Journal of Nonlinear Mathematical Physics

Volume 28, Issue 2, June 2021, Pages 254 - 275

Statistical de Rham Hodge Operators and the Kastler-Kalau-Walze Type Theorem for Manifolds With Boundary

Authors
Sining Wei, Yong Wang*
School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P.R. China
*Corresponding author: Email: wangy581@nenu.edu.cn
Corresponding Author
Yong Wang
Received 28 December 2020, Accepted 10 April 2021, Available Online 29 April 2021.
DOI
https://doi.org/10.2991/jnmp.k.210419.001How to use a DOI?
Keywords
Statistical de Rham Hodge operators, Lichnerowicz type formulas, Kastler-Kalau-Walze type theorem, noncommutative residue
Abstract

In this paper, we give the Lichnerowicz type formulas for statistical de Rham Hodge operators. Moreover, Kastler-Kalau-Walze type theorems for statistical de Rham Hodge operators on compact manifolds with (respectively without) boundary are proved.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

The noncommutative residue which is found in [5,19] plays a prominent role in noncommutative geometry. For this reason, it has been studied extensively by geometers. Connes derived a conformal 4-dimensional Polyakov action analogy by the noncommutative residue in [2]. Connes proved that the noncommutative residue on a compact manifold M coincided with the Dixmier’s trace on pseudodifferential operators of order −dimM in [3].

On the other hand, Connes had 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 gave a brute-force proof of this theorem in [8]. Kalau and Walze [7] proved this theorem in the normal coordinates system simultaneously. And then, for the Dirac operator D, Ackermann proved that the Wodzicki residue Wres(D−2) in turn was essentially the second coefficient of the heat kernel expansion of D2 in [1].

On the other hand, Wang generalized the Connes’ results to the case of manifolds with boundary in [13,14], and proved the Kastler-Kalau-Walze type theorems for the Dirac operator and the signature operator on lower-dimensional manifolds with boundary [15]. In [15,16], Wang computed Wres˜[π+D-1π+D-1] and Wres˜[π+D-2π+D-2] about symmetric operators and under the circumstances the boundary term vanished. Moreover, Wang got a nonvanishing boundary term for Wres˜[π+D-1π+D-3] in [17] and gave a theoretical explanation for the gravitational action on boundary. In others words, Wang provided a kind of method to study the Kastler-Kalau-Walze type theorem for manifolds with boundary.

In [6], Iochum and Levy computed heat kernel coefficients for Dirac operators with one-form perturbations and proved that there were no tadpoles for compact spin manifolds without boundary. Recently, we studied the Lichnerowicz-type formulas for modified Novikov operators. We proved Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) boundary in [18]. In [10], Barbara Opozda introduced statistical de Rham Hodge operators on manifolds with statistical structure. The aim of this paper is to prove the Kastler-Kalau-Walze type theorem for statistical de Rham Hodge operators on manifolds without (with) boundary, and also give the Lichnerowicz formulas about statistical de Rham Hodge operators.

The paper is organized as follows: in Section 2, we give the definition of statistical de Rham Hodge operators and their Lichnerowicz formulas. We also give the basic facts and formulas about the noncommutative residue for manifolds with boundary. In Section 3 and in Section 4, we give some expressions and symbols of operators associate with statistical de Rham Hodge operators. Moveover, we also prove the Kastler- Kalau-Walze type theorems for statistical de Rham Hodge operators on 4-dimensional and 6-dimensional manifolds with boundary.

2. STATISTICAL de RHAM HODGE OPERATORS AND THEIR LICHNEROWICZ FORMULAS

In this section, we prepare some basic notions about statistical de Rham Hodge operators. For details of the geometry of statistical manifolds, see [10].

Let M be a n-dimensional (n ≥ 3) Riemannian manifold with a positive definite Riemannian metric g, and ^ be a connection. We assume that M is oriented. Let ∇L be the Levi-Civita connection for g and VolM be the volume form determined by g.

For all X, Y, ZTxM, xM, if ^ is satisfying the following Codazzi condition:

(^Xg)(Y,Z)=(^Yg)(X,Z), (2.1)
we call a structure (g, ^) is a statistical structure, and a connection ^ is a statistical connection for g. Moreover, we define a statistical manifold with statistical structure by (M, g, ^).

We suppose that (M, g, ^) is statistical manifold. Let K be tensor field, where K: Γ(TM) × Γ(TM) → Γ(TM) and Γ(TM) denotes the algebra of smooth vector fields on M. Let K be the difference tensor between ^ and ∇L, that is

^XY=XLY+KXY. (2.2)

Let K(X, Y) stand for KXY. The de Rham derivative d is a differential operator on C(M; ∧*T*M). Then we have the de Rham coderivative δ = d* and the symmetric operator D = d + δ. The standard Hodge Laplacian is defined by

Δ=δd+dδ. (2.3)

For a statistical manifold (M, g, ^), we will investigate a (Lichnerowicz) Laplacian relative to the connection ^. If f is a function, then we set

Δ^f=-div^gradf. (2.4)

We now extend the definition (2.4) and set E = traceg K(·,·). For any differential form ν, we set

Δ^ν=(δ-l(E))dν+d(δ-l(E))ν, (2.5)
where l(E) is the contraction operator.

By the above definition and Lemma 6.4 in [10], we have

Δ^=(δ-l(E)+d)2. (2.6)

For v ∈ Γ(TM), we define the generalized statistical de Rham Hodge operators by

Di=d+δ+λil(v),(i=1,2);Di*=d+δ+λiɛ(v*),(i=1,2), (2.7)
where λi is a real number and v* = g(v, ·).

In the local coordinates {xi; 1 ≤ in} and the fixed orthonormal frame {e1, ···, en}, the connection matrix (ωs,t) is defined by

L(e1,,en)=(e1,,en)(ωs,t). (2.8)

Let ε(ej*), l(ej) be the exterior and interior multiplications respectively and c(ej) be the Clifford action. Write

c(ej)=ɛ(ej*)-l(ej),c¯(ej)=ɛ(ej*)+l(ej). (2.9)

Moreover, we assume that i is a natural local frame on TM and (gij)1≤i,jn is the inverse matrix associated with the metric matrix (gij)1≤i,j≤n on M. The statistical de Rham Hodge operators Di and Di*(i=1,2) are defined by

Di=d+δ+λil(v)=i=1nc(ei)[ei+14s,tωs,t(ei)[c¯(es)c¯(et)-c(es)c(et)]]+λil(v); (2.10)
Di*=d+δ+λiɛ(v*)=i=1nc(ei)[ei+14s,tωs,t(ei)[c¯(es)c¯(et)-c(es)c(et)]]+λiɛ(v*). (2.11)

Let gij = g(dxi, dxj), ξ = ∑j ξjdxj and iLj=kΓijkk, we denote that

σi=-14s,tωs,t(ei)c(es)c(et);ai=14s,tωs,t(ei)c¯(es)c¯(et);ξj=gijξi;Γk=gijΓijk;σj=gijσi;aj=gijai. (2.12)

Then the statistical de Rham Hodge operators Di and Di* can be written as

Di=i=1nc(ei)[ei+ai+σi]+λil(v),(i=1,2); (2.13)
Di*=i=1nc(ei)[ei+ai+σi]+λiɛ(v*),(i=1,2). (2.14)

On the other hand, we recall some basic facts and formulas about Boutet de Monvel’s calculus and the definition of the noncommutative residue for manifolds with boundary (for details see Section 2 in [15]).

Let UM be a collar neighborhood of ∂M which is diffeomorphic with ∂M × [0, 1). fC([0, 1)) means that there is f˜C((-ɛ,1)) such that f˜|[0,1)=f for a small positive number ε. Let h(xn) ∈ C([0, 1)) and h(xn) > 0. By the definition of h(xn) ∈ C([0, 1)) and h(xn) > 0, there exists h˜C((-ɛ,1)) such that h˜|[0,1)=h and h˜>0 for some sufficiently small ɛ > 0. Then there exists a metric g^ on M^=MMM×(-ɛ,0] which has the form on UMM × (−ε, 0]

g^=1h˜(xn)gM+dxn2, (2.15)
such that g^|M=g. We fix a metric g^ on the M^ such that g^|M=g.

Let

F:L2(Rt)L2(Rλ);F(u)(λ)=e-ivtu(t)dt

denote the Fourier transformation and φ(R+¯)=r+φ(R)(similarly define φ(R-¯)), where ϕ(R) denotes the Schwartz space and

r+:C(R)C(R+¯);ff|R+¯;R+¯={x0;xR}.

We define H+=F(φ(R+¯));H0-=F(φ(R-¯)) which are orthogonal to each other. We have the following property: hH+(H0-) if and only if hC (R) which has an analytic extension to the lower (upper) complex half-plane {Imξ < 0} ({Imξ > 0}) such that for all nonnegative integer l,

dlhdξl(ξ)k=1dldξl(ckξk),
as |ξ|→ +∞, Imξ ≤ 0 (Imξ ≥ 0).

Let H′ be the space of all polynomials and H-=H0-H;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˜,

π+h(ξ0)=12πilimu0-Γ+h(ξ)ξ0+iu-ξdξ, (2.16)
where Γ+ is a Jordan close curve included Im(ξ) > 0 surrounding all the singularities of h in the upper half-plane and ξ0R. Similarly, define π′ on H˜,
πh=12πΓ+h(ξ)dξ. (2.17)

So, π′(H) = 0. For hHL1(R), πh=12πRh(v)dv and for hH+L1(R), π′h = 0.

Next we give the basic notions of Laplace type operators. Let M be a smooth compact oriented Riemannian n-dimensional manifolds without boundary and V′ be a vector bundle on M. Any differential operator P of Laplace type has locally the form

P=-(gijij+Aii+B), (2.18)
where Ai and B are smooth sections of the endomorphism End(V′) on M. If P is a Laplace type operator with the form (2.18), then there is a unique connection ∇ on V′ and a unique endomorphism E′ such that
P=-[gij(ij-iLj)+E], (2.19)
where ∇L is the Levi-Civita connection on M. Moreover (with local frames of TM and V′), Δi = Δi + ωi and E′ are related to gij, Ai and B through
ωi=12gij(Ai+gklΓkljid), (2.20)
E=B-gij(i(ωj)+ωiωj-ωkΓijk), (2.21)
where Γklj is the Christoffel coefficient of ∇L.

By the above definitions, we establish the main theorem in this section. One has the following Lichneriowicz formulas.

Theorem 2.1.

The following equalities hold:

D1D2=-[gij(ij-iLj)]-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+14i[λ2c(ei)l(v)+λ1l(v)c(ei)]2-12[λ1ejTM(l(v))c(ej)-λ2c(ej)ejTM(l(v))], (2.22)
D2*D1*=-[gij(ij-iLj)]-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+14i[λ1c(ei)ɛ(v*)+λ2ɛ(v*)c(ei)]2-12[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(ɛ(v*))], (2.23)
D2*D1=-[gij(ij-iLj)]-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+λ1λ2ɛ(v*)l(v)+14i[λ1c(ei)l(v)+λ2ɛ(v*)c(ei)]2-12[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))], (2.24)
where s is the scalar curvature.

Proof. By (2.13), we note that

D1D2=(d+δ)2+λ2(d+δ)l(v)+λ1l(v)(d+δ)+λ1λ2[l(v)]2. (2.25)

By [20], the local expression of (d + δ)2 is

(d+δ)2=-Δ0-18ijklRijklc¯(ei˜)c¯(ej˜)c(ek˜)c(el˜)+14s. (2.26)

By [20] and [1], we have

-Δ0=Δ=-gij(iLjL-ΓijkkL). (2.27)
λ2(d+δ)l(v)+λ1l(v)(d+δ)=i,jgi,j[λ2c(i)l(v)+λ1l(v)c(i)]j+i,jgi,j[λ1l(v)c(i)(σi+ai)+λ2c(i)j(l(v))+λ2c(i)(σi+ai)l(v)]; (2.28)
[l(v)]2=0. (2.29)
then we obtain
D1D2=-i,jgi,j[ij+2σij+2aij-Γi,jkk+(iσj)+(iaj)+σiσj+σiaj+aiσj+aiaj-Γi,jkσk-Γi,jkak]+i,jgi,j[λ2c(i)l(v)+λ1l(v)c(i)]j-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+i,jgi,j[λ2c(i)j(l(v))+λ2c(i)(σi+ai)l(v)+λ1l(v)c(i)(σi+ai)]. (2.30)

Similarly, we have

D2*D1*=-i,jgi,j[ij+2σij+2aij-Γi,jkk+(iσj)+(iaj)+σiσj+σiaj+aiσj+aiaj-Γi,jkσk-Γi,jkak]+i,jgi,j[λ1c(i)ɛ(v*)+λ2ɛ(v*)c(i)]j-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+i,jgi,j[λ1c(i)j(ɛ(v*))+λ1c(i)(σi+ai)ɛ(v*)+λ2ɛ(v*)c(i)(σi+ai)]. (2.31)
and
D2*D1=-i,jgi,j[ij+2σij+2aij-Γi,jkk+(iσj)+(iaj)+σiσj+σiaj+aiσj+aiaj-Γi,jkσk-Γi,jkak]+i,jgi,j[λ1c(i)l(v)+λ2ɛ(v*)c(i)]j-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)+14s+λ1λ2×ɛ(v*)l(v)+i,jgi,j[λ1c(i)j(l(v))+λ1c(i)(σi+ai)l(v)+λ2ɛ(v*)c(i)(σi+ai)]. (2.32)

By (2.18), (2.20) and (2.30) we have

(ωi)D1D2=σi+ai-12[λ2c(i)l(v)+λ1l(v)c(i)], (2.33)
ED1D2=i,jgi,j[i(σj+aj)+σiσj+σiaj+aiσj-Γijkσk-Γijkak+aiaj-λ2c(i)j(l(v))-c(i)(σj+aj)λ2l(v)-λ1l(v)c(i)(σj+aj)]+18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)-14s-λ1λ2[l(v)]2-i,jgi,j{i(σj+aj)-12i[c(j)λ2l(v)+λ1l(v)c(j)]+[σi+ai-12[λ2c(i)l(v)+λ1l(v)c(i)]]×[σj+aj-12[λ2c(j)l(v)+λ1l(v)c(j)]]-[σk+ak-12[c(k)λ2l(v)+λ1l(v)c(k)]]Γijk}. (2.34)

Let c(Y) denote the Clifford action, where Y is a smooth vector field on M. Since E′ is globally defined on M, taking normal coordinates at x0, we have σi(x0) = 0, ai(x0) = 0, j[c(j)](x0) = 0, Γk(x0) = 0, gij(x0)=δij. By (2.21), then we have

ED1D2(x0)=18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)-14s-14i[λ2c(ei)l(v)+λ1l(v)c(ei)]2+12[λ1ejTM(l(v))c(ej)-λ2c(ej)ejTM(l(v))]. (2.35)

Similarly, we have

ED2*D1*(x0)=18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)-14s-14i[λ1c(ei)ɛ(v*)+λ2ɛ(v*)c(ei)]2+12[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(ɛ(v*))]. (2.36)
ED2*D1(x0)=18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)-14s-14i[λ1c(ei)l(v)+λ2ɛ(v*)c(ei)]2+12[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]-λ1λ2ɛ(v*)l(v). (2.37)
which, together with (2.19), we complete the proof.

The noncommutative residue of a generalized laplacian Δ is expressed by [1], as

(n-2)ϕ(Δ)=(4π)-n2Γ(n2)res˜(Δ-n2+1), (2.38)
where φ(Δ) denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of Δ. Since D1D2, D2*D1* and D2*D1 are generalized laplacian opeartors, we have
Wres(D1D2)-n-22=(n-2)(4π)n2(n2-1)!Mtrace(16s+ED1D2)dVolM, (2.39)
where Wres is the noncommutative residue.

Similarly, we have

Wres(D2*D1*)-n-22=(n-2)(4π)n2(n2-1)!Mtrace(16s+ED2*D1*)dVolM, (2.40)
Wres(D2*D1)-n-22=(n-2)(4π)n2(n2-1)!Mtrace(16s+ED2*D1)dVolM. (2.41)

By Theorem 2.1 and its proof, we have

Theorem 2.2.

For even n-dimensional compact oriented manifolds without boundary, the following equalities holds:

Wres(D1D2)-n-22=(n-2)(4π)n2(n2-1)!M2n(-112s-14(λ12+λ22)|v|2)dVolM, (2.42)
Wres(D2*D1*)-n-22=(n-2)(4π)n2(n2-1)!M2n(-112s-14(λ12+λ22)|v*|2)dVolM, (2.43)
Wres(D2*D1)-n-22=(n-2)(4π)n2(n2-1)!M[2n(-112s-λ12+λ22-2nλ1λ2+4λ1λ28|v|2)+12tr[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]]dVolM, (2.44)
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 for 4-dimensional oriented compact manifold with boundary about statistical de Rham Hodge Operators.

Denote by M a n-dimensional manifold with boundary ∂M. We assume that M is compact and oriented. Let ℬ be Boutet de Monvel’s algebra, we now recall the main theorem in [4,15].

Theorem 3.1.

[4](Fedosov-Golse-Leichtnam-Schrohe) Let X and ∂X be connected, dimX = n ≥ 3, A=(π+P+GKTS)∈ ℬ, and denote by p, b and s the local symbols of P, G and S respectively. Define:

Wres˜(A)=XStraceE[p-n(x,ξ)]σ(ξ)dx+2πXS{traceE[(trb-n)(x,ξ)]+traceF[s1-n(x,ξ)]}σ(ξ)dx. (3.1)

Then

a) Wres˜([A,B])=0, for any A, Bℬ;

b) It is a unique continuous trace on ℬ/ℬ−∞.

Definition 3.2.

[15] Lower dimensional volume of spin manifolds with boundary is defined by

Voln(p1,p2)MWres˜[π+D-p1π+D-p2]. (3.2)

By (2.1.4)-(2.1.8) in [15], we get

Wres˜[π+D-p1π+D-p2]=M|ξ|=1trace*T*M[σ-n(D-p1-p2)]σ(ξ)dx+MΦ, (3.3)
and
Φ=|ξ|=1-+j,k=0(-i)|α|+j+k+1α!(j+k+1)!×trace*T*M[xnjξαξnkσr+(D-p1)(x,0,ξ,ξn)×xαξnj+1xnkσl(D-p2)(x,0,ξ,ξn)]dξnσ(ξ)dx, (3.4)
where the sum is taken over r + lk − |α|− j − 1 = −n, r ≤ −p1, l ≤ −p2.

For any fixed point x0 ∈ ∂M, we choose the normal coordinates U of x0 in ∂M (not in M) and compute Φ(x0) in the coordinates U˜=U×[0,1)M and the metric 1h(xn)gM+dxn2. The dual metric of gM on U˜ is h(xn)gM+dxn2. Write gijM=gM(xi,xj);gMij=gM(dxi,dxj), then

[gi,jM]=[1h(xn)[gi,jM]001];[gMi,j]=[h(xn)[gMi,j]001], (3.5)
and
xsgijM(x0)=0,1i,jn-1;gijM(x0)=δij. (3.6)

We will give following three lemmas as computation tools.

Lemma 3.3.

[15] With the metric gM on M near the boundary

xj(|ξ|gM2)(x0)={0ifj<n,h(0)|ξ|gM2ifj=n; (3.7)
xj[c(ξ)](x0)={0ifj<n,xn(c(ξ))(x0)ifj=n, (3.8)
where ξ = ξ′ + ξndxn.

Lemma 3.4.

[15] With the metric gM on M near the boundary

ωs,t(ei)(x0)={ωn,i(ei)(x0)=12h(0)ifs=n,t=i,i<n;ωi,n(ei)(x0)=-12h(0)ifs=i,t=n,i<n;ωs,t(ei)(x0)=0othercases, (3.9)

where (ωs,t) denotes the connection matrix of Levi-Civita connectionL.

Lemma 3.5.

[15] When i < n, then

Γiin(x0)=12h(0);Γnii(x0)=-12h(0);Γini(x0)=-12h(0),
in other cases, Γsti(x0)=0.

By (3.3) and (3.4), we firstly compute

Wres˜[π+D2-1π+D1-1]=M|ξ|=1trace*T*M[σ-4(D1D2)-1]σ(ξ)dx+MΦ1, (3.10)
where
Φ1=|ξ|=1-+j,k=0(-i)|α|+j+k+1α!(j+k+1)!×trace*T*M[xnjξαξnkσr+(D2-1)(x,0,ξ,ξn)×xαξnj+1xnkσl(D1-1)(x,0,ξ,ξn)]dξnσ(ξ)dx, (3.11)
and the sum is taken over r + lkj − |α|= −3, r ≤ −1, l ≤ −1.

Locally we can use Theorem 2.2 (2.42) to compute the interior of Wres˜[π+D2-1π+D1-1], we have

M|ξ|=1trace*T*M[σ-4(D1D2)-1]σ(ξ)dx=32π2M(-43s-4(λ12+λ22)|v|2)dVolM. (3.12)

So we only need to compute ∫M Φ1. Let us now turn to compute the symbols of some operators. By (2.10)-(2.14), then we have the following symbols of some operators.

Lemma 3.6.

The following identities hold:

σ1(Dj)=σ1(Dj*)=ic(ξ),(j=1,2);σ0(Dj)=14i,s,tωs,t(ei)c(ei)c¯(es)c¯(et)-14i,s,tωs,t(ei)c(ei)c(es)c(et)+λjl(v),(j=1,2);σ0(Dj*)=14i,s,tωs,t(ei)c(ei)c¯(es)c¯(et)-14i,s,tωs,t(ei)c(ei)c(es)c(et)+λjɛ(v*),(j=1,2). (3.13)

Write

Dxα=(-i)|α|xα;σ(D)=p1+p0;σ(D-1)=j=1q-j. (3.14)

By the composition formula of pseudodifferential operators, we have

1=σ(DD-1)=α1α!ξα[σ(D)]Dxα[σ(D-1)]=(p1+p0)(q-1+q-2+q-3+)+j(ξjp1+ξjp0)(Dxjq-1+Dxjq-2+Dxjq-3+)=p1q-1+(p1q-2+p0q-1+jξjp1Dxjq-1)+, (3.15)
so
q-1=p1-1;q-2=-p1-1[p0p1-1+jξjp1Dxj(p1-1)]. (3.16)

By Lemma 3.6, we have some symbols of operators.

Lemma 3.7.

The following identities hold:

σ-1(Dj-1)=σ-1((Dj*)-1)=ic(ξ)|ξ|2,(j=1,2);σ-2(Dj-1)=c(ξ)σ0(Dj)c(ξ)|ξ|4+c(ξ)|ξ|6jc(dxj)[xj(c(ξ))|ξ|2-c(ξ)xj(|ξ|2)],(j=1,2);σ-2((Dj*)-1)=c(ξ)σ0(Dj*)c(ξ)|ξ|4+c(ξ)|ξ|6jc(dxj)[xj(c(ξ))|ξ|2-c(ξ)xj(|ξ|2)],(j=1,2). (3.17)

From the remark above, now we can compute Φ1(see formula (3.11) for the definition of Φ1). We use tr as shorthand of trace. Since n = 4, then tr T M [id]= 16, since the sum is taken over r + lkj − |α|= −3, r ≤ −1, l ≤ −1, we have the following five cases:

case 1) I) r = −1, l = −1, k = j = 0, |α|= 1.

By (3.11), we get

case1)I)=-|ξ|=1-+|α|=1tr[ξαπξn+σ-1(D2-1)×xαξnσ-1(D1-1)](x0)dξnσ(ξ)dx. (3.18)

By Lemma 3.3, for i < n, then

xi(ic(ξ)|ξ|2)(x0)=ixi[c(ξ)](x0)|ξ|2-ic(ξ)xi(|ξ|2)(x0)|ξ|4=0, (3.19)
so case 1) I) vanishes.

case 1) II) r = −1, l = −1, k = |α|= 0, j = 1.

By (3.11), we get

case1)II)=-12|ξ|=1-+tr[xnπξn+σ-1(D2-1)×ξn2σ-1(D1-1)](x0)dξnσ(ξ)dx. (3.20)

By Lemma 3.7, we have

ξn2σ-1(D1-1)(x0)=i(-6ξnc(dxn)+2c(ξ)|ξ|4+8ξn2c(ξ)|ξ|6); (3.21)
xnσ-1(D2-1)(x0)=ixnc(ξ)(x0)|ξ|2-ic(ξ)|ξ|2h(0)|ξ|4. (3.22)

By (2.16), (2.17) and the Cauchy integral formula we have

πξn+[c(ξ)|ξ|4](x0)||ξ|=1=πξn+[c(ξ)+ξnc(dxn)(1+ξn2)2]=12πilimu0-Γ+c(ξ)+ηnc(dxn)(ηn+i)2(ξn+iu-ηn)(ηn-i)2dηn=-(iξn+2)c(ξ)+ic(dxn)4(ξn-i)2. (3.23)

Similarly, we have

πξn+[ixnc(ξ)|ξ|2](x0)||ξ|=1=xn[c(ξ)](x0)2(ξn-i). (3.24)
then
πξn+xnσ-1(D2-1)||ξ|=1=xn[c(ξ)](x0)2(ξn-i)+ih(0)[(iξn+2)c(ξ)+ic(dxn)4(ξn-i)2]. (3.25)

By the relation of the Clifford action and trAB = trBA, we have the equalities:

tr[c(ξ)c(dxn)]=0;tr[c(dxn)2]=-16;tr[c(ξ)2](x0)||ξ|=1=-16;tr[xnc(ξ)c(dxn)]=0;tr[xnc(ξ)c(ξ)](x0)||ξ|=1=-8h(0);tr[c¯(ei)c¯(ej)c(ek)c(el)]=0(ij). (3.26)

By (3.26) and a direct computation, we have

h(0)tr[(iξn+2)c(ξ)+ic(dxn)4(ξn-i)2×(6ξnc(dxn)+2c(ξ)(1+ξn2)2-8ξn2[c(ξ)+ξnc(dxn)](1+ξn2)3)](x0)||ξ|=1=-16h(0)-2iξn2-ξn+i(ξn-i)4(ξn+i)3. (3.27)

Similarly, we have

-itr[(xn[c(ξ)](x0)2(ξn-i))×(6ξnc(dxn)+2c(ξ)(1+ξn2)2-8ξn2[c(ξ)+ξnc(dxn)](1+ξn2)3)](x0)||ξ|=1=-8ih(0)3ξn2-1(ξn-i)4(ξn+i)3. (3.28)

Then

case1)II)=-|ξ|=1-+4ih(0)(ξn-i)2(ξn-i)4(ξn+i)3dξnσ(ξ)dx=-4ih(0)Ω3Γ+1(ξn-i)2(ξn+i)3dξndx=-4ih(0)Ω32πi[1(ξn+i)3]|ξn=idx=-32πh(0)Ω3dx,

where Ω3 is the canonical volume of S3.

case 1) III) r = −1, l = −1, j = |α|= 0, k = 1.

By (3.11), we get

case1)III)=-12|ξ|=1-+tr[ξnπξn+σ-1(D2-1)×ξnxnσ-1(D1-1)](x0)dξnσ(ξ)dx. (3.29)

By Lemma 3.7, we have

ξnxnσ-1(D1-1)(x0)||ξ|=1=-ih(0)[c(dxn)|ξ|4-4ξnc(ξ)+ξnc(dxn)|ξ|6]-2ξnixnc(ξ)(x0)|ξ|4; (3.30)
ξnπξn+σ-1(D2-1)(x0)||ξ|=1=-c(ξ)+ic(dxn)2(ξn-i)2. (3.31)

Similar to case 1) II), we have

tr{c(ξ)+ic(dxn)2(ξn-i)2×ih(0)[c(dxn)|ξ|4-4ξnc(ξ)+ξnc(dxn)|ξ|6]}=8h(0)i-3ξn(ξn-i)4(ξn+i)3 (3.32)
and
tr[c(ξ)+ic(dxn)2(ξn-i)2×2ξnixnc(ξ)(x0)|ξ|4]=-8ih(0)ξn(ξn-i)4(ξn+i)2. (3.33)

So we have

case1)III)=-|ξ|=1-+h(0)4(i-3ξn)(ξn-i)4(ξn+i)3dξnσ(ξ)dx-|ξ|=1-+h(0)4iξn(ξn-i)4(ξn+i)2dξnσ(ξ)dx=-h(0)Ω32πi3![4(i-3ξn)(ξn+i)3](3)|ξn=idx+h(0)Ω32πi3![4iξn(ξn+i)2](3)|ξn=idx=32πh(0)Ω3dx. (3.34)

case 2) r = −2, l = −1, k = j = |α|= 0.

By (3.11), we get

case2)=-i|ξ|=1-+tr[πξn+σ-2(D2-1)×ξnσ-1(D1-1)](x0)dξnσ(ξ)dx. (3.35)

By Lemma 3.7, we have

σ-2(D2-1)(x0)=c(ξ)σ0(D2)(x0)c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)|ξ|M2], (3.36)
where
σ0(D2)(x0)=14s,t,iωs,t(ei)(x0)c(ei)c¯(es)c¯(et)-14s,t,iωs,t(ei)(x0)c(ei)c(es)c(et))+λ2l(v). (3.37)

Write

A(x0)=14s,t,iωs,t(ei)(x0)c(ei)c¯(es)c¯(et);B(x0)=-14s,t,iωs,t(ei)(x0)c(ei)c(es)c(et)). (3.38)

Then

πξn+σ-2(D2-1(x0))||ξ|=1=πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]+πξn+[λ2c(ξ)(l(v)(x0))c(ξ)(1+ξn2)2]+πξn+[c(ξ)B(x0)c(ξ)+c(ξ)c(dxn)xn[c(ξ)](x0)(1+ξn2)2-h(0)c(ξ)c(dxn)c(ξ)(1+ξn2)3]. (3.39)

By direct calculations, we have

πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]=πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]+πξn+[ξnc(ξ)A(x0)c(dxn)(1+ξn2)2]+πξn+[ξnc(dxn)A(x0)c(ξ)(1+ξn2)2]+πξn+[ξn2c(dxn)A(x0)c(dxn)(1+ξn2)2]=-c(ξ)A(x0)c(ξ)(2+iξn)4(ξn-i)2+ic(ξ)A(x0)c(dxn)4(ξn-i)2+ic(dxn)A(x0)c(ξ)4(ξn-i)2+-iξnc(dxn)A(x0)c(dxn)4(ξn-i)2. (3.40)

Since

c(dxn)A(x0)=-14h(0)i=1n-1c(ei)c¯(ei)c(en)c¯(en), (3.41)
by the relation of the Clifford action and trAB = trBA, we have the equalities:
tr[c(ei)c¯(ei)c(en)c¯(en)]=0(i<n);tr[Ac(dxn)]=0;tr[c¯(ξ)c(dxn)]=0; (3.42)

Since

ξnσ-1(Dv-1)=ξnq-1(x0)||ξ|=1=i[c(dxn)1+ξn2-2ξnc(ξ)+2ξn2c(dxn)(1+ξn2)2]. (3.43)

By (3.40), (3.42) and (3.43), we have

tr[πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]×ξnσ-1(D1-1)(x0)]||ξ|=1=12(1+ξn2)2tr[c(ξ)A(x0)]+i2(1+ξn2)2tr[c(dxn)A(x0)]=12(1+ξn2)2tr[c(ξ)A(x0)]. (3.44)

We note that i < n, ∫|ξ′|=1{ξi1 ξi2 ··· ξi2d+1}σ(ξ) = 0, so tr[c(ξ′)A(x0)] has no contribution for computing case 2).

By direct calculations, we have

πξn+[c(ξ)B(x0)c(ξ)+c(ξ)c(dxn)xn[c(ξ)](x0)(1+ξn2)2]-h(0)πξn+[c(ξ)c(dxn)c(ξ)(1+ξn)3]P1-P2, (3.45)
where
P1=-14(ξn-i)2[(2+iξn)c(ξ)b02(x0)c(ξ)+iξnc(dxn)b02(x0)c(dxn)+(2+iξn)c(ξ)c(dxn)xnc(ξ)+ic(dxn)b02(x0)c(ξ)+ic(ξ)b02(x0)c(dxn)-ixnc(ξ)] (3.46)
and
P2=h(0)2[c(dxn)4i(ξn-i)+c(dxn)-ic(ξ)8(ξn-i)2+3ξn-7i8(ξn-i)3[ic(ξ)-c(dxn)]]. (3.47)

By (3.43) and (3.46), we have

tr[P1×ξnσ-1(D1-1)]||ξ|=1=-6ih(0)(1+ξn2)2+2h(0)ξn2-iξn-2(ξn-i)(1+ξn2)2. (3.48)

By (3.43) and (3.47), we have

tr[P2×ξnσ-1(D1-1)]||ξ|=1=i2h(0)-iξn2-ξn+4i4(ξn-i)3(ξn+i)2tr[id]=8ih(0)-iξn2-ξn+4i4(ξn-i)3(ξn+i)2. (3.49)

By (3.48) and (3.49), we have

-i|ξ|=1-+tr[(P1-P2)×ξnσ-1(D1-1)](x0)dξnσ(ξ)dx=-Ω3Γ+8[-34h(0)](ξn-i)+ih(0)(ξn-i)3(ξn+i)2dξndx=92πh(0)Ω3dx. (3.50)

Similar to (3.44), we have

tr[πξn+[λ2c(ξ)l(v)(x0)c(ξ)(1+ξn2)2]×ξnσ-1(D1)-1)(x0)]||ξ|=1=12(1+ξn2)2tr[λ2c(ξ)l(v)(x0)]+i2(1+ξn2)2tr[λ2c(dxn)l(v)(x0)]. (3.51)

By the relation of the Clifford action and trAB = trBA, we have the equalities:

tr[c(dxn)l(v)]=8v,dxn;tr[c(ξ)l(v)]=8v,ξ; (3.52)

By (3.51) and (3.52), we have

-i|ξ|=1-+tr[πξn+[λ2c(ξ)(l(v))c(ξ)(1+ξn2)2]×ξnσ-1(D1-1)](x0)dξnσ(ξ)dx=2λ2πv,dxnΩ3dx. (3.53)

By (3.44), (3.50) and (3.53), we have

case2)=[92πh(0)+2λ2πv,dxn]Ω3dx. (3.54)

case 3) r = −1, l = −2, k = j = |α|= 0.

By (3.11), we get

case3)=-i|ξ|=1-+tr[πξn+σ-1(D2-1)×ξnσ-2(D1-1)](x0)dξnσ(ξ)dx. (3.55)

By (2.16) and Lemma 3.7, we have

πξn+σ-1(D2-1)||ξ|=1=c(ξ)+ic(dxn)2(ξn-i). (3.56)

By (3.36), (3.37) and (3.38), we have

ξnσ-2(D1-1)(x0)||ξ|=1=ξn{c(ξ)[A(x0)+B(x0)+(λ1l(v)(x0))]c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)]}=ξn{c(ξ)A(x0)]c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)]}+ξnc(ξ)B(x0)c(ξ)|ξ|4+λ1ξnc(ξ)(l(v)(x0))c(ξ)|ξ|4. (3.57)

By direct calculation, we have

ξnc(ξ)A(x0)c(ξ)|ξ|4=c(dxn)A(x0)c(ξ)|ξ|4+c(ξ)A(x0)c(dxn)|ξ|4-4ξnc(ξ)A(x0)c(ξ)|ξ|6; (3.58)
ξnc(ξ)(l(v)(x0))c(ξ)|ξ|4=c(dxn)(l(v)(x0))c(ξ)|ξ|4+c(ξ)(l(v)(x0))c(dxn)|ξ|4-4ξnc(ξ)(l(v)(x0))c(ξ)|ξ|4. (3.59)

Write

P3=c(ξ)B(x0)c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)],

then

ξn(P3)=1(1+ξn2)3[(2ξn-2ξn3)c(dxn)Bc(dxn)+(1-3ξn2)c(dxn)Bc(ξ)+(1-3ξn2)c(ξ)Bc(dxn)-4ξnc(ξ)Bc(ξ)+(3ξn2-1)xnc(ξ)-4ξnc(ξ)c(dxn)xnc(ξ)+2h(0)c(ξ)+2h(0)ξnc(dxn)]+6ξnh(0)c(ξ)c(dxn)c(ξ)(1+ξn2)4. (3.60)

By (3.56) and (3.58), we have

tr[πξn+σ-1(D2-1)×ξnc(ξ)Ac(ξ)|ξ|4](x0)||ξ|=1=-1(ξ-i)(ξ+i)3tr[c(ξ)A(x0)]+i(ξ-i)(ξ+i)3tr[c(dxn)A(x0)]. (3.61)

By (3.42), we have

tr[πξn+σ-1(D2-1)×ξnc(ξ)Ac(ξ)|ξ|4](x0)||ξ|=1=-1(ξ-i)(ξ+i)3tr[c(ξ)A(x0)]. (3.62)

We note that i < n, ∫|ξ′|=1{ξi1 ξi2 ··· ξi2d+1}σ(ξ′) = 0, so tr[c(ξ′)A(x0)] has no contribution for computing case 3).

By (3.56) and (3.60), we have

tr[πξn+σ-1(D2-1)×ξn(P3)](x0)||ξ|=1=12h(0)(iξn2+ξn-2i)(ξ-i)3(ξ+i)3+48h(0)iξn(ξ-i)3(ξ+i)4, (3.63)
then
-iΩ3Γ+[12h(0)(iξn2+ξn-2i)(ξn-i)3(ξn+i)3+48h(0)iξn(ξn-i)3(ξn+i)4]dξndx=-92πh(0)Ω3dx. (3.64)

By (3.56) and (3.59), we have

tr[πξn+σ-1(D2-1)×ξnc(ξ)(λ1l(v))c(ξ)|ξ|4](x0)||ξ|=1=-1(ξ-i)(ξ+i)3tr[c(ξ)(λ1l(v))(x0)]+i(ξ-i)(ξ+i)3tr[c(dxn)(λ1l(v))(x0)]. (3.65)

By (3.52), we have

-i|ξ|=1-+tr[πξn+σ-1(D2-1)×ξnc(ξ)(λ1l(v))c(ξ)|ξ|4](x0)dξnσ(ξ)dx=-i|ξ|=1-+i(ξ-i)(ξ+i)3[tr[c(dxn)(λ1l(v))(x0)]+itr[c(ξ)(λ1l(v))(x0)]]dξnσ(ξ)dx=-π4[tr[c(dxn)(λ1l(v))]+itr[c(ξ)(λ1l(v))(x0)]]Ω3dx=-2πλ1v,dxnΩ3dx. (3.66)

So we have

case3)=[-92πh(0)-2λ1πv,dxn]Ω3dx. (3.67)

Since Φ1 is the sum of the cases 1), 2) and 3), Φ1 = 2(λ2λ1)πv, dxn〉Ω3dx′.

Theorem 3.8.

Let M be a 4-dimensional oriented compact manifold with the boundary ∂M and the metric gM as above, Di (i = 1, 2) be statistical de Rham Hodge Operators on M^, then

Wres˜[π+D2-1π+D1-1]=32π2M(-43s-4(λ12+λ22)|v|2)dVolM+M2(λ2-λ1)πv,dxnΩ3dx, (3.68)

where s is the scalar curvature.

Let D = d + δ + l(v), D = d + δ + ε(v).

Corollary 3.9.

For 4-dimensional oriented compact manifold M with the boundary ∂M, when λ1 = λ2 = 1, we get

Wres˜[π+D-1π+D-1]=32π2M(-43s-8|v|2)dVolM,
where s is the scalar curvature.

On the other hand, we also prove the Kastler-Kalau-Walze type theorem for 4-dimensional manifolds with boundary associated with (Di*)2(i=1,2). By (3.3) and (3.4), we will compute

Wres˜[π+(D1*)-1π+(D2*)-1]=M|ξ|=1trace*T*M[σ-4((D2*D1*)-1)]σ(ξ)dx+MΦ2, (3.69)
where
Φ2=|ξ|=1-+j,k=0(-i)|α|+j+k+1α!(j+k+1)!×trace*T*M[xnjξαξnkσr+((D1*)-1)(x,0,ξ,ξn)×xαξnj+1xnkσl((D2*)-1)(x,0,ξ,ξn)]dξnσ(ξ)dx, (3.70)
and the sum is taken over r + lkj − |α|= −3, r ≤ −1, l ≤ −1.

Locally we can use Theorem 2.2 (2.43) to compute the interior of Wres˜[π+(D1*)-1π+(D2*)-1], we have

M|ξ|=1trace*T*M[σ-4((D2*D1*)-1)]σ(ξ)dx=32π2M(-43s-4(λ12+λ22)|v*|2)dVolM. (3.71)

So we only need to compute ∫∂M Φ2. From the remark above, now we can compute Φ2 (see formula (3.70) for the definition of Φ2). We use tr as shorthand of trace. Since n = 4, then tr T M[id] = 16, since the sum is taken over r + lkj− |α| = −3, r ≤ − 1, l ≤ −1, then we have the following five cases:

case a) I) r = −1, l = −1, k = j = 0, |α|= 1.

By (3.70), we get

casea)I)=-|ξ|=1-+|α|=1tr[ξαπξn+σ-1((D1*)-1)×xαξnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.72)

case a) II) r = −1, l = −1, k = |α|= 0, j = 1.

By (3.70), we get

casea)II)=-12|ξ|=1-+tr[xnπξn+σ-1((D1*)-1)×ξn2σ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.73)

case a) III) r = −1, l = −1, j = |α|= 0, k = 1

By (3.70), we get

casea)III)=-12|ξ|=1-+tr[ξnπξn+σ-1((D1*)-1)×ξnxnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.74)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2). By (3.19)-(3.34), so case a) vanishes.

case b) r = −2, l = −1, k = j = |α|= 0.

By (3.70), we get

caseb)=-i|ξ|=1-+tr[πξn+σ-2((D1*)-1)×ξnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.75)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2) and

σ-2((D1*)-1)(x0)=c(ξ)σ0(D1*)(x0)c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)|ξ|M2], (3.76)
where σ0(D1*)(x0)=A(x0)+B(x0)+λ1ɛ(v*). Then
πξn+σ-2((D1*)-1(x0))||ξ|=1=πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]+πξn+[c(ξ)(λ1ɛ(v*)(x0))c(ξ)(1+ξn2)2]+πξn+[c(ξ)B(x0)c(ξ)+c(ξ)c(dxn)xn[c(ξ)](x0)(1+ξn2)2-h(0)c(ξ)c(dxn)c(ξ)(1+ξn2)3]. (3.77)

By (3.40)-(3.50), we have

case b)=92πh(0)Ω3dx-i|ξ|=1-+trace[πξn+[c(ξ)(λ1ɛ(v*)(x0))c(ξ)(1+ξn2)2]×ξnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.78)

Similar to (3.51), we have

tr[πξn+[c(ξ)(λ1ɛ(v*)(x0))c(ξ)(1+ξn2)2]×ξnσ-1(D2*)-1)(x0)]||ξ|=1=12(1+ξn2)2tr[c(ξ)λ1ɛ(v*)(x0)]+i2(1+ξn2)2tr[c(dxn)λ1ɛ(v*)(x0)]. (3.79)

By the relation of the Clifford action and trAB = trBA, we have the equalities:

tr[c(dxn)ɛ(v*)]=-8v*,xn;tr[c(ξ)ɛ(v*)]=-8v*,g(ξ,); (3.80)

By (3.79) and (3.80), we have

-i|ξ|=1-+tr[πξn+[c(ξ)λ1ɛ(v*)c(ξ)(1+ξn2)2]×ξnσ-1(D2-1)](x0)dξnσ(ξ)dx=-2λ1πv*,xnΩ3dx. (3.81)

By (3.78) and (3.81), we have

caseb)=[92πh(0)-2λ1πv*,xn]Ω3dx. (3.82)

case c) r = −1, l = −2, k = j = |α|= 0.

By (3.70), we get

casec)=-i|ξ|=1-+tr[πξn+σ-1((D1*)-1)×ξnσ-2((D2*)-1)](x0)dξnσ(ξ)dx. (3.83)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2). Similar to (3.57), we have

ξnσ-2((D2*)-1)(x0)||ξ|=1=ξn{c(ξ)[A(x0)+B(x0)+(λ2ɛ(v*)(x0))]c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)]}=ξn{c(ξ)[A(x0)]c(ξ)|ξ|4+c(ξ)|ξ|6c(dxn)[xn[c(ξ)](x0)|ξ|2-c(ξ)h(0)]}+ξnc(ξ)B(x0)c(ξ)|ξ|4+ξnc(ξ)(λ2ɛ(v*)(x0))c(ξ)|ξ|4. (3.84)

By (3.58)-(3.65), we have

casec)=-92πh(0)-i|ξ|=1-+tr[πξn+σ-1((D1*)-1)×ξn(c(ξ)(λ2ɛ(v*))c(ξ)|ξ|4)](x0)dξnσ(ξ)dx. (3.85)

Similar to (3.66), we have

-i|ξ|=1-+tr[πξn+σ-1((D1*)-1)×ξnc(ξ)(λ2ɛ(v*))c(ξ)|ξ|4](x0)dξnσ(ξ)dx=-i|ξ|=1-+i(ξ-i)(ξ+i)3[tr[c(dxn)(λ2ɛ(v*))(x0)]+itr[c(ξ)(λ2ɛ(v*))(x0)]]dξnσ(ξ)dx=-π4[tr[c(dxn)(λ2ɛ(v*))]+itr[c(ξ)(λ2ɛ(v*))(x0)]]Ω3dx=2λ2πv*,xnΩ3dx. (3.86)

So, we have

casec)=[-92πh(0)+2λ2πv*,xn]Ω3dx. (3.87)

Since Φ2 is the sum of the cases a), b) and c), so Φ2=2(λ2-λ1)πv*,xnΩ3dx.

Theorem 3.10.

Let M be a 4-dimensional oriented compact manifold with the boundary ∂M and the metric gM as above, Di*(i=1,2) be statistical de Rham Hodge Operators on M^, then

Wres˜[π+(D1*)-1π+(D2*)-1]=32π2M(-43s-4(λ12+λ22)|v*|2)dVolM+M2(λ2-λ1)πv*,xnΩ3dx, (3.88)

where s is the scalar curvature.

Corollary 3.11.

For 4-dimensional oriented compact manifold M with the boundary ∂M, when λ1 = λ2 = 1, we get

Wres˜[π+(D*)-1π+(D*)-1]=32π2M(-43s-8|v*|2)dVolM,
where s is the scalar curvature.

Next, we prove the Kastler-Kalau-Walze type theorem for 4-dimensional manifolds with boundary associated with D2*D1. By (3.3) and (3.4), we will compute

Wres˜[π+D1-1π+(D2*)-1]=M|ξ|=1trace*T*M[σ-4((D2*D1)-1)]σ(ξ)dx+MΦ3, (3.89)
where
Φ3=|ξ|=1-+j,k=0(-i)|α|+j+k+1α!(j+k+1)!×trace*T*M[xnjξαξnkσr+(D1-1)(x,0,ξ,ξn)×xαξnj+1xnkσl((D2*)-1)(x,0,ξ,ξn)]dξnσ(ξ)dx (3.90)
and the sum is taken over r + lkj − |α| = −3, r ≤ −1, l ≤ −1.

Locally we can use Theorem 2.2 (2.44) to compute the interior of Wres˜[π+D1-1π+(D2*)-1], we have

M|ξ|=1trace*T*M[σ-4((D2*D1)-1)]σ(ξ)dx=32π2M[-43s-2(λ12+λ22-4λ1λ2)|v|2+12tr[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]]dVolM. (3.91)

So we only need to compute ∂M Φ3. From the remark above, now we can compute Φ3 (see formula (3.89) for the definition of Φ3). We use tr as shorthand of trace. Since n = 4, then trTM [id] = 16, since the sum is taken over r + lkj− |α| = −3, r ≤ −1, l ≤ − 1, then we have the following five cases:

case a) I) r = −1, l = −1, k = j = 0, |α|= 1.

By (3.89), we get

casea)I)=-|ξ|=1-+|α|=1tr[ξαπξn+σ-1(D1-1)×xαξnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.92)

case a) II) r = −1, l = −1, k = |α|= 0, j = 1.

By (3.89), we get

casea)II)=-12|ξ|=1-+tr[xnπξn+σ-1((D1-1))×ξn2σ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.93)

case a) III) r = −1, l = −1, j = |α|= 0, k = 1.

By (3.89), we get

casea)III)=-12|ξ|=1-+tr[ξnπξn+σ-1((D1-1))×ξnxnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.94)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2). By (3.19)-(3.34), so case a) vanishes.

case b) r = −2, l = −1, k = j = |α|= 0.

By (3.89), we get

caseb)=-i|ξ|=1-+tr[πξn+σ-2(D1-1)×ξnσ-1((D2*)-1)](x0)dξnσ(ξ)dx. (3.95)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2). By (3.35)-(3.54), we have

caseb)=[92πh(0)+2λ1πdxn,v]Ω3dx. (3.96)

case c) r = −1, l = −2, k = j = |α|= 0.

By (3.70), we get

casec)=-i|ξ|=1-+tr[πξn+σ-1(D1-1)×ξnσ-2((D2*)-1)](x0)dξnσ(ξ)dx. (3.97)

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1)(i=1,2). By (3.83)-(3.87), we have

casec)=[-92πh(0)+2λ2πv*,xn]Ω3dx. (3.98)

Since Φ3 is the sum of the cases a), b) and c), so Φ3 = 2(λ1 + λ2)πdxn, v〉Ω3dx′.

Theorem 3.12.

Let M be a 4-dimensional oriented compact manifold with the boundary ∂M and the metric gM as above, Di and Di*(i=1,2) be statistical de Rham Hodge Operators on M^, then

Wres˜[π+D1-1π+(D2*)-1]=32π2M[-43s-2(λ12+λ22-4λ1λ2)|v|2+12tr[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]]dVolM+M2(λ1+λ2)πdxn,vΩ3dx, (3.99)
where s is the scalar curvature.

Corollary 3.13.

For a 4-dimensional oriented compact manifold M with the boundary ∂M, when λ1 = λ2 = 1, we get

Wres˜[π+D-1π+(D*)-1]=32π2M[-43s+4|v|2+12tr[ejTM(ɛ(v*))c(ej)-c(ej)ejTM(l(v))]]dVolM+M4πdxn,vΩ3dx,
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 boundary. An application of (2.1.4) in [17] shows that

Wres˜[π+D1-1π+(D2*D1D2*)-1]=M|ξ|=1trace*T*M[σ-4((D2*D1)-2)]σ(ξ)dx+MΨ, (4.1)
where
Ψ=|ξ|=1-+j,k=0(-i)|α|+j+k+1α!(j+k+1)!×trace*T*M[xnjξαξnkσr+(D1-1)(x,0,ξ,ξn)×xαξnj+1xnkσl((D2*D1D2*)-1)(x,0,ξ,ξn)]dξnσ(ξ)dx (4.2)
and the sum is taken over r + kj − |α|− 1 = −6, r ≤ −1, ≤ −3.

Locally we can use Theorem 2.2 (2.44) to compute the interior term of (4.1), we have

M|ξ|=1trace*T*M[σ-4((D2*D1)-2)]σ(ξ)dx=128π3M[-163s-8(λ12+λ22-8λ1λ2)|v|2+12tr[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]]dVolM. (4.3)

So we only need to compute ∂M Ψ. Let us now turn to compute the specification of D2*D1D2*.

D2*D1D2*=i=1nc(ei)ei,dxl(-gijlij)+i=1nc(ei)ei,dxl{-(lgij)ij-gij(4(σi+ai)j-2Γijkk)l}+i=1nc(ei)ei,dxl{-2(lgij)(σi+ai)j+gij(lΓijk)k-2gij[(lσi)+(lai)]j+(lgij)Γijkk+j,k[l(c(ej)λ2ɛ(v*)+λ1l(v)c(ej))]×ej,dxkk+j,k(c(ej)λ2ɛ(v*)+λ1l(v)c(ej))[lej,dxk]k}+[(σi+ai)+λ2ɛ(v*)](-gijij)+i=1nc(ei)ei,dxl{2j,k(c(ej)λ2ɛ(v*)+λ1l(v)c(ej))×ei˜,dxk}lk+[(σi+ai)+λ2ɛ(v*)]{-i,jgi,j[2σij+2aij-Γi,jkk+(iσj)+14s+(iaj)+σiσj+σiaj+aiσj+aiaj-Γi,jkσk-Γi,jkak]+i,jgi,j(c(ej)λ2ɛ(v*)+λ1l(v)c(ej))j+i,jgi,j[λ1l(v)c(i)σi+λ1l(v)c(i)ai+c(i)λ2i(ɛ(v*))+c(i)σiλ2ɛ(v*)+c(i)aiλ2ɛ(v*)]+λ1λ2l(v)ɛ(v*)-18ijklRijklc¯(ei)c¯(ej)c(ek)c(el)}. (4.4)

Then, we obtain

Lemma 4.1.

The following identities hold:

σ2(D2*D1D2*)=i,j,lc(dxl)l(gi,j)ξiξj+c(ξ)(4σk+4ak-2Γk)ξk-2[λ1c(ξ)l(v)c(ξ)-|ξ|2λ2ɛ(v*)]+14|ξ|2s,t,lωs,t(el)[c(el)c¯(es)c¯(et)-c(el)c(es)c(et)]+|ξ|2λ2ɛ(v*);σ3(D2*D1D2*)=ic(ξ)|ξ|2. (4.5)

Write

σ(D2*D1D2*)=p3+p2+p1+p0;σ((D2*D1D2*)-1)=j=3q-j. (4.6)

By the composition formula of pseudodifferential operators, we have

1=σ((D2*D1D2*)(D2*D1D2*)-1)=α1α!ξα[σ(D2*D1D2*)]Dxα[(D2*D1D2*)-1]=(p3+p2+p1+p0)(q-3+q-4+q-5+)+j(ξjp3+ξjp2+ξjp1+ξjp0)(Dxjq-3+Dxjq-4+Dxjq-5+)=p3q-3+(p3q-4+p2q-3+jξjp3Dxjq-3)+, (4.7)
by (4.7), we have
q-3=p3-1;q-4=-p3-1[p2p3-1+jξjp3Dxj(p3-1)]. (4.8)

By Lemma 4.1, we have some symbols of operators.

Lemma 4.2.

The following identities hold:

σ-3((D2*D1D2*)-1)=ic(ξ)|ξ|4;σ-4((D2*D1D2*)-1)=c(ξ)σ2(D2*D1D2*)c(ξ)|ξ|8+ic(ξ)|ξ|8(|ξ|4c(dxn)xnc(ξ)-2h(0)c(dxn)c(ξ)+2ξnc(ξ)xnc(ξ)+4ξnh(0)). (4.9)

In the normal coordinate, gij(x0)=δij and xj(gαβ)(x0)=0, if j < n; xj(gαβ)(x0)=h(0)δβα, if j = n. So by Lemma A.2 in [15], we have Γn(x0)=52h(0) and Γk(x0) = 0 for k < n. By the definition of δk and Lemma 2.3 in [15], we have δn(x0) = 0 and δk=14h(0)c(ek)c(en) for k < n. By Lemma 4.2, we obtain

σ-4((D2*D1D2*)-1)(x0)||ξ|=1=c(ξ)σ2((D2*D1D2*)-1)(x0)||ξ|=1c(ξ)|ξ|8-c(ξ)|ξ|4jξj(c(ξ)|ξ|2)Dxj(ic(ξ)|ξ|4)=1|ξ|8c(ξ)(12h(0)c(ξ)k<nξkc(ek)c(en)-12h(0)c(ξ)k<nξkc¯(ek)c¯(en)-52h(0)ξnc(ξ)-14h(0)|ξ|2c(dxn)-2[c(ξ)λ1l(v)c(ξ)-|ξ|2λ2ɛ(v*)]+|ξ|2λ2ɛ(v*))c(ξ)+ic(ξ)|ξ|8(|ξ|4c(dxn)xnc(ξ)-2h(0)c(dxn)c(ξ)+2ξnc(ξ)xnc(ξ)+4ξnh(0)). (4.10)

From the remark above, now we can compute Ψ (see formula (4.2) for the definition of Ψ). We use tr as shorthand of trace. Since n = 6, tr∧∗ TM[id]= 64. Since the sum is taken over r + kj − |α|− 1 = −6, r ≤ −1, ≤ −3, we have the ∫∂M Ψ is the sum of the following five cases:

case (a) (I) r = −1, l = −3, j = k = 0, |α|= 1.

By (4.2), we get

case(a)(I)=-|ξ|=1-+|α|=1tr[ξαπξn+σ-1(D1-1)×xαξnσ-3((D2*D1D2*)-1)](x0)dξnσ(ξ)dx. (4.11)

By Lemma 4.2, for i < n, we have

xiσ-3((D2*D1D2*)-1)(x0)=xi[ic(ξ)|ξ|4](x0)=ixi[c(ξ)]|ξ|-4(x0)-2ic(ξ)xi[|ξ|2]|ξ|-6(x0)=0. (4.12)
so case (a) (I) vanishes.

case (a) (II) r = −1, l = −3, |α|= k = 0, j = 1.

By (4.2), we have

case(a)(II)=-12|ξ|=1-+tr[xnπξn+σ-1(D1-1)×ξn2σ-3((D2*D1D2*)-1)](x0)dξnσ(ξ)dx. (4.13)

By Lemma 4.2 and direct calculations, we have

ξn2σ-3((D2*D1D2*)-1)=i[(20ξn2-4)c(ξ)+12(ξn3-ξn)c(dxn)(1+ξn2)4]. (4.14)

Since n = 6, tr[−id]= −64. By the relation of the Clifford action and trAB = trBA, then

tr[c(ξ)c(dxn)]=0;tr[c(dxn)2]=-64;tr[c(ξ)2](x0)||ξ|=1=-64;tr[xn[c(ξ)]c(dxn)]=0;tr[xnc(ξ)c(ξ)](x0)||ξ|=1=-32h(0). (4.15)

By (3.31), (4.14) and (4.15), we get

tr[xnπξn+σ-1(D1-1)×ξn2σ-3((D2*D1D2*)-1)](x0)=64h(0)-1-3ξni+5ξn2+3iξn3(ξn-i)6(ξn+i)4. (4.16)

Then we obtain

case(a)(II)=-12|ξ|=1-+h(0)-8-24ξni+40ξn2+24iξn3(ξn-i)6(ξn+i)4dξnσ(ξ)dx=-152πh(0)Ω4dx, (4.17)
where Ω4 is the canonical volume of S4.

case (a) (III) r = −1, l = −3, |α|= j = 0, k = 1.

By (4.2), we have

case(a)(III)=-12|ξ|=1-+tr[ξnπξn+σ-1(D1-1)×ξnxnσ-3((D2*D1D2*)-1)](x0)dξnσ(ξ)dx. (4.18)

By Lemma 4.2 and direct calculations, we have

ξnxnσ-3((D2*D1D2*)-1)=-4iξnxnc(ξ)(x0)(1+ξn2)3+i12h(0)ξnc(ξ)(1+ξn2)4-i(2-10ξn2)h(0)c(dxn)(1+ξn2)4. (4.19)

Combining (3.31) and (4.19), we have

tr[ξnπξn+σ-1(D1-1)×ξnxnσ-3((D2*D1D2*)-1)](x0)||ξ|=1=8h(0)8i-32ξn-8iξn2(ξn-i)5(ξ+i)4. (4.20)

Then

case(a)III)=-12|ξ|=1-+8h(0)8i-32ξn-8iξn2(ξn-i)5(ξ+i)4dξnσ(ξ)dx=252πh(0)Ω4dx. (4.21)

case (b) r = −1, l = −4, |α|= j = k = 0.

By (4.2), we have

case(b)=-i|ξ|=1-+tr[πξn+σ-1(D1-1)×ξnσ-4((D2*D1D2*)-1)](x0)dξnσ(ξ)dx=i|ξ|=1-+tr[ξnπξn+σ-1(D1-1)×σ-4((D2*D1D2*)-1)](x0)dξnσ(ξ)dx. (4.22)

By (3.31) and (4.23), we have

tr[ξnπξn+σ-1(D1-1)×σ-4(D2*D1D2*)-1](x0)||ξ|=1=12(ξn-i)2(1+ξn2)4(34i+2+(3+4i)ξn+(-6+2i)ξn2+3ξn3+9i4ξn4)h(0)tr[id]+12(ξn-i)2(1+ξn2)4(-1-3iξn-2ξn2-4iξn3-ξn4-iξn5)tr[c(ξ)xnc(ξ)]-12(ξn-i)2(1+ξn2)4(12i+12ξn+12ξn2+12ξn3)tr[c(ξ)c¯(ξ)c(dxn)c¯(dxn)]+tr[πξn+σ-1(D1-1)×ξn(3c(ξ)λ2ɛ(v*)c(ξ)|ξ|6-2λ1l(v)|ξ|4)](x0)||ξ|=1 (4.23)

By direct calculations, we have

tr[πξn+σ-1(D1-1)×ξn(3c(ξ)λ2ɛ(v*)c(ξ)|ξ|6-2λ1l(v)|ξ|4)](x0)||ξ|=1=3(4iξn+2)i2(ξn+i)(1+ξn2)3tr[λ2ɛ(v*)c(ξ)]+3(4iξn+2)2(ξn+i)(1+ξn2)3tr[λ2ɛ(v*)c(dxn)]+4ξn2(ξn-i)(1+ξn2)3tr[λ1l(v)c(ξ)]+4ξni2(ξn-i)(1+ξn2)3tr[λ1l(v)c(dxn)]. (4.24)

By the relation of the Clifford action and trAB = trBA, then we have the following equalities:

tr[c(dxn)l(v)]=32dxn,v;tr[c(ξ)l(v)]=32ξ,v;tr[c(dxn)ɛ(v*)]=-32v*,xn;tr[c(ξ)ɛ(v*)]=-32v*,g(ξ,);tr[c(ei)c¯(ei)c(en)c¯(en)]=0(i<n). (4.25)

So

tr[c(ξ)c¯(ξ)c(dxn)c¯(dxn)]=i<n,j<ntr[ξiξjc(ei)c¯(ej)c(dxn)c¯(dxn)]=0. (4.26)

By (4.25), then we have

case(b)=ih(0)|ξ|=1-+64×34i+2+(3+4i)ξn+(-6+2i)ξn2+3ξn3+9i4ξn42(ξn-i)5(ξn+i)4dξnσ(ξ)dx+ih(0)|ξ|=1-+32×1+3iξn+2ξn2+4iξn3+ξn4+iξn52(ξn-i)2(1+ξn2)4dξnσ(ξ)dx-i|ξ|=1-+tr[πξn+σ-1(D1-1)×ξn(3c(ξ)λ2ɛ(v*)c(ξ)|ξ|6-2λ1l(v)|ξ|4)](x0)||ξ|=1dξnσ(ξ)dx=(-418i-1958)πh(0)Ω4dx+(4λ1+18λ2)πdxn,vΩ4dx. (4.27)

case (c) r = −2, l = −3, |α|= j = k = 0.

By (4.2), we have

case(c)=-i|ξ|=1-+tr[πξn+σ-2(D1-1)×ξnσ-3((D2*D1D2*)-1)](x0)dξnσ(ξ)dx. (4.28)

By (3.36) and (3.37), we have

πξn+σ-2(D1-1(x0))||ξ|=1=πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]+πξn+[c(ξ)(λ1l(v)(x0))c(ξ)(1+ξn2)2]+πξn+[c(ξ)B(x0)c(ξ)+c(ξ)c(dxn)xn[c(ξ)](x0)(1+ξn2)2-h(0)c(ξ)c(dxn)c(ξ)(1+ξn2)3]. (4.29)

By (3.40), we have

πξn+[c(ξ)A(x0)c(ξ)(1+ξn2)2]=-c(ξ)A(x0)c(ξ)(2+iξn)4(ξn-i)2+ic(ξ)A(x0)c(dxn)4(ξn-i)2+ic(dxn)A(x0)c(ξ)4(ξn-i)2+-iξnc(dxn)A(x0)c(dxn)4(ξn-i)2. (4.30)

By (3.45)-(3.47), we have

πξn+[c(ξ)B(x0)c(ξ)+c(ξ)c(dxn)xn[c(ξ)](x0)(1+ξn2)2]-h(0)πξn+[c(ξ)c(dxn)c(ξ)(1+ξn)3]P1-P2, (4.31)
where
P1=-14(ξn-i)2[(2+iξn)c(ξ)b02(x0)c(ξ)+iξnc(dxn)b02(x0)c(dxn)+(2+iξn)c(ξ)c(dxn)xnc(ξ)+ic(dxn)b02(x0)c(ξ)+ic(ξ)b02(x0)c(dxn)-ixnc(ξ)] (4.32)
and
P2=h(0)2[c(dxn)4i(ξn-i)+c(dxn)-ic(ξ)8(ξn-i)2+3ξn-7i8(ξn-i)3[ic(ξ)-c(dxn)]]. (4.33)

Similar to (4.30), we have

πξn+[c(ξ)(l(v)(x0))c(ξ)(1+ξn2)2]=-c(ξ)(l(v)(x0))c(ξ)(2+iξn)4(ξn-i)2+ic(ξ)(l(v)(x0))c(dxn)4(ξn-i)2+ic(dxn)(l(v)(x0))c(ξ)4(ξn-i)2+-iξnc(dxn)(l(v)(x0))c(dxn)4(ξn-i)2. (4.34)

On the other hand,

ξnσ-3((D2*D1D2*)-1)=-4iξnc(ξ)(1+ξn2)3+i(1-3ξn2)c(dxn)(1+ξn2)3. (4.35)

By the relation of the Clifford action and trAB = trBA, then we have equalities:

tr[Ac(dxn)]=0;tr[c¯(ξ)c¯(dxn)]=0. (4.36)

Then we have

tr[πξn+(c(ξ)A(x0)c(ξ)(1+ξn2)2)×ξnσ-3((D2*D1D2*)-1)(x0)]||ξ|=1=2-8iξn-6ξn24(ξn-i)2(1+ξn2)3tr[A(x0)c(ξ)], (4.37)

We note that i < n, |ξ′|=1{ξi1 ξi2 ··· ξi2d+1}σ(ξ′) = 0, so tr[A(x0)c(ξ′)] has no contribution for computing case c).

By (4.32) and (4.35), we have

tr[P1×ξnσ-3((D2*D1D2*)-1)(x0)]||ξ|=1=tr{14(ξn-i)2[52h(0)c(dxn)-5i2h(0)c(ξ)-(2+iξn)c(ξ)c(dxn)ξnc(ξ)+iξnc(ξ)]×-4iξnc(ξ)+(i-3iξn2)c(dxn)(1+ξn2)3}=8h(0)3+12iξn+3ξn2(ξn-i)4(ξn+i)3. (4.38)

By (4.33) and (4.35), we have

tr[P2×ξnσ-3((D2*D1D2*)-1)(x0)]||ξ|=1=tr{h(0)2[c(dxn)4i(ξn-i)+c(dxn)-ic(ξ)8(ξn-i)2+3ξn-7i8(ξn-i)3[ic(ξ)-c(dxn)]]×-4iξnc(ξ)+(i-3iξn2)c(dxn)(1+ξn2)3}=8h(0)4i-11ξn-6iξn2+3ξn3(ξn-i)5(ξn+i)3. (4.39)

By (4.34) and (4.35), we have

tr[πξn+(c(ξ)[λ1l(v)(x0)]c(ξ)(1+ξn2)2)×ξnσ-3((D2*D1D2*)-1)(x0)]||ξ|=1=-2-8iξn+6ξn24(ξn-i)2(1+ξn2)3tr[λ1l(v)(x0)c(ξ)]+-2i+8ξn+6iξn24(ξn-i)2(1+ξn2)3tr[λ1c(dxn)l(v)(x0)]. (4.40)

By (4.25), we have

case(c)=-ih(0)|ξ|=1-+8×-7i+26ξn+15iξn2(ξn-i)5(ξn+i)3dξnσ(ξ)dx-i|ξ|=1-+tr[πξn+(c(ξ)[λ1l(v)(x0)]c(ξ)(1+ξn2)2)×ξnσ-3((D2*D1D2*)-1)(x0)]||ξ|=1dξnσ(ξ)dx=-8ih(0)×2πi4![-7i+26ξn+15iξn2(ξn+i)3](5)|ξn=iΩ4dx+(9πi-4π)(λ1dxn,v-iλ1ξ,v)Ω4dx=[552πh(0)+(9πi-4π)λ1dxn,v]Ω4dx. (4.41)

Since Ψ is the sum of the cases a), b) and c),

Ψ=(65-41i)πh(0)8Ω4dx+(18πλ2+9λ1πi)dxn,vΩ4dx.

Theorem 4.3.

Let M be a 6-dimensional compact oriented manifold with the boundary ∂M and the metric gM as above, Di and Di*(i=1,2) be statistical de Rham Hodge Operators on M^, then

Wres˜[π+D1-1π+(D2*D1D2*)-1]=128π3M[-163s-8(λ12+λ22-8λ1λ2)|v|2+12tr[λ2ejTM(ɛ(v*))c(ej)-λ1c(ej)ejTM(l(v))]]dVolM+M[(65-41i)πh(0)8Ω4dx+(18πλ2+9λ1πi)dxn,v]Ω4dx, (4.42)

where s is the scalar curvature.

Corollary 4.4.

When λ1 = λ2 = 1, we get for a 6-dimensional oriented compact manifold M with the boundary ∂M

Wres˜[π+D-1π+(D*DD*)-1]=128π3M[-163s+48|v|2+12tr[ejTM(ɛ(v*))c(ej)-c(ej)ejTM(l(v))]]dVolM+M((65-41i)πh(0)8Ω4dx+(18π+9πi)dxn,vΩ4dx),
where s is the scalar curvature.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

AUTHORS’ CONTRIBUTION

SW contributed in study conceptualization and writing (review and editing) the manuscript, data curation, formal analysis and writing (original draft). YW contributed in funding acquisition and project administration, supervised the project.

FUNDING

This research was funded by National Natural Science Foundation of China: No. 11771070.

ACKNOWLEDGMENTS

This work was supported by NSFC. 11771070. The authors thank the referees for their careful reading and helpful comments.

Footnotes

Data availability statement: The authors confirm that the data supporting the findings of this study are available within the article.

REFERENCES

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[2]A Connes, Quantized calculus and applications, Internat Press, in 11th International Congress of Mathematical Physics (Paris, 1994) (Cambridge, MA, 1995), pp. 15-36.
[3]A Connes, The action functional in noncommutative geometry, Comm. Math. Phys., Vol. 117, 1998, pp. 673-683.
[8]D Kastler, The Dirac operator and gravitation, Comm. Math. Phys., Vol. 166, 1995, pp. 633-643.
[9]JAÁ López, YA Kordyukov, and E Leichtnam, Analysis on Riemannian foliations of bounded geometry, 2019. ariXiv:1905.12912.
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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
28 - 2
Pages
254 - 275
Publication Date
2021/04/29
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.210419.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Sining Wei
AU  - Yong Wang
PY  - 2021
DA  - 2021/04/29
TI  - Statistical de Rham Hodge Operators and the Kastler-Kalau-Walze Type Theorem for Manifolds With Boundary
JO  - Journal of Nonlinear Mathematical Physics
SP  - 254
EP  - 275
VL  - 28
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.210419.001
DO  - https://doi.org/10.2991/jnmp.k.210419.001
ID  - Wei2021
ER  -