Non-unitary minimal models, Bailey's lemma and N=1,2 superconformal algebras

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by  Lipika Deka

NON-UNITARY MINIMAL MODELS, BAILEY’S LEMMA AND N =1,2
SUPERCONFORMAL ALGEBRAS
LIPIKADEKAANDANNESCHILLING
5
0
0 ABSTRACT. Using the Bailey flow construction, we derive character identities for the
2 N =1superconformalmodelsSM(p′,2p+p′)andSM(p′,3p′−2p),andtheN =2
n superconformalmodelwithcentralchargec = 3(1− 2pp′)fromthenonunitaryminimal
a modelsM(p,p′).AnewRamondsectorcharacterformulaforrepresentationsofN =2
J superconformalalgebraswithcentralelementc=3(1− 2pp′)isgiven.

4
1

2
v 1. INTRODUCTION
4
8 Bailey’slemmaisapowerfulmethodtoproveq-seriesidentitiesoftheRogers–Ramanujan-
0 type[3]. OneofthekeyfeaturesofBailey’slemmaisitsiterativestructurewhichwasfirst
2
observedby Andrews[2] (see also [22]). This iterative structure called the Bailey chain
1
makesitpossibletostartwithoneseedidentityandderiveaninfinitefamilyofidentities
4
0 from it. The Bailey chain has been generalized to the Bailey lattice [1] which yields a
/ wholetreeofidentitiesfromasingleseed.
h
TherelevanceoftheAndrews–Baileyconstructiontophysicswasfirstrevealedinthe
p
- papersbyFodaandQuano[14,15]inwhichtheyderivedidentitiesfortheVirasorochar-
h
actersusingBailey’slemma.BytheapplicationofBailey’slemmatopolynomialversions
t
a ofthecharacteridentityofoneconformalfieldtheory,oneobtainscharacteridentitiesof
m
anotherconformalfieldtheory. Thisrelationbetweenthe twoconformalfieldtheoriesis
: calledBaileyflow. In[4]itwasdemonstratedthatthereisaBaileyflowfromtheminimal
v
modelsM(p−1,p)toN =1andN =2superconformalmodels. Moreprecisely,itwas
i
X shownthatthereisaBaileyflowfromM(p−1,p)toM(p,p+1),andfromM(p−1,p)
r totheN =1superconformalmodelSM(p,p+2)andtheunitaryN =2superconformal
a
modelwithcentralchargec=3(1−2). Intheconclusionsof[4]itwasmentionedthatthis
p
constructioncanalsobecarriedoutforthenonunitaryminimalmodelsM(p,p′)wherep
andp′ arerelativelyprime. Inthispaperweconsiderthenonunitarycase. We showthat
starting with character identities for the nonunitary minimal model M(p,p′) of [6, 26],
charactersoftheN = 1superconformalmodelsSM(p′,2p+p′),SM(p′,3p′−2p)and
oftheN =2superconformalmodelwithcentralelementc=3(1−2p)canbeobtainedvia
p′
theBaileyflow. WealsogiveanewRamondsectorcharacterformulaforarepresentation
oftheN =2superconformalmodelwithcentralelementc=3(1− 2p).
p′
ThecharacteridentitiesobtainedfromtheBaileyflowconstructionareofBose-Fermi
type.Thebosonicsideisassociatedwiththeconstructionofsingularvectorsoftheunder-
lyingconformalfieldtheory.Thefermionicsideisusuallymanifestlypositiveandreflects
thequasiparticlestructureofthemodel.
Date:December2004.
SupportedinpartbyNSFgrantDMS-0200774.
1
2 L.DEKAANDA.SCHILLING
The paperis organizedasfollows. In section 2 we providethe necessarybackground
aboutBaileypairsandfermionicformulasoftheM(p,p′)models.Thissectionisaddedto
makethispaperself-contained.Fordetailsthereadershouldconsult[4,6,7]. Insection3
thecharactersoftheN =1supersymmetricmodelsSM(2p+p′,p′)andSM(3p′−2p,p′)
arederivedusingtheBaileyflow. Explicitfermionicexpressionsforthesecharactersare
given. Insection4thebackgroundregardingN = 2superconformalmodelsisstatedand
anewcharacterfortheRamondsectorisderived. Thenitisdemonstratedhowtoobtain
the charactersof the N = 2 superconformalmodelwith centralelementc = 3(1− 2p)
p′
viathe Baileyflow alongwith theexplicitfermionicexpressionsforthese characters. In
section5weconcludewithsomeremarks.
Acknowledgment. SpecialthankstoProfessorGaberdielandHannoKlemmfortheirhelp
throughthejungleofliteratureregardingN =2characterformulasandtheirhelpregard-
ingthespectralflowofN = 2superconformalalgebras. Wearegratefultobothofthem
fortheire-mailcorrespondences.WewouldalsoliketothankProfessorDobrevforhelpful
discussions.
2. BAILEY’S LEMMA
InthissectionwesummarizeBailey’soriginallemma[2,3]andtheBose-Fermiidenti-
tiesfortheM(p,p′)minimalmodels[5,6,16,26].
2.1. Bilateral Bailey lemma. A pair (α ,β ) of sequences {α } and {β } is
n n n n≥0 n n≥0
calledaBaileypairwithrespecttoaif
n
α
(2.1) β = j
n
(q) (aq)
n−j n+j
j=0
X
where
n−1
(a) :=(a;q) = (1−aqk),
n n
k=0
Y
1
(a) :=(a;q) = .
−n −n n (1−aq−k)
k=1
Following[4],wearegoingtouseanextendedQdefinitioninthispapercalledthebilateral
Baileypair. Apair(αn,βn)ofsequences{αn}n∈Z and{βn}n∈Z issaidtobeabilateral
Baileypairwithrespecttoaif
n
α
(2.2) β = j .
n
(q) (aq)
n−j n+j
j=−∞
X
Theorem2.1(BilateralBaileylemma[2,3,4]). If(α ,β )isabilateralBaileypairthen
n n
∞
(ρ ) (ρ ) (aq/ρ ρ )nβ
1 n 2 n 1 2 n
n=−∞
(2.3) X
(aq/ρ ) (aq/ρ ) ∞ (ρ ) (ρ ) (aq/ρ ρ )nα
1 ∞ 2 ∞ 1 n 2 n 1 2 n
= .
(aq) (aq/ρ ρ ) (aq/ρ ) (aq/ρ )
∞ 1 2 ∞ n=−∞ 1 n 2 n
X
NON-UNITARYMINIMALMODELS,BAILEY’SLEMMAANDN=1,2SUPERCONFORMALALGEBRAS 3
ThislemmahasbeenusedwithvariousBaileypairsanddifferentspecializationsofthe
parametersρ andρ toprovemanyq-seriesidentities(seeforexample[1,4,15,25]). In
1 2
this paper the bilateral Bailey lemma is used to derive character identities for N = 1,2
superconformalalgebrasfromnonunitaryminimalmodelsM(p,p′).
AusefulwaytoobtainnewBaileypairsfromoldonesistheconstructionofdualBailey
pairs.If(α ,β )isabilateralBaileypairwithrespecttoa,thedualBaileypair(A ,B )
n n n n
isdefinedas
A (a,q)=anqn2α (a−1,q−1),
n n
(2.4)
B (a,q)=a−nq−n2−nβ (a−1,q−1).
n n
Then(A ,B )satisfies(2.2)withrespecttoa.
n n
2.2. BaileypairsfromtheminimalmodelsM(p,p′). AsshownbyFodaandQuano[15],
theBose-Fermicharacteridentities[5,6,16,26]fortheminimalmodelsM(p,p′)areof
theform
(2.5) B (L,b;q)=q−Nr(b),sF (L,b;q),
r(b),s r(b),s
withN asgivenin[6]and
r(b),s
∞
B (L,b;q)= qj(jpp′+r(b)p′−sp) L
r(b),s 1(L+s−b)−jp′
j=−∞ (cid:20) 2 (cid:21)q
(2.6) X
−q(jp−r)(jp′−s) L .
1(L−s−b)+jp′
(cid:20) 2 (cid:21)q!
Here
n (q)
(2.7) = n
j (q) (q)
(cid:20) (cid:21)q j n−j
istheq-binomialcoefficient. ThefunctionfermionicformulaF (L,b;q)willbedis-
r(b),s
cussed in the next section. For simplicity we are going to write r for r(b). Follow-
ing[15,4]theidentity(2.5)yieldsthebilateralBaileypairrelativetoa= qb−s+2x where
x= L−2n−b+s
2
qj(jpp′+rp′−sp) ifn=jp′−x
α = −q(jp−r)(jp′−s) ifn=jp′−b−x
n 
(2.8) 0 otherwise
β =q−Nr,sF(p,p′)(2n+b−s+2x,b;q).
n (aq) r,s
2n
ThedualBaileypairto(2.8)relativetoa=qb−s+2x is
qj2p′(p′−p)−jp′(r−b)−js(p′−p)−x(b+x−s) ifn=jp′−x
αˆ = −q(jp′−s)(j(p′−p)+r−b)−x(b+x−s) ifn=jp′−b−x
n 
(2.9) 0 otherwise
βˆ =qNr,s anqn2F(p,p′)(2n+b−s+2x,b;q−1).
n (aq) r,s
2n
4 L.DEKAANDA.SCHILLING
Inserting(2.8)and(2.9)intothebilateralBaileylemmayields
∞ (ρ ) (ρ ) (aq/ρ ρ )nq−Nr(b),sF(p,p′)(2n+b−s+2x,b;q)
1 n 2 n 1 2 (aq) r,s
n=0 2n
X
∞
= (aq/ρ1)∞(aq/ρ2)∞ (ρ1)jp′−x(ρ2)jp′−x (aq/ρ ρ )jp′−x
1 2
(aq)∞(aq/ρ1ρ2)∞ (aq/ρ1)jp′−x(aq/ρ2)jp′−x
(2.10) j=X−∞
×qj(jpp′+rp′−sp)− (ρ1)jp′−b−x(ρ2)jp′−b−x
(aq/ρ1)jp′−b−x(aq/ρ2)jp′−b−x
×(aq/ρ ρ )jp′−b−xq(jp−r)(jp′−s)
1 2
!
and
∞ (ρ ) (ρ ) (aq/ρ ρ )nqNr(b),sanqn2F(p,p′)(2n+b−s+2x,b;q−1)
1 n 2 n 1 2 (aq) r,s
n=0 2n
X
∞
= (aq/ρ1)∞(aq/ρ2)∞ (ρ1)jp′−x(ρ2)jp′−x (aq/ρ ρ )jp′−x
1 2
(aq)∞(aq/ρ1ρ2)∞ (aq/ρ1)jp′−x(aq/ρ2)jp′−x
(2.11) j=X−∞
×qj2p′(p′−p)−jp′(r−b)−js(p′−p)−x(b+x−s)− (ρ1)jp′−b−x(ρ2)jp′−b−x
(aq/ρ1)jp′−b−x(aq/ρ2)jp′−b−x
×(aq/ρ ρ )jp′−b−xq(jp′−s)(j(p′−p)+r−b)−x(b+x−s) .
1 2
!
Asin[4],wearegoingtoconsiderdifferentspecializationsoftheparametersρ andρ in
1 2
(2.10)and(2.11)togetcharacteridentitiesforN =1,2superconformalalgebras.
2.3. FermionicformulasforM(p,p′). Sofarwehaveonlyconsideredthebosonicside
of(2.5) explicitly. Itsufficesforthe purposeofthispaperto state the fermionicformula
for the case p < p′ < 2p with p and p′ relatively prime and r,s being pure Takahashi
length.Wefollow[7,Section4].Thefermionicformuladependsonthecontinuedfraction
decomposition
p′ 1
=1+ν + .
p′−p 0 1
ν +
1
1
ν +···
2
ν +2
n0
Definet = i−1ν for1≤i≤n +1andthefractionallevelincidencematrixI and
i j=0 j 0 B
correspondingCartanmatrixBas
P
δ +δ for1≤j <t ,j 6=t
j,k+1 j,k−1 n0+1 i
(I ) = δ +δ −δ forj =t ,1≤i≤n −δ
B j,k  j,k+1 j,k j,k−1 i 0 νn0,0
δj,k+1+δνn0,0δj,k forj =tn0+1
B =2I −I ,
tn0+1 B
whereI istheidentitymatrixofdimensionn. Recursivelydefine
n
y =y +(ν +δ +2δ )y , y =0, y =1,
m+1 m−1 m m,0 m,n0 m −1 0
y =y +(ν +δ +2δ )y , y =−1, y =1.
m+1 m−1 m m,0 m,n0 m −1 0
NON-UNITARYMINIMALMODELS,BAILEY’SLEMMAANDN=1,2SUPERCONFORMALALGEBRAS 5
ThentheTakahashilengthandtruncatedTakahashilengtharegivenby
ℓ =y +(j−t )y
j+1 m−1 m m fort <j ≤t +δ with0≤m≤n .
ℓ =y +(j−t )y m m+1 m,n0 0
j+1 m−1 m m
Forb = ℓ ,r(b) = ℓ witht < β ≤ t +δ ands = ℓ witht < σ ≤
β+1 β+1 ξ ξ+1 ξ,n0 σ+1 ζ
t +δ thefermionicformulaisgivenby
ζ+1 ζ,n0
(2.12) Fr(,ps,p′)(L,b;q)=qkb,s q14mtBm−21Au,vmtn0+1 njm+mj ′
m≡Qu,vX(mod2) jY=1 (cid:20) j (cid:21)q
wherekb,s isanormalizationconstantandn,m∈Ztn0+1 suchthat
1
(2.13) n+m= I m+u+v+Le
B 1
2
(cid:16) (cid:17)
with ei the standard i-th basis element of Ztn0+1, u = eβ − nk=0ξ+1etk, v = eσ −
n0 e andQ ,A asdefinedin[7,Section4.2].Theq-binomialisalsodefined
k=ζ+1 tk u,v u,v P
fornegativeentries
P
′
n+m (qn+1)
m
= .
m (q)
(cid:20) (cid:21)q m
Notethat
′ ′
n+m n+m
(2.14) =q−nm .
m m
(cid:20) (cid:21)q−1 (cid:20) (cid:21)q
Infactusing(2.14)we getthe followingdualformofthe fermionicformulathatwillbe
usefullateron
(2.15) F(p,p′)(L,b;q−1)=
r,s
q−kb,s q41mtBm−12Lm1+12Au,vm−21mt(u+v)tn0+1 nj +mj ′ .
m
m≡XQu,v jY=1 (cid:20) j (cid:21)q
3. N =1SUPERCONFORMAL CHARACTER FROMM(p,p′)
Inthissectionwearegoingtoconsiderthespecializationin(2.10)and(2.11)
(3.1) ρ −→∞, ρ =finite.
1 2
We will see that these give characters of the N = 1 superconformal model SM(p,p′)
givenby[10,17],
(3.2) χ˜(rp,s,p′)(q)=χ˜(pp−,pr′,p)′−s(q)= (−q(qǫr)−s)∞ ∞ qj(jpp′+2rp′−sp) −q(jp−r)2(jp′−s) ,
∞
j=X−∞(cid:16) (cid:17)
where1≤r ≤p−1,1≤s≤p′−1,pand(p′−p)/2arerelativelyprimeand
1 ifiiseven(NS-sector),
(3.3) ǫ = 2
i
(1 ifiisodd(R-sector).
Thecentralchargeisc= 3 − 3(p−p′)2.
2 pp′
6 L.DEKAANDA.SCHILLING
3.1. ThemodelSM(p′,2p+p′). Specializingρ1 −→∞andρ2 =−qb−2s+1 withx=0
in(2.10)wefindforb−seven(NSsector)
(3.4) χ˜(p′,2p+p′)(q)= q12(n2+nb−ns)(−q12)n+(b−s)/2q−Nr,sF(p,p′)(2n+b−s,b;q)
s,2r+b (q) r,s
2n+b−s
n≥0
X
andforb−sodd(R-sector)
(3.5)
χ˜(p′,2p+p′)(q)= q21(n2+nb−ns)(−q)n+(b−s−1)/2q−Nr,sF(p,p′)(2n+b−s,b;q).
s,2r+b (q) r,s
2n+b−s
n≥0
X
Toobtainanexplicitfermionicformulasetm =L=2n+b−sandinsert(2.12)into
0
(3.4). Thenusing
m0
(3.6) (−q21)m20 = 2 q12(m20−k)2 mk20
k=0 (cid:20) (cid:21)q
X
wefind
χ˜(p′,2p+p′)(q)=q−81(b−s)2−Nr,s+kb,s ∞ m20 q18m20+21(m20−k)2
s,2r+b
(3.7) mmX00e=ve0nXk=0m≡XQu,v
×q41mtBm−12Au,vm× 1 m20 tn0+1 nj +mj ′ .
(q)m0 (cid:20) k (cid:21)q j=1 (cid:20) mj (cid:21)q
Y
Settingp=(k,m0,m)∈Ztn0+1+2,(3.7)intheNS-sectorcanberewrittenas
χ˜(p′,2p+p′)(q)=q−81(b−s)2−Nr,s+kb,s q14ptB˜p−21A˜u,vp
s,2r+b
p∈ZXtn0+1+2
(3.8) pi≡(Q˜u,v)i,i≥2
1 tn0+1+2 1(I p+u˜+v˜) ′
× 2 B˜ j
(q)p2 j=1,j6=2(cid:20) pj (cid:21)q
Y
whereI =2I −B˜,
B˜ tn0+1+2
2 −1 0
B˜ = −1 1 1
 
0 −1 B
A˜ =(0,0,A ), 
(3.9) u,v u,v
u˜t =(0,0,ut),
v˜t =(0,0,vt),
Q˜t =(0,0,Qt ).
u,v u,v
Similarlysettingm =2n+b−sin(3.5)andusing
0
m0+1
(3.10) (−q)m02−1 = 21 k=20 q12(m02+1−k)(m02−1−k)(cid:20) m0k2+1 (cid:21)q
X
NON-UNITARYMINIMALMODELS,BAILEY’SLEMMAANDN=1,2SUPERCONFORMALALGEBRAS 7
wegetthefermionicformulaintheR-sector,
(3.11) χ˜(p′,2p+p′)(q)= 1q−18((b−s)2+1)−Nr,s+kb,s q41ptB˜p−21A˜u,vp
s,2r+b 2
p∈ZXtn0+1+2
pi≡(Q˜u,v)i,i≥2
1 tn0+1+2 1(I p+u˜+v˜) ′
× 2 B˜ j
(q)p2 j=1,j6=2(cid:20) pj (cid:21)q
Y
whereB˜,A˜,v˜ areasin(3.9)andu˜t =(1,0,ut),Q˜t =(0,1,Qt ).
u,v u,v
3.2. ThemodelSM(p′,3p′−2p). Similarlyusingthesamespecializationwiththedual
Baileypairin(2.11)wefindforb−sevenintheNS-sector
(3.12)
χ˜(p′,3p′−2p)(q)= q32n(n+b−s)(−q21)n+(b−s)/2qNr,sF(p,p′)(2n+b−s,b;q−1)
s,3b−2r (q) r,s
2n+b−s
n≥0
X
andforb−soddintheR-sector
(3.13)
χ˜(p′,3p′−2p)(q)= q32n(n+b−s)(−q)n+(b−s−1)/2qNr,sF(p,p′)(2n+b−s,b;q−1).
s,3b−2r (q) r,s
2n+b−s
n≥0
X
To obtain the fermionic formula, as before we are going to set m = 2n + b − s.
0
Inserting(3.10)and(2.15)into(3.13)wegetintheR-sector
χ˜(sp,3′,b3−p2′−r2p)(q)= 21q−81(3(b−s)2+1)+Nr,s−kb,s ∞ m02+1
mmX00=od0d kX=0 m≡XQu,v
(3.14) ×q21(m20+k2−m0k−m0m1)q14mtBm−12mt(u+v)+12Au,vm
× 1 m02+1 tn0+1 nj +mj ′ .
(q)m0 (cid:20) k (cid:21)q j=1 (cid:20) mj (cid:21)q
Y
Definep=(k,m0,m)∈Ztn0+1+2,sothat(3.14)intheR-sectorcanberewrittenas
(3.15) χ˜(sp,3′,b3−p2′−r2p)(q)= 21q−18(3(b−s)2+1)+Nr,s−kb,s q41ptB˜′p+21A˜u,vp
p∈ZXtn0+1+2
pi≡(Q˜′u,v)i,i≥2
1 tn0+1+2 1(I p+u˜+v˜) ′
× 2 B˜′ j
(q)p2 j=1,j6=2(cid:20) pj (cid:21)q
Y
whereI =2I −B˜′,v˜ asin(3.9),u˜t =(1,0,ut),(Q˜′ )t =(0,1,Qt ),and
B˜′ tn0+1+2 u,v u,v
2 −1 0
B˜′ = −1 2 −1
(3.16)  
0 −1 B
A˜ =(0,0,A −ut−vt).
u,v u,v
8 L.DEKAANDA.SCHILLING
Similarly,fortheNS-sectoritfollowsfrom(3.12)
(3.17) χ˜(sp,3′,b3−p2′−r2p)(q)=q−83(b−s)2+Nr,s−kb,s q41ptB˜′p+12A˜u,vp
p∈ZXtn0+1+2
pi≡(Q˜′u,v)i,i≥2
1 tn0+1+2 1(I p+u˜+v˜) ′
× 2 B˜′ j
(q)p2 j=1,j6=2(cid:20) pj (cid:21)q
Y
withB˜′andA˜ asin(3.16),(Q˜′ )t =(0,0,Qt ),u˜t =(0,0,ut)andv˜t =(0,0,vt).
u,v u,v u,v
4. N =2CHARACTER FORMULAS
4.1. N = 2 superconformal algebra and Spectral flow. The N = 2 superconformal
algebraAistheinfinitedimensionalLiesuperalgebra[13]withbasisL ,T ,G±,C and
n n r
(anti)-commutationrelationgivenby
C
[L ,L ]=(m−n)L + (m3−m)δ
m n m+n m+n,0
12
1
L ,G± =( m−r)G±
m r 2 m+r
(cid:2)[L ,T (cid:3)]=−nT
m n m+n
1
[T ,T ]= cmδ
m n m+n,0
3
T ,G± =±G±
m r m+r
C 1
{(cid:2)G+,G−}(cid:3)=2L +(r−s)T + (r2− )δ
r s r+s r+s 3 4 r+s,0
[L ,C]=[T ,C]= G±,C =0
m n r
{G+r,G+s}={G−r ,G−s}(cid:2)=0 (cid:3)
where n,m ∈ Z, but r,s are integers in R-sector and half-integer in NS-sector. The
elementC is the centralelementand its eigenvaluec is parametrizedas c = 3(1− 2p),
p′
wherep,p′arerelativelyprimepositiveintegers.
Itwasobservedin[18,24]thatthereexitsafamilyofouterautomorphismsα :A→A
η
whichmapstheN =2superconformalalgebrastoitself. Theseareexplicitlygivenby
α (G+)=Gˆ+ =G+
η r r r−η
α (G−)=Gˆ− =G−
η r r r+η
(4.1) c
α (L )=Lˆ =L −ηT + η2δ
η n n n n n,0
6
c
α (T )=Tˆ =T − ηδ
η n n n n,0
3
Thisfamilyofautomorphismsiscalledspectralflowandη ∈Riscalledtheflowparam-
eter. When η ∈ Z each sector of the algebrais mappedto itself. When η ∈ Z+ 1 the
2
Neveu-Schwarzsector is mappedto the Ramond sector and vice-versa. We are going to
usethespectralflowη =±1 tomaptheNS-sectortotheR-sector.
2
NON-UNITARYMINIMALMODELS,BAILEY’SLEMMAANDN=1,2SUPERCONFORMALALGEBRAS 9
4.2. Spectralflowandcharacters. WedenotetheVermamodulegeneratedfromahigh-
estweightstate |h,Q,ciwith L eigenvalueh, T eigenvalueQ and centralchargec by
0 0
V . Thecharacterχ ofahighestweightrepresentationV isdefinedas
h,Q Vh,Q h,Q
χ (q,z)=Tr (qL0−c/24zT0).
Vh,Q Vh,Q
Following[18]thecharactertransformsunderthespectralflowinthefollowingway
(4.2) Tr (qLˆ0−c/24zTˆ0)=Tr (qL0−c/24zT0),
Vh,Q Vhη,Qη
wherehη andQη aretheeigenvaluesofLˆ andTˆ ,respectively,asdefinedin(4.1). This
0 0
means the new character χ (q,z) which is the trace of the transformed operators
Vhη,Qη
over the original representationequals the character of the representationdefined by the
eigenvalueshη andQη ofLˆ andTˆ ,respectively.Sothenewcharacteristhecharacterof
0 0
therepresentationV .
hη,Qη
For η = 1 the spectral flow α takes a NS-sector character to an R-sector character.
2 12
LetχNS (q,z)beaNS-sectorcharactercorrespondingtotherepresentationV . Then
Vh,Q h,Q
by(4.2)and(4.1)thenewR-sectorcharacterχR (q,z)isderivedusing
Vhη,Qη
(4.3) χRVhη,Qη(q,z)=TrVh,Q(qLˆ0−c/24zTˆ0)=TrVh,Q(qL0−21T0+2c4−2c4zT0−6c)
=q2c4z−6cTrVh,Q(qL0−2c4(zq−12)T0)=q2c4z−6cχNVhS,Q(q,zq−21).
4.3. R-sectorcharacterfromNS-sectorcharacter. Tosimplifynotationwearegoingto
useaslightlydifferentnotationforcharacters. Sinceweareonlydealingwiththevacuum
character in the NS-sector for which h = 0,Q = 0, we write χˆNS(q,z). The R-sector
p,p′
characterisdenotedbyχˆR (q,z)withthecorresponding(h,Q)specifiedseparately.
p,p′
Following [9, 12, 13, 18, 19] the vacuum character for the N = 2 superconformal
algebrawithcentralelementc=3(1− 2p)intheNS-sectorisgivenby
p′
(4.4) χˆNp,pS′(q,z)=q−c/24 ∞ (1+zqn−(121)−(1q+n)2z−1qn−12)
n=1
Y
× 1− ∞ qp(n+1)(p′(n+1)−1)+ zqp′n(pn+1)+pn+21 + z−1qp′n(pn+1)+pn+12
(cid:16) nX=0(cid:0) 1+zqp′n+21 1+z−1qp′n+21 (cid:1)
+ ∞ qpn(p′n+1)+ zqp′n(pn+1)−pn−21 + z−1qp′n(pn+1)−pn−21 .
nX=1(cid:0) 1+zqp′n−21 1+z−1qp′n−21 (cid:1)(cid:17)
This formula can be verified using the embedding diagram for the vacuum character as
describedin[13,18]andcanberewrittenas(aswillbeusefullater)
(4.5) χˆNp,pS′(q,z)=q−c/24 ∞ (1+zqn−(121)−(1q+n)2z−1qn−12)
n=1
Y
× ∞ qpj(p′j+1) 1−q2p′j+1 .
j=−∞ (1+zqp′j+12)(1+z−1qp′j+21)
X
The unitary case p = 1 of these character formulas was given in [11, 20, 21, 23]. In
particularifweputz =1in(4.5)weobtainthefollowingformuladerivedin[13]
(4.6) χˆNp,pS′(q)=q−c/24n∞=1(1(1+−qnq−n)212)2 j=∞−∞qpj(p′j+1)11−+qqpp′′jj++2211.
Y X
10 L.DEKAANDA.SCHILLING
Let us apply (4.3) to the NS-sector vacuum character (4.5) to get a Ramond sector
character.From(4.1)itfollowsthat
1 c
Lˆ =L − T +
0 0 0
2 24
c
Tˆ =T − .
0 0
6
ForthevacuumcharacterintheNS-sector(h,Q) = (0,0),sotheneweigenvaluesare
(hη,Qη)=( c ,−c)intheR-sector. HencethenewcharacterintheR-sectorcorresponds
24 6
to(hη,Qη)andby(4.3)
(4.7) χˆRp,p′(q,z)=q2c4z−c6χˆNp,pS′(q,zq−12)
=z−6c(−z)∞((q−)2z−1q)∞ ∞ qpj(p′j+1)(1+zqp1′j−)(1q2+p′jz+−11qp′j+1).
∞ j=−∞
X
4.4. N = 2superconformalcharactersforc = 3(1− 2p). Usingr = 0andb = 1in
p′
(2.10)weobtain
(4.8) ∞ (ρ ) (ρ ) (aq/ρ ρ )nq−N0,sF(p,p′)(2n+1−s+2x,1;q)
1 n 2 n 1 2 (aq) 0,s
n=0 2n
X
∞
= (aq/ρ1)∞(aq/ρ2)∞ (ρ1)jp′−x(ρ2)jp′−x (aq/ρ ρ )jp′−x
1 2
(aq)∞(aq/ρ1ρ2)∞ (aq/ρ1)jp′−x(aq/ρ2)jp′−x
j=−∞
X
− (ρ1)jp′−1−x(ρ2)jp′−1−x (aq/ρ ρ )jp′−1−x qjp(jp′−s).
1 2
(aq/ρ1)jp′−1−x(aq/ρ2)jp′−1−x !
Inthissectionweconsiderthespecialization
ρ =finite, ρ =finite.
1 2
Takingthelimit aq −→1in(4.8),wefind
ρ1ρ2
(4.9) ∞ (ρ ) (ρ ) q−N0,sF(p,p′)(2n+1−s+2x,1;q)
1 n 2 n(aq) 0,s
n=0 2n
X
= (ρ1)∞(ρ2)∞ ∞ qjp(jp′−s) ρ1ρ2q2(jp′−x−1)−1 .
(ρ ρ ) (q) (1−ρ qjp′−x−1)(1−ρ qjp′−x−1)
1 2 ∞ ∞ 1 2
j=−∞
X
4.5. NS-sector characters. Let us set ρ1 = −zqx+21,ρ2 = −z−1qx+21 in (4.9), which
impliesa = q2x and s = 1. Making the variable change j −→ −j in (4.9) and setting
x=0weobtain
(4.10) ∞ (−zq21)n(−z−1q21)nq(−qN)0,1F0(,p1,p′)(2n,1;q)
n=0 2n
X
= (−zq21)∞(−z−1q21)∞ ∞ qjp(jp′+1) 1−q2jp′+1 .
(q)2∞ j=−∞ (1+zqjp′+12)(1+z−1qjp′+21)
X
Comparingwith(4.5),weobtain
(4.11) χˆNp,pS′(q,z)=q−2c4−N0,1 ∞ (−zq21)(nq()−z−1q12)nF0(,p1,p′)(2n,1;q).
n=0 2n
X