Noncommutative Residue and Dirac operators for Manifolds with the Conformal Robertson-Walker metric

0.23 MB·English

by  Jian Wang

Noncommutative Residue and Dirac operators for Manifolds
with the Conformal Robertson-Walker metric
Jian Wanga,b, Yong Wanga,∗
aSchool of Mathematics and Statistics, Northeast Normal University, Changchun, 130024, P.R.China
bChengde Petroleum College, Chengde, 067000, P.R.China
3
1 Abstract
0
2 In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin
manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds
n
a of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional
J manifolds with flat boundary. In particular, for 6-dimensional spin manifolds with boundary with the

2 conformal Robertson-Walker metric, we obtain the noncommutative residue of the composition of π+D−1
1 and π+D−3 is proportional to the Einstein-Hilbert action for manifolds with boundary.

] Keywords: lower-dimensionalvolumes; noncommutative residue; gravitationalaction; conformal
G
Robertson-Walker metric.
D
2000 MSC: 53G20, 53A30, 46L87
.
h
t
a
m
1. Introduction
[

1 The noncommutative residue plays a prominent role in noncommutative geometry [1][2]. In [3], Connes
v usedthenoncommutativeresiduetoderiveaconformal4-dimensionalPolyakovactionanalogy. Severalyears
2 ago,Connesmadeachallengingobservationthatthe noncommutativeresidueofthe squareofthe inverseof
5
the Dirac operator was proportional to the Einstein-Hilbert action, which was called Kastler-Kalau-Walze
6
Theorem now. In [4], Kastler gave a brute-force proof of this theorem. In [5], Kalau and Walze proved this
2
. theorem by the normal coordinates way simultaneously. In [6], Ackermann gave a note on a new proof of
1
this theorem by means of the heat kernel expansion.
0
On the other hand, Fedosov etc. defined a noncommutative residue on Boutet de Monvel’s algebra
3
1 and proved that it was a unique continuous trace in [7]. Wang generalized the Connes’ results to the
: case of manifolds with boundary in [8] [9] , and proved a Kastler-Kalau-Walze type theorem for the Dirac
v
i operator and the signature operator for lower-dimensional manifolds with boundary. In [10], we get the
X
Kastler-Kalau-Walze type theorem associated to nonminimal operators by heat equation asymptotics on
r compact manifolds without boundary. The purpose of papers [11] [12] is to prove the Kastler-Kalau-Walze
a
Theorem for manifolds associated with the metric near the boundary as follows: gM = 1 g∂M +dx2.
h(xn) n
In [13], Antoci considered the metric gM =e−2(a+1)xndx2 +e−2bxng∂M near the boundary and studied the
n
spectrum of the Laplace-Beltrami operator for p-forms. In [15], the authors dealed with a particular class
of warped products, i.e. when the pseudo- metric in the base is affected by a conformal change. Motivated
by Antoci and Dobarro etc., in this paper we consider the following metric near the boundary gM =
1 g∂M +ψ(x )dx2, which we call the conformal Robertson-Walker metric. We derive the gravitational
ϕ(xn) n n
action on boundary by the noncommutative residue associated with Dirac operator for the above metric.
]
For lower dimensional manifolds with boundary, we compute Wres[π+D−p1 π+D−p2] with the conformal
◦
∗Correspondingauthor.
Email address: [email protected] (YongWang)
Preprint submitted toElsevier January 15, 2013
Robertson-Walker metric gM on M , and we get a generalized Kastler-Kalau-Walze theorem for lower
dimensional spin manifolds. For 6-dimensional manifolds with boundary with the conformal Robertson-
]
Walker metric, we get Wres[π+D−1 π+D−3] is proportional to the Einstein-Hilbert action for manifolds
◦
withboundary. Whichgivesaoperatortheoreticexplanationsofthetotalgravitationalactionformanifolds
]
with boundary, i.e, I = −3 Wres[π+D−1 π+D−3].
Gr 80πΩ5 ◦
This paper is organized as follows: In Section 2, we define lower dimensional volumes of spin manifolds
with boundary. In Section 3, for 4-dimensional spin manifolds with boundary of the conformal Robertson-
Walkermetricandthe associatedDiracoperatorD ,we computethe lowerdimensionalvolumeVol(1,1) and
4
get a Kastler-Kalau-Walze type theorem in this case. In Section 4, for 6-dimensional spin manifolds with
boundaryofthe conformalRobertson-Walkermetric andthe associatedDirac operatorD , wecompute the
lowerdimensionalvolumeVol(2,2) andgetaKastler-Kalau-Walzetypetheoreminthiscase. InSection5,for
6
6-dimensional spin manifolds with boundary of the conformal Robertson-Walker metric and the associated
Dirac operator D and D3 , We compute the lower dimensional volume Vol(1,3) for 6-dimensional spin
6
manifoldswithboundaryandobtainthenoncommutativeresidueofthe compositionofπ+D−1 andπ+D−3
is proportional to the Einstein-Hilbert action for manifolds with boundary..
2. Lower dimensional volumes of spin manifolds with boundary
In order to define lower dimensional volumes of spin manifolds with boundary, we need some basic
facts and formulae about Boutet de Monvel’s calculus and the definition of the noncommutative residue for
manifolds with boundary. We can find them in Section 2,3 [9] and Section 2.1 [11].
Let M be a n-dimensional compact oriented spin manifold with boundary ∂M. We assume that the
metric gM on M has the following form near the boundary,
1
gM = g∂M +ψ(x )dx2, (2.1)
ϕ(x ) n n
n
where g∂M is the metric on ∂M, ϕ(x ),ψ(x ) > 0 and ϕ(0) = ψ(0) = 1. Let D be the Dirac operator
n n
associatedto g onthe spinorsbundle S(TM)[11]. Letp ,p be nonnegativeintegers andp +p n. From
1 2 1 2
≤
Section 2 in [11], we have
Definition 2.1. Lower dimensional volumes of spin manifolds with boundary are defined by
Vol(p1,p2)M :=W]res[π+D−p1 π+D−p2]. (2.2)
n ◦
Denote by σ (A) the l-order symbol of an operator A. By (2.1.4)-(2.1.8) in [11], we get
l
W]res[π+D−p1 π+D−p2]= trace [σ (D−p1−p2)]σ(ξ)dx+ Φ, (2.3)
S(TM) −n
◦
ZMZ|ξ|=1 Z∂M
and
+∞ ∞ ( i)|α|+j+k+1
Φ = Z|ξ′|=1Z−∞ j,k=0 α−!(j+k+1)! ×traceS(TM)[∂xjn∂ξα′∂ξknσr+(D−p1)(x′,0,ξ′,ξn)
X X
×∂xα′∂ξjn+1∂xknσl(D−p2)(x′,0,ξ′,ξn)]dξnσ(ξ′)dx′, (2.4)
where the sum is taken over r k α +l j 1= n, r p ,l p .
1 2
− −| | − − − ≤− ≤−
3. A Kastler-Kalau-Walze type theorem for 4-dimensional spin manifolds with boundary of
conformal warped product metric
Inthissection,We computethe lowerdimensionalvolumeVol(1,1) for4-dimensionalspinmanifolds with
4
boundary of conformal warped product metric and get a Kastler-Kalau-Walzetype theorem in this case.
2
Since [σ (D−2)] has the same expressionas σ (D−2) in the case of manifolds without boundary in
−4 M −4
|
[4], [5], [6] and [11], we have
Ω
tr[σ (D−2)]σ(ξ)dx= 4 sdvol , (3.1)
−4 M
− 3
ZMZ|ξ|=1 ZM
n
where Ω = 2π2 . So we only need to compute Φ.
n Γ(n) ∂M
2
Firstly, we compute the symbol σ(D−1) of D−1. Recall the definition of the Dirac operator D [7][16].
R
Let L denote the Levi-Civita connection about gM. In the local coordinates x ;1 i n and the fixed
i
∇ { ≤ ≤ }
orthonormalframe e , ,e , the connection matrix (ω ) is defined by
1 n s,t
{ ··· }
L(e , ,e )=(e , ,e )(ω ). (3.2)
e f ∇ 1 ··· n 1 ··· n s,t
The Dirac operator is defined by
e f e f
n
1
D = c(e ) e ω (e )c(e )c(e ) . (3.3)
i i s,t i s t
− 4
Xi=1 h Xs,t i
e e e e e
where c(e ) denotes the Clifford action.
i
Then,
1
e σ (D)=√ 1c(ξ); σ (D)= ω (e )c(e )c(e )c(e ), (3.4)
1 0 s,t i i s t
− −4
i,s,t
X
where ξ = n ξ dx denotes the cotangent vector. e e e e
i=1 i i
By Lemma 2.1 in [11] , we have
P
Lemma 3.1.
√ 1c(ξ)
σ (D−1) = − ; (3.5)
−1 ξ 2
| |
c(ξ)σ (D)c(ξ) c(ξ)
σ (D−1) = 0 + c(dx ) ∂ [c(ξ)]ξ 2 c(ξ)∂ (ξ 2) , (3.6)
−2 ξ 4 ξ 6 j xj | | − xj | |
| | | | Xj (cid:16) (cid:17)
where σ (D)= 1 ω (e )c(e )c(e )c(e ).
0 −4 s,t s,t i i s t
Since Φ is a globPal form on ∂M, so for any fixed point x ∂M, we can choose the normal coordinates
0
e e e e ∈
U of x in ∂M (not in M) and compute Φ(x ) in the coordinates U =U [0,1) M and the metric gM =
0 0
× ⊂
1 g∂M+ψ(x )dx2. ThedualmetricofgM onU isϕ(x )g∂M+ 1 dx2.WritegM =gM( ∂ , ∂ ); gij =
ϕ(xn) n n n ψ(xn) n ij ∂xi ∂xj M
gM(dx ,dx ), then e
i j
e
1 [g∂M] 0 ϕ(x )[gi,j ] 0
[giM,j]= 0ϕ(xn) i,j ψ(x ) ; [gMi,j]= 0 n ∂M 1 , (3.7)
h n i h ψ(xn) i
and
∂ g∂M(x )=0,1 i,j n 1; gM(x )=δ . (3.8)
xs ij 0 ≤ ≤ − ij 0 ij
Let n = 4 and e , ,e be an orthonormal frame field in U about g∂M which is parallel along
1 n−1
geodesics and e (x{) =···∂ (x ),}then e = ϕ(x )e , ,e] = ϕ(x )e ,e = 1 dx is
i 0 ∂xi 0 { 1 n 1 ··· n−1 n n−1 n √ψ(xn) n}
the orthonormal frame field in U aboutegM. pLocally S(TM)|Ue ∼= Up× ∧∗C(n2). Letf{f1,··· ,f4} be the
orthonormal basis of ∗(n). Take a spin frame field σ : U Spin(M) such that πσ = e , ,e ,
∧C 2 → { 1 ··· n}
where π : Spin(M) O(M) isea double covering, then [(σ,f )], 1 e i 4 is an orthonormal frame of
i
→ { ≤ ≤ }
S(TM)|Ue. In the following, since the global form Φ is indepeendent of the choice of the local freame, sofwe
3
cancompute tr in the frame [(σ,f )], 1 i 4 .Let E , ,E be the canonicalbasisofRn and
S(TM) i 1 n
c(Ei)∈clC(n)∼=Hom(∧∗C(n2),∧∗C({n2)) be the C≤liffo≤rd}action{. By·[·1·1], [12}] and [16] , we have
∂ ∂
c(e )=[(σ,c(E ))]; c(e )[(σ,f )]=[(σ,c(E )f )]; =[(σ, )], (3.9)
i i i i i i
∂x ∂x
i i
then we have ∂ c(e )=e 0 in the above framee. Therefore, we obtain
∂xi i
Lemma 3.2. For n-dimensional spin manifolds with boundary,
e
0, if j <n;
∂ (ξ 2 )(x ) = (3.10)
xj | |gM 0 ( ϕ′(0)|ξ′|2g∂M −ψ′(0)ξn2, if j =n.
0, if j <n;
∂ [c(ξ)](x ) = (3.11)
xj 0 ∂ [c(ξ′)](x )+ξ ∂ [c(dx ](x ), if j =n.
(cid:26) xn 0 n xn n 0
where ξ =ξ′+ξ dx .
n n
Proof. By the equality ∂ (ξ 2 )(x ) = ∂ ϕ(x )gl,m(x′)ξ ξ + ψ(x )ξ2 (x ) and (3.7), then (3.9) is
xj | |gM 0 xj n ∂M l m n n 0
correct. By Lemma A.1 in [11], (3.10) is correct.
(cid:0) (cid:1)
In order to compute σ (D)(x ), we need to compute ω (e )(x ). By Appendix in [11], we have
0 0 s,t i 0
Lemma 3.3. For n-dimensional spin manifolds with boundary, When i < n , ω (e )(x ) = 1ϕ′(0); and
e n,i i 0 2
ω (e )(x )= 1ϕ′(0). In other cases, ω (e )(x )=0.
i,n i 0 −2 s,t i 0
e
Combining (3.3) and Lemma 3.3, we obtain
e e
Lemma 3.4. For 4-dimensional spin manifolds with boundary,
3
σ (D)(x )= ϕ′(0)c(dx ). (3.12)
0 0 n
−4
Now we can compute Φ (see formula (2.4) for the definition of Φ), since the sum is taken over r l+
− −
k+j+ α =3, r, l 1, then we have the following five cases:
| | ≤−
case a) I) r= 1, l= 1, k =j =0, α =1
− − | |
From (2.4) we have
+∞
case a) I)=−Z|ξ′|=1Z−∞ |αX|=1trace[∂ξα′πξ+nσ−1(D−1)×∂xα′∂ξnσ−1(D−1)](x0)dξnσ(ξ′)dx′. (3.13)
By Lemma 3.2, for i<n, then
√ 1c(ξ)
∂ σ (D−1)(x ) = ∂ − (x )
xi −1 0 xi ξ 2 0
(cid:18) | | (cid:19)
√ 1∂ [c(ξ)](x ) √ 1c(ξ)∂ (ξ 2)(x )
= − xi 0 − xi | | 0
ξ 2 − ξ 4
| | | |
= 0. (3.14)
Then case a) I) vanishes.
case a) II) r = 1, l= 1, k = α =0, j =1
− − | |
From (2.4) we have
1 +∞
case a) II)= trace[∂ π+σ (D−1) ∂2 σ (D−1)](x )dξ σ(ξ′)dx′. (3.15)
−2Z|ξ′|=1Z−∞ xn ξn −1 × ξn −1 0 n
4
By Lemma 3.1 and Lemma 3.2, we have
6ξ c(dx )+2c(ξ′) 8ξ2c(ξ)
∂2 σ (D−1) = √ 1 n n + n
ξn −1 − − ξ 4 ξ 6
(cid:16) | | | | (cid:17)
6iξ2 2i 2iξ3 6iξ
= n− c(ξ′)+ n− nc(dx ), (3.16)
(1+ξ2)3 (1+ξ2)3 n
n n
and
√ 1∂ [c(ξ)](x ) √ 1c(ξ)∂ (ξ 2)(x )
∂ σ (D−1)(x ) = − xn 0 − xn | | 0
xn −1 0 ξ 2 − ξ 4
| | | |
i∂ [c(ξ′)](x ) iξ ∂ [c(dx )](x ) iϕ′(0) iξ2ψ′(0) c(ξ)
= xn 0 + n xn n 0 − n . (3.17)
ξ 2 ξ 2 − ξ 4
| | | | (cid:0) | | (cid:1)
By (2.1.1) in [11], (3.3) in [10] and the Cauchy integral formula, then
c(ξ) c(ξ′)+ξ c(dx )
πξ+n ξ 4 (x0)||ξ′|=1 = πξ+n (1+nξ2)2 n
(cid:20)| | (cid:21) (cid:20) n (cid:21)
= 1 lim (ηnc(+ξi′))2+(ηξnn+c(idux−nη)n)dη
2πiu→0−ZΓ+ (ηn−i)2 n
c(ξ′)+η c(dx ) (1)
n n
=
(η +i)2(ξ η ) |ηn=i
(cid:20) n n− n (cid:21)
2 iξ i
= − − n c(ξ′) c(dx ). (3.18)
4(ξ i)2 − 4(ξ i)2 n
n n
− −
Similarly,
i∂ [c(ξ′)] ∂ [c(ξ′)](x )
πξ+n(cid:20) xn|ξ|2 (cid:21)(x0)||ξ′|=1 = xn2(ξn−i)0 , (3.19)
iξ ∂ [c(dx )] i∂ [c(dx )](x )
πξ+n(cid:20) n xn|ξ|2 n (cid:21)(x0)||ξ′|=1 = xn2(ξn−ni) 0 , (3.20)
and
ξ2c(ξ) iξ 2ξ i
πξ+n(cid:20) n|ξ|4 (cid:21)(x0)||ξ′|=1 = 4(ξ−n−ni)2c(ξ′)+ 4(ξnn−−i)2c(dxn). (3.21)
Combining (3.17)-(3.21),we obtain
∂ [c(ξ′)](x ) i∂ [c(dx )](x )
∂xnπξ+nσ−1(D−1)(x0)||ξ′|=1 = xn2(ξn i)0 + xn2(ξn ni) 0
− −
(2i ξ )ϕ′(0)+ξ ψ′(0)
+ − n n c(ξ′)
4(ξ i)2
n
−
ϕ′(0)+(1+2iξ )ψ′(0)
+− n c(dx ). (3.22)
4(ξ i)2 n
n
−
Since n = 4, tr [id] = dim( ∗(2)) = 4. By the relation of the Clifford action and trAB = trBA, we
S(TM)
∧
have the equalities:
tr[c(ξ′)c(dxn)]=0; tr[c(dxn)2]= 4; tr[c(ξ′)2](x0)|ξ′|=1 = 4;
− | −
tr[∂xn[c(ξ′)]c(dxn)]=0; tr[∂xn[c(ξ′)]c(ξ′)](x0)||ξ′|=1 =−2ϕ′(0);
tr[∂xn[c(dxn)]c(ξ′)]=0; tr[∂xn[c(dxn)]c(dxn)](x0)||ξ′|=1 =2ψ′(0). (3.23)
5
From (3.16), (3.22)and (3.23), we have
∂ [c(ξ′)](x ) i∂ [c(dx )](x ) (2i ξ )ϕ′(0)+ξ ψ′(0) ϕ′(0)+(1+2iξ )ψ′(0)
tr xn 0 + xn n 0 + − n n c(ξ′)+ − n c(dx )
2(ξ i) 2(ξ i) 4(ξ i)2 4(ξ i)2 n
n n n n
nh − − − − i
6iξ2 2i 2iξ3 6iξ
× (1+n−ξ2)3c(ξ′)+ (1n+−ξ2)3nc(dxn) (x0)||ξ′|=1
n n
h io
2 iϕ′(0)+ξ ψ′(0)
n
= . (3.24)
(ξ i)2(ξ +i)3
(cid:0)n− n (cid:1)
Substituting (3.24) into (3.15), we have
+∞ iϕ′(0)+ξ ψ′(0)
case a) II) = n dξ σ(ξ′)dx′
−Z|ξ′|=1Z−∞ (ξn−i)2(ξn+i)3 n
iϕ′(0)+ξ ψ′(0)
= Ω n dξ dx′
− 3 (ξ i)2(ξ +i)3 n
ZΓ+ n− n
iϕ′(0)+ξ ψ′(0) (1)
= Ω 2πi n dx′
− 3 (ξ +i)3 |ξn=i
n
h i
1
= 3ϕ′(0)+ψ′(0) πΩ dx′, (3.25)
3
−8
(cid:0) (cid:1)
where Ω is the canonical volume of S4.
4
case a) III) r = 1, l = 1, j = α =0, k =1
− − | |
From (2.4) we have
1 +∞
case a) III)= trace[∂ π+σ (D−1) ∂ ∂ σ (D−1)](x )dξ σ(ξ′)dx′. (3.26)
−2Z|ξ′|=1Z−∞ ξn ξn −1 × ξn xn −1 0 n
By (2.2.29) in [11], we have
c(ξ′) ic(dx )
∂ξnπξ+nσ−1(D−1)(x0)||ξ′|=1 = 2(−ξn i)2 + 2−(ξn in)2. (3.27)
− −
From (3.5) we have
2iξ i iξ2
∂ξn∂xnσ−1(D−1)(x0)||ξ′|=1 = (1−+ξ2n)2∂xn[c(ξ′)](x0)+ (1−+ξ2n)2∂xn[c(dxn)](x0)
n n
(3iξ2 i)ϕ′(0)+(3iξ2 iξ4)ψ′(0)
+ n− n− n c(dx )
(1+ξ2)3 n
n
4iξ ϕ′(0)+(2iξ 2iξ3)ψ′(0)
+ n n− n c(ξ′). (3.28)
(1+ξ2)3
n
Combining (3.27) and (3.28), we obtain
c(ξ′) ic(dx ) 2iξ i iξ2
tr − − n − n ∂ [c(ξ′)](x )+ − n ∂ [c(dx )](x )
2(ξ i)2 × (1+ξ2)2 xn 0 (1+ξ2)2 xn n 0
nh n− i h n n
(3iξ2 i)ϕ′(0)+(3iξ2 iξ4)ψ′(0) 4iξ ϕ′(0)+(2iξ 2iξ3)ψ′(0)
+ n− (1+ξ2)3n− n c(dxn)+ n (1+ξn2−)3 n c(ξ′) (x0)||ξ′|=1
n n
io
2iϕ′(0)+(ξ i)ψ′(0)
= − n− . (3.29)
(ξ i)2(ξ +i)3
n n
−
6
Substituting (3.29) into (3.26), one sees that
1 +∞ 2iϕ′(0)+(ξ i)ψ′(0)
case a) III) = − n− dξ σ(ξ′)dx′
−2Z|ξ′|=1Z−∞ (ξn−i)2(ξn+i)3 n
1 2iϕ′(0)+(ξ i)ψ′(0) (1)
= 2πi − n− Ω dx′
−2 × (ξ +i)3 |ξn=i 3
n
h i
1
= 3ϕ′(0)+ψ′(0) πΩ dx′. (3.30)
3
8
(cid:0) (cid:1)
case b) r = 2, l = 1, k =j = α =0
− − | |
From (2.4) we have
+∞
case b)= i trace[π+σ (D−1) ∂ σ (D−1)](x )dξ σ(ξ′)dx′. (3.31)
− Z|ξ′|=1Z−∞ ξn −2 × ξn −1 0 n
By Lemma 3.1 and Lemma 3.2, we have
2iξ i iξ2
∂ σ (D−1)= − n c(ξ′)+ − n c(dx ), (3.32)
ξn −1 (1+ξ2)2 (1+ξ2)2 n
n n
and
c(ξ)σ (D)(x )c(ξ) 1
σ (D−1)(x ) = 0 0 + c(ξ)c(dx ) ∂ [c(ξ′)](x )+ξ ∂ [c(dx )](x )
−2 0 ξ 4 ξ 4 n xn 0 n xn n 0
| | | | h i
1
c(ξ)c(dx )c(ξ) ϕ′(0) ξ2ψ′(0) . (3.33)
− ξ 6 n − n
| | h i
Then
c(ξ)σ (D)(x )c(ξ)
πξ+nσ−2(D−1)(x0)||ξ′|=1 = πξ+n 0 ξ 4 0
h | | i
1
+π+ c(ξ)c(dx ) ∂ [c(ξ′)](x )+ξ ∂ [c(dx )](x )
ξn ξ 4 n xn 0 n xn n 0
h|1| (cid:2) (cid:3)i
+π+ c(ξ)c(dx )c(ξ) ξ2ψ′(0) ϕ′(0)
ξn ξ 6 n n −
:= A+Bh|+|C. (cid:2) (cid:3)i (3.34)
Similarly to (3.18), by Lemma 3.1 we have
1 3ϕ′(0) 3iξ ϕ′(0) 3iϕ′(0)
A= − − (2+iξ )c(ξ′)c(dx )c(ξ′) − n c(dx )+ c(ξ′) . (3.35)
4(ξ i)2 4 n n − 4 n 2
n
− h i
And
1
B = − (2+iξ )c(ξ′)c(dx )∂ [c(ξ′)](x )+ic(ξ′)c(dx )∂ [c(dx )](x )
4(ξ i)2 n n xn 0 n xn n 0
n
− h
i∂ [c(ξ′)](x ) iξ ∂ [c(dx )](x ) , (3.36)
− xn 0 − n xn n 0
i
ξ
C = n 3ϕ′(0) iξ ϕ′(0)+ψ′(0)+3iξ ψ′(0) c(dx )
16(ξ i)3 − − n n n
n
− h i
1
+ 3ϕ′(0) iξ ϕ′(0)+ψ′(0)+3iξ ψ′(0) c(ξ′)
8(ξ i)3 − − n n
n
− h i
1
+ 8iϕ′(0)+9ξ ϕ′(0)+3iξ2ϕ′(0) 3ξ ψ′(0) iξ2ψ′(0) c(ξ′)c(dx )c(ξ′).(3.37)
16(ξ i)3 − n n − n − n n
n
− h i
7
By (3.32) and (3.35)-(3.37), we obtain
3iϕ′(0)
tr[A×∂ξnq−1(x0)]||ξ′|=1 = 2(1+ξ2)2; (3.38)
n
2ϕ′(0) iξ ϕ′(0)+ξ2ϕ′(0)+iξ ψ′(0)+ξ2ψ′(0)
tr[B×∂ξnq−1(x0)]||ξ′|=1 = − − n 2(ξ ni)3(ξ +i)2n n ; (3.39)
n n
−
4ϕ′(0)+iξ ϕ′(0) ξ2ϕ′(0) 3iξ ψ′(0) ξ2ψ′(0)
tr[C×∂ξnq−1(x0)]||ξ′|=1 = n 2(ξ− ni)3(ξ−+i)2n − n . (3.40)
n n
−
Combining (3.31), (3.38), (3.39) and (3.40), we obtain
+∞ 5iϕ′(0)+3ξ ϕ′(0) 2ξ ψ′(0)
case b) = i − n − n dξ σ(ξ′)dx′
− Z|ξ′|=1Z−∞ −2i(ξn−i)3(ξn+i)2 n
5iϕ′(0)+3ξ ϕ′(0) 2ξ ψ′(0)
= iΩ − n − n dξ dx′
− 3 2i(ξ i)3(ξ +i)2 n
ZΓ+ − n− n
2πi 5iϕ′(0)+3ξ ϕ′(0) 2ξ ψ′(0) (2)
= iΩ − n − n dx′
− 3 2! 2i(ξ +i)2 |ξn=i
n
h − i
1
= 9ϕ′(0) ψ′(0) πΩ dx′. (3.41)
3
8 −
case c) r = 1, l= 2, k =(cid:0)j = α =0 (cid:1)
− − | |
From (2.4) we have
+∞
case c)= i trace[π+σ (D−1) ∂ σ (D−1)](x )dξ σ(ξ′)dx′. (3.42)
− Z|ξ′|=1Z−∞ ξn −1 × ξn −2 0 n
By (2.2.44) in [11], we have
c(ξ′)+ic(dx )
πξ+nσ−1(D−1)(x0)||ξ′|=1 = 2(ξn i) n ; (3.43)
−
By (3.33) we have
1
∂ξnσ−2(D−1)(x0)||ξ′|=1 = (1+ξ2)3 (3ξn2 −1)∂xn[c(ξ′)](x0)+(2ξn3 −2ξn)∂xn[c(dxn)](x0)
n
h
+(1 3ξ2)c(ξ′)c(dx )∂ [c(dx )](x ) 4ξ c(ξ′)c(dx )∂ [c(ξ′)](x )
− n n xn n 0 − n n xn 0
1 i
+ ξ 9ϕ′(0)+3ξ2ϕ′(0)+2ψ′(0) 4ξ2ψ′(0) c(ξ′)c(dx )c(ξ′)
(1+ξ2)4 n n − n n
n
h (cid:16) (cid:17)
1
7ϕ′(0)+26ξ2ϕ′(0)+9ξ4ϕ′(0)+12ξ2ψ′(0) 12ξ4ψ′(0) c(ξ′)
−2 − n n n − n
ξ(cid:16) (cid:17)
n 7ϕ′(0)+8ξ2ϕ′(0)+3ξ4ϕ′(0)+8ξ2ψ′(0) 4ξ4ψ′(0) c(dx ) .
− 2 − n n n − n n
(cid:16) (cid:17) i(3.44)
Then similarly to computations of the case b), we have
trace[πξ+nσ−1(D−1)×∂ξnσ−2(D−1)](x0)||ξ′|=1
1
= 6iϕ′(0)+3ξ ϕ′(0)+iψ′(0) 2ξ ψ′(0) (3.45)
i(ξ i)(ξ +i)4 n − n
n n
− − h i
Combining (3.42) and (3.45), we obtain
1
case c)= 9ϕ′(0) ψ′(0) πΩ dx′. (3.46)
3
−8 −
Since Φ is the sum of the cases a), b) and c),(cid:0)so Φ=0. Ther(cid:1)efore
8
Theorem 3.5. Let M be a 4-dimensional compact spin manifold with the boundary ∂M and the metric gM
as above and D be the Dirac operator on M, then
Vol(1,1) =W]resc[π+D−1 π+D−1]= Ω4 sdvol . (3.47)
4 ◦ − 3 M
ZM
Now, we recall the Einstein-Hilbert action for manifolds with boundary in [11],
1
I = sdvol +2 Kdvol :=I +I , (3.48)
Gr 16π M ∂M Gr,i Gr,b
ZM Z∂M
where
K = K gi,j ; K = Γn , (3.49)
i,j ∂M i,j − i,j
1≤i,j≤n−1
X
and K is the second fundamental form, or extrinsic curvature. Take the metric in Section 2, then by
i,j
Lemma A.2 in [11], K (x ) = Γn (x ) = 1ϕ′(0), when i = j < n, otherwise is zero. For n = 4, we
i,j 0 − i,j 0 −2
obtain
3
3
K(x )= K (x )gi,j (x )= K (x )= ϕ′(0). (3.50)
0 i.j 0 ∂M 0 i,i 0 −2
i,j i=1
X X
So
I = 3ϕ′(0)Vol . (3.51)
Gr,b ∂M
−
Let M be a 4-dimensional manifold with boundary and P,P′ be two pseudodifferential operators with
transmission property (see [8]) on M. From (4.4) in [11], we have
cπ+P π+P′ =π+(PP′)+L(P,P′) (3.52)
◦
andL(P,P′)isleftovertermwhichrepresentsthedifferencebetweenthecompositionπ+P π+P′ inBoutet
◦
de Monvel algebra and the composition PP′ in the classical pseudodifferential operators algebra. By (2.4),
we define locally
1 +∞
res (P,P′):= trace[∂ π+σ (P) ∂2 σ (P′)]dξ σ(ξ′)dx′; (3.53)
1,1 −2Z|ξ′|=1Z−∞ xn ξn −1 × ξn −1 n
+∞
res (P,P′):= i trace[π+σ (P) ∂ σ (P′)]dξ σ(ξ′)dx′. (3.54)
2,1 − Z|ξ′|=1Z−∞ ξn −2 × ξn −1 n
Hence, they represent the difference between the composition π+P π+P′ in Boutet de Monvel algebra
◦
and the composition PP′ in the classical pseudodifferential operators algebra partially. Then
case a) II)=res (D−1,D−1); case b)=res (D−1,D−1). (3.55)
1,1 2,1
Now, we assume ∂M is flat , then dx = e , g∂M = δ , ∂ g∂M = 0. So res (D−1,D−1) and
{ i i} i,j i,j xs i,j 1,1
res (D−1,D−1) are two global forms locally defined by the aboved oriented orthonormal basis dx . Let
2,1 i
{ }
ψ′(0)=ϕ′(0), from case a) II) and case b), then we obtain:
Theorem 3.6. Let M be a 4-dimensional flat compact connected foliation with the boundary ∂M and the
metric gM as above , and D be the Dirac operator on M, then
π
res (D−1,Dc−1)= Ω I ; (3.56)
1,1 3 Gr,b
6
Z∂M
π
res (D−1,D−1)= Ω I . (3.57)
2,1 3 Gr,b
−3
Z∂M
9
(1,1)
Nextly, for 3-dimensional spin manifolds with boundary, we compute Vol . By Section 5 in [11], we
3
have
W]res[π+D−1 π+D−1]= Φ. (3.58)
◦
Z∂M
By (2.4), whenn=3,we have r k α +l j 1= 3, r 1,l 3,sowe getr = 1, l= 1, k=
− −| | − − − ≤− ≤− − −
α =j =0, then
| |
+∞
Φ= trace [σ+ (D−1)(x′,0,ξ′,ξ ) ∂ σ (D−3)(x′,0,ξ′,ξ )]dξ σ(ξ′)dx′. (3.59)
Z|ξ′|=1Z−∞ S(TM) −1 n × ξn −1 n n
By (2.2.44) in [11], we have
c(ξ′)+ic(dx )
πξ+nσ−1(D−1)(x0)||ξ′|=1 = 2(ξn i) n . (3.60)
−
From (3.5) we have
2iξ i iξ2
∂ σ (D−1)= − n c(ξ′)+ − n c(dx ). (3.61)
ξn −1 (1+ξ2)2 (1+ξ2)2 n
n n
Since n = 3, tr(id) = dim(S(TM)) = 2. By the relation of the Clifford action and trAB = trBA, we
have the equalities:
tr[c(ξ′)c(dxn)]=0; tr[c(dxn)2]= 2; tr[c(ξ′)2](x0)|ξ′|=1 = 2. (3.62)
− | −
Hence from (3.59), (3.60) and (3.61), we have
1
trace[σ−+1(D−1)×∂ξnσ−1(D−1)](x0)||ξ′|=1 = (ξ +i−)2(ξ i). (3.63)
n n
−
Then
iπ
Φ= Ω vol . (3.64)
2 ∂M
2
where vol denotes the canonical volume form of ∂M. Then
∂M
Theorem 3.7. Let M be a 3-dimensional compact spin manifold with the boundary ∂M and the metric gM
as in Section 2 and D be the Dirac operator on M, then
iπ
Vol(1,1c) = Ω vol . (3.65)
3 2 2 ∂M
where Vol denotes the canonical volume of ∂M.
∂M
Remark 3.8. When ψ(x )=1, we get Theorem 2.5 and Theorem 5.1 in [11].
n
4. A Kastler-Kalau-Walze type theorem for 6-dimensional spin manifolds with boundary of
conformal warped product metric associated with D2
In this section, We compute the lower dimensional volume Vol(2,2) for 4-dimensional spin manifolds
6
with boundary of warped product metric gM = 1 g∂M +ψ(x )dx2 and get a Kastler-Kalau-Walzetype
ϕ(xn) n n
theorem in this case.
Since [σ (D−4)] has the same expressionas σ (D−4) in the case of manifolds without boundary in
−6 M −6
|
[12], we have
5Ω
tr[σ (D−4)]σ(ξ)dx= 6 sdvol , (4.1)
−6 M
− 3
ZMZ|ξ|=1 ZM
n
where Ω = 2π2 . So we only need to compute Φ.
n Γ(n) ∂M
2
By Lemma 1 in [12] , we have
R
10