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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.

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

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

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.

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