Laplaian on Forms and Anomalies in
Closed Hyperboli Manifolds
A.A. Bytsenko
A. E. Gonalves, and M. Sim~oes
Departamentode Fsia, UniversidadeEstadualde Londrina,
CaixaPostal6001,CEP86051-970,Londrina,PR,Brazil
Reeived1stJune,1999
TheglobalmultipliativepropertiesoftheLaplaianonj formsandrelatedzetafuntionsare
ana-lyzed. Theexpliitformofthemultipliativeandonformalanomaliesinlosedorientedhyperboli
manifolds H d
arederived.
I Introdution
The multipliative properties of (pseudo-) dierential
operators as well as properties of their determinants
have been studied atively during reent years in the
mathematial and physial literature. The anomaly
assoiated with produt of regularized determinants
of operators an be expressed by means of the
non-ommutativeresidue,theWodzikiresidue[1℄(seealso
Refs. [2,3℄). TheWodzikiresidue,whihistheunique
extension of the Dixmier trae to the wider lass of
(pseudo-)dierentialoperators[4,5℄,hasbeen
onsid-eredwithinthenon-ommutativegeometrialapproah
to the standardmodel of theeletroweak interations
[6,7,8℄andtheYang-Millsationfuntional. The
prod-ut of two (or more) dierential operators of Laplae
type an arise in higher derivative eld theories (for
example,in higherderivativequantumgravity).
Somereentpapersalongtheselinesanbefoundin
Refs. [9,10,11,12℄. Thezetafuntion assoiatedto the
produtofLaplaetypeoperatorsatinginirreduible
rank 1 symmetri spaes and the expliit form of the
multipliativeanomalyhavebeenderivedin[11℄.
Under suh irumstanesweshould note that the
onformaldeformationofametriandthe
orrespond-ing onformal anomaly an also play an important
role in quantum theorieswith higherderivatives. Itis
well known that evaluation of the onformalanomaly
is atually possible only for even dimensional spaes
and up to nowits omputation is extremely involved.
Thegeneralstrutureofsuhananomalyin urved
d-dimensional spaes (d even) has been studied in [13℄.
We briey mention here analysis related to this
phe-nomenonforonstanturvaturespaes. Theonformal
anomalyalulationforthed dimensionalspherean
befound, forexample,in Ref. [14℄. Theexpliit
om-putation of the anomaly (of the stress-energy tensor)
inirreduiblerank1symmetrispaeshasbeenarried
outin[15,16,17℄usingthezeta-funtionregularization
andtheSelbergtraeformula.
The purpose of the present paper is to investigate
thespetralzetafuntionsassoiatedwithaprodutof
Laplaiansonj formsandto alulatein anexpliit
formthemultipliativeandonformalanomaliesford
dimensionallosedorientedhyperbolimanifolds =H d
.
II The spetral zeta funtion
and the trae formula
Weshall be working with irreduible rank1
symmet-ri spaes X = G=K of non-ompat type. Thus G
will be a onneted non-ompat simple split rank 1
Lie group with nite enter and K G will be a
maximalompat subgroup. Up toloal isomorphism
we hoose X = SO
1
(d;1)=SO(d). Thus the isotropy
groupK of thebase point(1;0;:::0)is SO(d); X an
beidentiedwithhyperbolid spaeH d
, d=dimX.
It is possible to view H d
, for example, as one sheet
of the hyperboloid of two sheets in R d+1
given by
q(x) = x 2
0 +x
2
1
+:::+x 2
d
= 1;x
0
> 0 with the
metriinduedbythequadratiform q(x). Let G
be a disrete, o-ompat, torsion free subgroup, and
let () = trae(()) be the harater of a
nite-dimensional unitary representation of for 2 .
LetL (j)
4
(j)
bethe Laplaianon j forms ating
onthevetorbundleV(X )overX = nG=Kindued
by . Note that the non-twistedj forms on X are
obtained by taking = 1. One an dene the heat
kerneloftheelliptioperatorL (j)
=L (j)
+b (j)
by
Tr
e tL
(j)
= 1
2i Tr
Z
C0 e
zt
(z L (j)
) 1
dz, (2:1)
whereC
0
isanarin theomplexplaneC; theb (j)
are
endomorphismsofthevetorbundleV(X ). By
stan-dardresultsinoperatortheorythereexist";Æ>0suh
that for0<t<Ætheheatkernelexpansionholds
! (j)
(t;b (j)
)= 1
X
`=0 n
` ()e
( (j)
` +b
(j)
)t
= X
0``0 a
` (L
(j)
)t `
+O(t "
), (2:2)
d
wheref (j)
` g
1
`=0
isthesetofeigenvaluesofoperatorL (j)
andn
`
()denotethemultipliityof (j)
`
. Eventuallywe
would likealso to take b (j)
= 0, but for now we
on-sideronlynon-zeromodes: b (j)
+ (j)
`
>0,8`: (j)
0 =0,
b (j)
>0.
Leta
0 ;n
0
denotetheLiealgebrasofA;N inan
Iwa-sawadeomposition G = KAN. Sine the rankof G
is 1, dima
0
= 1 by denition, say a
0
= RH
0 for a
suitablebasisvetorH
0
. Oneannormalizethehoie
of H
0
by (H
0
) = 1, where : a
0
! R is the
pos-itive root whih denes n
0
; for more detail see Ref.
[18℄. Sine is torsion free, eah 2 f1g an
be representeduniquely assome power of aprimitive
element Æ : = Æ j()
where j() 1 is an integer
and Æ annot bewritten as j
1 for
1
2 , j >1 an
integer. Taking 2 , 6= 1, one an nd t
> 0
and m
2 M
def
= fm
2 Kjm
a = am
;8a 2 Ag
suh that is G onjugate to m
exp(t
H
0
), namely
for some g 2 G;g g 1
= m
exp(t
H
0
). Besides let
(m) = trae((m)) be the harater of , for a
nite-dimensionalrepresentationofM.
II.1 Fried's trae formula [19℄
For0jd 1,
Tr
e tL
(j)
=I (j)
(t;b (j)
)+I (j 1)
(t;b (j 1)
)+H (j)
(t;b (j)
)+H (j 1)
(t;b (j 1)
), (2:3)
where
I (j)
(t;b (j)
) def
=
(1)Vol( nG)
4 Z
R
j (r)e
t[r 2
+b (j)
+(
0 j)
2
℄
dr, (2:4)
H (j)
(t;b (j)
) def
= 1
p
4t X
2C f1g ()t
j()
1
C()
j (m
)
exp ( "
b (j)
t+(
0 j)
2
t+ t
2
4t # )
, (2:5)
0
=(d 1)=2,andthefuntion C(), 2 ,dened on f1gby
C() def
= e 0t
jdet
n
0 Ad(m
e
tH0
) 1
1
j 1
. (2:6)
ForAddenotingtheadjointrepresentationofGon
itsomplexiedLiealgebra,oneanomputet
as
fol-lows[20℄
t
Here C is aompleteset ofrepresentativesin ofits
onjugay lasses;Haar measureonG issuitably
nor-malized. InouraseK'SO(d);M'SO(d 1). For
bun-dlesetions)themeasure
0
(r)orrespondstothe
triv-ial representationof M. Forj 1 thereis ameasure
(r) orresponding to ageneral irreduible
represen-tation ofM. Let
j
isthestandardrepresentationof
M = SO(d 1) on j
C (d 1)
. If d =2n iseven then
j
(0j d 1) isalwaysirreduible;if d=2n+1
theevery
j
isirreduibleexeptforj=(d 1)=2=n,
in whih ase
n
is thediret sumof two(1=2) spin
representations
:
n =
+
. For j = n the
representation
n
ofK=SO(2n)on n
C 2n
isnot
irre-duible,
n =
+
n
n
isthediretsumof(1=2) spin
representations.
II.2 The Harish-Chandra Planherel
measure
LetthegroupG=SO
1
(2n;1). Then
j (r)=
2n 1
j
r
2 4n 4
(n) 2
j+1
Y
i=2
r 2
+(n+ 3
2 i)
2
n
Y
i=j+2
r 2
+(n+ 1
2 i)
2
tanh(r) for 0jn 1, (2:8)
j (r)=
2n 1
j
r
2 4n 4
(n) 2
2n j
Y
i=2
r 2
+(n+ 3
2 i)
2
n
Y
i=2n j+1
r 2
+(n+ 1
2 i)
2
tanh(r) for nj2n 1, (2:9)
and
j (r)=
2n
j 1 (r).
ForthegroupG=SO
1
(2n+1;1)onehas
j (r)=
2n
j
2 4n 2
(n+ 1
2 )
2 j+1
Y
i=1
r 2
+(n+1 i) 2
n
Y
i=j+2
r 2
+(n i) 2
for 0j<n, (2:10)
j (r)=
2n
j
2 4n 2
(n+ 1
2 )
2 2n j+1
Y
i=1
r 2
+(n+1 i) 2
n
Y
i=2n j+2
r 2
+(n i) 2
for n+1j 2n 1. (2:11)
Weshould note that the reasonfor the pairof terms fI (j)
;I (j 1)
g, fH (j)
;H (j 1)
g in the traeformula Eq.
(2.3)isthat
j
satises
j j
M =
j
j 1 .
FinallyusingEqs. (2.8)-(2.11)wehave
j
(r)=C (j)
(d)P(r;d)
tanh(r) ford=2n
1 ford=2n+1
=C (j)
(d) 8
>
<
>
: P
d=2 1
`=0 a
(j)
2` (d)r
2`+1
tanh(r) ford=2n
P
(d 1)=2
`=0 a
(j)
2` (d)r
2`
ford=2n+1
, (2:12)
C (j)
(d)=
d 1
j
2d 4 2
where the P(r;d) are even polynomials (withsuitable
oeÆientsa (j)
2`
(d))ofdegreed 1forG6=SO(2n+1;1),
andofdegreed=2n+1forG=SO
1
(2n+1;1)[21,18℄.
II.3 Case of the trivial representation
For j = 0 we take I ( 1)
= H ( 1)
= 0. Sine
0
is thetrivialrepresentation
0 (m
)=1. Inthis ase
Fried's formulaEq. (2.3) redues exatlyto the trae
formulaforj=0[20, 22℄:
! (0)
(t;b (0)
)=
(1)vol( nG)
4 Z
R
0 (r)e
(r 2
+b (0)
+ 2
0 )t
dr+H (0)
(t;b (0)
), (2:14)
where
0
is assoiatedwiththepositiverestrited(real)rootsofG(withmultipliity)with respetto anilpotent
fatorN of Gin anIwasawadeompositionG=KAN. ThefuntionH (0)
(t;b (0)
)hastheform
H (0)
(t;b (0)
)= 1
p
4t X
2C f1g ()t
j()
1
C()e [b
(0)
t+ 2
0 t+t
2
=(4t)℄
. (2:15)
III Case of zero modes.
Itanbeshown[23℄ thattheMellintransformofH (0)
(t;0)(b (0)
=0,i.e. thezeromodesase)
H (0)
(s) def
= Z
1
0 H
(0)
(t;0)t s 1
dt, (2:16)
isaholomorphifuntiononthedomainRes<0. ThenusingtheresultofRefs. [21,18℄oneanobtainonRes<0,
H (0)
(s)= X
2C f1g ()t
j()
1
C() Z
1
0 e
( 2
0 t+t
2
=(4t))
p
4t t
s 1
dt
= (2
0 )
1
2 s
p
X
2C f1g ()t
j()
1
C()t s+
1
2
K1
2 s
(t
0
), (2:17)
whereK
(s)isthemodiedBesselfuntion,andnally
H (0)
(s)=
sin(s)
(s)
Z
1
0
(t+2
0 ;)(2
0 t+t
2
) s
dt. (2:18)
d
Here (s;) d(logZ (s;))=ds, and Z (s;) is a
meromorphisuitablynormalizedSelbergzetafuntion
[24-29,22,30,21℄.
III The multipliative anomaly
Inthissetiontheprodutoftheoperatorsonj forms
N
L (j)
p ;L
(j)
p = L
(j)
+b (j)
p
,p=1;2will beonsidered.
Weareinterestedinmultipliativepropertiesof
deter-minants,themultipliativeanomaly [2,3℄. The
multi-pliativeanomalyF(L (j)
1 ;L
(j)
2
)reads
F(L (j)
1 ;L
(j)
2
)=det
[
O
p L
(j)
p ℄[det
(L
(j)
1 )det
(L
(j)
2 )℄
1
whereweassumeazeta-regularizationofdeterminants,
i.e.
det
(L
(j)
p )
def
= exp
s (sjL
(j)
p )j
s=0
. (3:2)
Generally speaking, if the anomaly related to
el-lipti operators is nonvanishing then the relation
logdet( N
L (j)
p
)=Trlog( N
L (j)
p
)doesnothold.
II.1 The zeta funtion of the produt of
Laplaians
Thespetralzetafuntionassoiatedwiththe
prod-ut N
L (j)
p
hastheform
(sj O
p L
(j)
p )=
X
`0 n
` 2
Y
p (
(j)
` +b
(j)
p )
s
. (3:3)
Weshallalwaysassumethatb (j)
1 6=b
(j)
2 ,sayb
(j)
1 >b
(j)
2 .
If b (j)
1 = b
(j)
2
then (sj N
L (j)
p
) = (2sjL (j)
) is a
well-known funtion. For b (j)
1 ;b
(j)
2
2 R, set b
+ def
= (b (j)
1 +
b (j)
2 )=2; b
def
= (b (j)
1 b
(j)
2
)=2, thus b (j)
1 =b
+
+b and
b (j)
2 =b
+ b .
Thespetralzetafuntionanbewrittenasfollows
[11℄:
(sj O
p L
(j)
p
)=(2b ) 1
2 s
p
(s) Z
1
0 !
(j)
(t;b
+ )I
s 1
2
(b t)dt, (3:4)
d
where theintegralonvergesabsolutelyforR es>d=4.
This formula is a main starting point to study the
zeta funtion. It expresses (sj N
L (j)
p
) in terms of
the Bessel funtion I
s 1
2
(b t) and ! (j)
(t;b
+
), where
the trae formula applies to ! (j)
(t;b
+
). Let B
p (j) =
(
0
(p) j) 2
+b (j)
p
andA def
= (1)vol( nG)C (j)
(d)=4.
ForRes>d=4the expliitmeromorphi
ontinua-tionholds:
(sj O
p L
(j)
p )=A
d
2 1
X
`=0 h
a (j)
2` (d)
F (j)
`
(s) E (j)
` (s)
+a (j 1)
2` (d)
F (j 1)
`
(s) E (j 1)
` (s)
i
+I (j)
(s)+I (j 1)
(s), (3:5)
where
E (j)
` (s)
def
= 4 Z
1
0 drr
2j+1
1+e 2r
Y
p (r
2
+B
p (j))
s
, (3:6)
whihisanentirefuntion ofs,
F (j)
` (s)
def
= (B
1 (j)B
2 (j))
s
`!
2B1(j)B2(j)
B
1 (j)+B
2 (j)
`+1
(2s 1)(2s 2):::(2s (`+1))
F
`+1
2 ;
`+2
2 ;s+
1
2 ;
B
1 (j) B
2 (j)
B
1 (j)+B
2 (j)
2 !
, (3:7)
I (j)
(s) def
= (2b ) 1
2 s
p
(s) Z
1
0 H
(j)
(t;b
+ )I
s 1
2 (b t)t
s 1
2
dt. (3:8)
F (j)
` (0)=
( 1) `+1
`+1
2B
1 (j)
B
1 (j)+B
2 (j)
`+1
F
`+1
2 ;
`+2
2 ;
1
2 ;
B
1 (j) B
2 (j)
B
1 (j)+B
2 (j)
2 !
= ( 1)
`+1
2(`+1) 2
X
p B
p (j)
`+1
, (3:9)
E (j)
`
(0)=4 Z
1
0 drr
2`+1
1+e 2r
= ( 1)
`
`+1 (1 2
2` 1
)B
2`+2
, (3:10)
I (j)
(0)=0, (3:11)
whereB
2n
aretheBernoullinumbers.
Apreliminaryformofthezetafuntion (sj N
p L
(j)
p
)ats=0is
(0j O
p L
(j)
p )=A
d
2 1
X
`=0 ( 1)
`+1
2(`+1) "
X
p
a (j)
2` (d)B
p (j)
`+1
+a (j 1)
2`
(d)B
p (j 1)
`+1
+(2 2 2`
)B
2`+2
a (j)
2` (d)+a
(j 1)
2` (d)
i
. (3:12)
Thederivativeofthezetafuntion ats=0hastheform:
0
(0j O
p L
(j)
p )=A
d
2 1
X
`=0 "
4
X
m
a (j)
2` (d)E
(j)
m +a
(j 1)
2` (d)E
(j 1)
m
#
, (3:13)
where
E (j)
1
=`! B
1 (j)
`+1
+B
2 (j)
`+1
`
X
k =0
( 1) k +1
k!(` k)!(j+1 k)!
, (3:14)
E (j)
2 =B
2 (j)
`+1
B
1 (j) B
2 (j)
2B
1 (j)
( 1) `
(`+1)! 1
X
k =1
(`+k+1)!
(k+1)!
n
B
1 (j) B
2 (j)
B
1 (j)
k
, (3:15)
E (j)
3
=log(B
1 (j)B
2 (j))
( 1) `
2(`+1) (B
1 (j)
`+1
+B
2 (j)
`+1
)
4 Z
1
0 r
2`+1
log
r 2
+B1(j)
r 2
+B
2 (j)
dr
1+e 2r
, (3:16)
E (j)
4
d
ds I
(j)
(s)j
s=0 =
Z
1
0 h
H (j)
(t;b (j)
1 )+H
(j)
(t;b (j)
2 )
i
t 1
dt, (3:17)
and
n def
= P
n
k =1 k
1
.
III.2 The one-loop eetive ation
Afterastandardintegration,theontributiontotheEulideanone-loopeetiveationanbewrittenasfollows:
W (1)
= 1
2 log det(
O
p L
(j)
p
2
)= 1
2 "
0
(0j O
p L
(j)
p
)+log 2
(0j O
p L
(j)
p )
#
, (3:18)
W (1)
= 1
2 A
d
2 1
X
`=0 "
a (j)
2` (d)
4
X
m E
(j)
m
+log 2
(F (j)
`
(0) E (j)
` (0))
!
+a (j 1)
2` (d)
4
X
m E
(j 1)
m
+log 2
(F (j 1)
`
(0) E (j 1)
`
(0)) !#
, (3:19)
where F (j)
` (0);E
(j)
`
(0)andE (j)
m
aregivenbytheformulae(3.9),(3.10) and(3.14)-(3.17)respetively.
d
III.3 The residue formula and the
multi-pliative anomaly
ThevalueofF(L
1 ;L
2
)anbeexpressed bymeans
of the non-ommutative Wodziki residue [1℄. Let
O
p
; p = 1;2; be invertible ellipti (pseudo-)
dieren-tialoperatorsofrealnon-zeroordersandsuhthat
+ 6= 0. Even if the zeta funtions for operators
O
1 ;O
2 andO
1 N
O
2
arewelldenedandiftheir
prin-ipal symbols satisfy the Agmon-Nirenberg ondition
(withappropriatespetrauts)onehasingeneralthat
F(O
1 ;O
2
) 6= 1. For suh invertible ellipti operators
the formula for the anomaly of ommuting operators
holds:
A(O
1 ;O
2
)=A(O
2 ;O
1
)=log(F(O
1 ;O
2 ))=
res h
(log(O
1 N
O
2 ))
2 i
2(+)
. (3:20)
More general formulaehave been derived in Refs. [2, 3℄. Furthermore the anomaly anbe iteratedonsistently.
Indeed,usingEq. (3.20)wehave
A(O
1 ;O
2 )=
0
(0jO
1 O
2
)
0
(0jO
1
)
0
(0jO
2 ),
A(O
1 ;O
2 ;O
3 )=
0
(0j 3
O
j O
j )
3
X
j
0
(0jO
j
) A(O
1 ;O
2 ),
: : : : : : : : : : :
A(O
1 ;O
2 ;:::;O
n )=
0
(0j n
O
j O
j )
n
X
j
0
(0jO
j
) A(O
1 ;O
2 )
A(O
1 ;O
2 ;O
3
)::: A(O
1 ;O
2 ;:::;O
n 1
). (3:21)
III.4 The expliit formula for the multipliative anomaly
Inpartiular, forn=2andO
p L
(j)
p
theanomaly isgivenbythefollowingformula
A(L (j)
1 ;L
(j)
2 )=A
d
2 1
X
`=0 h
(j)
` +
(j 1)
` i
, (3:22)
where
(j)
` =
a (j)
2`
(d)( 1) `
2
`
2 (B
1 (j) B
2 (j))
2
B
2 (j)
` 1
+
`(` 1)
4 (B
1 (j) B
2 (j))
3
B
2 (j)
` 2
+ `
X
p=3
`!
(p+1)p!(` p)!
1
p +
1
p 1 +
p 2
X
q=1 1
p q 1
!
(B
1 (j) B
2 (j))
p+1
B
2 (j)
` p #
. (3:23)
A(L (j)
1 ;L
(j)
2
)= A (j)
G
b (j)
1 b
(j)
1
2
A (j 1)
G
b (j 1)
1 b
(j 1)
1
2
, (3:24)
whihalsofollowsfromWodziki'sformula(3.20), whereweshouldset A (j)
G =Aa
(j)
21 (4)=4.
d
IV The onformal anomaly and
assoiated operator
prod-uts.
In this setionwestart with aonformal deformation
of a metri and the onformal anomaly of theenergy
stresstensor. Itis wellknownthat(pseudo-)
Rieman-nian metris g
(x) and ~g
(x) on a manifold X are
(pointwise)onformalifg~
(x)=exp(2f)g
(x); f 2
C 1
(R). Foronstantonformaldeformationsthe
vari-ationoftheonnetedvauumfuntional(theeetive
ation) an be expressed in terms of the generalized
zetafuntionrelatedtoanelliptiself-adjointoperator
O[31℄:
ÆW = (0jO)log 2
= Z
X <T
(x)>Æg
(x)dx, (4:1)
where <T
(x)>meansthat allonnetedvauumgraphsofthestress-energytensorT
(x)areto beinluded.
ThereforeEq. (4.1)leadsto
<T
(x)>=(Vol(X )) 1
(0jO). (4:2)
Theformulae(3.5),(3.9),(3.10)and (3.11)giveanexpliitresultfortheonformalanomaly,namely
<T
(x)>
(O= N
L (j)
p )
=
1
(4) d=2
(d=2) d
2 1
X
`=0 ( 1)
`+1
2(`+1) (
X
p h
a (j)
2` (d)B
p (j)
`+1
+a (j 1)
2`
(d)B
p (j 1)
`+1 i
+(2 2 2`
)B
2`+2
a (j)
2` (d)+a
(j 1)
2` (d)
o
, (4:3)
wherediseven. ForB
1;2
(j)=B(j);B
1;2
(j 1)=B(j 1)theanomaly(4.3)hastheform
<T
(x)>
(L (j)
N
L (j)
) =
1
(4) d=2
(d=2) d
2 1
X
`=0 ( 1)
`+1
2(`+1) n h
a (j)
2`
(d)B(j) `+1
+a (j 1)
2`
(d)B(j 1) `+1
i
+(2 2 2`
)B
2`+2
a (j)
2` (d)+a
(j 1)
2` (d)
o
. (4:4)
Notethatforaminimallyoupledsalareldofmassm,B(0)= 2
0 +m
2
. Thesimplestaseis,forexample,G=
SO
1
(2;1)'SL(2;R);besidesX=H 2
isatwo-dimensionalrealhyperbolispae. Thenwehave 2
0
=1=4; a (0)
20 =1,
andnally
<T
(x2 nH 2
)>
(L (0)
N
L (0)
) =
1
4
b+ 1
3
. (4:5)
Forreald-dimensionalhyperbolispaethesalarurvatureisR (x)= d(d 1). Intheaseoftheonformally
invariantsalareld wehaveB(0)= 2
o
+R (x)(d 2)=[4(d 1)℄. Asaonsequene,B(0)=1=4and
<T
(x2 nH d
)>
(L (0)
N
L (0)
) =
1
(4) d=2
(d=2) d
2 1
X
`=0 ( 1)
`+1
`+1 a
(0)
2` (d)
2 2` 2
+(1 2 2` 1
Thus in onformally invariant salar theory the
anomaly ofthestresstensoroinideswithone
assoi-ated withoperatorprodut. This statement holdsnot
onlyforhyperbolispaesonsideredabovebut forall
onstanturvaturemanifoldsaswell[16℄.
V Conluding remarks
Inthispapertheone-loopontributiontotheeetive
ation(3.19),themultipliativeanomaly(3.22)andthe
onformalanomalyofthestress-energymomentum
ten-sor (4.4), related to the operator produt, have been
evaluated expliitly. In addition we have onsidered
the produt N
p L
(j)
p
of Laplae operators L (j)
p
ating
inlosedrealhyperbolimanifolds. Notethatthe
mul-tipliativeanomaly is equalto zero for d =2 and for
the odd dimensional ases. It seems to us that the
expliitresultsfortheanomaliesarenotonly
interest-ingasmathematialresultsbutareofphysialinterest.
We hope that proposed disussion will be interesting
in viewof future appliations to onrete problems in
quantum eldtheory.
Aknowledgments
WethankF.L.Williamsforusefuldisussion. First
author partially supported bya CNPq grant(Brazil),
RFFI grant (Russia) No98-02-18380-a,and by
GRA-CENASgrant(Russia)No6-18-1997.
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