A Kastler–Kalau–Walze type theorem for 7-dimensional spin manifolds with boundary about Dirac operators with torsion

Journal of Geometry and Physics 110 (2016) 213–232

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A Kastler–Kalau–Walze type theorem for 7-dimensional spin manifolds with boundary about Dirac operators with torsion Kai Hua Bao a,∗ , Ai Hui Sun b , Jian Wang c a

School of Mathematics, Ineer Mongolia University for Nationalities, TongLiao, 028005, PR China

b

School of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, PR China

c

School of Science, Tianjin University of Technology and Education, Tianjin, 300222, PR China

article

info

Article history: Received 21 May 2016 Received in revised form 8 August 2016 Accepted 9 August 2016 Available online 24 August 2016

abstract In this paper, we give a brute-force proof of the Kastler–Kalau–Walze type theorem for 7-dimensional manifolds with boundary about Dirac operators with torsion. © 2016 Elsevier B.V. All rights reserved.

Keywords: Dirac operators with torsion Noncommutative residue for manifolds with boundary Lower dimensional volumes

1. Introduction The noncommutative residue found in [1,2] plays a prominent role in noncommutative geometry. For one-dimensional manifolds, the noncommutative residue was discovered by Adler [3] in connection with geometric aspects of nonlinear partial differential equations. For arbitrary closed compact n-dimensional manifolds, the noncommutative residue was introduced by Wodzicki in [2] using the theory of zeta functions of elliptic pseudodifferential operators. In [4], Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy. Furthermore, Connes made a challenging observation that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein–Hilbert action in [5]. In [6], Kastler gave a brute-force proof of this theorem. In [7], Kalau and Walze proved this theorem in the normal coordinates system simultaneously. And then, Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator Wres(D−2 ) in turn is essentially the second coefficient of the heat kernel expansion of D2 in [8]. Recently, Ponge defined lower dimensional volumes of Riemannian manifolds by the Wodzicki residue [9]. Fedosov et al. defined a noncommutative residue on Boutet de Monvel’s algebra and proved that it was a unique continuous trace in [10]. In [11], Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary. In [12], Wang generalized the Connes results to the cases of 3, 4-dimensional spin manifolds with boundary and proved a Kastler–Kalau–Walze type theorem. In [13,14], Wang computed the lower dimensional volume Vol(2, 2) for 5-dimensional and 6-dimensional spin manifolds with boundary and also got a Kastler–Kalau–Walze type theorem in this case. In [15], [(π + D−2 )2 ] for 7-dimensional manifolds with boundary, and proved a Kastler–Kalau–Walze type authors computed Wres

∗

Corresponding author. E-mail address: [email protected] (K.H. Bao).

http://dx.doi.org/10.1016/j.geomphys.2016.08.005 0393-0440/© 2016 Elsevier B.V. All rights reserved.

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theorem. In [16], Ackermann and Tolksdorf proved a generalized version of the known Lichnerowicz formula for the square of the most general Dirac operator with torsion DT on an even dimensional spin manifolds associated to a metric connection with torsion. Recently, Pfäffle and Stephan considered compact Riemannian spin manifolds without boundary equipped with orthogonal connection, and investigated the induced Dirac operators in [17]. In [18], Pfäffle and Stephan considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds, and for the induced Dirac operators, twisted operators and Dirac operators of Chamseddine–Connes type they computed the spectral action. In [19], authors generalized the results in [12,17] and got a Kastler–Kalau–Walze type theorems associated with Dirac operators with torsion on compact Riemannian manifolds with boundary, and derived the gravitational action on boundary by the noncommutative residue associated with Dirac operators with torsion. The motivation of this paper is to generalize the results of [19,15]. That is, we want to establish a Kastler–Kalau–Walze type theorem associated with Dirac operators with torsion for 7-dimensional manifolds with boundary.

 be an orthogonal connection Main Theorem: Let M be a 7-dimensional compact manifold with the boundary ∂ M, and ∇  with torsion. Then we get the volumes associated to D∗T DT with torsion on M. [π + (D∗ )−2 ◦ π + D−2 ] = Wres T T



 1475  2 25 ′′ 1 77 h′ (0) + h (0) − s − s∂M −

384 8 2 192   3 99 ′ + ∂x′ Akki + h (0) Aiin π Ω5 dx′ . 8 k 32 i ∂M

This paper is organized as follows: In Section 2, we define lower dimensional volumes of compact Riemannian manifolds [π + (D∗ )−2 ◦ π + D−2 ] for 7-dimensional with boundary about Dirac operators with torsion. In Section 3, we compute Wres T T spin manifolds with boundary and the associated Dirac operators with torsion, and get a Kastler–Kalau–Walze type theorem in this case. In Section 4, we derive the gravitational action on boundary by the noncommutative residue associated with Dirac operators with torsion. 2. Lower-dimensional volumes of spin manifolds with boundary about Dirac operators with torsion In this section we consider an n-dimensional (n ≥ 3) oriented Riemannian manifold (M , g M ) with boundary ∂M equipped with a fixed spin structure. The Levi-Civita connection ∇ : Γ (TM ) → Γ (T ∗ M ⊗ TM ) on M induces a connection ∇ S : Γ (S ) → Γ (T ∗ M ⊗ S ). By adding an additional torsion term t ∈ Ω 1 (M , EndTM ), we obtain a new covariant derivative  := ∇ + t on the tangent bundle TM. Since t is really a one-form on M with values in the bundle of skew endomorphism ∇ S := ∇ S + T Sk(TM ) in [20], ∇ is in fact compatible with the Riemannian metric g and therefore also induces a connection ∇ on the spinor bundle. Here T ∈ Ω 1 (M , EndS ) denotes the lifted torsion term t ∈ Ω 1 (M , EndTM ). Y := ∇ Y + A(X , Y ) with the Levi-Civita We will recall the construction of this connection by [19,17]. We write ∇ X X connection ∇ . For any X ∈ Tp M the endomorphism A(X , ·) is skew-adjoint and hence it is an element of so(Tp M ), we can express it as A(X , ·) =



αij Ei ∧ Ej .

(2.1)

i
Here Ei ∧ Ej is meant as the endomorphism of Tp M defined by Ei ∧ Ej . For any X ∈ Tp M one determines the coefficients in (2.1) by

αij = ⟨A(X , Ei ), Ej ⟩ = AXEi Ej .

(2.2)

 is locally given by Each Ei ∧ Ej lifts to 12 Ei Ej in spin(n), and the spinor connection induced by ∇  1 X ψ = ∇X ψ + 1 ∇ αij Ei · Ej ψ = ∇X ψ + AXEi Ej Ei · Ej ψ. 2 i
(2.3)

2 i
The connection given by (2.3) is compatible with the metric on the spinors and with Clifford multiplication. Then, the Dirac operator associated to the spinor connection form (2.3) is defined as

T ψ = ∇

n 

E ψ = Dψ + Ei ∇ i

i=1

= Dψ +

n 1 

4 i,j,k=1

n 1 

2 i=1 i
AEi Ej Ek Ei · Ei · Ek ψ

AEi Ej Ek Ei · Ei · Ek ψ.

(2.4)

Where D is Dirac operator induced by the Levi-Civita connection and ‘‘·’’ is the Clifford multiplication. By [18] this Dirac operator with torsion can be written as : DT ψ = Dψ +

3 2

T ·ψ −

n−1 2

V · ψ,

D∗T ψ = Dψ +

3 2

T ·ψ −

n−1 2

V · ψ.

(2.5)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

215

As Clifford multiplication by any 3-form is self-adjoint and by vector field V is skew-adjoint, using the Hermitian product on the spinor bundle one observes that DT is symmetric with respect to the natural L2 -scalar product on the spinors if and only if the vectorial component of the torsion vanishes, V ≡ 0. Note that the Cartan type torsion S does not contribute to the Dirac operator DT . As DT D∗T is generalized Laplacian, one has the following Lichnerowicz formula.

, we have Theorem 2.1 ([18]). For the Dirac operator DT associated to the orthogonal connection ∇ D∗T DT ψ = △ψ +

1 4

Rg ψ +

3 2

dT · ψ −

3 4

∥T ∥2 ψ +

n−1 2

div g (V )ψ +



n−1

+ 3(n − 1)(T · V · ψ + (V ⌋T ) · ψ),

2

2

(2 − n)|V |2 ψ (2.6)

for any spinor field ψ , where △ is the Laplacian associated to the connection

X ψ = ∇X ψ + 3 (X ⌋T ) · ψ − n − 1 V · X · ψ − n − 1 ⟨V , X ⟩ψ. ∇ 2

2

(2.7)

2

We assume that the metric g M on M has the following form near the boundary gM =

1 h(xn )

g ∂ M + dx2n ,

(2.8)

where g ∂ M is the metric on ∂ M. Let U ⊂ M be a collar neighborhood of ∂ M which is diffeomorphic with ∂ M × [0, 1). By the definition of h(xn ) ∈ C ∞ ([0, 1)) and h(xn ) > 0, there exists h˜ ∈ C ∞ ((−ε, 1)) such that h˜ |[0,1) = h and h˜ > 0 for some sufficiently small ε > 0.   =M Then there exists a metric  g on M ∂ M ∂ M × (−ε, 0] which has the form on U ∂ M ∂ M × (−ε, 0]

 g =

1 h˜ (xn )

g ∂ M + dx2n ,

(2.9)

 such that  such that  g |M = g. We fix a metric  g on the M g |M = g. Note DT is the most general Dirac operator on the spinor  on TM. bundle S corresponding to a metric connection ∇ To define the lower dimensional volume, some basic facts and formulae about Boutet de Monvel’s calculus which can be found in Sec. 2 in [12] are needed. Let F : L2 (Rt ) → L2 (Rv ); F (u)(v) =



e−iv t u(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+ ); f → f |R+ ; R+ = {x ≥ 0; x ∈ R}. We define H + = F (Φ (R+ )); H0− = F (Φ (R− )) which are orthogonal to each other. We have the following property: h ∈ H + (H0− ) iff h ∈ C ∞ (R) which has an analytic extension to the lower (upper) complex half-plane {Imξ < 0} ({Imξ > 0}) such that for all nonnegative integer l, dl h dξ

(ξ ) ∼ l

  ∞  dl ck , l dξ ξk k=1

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

˜ is a projection on H + (H − ). For calculations, we take H =  H = {rational functions having no poles on the real axis} (H ˜ dense set in the topology of H). Then on H, π + h(ξ0 ) =

h(ξ )



1 2π i

lim

u→0−

Γ+

ξ0 + iu − ξ

dξ ,

(2.10)

where Γ + is a Jordan close curve included Im(ξ ) > 0 surrounding all the singularities of h in the upper half-plane and ˜ ξ0 ∈ R. Similarly, define π ′ on H,

π ′h =

1 2π

 Γ+

h(ξ )dξ .

(2.11)

So, π ′ (H − ) = 0. For h ∈ H L1 (R), π ′ h = 21π R h(v)dv and for h ∈ H + L1 (R), π ′ h = 0. 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].







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Theorem 2.2 (Fedosov–Golse–Leichtnam–Schrohe). Let X and ∂ X be connected, dim X = n ≥ 3, A = denote by p, b and s the local symbols of P , G and S respectively. Define:

(A) = Wres

 

trE [p−n (x, ξ )] σ (ξ )dx + 2π



S

X

T

K S



∈ B , and

′

 ∂X

π +P + G





trE (trb−n )(x′ , ξ ′ ) + trF s1−n (x′ , ξ ′ )









σ (ξ ′ )dx′ ,

(2.12)

S

([A, B]) = 0, for any A, B ∈ B ; (b) It is a unique continuous trace on B /B −∞ . Then (a) Wres Let p1 , p2 be nonnegative integers and p1 + p2 ≤ n. From Sec. 2.1 of [12], we have the following definition. Definition 2.3 ([19]). Lower-dimensional volumes of spin manifolds with boundary with torsion are defined by

[π + (D∗ )−p1 ◦ π + D−p2 ]. Voln(p1 ,p2 ) M := Wres T T

(2.13)

Denote by σl (A) the l-order symbol of an operator A. An application of (2.1.4) in [21] shows that

[π + (D∗ )−p1 ◦ π + D−p2 ] = Wres T T

  M

Φ =

−p2

traceS (TM ) [σ−n ((D∗T )−p1 ◦ DT

)]σ (ξ )dx +

|ξ |=1

 ∂M

Φ,

  (−i)|α|+j+k+1 traceS (TM ) ∂xj n ∂ξα′ ∂ξkn σr+ (D∗T )−p1 (x′ , 0, ξ ′ , ξn ) α!(j + k + 1)! |ξ ′ |=1 −∞ j,k=0  −p2 ′ 1 k ′ × ∂xα′ ∂ξj+ ∂ σ D ( x , 0 , ξ , ξ ) dξn σ (ξ ′ )dx′ , l n xn T n



(2.14)

∞ +∞ 



(2.15)

and the sum is taken over r − k + |α| + ℓ − j − 1 = −n, r ≤ −p1 , ℓ ≤ −p2 . The following proposition is the key of the computation of lower-dimensional volumes of spin manifolds with boundary. Proposition 2.4 ([13]). The following identity holds:

(1) When p1 + p2 = n, then, Vol(np1 ,p2 ) M = c0 VolM ;  (2) when p1 + p2 ≡ n mod 1, Vol(np1 ,p2 ) M = Φ.

(2.16) (2.17)

∂M

(2,2)

Nextly, for 7-dimensional spin manifolds with boundary, we compute Vol7

[π + (D∗ )−p1 ◦ π + D−p2 ] = Wres T T

 ∂M

. By Proposition 2.4, we have

Φ.

(2.18)

So we only need to compute ∂ M Φ .



3. A Kastler–Kalau–Walze type theorem for 7-dimensional spin manifolds with boundary In this section, we compute the lower dimensional volume for 7-dimensional compact manifolds with boundary and get a Kastler–Kalau–Walze type formula in this case. From now on we always assume that M carries a spin structure so that the spinor bundle and √ the Dirac operator are defined on M. Let us give the expression of Dirac operators near the boundary. Set  Ej = h(xn )Ej (1 ≤ j ≤ n − 1), where {E1 , . . . , En−1 } are orthonormal basis of T ∂M . Let ∇ L denote the Levi-Civita En = ∂∂x ,  n

connection about g M . In the local coordinates {xi ; 1 ≤ i ≤ n} and the fixed orthonormal frame { E1 , . . . ,  En }, the connection matrix (ωs,t ) is defined by

∇ L ( e1 , . . . , en )t = (ωs,t )( e1 , . . . , en ) t .

(3.1)

Since Φ is a global form on ∂ M, so for any fixed point x0 ∈ ∂ M, we can choose the normal coordinates U of x0 in ∂ M(not in M) and compute Φ (x0 ) in the coordinates  U = U × [0, 1) and the metric h(1x ) g ∂ M + dx2n . The dual metric of g M on  U is

h(xn )g ∂ M + dx2n . Write gijM = g M ( ∂∂x , ∂∂x ); gM = g M (dxi , dxj ), then i j

n

ij



1

[giM,j ] =  h(xn )

[gi∂,jM ]

0

 0

;

 i ,j [gMi,j ] = h(xn )[g∂ M ]

1

0



0 , 1

(3.2)

and

∂xs gij∂ M (x0 ) = 0,

1 ≤ i, j ≤ n − 1;

giM ,j (x0 ) = δij .

(3.3)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

217

Let {e1 , . . . , en−1 } be an orthonormal frame field in U about g ∂ M which is parallel along geodesics and ei = ∂∂x (x0 ), i √ √ then { e1 = h(xn )e1 , . . . , e h(xn )en−1 , en = dxn } is the orthonormal frame field in  U about g M . Locally n−1 = ∗ n ∗ n ∼  S (TM )| U = U ×∧C ( 2 ). Let {f1 , . . . , fn } be the orthonormal basis of ∧C ( 2 ). Take a spin frame field σ : U → Spin(M ) such that πσ = { e1 , . . . , en }, where π : Spin(M ) → O(M ) is a double covering, then {[σ , fi ], 1 ≤ i ≤ 6} is an orthonormal frame of S (TM )| U . In the following, since the global form Φ is independent of the choice of the local frame, so we can compute trS (TM ) in the frame {[σ , fi ], 1 ≤ i ≤ 6}. Let {ˆe1 , . . . , eˆ n } be the canonical basis of Rn and c (ˆei ) ∈ clC (n) ∼ = Hom(∧∗C ( 2n ), ∧∗C ( 2n )) be the Clifford action. By [12], then c ( ei ) = [(σ , c (ˆei ))];

c ( ei )[(σ , fi )] = [σ , c (ˆei )fi ];

∂ = ∂ xi

  ∂ σ, , ∂ xi

(3.4)

then we have ∂∂x c ( ei ) = 0 in the above frame. The Dirac operator is defined by i

D=

n 



c ( ej )  ej +

j =1

1 4

s,t

ωs,t ( ej )c ( es )c ( et )



(3.5)

where c ( ei ) denote the Clifford action. As Clifford multiplication by any 3-form is self-adjoint, Clifford multiplication by the vector field V is skew-adjoint, and combining these facts with Eq. (2.4), in [19] authors got Lemma 3.1 ([19]). DT =

n 



c ( ei )  ei −

i=1

1 4

s,t

 1  ωs,t ( ei )c ( es )c ( et ) + Aist c ( ei )c ( es )c ( et ). 4 i= ̸ s̸=t

1 + [−Aiit c ( et ) + Aisi c ( es ) − Aiss c ( ei ) + 2Aiii c ( ei )].

(3.6)

4 i,s,t

D∗T =

n 



c ( ei )  ei −

i=1

1 4

s,t

 1  ωs,t ( ei )c ( es )c ( et ) + Aist c ( ei )c ( es )c ( et ) 4 i= ̸ s̸=t

1 − [−Aiit c ( et ) + Aisi c ( es ) − Aiss c ( ei ) + 2Aiii c ( ei )].

(3.7)

4 i,s,t

Define Γ k = i ,j < n [13], one has following.





l
g ij g lk ⟨∇∂Li ∂j , ∂l ⟩ +



l
g lk ⟨∇∂Ln ∂n , ∂l ⟩ and σ k (x0 ) =

1 4



s,t

ωs,t (∂i )c ( es )c ( et ). By Lemma 1 in

Lemma 3.2 ([19]). The symbols of the Dirac operators with torsion

σ−2 ((D∗T )−2 ) = σ−2 ((DT )−2 ) = |ξ |−2 ; √ √ √ σ−3 ((D∗T )−2 ) = − −1|ξ |−4 ξk (Γ k − 2σ k ) − −1|ξ |−6 2ξ j ξα ξβ ∂j g αβ − 2 −1|ξ |−4 (u + v)c (ξ ); √ √ √ σ−3 ((DT )−2 ) = − −1|ξ |−4 ξk (Γ k − 2σ k ) − −1|ξ |−6 2ξ j ξα ξβ ∂j g αβ − 2 −1|ξ |−4 (u − v)c (ξ ).

(3.8)

Where u =

v =

1  4 i= ̸ s̸=t

Aist c ( ei )c ( es )c ( et ),

1 4 i,s,t

[−Aiit c ( et ) + Aisi c ( es ) − Aiss c ( ei ) + 2Aiii c ( ei )].

(3.9)

Since the equation (4.16) of [7] is ˜ −1 △

σ−4 (x, ξ ) = σ2−1 (γ−2 1 + γ−2 ) + iσ2−2 ∂ξµ σ2 ∂xµ γ−1 ,

(3.10)

where ˜1

σ △ (x, ξ ) = σ2 + σ1 + σ0 , γ−1 (x, ξ ) = σ2−1 σ1 − iσ2−2 ∂ξµ σ2 ∂xµ σ2 ,   1 −1 −2 γ−2 (x, ξ ) = −σ2 σ0 − σ2 i∂ξµ σ1 ∂xµ σ2 + ∂ξµ ∂ξν σ2 ∂xµ ∂xν σ2 + σ2−3 ∂ξµ ∂ξν σ2 ∂xµ σ2 ∂xν σ2 . 2

(3.11)

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K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

In the following we still use Aiij to express Aiij (x0 ), use ∂xk (u − v) to express ∂xk (u − v)(x0 ). Then by (3.10) and some calculations we get Lemma 3.3.

  2 σ−4 (D− T )(x0 )

|ξ ′ |=1

  = σ−4 (D−2 )(x0 ) − + −

1

(1 + ξ ) 1

2 3 n

(1 + ξn2 )3

|ξ ′ |=1

[(u + v)c (ξ )(u + v)c (ξ )] −

2 3  (1 + ξn )

6

(1 + ξ )

2 3 n

1

(1 + ξn2 )3

[c (ξ )(u + v)c (ξ )(u + v)]

[c (ξ )(u + v)(u + v)c (ξ )]

6

−

1

4

+

(1 + ξn2 )4 4

+

(1 + ξ )

2 4 n

+ +



2

(1 + ξn2 )2  2

(1 + ξn2 )2

h′ (0)ξn (u + v)c (ξ ) h′ (0)ξn c (ξ )(u + v) 1

h′ (0)(u + v)c (ξ )c (ξ )c (dxn ) 2(1 + ξn2 )3 3 1 − c (dxk )∂xk (u + v) − h′ (0) (u + v) (1 + ξn2 )2 k=1 2 (1 + ξn2 )2 1 1 + h′ (0)(u + v)c (dxn ) + h′ (0)c (dxn )(u + v) (1 + ξn2 )3 (1 + ξn2 )3 1 1 +2 h′ (0)(u + v)∂xn c (ξ ′ ) + 2 h′ (0)∂xn c (ξ ′ )(u + v) (1 + ξn2 )2 ( 1 + ξn2 )2    

−

+2   σ−4 ((D∗T )−2 )(x0 )

|ξ ′ |=1

h′ (0)c (ξ )c (dxn )(u + v)c (ξ ) −

2(1 + ξ ) n  i 2 3 n

1

(1 + ξ )

2 2 n

n 

c (ξ )

ξk ∂xk (u + v) + 2

k=1

n 

1

(1 + ξ )

2 2 n

ξk ∂xk (u + v) c (ξ );

(3.12)

k=1

  = σ−4 (D−2 )(x0 )

|ξ ′ |=1

− + −

1

[(u − v)c (ξ )(u − v)c (ξ )] −

(1 + ξ ) 1

2 3 n

6

2 3  (1 + ξn )

6

(1 + ξ ) 1

(1 + ξn2 )3

[c (ξ )(u − v)c (ξ )(u − v)]

[c (ξ )(u − v)(u − v)c (ξ )]

(1 + ξn2 )3

−

1

2 3 n

+ +

4

(1 + ξn2 )4 4

(1 + ξ )

2 4 n

+ +

2



(1 + ξn2 )2  2

(1 + ξn2 )2

h′ (0)ξn (u − v)c (ξ ) h′ (0)ξn c (ξ )(u − v) 1

h′ (0)(u − v)c (ξ )c (ξ )c (dxn ) 2(1 + ξn2 )3 3 1 (u − v) − c (dxk )∂xk (u − v) − h′ (0) (1 + ξn2 )2 k=1 2 (1 + ξn2 )2 1 1 + h′ (0)(u − v)c (dxn ) + h′ (0)c (dxn )(u − v) 2 3 (1 + ξn ) (1 + ξn2 )3 1 1 h′ (0)(u − v)∂xn c (ξ ′ ) + 2 h′ (0)∂xn c (ξ ′ )(u − v) +2 (1 + ξn2 )2 ( 1 + ξn2 )2    

−

+2  

Where σ−4 (D−2 )(x0 )

|ξ ′ |=1

  σ−4 (D−2 )(x0 )

|ξ ′ |=1

=

h′ (0)c (ξ )c (dxn )(u − v)c (ξ ) −

2(1 + ξ ) n  i 2 3 n

1

c (ξ ) 2

(1 + ξn2 )

n  k=1

ξk ∂xk (u − v) + 2

1

n 

(1 + ξn2 )2

k=1

has the following expression by the equation (115) in [15]

 2  2 − h′ (0) 9 h′ (0)  c ( e ) c ( e ) c ( e ) c ( e ) − ξ 3 ξµ ξν µ n ν n 4(1 + ξn2 )3 (1 + ξn2 )3 n  ′ 2 h (0) 1 + ξµ ξν c (eµ )c ( en )c ( eν )c ( en ) − s(x0 ) 2 2 4(1 + ξn ) 4(1 + ξn2 )2

ξk ∂xk (u − v) c (ξ ).

(3.13)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

−

5 3(1 + ξn2 )3

ξµ ξν

 i
∂

RiµMiν (x0 ) −

6

(1 + ξn2 )3

219

h′′ (0)ξn2

  ∂ 4h′′ (0) 2 ∂M M ξ ξ ξ ξ R ( x ) + R ( x ) + ξ µ ν γ δ 0 0 µγ νδ νγ µδ 2 4 3(1 + ξn ) (1 + ξn2 )4 n γ ,δ
−

+

2 + 3ξn + 10ξn2 + 12ξn3 − 4ξn4 + 9ξn5  ′ 2 h (0) . (1 + ξn2 )5

(3.14)

By Lemma 2.2 in [12], we can get Lemma 3.4. With the metric g M on M near the boundary 0, ∂xj (|ξ | )(x0 ) = h′ (0)|ξ ′ |2 , g∂M  0, ∂xj [c (ξ )](x0 ) = ∂ xn (c (ξ ′ ))(x0 ),



2 gM

if j < n; if j = n.

(3.15)

if j < n; if j = n,

(3.16)

where ξ = ξ ′ + ξn dxn . By Proposition 1.28 in [22], and some calculations one has following Lemma 3.5 ([23]). With the metric g M on M near the boundary

  ∂xi ∂xj (|ξ |2g M )(x0 )

|ξ ′ |=1

  ∂xi ∂xj [c (ξ )](x0 )

|ξ ′ |=1

 0, if i < n, j = n; or i = n, j < n;     1  ∂ ∂ M if i, j < n; Riα jβ (x0 ) + RiβMjα (x0 ) ξα ξβ , = −3  α,β
2

(3.17)

(3.18)

j
where ξ = ξ + ξn dxn . ′

By some calculations, we get Lemma 3.6.

Γ (x0 ) = k

if k < n; if k = n.



0 3h′ (0)

 σ (x0 ) = k

1 ′ h (0)c ( ek )c ( en )

(3.19)

if k < n;

4 0

(3.20)

if k = n.

Lemma 3.7 ([15]). Let g M be the metric on the 7-dimensional spin manifolds M near the boundary, then

  ∂xγ Γ k (x0 )

|ξ ′ |=1

  5 ∂   RiγMik (x0 ),  6   i
if γ < n, k < n; if γ < n, k = n; if γ = n, k < n; if γ = n, k = n

(3.21)

220

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

  ∂xγ σ k (x0 )

|ξ ′ |=1

1  ∂  RkγMst (x0 )c ( es )c ( et ),    8  s̸=t
if γ < n, k < n; if γ < n, k = n; (3.22)

if γ = n, k < n; if γ = n, k = n.

8 t
By relative equation in [19] and some calculations, we get Lemma 3.8. When γ < n,

    2 ∂xγ σ−3 (D− T ) (x0 )

|ξ ′ |=1

=−

ξk

5i



6 (1 + ξn2 )2 i
+

1

∂

RiγMik (x0 ) +



3 (1 + ξ )



1

(1 + ξn2 )2



∂

∂

∂

RkγMst (x0 )c ( es )c ( et )



RiαMjβ (x0 ) + RiβMjα (x0 ) ξj ξα ξβ

2 3 n α,β
− 2i

ξk

i

4 (1 + ξn2 )2 s= ̸ t
 ∂xγ (u + v) c (ξ ).

(3.23)

When γ = n,

    2 ∂xn σ−3 (D− T ) ( x0 ) 

|ξ ′ |=1

=

2ih′ (0)



(1 + ξn2 )3

2



ξk c ( ek )c ( en ) + 3h′ (0)ξn



k
   9 ′ ′′ 2 ξ 3h ( 0 ) − ( h ( 0 )) n 2

i

−

1

− h′ (0)

(1 + ξn2 ) 2      3 ′ 1 1  c ( es )c ( et ) − 2ξk (h (0))2 − h′′ (0) c ( en )c ( et ) − ξn (h′ (0))2 − h′′ (0) 8

4

4i

4

t
s̸=t
  h′ (0) (u + v)(x0 ) c (ξ ) 3

+

(1 + ξn2 )

−

(1 + ξn2 )2

2i

  ∂xn (u + v)(x0 ) c (ξ ) −

2i

(1 + ξn2 )2

  (u + v) ∂xn c (ξ ′ )(x0 ) .

(3.24)

Next we will compute Φ (seeformula (2.15) for definition of Φ ). Since the sum is taken over −r − ℓ + 1 + k + j + |α| = 7, r , ℓ ≤ −2, then we have the ∂ Φ is the sum of the following fifteen cases: M Case (1): r = −2, ℓ = −2, k = 0, j = 1, |α| = 1. From (2.15), we have Case (1) =

i





+∞





2 |ξ ′ |=1 −∞ |α|=1



2 ′ ′ trace ∂xn ∂ξα′ πξ+n σ−2 ((D∗T )−2 )∂xα′ ∂ξ2n σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.25)

By Lemma 3.4, for i < n, we have

 −2  2 ∂xi σ−2 (D− (x0 ) = 0. T )(x0 ) = ∂xi |ξ |

(3.26)

So Case (1) vanishes. Case (2): r = −2, ℓ = −2, k = 0, j = 2, |α| = 0. From (2.15), we have Case (2) =

i





+∞



6 |ξ ′ |=1 −∞





2 ′ ′ trace ∂x2n πξ+n σ−2 ((D∗T )−2 )∂ξ3n σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.27)

j =2

By Lemma 3.2 and a simple calculation, we get

  2 ∂ξ3n σ−2 (D− )( x )  0 T

|ξ ′ |=1

=

24ξn − 24ξn3

(1 + ξn2 )4

,

(3.28)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

221

and 2(h′ (0))2

∂x2n σ−2 ((D∗T )−2 )(x0 ) =

(1 + ξ )

2 3 n

−

h′′ (0)

(1 + ξn2 )2

.

(3.29)

By (2.10) and the Cauchy integral formula, we get

πξ+n

c (ξ )



 =

(1 + ξn2 )2

lim 2π i u→0−

πξ+n πξ+n

=

(1 + ξn2 )2 

c (ξ ′ ) + ic (dxn )

−

.

4(ξn − i)2

.

−3iξn2 − 9ξn + 8i . 16(ξn − i)3

=

(1 + ξn2 )3

4(ξn − i)

4(ξn − i)2



1

Γ+

ic (ξ ′ )

−2 − iξn



1

dηn

(ηn − i)2  (1)   c (ξ ′ ) + ηn c (dxn )  =  2 (ηn + i) (ξn − ηn ) ηn =i =−



c (ξ ′ )+ηn c (dxn ) (ηn +i)2 (ξn +iu−ηn )



1

(3.30)

By (3.29) and (3.30), we get

  ∂x2n πξ+n σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

2 + iξn ′′ −3iξn2 − 9ξn + 8i ′ (h (0))2 + h (0). 3 8(ξn − i) 4(ξn − i)2

(3.31)

Note that tr[id] = 8, then by (3.28), (3.31) and some direct computations, we obtain





 

2 trace ∂x2n πξ+n σ−2 ((D∗T )−2 )∂ξ3n σ−2 (D− T ) ( x0 ) 

|ξ ′ |=1

 2 (−3iξn2 − 9ξn + 8i)(24ξn − 24ξn3 ) (4 + 2iξn )(24ξn − 24ξn3 ) + h′′ (0) . = h′ (0) 3 2 4 (ξn − i) (1 + ξn ) (ξn − i)2 (1 + ξn2 )4

(3.32)

Therefore +∞ i  ′ 2 (−3iξn2 − 9ξn + 8i)(24ξn − 24ξn3 ) Case (2) = h (0) dξn σ (ξ ′ )dx′ 6 (ξn − i)3 (1 + ξn2 )4 |ξ ′ |=1 −∞





+∞ i (4 + 2iξn )(24ξn − 24ξn3 ) + h′′ (0) dξn σ (ξ ′ )dx′ 6 (ξn − i)2 (1 + ξn2 )4 |ξ ′ |=1 −∞





i  ′ 2 2π i (−3iξn2 − 9ξn + 8i)(24ξn − 24ξn3 ) h (0) 6 6! (ξn + i)4



=

i 2π i (4 + 2iξn )(24ξn − 24ξn3 ) + h′′ (0) 6 5! (ξn2 + i)4



=

7 8

2

h′ (0)

(5)    

ξn = i

(6)    

ξn = i

Ω5 dx′

Ω5 dx′

 3 − h′′ (0) π Ω5 dx′ ,

(3.33)

8

where Ω5 is the canonical volume of S 5 . Case (3): r = −2, ℓ = −2, k = 0, j = 0, |α| = 2. From (2.15), we have Case (3) =

i





+∞



2 |ξ ′ |=1 −∞ |α|=2





2 ′ ′ trace ∂ξα′ πξ+n σ−2 ((D∗T )−2 )∂xα′ ∂ξn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.34)

By Lemma 3.5, a simple calculation shows

  ∂ξα′ σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

  = ∂ξj ∂ξi σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

j i 2 2 n

8 −2δ + ξi ξj . (1 + ξ ) (1 + ξn2 )3

(3.35)

222

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

By (3.30) and (3.35), we obtain

  πξ+n ∂ξα′ σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

(2 + iξn )δij −3iξn2 − 9ξn + 8i + ξi ξj . 2 2(ξn − i) 2(ξn − i)3

(3.36)

On the other hand, by Lemmas 3.2 and 3.5, we obtain

  2 ∂xα′ σ−2 (D− T )(x0 )

|ξ ′ |=1

−1 2 ∂x ∂x (|ξ |2 )(x0 ) + ∂x (|ξ |2 )∂xi (|ξ |2 )(x0 ) (1 + ξn2 )2 i j (1 + ξn2 )3 j  2   ∂ 2 h′ (0) 1 ∂M M = R (x0 ) + Riβ jα (x0 ) ξα ξβ + . 3(1 + ξn2 )2 α,β
(3.37)

Hence in this case,

 2   ∂ −12ξn h′ (0) −4ξn ∂M M = R (x0 ) + Riβ jα (x0 ) ξα ξβ + . 3(1 + ξn2 )3 α,β
  ∂x′ ∂ξn σ−2 (DT )(x0 ) α

−2

|ξ ′ |=1

(3.38)

By (3.36), (3.38) and some direct computations, we obtain

    ∂ −4ξn − 2iξn2 ∂ 2 trace ∂ξα′ πξ+n σ−2 ((D∗T )−2 )∂xα′ ∂ξn σ−2 (D− RiαMjβ (x0 ) + RiβMjα (x0 ) ξα ξβ T ) (x0 ) = 2 2 3 3(ξn − i) (1 + ξn ) α,β
 −16iξn + 18ξn2 + 6iξn3   ∂M ∂ Riα jβ (x0 ) + RiβMjα (x0 ) ξi ξj ξα ξβ 3 2 3 3(ξn − i) (1 + ξn ) α,β
 2 + h′ (0)

 ′ 2 −48iξn + 54ξn2 + 18iξn3 −12ξn − 6iξn2 + h (0) . (ξn − i)2 (1 + ξn2 )4 (ξn − i)3 (1 + ξn2 )4

(3.39)

Similar to (16) in [6], we have



1 ξ ξ = [µν ], 6 µ ν



ξ µ ξ ν ξ α ξ β = c0 [µναβ ],

(3.40)

αβ where [µναβ ] stands for the sum of products of of µναβ and c0 is a constant. Using the   g 5 determined by all ‘‘pairings’’ 1 5 integration over S and the shorthand = π 3 S 5 d ν , we obtain Ω5 = π 3 . Let s∂M is the scalar curvature ∂M , then

 i,α,j,β
∂

RiαMjβ (x0 )

 |ξ ′ |=1

ξα ξβ ξi ξj σ (ξ ′ ) = c π 3

∂

 i,α,j,β


RiαMjβ (x0 ) δαβ δi + δαi δβ + δαj δβi j

j



= 0,

(3.41)

where c is a constant. Therefore

   ′ 2 +∞ 4iξn − 9ξn2 − 3iξn3 −4ξn − 2iξn2 ′ d ξ + h ( 0 ) d ξ n n dx 2 2 3 3 2 4 2 −∞ 9(ξn − i) (1 + ξn ) −∞ (ξn − i) (1 + ξn ) (4)          ′ 2 2π i 4iξn − 9ξn2 − 3iξn3 (6)  i 2π i −4ξn − 2iξn2   = Ω5 s∂M dx′ + h (0)   2 4! 9(ξn + i)3 6! (ξn + i)4 ξn = i ξn =i

Case (3) =

=

i

π 6

  Ω5 s∂M

+∞

s∂M Ω5 dx′ +

11π  ′ 2 h (0) Ω5 dx′ , 128

(3.42)

∂

M (x0 ) is the scalar curvature s∂M . where t ,l


Case (4) =

i





+∞

6 |ξ ′ |=1 −∞





2 ′ ′ trace ∂xn ∂ξn πξ+n σ−2 ((D∗T )−2 )∂ξ2n ∂xn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.43)

By (3.30), we obtain

∂xn ∂ξn πξ+n σ−2 ((D∗T )−2 )(x0 )||ξ ′ |=1 = h′ (0)

−3 − iξn 4(ξn − i)3

.

(3.44)

By Lemma 3.2 and some calculations, we obtain 2 ′ ′ ∂ξ2n ∂xn σ−2 (D− T )(x0 )||ξ |=1 = h (0)

4 − 20ξn2

(1 + ξn2 )4

.

(3.45)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

223

Note that tr[id] = 8, then by (3.44), (3.45) and some direct computations, we obtain +∞ −24 − 8iξn + 120ξn2 + 40iξn3 i  ′ 2 h (0) dξn σ (ξ ′ )dx′ 6 (ξn − i)3 (1 + ξn2 )4 |ξ ′ |=1 −∞



Case (4) =



i  ′ 2 2π i −24 − 8iξn + 120ξn2 + 40iξn3 h (0) 6 6! (ξn + i)4



=

(6)    

ξn = i

Ω5 dx′

2 5 = − h′ (0) π Ω5 dx′ . 8

(3.46)

Case (5): r = −2, ℓ = −2, k = 1, j = 0, |α| = 1 By (2.15), we have Case (5) =



i



+∞



2 |ξ ′ |=1 −∞ |α|=1





2 ′ ′ trace ∂ξα′ ∂ξn πξ+n σ−2 ((D∗T )−2 )∂xα′ ∂ξn ∂xn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.47)

By Lemmas 3.2 and 3.5, for i < n, we obtain

  2 ∂x′ ∂xn σ−2 (D− T )(x0 )

=

|ξ ′ |=1

−1 2 ∂x ∂x (|ξ |2 )(x0 ) + ∂x (|ξ |2 )∂xi (|ξ |2 )(x0 ) (1 + ξn2 )2 i n (1 + ξn2 )3 n

= 0.

(3.48)

Therefore Case (5) vanishes. Case (6): r = −2, ℓ = −2, k = 2, j = 0, |α| = 0 By (2.15), we have Case (6) =



i



+∞



6 |ξ ′ |=1 −∞





2 ′ ′ trace ∂ξ2n πξ+n σ−2 ((D∗T )−2 )∂ξn ∂x2n σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.49)

k=2

By Lemma 3.2 and some calculations, we have

  ∂ξ2n πξ+n σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

−i , (ξn − i)3

(3.50)

and

  2 ∂ξn ∂x2n σ−2 (D− T )(x0 )

=

|ξ ′ |=1

4ξn h′′ (x0 )

+

(1 + ξn2 )3

−12ξn (h′ (0))2 . (1 + ξn2 )4

(3.51)

Therefore +∞ i −32iξn Case (6) = h′′ (0) dξn σ (ξ ′ )dx′ ′ 6 (ξ − i)3 (1 + ξn2 )3 n |ξ |=1 −∞





+∞ 2 96iξn i dξn σ (ξ ′ )dx′ + h′ (0) ′ 6 (ξ − i )3 (1 + ξn2 )4 n |ξ |=1 −∞





i ′′ 2π i −32iξn h (0) 6 5! (ξn + i)3



=

(5)    

ξn = i

Ω5 dx′ +

i  ′ 2 2π i 96iξn h (0) 6 6! (ξn + i)4



(6)    

ξn = i

 3 2  7 = − h′′ (0) + h′ (0) π Ω5 dx′ . 8

Ω5 dx′ (3.52)

8

Case (7): r = −2, ℓ = −3, k = 0, j = 1, |α| = 0 From (2.15) and the Leibniz rule, we obtain Case (7) =

1





+∞

2 |ξ ′ |=1 −∞

=−

1









2 ′ ′ trace ∂ξn ∂xn πξ+n σ−2 ((D∗T )−2 )∂ξn σ−3 (D− T ) (x0 )dξn σ (ξ )dx

+∞

2 |ξ ′ |=1 −∞





2 ′ ′ trace ∂ξ2n ∂xn πξ+n σ−2 ((D∗T )−2 )σ−3 (D− T ) (x0 )dξn σ (ξ )dx .

(3.53)

By Lemma 3.2 and some calculations, we have

πξ+n ∂xn σ−2 ((D∗T )−2 )(x0 )||ξ ′ |=1 = h′ (0)

2 + iξn 4(ξn − i)2

.

(3.54)

224

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

Then 2 ′ ′ ∂ξ2n πξ+n ∂xn σ−2 (D− T )(x0 )||ξ |=1 = h (0)

4 + iξn 2(ξn − i)4

.

(3.55)

j In the normal coordinate, g ij (x0 ) = δi and ∂xj (g αβ )(x0 ) = 0, if j < n; = h′ (0)δβα , if j = n. By Lemma A.2 in [12] and Lemma 3.2, we obtain

σ−3 (D−2 )(x0 )||ξ ′ |=1 = −i|ξ |−4 ξk (Γ k − 2δ k )(x0 )||ξ ′ |=1 − i|ξ |−6 2ξ j ξα ξβ ∂j g αβ (x0 )||ξ ′ |=1 − 2i|ξ |−4 (u + v)c (ξ )  2ih′ (0)ξ  1  −i 2i(u + v)c (ξ ) n ′ ′ − = − h ( 0 ) ξ c ( e ) c ( e ) + 3h ( 0 )ξ − . k k n n 2 2 2 3 (1 + ξn ) 2 (1 + ξn ) (1 + ξn2 )2 k
(3.56)

By the relation of the Clifford action and trAB = trBA, we have the equalities: trace[(u + v)c (ξ )](x0 ) = −trace[(u − v)c (ξ )](x0 ) = −4



ξj Aiij − 4



j
trace[(u + v)c (dxn )](x0 ) = −trace[(u − v)c (dxn )](x0 ) = −4

ξn Aiin ,

i



Aiin .

(3.57)

i

We note that

 |ξ ′ |=1

ξ1 · · · ξ2q+1 σ (ξ ′ ) = 0.

(3.58)

So the first term in (3.56) has no contribution for computing case (7). Combining (3.55), (3.56) and some direct computations, we obtain





2 ′ 2 trace ∂ξ2n ∂xn πξ+n σ−2 ((D∗T )−2 )σ−3 (D− T ) (x0 ) = (h (0))

− h′ (0)

−80iξn + 20ξn2 − 48iξn3 + 12ξn4 (ξn − i)4 (1 + ξn2 )3 4i − ξn

(ξn − i)4 (1 + ξn2 )2

trace[(u + v)c (ξ )].

(3.59)

Therefore +∞ 2 −80iξn + 20ξn2 − 48iξn3 + 12ξn4 1 Case (7) = − h′ (0) 2 (ξn − i)4 (1 + ξn2 )3 |ξ ′ |=1 −∞







4i − ξn

− h′ (0)



trace[(u + v)c (ξ )] dξn σ (ξ ′ )dx′ 2

(ξn − i)4 (1 + ξn2 )    2 2π i −80iξn + 20ξn2 − 48iξn3 + 12ξn4 (6)  1  Ω5 dx′ = − h′ (0)  2 6! (ξn + i)3 ξn = i   2 (5)  2 π i 4i ξ − ξ n n + 4h′ (0) Aiin 5! (ξn + i)2 i    21  ′ 2 9 = h (0) + h′ (0) Aiin π Ω5 dx′ . 8

8

(3.60)

i

Case (8): r = −2, ℓ = −3, k = 0, j = 0, |α| = 1 From (2.15) and the Leibniz rule, we obtain





+∞

Case (8) = − |ξ ′ |=1





−∞ |α|=1 +∞





2 ′ ′ trace ∂ξα′ πξ+n σ−2 ((D∗T )−2 )∂xα′ ∂ξn σ−3 (D− T ) (x0 )dξn σ (ξ )dx





= |ξ ′ |=1



−∞ |α|=1



2 ′ ′ trace ∂ξn ∂ξα′ πξ+n σ−2 ((D∗T )−2 )∂xα′ σ−3 (D− T ) (x0 )dξn σ (ξ )dx .

(3.61)

By Lemma 3.2 and some calculations, we get

  ∂ξα′ σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

 i
  ∂ξi σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

−2ξi . (1 + ξn2 )2

(3.62)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

225

From (3.30) and some calculations, we obtain

  ∂ξn ∂ξα′ πξ+n σ−2 ((D∗T )−2 )(x0 )

=

|ξ ′ |=1

−3 − iξn 2(ξn − i)3

ξi .

(3.63)

By Lemma 3.8 and (3.63) and some direct computations, we obtain



+

8ξn − 24i



2 trace ∂ξn ∂ξα′ πξ+n σ−2 (D−2 )∂xα′ σ−3 (D− T ) ( x0 ) =

30i − 10ξn



3(ξn − i)3 (1 + ξn2 )2 i
∂ RiγMik (x0 )ξγ ξk



3(ξn − i) (1 + ξ ) 3

2 3 n α,β
∂

∂



RiαMjβ (x0 ) + RiβMjα (x0 ) ξi ξj ξα ξβ

  (3i − ξn )ξi trace ∂xγ (u + v)c (ξ ) + . (ξn − i)3 (1 + ξn2 )2

(3.64)

By (3.40), (3.58), (3.64) and some calculations, we obtain





Case (8) = |ξ ′ |=1

 =

1 9

 =

1 9

30i − 10ξn

+∞ 

3(ξn − i)3 (1 + ξn2 )2 i
−∞

s ∂ M Ω5 s ∂ M Ω5



15i − 5ξn

+∞



−∞ +∞ −∞

  (3i − ξn )ξi trace ∂x′ (u + v)c (ξ ) + (ξn − i)3 (1 + ξn2 )2    (−12i + 4ξn )ξi ξl ∂x′ Akkl 

∂ RiγMik (x0 )ξγ ξk



+∞



l
dξn dx′ (ξn − i)3 (1 + ξn2 )2 (ξn − i)3 (1 + ξn2 )2 −∞   +∞ (12i − 4ξn ) 15i − 5ξn 1 ′ A − ∂ dξn dx′ x kkl 3 (1 + ξ 2 )2 (ξn − i)3 (1 + ξn2 )2 6 k (ξ − i ) n −∞ n +

(4)   (4)     2π i 12i − 4ξn  Ω5 dx′ − 1  Ω5 dx′ ′ ∂ A x kki   2 2 9 4! (ξn + i) 6 4 ! (ξ + i ) n ξn = i ξn = i k    3 5 s∂M − ∂x′ Akki π Ω5 dx′ . = =

1

2π i 15i − 5ξn



s∂M

16

8

(3.65)

k

Case (9): r = −2, ℓ = −3, k = 1, j = 0, |α| = 0 From (2.15) and the Leibniz rule, we obtain Case (9) = −

=

1

1





+∞

2 |ξ ′ |=1 −∞ |α|=1





+∞



2 |ξ ′ |=1 −∞ |α|=1







2 ′ ′ trace ∂ξn πξ+n σ−2 ((D∗T )−2 )∂ξn ∂xn σ−3 (D− T ) (x0 )dξn σ (ξ )dx





2 ′ ′ trace ∂ξ2n πξ+n σ−2 ((D∗T )−2 )∂xn σ−3 (D− T ) (x0 )dξn σ (ξ )dx .

(3.66)

−i . (ξn − i)3

(3.67)

From (3.51), we have

  ∂ξ2n πξ+n σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

Combining Lemma 3.8 and (3.67), we obtain

2

h′ (0) (84ξn + 36ξn3 )

−24ξn h′′ (0) + |ξ ′ |=1 (ξn − i)3 (1 + ξn2 )3 (ξn − i)3 (1 + ξn2 )2     ′ 4h (0) 2 + trace ( u + v) c (ξ )( x ) − trace ∂ ( u + v) c (ξ )( x ) 0 xn 0 (1 + ξn2 )2 (ξn − i)3 (1 + ξn2 )3 (ξn − i)3   2 − trace (u + v)∂xn c (ξ ′ )(x0 ) . 2 (1 + ξn )2 (ξn − i)3 

trace

2 ∂ξn πξ+n σ−2 ((D∗T )−2 )∂xn σ−3 (D− T ) 2

   (x0 )



=

(3.68)

Since

∂xn c (ξ ′ )(x0 ) =

 j
∂xn ξj c (dxj ) =

 j
√ ξj ∂xn ( h < dxj , el >∂ M )c ( el ).

(3.69)

226

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

By (3.58), we obtain +∞ (84ξn + 36ξn3 ) 1  ′ 2 h (0) dξn σ (ξ ′ )dx′ 3 2 3 2 |ξ ′ |=1 −∞ (ξn − i) (1 + ξn )



Case (9) =



−24ξn dξn σ (ξ ′ )dx′ 3 2 2 2 −∞ (ξn − i) (1 + ξn )     +∞ 1 4h′ (0) + trace (u + v)c (ξ )(x0 ) 2 |ξ ′ |=1 −∞ (1 + ξn2 )3 (ξn − i)3     +∞ 2 1 trace ∂ ( u + v) c (ξ )( x ) dξn σ (ξ ′ )dx′ − x 0 n 2 |ξ ′ |=1 −∞ (1 + ξn2 )3 (ξn − i)3  (5)   (4)    1  ′ 2 2π i 84ξn + 36ξn3  Ω5 dx′ + 1 h′′ (0) 2π i −24ξn  Ω5 dx′ = h (0)   3 2 2 5! (ξn + i) 2 4 ! (ξ + i ) n ξn = i ξn = i  (5)  (4)     ξ ξ 2 π i 2 π i n n +4 ∂xn Aiin − 8h′ (0) Aiin 3 5 ! (ξ + i ) 4 ! (ξ + i)2 n n i i 9   3  27  ′ 2 9 ′ Aiin − = h′′ (0) − h (0) + h (0) ∂xn Aiin π Ω5 dx′ . 1

+ h′′ (0)





+∞

|ξ ′ |=1

8

8

16

8

i

(3.70)

i

Case (10): r = −3, ℓ = −2, k = 0, j = 1, |α| = 0 From (2.15), we have Case (10) = −

1





+∞

2 |ξ ′ |=1 −∞





2 ′ ′ trace ∂xn πξ+n σ−3 ((D∗T )−2 )∂ξ2n σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.71)

By the Leibniz rule, trace property and ‘‘++’’ and ‘‘− −’’ vanishing after the integration over ξn in [10], then



+∞





2 trace ∂xn πξ+n σ−3 ((D∗T )−2 )∂ξ2n σ−2 (D− T ) dξn

−∞

+∞







2 trace ∂xn σ−3 ((D∗T )−2 )∂ξ2n σ−2 (D− T ) dξn −

= −∞



+∞ −∞





2 trace ∂xn σ−3 ((D∗T )−2 )∂ξ2n πξ+n σ−2 (D− T ) dξn .

(3.72) Similar to case (9), we obtain 1





+∞



2 |ξ ′ |=1 −∞ |α|=1

 =





2 ∗ −2 trace ∂ξ2n πξ+n σ−2 (D− (x0 )dξn σ (ξ ′ )dx′ T )∂xn σ−3 (DT )



  9 ′′ 27  ′ 2 9 ′ 3  h (0) − h (0) − h (0) Aiin (x0 ) + ∂xn Aiin (x0 ) π Ω5 dx′ . 8 8 16 8 i i

(3.73)

By Lemma 3.2, a simple computation we obtain

  2 ∂ξ2n σ−2 (D− )( x ) 0  T

|ξ ′ |=1

=

6ξn2 − 2

(1 + ξn2 )3

.

(3.74)

Combining Lemma 3.8 and (3.74), we obtain





2 ′ 2 trace ∂xn σ−3 ((D∗T )−2 )∂ξ2n σ−2 (D− T ) (x0 ) = (h (0))

+ h′ (0) −

24iξn2 − 8i

(1 + ξn2 )6

12iξn2 − 4i

(1 + ξn2 )5

trace[(u + v)c (ξ )(x0 )] −

trace[(u + v)∂xn c (ξ ′ )(x0 )].

(84iξn + 36iξn3 )(−2 + 6ξn2 ) 24iξn (2 − 6ξn2 ) ′′ + h ( 0 ) . (1 + ξn2 )6 (1 + ξn2 )5

12iξn2 − 4i

(1 + ξn2 )5

trace[∂xn (u + v)c (ξ )(x0 )] (3.75)

By simple calculation, we obtain



+∞ −∞

(84iξn + 36iξn3 )(−2 + 6ξn2 ) dξn = 0, (1 + ξn2 )6

(3.76)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

227

and



24iξn (2 − 6ξn2 )

+∞

(1 + ξn2 )5

−∞

dξn = 0.

(3.77)

By (3.57), (3.58) we obtain 1





h′ (0)

2 |ξ ′ |=1 −∞ 1

−

24iξn3 − 8iξn

+∞





(1 + ξn2 )6

+∞





trace (u + v)c (ξ )(x0 )

12iξn3 − 4iξn





trace ∂xn (u + v)c (ξ )(x0 ) dξn σ (ξ ′ )dx′

(1 + ξn2 )5  (5)      2π i 12iξn3 − 4iξn (4) 2π i 24iξn3 − 8iξn ′ = −2h (0) Aiin +2 ∂xn Aiin 5! (ξn + i)6 4! (ξn + i)5 i i 2 |ξ ′ |=1 −∞

= 0.

(3.78)

Therefore Case (10) =

9 8

h′′ (0) −

  3  27  ′ 2 9 ′ Aiin + h (0) − h (0) ∂xn Aiin π Ω5 dx′ . 8 16 8 i i

(3.79)

Case (11): r = −3, ℓ = −2, k = 0, j = 0, |α| = 1 From (2.15), we have





+∞



Case (11) = − |ξ ′ |=1

−∞ |α|=1





2 ′ ′ trace ∂ξα′ πξ+n σ−3 ((D∗T )−2 )∂xα′ ∂ξn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.80)

By Lemmas 3.2 and 3.4, for i < n, we have

 −2  2 ∂xi σ−2 (D− (x0 ) = 0. T )(x0 ) = ∂xi |ξ |

(3.81)

So Case (11) vanishes. Case (12): r = −3, ℓ = −2, k = 1, j = 0, |α| = 0 From (2.15) and the Leibniz rule, we have Case (12) = −

=

1

1





+∞

2 |ξ ′ |=1 −∞





+∞

2 |ξ ′ |=1 −∞





2 ′ ′ trace ∂xn πξ+n σ−3 ((D∗T )−2 )∂ξn ∂xn σ−2 (D− T ) (x0 )dξn σ (ξ )dx





2 ′ ′ trace πξ+n σ−3 ((D∗T )−2 )∂ξ2n ∂xn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.82)

By the Leibniz rule, trace property and ‘‘++’’ and ‘‘− −’’ vanishing after the integration over ξn in [10], then



+∞





2 trace πξ+n σ−3 ((D∗T )−2 )∂ξ2n ∂xn σ−2 (D− T ) dξn

−∞



+∞

= −∞





2 trace σ−3 ((D∗T )−2 )∂ξ2n ∂xn σ−2 (D− T ) dξn −



+∞ −∞





2 trace σ−3 ((D∗T )−2 )∂ξ2n ∂xn πξ+n σ−2 (D− T ) dξn .

(3.83)

Similar to Case (7), we get the second term (II) 1





+∞

2 |ξ ′ |=1 −∞





2 trace σ−3 ((D∗T )−2 )∂ξ2n ∂xn πξ+n σ−2 (D− T ) dξn =





 21  ′ 2 9 h (0) − h′ (0) Aiin (x0 ) π Ω5 dx′ . 8 8 i

(3.84)

From some direct computations, we obtain 2 ′ ∂ξ2n ∂xn σ−2 (D− T )(x0 )||ξ |=1 =

4 − 20ξn2 ′ h (0). (1 + ξn2 )4

(3.85)

Combining Lemma 3.8 and (3.85), we obtain





2 ′ 2 trace σ−3 ((D∗T )−2 )∂ξ2n ∂xn σ−2 (D− T ) (x0 ) = (h (0))

+ h′ (0)

−20iξn + 88iξn3 + 60iξn5 (1 + ξn2 )7

−8i + 40iξn2 trace[(u + v)c (ξ )(x0 )]. (1 + ξn2 )6

(3.86)

228

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

From some direct computations, we obtain

 (6)  −20iξn + 88iξn3 + 60iξn5 2π i −20iξn + 88iξn3 + 60iξn5  d ξ = = 0.  n ξn = i (1 + ξn2 )7 6! (ξn + i)7 −∞  (5)   +∞  −8i + 40iξn2 2π i −8iξn + 40iξn3  ′ ′ h (0) trace[(u + v)c (ξ )(x0 )]dξn = h (0) Aiin (x0 ) = 0.  ξn = i (1 + ξn2 )6 5! (ξn + i)7 −∞ i 

+∞

(3.87)

(3.88)

Therefore

 Case (12) =



 21  ′ 2 9 h (0) − h′ (0) Aiin π Ω5 dx′ . 8 8 i

(3.89)

In the following, since the related traces are complicated, so we just display the valid parts of these traces in the calculation of the Case (13) to Case (15). By the relation of the Clifford action and trAB = trBA, we have following equalities:



trace[uc (dxn )uc (dxn )](x0 ) = s1 =







Aiin Ajjn − 2

Aiit Ajjt ,

t
j ,i

trace[uc (ξ ′ )uc (ξ ′ )](x0 ) = s3 =

[Akts + Askt + Atsk ]2 −





Aist [Aist + Atis + Asti ],

i̸=s̸=t
k̸=s̸=t

trace[uu](x0 ) = s4 =

Aits [Aits + Atis + Asti ],

i̸=s̸=t
n̸=s̸=t

trace[v c (dxn )v c (dxn )](x0 ) = s2 = 2



[Ants + Atsn + Asnt ]2 +

Aist [Aist + Atis + Asti ],

i̸=s̸=t

trace[vv](x0 ) = s5 = −2



Aiit Ajjt .

(3.90)

t ,j ,i

Other cases are zeros. Case (13): r = −3, ℓ = −3, k = 0, j = 0, |α| = 0 From (2.15) and the Leibniz rule, we have





+∞

|ξ ′ |=1





−∞ +∞

=i |ξ ′ |=1



−∞



2 ′ ′ trace πξ+n σ−3 ((D∗T )−2 )∂ξn σ−3 (D− T ) (x0 )dξn σ (ξ )dx

Case (13) = −i





2 ′ ′ trace ∂ξn πξ+n σ−3 ((D∗T )−2 )σ−3 (D− T ) (x0 )dξn σ (ξ )dx .

(3.91)

By Lemma 3.2, we obtain

σ−3 ((D∗T )−2 )(x0 )||ξ ′ |=1 = 2 ′ σ−3 (D− T )(x0 )||ξ |=1 =

i 2(1 + ξ )

2 2 n

i 2(1 + ξ )

2 2 n

h′ (0)

h′ (0)



ξk c ( ek )c ( en ) + h′ (0)

k


ξk c ( ek )c ( en ) + h′ (0)

k
2i −5iξn − 3iξn3 − (u − v)c (ξ ). (1 + ξn2 )3 (1 + ξn2 )2

−5iξn − 3iξn3 2i − (u + v)c (ξ ). (1 + ξn2 )3 (1 + ξn2 )2

(3.92)

(3.93)

Similar to the calculations of (3.30), we obtain

 9i − 7ξn −5iξn − 3iξn3 = . (1 + ξn2 )3 8(ξn − i)3   2i c (ξ ′ ) c (ξ ′ ) + ic (dxn ) πξ+n ( u + v) c (ξ ) = −( u − v) . + ( u − v) i (1 + ξn2 )2 2(ξn − i) 2(ξn − i)2

πξ+n



(3.94)

(3.95)

Then we obtain

∂ξn πξ+n σ−3 ((D∗T )−2 )(x0 )||ξ ′ |=1 =

3i − ξn

h′ (0) 3

4(ξn − i)

+ (u − v)c (ξ ′ )



ξk c ( ek )c ( en ) + h′ (0)

k
(ξn − 3i) 2(ξn − i)3

7ξn − 10i 4(ξn − i)4

+ (u − v)c (dxn )

1

(ξn − i)3

.

(3.96)

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

229

By (3.90) and some calculations, we get

 tr

 



ξk c ( ek )c ( en )(u + v)c (ξ ) (x0 ) = 4 ′

k
Aiin , tr[(u + v)c (dxn )](x0 ) = −4



i

Aiin ,

i

tr[(u − v)c (dxn )(u + v)c (ξ )](x0 ) = ξn (s1 − s2 ),

tr[(u − v)c (ξ ′ )(u + v)c (ξ )](x0 ) = s3 .

(3.97)

Then

  (−3 − iξn )h′ (0)  ξj ξk trace c ( ej )c ( en )c ( ek )c ( en ) 3 2 2 8(ξn − i) (1 + ξn ) k




2 trace ∂ξn πξ+n σ−3 ((D∗T )−2 )σ−3 (D− T ) (x0 ) =

+ (h′ (0))2

2(10i − 7ξn )(5iξn + 3iξn3 )

(ξn − i)4 (1 + ξn2 )3

   (ξn − 3i)i ′ + h (0) trace (u − v)c (ξ ) ξk c ( ek )c ( en ) 4(1 + ξn2 )2 (ξn − i)3 k
−5iξn − 3iξn3 trace[(u − v)c (dxn )] 2(ξn − i)3 (1 + ξn2 )3    3 + iξn ′ + h (0) ξk c ( ek )c ( en )(u + v)c (ξ ) trace 2(ξn − i)3 (1 + ξn2 )2 k
+ h′ (0)

− h′ (0) − −

7iξn + 10 2(ξn − i)4 (1 + ξn2 )2

trace[(u + v)c (ξ )]

(iξn + 3) trace[(u − v)c (ξ ′ )(u + v)c (ξ )] (ξn − i)3 (1 + ξn2 )2 2i

(ξn − i)3 (1 + ξn2 )2

trace[(u − v)c (dxn )(u + v)c (ξ )].

(3.98)

By (3.58), we obtain

 Case (13) =

9 8

π h′ (0)



Aiin +

i

3

27 16

9

π h′ (0)



Aiin −

i

9 16

57

 π (h (0)) Ω5 dx′

16

16

π h′ (0)



Aiin +

i

27 32

π h′ (0)



Aiin

i

− π s0 − π s1 − 16 16 8    3 9 57 99 Aiin − π h′ (0) π (s1 − s2 ) − π (s3 − s4 ) − π (h′ (0))2 Ω5 dx′ . = 32

i

′

2

8

(3.99) Case (14): r = −2, ℓ = −4, k = 0, j = 0, |α| = 0 From (2.15) and the Leibniz rule, we have





+∞

Case (14) = −i |ξ ′ |=1





−∞ +∞

=i |ξ ′ |=1



−∞



2 ′ ′ trace πξ+n σ−2 ((D∗T )−2 )∂ξn σ−4 (D− T ) (x0 )dξn σ (ξ )dx





2 ′ ′ trace ∂ξn πξ+n σ−2 ((D∗T )−2 )σ−4 (D− T ) (x0 )dξn σ (ξ )dx .

(3.100)

From (3.30), we have

  ∂ξn πξ+n σ−2 ((D∗T )−2 )(x0 )

|ξ ′ |=1

=

i 2(ξn − i)2

.

(3.101)

By (3.90), we obtain trace[(u + v)c (ξ )(u + v)c (ξ )](x0 ) = trace[(u − v)c (ξ )(u − v)c (ξ )](x0 ) = s3 + ξn2 (s1 + s2 ), trace[(u + v)(u + v)](x0 ) = trace[(u − v)(u − v)](x0 ) = s4 + s5 , trace[c (ξ )(u + v)(u + v)c (ξ )](x0 ) = trace[c (ξ )(u − v)(u − v)c (ξ )](x0 ) = −(1 + ξn2 )(s4 + s5 ), trace[c (ξ ′ )c (dxn )(u + v)c (ξ ′ )](x0 ) = −trace[c (ξ ′ )c (dxn )(u − v)c (ξ ′ )](x0 ) = 4

 i̸=n

Aiin ,

230

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

trace

 n 

 c (dxk )∂xk (u + v) (x0 ) = −trace

c (dxk )∂xk (u − v) (x0 ) = −4



 n  k=1





∂k Aiik ,

i,k

k=1

k=1

trace



 n 



 n 



ξk ∂xk (u − v) c (ξ ) (x0 ) ξk ∂xk (u + v) c (ξ ) (x0 ) = −trace   k=1 = −4 ∂k Aiik − 4ξn2 ∂n Aiin . k
(3.102)

i

By Lemma 3.3, (3.101), (3.102) and (118) in [15], we obtain

 Case (14) =

−1 4

35

s(x0 ) −

− h′ (0)



96

Aiin

s∂M (x0 ) +

i15π

15π 3i  ′ 2 13 ′′ π − h (0) + h (0) − s3 − (s1 + s2 ) 192 2 16 64 64



343



+ h′ (0)

8

i



Aiin

15π 32

i

− ( s4 + s5 )

π 8

+

     15π π Aiin ∂xk Aiik 2π  − + h′ (0) Aiin − h′ (0) 4 16 =

−1 4

i,,k

i

i



35

s(x0 ) −

− ( s4 + s5 )

96

π 8

s∂M (x0 ) +

 −

i15π



+

8



∂xk Aiik

i,k

iπ 2

Ω5 dx′ ξn = i

3i  ′ 2 13 ′′ 15π π − h (0) + h (0) − s3 − (s1 + s2 ) 192 2 16 64 64



343

53π



32

h (0) ′



 Aiin +

iπ 2

i

− 2π





∂xk Aiik Ω5 dx′ .

(3.103)

i ,k

Case (15): r = −4, ℓ = −2, k = 0, j = 0, |α| = 0 From (2.15), we have





+∞

Case (15) = −i |ξ ′ |=1

−∞





2 ′ ′ trace πξ+n σ−4 ((D∗T )−2 )∂ξn σ−2 (D− T ) (x0 )dξn σ (ξ )dx .

(3.104)

By the Leibniz rule, trace property and ‘‘++’’ and ‘‘− −’’ vanishing after the integration over ξn in [10], then +∞







2 trace πξ+n σ−4 ((D∗T )−2 )∂ξn σ−2 (D− T ) dξn

−∞

+∞







2 trace σ−4 ((D∗T )−2 )∂ξn σ−2 (D− T ) dξn −

=



−∞

+∞ −∞





2 trace σ−4 ((D∗T )−2 )∂ξn πξ+n σ−2 (D− T ) dξn .

(3.105)

Similar to the calculation of Case (14), we obtain



+∞



i |ξ ′ |=1

 =

−∞

−1 4





2 ′ ′ trace σ−4 ((D∗T )−2 )∂ξn πξ+n σ−2 (D− T ) dξn σ (ξ )dx

s(x0 ) −

− (s4 + s5 )

π 8

35 96

s∂M (x0 ) +

+ h (0) ′





343 192

 Aiin

i15π 8

i̸=n

3i  ′ 2 13 ′′ 15π π h (0) + h (0) − s3 − ( s1 + s2 ) 2 16 64 64



−

+

53π 32

 −



∂xk Aiik

i̸=k



iπ

  − 2π  Ω5 dx′ . 2 ξn = i

(3.106)

By Lemma 3.2, we obtain

  2 ∂ξn σ−2 (D− T )(x0 )

|ξ ′ |=1

=

−2ξn . (1 + ξn2 )2

(3.107)

By Lemma 3.8 and (3.107), we obtain





 

2 trace σ−4 ((D∗T )−2 )∂ξn σ−2 (D− T ) (x0 )

|ξ ′ |=1

    = trace σ−4 (D−2 )∂ξn σ−2 (D−2 ) (x0 )

2ξn trace[−2|ξ |−6 (u + v)c (ξ )(u + v)c (ξ )] (1 + ξn2 )2 (1 + ξn2 )3 2ξn − trace[−2|ξ |−6 c (ξ )(u + v)(u + v)c (ξ )] 2 (1 + ξn )2 (1 + ξn2 )3

+

|ξ ′ |=1

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

+ h′ (0)

12ξn2



8ξn2

+

4ξn2

+



(1 + ξ ) (1 + ξ ) 2(ξn − i) (1 + ξ ) (1 + ξ ) (1 + ξ ) 2 ξ n + h′ (0) trace[c (ξ )c (dxn )(u + v)c (ξ )] 2(1 + ξn2 )2 (1 + ξn2 )3   n 2ξn + ih′ (0) trace c ( dx )∂ ( u + v) k x k (1 + ξn2 )2 (1 + ξn2 )2 k=1 + h′ (0)

2 2 n

2 3 n

6ξn 2(1 + ξ ) (1 + ξ ) 2 2 n

2 2 n

2

2 4 n

trace[(u + v)] − h′ (0)

231

2 2 n

2 2 n

2ξn

(1 + ξ ) (1 + ξn2 )3 2 2 n

trace[2(u + v)c (ξ )]

trace[2(u + v)c (dxn )]

4ξn trace[2(u + v)∂xn c (ξ ′ )] (1 + ξn2 )2 (1 + ξn2 )2     n 4ξn − ( u + v) c (ξ ) . trace 2 ξ ∂ k xk (1 + ξn2 )2 (1 + ξn2 )2 k=1

− h′ (0)

(3.108)

By some calculations, we get





+∞

|ξ ′ |=1





2 ′ 2 ′ trace σ−4 ((D∗T )−2 )∂ξn σ−2 (D− T ) dξn = 3i(h (0)) π Ω5 dx .

−i

(3.109)

−∞

Therefore

 Case (15) =

−1 4

s(x0 ) −

− ( s1 + s2 )

35 96

π



s∂M (x0 ) +

− (s4 + s5 )

64

π 8

3i  ′ 2 13 ′′ 15π + h (0) + h (0) − s3 192 2 16 64



343

 +

i15π 8

+

53π



32

h (0) ′



 Aiin −

iπ

i

− 2π

2



 ∂xk Aiik . Ω5 dx′ .

i,k

(3.110) Now Φ is the sum of the case (1, 2, . . . , 15), so 15 

Φ=

 1475 

case I = −

I =1

h′ (0)

384

2

  25 ′′ 1 77 3 99 Aiin π Ω5 dx′ . (3.111) h (0) − s − s∂M + ∂x′ Akki + h′ (0) 8 2 192 8 k 32 i

+

Hence we have the conclusion as follows.

 be an orthogonal connection with Theorem 3.9. Let M be a 7-dimensional compact manifold with the boundary ∂ M, and ∇  torsion. Then we get the volumes associated to D∗T DT with torsion on M. [π + (D∗ )−2 ◦ π + D−2 ] = Wres T T

 ∂M

+

 1475  2 25 ′′ 1 77 − h′ (0) + h (0) − s − s∂M 384

3 8

8

∂x′ Akki +

k

99 ′ h (0)

2



32

192



Aiin π Ω5 dx′ .

(3.112)

i

4. The gravitational action for 7-dimensional manifolds with boundary Firstly, we recall the Einstein–Hilbert action for manifolds with boundary (see [12] or [13]), IGr =

1 16π



  RdvolM + 2

M

∂M

 K dvol∂M := IGr,i + IGr,b ,

(4.1)

where

 1475    2 25 ′′ 1 77 3 99  K =K+ h′ (0) − h (0) + s + s∂M − ∂x′ Akki − h′ (0) Aiin , 384

K =

 1≤i,j≤n−1

8

i,j

Ki,j g∂ M ;

Ki,j = −Γi,nj

2

192

8

k

32

(4.2)

i

(4.3)

232

K.H. Bao et al. / Journal of Geometry and Physics 110 (2016) 213–232

and Ki,j is the second fundamental form, or extrinsic curvature. we assume that ∂ M is flat, then by Proposition 2.4 we obtain IGr,i = 0. Then

 3 1475  ′ 2 25 ′′ 1 77 99  K ( x0 ) = h (0) − h (0) + s + s∂M − Aiin , ∂x′ Akki − h′ (0) 384

8

2

192

8

k

32

K (x0 ) = 0.

(4.4)

i

Let

[π + (D∗ )−2 ◦ π + D−2 ] = Wres i [π + (D∗ )−2 ◦ π + D−2 ] + Wres b [π + (D∗ )−2 ◦ π + D−2 ]. Wres T T T T T T

(4.5)

Combining (3.112), (4.1) and (4.4), we obtain

 be an orthogonal connection with Theorem 4.1. Let M be a 7-dimensional compact manifold with the boundary ∂ M, and ∇ torsion. Then we get IGr,b =

−2  + ∗ − 2 2 Wresb [π (DT ) ◦ π + D− T ]. π Ω5

(4.6)

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