The boundary term from the Analytic Torsion of a cone over a $m$-dimensional sphere

0.26 MB·English

by  Luiz Hartmann

THE BOUNDARY TERM FROM THE ANALYTIC TORSION OF A CONE
OVER A m-DIMENSIONAL SPHERE.
L.HARTMANN
3
1
0
2
Abstract. We present a direct proof that the Anomaly Boundary term of J. Bru¨nning and X.
n
Ma[2,3]generalizestothecasesoftheconeoveram-dimensionalsphere.
a
J
9 1. Introduction
1
The Analytic torsion was defined by D. B. Ray and I. M. Singer answering the question as to

] how describe the Reidemeister torsion, which is a manifold invariant, in analytic terms. In [24],
G they presentedthe manifold invariantandthey conjecturedthe equality ofboth torsionin the case
D of a closed Riemannian manifold. A few years latter, J. Cheeger [6] and W. Mu¨ller [22] proved
. this conjecture with different approaches. J. Cheeger used surgerytheory to reducedto the case of
h
spheres and W. Mu¨ller used Hodge’s combinatory theory. This equality between the two torsions
t
a is the celebrated Cheeger-Mu¨ller theorem. After that many generalizations of this theorem arises
m
(see [1] and references therein). A natural question about the Cheeger-Mu¨ller theorem is your
[ extension to manifolds with boundary. W. Lu¨ck in [20] studied this situation, but in the case that

the metric of the manifold has a product structure neat the boundary and he proved a Cheeger-
1
v Mu¨llertheoremwiththisadditionalhypothesis. Inthisformula,theAnalytictorsionisequaltothe
5 ReidemeistertorsionplustheEulercharacteristicoftheboundary. Recently,J.Bru¨ningandX.Ma
6 [2, 3] proved the extension of Cheeger-Mu¨ller theorem for Riemannian manifolds with boundary.
5
In this situation, a new term appears in the equality of the two torsions, and this term is called
4
anomaly boundary term. Another possible extension of Cheeger-Mu¨ller theorem is to manifold
.
1 with conical singularities. Manifolds with conical singularities was studied by J. Cheeger [7, 8, 9].
0
Recently, many authorspresentednew informationsfor this problem. In[13], M. Spreaficoand the
3
author,presentedaqualitativeresultabouttheextensionofCheeger-Mu¨llertheoremforaconeover
1
: asphereofdimension1,2and3,andweconjecturedthattheanomalyboundarytermpresentedby
v
Bru¨ningandMaisthesameinthecaseoftheconeoverasphere. Thisconjecturewasansweredfor
i
X ageneralclosedRiemannianmanifoldbythe authorandM.Spreaficoin[14]andindependently by
r B.Vertmanin[27],butinbothcasesthe proofsareby anindirectargument. The mainmotivation
a
of this paper is present another approach of this fact with a direct argument, i.e, we calculate the
Analytic torsion over a m-dimensional sphere and prove that the contribution of the boundary in
the analytic torsion is the anomaly boundary term of Bru¨ning and Ma. Recently, the author and
M.Spreaficoprovedthe extensionofCheeger-Mu¨llertheoremfor the coneoveranodddimensional
Riemannian manifold [16]. The even dimensional case still open.
The paper is organized as follows. In section 2 we present the fundamental terminology and
notation, in section 3 we discuss the Laplacian operator in a finite metric cone, in section 4 we
2010Mathematics Subject Classification: 58J52.
1
2 L.HARTMANN
presentallfactsaboutthe calculationofthe AnalyticTorsionofafinite metricconeandinthelast
section, we prove the following main results of this paper.
Theorem 1.1. If Ss2ipn−α1 is the odd dimensional sphere (of radius sinα), with the standard induced
Euclidean metric, then the Anomaly Boundary contribution in the Analytic Torsion of ClSs2ipn−α1 is
theAnomalyBoundarytermofBru¨ningandMa, namelyABM(∂ClSs2ipn−α1). Inthiscase, theformula
for the analytic torsion reads
1
logTabs(ClSs2ipn−α1)= 2logVol(ClSs2ipn−α1)+ABM(∂ClSs2ipn−α1),
where
ABM(∂ClSs2ipn−α1)=4(p2(pp−11))!! p−1(p 1 k1)!(2k+1) k (k(−1j))k!(−2jj2+j+11)!!sin2k+1α.
− k=0 − − j=0 −
X X
Theorem 1.2. If S2p is the even dimensional sphere (of radius sinα), with the standard induced
sinα
Euclidean metric, then the Anomaly Boundary contribution in the Analytic Torsion of C S2p is
l sinα
the Anomaly Boundary term of Bru¨ning and Ma, namely ABM(∂ClSs2ipn−α1), i.e.,
A (∂CS2p )=sin2pαp−1 1 j j (−1)h2sin2(h−j)α.
BM l sinα 8 j!(p j)! h p j+h
j=0 − h=0(cid:18) (cid:19) −
X X
2. Preliminary and notation
InthissectionwewillrecallsomebasicresultsinRiemannianGeometry,HodgedeRhamtheory,
GlobalAnalysisandthedefinitionsofthemainobjectswewilldealwithinthiswork. Alltheresults
are contained in [8, 14, 24].
2.1. Some Riemannian geometry and Hodge theory. Let (W,g) be an orientable compact
connectedRiemannianmanifoldofdimensionmwithoutboundary,wheregdenotestheRiemannian
structure. We denote by TW the tangent bundle over W, and by T W the dual bundle.
∗
Let ρ : π (W) O(k,R) be a representation of the fundamental group of W in the real
1
→
orthogonal group of dimension k, and let E = W Rk be the associated vector bundle over W
ρ ρ
×
with fibre Rk and group O(k,R). We denote by Ω(W,E ) be the graded linear space of q-smooth
ρ
forms on W with values in E , namely Ω(W,Ef) = Ω(W) E . The exterior differential on W
ρ ρ ρ
⊗
defines the exterior differential on Ωq(W,E ), d : Ωq(W,E ) Ωq+1(W,E ) and g defines the
ρ ρ ρ
→
Hodge operator on W, and hence on Ωq(W,E ), ⋆ : Ωq(W,E ) Ωm q(W,E ). Using the inner
ρ ρ − ρ
→
product , in E , an inner product on Ωq(W,E ) is defined by
ρ ρ
h i
(2.1) (ω,η)= ω ⋆η .
h ∧ i
ZW
The closure of Ωq(W;E ) with respect to this inner product is the Hilbert space of L2 q-forms
ρ
on W with values in E . The de Rham complex with this product is an elliptic complex. The dual
ρ
of the exterior derivative d , defined by (α,dβ) = (d α,β), satisfies d = ( 1)mq+m+1 ⋆d⋆. The
† † †
−
Laplace operator is ∆ = (d+d )2. It satisfies: 1) ⋆∆ = ∆⋆, 2) ∆ is self adjoint, and 3) ∆ω = 0
†
if and only if dω = d ω = 0. Let q(W;E ) = ω Ω(q)(W;E ) ∆ω = 0 , be the space of the
† ρ ρ
H { ∈ | }
q-harmonic forms with values in E . Then, we have the Hodge decomposition
ρ
(2.2) Ωq(W,E )= q(W,E ) dΩq 1(W,E ) d Ωq+1(W,E ).
ρ ρ − ρ † ρ
H ⊕ ⊕
ANALYTIC TORSION OF CONES OVER SPHERES 3
This induces a decomposition of the eigenspace of a given eigenvalue λ = 0 of ∆(q) into the
6
spaces of closed forms and coclosed forms: (q) = (q) (q) , where
Eλ Eλ,cl⊕Eλ,ccl
(q) = ω Ωq(W,E ) ∆ω =λω, dω =0 , (q) = ω Ωq(W,E ) ∆ω =λω, d ω =0 .
Eλ,cl { ∈ ρ | } Eλ,ccl { ∈ ρ | † }
Defining exact forms and coexact forms by
Eλ(q,e)x ={ω ∈Ωq(W,Eρ) | ∆ω =λω, ω =dα}, Eλ(q,c)ex ={ω ∈Ωq(W,Eρ) | ∆ω =λω, ω =d†α}.
Note that, if λ=0, then (q) = (q) , and (q) = (q) , and we have an isometry
6 Eλ,cl Eλ,ex Eλ,ccl Eλ,cex
1
(2.3) φ:Eλ(q,c)l →Eλ(q,c−e1x), φ:ω 7→ √λd†ω,
whose inverse is 1 d. Also, the restriction of the Hodge star defines an isometry
√λ
⋆:d Ω(q+1)(W) dΩ(m q 1)(W),
† − −
→
and that composed with the previous one gives the isometries:
1 1
(2.4) √λd⋆:Eλ(q,c)l →Eλ(m,ce−xq+1), √λd†⋆:Eλ(q,c)cl →Eλ(m,ex−q−1).
2.2. Manifolds with boundary. An orientable compact connected riemannian n-manifold M
with boundary ∂M. Following [24], let ∂ denotes the outward pointing unit normal vector to
x
the boundary, and dx the corresponding one form. The smooth forms on M near the boundary
decomposeasω =ω +ω ,whereω istheorthogonalprojectiononthesubspacegenerated
tan norm norm
by dx and ω is in Ω(∂M). We write ω =ω +dx ω , where ω Ω(∂M), and
tan 1 2 j
∧ ∈
(2.5) ⋆ω = dx ⋆ω.
2
− ∧
Define absolute boundary conditions by
B (ω)=ω =ω =0
abs norm ∂M 2 ∂M
| |
and relative boundary conditions by
B (ω)=ω =ω =0.
rel tan ∂M 1 ∂M
| |
Note that, if ω Ωq(M), then B (ω) = 0 if and only if B (⋆ω) = 0, B (ω) = 0 implies
abs rel rel
∈
B (dω) = 0, and B (ω) = 0 implies B (d ω) = 0. Let (ω) = B(ω) B((d+d )(ω)). Then
rel abs abs † †
B ⊕
the operator ∆ = (d+d )2 with boundary conditions (ω) = 0 is self adjoint, and if (ω) = 0,
†
B B
then ∆ω =0 if and only if (d+d )ω =0. Note that correspond to
†
B
ω =0,
(2.6) Babs(ω)=0 if and only if (dnωor)mno|∂rmM∂M =0,
(cid:26) |
ω =0,
(2.7) Brel(ω)=0 if and only if (dt†anω|)∂tMan ∂M =0,
(cid:26) |
Let
q (M,E )= ω Ωq(M,E ) ∆(q)ω =0,B (ω)=0 ,
Habs ρ { ∈ ρ | abs }
q (M,E )= ω Ωq(M,E ) ∆(q)ω =0,B (ω)=0 ,
Hrel ρ { ∈ ρ | rel }
4 L.HARTMANN
be the spaces of harmonic forms with boundary conditions. Then the Hodge decomposition reads
Ωqabs(M,Eρ)=Haqbs(M,Eρ)⊕dΩaq−bs1(M,Eρ)⊕d†Ωqa+bs1(M,Eρ),
Ωqrel(M,Eρ)=Hrqel(M,Eρ)⊕dΩrqe−l1(M,Eρ)⊕d†Ωqre+l1(M,Eρ).
2.3. Analytic torsion. The analytic torsion is defined starting with a manifold (M,g) without
boundary , as previously, with twisted coefficients in E . The operator ∆(q) is symmetric, positive
ρ
and has pure point spectrum. The zeta function of the Laplace operator ∆(q) on q forms in
Ωq(M,E ) is defined by the meromorphic extension (analytic at s=0) of the series
ρ
ζ(s,∆(q))= λ s,
−
λ∈SXp+∆(q)
convergent for Re(s) > n, and where Sp denotes the positive part of the spectrum. If ∂M = ,
2 + ∅
the analytic torsion of (M,g) is
n
1
(2.8) logT((M,g);ρ)= ( 1)qqζ (0,∆(q)).
′
2 −
q=1
X
If M has a boundary, we denote by T ((M,g);ρ) the number defined by equation (2.8) with
abs
∆ satisfying absolute BC, and by T ((M,g);ρ) the number defined by the same equation with ∆
rel
satisfying relative BC.
2.4. The Cheeger Mu¨ller theorem for manifolds with boundary. Using recent works of J.
Bru¨ning andX.Ma [2,3], andclassicthe workof W.Lu¨ck [20], the CheegerMu¨ller theoremforan
oriented compact connected Riemannian n-manifold (M,g) with boundary reads [3] Theorem 3.4
(see [13, Section 6] or [14, Section 2.3] for details on our notation)
rk(ρ)
logT ((M,g);ρ)=logτ ((M,g);ρ)+ χ(∂M)log2+rk(ρ)A (∂M),
abs R BM,abs
4
rk(ρ)
logT ((M,g);ρ)=logτ ((M,∂M,g);ρ)+ χ(∂M)log2+rk(ρ)A (∂M),
rel R BM,rel
4
where ρ is an orthogonalrepresentationof the fundamental group, and where the boundary anom-
aly term of Bru¨ning and Ma is defined as follows. Using the notation of [2] (see [14, Section
2.2] for more details) for Z/2 graded algebras, we identify an antisymmetric endomorphism φ
of a finite dimensional vector space V (over a field of characteristic zero) with the element φˆ =
1 n φ(v ),v vˆ vˆ , of Λ2V. For the elements φ(v ),v are the entries of the tensor rep-
2 j,k=1h j ki j ∧ k h j ki
resenting φ in the base v , and this is an antisymmetric matrix. Now assume that r is an
k
P { }
antisymmetricendomorphismodfV withvaluesinΛ2V. Then, (R = r(v ),v )isatensoroftwo
jk j k
h i
forms in Λ2V. We extend the above construction identifying R with the element
n
1
Rˆ = r(v ),v vˆ vˆ ,
j k j k
2 h i∧ ∧
j,k=1
X
ofΛ2V Λ2V. Thiscanbegeneralizedtohigherdimensions. Inparticular,alltheconstructioncan
∧
be done taking the dual V instead of V. Accordingly to [2], we define the following forms (where
∗
d
ANALYTIC TORSION OF CONES OVER SPHERES 5
i:∂M M denotes the inclusion)
→
n 1
1 −
= (i ω i ω ) eˆ
S 2 ∗ − ∗ 0 0k∧ ∗k
k=1
X
n 1 n 1
i∗Ω= 21 − i∗Ωk,h∧eˆ∗k∧eˆ∗h, Θˆ = 12 − Θk,h∧eˆ∗k∧eˆ∗h.
k,h=1 k,h=1
X X
d
Here,ωandω aretheconnectiononeformsassociatedtothemetricsg andg =g,respectively,
0 0 1
whereg is asuitable deformationofg thatis aproductnearthe boundary. Ω is the curvaturetwo
0
formof g, Θ is the curvature two formof the boundary (with the metric induced by the inclusion),
and {ek}kn=−01 is an orthonormalbase of TM (with respect to the metric g). Then, setting
B = 21 1 Be−21Θˆ−u2S2 ∞ Γ k1+1 uk−1Skdu,
Z0 Z k=1 2
X
the Anomaly Boundary term is (cid:0) (cid:1)
1
A (∂M)=( 1)n+1A (∂M)= B.
BM,abs BM,rel
− 2
Z∂M
3. Geometric setting and Laplace operator
3.1. Thefinitemetriccone. Let(W,g˜)beanorientablecompactconnectedRiemannianmanifold
of finite dimension m without boundary and with Riemannian structure g˜. The metric cone CW
is the space (0,+ ) W with the metric
∞ ×
(3.1) g =dx dx+x2g˜.
⊗
The finite metric cone is C W = (x,p) CW 0 < x l with the Riemannian metric g and
(0,l]
{ ∈ | ≤ }
the completed finite metric cone over W is the compact space C W =C (W). The boundary of
l (0,l]
C W is the subspace l W of C W which is isometric to W with the metric l2g˜. We will call
l l
{ }×
(W,g˜) the section of the cone and operations on the section will be denoted with tilde.
In [7, 8, 9], J. Cheeger extended all the Hodge theory and the Laplace operator for this spaces,
in particular all results of section 2.1 are valid. Given a local coordinate system y on W, then
(x,y) is a local coordinate system on the cone. We present the explicit form of ⋆, d and ∆. If
†
ω Ωq(C W), set
(0,l]
∈
ω(x,y)=f (x)ω (y)+f (x)dx ω (y),
1 1 2 2
∧
with smooth functions f and f , and ω Ω(W), then
1 2 j
∈
(3.2) ⋆ω(x,y)=xm 2q+2f (x)˜⋆ω (y)+( 1)qxm 2qf (x)dx ˜⋆ω (y),
− 2 2 − 1 1
− ∧
dω(x,y)=f (x)d˜ω (y)+∂ f (x)dx ω (y) f (x)dx dω (y),
1 1 x 1 1 2 2
∧ − ∧
(3.3) d†ω(x,y)=x−2f1(x)d˜†ω1(y) (m 2q+2)x−1f2(x)+∂xf2(x) ω2(y)
− −
x−2f2(x)dx d˜†ω2(y(cid:0)), (cid:1)
− ∧
6 L.HARTMANN
∆ω(x,y)= ∂2f (x) (m 2q)x 1∂ f (x) ω (y)+x 2f (x)∆˜ω (y) 2x 1f (x)d˜ω (y)
− x 1 − − − x 1 1 − 1 1 − − 2 2
(3.4) +d(cid:0)x∧ x−2f2(x)∆˜ω2(y)+ω2(y) −(cid:1)∂x2f2(x)−(m−2q+2)x−1∂xf2(x)
+(m (cid:16)2q+2)x 2f (x) 2x 3f(cid:0)(x)d˜ω (y) .
− 2 − 1 † 1
− −
(cid:1) (cid:17)
3.2. The Laplace operator on the cone and its spectrum. TheLaplaceoperatoronformson
thespaceCW wasstudiedby[5]. Thedefinitionsofthisoperatorstartswiththeformaldifferential
l
operator defined by equation (3.4) acting on Ωq (C W) . This define a unique self adjoint
abs/rel (0,l]
semi bounded operator with pure point spectrum ∆ acting on L2(C W,Ω(q)C W), such that
abs/rel l l
∆ ω = ω,if ω dom∆ . All the solutionsofthe eigenvaluesequationfor is presented
abs/rel abs/rel
L ∈ L
in [8]. In particular, imposing the boundary conditions we obtain the spectrum of ∆ . More
abs/rel
precisely, let J be the Bessel function of index ν. Define
ν
1
α = (1+2q m), andµ = λ +α2,
q 2 − q,n q,n q
q
where λ is the eigenvalue of a q co-exact eigenform of W.
q,n
Lemma 3.1. The positive part of the spectrum of the Laplace operator on forms on CW, with
l
absolute boundary conditions on ∂C W is:
l
Sp ∆(q) = m :ˆj2 /l2 ∞ m :ˆj2 /l2 ∞
+ abs cex,q,n µq,n,αq,k n,k=1∪ cex,q−1,n µq−1,n,αq−1,k n,k=1
n o n o
m :j2 /l2 ∞ m :j2 /l2 ∞
∪ cex,q−1,n µq−1,n,k n,k=1∪ q−2,n µq−2,n,k n,k=1
n o n o
m :ˆj2 /l2 ∞ m :ˆj2 /l2 ∞ .
∪ har,q,0 |αq|,αq,k k=1∪ har,q−1,0 |αq−1|,αq,k k=1
n o n o
With relative boundary conditions:
Sp+∆r(qel) = mcex,q,n :jµ−q2,ns,k/l−2s ∞n,k=1∪ mcex,q−1,n :jµ−q2−s1,n,k/l−2s ∞n,k=1
n o n o
∪ mcex,q−1,n :ˆjµ−q2−s1,n,−αq−1,k/l−2s ∞n,k=1∪ mcex,q−2,n :ˆjµ−q2−s1,n,−αq−2,k/l−2s ∞n,k=1
n o n o
∪ mhar,q :j|αq|,k/l−2s ∞k=1∪ mhar,q−1 :j|αq−1|,k/l−2s ∞k=1,
where the j (cid:8)are the zeros of the(cid:9)Besse(cid:8)l function J (x), the ˆj (cid:9) are the zeros of the function
µ,k µ µ,c,k
Jˆ (x)=cJ (x)+xJ (x), c R.
µ,c µ µ′ ∈
Proof. See [14] (cid:3)
For the harmonic forms of ∆ we have,
abs/rel
Lemma 3.2. If dimW =2p 1 is odd. Then
−
q(W), 0 q p 1,
q (C W)= H ≤ ≤ −
Habs l ( 0 , p q 2p.
{ } ≤ ≤
0 , 0 q p 1,
q (C W)= { } ≤ ≤ −
Hrel l ( x2αq−1dx ϕ(q−1),ϕ(q−1) q−1(W) , p q 2p.
∧ ∈H ≤ ≤
(cid:8) (cid:9)
ANALYTIC TORSION OF CONES OVER SPHERES 7
If dimW =2p is even. Then
q(W), 0 q p,
q (C W)= H ≤ ≤
Habs l ( 0 , p+1 q 2p+1.
{ } ≤ ≤
0 , 0 q p,
q (C W)= { } ≤ ≤
Hrel l ( x2αq−1dx ϕ(q−1),ϕ(q−1) q−1(W) , p+1 q 2p+1.
∧ ∈H ≤ ≤
Proof. See [14] for the odd(cid:8)case. The even case follows by the sam(cid:9)e argument.
(cid:3)
3.3. Torsion zeta function. Using the description of the spectrum of the Laplace operator on
forms ∆(q) givenin the last section, we define the zeta function on q-forms as in Section 2.3, by
abs/rel
ζ(s,∆(q) )= λ s,
abs/rel −
λ∈Sp+X∆(aqb)s/rel
for Re(s)> m+1. This function possibly have a simple pole in s=0, but A. Dar [10] proved
2
Theorem 3.1. The torsion zeta function with absolute/relative boundary conditions, defined by
m+1
1
t (s)= ( 1)qqζ(s,∆(q) ),
abs/rel 2 − abs/rel
q=1
X
is regular in s=0.
Then the analytic torsion of C W is defined and
l
logTabs/rel(ClW)=t′abs/rel(0).
4. The analytic torsion of CW
l
In this section we present all principal facts about the calculation of the Analytic Torsion of
C W. For more details see [14]. As the Poincar´e Duality holds for the Analytic Torsion of C W,
l l
i.e,
logT (C W)=( 1)dimW logT (C W),
abs l rel l
−
for now on we use the absolute boundary conditions and we will omit the subscript abs. With
lemma 3.1, after some simplification, the torsion zeta function is
l2s p−2 ∞
t(s)= 2 (−1)q mcex,q,n 2jµ−q2,ns,k−ˆjµ−q2,ns,αq,k−ˆjµ−q2,ns,−αq,k 
Xq=0 nX,k=1 (cid:16) (cid:17)
 
l2s ∞
+(−1)p−1 2  mcex,p−1,n jµ−p2−s1,n,k−(jµ′p−1,n,k)−2s 
nX,k=1 (cid:16) (cid:17)
l2s p−1  ∞ 
− 2 (−1)qrkHq(∂ClW;Q) j−−α2sq−1,k−j−−α2sq,k .
Xq=0 Xk=1(cid:16) (cid:17)
8 L.HARTMANN
when dimW =2p 1 is odd and
−
l2s p−1 ∞
t(s)= 2 (−1)q mcex,q,n ˆjµ−q2,ns,−αq,k−ˆjµ−q2,ns,αq,k
Xq=0 nX,k=1 (cid:16) (cid:17)
l2s p−1 ∞
+ 2 (−1)q+1rkHq(∂ClW;Q) j−−α2sq−1,k+j−−α2sq,k
Xq=0 Xk=1(cid:16) (cid:17)
l2s ∞
+(−1)p+1 4 rkHp(∂ClW;Q) j−1,2ks+j−21s,k .
2 −2
Xk=1 (cid:16) (cid:17)
when dimW =2p is even.
So the Analytic Torsion of C W is described by the following two theorems. For the proof of
l
Theorem 4.1 see [14] and for the Theorem 4.2 see [15](compare with [27])
Theorem 4.1. If dimension of W is odd, dimW =2p 1 then the Analytic Torsion of C W is
l
−
p 1
1 1 − l
logT(CW)= logT(W,l2g˜)+ ( 1)qr log
l q
2 2 − 2(p q)
q=0 −
X
p 1 p 1
1 − −
+ ( 1)q Res Φodd (s) Res ζ s,∆˜(q)+α2
2Xq=0 − Xj=1 s=00 2j+1,q s=j+112 cex(cid:16) q(cid:17)
where the functions Φ (s) are some universal functions explicitly known by some recursive
2j+1,q
relations, and ∆˜ is the Laplace operator on forms on the section of the cone.
Theorem 4.2. If dimension of W is even, dimW =2p then the Analytic Torsion of C W is
l
logT(CW)=p−1( 1)qrq log l2p−2q+1 +( 1)prp logl+ 1χ(W)log2+ 1p−1( 1)q+1 (0)
l 0,0,q
− 2 2p 2q+1 − 4 2 2 − A
q=0 − q=0
X X
p 1 p 1 p
− 1 −
+ ( 1)q+1r log(2p 2q 1)!!+ ( 1)q Res Φeven(s)Res ζ s,∆˜(q)+α2 ,
− q − − 2 − s=00 2j,q s=j1 cex q
Xq=0 Xq=0 Xj=1 (cid:16) (cid:17)
where the functions Φ (s) are some universal functions explicitly known by some recursive rela-
2j,q
tions, ∆˜ is the Laplace operator on forms on the section of the cone and
∞ αq αq mq,n
(s)= log 1 log 1+ .
A0,0,q − µ − µ µ2s
n=1(cid:18) (cid:18) q,n(cid:19) (cid:18) q,n(cid:19)(cid:19) q,n
X
5. The proof of Theorem 1.1 and Theorem 1.2
On order to prove Theorem 1.1 and Theorem 1.2 we define,
Definition5.1. TheAnomalyBoundarycontributionintheanalytictorsionofaconeoveraclosed
manifold W, denoted by logT (C W), is
AB l
p 1 p 1
1 − −
( 1)q Res Φodd (s) Res ζ s,∆˜(q)+α2 ,
2Xq=0 − Xj=1 s=00 2j+1 s=j+112 cex(cid:16) q(cid:17)
ANALYTIC TORSION OF CONES OVER SPHERES 9
if dimW =2p 1 and
−
p 1 p
1 −
( 1)q Res Φeven(s)Res ζ s,∆˜(q)+α2 ,
2 − s=00 2j s=j1 cex q
Xq=0 Xj=1 (cid:16) (cid:17)
if dimW =2p.
Recall that we are considering the absolute BC case, we will calculate the analytic torsion of
Cl(W)inthecaseW =Ss2ipn−α1usingthetheorem4.1andtheAnomalyBoundarycontributioninthe
logT(CS2p ). Our strategy is by direct calculation, i.e, we will determine all terms necessary for
l sinα
the proof of theorem 1.1 and 1.2. With this in mind, first we determine the term logT(Ss2ipn−α1,l2g˜)
andthentheAnomalyBoundarycontribution,whichrequiresmorework,andthatwillbedeveloped
in the following subsections. In fact, the Anomaly Boundary contribution are similar in dimension
odd and dimension even. So, we will determine the odd case and present the equations for the
even case to be concise. Here we present the underlying geometric setting. Let Sm be the sphere
b
of radius b > 0 in Rm+1, Sm = x Rm+1 x = b (we simply write Sm for Sm). Let C Sm
b { ∈ | | | } 1 l sinα
denotes the cone of angle α over Sm in Rm+2. We embed C Sm in Rm+2 as the subset of the
sinα l sinα
segments joining the originto the sphere Sm (0,...,0,lcosα) . We parametrize the cone by
lsinα×{ }
x = rsinαsinθ sinθ sinθ sinθ cosθ
1 m m 1 3 2 1
− ···
 x = rsinαsinθ sinθ sinθ sinθ sinθ
2 m m 1 3 2 1
ClSsminα = x3 =... rsinαsinθmsinθm−−1······sinθ3cosθ2
x = rsinαcosθ
with r [0,l], θ1 [0,2π], xθ2mm,++..12.,θ=m rc[0o,sπα], andmwhere α is a fixed positive real number and
∈ ∈ ∈
0< 1 =sinα 1. The induced metric is (r >0)
ν ≤
gE =dr dr+r2gSm
⊗ sinα
m 1 m
−
=dr dr+r2sin2α sin2θ dθ dθ +dθ dθ ,
j i i m m
⊗    ⊗ ⊗ 
i=1 j=i+1
X Y
   
and detg =(rsinα)m(sinθ )m 1(sinθ )m 2 (sinθ )2(sinθ ).
E m − m 1 − 3 2
| | − ···
5.1. pThe Analytic Torsion of an odd dimensional sphere.
Proposition 5.1.
p 1
− l
logT(Ss2ipn−α1,l2g˜)=logVol(ClSs2ipn−α1)− (−1)qrqlog2(p q).
q=0 −
X
Proof. By [21], logT(Ss2ipn−α1,l2g˜)=logVol(Sl2spi−n1α), and this proves the proposition since, if W has
metric g˜ and dimension m, then
l lm+1
Vol(C W)= det(x2g)dx dvol = xm dvol = Vol(W),
l g˜ g˜
∧ m+1
ZClW Z0 ZW
p
10 L.HARTMANN
and
Vol(Sm)= 2πm2+1bm.
b Γ m+1
2
(cid:0) (cid:1) (cid:3)
5.2. The anomalyboundary contribution. Assumingthattheformulafortheanomalybound-
ary term A (∂C W) of Bru¨ning and Ma [2] is valid in the case of C Sm , we computed in [13]
BM l l sinα
(note the slight different notation), by applying the definition given equation (2.11) of [14], that
ABM(∂ClSs2ipn−α1)=p−1j!(2(p2p−jj) 1)!! j hj ((−21(p)hν−j2+(p−hj)+h)1+)1 4(p2(pp−11))!!,
j=0 − − h=0(cid:18) (cid:19) − − −
X X
A (∂C S2p )= 1 p−1 1 j j (−1)h2ν2(j−h).
BM l sinα 8ν2p j!(p j)! h p j+h
j=0 − h=0(cid:18) (cid:19) −
X X
Our purpose now is to prove that
(5.1) logTAB(ClSs2ipn−α1)=ABM(∂ClSs2ipn−α1)and logTAB(ClSs2ipnα)=ABM(∂ClSs2ipnα).
For it is convenient to rewrite the second terms as follows:
ABM(∂ClSs2ipn−α1)=p−1j!(2(p2p−jj) 1)!! j hj ((−21(p)hν−j2+(p−hj)+h)1+)14(p2(pp−11))!!
j=0 − − h=0(cid:18) (cid:19) − − −
X X
(2p 1)! p−1 1 k ( 1)k−j2j+1 1
= − − ,
4p(p 1)! (p 1 k)!(2k+1) (k j)!(2j+1)!!ν2k+1
− k=0 − − j=0 −
X X
A (∂C S2p )= 1 p−1 1 j j (−1)h2ν2(j−h)
BM,abs l sinα 8ν2p j!(p j)! h p j+h
j=0 − h=0(cid:18) (cid:19) −
X X
p 1 k
1 − 1 p p 1 j 1
= ( 1)k−j − − .
2p! 2(k+1) − p 1 j k j ν2(k+1)
k=0 j=0 (cid:18) − − (cid:19)(cid:18) − (cid:19)
X X
5.3. The eigenvalues of the Laplacian over C Sm . Let ∆ be the self adjoint extension of
l sinα
the formal Laplace operator on CSm as defined in section 3.2. Then, the positive part of the
l sinα
spectrum of ∆ (with absolute BC) is given in Lemma 3.1, once we know the eigenvalues of the
restriction of the Laplacian on the section and their coexact multiplicity, according to Lemma
3.1. These information are available by work of Ikeda and Taniguchi [18]. The eigenvalues of the
Laplacian on q-forms on Ss2ipn−α1 are
λ =ν2n(n+2p 2),
0,n
−
λ =ν2(n+q)(n+2p q 2), 1 q <p 2,
q,n
 − − ≤ −
λ =ν2((n 1+p)2 1),
 λp−2,n =ν2(n −1+p)2, −
p 1,n
− −
