On the Quantum Invariant for the Brieskorn Homology Spheres

0.24 MB·English

by  Kazuhiro Hikami

ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES
KAZUHIROHIKAMI
ABSTRACT. We study an exactasymptoticbehaviorof the Witten–Reshetikhin–TuraevSU(2) in-
variantfortheBrieskornhomologyspheresΣ(p1,p2,p3)byuseofpropertiesofthemodularform
followinga methodproposedbyR. LawrenceandD.Zagier. Keyobservationisthattheinvariant
5 coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We
0 show thatthe Casson invariantis related to the numberof the Eichlerintegralswhich do notvan-
0
ish in a limit τ N Z. Correspondingly there is a one-to-one correspondence between the
2 → ∈
non-vanishingEichlerintegralsandtheirreduciblerepresentationofthefundamentalgroup,andthe

n Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the
a
Ohtsuki invariantfollows from a nearly modular property of the Eichler integral, and we give an
J
explicitformintermsoftheL-function.
1

2
v
8
2
0 1. INTRODUCTION
5
0
4 The quantum invariant for the 3-manifold was introduced as a path integral on by Wit-
0 M M
ten [43];as theSU(2)invariantwehave
/
h
p
- Zk( ) = exp 2πik CS(A) A (1.1)
h M D
t Z (cid:16) (cid:17)
a wherek Z, and CS(A) istheChern–Simonsfunctionaldefined by
m ∈
: 1 2
v CS(A) = Tr A dA+ A A A (1.2)
i 8π2 ∧ 3 ∧ ∧
X Z (cid:18) (cid:19)
M
r
a Sincethiswork,studiesofthequantuminvariantsofthe3-manifoldshavebeen extensivelydevel-
oped, and a construction of the 3-manifold invariant was reformulated combinatorially and rigor-
ously in Refs. 15,33 using a surgery description of and the colored Jones polynomial defined
M
inRef. 14.
By applying a stationary phase approximation, an asymptotic behavior of the Witten invariant
inlarge k is expectedto be[6,43](seealso Ref. 2)
→ ∞
1
Zk( ) e−34πi Tα( )e−2πiIα/4e2πi(k+2)CS(A) (1.3)
M ∼ 2 M
α
Xp
Here the sum runs over a flat connection α, and T and I are the Reidemeister torsion and the
α α
spectral flowdefined modulo8 respectively.
Date:May2,2004.AcceptedonDecember21,2004.
1
In this article we consider the Brieskorn homology spheres Σ(p ,p ,p ) where p are pairwise
1 2 3 i
coprimepositiveintegers. Thisistheintersectionofthesingularcomplexsurface
z p1 +z p2 +z p3 = 0
1 2 3
incomplexthree-space withtheunitfive-sphere z 2 + z 2 + z 2 = 1. ThemanifoldΣ(2,3,5)
1 2 3
| | | | | |
isthePoincare´ homologysphere. Thesemanifoldshavearationalsurgery descriptionasinFig.1,
and thefundamentalgroup hasthepresentation
π Σ(p ,p ,p ) = x ,x ,x ,h hcenter,xpk = h−qk fork = 1,2,3,x x x = 1 (1.4)
1 1 2 3 1 2 3 k 1 2 3
D E
where(cid:0)q Zsuch(cid:1)that (cid:12)
k ∈ (cid:12)
3
q
P k = 1 (1.5)
p
k
k=1
X
Hereand hereafter weuse
P = P(p ,p ,p ) = p p p (1.6)
1 2 3 1 2 3
0
p /q p /q
1 1 3 3
p /q
2 2
Figure1: RationalsurgerydescriptionoftheBrieskornhomologysphereΣ(p1,p2,p3)
AsymptoticbehaviorofthequantuminvariantforthehomologysphereswasstudiedinRefs.19,
20,22,34–39. Our purpose here is to reformulate these results number theoretically by use of
propertiesofthemodularformfollowingamethodofLawrenceandZagier[23]. Akeyobservation
isafactthattheWitten–Reshetikhin–Turaev(WRT)invariantfortheBrieskornhomologyspheres
is regarded as a limit value of the Eichler integral of the modular form with weight 3/2 as was
suggested in Ref. 23. Using a nearly modular property of the Eichler integral, we can derive an
exactasymptoticbehavioroftheWRTinvariant. Correspondingly,wecanfindaninterpretationfor
topological invariants such as the Chern–Simons invariant, the Casson invariant [42], the Ohtsuki
invariant [27–30], and the Ray–Singer–Reidemeister torsion from the viewpoint of the modular
form.
Thispaperisorganizedasfollows. Insection2weconstructtheWRTinvariantfortheBrieskorn
homologyspheresΣ(p ,p ,p )followingRef.22. Weuseasurgerydescriptionofthe3-manifold,
1 2 3
andapplyaformulainRef. 13. In section3 weintroducethemodularformwithweight3/2. This
2
givesa(p 1)(p 1)(p 1)/4-dimensionalrepresentationofthemodulargroupPSL(2;Z).
1 2 3
− − −
We consider the Eichler integral thereof in section 4. We study a limiting value of the Eichler
integral at τ Q, and showthat the numberofthe Eichlerintegralswhich havenon-zero valuein
∈
a limit τ N Z is related to the Casson invariant of the Brieskorn homology spheres. It will
→ ∈
be discussed that we have a one-to-one correspondence with the irreducible representation of the
fundamental group, and that the Chern–Simons invariant is given from this limiting value of the
Eichler integral. We further give a nearly modular property of the Eichler integral. In section 5
werevealakeyidentitythattheWRTinvariantfortheBrieskornspherescoincideswithalimiting
value of the Eichler integral at τ 1/N. This was suggested in Ref. 23 where a correspondence
→
was proved only for a case of the Poincare´ homologysphere. Combining with results in section 4
weobtainanexactasymptoticexpansionoftheWRTinvariantwhichisconstitutedfromtwoterms;
one is a sum of dominating exponential term which gives the Chern–Simons term, and another is
a “tail” part which may be regarded as a contribution from a trivial connection. In section 6 we
provethat theS-matrixof themodulartransformationgivesboththeReidemeistertorsionand the
spectralflow. Weshowinsection7thatatailpartgivestheOhtsukiinvariant. Explicitlycomputed
is the n-th Ohtsuki invariant in terms of the L-function. The last section is devoted to concluding
remarks.
2. THE WITTEN–RESHETIKHIN–TURAEV INVARIANT FOR THE BRIESKORN
HOMOLOGY SPHERES
We introducethe Reshetikhin–Turaevinvariant τ ( ) [33] for N Z . This is related to the
N +
M ∈
WitteninvariantZ ( ) defined ineq. (1.1)as
k
M
τ ( )
Z ( ) = k+2 M (2.1)
k
M τ (S2 S1)
k+2
×
Heretheinvariantisnormalizedto be
τ (S3) = 1
N
and wehave
N 1
τ (S2 S1) =
N
× 2 sin(π/N)
r
Whenthe3-manifold isconstructedbytherationalsurgeriesp /q onthej-thcomponentof
j j
M
n-componentlink ,it was shown[13,33]thattheinvariantτ ( )isgivenby
N
L M
N−1 n
τN(M) = eπ4iNN−2( nj=1Φ(U(pj,qj))−3sign(L)) Jk1,...,kn(L) ρ(U(pj,qj))kj,1 (2.2)
P
k1,.X..,kn=1 Yj=1
3
Herethesurgery p /q isdescribed by anSL(2;Z) matrix
j j
p r
U(pj,qj) = j j
q s
j j
(cid:18) (cid:19)
and Φ(U) istheRademacherΦ-function defined by (see, e.g.,Ref. 31)
p+s
12s(p,q) forq = 0
p r q − 6
Φ = (2.3)
q s r
(cid:20)(cid:18) (cid:19)(cid:21)  forq = 0

s
wheres(b,a) denotes theDedekind sum(see, e.g.,Ref. 32, andalso Ref. 16)

|a|−1
k kb
s(b,a) = sign(a) (2.4)
a · a
Xk=1(cid:16)(cid:16) (cid:17)(cid:17) (cid:16)(cid:16) (cid:17)(cid:17)
with
x x 1 ifx Z
−⌊ ⌋− 2 6∈
((x)) =

0 ifx Z
 ∈
and x isthegreatestintegernot exceedingx. It isknownthattheDedekindsumis rewrittenas
⌊ ⌋ 
a−1
1 k kb
s(b,a) = cot π cot π
4a a a
k=1 (cid:18) (cid:19) (cid:18) (cid:19)
X
An n n matrix L is a linking matrix L = lk +p /q δ , and sign(L) is a signature of L,
j,k j,k j j j,k
× ·
i.e.,thedifferencebetween thenumberofpositiveandnegativeeigenvaluesofL. Thepolynomial
J ( )isthecoloredJonepolynomialforlink withthecolork forthej-thcomponentlink,
k1,...,kn L L j
and ρ(U(p,q)) isarepresentationρ ofPSL(2;Z);
ρ(U(p,q))
a,b
sign(q)
= i e−π4iΦ(U(p,q))e2πNiqsb2 e2Nπiqpγ2 eNπiqγb e−Nπiqγb (2.5)
− 2N q −
| | γ mXod2Nq (cid:16) (cid:17)
γ=a mod2N
p
for1 a,b N 1 [13], andwehave
≤ ≤ −
2 abπ
ρ(S) = sin
a,b
N N
r (cid:18) (cid:19) (2.6)
ρ(T)a,b = e2πNia2−π4i δa,b
with
0 1 1 1
S = − T = (2.7)
1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19)
satisfying
S2 = (ST)3 = 1
4
Weshouldnote[32]thattheDedekindsumsatisfies
s( b,a) = s(b,a) (2.8)
− −
s(b,a) = s(b′,a) forbb′ 1 (mod a) (2.9)
≡
and thattheRademacher Φ-functionfulfills
Φ(SU(p,q)) = Φ(U(p,q)) 3 sign(pq) (2.10)
−
Proposition1( [22]). TheWRT invariantfortheBrieskornhomologyspheres isgiven by
e2Nπi(φ4−21) e2Nπi 1 τN Σ(p1,p2,p3)
−
(cid:16) (cid:17) (cid:0) = eπi(cid:1)/4 2PN−1e−2P1Nn2πi 3j=1 eNnpjπi −e−Nnpjπi (2.11)
2√2P N Q (cid:16)eNnπi e−Nnπi (cid:17)
n=0 −
NX∤n
where
1
φ = φ(p ,p ,p ) = 3 +12 s(p p ,p )+s(p p ,p )+s(p p ,p ) (2.12)
1 2 3 2 3 1 1 3 2 1 2 3
− P
(cid:0) (cid:1)
Proof. This was proved in Refs. 22,38 for a general n-fibered manifold, but we give a proof here
againforcompletion.
TheJones polynomialforalink depicted inFig. 1 isgivenby
L
1 3 sin(k k π/N)
j=1 0 j
J ( ) =
k0,k1,k2,k3 L sin(π/N) · sin2(k π/N)
Q 0
where k is a color of an unknotted component whose linking number with other components is
0
1, and k (for j = 1,2,3) denotes a color of a component of link which is to be p /q -surgery.
j j j
L
Withthissettingwehave
3
q
sign(L) = sign j 1
p −
j=1 (cid:18) j(cid:19)
X
From (2.2)wegetthequantuminvariantas
eπi/N e−πi/N τ Σ(p ,p ,p )
N 1 2 3
− ·
N−1
(cid:0) (cid:1) =(cid:0)eπ4iNN−2(3+ 3j(cid:1)=1Φ(SU(pj,qj))) ρ(S)k0,1
2
P kX0=1 eπNik0 −e−πNik0
3 N(cid:16)−1 (cid:17)
× ρ(U(pj,qj))kj,1 eπNik0kj −e−πNik0kj
Yj=1kXj=1 (cid:16) (cid:17)
In thisexpressionwehavebydefinition
i
ρ(S)k0,1 = √−2N eπNik0 −e−πNik0
5(cid:16) (cid:17)
and forℓ Zwehave
∈
N−1
ρ(U(p,q))k,1 eπNiℓk e−πNiℓk
−
Xk=1 (cid:16) (cid:17)
N−1
sign(q)
= ie−π4iΦ(U(p,q))+2πNisq e2πNipqγ2 eNπiqγ e−Nπiqγ eπNiℓγ e−πNiℓγ
− 2N q − −
| | Xk=1 γ mXod2Nq (cid:16) (cid:17) (cid:16) (cid:17)
γ≡k mod2N
p
= ie−π4iΦ(U(p,q))+2πNisq sign(q) e2Nπiqpγ2 e2πiq2ℓN+q1γ e2πiq2ℓN−q1γ
− 2N q −
| | γ mXod2Nq (cid:16) (cid:17)
= sign(p) e−π4iΦ(SU(pp,q))+2πNisq e−2pπqiN(2Nqn+qℓ+1)2 e−2pπqiN(2Nqn+qℓ−1)2
− p −
| | n Xmodp(cid:16) (cid:17)
p
Here in the first equality we have used γ = k (mod 2N), and applied a symmetry under γ
→
2N q γ in the second equality. In the last equality, we have used an identity eπi(1−sign(p)) =
2
−
sign(p),and theGausssumreciprocityformula[13]
N
eπNiMn2+2πikn = eπ4isign(NM) e−MπiN(n+k)2 (2.13)
s M
n modN (cid:12) (cid:12) n modM
X (cid:12) (cid:12) X
(cid:12) (cid:12)
whereN,M ZwithN k Zand N M b(cid:12)eing(cid:12)even.
∈ ∈
Acombinationoftheseresultsreduces to
eπi/N e−πi/N τ Σ(p ,p ,p )
N 1 2 3
− ·
N−1
(cid:0) (cid:1) i sig(cid:0)n(P) 3πi+π(cid:1)i 1 −πiφ 1
= e4 2N j pjqj 2N
2 P N P eπNik0 e−πNik0
| | kX0=1nj Xmodpj −
p 3
e−2Nπpijqj(2Nqjnj+k0qj+1)2 e−2Nπpjiqj(2Nqjnj+k0qj−1)2
× −
Yj=1(cid:16) (cid:17)
The summand is invariant under (i) k k + 2N and n n 1, (ii) n n + p . Using
0 0 j j j j j
→ → − →
this symmetry and recalling that p are pairwise coprime integers, the sum, N−1 , is
j k0=1 nj modpj
transformedintoasum, , withsettingalln = 0. Asa result,wefind
k0=a+2Nn j P P
1≤a≤N−1
0≤n≤P−1
P
eπi/N e−πi/N τ Σ(p ,p ,p )
N 1 2 3
− ·
(cid:0) (cid:1) (cid:0) = i (cid:1) e34πi−2πNiφ e−2NπiPk02 3j=1 eNπpijk0 −e−Nπpijk0
−√2P N Q (cid:16)eπNik0 e−πNik0 (cid:17)
k0=Xa+2Nn −
1≤a≤N−1
0≤n≤P−1
Settingk 2P N k , weobtainastatementoftheproposition. (cid:3)
0 0
→ −
6
3. MODULAR FORMS
Wedefinetheoddperiodicfunctionχ(ℓ1,ℓ2,ℓ3)(n) withmodulus2P by
2P
3
ℓ
1 forn = P 1+ ε j mod 2P whereε ε ε = 1
j 1 2 3
 pj −
 (cid:16) Xj=1 (cid:17)
χ(2ℓP1,ℓ2,ℓ3)(n) =  1 forn = P 1+ 3 ε ℓj mod 2P whereε ε ε = 1 (3.1)
 j 1 2 3
− p
j
(cid:16) Xj=1 (cid:17)
0 others






Here P = p p p with pairwise coprime positiveintegers p , and we mean ε = 1. Integers ℓ
1 2 3 j j j
±
are
1 ℓ p 1 (3.2)
j j
≤ ≤ −
Thereexistsasymmetryoftheperiodicfunction
χ(ℓ1,ℓ2,ℓ3)(n) = χ(p1−ℓ1,p2−ℓ2,ℓ3)(n)
2P 2P
= χ(p1−ℓ1,ℓ2,p3−ℓ3)(n) = χ(ℓ1,p2−ℓ2,p3−ℓ3)(n) (3.3)
2P 2P
With this periodic function, we define the function Φ(ℓ1,ℓ2,ℓ3)(τ) for τ in the upper half plane,
p
τ H, by
∈
Φ(pℓ1,ℓ2,ℓ3)(τ) = 21 nχ(2ℓP1,ℓ2,ℓ3)(n)q4nP2 (3.4)
n∈Z
X
whereas usual
q = exp(2πiτ)
Eq.(3.3)makes thenumberoftheindependentfunctionsΦ(ℓ1,ℓ2,ℓ3)(τ) tobe
p
1
D = D(p ,p ,p ) = (p 1)(p 1)(p 1) (3.5)
1 2 3 1 2 3
4 − − −
Proposition2. ThefunctionΦ(ℓ1,ℓ2,ℓ3)(τ)isamodularformwithweight3/2. NamelyundertheS-
p
andT-transformations(2.7)we have
3/2
Φ(ℓ1,ℓ2,ℓ3)(τ) = i Sℓ′1,ℓ′2,ℓ′3Φ(ℓ′1,ℓ′2,ℓ′3)( 1/τ) (3.6)
p τ ℓ1,ℓ2,ℓ3 p −
(cid:18) (cid:19) ℓ′,ℓ′,ℓ′
1X2 3
Φ(ℓ1,ℓ2,ℓ3)(τ +1) = Tℓ1,ℓ2,ℓ3Φ(ℓ1,ℓ2,ℓ3)(τ) (3.7)
p p
7
where the sum runs over D(p ,p ,p ) distinct triples. A D D matrix S and diagonal matrix T
1 2 3
×
arerespectivelygiven by
Sℓ′1,ℓ′2,ℓ′3 = 32 ( 1)1+P+P 3j=1 ℓjp+jℓ′j+(ℓ×ℓ′)·p 3 sin P ℓjℓ′j π (3.8)
ℓ1,ℓ2,ℓ3 rP − P j=1 (cid:18) pj2 (cid:19)
Y
3
πi ℓ 2
Tℓ1,ℓ2,ℓ3 = exp P 1+ j (3.9)
2 p
j !
(cid:16) Xj=1 (cid:17)
We omit a proof as it is tedious but straightforward. We only need the Poisson summation
formula
∞
f(n) = e−2πitnf(t)dt (3.10)
n∈Z n∈ZZ−∞
X X
4. THE EICHLER INTEGRAL AND THE CHERN–SIMONS INVARIANT
The Eichler integral was originally defined as a k 1-fold integration of a modular form with
−
integral weight k Z (see, e.g., Ref. 18). Following Refs. 23,44 (see also Refs. 9–11), we
≥2
∈
definetheEichlerintegralofthemodularform Φ(ℓ1,ℓ2,ℓ3)(τ) withhalf-integralweight3/2by
p
∞
Φ(pℓ1,ℓ2,ℓ3)(τ) = χ(2ℓP1,ℓ2,ℓ3)(n)q4nP2 (4.1)
n=0
X
e
We should remark that there are D(p ,p ,p ) independent Eichler integrals due to the symme-
1 2 3
try (3.3).
Proposition3. ThefunctionΦ(ℓ1,ℓ2,ℓ3)(τ) hasa limitingvalueinτ 1/N forN Zas
p
→ ∈
PN
Φ(pℓ1,ℓ2,ℓ3)e(1/N) = χ(2ℓP1,ℓ2,ℓ3)(n) 1− PnN e2P1Nn2πi (4.2)
Xn=0 (cid:16) (cid:17)
e
Wealsohave
Φ(ℓ1,ℓ2,ℓ3)(N) = 1 2P nχ(ℓ1,ℓ2,ℓ3)(n) eπ2iPN 1+ j pℓjj 2 (4.3)
p −2P 2P ! (cid:18) P (cid:19)
n=1
X
e 2P
1
= nχ(ℓ1,ℓ2,ℓ3)(n) Tℓ1,ℓ2,ℓ3 N
−2P 2P
!
n=1
X (cid:0) (cid:1)
Toprovethisproposition,weusethefollowingformulaforasymptoticexpansions(seeRefs.23,
44);
8
Proposition4. Let C (n)beaperiodicfunctionwithmeanvalue0andmodulusf. Thenwehave
f
an asymptoticexpansionast 0;
ց
∞ ∞ ( t)k
C (n)e−nt L( k,C ) − (4.4)
f f
≃ − k!
n=1 k=0
X X
∞ ∞ ( t)k
C (n)e−n2t L( 2k,C ) − (4.5)
f f
≃ − k!
n=1 k=0
X X
HereL(k,C )is theDirichletL-functionassociatedwith C (n), andis givenby
f f
fk f n
L( k,C ) = C (n)B
f f k+1
− −k +1 f
n=1 (cid:18) (cid:19)
X
where B (x) isthen-th Bernoullipolynomialdefined from
n
text ∞ B (x)
= n tn
et 1 n!
− n=0
X
See, e.g.,Ref. 23foraproof.
Proofof Proposition3. We assumeM and N are coprime integers,and N > 0. By definition, we
have
∞
M y
Φ(pℓ1,ℓ2,ℓ3) N +i 2π = C2(ℓP1N,ℓ2,ℓ3)(n)e−4yPn2
(cid:18) (cid:19) n=0
X
wherey > 0and e
C(ℓ1,ℓ2,ℓ3)(n) = χ(ℓ1,ℓ2,ℓ3)(n)e2PMNn2πi
2PN 2P
We see that C(ℓ1,ℓ2,ℓ3)(n + 2P N) = C(ℓ1,ℓ2,ℓ3)(n), and C(ℓ1,ℓ2,ℓ3)(2P N n) = C(ℓ1,ℓ2,ℓ3)(n).
2PN 2PN 2PN − − 2PN
Then wecan applyProp. 4, and wehavean asymptoticexpansioniny 0 as
ց
M y ∞ L( 2k,C(ℓ1,ℓ2,ℓ3)) y k
Φ(ℓ1,ℓ2,ℓ3) +i − 2PN
p N 2π ≃ k! −4P
(cid:18) (cid:19) Xk=0 (cid:16) (cid:17)
whichgivesalimiteingvalueas
Φ(ℓ1,ℓ2,ℓ3)(M/N) = L(0,C(ℓ1,ℓ2,ℓ3)) (4.6)
p 2PN
Using a fact that χ(ℓ1,ℓ2,ℓ3)(2P n) = χ(ℓ1,ℓ2,ℓ3)(n) and that an explicit form of the Bernoulli
2P e− − 2P
polynomialisB (x) = x 1,we get
1 − 2
PN
n
Φp(ℓ1,ℓ2,ℓ3)(M/N) = χ(2ℓP1,ℓ2,ℓ3)(n) 1− P N e2PMNn2πi
Xn=0 (cid:16) (cid:17)
Eq. (4.2) directly foellows from this formula. Eq. (4.3) can also be given from the above formula
when werecall 2P χ(ℓ1,ℓ2,ℓ3)(n) = 0. (cid:3)
n=1 2P
9
P
We can see that, though we have D(p ,p ,p ) independent Eichler integrals, the limiting value
1 2 3
Φ(ℓ1,ℓ2,ℓ3)(N)atN Zcomputedineq.(4.3)becomesidenticallyzeroforsometriples(ℓ ,ℓ ,ℓ ).
p 1 2 3
∈
Peroposition 5. Let γ(p ,p ,p ) be the number of independent Eichler integrals such that
1 2 3
Φ(ℓ1,ℓ2,ℓ3)(N) 0 forN Z, namely
p
6≡ ∈
2P
e nχ(ℓ1,ℓ2,ℓ3)(n) = 0 (4.7)
2P 6
n=1
X
Wethenhave
γ(p ,p ,p ) = s(p p ,p )+s(p p ,p )+s(p p ,p )
1 2 3 1 2 3 2 3 1 1 3 2
P 1 1 1 1 1
+ 1 + (4.8)
12 − p 2 − p 2 − p 2 − 12P 4
(cid:18) 1 2 3 (cid:19)
where s(b,a) istheDedekindsum defined ineq. (2.4).
Proof. ForasakeofourbrevitywesetA = ℓi + ℓj ℓk fori = j = k = iandi,j,k 1,2,3 .
k pi pj − pk 6 6 6 ∈ { }
As wehave0 < ℓk < 1 bydefinition,wehave0 < ℓj < 3 and 1 < A < 2.
pk j pj − k
When 0 < ℓj < 1, we have 0 < A +P1 < 2 1 ℓk < 2. Then in a domain
j pj k − pk
0 < n < 2PP, the periodic function χ(ℓ1,ℓ2,ℓ3)(n) defined(cid:16)in eq.(cid:17)(3.1) takes a value 1 when
2P
n = P 1 ℓj ,P(1 + A ), while it is 1 when n = P 1+ ℓj ,P (1 A ). Then
− j pj k − j pj − k
wefind(cid:16) 2P Pnχ(ℓ1,(cid:17)ℓ2,ℓ3)(n) = 0, andit isinconsistentwitheq. ((cid:16)4.7). P (cid:17)
n=1 2P
We thPus have a condition 1 < ℓj < 3 to fulfill eq. (4.7), because it is impossible to have
j pj
ℓj Z. UnderthisconditiontherearetwopossibilitiesforaconditionofA ;(i) 1 < A < 1
j pj ∈ P k − k
forall k, or(ii) 1 < A < 1 fortwo k’s and 1 < A < 2 for anotheri. By thesamecomputation
P − k i
wecan check
2P
nχ(ℓ1,ℓ2,ℓ3)(n) = 4P (4.9)
2P
n=1
X
for the former case, while we have 2P nχ(ℓ1,ℓ2,ℓ3)(n) = 0 for the latter. To conclude, a condi-
n=1 2P
tion(4.7)isfulfilledwhen thetripleofintegerssatisfies
P
ℓ ℓ ℓ ℓ ℓ ℓ
1 2 3 1 2 3
1 < + + < 3 1 < + < 1
p p p − p p − p
1 2 3 1 2 3
(4.10)
ℓ ℓ ℓ ℓ ℓ ℓ
1 2 3 1 2 3
1 < + < 1 1 < + + < 1
− p − p p − −p p p
1 2 3 1 2 3
This constraint is depicted as the numberof theintegral lattice pointsof an interiorof the tetrahe-
dron (seeFig. 2).
10