A note on the "logarithmic-W_3" octuplet algebra and its Nichols algebra

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by  A. M. Semikhatov

A NOTE ON THE “LOGARITHMIC-W ” OCTUPLET ALGEBRA AND ITS
3
NICHOLS ALGEBRA
AMSEMIKHATOV
3
1 ABSTRACT. WedescribeaNichols-algebra-motivatedconstructionofanoctupletchiral
0 algebrathat is a “W -counterpart”of the tripletalgebra of p,1 logarithmicmodelsof
3
2 p q
two-dimensionalconformalfieldtheory.

n
a
J

0
1 1. INTRODUCTION

]
A Logarithmicmodelsof two-dimensionalconformal field theory can be defined as cen-
Q tralizers ofNicholsalgebras [1, 2]. Forthis,thegenerators F of agivenNichols algebra
i
h. B X with diagonal braiding [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] are to be
t p q
a realized as
m F ea i.j , 1 i rank q ,
i
[ “ ď ď ”
¿
1 wherej z isaq -pletofscalarfieldsanda Cq arechosensoastoreproducethegiven
i
v p q P
braidingcoefficients q in
7 i,j
2
Y :F F q F F, 1 i, j q .
2 i j i,j j i
b ÞÑ b ď ď
2
1. Thecoefficientsarestandardlyarrangedintoabraidingmatrixpqi,jq11ďijďrraannkk. Therelation
0 between the braiding matrix and the screening momenta is postulateďdď[2] in the form of
3
1 equations
v: q eipa j.a j, q q e2ipa j.a k
j,j j,k k,j
i “ “
X
and thelogical-“or”conditions
r
a a a .a 2a .a , 1 a a .a 2
i,j i i i j i,j i i
“ p ´ q “
imposedforeach pairi j andinvolvingthŽeCartanmatrix ai,j associatedwiththegiven
‰
braidingmatrix(see, e.g., [18]and thereferences therein).
In this note, we describe some details related to the construction of the octuplet alge-
bra [2] that can be considered a “logarithmicextension”of the W algebra [19] similarly
3
tohowthetripletalgebra[21,22,23]isa“logarithmicextension”oftheVirasoroalgebra.
The starting point is a particular item in Heckenberger’s list of rank-2 Nichols algebras
withdiagonalbraiding(which isitem 5.7(1)in[20])—thebraidingmatrix
q2 q 1
(1.1) q ´ ,
ij “ q 1 q2
˜ ´ ¸
2 SEMIKHATOV
where q2 is aprimitive2pth rootofunity. Wechoose
ip
(1.2) q ep
“
with p 2,3,.... This choice leads to p,1 -type logarithmic CFT models [21, 22, 23,
“ p q
ip p1
24, 25, 26, 27], in contrast to p,p models that follow if q is chosen as e p instead.
1
p q
Themainexpectationassociatedwith p,1 -typemodelsisthattheirrepresentationcate-
p q
goriesare“verycloselyrelated”[25,28,29]toanappropriaterepresentationcategoryon
the algebraic side, which in the braided case is some category of Yetter–Drinfeld B X -
p q
modules(cf.[30]). Inthispaper,wethereforeproceedalongtworoutes: (i)describingthe
structureoftheB X algebraassociatedwith(1.1)(solelywiththechoicein(1.2))andits
p q
suitable Yetter–Drinfeld modules, and (ii) discussing some properties of the octuplet al-
gebrathatcentralizesthisB X . Noneofthetwodirectionsispursuedtothepointwhere
p q
they actually meet (which would mean constructing a functor), but the results presented
herehopefullybringus somewhatclosertothat point.
2. THE NICHOLS ALGEBRA
2.1. Presentation for B X . We first recall the presentation of the relevant Nichols al-
p q
gebra,asaquotientofthetensoralgebra. Ourstartingpointisatwo-dimensionalbraided
vectorspaceX withthepreferred basis F ,F andtheabovebraidingmatrixinthisbasis.
1 2
TheNicholsalgebra B X isthequotientby agraded ideal I[16, 20],
p q
(2.1) B X T X F , F ,F , F , F ,F , Fp, F ,F p, Fp , dimB X p3,
p q“ p q{ r 1 r 1 2ss r 2 r 2 1ss 1 r 2 1s 2 p q“
If p 2, the double-b`racket generators of the ideal are absent. The˘brackets here denote
“
q-commutators determined by the braiding matrix: F ,F F F q 1F F , F ,F
1 2 1 2 ´ 2 1 2 1
r s“ ´ r s“
F F q 1F F , and so on by multiplicativity of the “q”-factor, whence the two double
2 1 ´ 1 2
´
commutatorsin theideal areexplicitlygivenby
F , F ,F F2F q q 1 F F F F F2,
1 1 2 1 2 ´ 1 2 1 2 1
r r ss“ ´p ` q `
F , F ,F F2F q q 1 F F F F F2.
2 2 1 2 1 ´ 2 1 2 1 2
r r ss“ ´p ` q `
A PBWbasisinB X isgivenbyFrFtFs, 0 r,s,t p 1 [20], where
1 3 2
p q ď ď ´
F F ,F .
3 2 1
“r s
Thedouble-bracketrelationsintheidealcanalsoberewrittenasF F qF F andF F
2 3 3 2 3 1
“ “
qF F .
1 3
Multiplicationin B X T X I is the one induced by “concatenation” in the tensor
p q“ p q{
algebra, X m X n X m n , x ,...,x y ,...,y x ,...,x ,y ,...,y . It
b b bp ` q 1 m 1 n 1 m 1 n
b Ñ p qbp q ÞÑ p q
is then relatively straightforward to show that the multiplication table of the PBW basis
elementsis
ANOTEONTHE“LOGARITHMIC-W ”OCTUPLETALGEBRAANDITSNICHOLSALGEBRA 3
3
r t s r t s
(2.2) F 1F 1F 1 F 2F 2F 2
1 3 2 1 3 2
p qp q“
min s ,r
1 2
p qqt1pr2´iq`t2ps1´iq´s1r2`ip1`iq{2xiy! si1 ri2 F1r1`r2´iF3t1`t2`iF2s1`s2´i.
i 0 B FB F
ÿ“
Comultiplication is by “deshuffling,” determined by the defining property of a braided
Hopfalgebraand thefact that F and F are primitive.
1 2
2.2. B X as a subalgebra in T X . For any Nichols algebra B X , the graded ideal I
p q p q p q
suchthatB X T X Iisknowntobethekernelofthetotalbraidedsymmetrizermap
p q“ p q{
in each grade, S : X n X n. Mapping by S in each grade therefore results in an
n b b n
Ñ
equivalentdescriptionof B X withmultiplicationgivenbytheshuffleproduct
p q
: x ,...,x y ,...,y X x ,...,x ,y ,...,y ,
1 m 1 n m,n 1 m 1 n
˚ p qbp qÞÑ p q
and comultiplication by deconcatenation (see [1] for the definition of shuffles and the
braided symmetrizer; the only notational difference is that is not used for the shuffle
˚
productthere).
We let B r,t,s be the image of FrFtFs under the map by the braided symmetrizer, or
1 3 2
p q
moreprecisely,
1
Bpr,t,sq“ r ! s ! t ! 1 q2 tSr`2t`spF1rF3tF2sq.
x y x y x y p ´ q
In particular,
B 1,0,0 F1, B 2,0,0 F1F1,
p q“ p q“
B 0,0,1 F2, B 1,0,1 F1F2 q´1F2F1,
p q“ p q“ `
B 0,0,2 F2F2,
p q“
B 0,1,0 q´2F2F1.
p q“´
2.2.1. Theshuffleproduct of B r1,t1,s1 and B r2,t2,s2 followsfrom (2.2):
p q p q
(2.3) B r1,t1,s1 B r2,t2,s2
p q˚ p q“
min s ,r
p 1 2q r1`r2´i s1`s2´i p1´q2qixt1`t2`iy! qt1pr2´iq`t2ps1´iq´s1r2`ipi`1q{2
r s t ! t ! i !
i 0 B 1 FB 2 F x1y x2y x y
ÿ“
B r1 r2 i,t1 t2 i,s1 s2 i .
ˆ p ` ´ ` ` ` ´ q
and thecoproduct is
r s t k
(2.4) D :B r,t,s 1 iq´ipi`3q{2`pk´m´2iqj`mpt´i´kq
p qÞÑ p´ q
j 0m 0k 0i 0
ÿ“ ÿ“ ÿ“ ÿ“
i j i m
` ` i !B r j,k i,i m B j i,t k,s m ,
ˆ i i x y p ´ ´ ` qb p ` ´ ´ q
B FB F
4 SEMIKHATOV
whereterms withthelowestgrades inthefirst tensorfactorare
1 B r,t,s F1 B r 1,t,s
“ b p q` b p ´ q
qt´rF2 B r,t,s 1 q´r´2 r 1 F2 B r 1,t 1,s
` b p ´ q´ x ` y b p ` ´ q
...
`
(thedotsstandforterms B r1,t1,s1 B r2,t2,s2 withr1 2t1 s1 2).
p qb p q ` ` ě
2.2.2. Remark. Although this is obvious, we note explicitly that the “Serre relations”—
thedoubleq-commutatorsintheideal—areresolvedintermsoftheshuffleproductinthe
sensethattherelations
F F F q q 1 F F F F F F 0,
1 1 2 ´ 1 2 1 2 1 1
˚ ˚ ´p ` q ˚ ˚ ` ˚ ˚ “
F F F q q 1 F F F F F F 0
2 2 1 ´ 2 1 2 1 2 2
˚ ˚ ´p ` q ˚ ˚ ` ˚ ˚ “
holdidenticallyfortheshuffleproductdefined by thebraidingmatrix(1.1).
2.2.3. TheactionoftheantipodeonthePBWbasiselementsisdefinedbytheformulas
S B r,0,0 1 rqrpr´1qB r,0,0 ,
p p qq“p´ q p q
t
S B 0,t,0 1 tq21ipi´1q´pi`3qt`t2 i !B i,t i,i ,
p p qq“ p´ q x y p ´ q
i 0
ÿ“
S B 0,0,s 1 sqsps´1qB 0,0,s
p p qq“p´ q p q
and bythefact that S isa braidedantiautomorphism:
S B r,t,s qrt´rs`tsS B 0,0,s S B 0,t,0 S B r,0,0 .
p p qq“ p p qq˚ p p qq˚ p p qq
2.3. VertexoperatorsandYetter–DrinfeldB X modules. MultivertexB X module
p q p q
comodules, which are Yetter–Drinfeld modules, were defined in [1]. We here realize
simpleYetter–Drinfeldmodulesofour B X in termsofone-vertexmodules.
p q
2.3.1. TheY spaces. LetY n1,n2 be a one-dimensional vector space with basisV n1,n2
t u t u
and braiding y : B X Y n1,n2 Y n1,n2 B X and Y n1,n2 B X B X
t u t u t u
p qb Ñ b p q b p q Ñ p qb
Y n1,n2 defined by
t u
y F V n1,n2 q1 niV n1,n2 F,
i t u ´ t u i
p b q“ b
y V n1,n2 F q1 niF V n1,n2 ,
t u i ´ i t u
p b q“ b
i 1,2. Every space B X Vtn11,n12u B X Vtn21,n22u ... VtnN1,nN2u is a Yetter–
“ p qb b p qb b b
j
Drinfeld B X module. Taking the a to be generic leads to continuum families of such
i
p q
modules,leavinguswithnochanceofanicecorrespondencewithanytypeof“reasonably
j
rational”CFTmodel. Thechoiceofthepossiblea valuesisgovernedbytherequirement
i
that all of them (and the braided vector space X itself) be objects of a suitable HYD
H
ANOTEONTHE“LOGARITHMIC-W ”OCTUPLETALGEBRAANDITSNICHOLSALGEBRA 5
3
category of Yetter–Drinfeld modules over a nonbraided Hopf algebra H. In the case of
diagonal braiding, more specifically, H kG for an Abelian group G , which can then be
“
j
considered the origin of the appropriate discreteness in the a values. We do not pursue
i
j
thislinein thispaper, andsimplyassumethatthe a takeintegervalues.
i
Weconsiderone-vertexmodules B X V n1,n2 and forbrevitywrite
t u
p qb
B r,t,s tn1,n2u B r,t,s Vtn1,n2u B X Ytn1,n2u,
p q “ p qb P p qb
and, inparticular,
Fitn1,n2u Fi Vtn1,n2u B X Ytn1,n2u
“ b P p qb
(but B 0,0,0 tn1,n2u 1 Vtn1,n2u isnormallywrittenasVtn1,n2u).
p q “ b
2.3.2. Left adjoint action. The formulas for the product, coproduct, and antipode in
2.2.1–2.2.3allowcalculatingtheleftadjointactionoftheB X generatorsonone-vertex
p q
modules:
F1§B r,t,s tn1,n2u r 1 1 q2pr´s`t`1´n1q B r 1,t,s tn1,n2u
p q “x ` yp ´ q p ` q
q2r´2s`t´2n1`3 t 1 1 q2 B r,t 1,s 1 tn1,n2u
´ x ` yp ´ q p ` ´ q
and
F2§B r,t,s tn1,n2u q1´r t 1 1 q2 B r 1,t 1,s tn1,n2u
p q “ x ` yp ´ q p ´ ` q
qt´r s 1 1 q2ps`1´n2q B r,t,s 1 tn1,n2u.
` x ` yp ´ q p ` q
Theseformulasdependonn andn onlythrough a modp . TheB X coactionisgiven
1 2 i
p q p q
byliterallyapplyingformula(2.4)toB r,t,s Vtn1,n2u(andisentirelyindependentofai).
p qb
2.3.3. SimpleYetter–Drinfeldmodules. AsimpleYetter–DrinfeldB X -moduleY
n ,n
p q 1 2
isgenerated fromV n1,n2 undertheaction ofB X ;itsdimensionis givenby
t u
p q
d n ,n , n n p,
1 2 1 2
d p,n ,n p q ` ď
1 2
p q“#d n1,n2 d p n1,p n2 , n1 n2 p 1,
p q´ p ´ ´ q ` ě `
where d n ,n 1n n n n and
p 1 2q“ 2 1 2p 1` 2q
p, xmod p 0,
x p q“
“#xmod p, otherwise.
3. THE OCTUPLET ALGEBRA CENTRALIZING B X
p q
WenextdiscussaCFT constructionrelated toour B X .
p q
6 SEMIKHATOV
3.1. Screenings andtheirzero-momentum centralizer. WeidentifytheB X genera-
p q
torswithtwo screenings
j j
(3.1) Fa F1 e a , Fb F2 e b ,
“ “ “ “
¿ ¿
where j a z and j b z are two scalarfields whose OPEs are defined in accordance with
p q p q
thebraidingmatrixas follows:
2 1
j a z j a w log z w , j a z j b w log z w ,
p q p q“ p p ´ q p q p q“´p p ´ q
2
j b z j b w log z w .
p q p q“ p p ´ q
It follows from the formulas in [2] that the centralizer (“kernel”) of screenings (3.1)
containsaVirasoro algebrawiththecentral charge
24 2 3p 4 4p 3
(3.2) c 50 24p p ´ qp ´ q.
“ ´ p ´ “´ p
ThisVirasoro algebraisrepresented by theenergy–momentumtensor
p p p
T z j a j a z j a j b z j b j b z p 1 2j a z p 1 2j b z .
p q“ 3B B p q` 3B B p q` 3B B p q´p ´ qB p q´p ´ qB p q
In additiontotheVirasoro algebra,thekernelofthescreenings containsthedimension-3
Virasoro primaryfield (omittingtheconventional z argumentsoffields)
p q
3 3
(3.3) W z j a j a j a j a j a j b j a j b j b j b j b j b
p q“B B B `2B B B ´2B B B ´B B B
9 p 1 9 p 1 9 p 1 9 p 1
2j j 2j j 2j j 2j j
p ´ q a a p ´ q a b p ´ q b a p ´ q b b
´ 2p B B ´ 4p B B ` 4p B B ` 2p B B
9 p 1 2 9 p 1 2
p ´ q 3j a p ´ q 3j b .
` 4p2 B ´ 4p2 B
Theoperatorproductofthisfield withitselfisgivenby
81 3p 5 3p 4 4p 3 5p 3 1 243 3p 5 5p 3 T w
W z W w p ´ qp ´ qp ´ qp ´ q p ´ qp ´ q p q
p q p q“ 4p5 z w 6 ´4p4 z w 4
p ´ q p ´ q
243 3p 5 5p 3 T w 243 8pTT w 9 p 1 2 2T w
p ´ qp ´ qB p q p q´ p ´ q B p q
´8p4 z w 3 `16p4 z w 2
p ´ q p ´ q
243 4p T T w p 1 2 3T w
pB q p q´p ´ q B p q,
`8p4 z w
´
where TT w is the normal-ordered product T w T w (and similarly for T T w ).
p q p q p q pB q p q
ThisOPE defines theW algebra[19] (alsosee[31]).
3
In an equivalent description, the W algebra relations for the modes introduced as
3
T z L z n 2 andW z W z n 3 are
n Z n ´ ´ n Z n ´ ´
p q“ p q“
P P
1 24
L ,Lř m n L 50 ř 24p m 1 m m 1 d ,
m n m n m n,0
r s“p ´ q ` `12p ´ p ´ qp ´ q p ` q `
L ,W 2m n W ,
m n m n
r s“p ´ q `
81 3p 5 5p 3 m n 3 m n 2 m 2 n 2
rWm,Wns“´ p ´8pq4p ´ qpm´nq p ` ` q5p ` ` q´p ` q2p ` q Lm`n
´ ¯
ANOTEONTHE“LOGARITHMIC-W ”OCTUPLETALGEBRAANDITSNICHOLSALGEBRA 7
3
243 27 3p 5 3p 4 4p 3 5p 3
`4p3pm´nqL m`n` p ´ qp 1´60qpp5 ´ qp ´ qmpm2´1qpm2´4qd m`n,0,
where
3
L L L L L m 3 m 2 L .
m n m n m n n m
“ ´ ` ´ ´10p ` qp ` q
n 2 n 1
ďÿ´ ěÿ´
3.2. Long screenings. TheW algebraisalso centralized bytwo“long”screenings
3
(3.4) Ea e´pj a and Eb e´pj b .
“ “
¿ ¿
Because
F,E 0,
i j
r s“
the long screenings act on the kernel of the Fa and Fb , and are therefore a useful tool in
studyingthat kernel.
j a j
3.3. Remark. We note that, generally, given the screenings F e i e i , i
i ¨
1,...,q ,theVirasoro dimensionofa vertex em .j z with m q c“a is “ “
p q “ i 1 i űi ű
“
q q
a .a 1 ř
D c c 1 i i cc a .a .
i i j i j
p q“ ´ 2 `2
i 1 i,j 1
ÿ“ ` ˘ ÿ“
We list the generators of the ideal in (2.1) togetherwith the vertex operators that naively
(by momentumcounting)correspondtothem,and withtheVirasoro dimensionsofthese
vertices:
F , F ,F , F , F ,F , Fp, F ,F p, Fp,
r 1 r 1 2ss r 2 r 2 1ss 1 r 1 2s 2
(3.5) e2j a z j b z , ej a z 2j b z , epj a z , epj a z pj b z , epj b z ,
p q` p q p q` p q p q p q` p q p q
3, 3, 2p 1, 3p 2, 2p 1.
´ ´ ´
3.4. The octuplet algebra. Thefield
W z epj a z pj b z ,
p q` p q
p q“
whichisthetop-dimensionfieldin(3.5),isinthekernelofFa andFb andisaW3-primary
field of dimensionD 3p 2 and theW eigenvaluezero. To describe how it is mapped
0
“ ´
by thelongscreenings, weneed areminderon W singularvectors.
3
3.4.1. Singular vectors in W Verma modules. We recall from [32] (also see [31] and
3
thereferences therein) that highest-weightvectors of the W algebra can be conveniently
3
parameterized by x,y suchthat
p q
L x,y 0, m 1,
m
“ ě
Wmˇx,yD 0, m 1,
ˇ “ ě
L0ˇˇx,yD x2`y2`xy pp´1q2 x,y ,
“ 3 ´ p
´ ¯
ˇ D ˇ D
ˇ ˇ
8 SEMIKHATOV
1
W x,y x y 2x y x 2y x,y .
0
“ 2p3 2p ´ qp ` qp ` q
{
The two numbers x and yˇ areDdefined not uniquely but up toˇa WDeyl transformation; the
ˇ ˇ
Weyl group orbit of x,y also contains x,x y , x y, y , y, x y , x y,x ,
p q . p´ ` q p ` ´ q p ´ ´ q p´ ´ q
and y, x . We write V z x,y for any field state V z that satisfies the above
p´ ´ q p q “ { p q
conditions.
ˇ D
ˇ
In what follows, we use the conditions for the existence of singular vectors in Verma
modulesoftheW algebra[33,34,32]. Wheneverastatecanberepresentedas x,y with
3
c
x a?p forintegera and c such that ac 0, thereis asingularvectoron thelevel
“ ´?p ą ˇ D
ˇ
acbuiltonthatstate. Thesingularvectorhasthehighest-weightparameters x ,y x
1 1
p q“p ´
d
2a?p,y a?p . Similarly, if y b?p with bd 0, then a singularvectoroccurs
` q “ ´ ?p ą
on thelevelbd andhas thehighest-weightparameters x ,y x b?p,y 2b?p .
2 2
p q“p ` ´ q
3.4.2. It followsthat
W z epj a z pj b z . 2?p 1 ,2?p 1 ,
p q` p q
p q“ “ ´?p ´?p
andhencethecorrespondingVerma-modulesˇtatehastwosingularveDctorsatlevel2. Both
ˇ
of them vanish in our free-field realization. Of the two fields Ea W z and Eb W z , we
p q p q
concentrateon thesecond; itlandsinthemodulegenerated from
epj a z . 3?p 1 , 1 .
p q
“ ´?p ´?p
Thecorrespondinghighest-weightstateiˇntheVermamodDulehassingularvectorsatlevels
ˇ
3 and p 1. The first of thesevanishesin the free-boson realization, but thesecond does
´
not,yieldingjustthefield W z E W z ,as weshowinFig. 1. Wenotethat
b b
p q“ p q
Wb z Prbp´1s j z epj a pzq
p q“ pB p qq
with a differential polynomials in j a z , j b z in front of the exponential; here and
B p q B p q
hereafter, weindicatethedegree d ofadifferentialpolynomialas P d .
r s
Totallysimilarly,
Wa z Ea W z Parp´1s j z epj b pzq
p q“ p q“ pB p qq
isa descendantof
epj b z . 1 ,3?p 1 .
p q
“ ´?p ´?p
ˇ D
The maps of Wa z by Eb and of Wb z ˇby Ea are differential polynomials (not in-
p q p q
volving exponentials). They are not descendants of the unit operator, however. We have
.
1 ?p 1 ,?p 1 , which implies singular vectors at levels 1, 1, 4, 2p 1, and
“ ´ ?p ´ ?p ´
2p 1. All of these vanish in the free-field realization. In each of the grades where
a le´vˇˇel- 2p 1 singularDvector vanishes, another state is produced as Ea epj a pzq and
p ´ q p q
e pj a pj b
´ ´ ˝
A
N
O
T
E
O
1 N
˝ p 1 T
´ H
e pj b E
´ ˝ Ea ‚ ‚ “L
Eb OG
2 A
R
I
T
4 ˆˆˆˆˆˆ ˆˆˆ ˆˆˆ H
M
I
C
ˆˆˆ Eb 2p 2 -W
2p 1 2p 2 ´ 3”
´ Eb ´ O
epj a CT
˝ Ea ˝ˆˆˆ ˝ˆˆˆ Eb ˝ˆˆˆ ˝ˆˆˆ Ea ˝ˆˆˆ ˝ˆˆˆ UP
Eb Ea Eb LE
3 T
A
p 1
L
ˆˆˆ ´ G
p´1 p´1 p´1 p´1 EBR
A
W Eb Wb Wab Wba Wbab Waabb A
‚ ‚ Ea ‚ ‚ Eb ‚ ‚ N
D
Eb Ea I
Eb Eb TS
N
I
C
H
FIGURE 1. Maps by the long screenings Ea and Eb . Crosses and downward arrows leading tothem show W3 singular vectors that vanish in OL
S
the free-boson realization. Bullets (and downward arrows) show nonvanishing states in the same grades; the relative levels of singular vectors
A
are indicated at the arrows. An open circle superimposed with a cross shows a vanishing W singular vector and a (W -primary) state in the LG
3 3
E
same grade, but not in the same W -module (and downward arrows drawn from such show singular vectors built on those primary states). B
3 R
Twomoremodules—those with epj b ande´pj a atthetop—are not shownhere; theirs˝tˆˆˆructure repeats that ofthe“epj a ”and“e´pj b ”modules A
witha b . Dottedarrowsshowthemapsby Ea andEb fromthemissingmodules.
Ø 9
10 SEMIKHATOV
Eb epj b pzq . This is shown in Fig. 1 with the symbols (“a state superimposed with a
p q ˝ˆ
vanishing singular vector”). Next, Ea epj a pzq and Eb epj b pzq have singular-vector de-
p q p q
scendantsontherelativelevel p 1,whichareofcoursetherespectiveimagesof Wb z
´ p q
and Wa z underEa and Eb ,
p q
Wab z Ea Wb z Prab3p´2s j z , Wba z Eb Wa z Prba3p´2s j z .
p q“ p q“ pB p qq p q“ p q“ pB p qq
Furthermapsbythelongscreeningsdonotproduce W -descendantsofthecorrespond-
3
ingexponentialseither. WeconsiderE W z andE W z . Inthemoduleassociated
b ab b ba
p q p q
with
e pj b z . 2?p 1 , ?p 1 ,
´ p q
“ ´?p ´ ´?p
two singular vectors at level 2 and ˇtwo at level 2p 2 vaDnish; located at the grades
of the last two are Eb Ea epj a pzq (theˇ maps shown in´Fig. 1) and Eb Eb epj b pzq.1 Now,
Eb Ea epj a pzq and Eb Eb epj b pzq have a level- p 1 singular-vector descendant each. In
p ´ q
our free-field realization, these two singular vectors evaluate the same up to a nonzero
overallfactor, thusproducinga W -primaryfield
3
Wbab z Eb Wab z Pbrab3p´3s j z e´pj b pzq.
p q“ p q“ pB p qq
Everythingwiththereplacement a b applies tothefield
Ø
Waba z Ea Wba z Parba3p´3s j z e´pj a pzq.
p q“ p q“ pB p qq
Finally,mappingbythelongscreenings onceagaingivesafield
Waabb z Ea Wbab z Praabb4p´4s j z e´pj a pzq´pj b pzq
p q“ p q“ pB p qq
(which is also Eb Waba z up to a factor), which is not in the module associated with
p q
e pj a z pj b z ,however. IntheVermamoduleassociatedwiththehighest-weightvector
´ p q´ p q
e pj a z pj b z . 1 , 1 ,
´ p q´ p q
“ ´?p ´?p
there are two singular vectors at level p 1, bˇoth of whichDare nonvanishing in the free-
field realization and are in fact the image´s of eˇ pj b z (and e pj a z ; see Fig. 1). Each of
´ p q ´ p q
thesesingularvectorstherefore has two level- 2p 2 singularvectors,which are infact
p ´ q
the same pair of singular vectors. These two next-generation singular vectors vanish in
1We illustrate the use of 3.4.1. In the Verma module with the highest-weightvector x,y 2?p
?1p,´?p´?1p associatedwithe´pj b pzq,oneofthelevel-p2p´2qsingularvectorsexistˇsduDe“toˇtherep´-
resentationy“Dp?´p1´2?p,andthereforethesingularvectorhasthehighest-weightparamˇeterspx2ˇ,y2q“
p´?1p,3?p´?1pq, i.e., those of epj b pzq. The otherlevel-p2p´2qsingularvectoris seen immediatelyif
weWeyl-reflectthehighest-weightparameterstopx˜,y˜q“p´x,x`yq. Wethenhavey˜“ 2p?p´p1q´?p,and
hencethesingularvectorhastheparameters 3?p 1 ,3?p 2 . AfterthesameWeylreflection,the
p´ `?p ´?pq
parametersp3?p´?1p,´?1pqcorrespondtoepj a pzq.