk=gijΓijk,σj=gijσi,aj=gijai.Then, the modified Novikov operatorsD^ and D^∗ can be written as
(14)D^=∑i=1ncei~ei~+ai+σi+c¯θ+cθ′,D^∗=∑i=1ncei~ei~+ai+σi+c¯θ−cθ′.By [7, 18], we have
(15)−△0=Δ=−gij∇iL∇jL−Γijk∇kL.We note that(16)D^∗D^=d+δ+c¯θ2+d+δcθ′+c¯θcθ′−cθ′d+δ−cθ′c¯θ+θ′2,−cθ′d+δ+d+δcθ′=∑i,jgi,jc∂icθ′−cθ′c∂i∂j−∑i,jgi,jcθ′c∂iσi+cθ′c∂iai+c∂i∂icθ′+c∂iσicθ′+c∂iaicθ′,then we obtain
(17)D^∗D^=−∑i,jgi,j∂i∂j+2σi∂j+2ai∂j−Γi,jk∂k+∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak+∑i,jgi,jc∂icθ′−cθ′c∂i∂j−∑i,jgi,jcθ′c∂iσi+cθ′c∂iai−c∂i∂icθ′−c∂iσicθ′−c∂iaicθ′−18∑ijklRijklc¯ei~c¯ej~cek~cel~+14s+∑icei~c¯∇ei~TMθ+θ2+θ′2+c¯θcθ′−cθ′c¯θ.Similarly, we have(18)D^2=−∑i,jgi,j∂i∂j+2σi∂j+2ai∂j−Γi,jk∂k+∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak+∑i,jgi,jc∂icθ′+cθ′c∂i∂j+∑i,jgi,jcθ′c∂iσi+cθ′c∂iai+c∂i∂icθ′+c∂iσicθ′+c∂iaicθ′−18∑ijklRijklc¯ei~c¯ej~cek~cel~+14s+∑icei~c¯∇ei~TMθ+θ2−θ′2+c¯θcθ′+cθ′c¯θ.By (8), (9), (10), and (17), we have
(19)ωiD^∗D^=σi+ai−12c∂icθ′−cθ′c∂i,ED^∗D^=−c∂iσicθ′−c∂iaicθ′+18∑ijklRijklc¯ei~c¯ej~cek~cel~−∑icei~c¯∇ei~TMθ−θ2−θ′2−14s+cθ′c¯θ+cθ′c∂iσi+cθ′c∂iai−c∂i∂icθ′+12∂jc∂jcθ′−cθ′c∂j−12c∂jcθ′−cθ′c∂jσj+aj−gij4c∂icθ′−cθ′c∂ic∂jcθ′−cθ′c∂j−12Γkc∂kcθ′−cθ′c∂k−c¯θcθ′−12σj+ajc∂jcθ′−cθ′c∂j.For a smooth vector fieldY on M, let cY denote the Clifford action. Since E is globally defined on M, taking normal coordinates at x0, we have σix0=0, aix0=0, ∂jc∂jx0=0, Γkx0=0, and gijx0=δij, so that
(20)ED^∗D^x0=18∑ijklRijklc¯ei~c¯ej~cek~cel~−∑icei~c¯∇ei~TMθ−14s+cθ′c¯θ−c¯θcθ′−θ2−θ′2−14∑icei~cθ′−cθ′cei~2−12cej~ej~cθ′+ej~cθ′cej~=18∑ijklRijklc¯ei~c¯ej~cek~cel~−∑icei~c¯∇ei~TMθ−14s+cθ′c¯θ−c¯θcθ′−θ2−θ′2−14∑icei~cθ′−cθ′cei~2+gej~,∇ej~TMθ′.Similarly, we have(21)ED^2x0=18∑ijklRijklc¯ei~c¯ej~cek~cel~−∑icei~c¯∇ei~TMθ−14s−cθ′c¯θ−c¯θcθ′−θ2+θ′2−14∑icei~cθ′+cθ′cei~2+12c∇ej~TMθ′cej~−cej~c∇ej~TMθ′,which, together with (7), yields Theorem 1.The noncommutative residue of a generalized LaplacianΔ~ is expressed as by [7]
(22)n−2Φ2Δ~=4π−n/2Γn2res~Δ~−n/2+1,where Φ2Δ~ denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of Δ~. Now let Δ~=D^∗D^. Since D^∗D^ is a generalized Laplacian, we can suppose D^∗D^=Δ−E, then we have
(23)WresD^∗D^−n−2/2=n−24πn/2n/2−1!∫Mtr16s+ED^∗D^dVolM,where Wres denotes the noncommutative residue.Similarly, we have(24)WresD^2−n−2/2=n−24πn/2n/2−1!∫Mtr16s+ED^2dVolM,where Wres denotes the noncommutative residue.Theorem 2.
For evenn-dimensional compact-oriented manifolds without a boundary, the following equalities hold:
(25)WresD^∗D^−n−2/2=n−24πn/2n/2−1!∫M2n−112s−θ2+n−2θ′2+gej~,∇ej~TMθ′dVolM,(26)WresD^2−n−2/2=n−24πn/2n/2−1!∫M2n−112s−θ2dVolM,where s 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 for4-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]).LetM be a 4-dimensional compact-oriented manifold with boundary ∂M. We assume that the metric gM on M has the following form near the boundary,
(27)gM=1hxng∂M+dxn2,where g∂M is the metric on ∂M. hxn∈C∞0,1=h~0,1h~∈C∞−ε,1 for some ε>0 and satisfies hxn>0, h0=1 where xn denotes the normal directional coordinate. Let U⊂M be a collar neighborhood of ∂M which is diffeomorphic with ∂M×0,1. By the definition of hxn∈C∞0,1 and hxn>0, there exists h~∈C∞−ε,1 such that h~∣0,1=h and h~>0 for some sufficiently small ε>0. Then, there exists a metric g^ on M^=M⋃∂M∂M×−ε,0 which has the form on U⋃∂M∂M×−ε,0
(28)g^=1h~xng∂M+dxn2,such that g^M=g. We fix a metric g^ on the M^ such that g^M=g.Let(29)F:L2Rt⟶L2Rv,Fuv=∫e−ivtutdt,denote the Fourier transformation and ΦR+¯=r+ΦR (similarly define ΦR−¯), where ΦR denotes the Schwartz space and
(30)r+:C∞R⟶C∞R+¯,f⟶fR+¯;R+¯=x≥0;x∈R.We defineH+=FΦR+¯ and H0−=FΦR−¯ which are orthogonal to each other. We have the following property: h∈H+H0− if and only if h∈C∞R which has an analytic extension to the lower (upper) complex half-plane Imξ<0Imξ>0 such that for all nonnegative integer l,
(31)dlhdξlξ~∑k=1∞dldξlckξk,as ∣ξ∣⟶+∞, Imξ≤0Imξ≥0.LetH′ be the space of all polynomials and H−=H0−⊕H′ and H=H+⊕H−. Denote by π+π−, respectively, the projection on H+H−. For calculations, we take H=H~=rationalfunctionshavingnopolesontherealaxis (H~ is a dense set in the topology of H). Then, on H~,
(32)π+hξ0=12πilimu→0−∫Γ+hξξ0+iu−ξdξ,where Γ+ is a Jordan close curve including Imξ>0 surrounding all the singularities of h in the upper half-plane and ξ0∈R. Similarly, define π′ on H~,
(33)π′h=12π∫Γ+hξdξ.So,π′H−=0. For h∈H∩L1R, π′h=1/2π∫Rhvdv, and for h∈H+∩L1R, π′h=0.An operator of orderm∈Z and type d is a matrix
(34)A=π+P+GKTS:C∞X,E1⊕C∞∂X,F1⟶C∞X,E2⊕C∞∂X,F2.where X is a manifold with boundary ∂X and E1,E2F1,F2 are vector bundles over X∂X. Here, P:C0∞Ω,E1¯⟶C∞Ω,E2¯ is a classical pseudodifferential operator of order m on Ω, where Ω is an open neighborhood of X and Ei¯∣X=Ei i=1,2. P has an extension: E′Ω,E1¯⟶D′Ω,E2¯, where E′Ω,E1¯D′Ω,E2¯ is the dual space of C∞Ω,E1¯C0∞Ω,E2¯. Let e+:C∞X,E1⟶E′Ω,E1¯ denote extension by zero from X to Ω and r+:D′Ω,E2¯⟶D′Ω,E2 denote the restriction from Ω to X, then define
(35)π+P=r+Pe+:C∞X,E1⟶D′Ω,E2.In addition,P is supposed to have the transmission property; this means that, for all j,k,α, the homogeneous component pj of order j in the asymptotic expansion of the symbol p of P in local coordinates near the boundary satisfies
(36)∂xnk∂ξ′αpjx′,0,0,+1=−1j−∣α∣∂xnk∂ξ′αpjx′,0,0,−1,then π+P:C∞X,E1⟶C∞X,E2 by [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 Bm,d the collection of all operators of order m and type d, and B is the union over all m and d.RecallBm,d is a Fréchet space. The composition of the above operator matrices yields a continuous map: Bm,d×Bm′,d′⟶Bm+m′,maxm′+d,d′. Write
(37)A=π+P+GKTS∈Bm,d,A′=π+P′+G′K′T′S′∈Bm′,d′.The compositionAA′ 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
(38)π+P∘π+P′=π+PP′+LP,P′.Here,PP′ is the usual composition of pseudodifferential operators, and LP,P′ 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∞ .LetM 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).
LetX and ∂X be connected, dimX=n≥3,
(39)A=π+P+GKTS∈B,and denote by p, b, and s the local symbols of P,G, and S, respectively. Define:
(40)Wres~A=∫X∫StrEp−nx,ξσξdx+2π∫∂X∫S′trEtrb−nx′,ξ′+trFs1−nx′,ξ′σξ′dx′.
Then, (a)Wres~A,B=0, for any A,B∈B; (b) it is a unique continuous trace on B/B−∞.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.By (2.1.4)–(2.1.8) in [10], we get
(41)Wres~π+D−p1∘π+D−p2=∫M∫∣ξ∣=1trace∧∗T∗Mσ−nD−p1−p2σξdx+∫∂MΦ,(42)Φ=∫ξ′=1∫−∞+∞∑j,k=0∞∑−iα+j+k+1α!j+k+1!×trace∧∗T∗M∂xnj∂ξ′α∂ξnkσr+D−p1x′,0,ξ′,ξn×∂x′α∂ξnj+1∂xnkσlD−p2x′,0,ξ′,ξndξnσξ′dx′,where the sum is taken over r+l−k−α−j−1=−n, r≤−p1, l≤−p2. D denotes the de Rham operator d+δ. In fact, for a general one-order elliptic differential operator, (41) and (42) are also correct.Sinceσ−nD−p1−p2M has the same expression as σ−nD−p1−p2 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 pointx0∈∂M, we choose the normal coordinates U of x0 in ∂M (not in M) and compute Φx0 in the coordinates U~=U×0,1⊂M and the metric 1/hxng∂M+dxn2. The dual metric of gM on U~ is hxng∂M+dxn2. Write gijM=gM∂/∂xi,∂/∂xj and gMij=gMdxi,dxj, then
(43)gi,jM=1hxngi,j∂M001,gMi,j=hxng∂Mi,j001,∂xsgij∂Mx0=0,1≤i,j≤n−1,gijMx0=δij.For a general Clifford module, the conclusion of Section2 and the Appendix in [10] is true. In our case, for ∧∗T∗M, cdxj=dxj∧−dxj∧∗ 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 metricgM on M near the boundary
(44)∂xjξgM2x0=0,ifj<n,h′0ξ′g∂M2,ifj=n,∂xjcξx0=0,ifj<n,∂xncξ′x0,ifj=n,where ξ=ξ′+ξndxn.Lemma 5 (see [10]).
With the metricgM on M near the boundary
(45)ωs,tei~x0=ωn,iei~x0=12h′0,ifs=n,t=i,i<n,ωi,nei~x0=−12h′0,ifs=i,t=n,i<n;ωs,tei~x0=0,othercases,where ωs,t denotes the connection matrix of Levi-Civita connection ∇L.Lemma 6 (see [10]).
Wheni<n, then
(46)Γiinx0=12h′0;Γniix0=−12h′0;Γinix0=−12h′0,in other cases, Γstix0=0.By (41) and (42), we firstly compute
(47)Wres~π+D^−1∘π+D^∗−1=∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^∗D^−1σξdx+∫∂MΦ,where
(48)Φ=∫ξ′=1∫−∞+∞∑j,k=0∞∑−iα+j+k+1α!j+k+1!×trace∧∗T∗M∂xnj∂ξ′α∂ξnkσr+D^−1x′,0,ξ′,ξn×∂x′α∂ξnj+1∂xnkσlD^∗−1x′,0,ξ′,ξndξnσξ′dx′,and the sum is taken over r+l−k−j−α=−3, r≤−1, l≤−1, D^ denotes the modified Novikov operators.Locally, we can use Theorem2 (25) to compute the interior of Wres~π+D^−1∘π+D^∗−1; we have
(49)∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^∗D^−1σξdx=32π2∫M16gej~,∇ej~TMθ′−43s−16θ2+32θ′2dVolM.So we only need to compute∫∂MΦ. 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:(50)σ1D^=σ1D^∗=icξ,σ0D^=14∑i,s,tωs,tei~cei~c¯es~c¯et~−14∑i,s,tωs,tei~cei~ces~cet~+c¯θ+cθ′,σ0D^∗=14∑i,s,tωs,tei~cei~c¯es~c¯et~−14∑i,s,tωs,tei~cei~ces~cet~+c¯θ−cθ′.Write(51)Dxα=−iα∂xα,σD^=p1+p0,σD^−1=∑j=1∞q−j.By the composition formula of pseudodifferential operators, we have(52)1=σD^∘D^−1=∑α1α!∂ξασD^DxασD^−1=p1+p0q−1+q−2+q−3+⋯+∑j∂ξjp1+∂ξjp0Dxjq−1+Dxjq−2+Dxjq−3+⋯=p1q−1+p1q−2+p0q−1+∑j∂ξjp1Dxjq−1+⋯,so
(53)q−1=p1−1,q−2=−p1−1p0p1−1+∑j∂ξjp1Dxjp1−1.By Lemma7, we have some symbols of operators.Lemma 8.
The following identities hold:(54)σ−1D^−1=σ−1D^∗−1=icξξ2,σ−2D^−1=cξσ0D^−1cξξ4+cξξ6∑jcdxj∂xjcξξ2−cξ∂xjξ2,σ−2D^∗−1=cξσ0D^∗−1cξξ4+cξξ6∑jcdxj∂xjcξξ2−cξ∂xjξ2.From the remark above, we can now computeΦ (see formula (48) for the definition of Φ). We use tr as shorthand of trace. Since n=4, then tr∧∗T∗Mid=dim∧∗4=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.By (48), we get
(55)Case1i=−∫ξ′=1∫−∞+∞∑α=1tr∂ξ′απξn+σ−1D^−1×∂x′α∂ξnσ−1D^∗−1x0dξnσξ′dx′.By Lemma4, for i<n, then
(56)∂xiicξξ2x0=i∂xicξx0ξ2−icξ∂xiξ2x0ξ4=0,so Case 1 (i) vanishes.Case 1.
(ii)r=−1, l=−1, k=∣α∣=0, and j=1.By (48), we get
(57)Case1ii=−12∫ξ′=1∫−∞+∞trace∂xnπξn+σ−1D∧−1×∂ξn2σ−1D∧∗−1x0dξnσξ′dx′.By Lemma8, we have
(58)∂ξn2σ−1D^∗−1x0=i−6ξncdxn+2cξ′ξ4+8ξn2cξξ6,(59)∂xnσ−1D^−1x0=i∂xncξ′x0ξ2−icξξ′2h′0ξ4.By (32), (33), and the Cauchy integral formula, we have
(60)πξn+cξξ4x0ξ′=1=πξn+cξ′+ξncdxn1+ξn22=12πilimu⟶0−∫Γ+cξ′+ηncdxn/ηn+i2ξn+iu−ηnηn−i2dηn=−iξn+2cξ′+icdxn4ξn−i2.Similarly, we have,(61)πξn+i∂xncξ′ξ2x0ξ′=1=∂xncξ′x02ξn−i.By (59), then
(62)πξn+∂xnσ−1D^−1ξ′=1=∂xncξ′x02ξn−i+ih′0iξn+2cξ′+icdxn4ξn−i2.By the relation of the Clifford action andtrAB=trBA, we have the equalities:
(63)trcξ′cdxn=0,trcdxn2=−16,trcξ′2x0ξ′=1=−16,tr∂xncξ′cdxn=0,tr∂xncξ′cξ′x0ξ′=1=−8h′0,trc¯ei~c¯ej~cek~cel~=0i≠j.By (61) and a direct computation, we have
(64)h′0triξn+2cξ′+icdxn4ξn−i2×6ξncdxn+2cξ′1+ξn22−8ξn2cξ′+ξncdxn1+ξn23x0ξ′=1=−16h′0−2iξn2−ξn+iξn−i4ξn+i3.Similarly, we have(65)−itr∂xncξ′x02ξn−i×6ξncdxn+2cξ′1+ξn22−8ξn2cξ′+ξncdxn1+ξn23x0ξ′=1=−8ih′03ξn2−1ξn−i4ξn+i3.Then,(66)Case1ii=−∫ξ′=1∫−∞+∞4ih′0ξn−i2ξn−i4ξn+i3dξnσξ′dx′=−4ih′0Ω3∫Γ+1ξn−i2ξn+i3dξndx′=−4ih′0Ω32πi1ξn+i31ξn=idx′=−32πh′0Ω3dx′,where Ω3 is the canonical volume of S3.Case 1.
(iii)r=−1, l=−1, j=∣α∣=0, and k=1.By (48), we get
(67)Case1iii=−12∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×∂ξn∂xnσ−1D^∗−1x0dξnσξ′dx′.By Lemma8, we have
(68)∂ξn∂xnσ−1D^∗−1x0ξ′=1=−ih′0cdxnξ4−4ξncξ′+ξncdxnξ6−2ξni∂xncξ′x0ξ4,(69)∂ξnπξn+σ−1D^−1x0ξ′=1=−cξ′+icdxn2ξn−i2.Similar to Case 1 (ii), we have(70)trcξ′+icdxn2ξn−i2×ih′0cdxnξ4−4ξncξ′+ξncdxnξ6=8h′0i−3ξnξn−i4ξn+i3,trcξ′+icdxn2ξn−i2×2ξni∂xncξ′x0ξ4=−8ih′0ξnξn−i4ξn+i2.So we have(71)Case1iii=−∫ξ′=1∫−∞+∞h′04i−3ξnξn−i4ξn+i3dξnσξ′dx′−∫ξ′=1∫−∞+∞h′04iξnξn−i4ξn+i2dξnσξ′dx′=−h′0Ω32πi3!4i−3ξnξn+i33ξn=idx′+h′0Ω32πi3!4iξnξn+i23ξn=idx′=32πh′0Ω3dx′.Case 2.
r=−2, l=−1, and k=j=∣α∣=0.By (48), we get
(72)Case2=−i∫ξ′=1∫−∞+∞traceπξn+σ−2D^−1×∂ξnσ−1D^∗−1x0dξnσξ′dx′.By Lemma8, we have
(73)σ−2D^−1x0=cξσ0D^x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0ξ∂M2,where
(74)σ0D^x0=14∑s,t,iωs,tei~x0cei~c¯es~c¯et~−14∑s,t,iωs,tei~x0cei~ces~cet~+c¯θ+cθ′.We denote(75)b01x0=14∑s,t,iωs,tei~x0cei~c¯es~c¯et~,b02x0=−14∑s,t,iωs,tei~x0cei~ces~cet~.Then,(76)πξn+σ−2D^−1x0ξ′=1=πξn+cξb01x0cξ1+ξn22+πξn+cξc¯θ+cθ′x0cξ1+ξn22+πξn+cξb02x0cξ+cξcdxn∂xncξ′x01+ξn22−h′0cξcdxncξ1+ξn23.By direct calculation, we have(77)πξn+cξb01x0cξ1+ξn22=πξn+cξ′p01x0cξ′1+ξn22+πξn+ξncξ′b01x0cdxn1+ξn22+πξn+ξncdxnb01x0cξ′1+ξn22+πξn+ξn2cdxnb01x0cdxn1+ξn22=−cξ′b01x0cξ′2+iξn4ξn−i2+icξ′b01x0cdxn4ξn−i2+icdxnb01x0cξ′4ξn−i2+−iξncdxnb01x0cdxn4ξn−i2.Since(78)cdxnb01x0=−14h′0∑i=1n−1cei~c¯ei~cen~c¯en~,then by the relation of the Clifford action and trAB=trBA, we have the equalities:
(79)trcei~c¯ei~cen~c¯en~=0i<n,trb01cdxn=0,trc¯θcdxn=0,trcθ′cdxn=−16gθ′,dxn,trc¯ξ′c¯dxn=0.Since(80)∂ξnσ−1D^∗−1=∂ξnq−1x0ξ′=1=icdxn1+ξn2−2ξncξ′+2ξn2cdxn1+ξn22.By (77) and (80), we have
(81)trπξn+cξb01x0cξ1+ξn22×∂ξnσ−1D^∗−1x0ξ′=1=121+ξn22trcξ′b01x0+i21+ξn22trcdxnb01x0=121+ξn22trcξ′b01x0.We note that∫ξ′=1ξi1ξi2⋯ξi2d+1σξ′=0, i<n, so trcξ′b01x0 has no contribution for computing Case 2.By direct calculation, we have(82)πξn+cξb02x0cξ+cξcdxn∂xncξ′x01+ξn22−h′0πξn+cξcdxncξ1+ξn3≔B1−B2,where
(83)B1=−14ξn−i22+iξncξ′b02x0cξ′+iξncdxnb02x0cdxn+2+iξncξ′cdxn∂xncξ′+icdxnb02x0cξ′+icξ′b02x0cdxn−i∂xncξ′,(84)B2=h′02cdxn4iξn−i+cdxn−icξ′8ξn−i2+3ξn−7i8ξn−i3icξ′−cdxn.By (80) and (84), we have
(85)trB2×∂ξnσ−1D^∗−1ξ′=1=i2h′0−iξn2−ξn+4i4ξn−i3ξn+i2trid=8ih′0−iξn2−ξn+4i4ξn−i3ξn+i2.By (80) and (83), we have
(86)trB1×∂ξnσ−1D^∗−1ξ′=1=−8ic01+ξn22+2h′0ξn2−iξn−2ξn−i1+ξn22,where b02=c0cdxn and c0=−3/4h′0.By (86) and (85), we have
(87)−i∫ξ′=1∫−∞+∞traceB1−B2×∂ξnσ−1D^∗−1x0dξnσξ′dx′=−Ω3∫Γ+8c0ξn−i+ih′0ξn−i3ξn+i2dξndx′=92πh′0Ω3dx′.Similar to (81), we have
(88)trπξn+cξc¯θx0cξ1+ξn22×∂ξnσ−1D^∗−1x0ξ′=1=121+ξn22trcξ′c¯θx0+i21+ξn22trcdxnc¯θx0=121+ξn22trcξ′c¯θx0.Similar to (83), we have
(89)trπξn+cξcθ′x0cξ1+ξn22×∂ξnσ−1D^∗−1x0ξ′=1=i21+ξn22trcdxncθ′x0.By (88) and (89), we have
(90)−i∫ξ′=1∫−∞+∞traceπξn+cξc¯θ+cθ′cξ1+ξn22×∂ξnσ−1D^∗−1x0dξnσξ′dx′=π4trcdxncθ′Ω3dx′=−4πgθ′,dxnΩ3dx′.By (87) and (90), we have
(91)Case2=92πh′0Ω3dx′−4πgθ′,dxnΩ3dx′.Case 3.
r=−1, l=−2, and k=j=∣α∣=0.By (48), we get
(92)Case3=−i∫ξ′=1∫−∞+∞traceπξn+σ−1D^−1×∂ξnσ−2D^∗−1x0dξnσξ′dx′.By (32) and (33) and Lemma 8, we have
(93)πξn+σ−1D^−1ξ′=1=cξ′+icdxn2ξn−i.Since(94)σ−2D^∗−1x0=cξσ0D^∗x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0ξ∂M2,where
(95)σ0D^∗x0=14∑s,t,iωs,tei~x0cei~c¯es~c¯et~−14∑s,t,iωs,tei~x0cei~ces~cet~+c¯θ−cθ′x0=b01x0+b02x0+c¯θ−cθ′x0,then
(96)∂ξnσ−2D^∗−1x0ξ′=1=∂ξncξb01x0+b02x0+c¯θ−cθ′x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0=∂ξncξb01x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0+∂ξncξb02x0cξξ4+∂ξncξc¯θ−cθ′x0cξξ4.By direct calculation, we have(97)∂ξncξb01x0cξξ4=cdxnb01x0cξξ4+cξb01x0cdxnξ4−4ξncξb01x0cξξ6,(98)∂ξncξc¯θ−cθ′x0cξξ4=cdxnc¯θ−cθ′x0cξξ4+cξc¯θ−cθ′x0cdxnξ4−4ξncξc¯θ−cθ′x0cξξ4.We denote(99)q−21=cξb02x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0,then
(100)∂ξnq−21=11+ξn232ξn−2ξn3cdxnb02cdxn+1−3ξn2cdxnb02cξ′+1−3ξn2cξ′b02cdxn−4ξncξ′b02cξ′+3ξn2−1∂xncξ′−4ξncξ′cdxn∂xncξ′+2h′0cξ′+2h′0ξncdxn+6ξnh′0cξcdxncξ1+ξn24.By (93) and (97), we have
(101)trπξn+σ−1D^−1×∂ξncξb01cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′b01x0+iξ−iξ+i3trcdxnb01x0.By (79), we have
(102)trπξn+σ−1D^−1×∂ξncξb01cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′b01x0.We note that∫ξ′=1ξi1ξi2⋯ξi2d+1σξ′=0, i<n, so trcξ′b01x0 has no contribution for computing Case 3.By (93) and (100), we have
(103)trπξn+σ−1D^−1×∂ξnq−21x0ξ′=1=12h′0iξn2+ξn−2iξ−i3ξ+i3+48h′0iξnξ−i3ξ+i4,then
(104)−iΩ3∫Γ+12h′0iξn2+ξn−2iξn−i3ξn+i3+48h′0iξnξn−i3ξn+i4dξndx′=−92πh′0Ω3dx′.By (93) and (98), we have
(105)trπξn+σ−1D^−1×∂ξncξc¯θ−cθ′cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′c¯θ−cθ′x0+iξ−iξ+i3trcdxnc¯θ−cθ′x0.By∫ξ′=1ξ1⋯ξ2d+1σξ′=0 and (81), we have
(106)−i∫ξ′=1∫−∞+∞trπξn+σ−1D^−1×∂ξncξc¯θ−cθ′cξξ4x0dξnσξ′dx′=−i∫ξ′=1∫−∞+∞iξ−iξ+i3trcdxnc¯θ−cθ′x0dξnσξ′dx′=−π4trcdxnc¯θ−cθ′Ω3dx′=−4πgθ′,dxnΩ3dx′.So we have(107)Case3=−92πh′0Ω3dx′−4πgθ′,dxnΩ3dx′.SinceΦ is the sum of Cases 1–3, so Φ=−8πgθ′,dxnΩ3dx′.Theorem 9.
LetM be 4-dimensional compact-oriented manifolds with the boundary ∂M and the metric gM as above, D^ and D^∗ be modified Novikov operators on M^, then
(108)Wres~π+D^−1∘π+D^∗−1=32π2∫M16gej~,∇ej~TMθ′−43s−16θ2+32θ′2dVolM−8π∫∂Mgθ′,dxnΩ3dx′.where s is the scalar curvature.On the other hand, we also prove the Kastler-Kalau-Walze-type theorem for4-dimensional manifolds with a boundary associated to D^2. By (41) and (42), we will compute
(109)Wres~π+D^−1∘π+D^−1=∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^−2σξdx+∫∂MΦ^,where
(110)Φ^=∫ξ′=1∫−∞+∞∑j,k=0∞∑−iα+j+k+1α!j+k+1!×trace∧∗T∗M∂xnj∂ξ′α∂ξnkσr+D^−1x′,0,ξ′,ξn×∂x′α∂ξnj+1∂xnkσlD^−1x′,0,ξ′,ξndξnσξ′dx′,and the sum is taken over r+l−k−j−∣α∣=−3, r≤−1, l≤−1.Locally, we can use Theorem2 (26) to compute the interior of Wres~π+D^−1∘π+D^−1; we have
(111)∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^−2σξdx=32π2∫M−43s−16θ2dVolM.So we only need to compute∫∂MΦ^. From the remark above, now we can compute Φ^ (see formula (110) for the definition of Φ^). We use tr as shorthand of trace. Since n=4, then tr∧∗T∗Mid=dim∧∗4=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.By (110), we get
(112)Case1i=−∫ξ′=1∫−∞+∞∑∣α∣=1tr∂ξ′απξn+σ−1D^−1×∂x′α∂ξnσ−1D^−1x0dξnσξ′dx′.Case 1.
(ii)r=−1, l=−1, k=∣α∣=0, and j=1.By (110), we get
(113)Case1ii=−12∫ξ′=1∫−∞+∞trace∂xnπξn+σ−1D^−1×∂ξn2σ−1D^−1x0dξnσξ′dx′.Case 1.
(iii)r=−1, l=−1, j=∣α∣=0, and k=1.By (110), we get
(114)Case1iii=−12∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×∂ξn∂xnσ−1D^−1x0dξnσξ′dx′.By Lemma8, we have σ−1D^−1=σ−1D^∗−1. By (55)-(71), so Case 1 vanishes.Case 2.
r=−2, l=−1, and k=j=∣α∣=0.By (110), we get
(115)Case2=−i∫ξ′=1∫−∞+∞traceπξn+σ−2D^−1×∂ξnσ−1D^−1x0dξnσξ′dx′.By Lemma8, we have σ−1D^−1=σ−1D^∗−1. By (72)–(91), we have
(116)Case2=92πh′0Ω3dx′−4πgθ′,dxnΩ3dx′,where Ω4 is the canonical volume of S4.Case 3.
r=−1, l=−2, and k=j=∣α∣=0.By (110), we get
(117)Case3=−i∫ξ′=1∫−∞+∞traceπξn+σ−1D^−1×∂ξnσ−2D^−1x0dξnσξ′dx′.By (33) and (32) and Lemma 8, we have
(118)πξn+σ−1D^−1ξ′=1=cξ′+icdxn2ξn−i.Since(119)σ−2D^−1x0=cξσ0D^x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0ξ∂M2,where
(120)σ0D^x0=14∑s,t,iωs,tei~x0cei~c¯es~c¯et~−14∑s,t,iωs,tei~x0cei~ces~cet~+c¯θ+cθ′x0=b01x0+b02x0+c¯θ+cθ′x0,then
(121)∂ξnσ−2D^−1x0ξ′=1=∂ξncξb01x0+b02x0+c¯θ+cθ′x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0=∂ξncξb01x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0+∂ξncξb02x0cξξ4+∂ξncξc¯θ+cθ′x0cξξ4.By direct calculation, we have(122)∂ξncξb01x0cξξ4=cdxnb01x0cξξ4+cξb01x0cdxnξ4−4ξncξb01x0cξξ6,(123)∂ξncξc¯θ+cθ′x0cξξ4=cdxnc¯θ+cθ′x0cξξ4+cξc¯θ+cθ′x0cdxnξ4−4ξncξc¯θ+cθ′x0cξξ4.We denote(124)q−21=cξb02x0cξξ4+cξξ6cdxn∂xncξ′x0ξ2−cξh′0,then
(125)∂ξnq−21=11+ξn232ξn−2ξn3cdxnb02cdxn+1−3ξn2cdxnb02cξ′+1−3ξn2cξ′b02cdxn−4ξncξ′b02cξ′+3ξn2−1∂xncξ′−4ξncξ′cdxn∂xncξ′+2h′0cξ′+2h′0ξncdxn+6ξnh′0cξcdxncξ1+ξn24.By (118) and (122), we have
(126)trπξn+σ−1D^−1×∂ξncξb01cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′b01x0+iξ−iξ+i3trcdxnb01x0.By (79), we have
(127)trπξn+σ−1D^−1×∂ξncξb01cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′b01x0.We note that∫ξ′=1ξi1ξi2⋯ξi2d+1σξ′=0, i<n, so trcξ′b01x0 has no contribution for computing Case 3.By (118) and (125), we have
(128)trπξn+σ−1D^−1×∂ξnq−21x0ξ′=1=12h′0iξn2+ξn−2iξ−i3ξ+i3+48h′0iξnξ−i3ξ+i4,then
(129)−iΩ3∫Γ+12h′0iξn2+ξn−2iξn−i3ξn+i3+48h′0iξnξn−i3ξn+i4dξndx′=−92πh′0Ω3dx′.By (118) and (123), we have
(130)trπξn+σ−1D^−1×∂ξncξc¯θ+cθ′cξξ4x0ξ′=1=−1ξ−iξ+i3trcξ′c¯θ+cθ′x0+iξ−iξ+i3trcdxnc¯θ+cθ′x0.By∫ξ′=1ξ1⋯ξ2d+1σξ′=0 and (79), we have
(131)−i∫ξ′=1∫−∞+∞trπξn+σ−1D^−1×∂ξncξc¯θ+cθ′cξξ4x0dξnσξ′dx′=−i∫ξ′=1∫−∞+∞iξ−iξ+i3trcdxnc¯θ+cθ′x0dξnσξ′dx′=−π4trcdxnc¯θ+cθ′Ω3dx′=4πgθ′,dxnΩ3dx′.So we have(132)Case3=−92πh′0Ω3dx′+4πgθ′,dxnΩ3dx′.SinceΦ^ is the sum of Cases 1–3, so Φ^=0.Theorem 10.
LetM be a 4-dimensional compact-oriented manifold with the boundary ∂M and the metric gM as above and D^ be a modified Novikov operator on M^, then
(133)Wres~π+D^−1∘π+D^−1=32π2∫M−43s−16θ2dVolM.where s 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
(134)Wres~π+D^−1∘π+D^∗D^D^∗−1=∫M∫ξ=1trace∧∗T∗Mσ−4D^∗D^−2σξdx+∫∂MΨ,where
(135)Ψ=∫ξ′=1∫−∞+∞∑j,k=0∞∑−iα+j+k+1α!j+k+1!×trace∧∗T∗M∂xnj∂ξ′α∂ξnkσr+D^−1x′,0,ξ′,ξn×∂x′α∂ξnj+1∂xnkσlD^∗D^D^∗−1x′,0,ξ′,ξndξnσξ′dx′,and the sum is taken over r+ℓ−k−j−∣α∣−1=−6, r≤−1, ℓ≤−3.Locally, we can use Theorem2 (25) to compute the interior term of (134); we have
(136)∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^∗D^−2σξdx=128π3∫M64gej~,∇ej~TMθ′−163s−64θ2+256θ′2dVolM.So we only need to compute∫∂MΨ. Let us now turn to compute the specification of D^∗D^D^∗.
(137)D^∗D^D^∗=∑i=1ncei~ei~,dxl−gij∂l∂i∂j+∑i=1ncei~ei~,dxl−∂lgij∂i∂j−gij4σi+ai∂j−2Γijk∂k∂l+∑i=1ncei~ei~,dxl−2∂lgijσi+ai∂j+gij∂lΓijk∂k−2gij∂lσi+∂lai∂j+∂lgijΓijk∂k+∑j,k∂lcθ′cej~−cej~cθ′ej~,dxk∂k+∑j,kcθ′cej~−cej~cθ′∂lej~,dxk∂k+∑i=1ncei~ei~,dxl∂l−gij∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak+∑i,jgi,jcθ′c∂iσi+cθ′c∂iai−c∂i∂icθ′−c∂iσicθ′−c∂iaicθ′+14s−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TMθ+θ2+θ′2−c¯θcθ′+cθ′c¯θ+σi+ai+c¯θ−cθ′−gij∂i∂j+∑i=1ncei~ei~,dxl2∑j,kcθ′cej~−cej~cθ′×ei~,dxk∂l∂k+σi+ai+c¯θ−cθ′−∑i,jgi,j2σi∂j+2ai∂j−Γi,jk∂k+∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak−∑i,jgi,jc∂icθ′−cθ′c∂i∂j+∑i,jgi,jcθ′c∂iσi+cθ′c∂iai−c∂i∂icθ′−c∂iσicθ′−c∂iaicθ′+14s+θ′2−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TMθ+θ2−c¯θcθ′+cθ′c¯θ.Then, we obtainLemma 11.
The following identities hold:(138)σ2D^∗D^D^∗=∑i,j,lcdxl∂lgi,jξiξj+cξ4σk+4ak−2Γkξk−2cξcθ′cξ+ξ2cθ′+14ξ2∑s,t,lωs,tel~cel~c¯es~c¯et~−cel~ces~cet~+ξ2c¯θ−cθ′,σ3D^∗D^D^∗=icξξ2.Write(139)σD^∗D^D^∗=p3+p2+p1+p0,σD^∗D^D^∗−1=∑j=3∞q−j.By the composition formula of pseudodifferential operators, we have(140)1=σD^∗D^D^∗∘D^∗D^D^∗−1=∑α1α!∂ξασD^∗D^D^∗DxαD^∗D^D^∗−1=p3+p2+p1+p0q−3+q−4+q−5+⋯+∑j∂ξjp3+∂ξjp2+∂ξjp1+∂ξjp0Dxjq−3+Dxjq−4+Dxjq−5+⋯=p3q−3+p3q−4+p2q−3+∑j∂ξjp3Dxjq−3+⋯,by (140), we have
(141)q−3=p3−1,q−4=−p3−1p2p3−1+∑j∂ξjp3Dxjp3−1.By Lemma11, we have some symbols of operators.Lemma 12.
The following identities hold:(142)σ−3D^∗D^D^∗−1=icξξ4,σ−4D^∗D^D^∗−1=cξσ2D^∗D^D^∗cξξ8+icξξ8ξ4cdxn∂xncξ′−2h′0cdxncξ+2ξncξ∂xncξ′+4ξnh′0.From the remark above, now we can computeΨ (see formula (135) for the definition of Ψ). We use tr as shorthand of trace. Since n=6, then tr∧∗T∗Mid=64. Since the sum is taken over r+ℓ−k−j−∣α∣−1=−6, r≤−1, ℓ≤−3, then we have the ∫∂MΨ as the sum of the following five cases:Case 1.
(i)r=−1, l=−3, j=k=0, and ∣α∣=1.By (135), we get
(143)Case1i=−∫ξ′=1∫−∞+∞∑∣α∣=1trace∂ξ′απξn+σ−1D^−1×∂x′α∂ξnσ−3D^∗D^D^∗−1x0dξnσξ′dx′.By Lemma12, for i<n, we have
(144)∂xiσ−3D^∗D^D^∗−1x0=∂xiicξξ4x0=i∂xicξξ−4x0−2icξ∂xiξ2ξ−6x0=0.so Case 1 (i) vanishes.Case 1.
(ii)r=−1, l=−3, ∣α∣=k=0, and j=1.By (135), we have
(145)Case1ii=−12∫ξ′=1∫−∞+∞trace∂xnπξn+σ−1D^−1×∂ξn2σ−3D^∗D^D^∗−1x0dξnσξ′dx′.By Lemma12 and direct calculations, we have
(146)∂ξn2σ−3D^∗D^D^∗−1=i20ξn2−4cξ′+12ξn3−ξncdxn1+ξn24.Sincen=6, tr−id=−64. By the relation of the Clifford action and trAB=trBA, then
(147)trcξ′cdxn=0,trcdxn2=−64,trcξ′2x0ξ′=1=−64,(148)tr∂xncξ′cdxn=0,tr∂xncξ′cξ′x0ξ′=1=−32h′0.By (62), (146), and (147), we get
(149)trace∂xnπξn+σ−1D^−1×∂ξn2σ−3D^∗D^D^∗−1x0=64h′0−1−3ξni+5ξn2+3iξn3ξn−i6ξn+i4.Then, we obtain(150)Case1ii=−12∫ξ′=1∫−∞+∞h′0dimF−8−24ξni+40ξn2+24iξn3ξn−i6ξn+i4dξnσξ′dx′=8h′0Ω4∫Γ+4+12ξni−20ξn2−122iξn3ξn−i6ξn+i4dξndx′=h′0Ω4πi5!8+24ξni−40ξn2−24iξn3ξn+i45ξn=idx′=−152πh′0Ω4dx′,where Ω4 is the canonical volume of S4.Case 1.
(iii)r=−1, l=−3, ∣α∣=j=0, and k=1.By (135), we have
(151)Case1iii=−12∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×∂ξn∂xnσ−3D^∗D^D^∗−1x0dξnσξ′dx′.By Lemma12 and direct calculations, we have
(152)∂ξn∂xnσ−3D^∗D^D^∗−1=−4iξn∂xncξ′x01+ξn23+i12h′0ξncξ′1+ξn24−i2−10ξn2h′0cdxn1+ξn24.Combining (69) and (152), we have
(153)trace∂ξnπξn+σ−1D^−1×∂ξn∂xnσ−3D^∗D^D^∗−1x0ξ′=1=8h′08i−32ξn−8iξn2ξn−i5ξ+i4.Then,(154)Case1iii=−12∫ξ′=1∫−∞+∞8h′08i−32ξn−8iξn2ξn−i5ξ+i4dξnσξ′dx′=−12h′08Ω4∫Γ+8i−32ξn−8iξn2ξn−i5ξ+i4dξndx′=−8h′0Ω4πi4!8i−32ξn−8iξn2ξ+i44ξn=idx′=252πh′0Ω4dx′.Case 2.
r=−1, l=−4, and ∣α∣=j=k=0.By (135), we have
(155)Case2=−i∫ξ′=1∫−∞+∞traceπξn+σ−1D^−1×∂ξnσ−4D^∗D^D^∗−1x0dξnσξ′dx′=i∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×σ−4D^∗D^D^∗−1x0dξnσξ′dx′.In the normal coordinate,gijx0=δij and ∂xjgαβx0=0, if j<n; ∂xjgαβx0=h′0δβα, if j=n. So by Lemma A.2 in [10], we have Γnx0=5/2h′0 and Γkx0=0 for k<n. By the definition of δk and Lemma 2.3 in [10], we have δnx0=0 and δk=1/4h′0cek~cen~ for k<n. By Lemma 12, we obtain
(156)σ−4D^∗D^D^∗−1x0ξ′=1=cξσ2D^∗D^D^∗−1x0∣ξ′∣=1cξξ8−cξξ4∑j∂ξjcξξ2Dxjicξξ4=1ξ8cξ12h′0cξ∑k<nξkcek~cen~−12h′0cξ∑k<nξkc¯ek~c¯en~−52h′0ξncξ−14h′0ξ2cdxn−2cξcθ′cξ+ξ2cθ′+ξ2c^θ−cθ′cξ+icξξ8ξ4cdxn∂xncξ′−2h′0cdxncξ+2ξncξ∂xncξ′+4ξnh′0.By (69) and (156), we have
(157)tr∂ξnπξn+σ−1D^−1×σ−4D^∗D^D^∗−1x0ξ′=1=12ξn−i21+ξn2434i+2+3+4iξn+−6+2iξn2+3ξn3+9i4ξn4h′0trid+12ξn−i21+ξn24−1−3iξn−2ξn2−4iξn3−ξn4−iξn5trcξ′∂xncξ′−12ξn−i21+ξn2412i+12ξn+12ξn2+12ξn3trcξ′c¯ξ′cdxnc¯dxn+−ξni+32ξn−i4i+ξn3trcθ′cdxn−3ξn+i2ξn−i4i+ξn3trcθ′cξ′.By direct calculation and the relation of the Clifford action andtrAB=trBA, we then have equalities:
(158)trcθ′x0cdxn=−64gθ′,dxn,trcθ′x0cξ′=−64gθ′,ξ′,(159)trcei~c¯ei~cen~c¯en~=0i<n.Then,(160)trcξ′c¯ξ′cdxnc¯dxn=∑i<n,j<ntrξiξjcei~c¯ej~cdxnc¯dxn=0.So, we have(161)Case2=ih′0∫ξ′=1∫−∞+∞64×3/4i+2+3+4iξn+−6+2iξn2+3ξn3+9i/4ξn42ξn−i5ξn+i4dξnσξ′dx′+ih′0∫ξ′=1∫−∞+∞32×1+3iξn+2ξn2+4iξn3+ξn4+iξn52ξn−i21+ξn24dξnσξ′dx′+i∫ξ′=1∫−∞+∞ξn−i−2ξni+12ξn−i4i+ξn3trcθ′cdxndξnσξ′dx′−i∫ξ′=1∫−∞+∞3ξn+i2ξn−i4i+ξn3trcθ′cξ′dξnσξ′dx′=−194i−15πh′0Ω4dx′+−38i−758πh′0Ω4dx′+120iπgdxn,θ′Ω4dx′=−418i−1958πh′0Ω4dx′+120iπgdxn,θ′Ω4dx′.Case 3.
r=−2, l=−3, and ∣α∣=j=k=0.By (135), we have
(162)Case3=−i∫ξ′=1∫−∞+∞traceπξn+σ−2D^−1×∂ξnσ−3D^∗D^D^∗−1x0dξnσξ′dx′.By Lemmas11 and 12, we have
(163)σ−2D^−1x0=cξσ0D^cξξ4x0+cξξ6∑jcdxj∂xjcξξ2−cξ∂xjξ2x0,where
(164)σ0D^=14∑i,s,tωs,tei~cei~c¯es~c¯et~−14∑i,s,tωs,tei~cei~ces~cet~+c^θ+cθ′.On the other hand,(165)∂ξnσ−3D^∗D^D^∗−1=−4iξncξ′1+ξn23+i1−3ξn2cdxn1+ξn23.By (163), (28), and (32), we have
(166)πξn+σ−2D^−1x0ξ′=1=πξn+cξσ0D^x0cξ+cξcdxn∂xncξ′x01+ξn22−h′0πξn+cξcdxncξ1+ξn23.We denote(167)σ0D^x0ξn=i=b0x0=b01x0+b02x0+c¯θ+cθ′.Then, we obtain(168)πξn+σ−2D^−1x0ξ′=1=πξn+cξb02x0cξ+cξcdxn∂xncξ′x01+ξn22−h′0cξcdxncξ1+ξn23+πξn+cξb01x0cξx01+ξn22+πξn+cξc¯θ+cθ′cξx01+ξn22.Furthermore,(169)πξn+cξc¯θ+cθ′x0cξ1+ξn22=πξn+cξ′c¯θ+cθ′x0cξ′1+ξn22+πξn+ξncξ′c¯θ+cθ′x0cdxn1+ξn22+πξn+ξncdxnc¯θ+cθ′x0cξ′1+ξn22+πξn+ξn2cdxnc¯θ+cθ′x0cdxn1+ξn22=−cξ′c¯θ+cθ′x0cξ′2+iξn4ξn−i2+icξ′c¯θ+cθ′x0cdxn4ξn−i2+icdxnc¯θ+cθ′x0cξ′4ξn−i2+−iξncdxnc¯θ+cθ′x0cdxn4ξn−i2,πξn+cξb01x0cξ1+ξn22=πξn+cξ′p01x0cξ′1+ξn22+πξn+ξncξ′b01x0cdxn1+ξn22+πξn+ξncdxnb01x0cξ′1+ξn22+πξn+ξn2cdxnb01x0cdxn1+ξn22=−cξ′b01x0cξ′2+iξn4ξn−i2+icξ′b01x0cdxn4ξn−i2+icdxnb01x0cξ′4ξn−i2+−iξncdxnb01x0cdxn4ξn−i2.By the relation of the Clifford action andtrAB=trBA, we then have equalities:
(170)trb01cdxn=0,trc¯ξ′c¯dxn=0,trc¯θcξ′=0.Then, we have(171)trπξn+cξb01x0cξ1+ξn22×∂ξnσ−3D^∗D^D^∗−1x0ξ′=1=2−8iξn−6ξn24ξn−i21+ξn23trb01x0cξ′,By direct calculation, we have(172)πξn+cξb02x0cξ+cξcdxn∂xncξ′x01+ξn22−h′0πξn+cξcdxncξ1+ξn3≔B1−B2,where
(173)B1=−14ξn−i22+iξncξ′b02cξ′+iξncdxnb02cdxn+2+iξncξ′cdxn∂xncξ′+icdxnb02cξ′+icξ′b02cdxn−i∂xncξ′=14ξn−i252h′0cdxn−5i2h′0cξ′−2+iξncξ′cdxn∂ξncξ′+i∂ξncξ′,(174)B2=h′02cdxn4iξn−i+cdxn−icξ′8ξn−i2+3ξn−7i8ξn−i3icξ′−cdxn.By (165) and (174), we have
(175)trB2×∂ξnσ−3D^∗D^D^∗−1x0ξ′=1=trh′02cdxn4iξn−i+cdxn−icξ′8ξn−i2+3ξn−7i8ξn−i3icξ′−cdxn×−4iξncξ′+i−3iξn2cdxn1+ξn23=8h′04i−11ξn−6iξn2+3ξn3ξn−i5ξn+i3.Similarly, we have(176)trB1×∂ξnσ−3D^∗D^D^∗−1x0ξ′=1=tr14ξn−i252h′0cdxn−5i2h′0cξ′−2+iξncξ′cdxn∂ξncξ′+i∂ξncξ′×−4iξncξ′+i−3iξn2cdxn1+ξn23=8h′03+12iξn+3ξn2ξn−i4ξn+i3,trπξn+cξc¯θ+cθ′x0cξ1+ξn22×∂ξnσ−3D^∗D^D^∗−1x0ξ′=1=2−8iξn−6ξn24ξn−i21+ξn23trc¯θ+cθ′x0cξ′=2−8iξn−6ξn24ξn−i21+ξn23trcθ′x0cξ′=2−8iξn−6ξn24ξn−i21+ξn23−gθ′,ξ′trid.By∫∣ξ′∣=1ξ1⋯ξ2q+1σξ′=0, we have
(177)Case3=−ih′0∫ξ′=1∫−∞+∞8×−7i+26ξn+15iξn2ξn−i5ξn+i3dξnσξ′dx′−i∫ξ′=1∫−∞+∞2−8iξn−6ξn24ξn−i21+ξn23−gθ′,ξ′tridξ′=1dξnσξ′dx′=−8ih′0×2πi4!−7i+26ξn+15iξn2ξn+i35ξn=iΩ4dx′=552πh′0Ω4dx′.NowΨ is the sum of Cases 1–3, then
(178)Ψ=658−418iπh′0Ω4dx′+120iπgdxn,θ′Ω4dx′.Theorem 13.
LetM be a 6-dimensional compact-oriented manifold with the boundary ∂M and the metric gM as above and D^ and D^∗ be modified Novikov operators on M^, then
(179)Wres~π+D^−1∘π+D^∗D^D^∗−1=128π3∫M64gej~,∇ej~TMθ′−163s−64θ2+256θ′2dVolM+∫∂M658−418iπh′0+120iπgdxn,θ′Ω4dx′.where s is the scalar curvature.On the other hand, we prove the Kastler-Kalau-Walze-type theorem for a6-dimensional manifold with a boundary associated with D^3. An application of (2.1.4) in [12] shows that
(180)Wres~π+D^−1∘π+D^−3=∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^−4σξdx+∫∂MΨ^,where Wres~ denotes a noncommutative residue on manifolds with a boundary,
(181)Ψ^=∫ξ′=1∫−∞+∞∑j,k=0∞∑−iα+j+k+1α!j+k+1!×trace∧∗T∗M∂xnj∂ξ′α∂ξnkσr+D^−1x′,0,ξ′,ξn×∂x′α∂ξnj+1∂xnkσlD^−3x′,0,ξ′,ξndξnσξ′dx′,and the sum is taken over r+ℓ−k−j−∣α∣−1=−6, r≤−1, ℓ≤−3.Locally, we can use Theorem2 (26) to compute the interior term of (181); we have
(182)∫M∫∣ξ∣=1trace∧∗T∗Mσ−4D^−4σξdx=128π3∫M−163s−64θ2dVolM.So we only need to compute∫∂MΨ^. Let us now turn to compute the specification of D^3.
(183)D^3=∑i=1ncei~ei~,dxl−gij∂l∂i∂j+∑i=1ncei~ei~,dxl−∂lgij∂i∂j−gij4σi+ai∂j−2Γijk∂k∂l+∑i=1ncei~ei~,dxl−2∂lgijσi+ai∂j+gij∂lΓijk∂k−2gij∂lσi+∂lai∂j+∂lgijΓijk∂k+∑j,k∂lcθ′cej~+cej~cθ′ej~,dxk∂k+∑j,kcθ′cej~+cej~cθ′∂lej~,dxk∂k+∑i=1ncei~ei~,dxl∂l−gij∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak+∑i,jgi,jcθ′c∂iσi+cθ′c∂iai+c∂i∂icθ′+c∂iσicθ′+c∂iaicθ′+14s−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TM+θ2−θ′2+c¯θcθ′+cθ′c¯θ+σi+ai+c¯θ+cθ′−gij∂i∂j+∑i=1ncei~ei~,dxl2∑j,kcθ′cej~+cej~cθ′×ei~,dxk∂l∂k+σi+ai+c¯θ+cθ′−∑i,jgi,j2σi∂j+2ai∂j−Γi,jk∂k+∂iσj+∂iaj+σiσj+σiaj+aiσj+aiaj−Γi,jkσk−Γi,jkak+∑i,jgi,jc∂icθ′+cθ′c∂i∂j+∑i,jgi,jcθ′c∂iσi+cθ′c∂iai+c∂i∂icθ′+c∂iσicθ′+c∂iaicθ′+14s−θ′2−18∑ijklRijklc¯ei~c¯ej~cek~cel~+∑icei~c¯∇ei~TMθ+θ2+c¯θcθ′+cθ′c¯θ.Then, we obtainLemma 14.
The following identities hold:(184)σ2D^3=∑i,j,lcdxl∂lgi,jξiξj+cξ4σk+4ak−2Γkξk−2cξcθ′cξ−ξ2cθ′+14ξ2∑s,t,lωs,tel~cel~c¯es~c¯et~−cel~ces~cet~+ξ2c¯θ+cθ′,σ3D^3=icξξ2.Write(185)σD^3=p3+p2+p1+p0,σD^−3=∑j=3∞q−j.By the composition formula of pseudodifferential operators, we have(186)1=σD^3∘D^−3=∑α1α!∂ξασD^3DxασD^−3=p3+p2+p1+p0q−3+q−4+q−5+⋯+∑j∂ξjp3+∂ξjp2+∂ξjp1+∂ξjp0Dxjq−3+Dxjq−4+Dxjq−5+⋯=p3q−3+p3q−4+p2q−3+∑j∂ξjp3Dxjq−3+⋯,by (186), we have
(187)q−3=p3−1,q−4=−p3−1p2p3−1+∑j∂ξjp3Dxjp3−1.By (183)–(187), we have some symbols of operators.Lemma 15.
The following identities hold:(188)σ−3D^−3=icξξ4,σ−4D^−3=cξσ2D^3cξξ8+icξξ8ξ4cdxn∂xncξ′−2h′0cdxncξ+2ξncξ∂xncξ′+4ξnh′0.From the remark above, we can now computeΨ^ (see formula (181) for the definition of Ψ^). We use tr as shorthand of trace. Since n=6, then tr∧∗T∗Mid=64. Since the sum is taken over r+ℓ−k−j−∣α∣−1=−6, r≤−1, ℓ≤−3, then we have the ∫∂MΨ^ as the sum of the following five cases:Case 1.
(i)r=−1, l=−3, j=k=0, and ∣α∣=1.By (181), we get
(189)Case1i=−∫ξ′=1∫−∞+∞∑∣α∣=1trace∂ξ′απξn+σ−1D^−1×∂x′α∂ξnσ−3D^−3x0dξnσξ′dx′.Case 1.
(ii)r=−1, l=−3, ∣α∣=k=0, and j=1.By (181), we have
(190)Case1ii=−12∫ξ′=1∫−∞+∞trace∂xnπξn+σ−1D^−1×∂ξn2σ−3D^−3x0dξnσξ′dx′.Case 1.
(iii)r=−1, l=−3, ∣α∣=j=0, and k=1.By (181), we have
(191)Case1iii=−12∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×∂ξn∂xnσ−3D^−3x0dξnσξ′dx′.By Lemmas12 and 15, we have σ−3D^∗D^D^∗−1=σ−3D^−3; by (143)–(154), we obtain
(192)Case1=5πh′0Ω4dx′,where Ω4 is the canonical volume of S4.Case 2.
r=−1, l=−4, ∣α∣=j=k=0.By (181), we have
(193)Case2=−i∫ξ′=1∫−∞+∞traceπξn+σ−1D^−1×∂ξnσ−4D^−3x0dξnσξ′dx′=i∫ξ′=1∫−∞+∞trace∂ξnπξn+σ−1D^−1×σ−4D^−3x0dξnσξ′dx′.In the normal coordinate,gijx0=δij and ∂xjgαβx0=0, if j<n; ∂xjgαβx0=h′0δβα, if j=n. So by Lemma A.2 in [10], we have Γnx0=5/2h′0 and Γkx0=0 for k<n. By the definition of δk and Lemma 2.3 in [10], we have δnx0=0 and δk=1/4h′0cek~cen~ for k<n. By Lemma 15, we obtain
(194)σ−4D^−3x0ξ′=1=cξσ2D^−3x0ξ′=1cξξ8−cξξ4∑j∂ξjcξ∣ξ2Dxjicξξ4=1ξ8cξ12h′0cξ∑k<nξkcek~cen~−12h′0cξ∑k<nξkc¯ek~c¯en~−52h′0ξncξ−14h′0ξ2cdxn−2cξcθ′cξ−ξ2cθ′+ξ2c^θ+cθ′cξ+icξξ8ξ4cdxn∂xncξ′−2h′0cdxncξ+2ξncξ∂xncξ′+4ξnh′0.By (69) and (194), we have
(195)tr∂ξnπξn+σ−1D^−1×σ−4D^−3x0ξ′=1=12ξn−i21+ξn2434i+2+3+4iξn+−6+2iξn2+3ξn3+9i4ξn4h′0trid+12ξn−i21+ξn24−1−3iξn−2ξn2−4iξn3−ξn4−iξn5trcξ′∂xncξ′−12ξn−i21+ξn2412i+12ξn+12ξn2+12ξn3trcξ′c¯ξ′cdxnc¯dxn+−3ξni+12ξn−i4i+ξn3trcθ′cdxn−ξn+3i2ξn−i4i+ξn3trcθ′cξ′.By (158) and (160), we have
(196)Case2=ih′0∫ξ′=1∫−∞+∞64×3/4i+2+3+4iξn+−6+2iξn2+3ξn3+9i/4ξn42ξn−i5ξn+i4dξnσξ′dx′+ih′0∫ξ′=1∫−∞+∞32×1+3iξn+2ξn2+4iξn3+ξn4+iξn52ξn−i21+ξn24dξnσξ′dx′+i∫ξ′=1∫−∞+∞−3ξni+12ξn−i4i+ξn3trcθ′cdxndξnσξ′dx′−i∫ξ′=1∫−∞+∞ξn+3i2ξn−i4i+ξn3trcθ′cξ′dξnσξ′dx′=−418i−1958πh′0Ω4dx′.Case 3.
r=−2, l=−3, ∣α∣=j=k=0.By (181), we have
(197)Case3=−i∫ξ′=1∫−∞+∞traceπξn+σ−2D^−1×∂ξnσ−3D^−3x0dξnσξ′dx′.By Lemmas12 and 15, we have σ−3D^∗D^D^∗−1=σ−3D^−3; by (162)–(177), we obtain
(198)Case3=552πh′0Ω4dx′.NowΨ^ is the sum of Cases 1–3, then
(199)Ψ^=658−418iπh′0Ω4dx′.Theorem 16.
LetM be a 6-dimensional compact-oriented manifold with the boundary ∂M and the metric gM as above and D^ be a modified Novikov operator on M^, then
(200)Wres~π+D^−1∘π+D^−3=128π3∫M−163s−64θ2dVolM+∫∂M658−418iπh′0Ω4dx′.where s is the scalar curvature.


## 5. The Spectral Action for Witten Deformation

In this section, we will compute the spectral action for the Witten deformation. LetM,gM be an n-dimensional compact-oriented Riemannian manifold. Now we will recall the definition of the Witten deformation Dθ (see details in [17]).Let∇L denote the Levi-Civita connection about gM which is a Riemannian metric of M. In the local coordinates xi,1≤i≤n and the fixed orthonormal frame e1~,⋯,en~, the connection matrix ωs,t is defined by
(201)∇Le1~,⋯,en~=e1~,⋯,en~ωs,t.Letεej∗~ and ιej∗~ be the exterior and interior multiplications, respectively. The Witten deformation is defined by
(202)Dθ=d+δ+c¯θ=∑i=1ncei~ei~+14∑s,tωs,tei~c¯es~c¯et~−ces~cet~+c¯θ.By Proposition 4.6 of [17], we have
(203)Dθ2=d+δ2+∑icei~c¯∇ei~TMθ+θ2.Letgij=gdxi,dxj, ξ=∑kξjdxj, and ∇∂iL∂j=∑kΓijk∂k, we denote
(204)σi=−14∑s,tωs,tei~ces~cet~,ai=14∑s,tωs,tei~c¯es~c¯et~,ξj=gijξi,Γk=gijΓijk,σj=gijσi,aj=gijai.For a smooth vector fieldX on M, let cX denote the Clifford action. Since E is globally defined on M, we can perform computations of E in normal coordinates. Taking normal coordinates about x0, then σix0=0, aix0=0, ∂jc∂jx0=0, Γkx0=0, and gijx0=δij, so that
(205)Ex0=18∑ijklRijklc¯ei~c¯ej~cek~cel~−14s−∑icei~c¯∇ei~TMθ−θ2.For the Witten deformationDθ, 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 Dθ in the interval −∧,∧ with ∧∈ℝ+. It is expressed as
(206)I=trF^Dθ2∧2.Here,tr denotes the operator trace in the L2 completion of ΓM,STM and F^:ℝ+⟶ℝ+ is a cut-off function with support in the interval 0,1 which is constant near the origin. By Lemma 1.7.4 in [21], we have the heat trace asymptotics, for t⟶0,
(207)tre−tDθ2~∑m≥0tm−n/2a2mDθ2.One uses the Seeley-DeWitt coefficientsa2mDθ2 and t=∧−2 to obtain asymptotics for the spectral action when dimM=4,
(208)I=trF^Dθ2∧2~∧4F4a0Dθ2+∧2F2a2Dθ2+∧0F0a4Dθ2as∧⟶∞,with the first three moments of the cut-off function which are given by F4=∫0∞sF^sds, F2=∫0∞F^sds, and F0=F^0.We use Theorem 4.1.6 in [17] to obtain the first three coefficients of the heat trace asymptotics:
(209)a0Dθ2=4π−4/2∫Mtriddvol,a2Dθ2=4π−4/2∫Mtrs6+Edvol,a4Dθ2=4π−4/2360∫Mtr−12Rijij,kk+5RijijRklkl−2RijikRljlk+2RijklRijkl−60RijijE+180E2+60E,kk+30ΩijΩijdvol.By the Clifford action and cyclicity of the trace, we have(210)trcei=0,trceicej=0i≠j,trc¯ei~c¯ej~cek~cel~=0i≠j,tr∑icei~c¯∇ei~TMθ=0.So we obtain(211)a0Dθ2=4π−4/2∫Mtriddvol=π−2∫Mdvol,a2Dθ2=4π−4/2∫Mtrs6+Edvol=4π−4/2∫Mtrs6+18∑ijklRijklc¯ei~c¯ej~cek~cel~−14s−∑icei~c¯∇ei~TMθ−θ2dvol=−∫Ms12π2+θ2π2dvol.And we have(212)∫Mtr5RijijRklkl−60RijijE+180E2−12Rijij,kk+60Ekkdvol=∫Mtr5s2+60sE+180E2−tr12Δs+60−ΔtrEdvol=∫Mtrace54s2+180∑icei~c¯∇ei~TMθ2+4516∑ijklRijklc¯ei~c¯ej~cek~cel~2+30sθ2+180θ4−45∑ijklRijklc¯ei~c¯ej~cek~cel~∑pcep~c¯∇ep~TMθ+48Δs+960Δθ2dvol=∫M20s2+2880∑i∇ei~TMθ2+θ4+180∑ijklRijkl2+480sθ2dvol.AndtrΩijΩij is globally defined, so we only compute it in normal coordinates about x0 and the local orthonormal frame ei obtained by parallel transport along geodesics from x0. Then,
(213)ωs,tx0=0,∂icej~=0,ei~,ej~x0=0.Then, we have(214)Ωei~,ej~x0=∇e~i∧∗T∗M∇e~j∧∗T∗M−∇e~j∧∗T∗M∇e~i∧∗T∗M−∇e~i,e~j∧∗T∗M=−14∑s,t=1nRijstMc¯es~c¯et~−ces~cet~.So we have(215)trΩijΩijx0=tr116∑s,t,s1,t1=1RijstMRijs1t1Mc¯es~c¯et~−ces~cet~c¯es1~c¯et1~−ces1~cet1~=tr116∑s,t,s1,t1=1RijstMRijs1t1Mc¯es~c¯et~c¯es1~c¯et1~−c¯es~c¯et~ces1~cet1~−ces~cet~c¯es1~c¯et1~+ces~cet~ces1~cet1~=−4∑ijstRijstM2.Proposition 17.
The following equality holds:(216)a4Dθ2=15760π2∫M20s2+2880∑i∇ei~TMθ2+θ4+180∑ijklRijkl2+480sθ2−32RijikRljlk+32Rijik2−1920∑i,j,s,tRijstM2dvol,where s is the scalar curvature.



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*Source: 9090656-2020-03-06.xml*