FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 1. INTRODUCTION

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1 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA DAVID RIDOUT AND SIMON WOOD ABSTRACT. I ths ote, a deep coecto betwee free feld realsatos of coforal feld theores ad syetrc polyoals s preseted. We gve a bref troducto to the ecessary prerequstes of both free feld realsatos ad syetrc polyoals, partcular Jack syetrc polyoals. The we cobe these two felds to classfy the rreducble represetatos of the al odel vertex operator algebras as a lluatg exaple of the power of these ethods. Whle these results o the represetato theory of the al odels are all kow, ths ote explots the full power of Jack polyoals to preset sgfcat splfcatos of the orgal proofs the lterature. 1. INTRODUCTION Free feld theores have log bee of great terest to coforal feld theory. Not oly are they elegat tractable coforal feld theores ther ow rght, but they are also a versatle tool for realsg ore coplcated coforal feld theores ad akg the tractable. The purpose of ths ote s to preset, a sple ad falar settg, a deep coecto betwee free feld theores ad Jack syetrc polyoals. The syetrc polyoal ethods wll the be appled to the well kow free feld realsatos of the Vrasoro al odels. However, t s portat to stress that these ethods work far ore geerally. We have sply chose to dscuss applcatos to the al odels for pedagogcal purposes. A dfferet exaple, where Jack syetrc polyoals have recetly garered a lot of atteto, s the uch celebrated AGT cojecture [1], whch relates coforal feld theores to the stato calculus of Yag-Mlls theores. The appearace of syetrc polyoals s due to the coforal feld theores questo beg resolved by Coulob gas free feld theores, just as the coforal feld theores ths ote are. Cotrary to what was tally beleved, t does ot see that Jack syetrc polyoals for the ost atural bass for uderstadg the AGT cojecture. Rather, a geeralsato of Jack syetrc polyoals sees to be eeded [2]. The two a results dscussed ths ote are Theores 5 ad 6. Theore 5, whch s orgally due to Mach ad Yaada [3], gves elegat forulae for Vrasoro sgular vectors Fock odules, whle Theore 6, whch s orgally due to Wag [4], deteres the coforal hghest weghts of the rreducble represetatos of the al odel vertex operator algebras also called chral algebras. The orgal proofs, pressve though they are, are rather coplcated ad ths ote gves ovel, drastcally shorteed ad strealed proofs by usg syetrc polyoals, ther er products ad the specalsato ap. These ethods also have the advatage of beg applcable far greater geeralty, as s evdeced by the fact that they were developed [5] whle classfyg the rreducble represetatos of certa logarthc extesos of the al odels. Ths forals also geeralses to the ore volved case of adssble level ŝl2 theores [6]. We equate the al odels wth the sple vertex operator algebras obtaed by takg the quotet of the uversal Vrasoro vertex operator algebras by ther axal deals at specal values of the cetral charge. 1 The represetato theory of the al odels ca thus be obtaed fro that of the uversal Vrasoro vertex operator algebras. Mal odel represetatos are just the uversal Vrasoro vertex operator algebra represetatos that are ahlated by the axal deal. Ths elegat approach to classfyg the represetato theory of the al odels sees to have frst bee cosdered by Feg, Nakash ad Oogur [7] who appled t to a subset of the al odels, because, geeral, havg full coputatoal cotrol over the axal deal s a very hard proble. However, free feld realsatos ad syetrc polyoals are exactly the tools oe eeds to solve ths proble ad Theore 6 exteds the ethods of ahlatg deals [7] to all al odels. 1 Uversal eas that we assue o relatos o the defg feld T other tha those requred by the axos of vertex operator algebras. 1

2 2 D RIDOUT AND S WOOD Ths ote s orgased as follows. Secto 2 gves a overvew of the free boso, the splest exaple of a free feld theory, as well as vertex operators ad screeg operators. Secto 3 troduces syetrc polyoals ad, partcular, gves a overvew of a oe-paraeter faly of bases called the Jack syetrc polyoals. The propertes of these Jack polyoals are what yeld such explct coputatoal cotrol that the represetato theory of the al odel represetatos ca be classfed. Secto 3 eds wth explct forulae for sgular vectors ters of Jack polyoals. These forulae are orgally due to Mach ad Yaada [3], though we gve the ew, uch spler proof of [5]. I Secto 4, the ateral of Sectos 2 ad 3 s cobed to classfy the hghest weghts of the rreducble al odel represetatos, a slar aer to the ethods of Feg, Nakash ad Oogur [7]. Ths s the used to prove the coplete reducblty of the represetato theory wthout recourse to the perhaps less falar ethods of Zhu [8]. Ackowledgeets. The secod author would lke to thak Akhro Tsuchya for troducg h to the fascatg topcs of free feld realsatos ad syetrc polyoals. The secod author would also lke to thak Jaes Lepowsky ad Sddhartha Sah for terestg dscussos ad Johaes Schude for hs extraordary efforts lterature searchg. Both authors thak Perre Matheu for terestg dscussos ad hs strog ecourageet to wrte ths ote. The frst author s research s supported by the Australa Research Coucl Dscovery Project DP The secod author s research s supported by the Australa Research Coucl Dscovery Early Career Researcher Award DE THE FREE BOSON The free boso chral algebra or Heseberg vertex algebra s geerated by a sgle feld a whch satsfes the operator product expaso The Fourer expaso of the feld a s aaw 1 w a = a 1, 2.2 Z thus the operator product expaso ples the followg coutatos relatos: [a,a ] = δ, The Heseberg Le algebra H s the fte desoal Le algebra geerated by the a ad the cetral eleet 1. We detfy the eleet 1 wth the ut of the uversal evelopg algebra UH of H ad assue that 1 acts as the detty o ay H represetato. 2 The Heseberg Le algebra adts a tragular decoposto H = H H 0 H, H 0 = Ca 0 C1, H ± = 1 Ca ±, H = H 0 H. 2.4 The Vera odules F, C, wth respect to ths decoposto are called Fock odules. They are geerated by a hghest weght vector o whch H acts by a = δ,0, [Throughout ths ote, kets wll be reserved for the hghest weght vectors of Fock odules ad wll deote the hghest weght.] The F are the duced fro by F = UH UH C Ths s oly a or restrcto, sce a sple rescalg of the geerators a allows oe to have the cetral eleet act as ultplcato by ay o-ero uber.

3 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 3 The paraeter s called the Heseberg weght. As s well kow, the F are all rreducble. As a vector space, F = UH = C[a 1,a 2,...]. 2.7 As a represetato over tself, the Heseberg vertex algebra s detfed wth F 0 ad the operator state correspodece s gve by 0 1, a 10 a, a 1 1 a : a a : !! The Heseberg vertex algebra ca be edowed wth the structure of a vertex operator algebra by choosg a eergy-oetu tesor. Ths choce s ot uque; there s a oe paraeter faly of choces: T = 1 2 : α2 : α 0 2 a, α 0 C. 2.9 The paraeter α 0 deteres the cetral charge of the eergy oetu tesor: c = 1 3α The coeffcets of the Fourer expaso of the eergy oetu tesor are, by defto, the geerators L of the Vrasoro algebra. Forula 2.9 detfes the Vrasoro geerators wth fte sus of eleets of the uversal evelopg algebra UH of the Heseberg Le algebra: T = L 2 = 1 2 : a a : 2 α 0 2 1a 2, Z, Z Z L = 1 2 : a a : α 0 2 1a. Z 2.11 Ths detfcato gves a acto of the Vrasoro algebra o the Fock odules F. The Fock odules thus becoe Vrasoro hghest weght represetatos, that s, L = h δ,0, 0, h = 1 2 α Though the Fock odules are rreducble as Heseberg represetatos, they eed ot be so as Vrasoro represetatos. The Heseberg weghts for whch the coforal weght h s 1 play a specal role as we shall see below. These weghts are roots of the degree 2 polyoal h 1 ad we deote the by α,α. They satsfy the relatos Theore 1 Feg-Fuchs [9]. Let α ± = α 0 ± α0 2 8, 2 α α = α 0, α α = α r,s = 1 r 2 α 1 s 2 α, r,s Z For α 2 C or equvaletly for α 2 C, the Fock odule F s reducble as a Vrasoro represetato f = α r,s for soe r,s Z, rs > 0. 2 If α 2 s o-ratoal or equvaletly f α 2 s o-ratoal, the the Fock odule F s reducble as a Vrasoro represetato f ad oly f = α r,s for soe r,s Z, rs > 0. 3 If α 2 s postve ratoal or equvaletly f α 2 s postve ratoal, the the Fock odule F s reducble as a Vrasoro represetato f ad oly f = α r,s for soe r,s Z. We ot the correspodg result for egatve ratoal α 2 ± as the applcato to the al odels does ot requre t. We reark that Feg ad Fuchs also detered the precse structure of Fock odules as Vrasoro represetatos [9]. For a coprehesve accout of Vrasoro represetato theory, we recoed the book by Iohara ad Koga [10].

4 4 D RIDOUT AND S WOOD The work of Feg ad Fuchs shows that oe ca realse the uversal Vrasoro vertex operator algebra at arbtrary cetral charge c as a vertex operator subalgebra of the Heseberg vertex operator algebra. Ths free feld realsato s called the Coulob gas the physcs lterature. The Fock odules F wth 0 ca be gve a geeralsed vertex algebra structure, that s, a operator state correspodece ca also be defed for the states of F, though the operator product expasos of these felds are geerally ot local. The operators correspodg to the geeratg states F are called vertex operators the physcs lterature. These should ot be cofused wth the felds called chral felds of the vertex operator algebra. Before we ca defe vertex operators, we eed to troduce a operator â whose coutato relatos wth the Heseberg algebra are The expoetal of â shfts weghts, that s, for,µ C, [a,â] = δ,0 1, [1,â] = a 0 e µâ = µe µâ e µâ a 0 = µ e µâ We detfy e µâ = µ. Note that e µâ does ot defe a hooorphs of H represetatos, sce t does ot coute wth H. The vertex operator V correspodg to the state s V = e â a 0 exp a The vertex operators are therefore lear aps The V are ofte defed as the orally ordered expoetals of a feld exp a V : F µ F µ [[, 1 ]] µ a φ = â a o log, V =: e φ : Clearly φ = a, whch tur ples that the operator product expasos of φ wth tself ad wth a are aφw 1, φφw log w w Usg these operator product expasos, oe ca verfy that the vertex operators V are coforal prares of coforal weght h, that s, that I partcular, for h α± = 1, T V α± w T V w h w 2V w 1 w V w w 2V α ± w 1 w V V α± w α ± w = w w, 2.22 that s, the sgular ters of these operator product expasos costtute total dervatves. Vertex operators wth coforal weght 1 are called screeg operators ad were troduced by Dotseko ad Fateev [11]. The coforal weght beg 1 ples that the resdue Q ± = 1 2π V α± wdw 2.23 s a Vrasoro hooorphs, because of I other words, [T,Q ± ] = 1 T V α± wdw = π

5 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 5 Ths resdue s, of course, oly well defed whe the expoet of α ±a 0 s a teger. Thus, for α ± µ Z, the resdue of the vertex operator V α± defes a Vrasoro hooorphs Q ± : F µ F µα± A atural questo that oe ca ask ths cotext s whether these resdues ca be geeralsed to obta ore Vrasoro hooorphss. The aswer s yes, at least for sutable Heseberg weghts µ. The soluto to geeralsg these Vrasoro hooorphss les coposg screeg operators. The coposto of vertex operators V µ, = 1,...,, s gve by V µ1 1 V µ = eâ µ j µ µ j 1 < j µ a 0 exp a µ exp a µ Ths forula s derved by usg the operator product expasos above or by usg the coutato relatos of the Heseberg Le algebra. If we set µ = α ±, = 1,...,, the the above forula splfes to V α± 1 V α± = e α ±â 1 < j j α2 ± α ±a 0 exp α ± a a exp α ± We take the opportuty to troduce a faly of syetrc polyoals, called power sus, to splfy otato: p =, p = Up to a phase factor, whch we suppress, the secod factor of 2.26 ca be rewrtte as j κ ±, κ ± = α2 ± j If we evaluate the product of these screeg operators o a Fock odule F µ, the the a 0 geerator acts by ultplcato wth µ ad therefore V α± 1 V α± Let Fµ = e α ±â 1 j j κ ± c κ ± = α ±µ 2π 1! 1 j=1 a exp α ± p a exp α ± p Γ1 j 1κ ± Γ jκ ± Γκ ± 1 Theore 2 Tsuchya-Kae [12]. If dd 1κ / Z ad d dκ / Z, for all tegers d satsfyg 1 d 1, the for each Heseberg weght α,k, k Z, there exsts a cycle such that Q [] = 1 V α 1 V α d 1 d 2.32 c κ s a o-trval Vrasoro hooorphs Q [] : F α,k F α,k Lkewse, f dd 1κ / Z ad d dκ / Z, for all tegers d satsfyg 1 d 1, the for each Heseberg weght α k,,k Z, there exsts a cycle such that Q [] = 1 V α 1 V α d 1 d 2.34 c κ

6 6 D RIDOUT AND S WOOD s a o-trval Vrasoro hooorphs I partcular, 1 f k 1, the there exst vectors v F α,k ad w F αk, such that Q [] : F αk, F αk, Q [] v = α,k Fα,k, Q [] w = αk, Fαk,, 2.36 whle α,k ad αk, are ahlated by Q [] ad Q [], respectvely, 2 f k 0, the Q [] α,k 0 Fα,k, Q [] αk, 0 Fαk, The explct costructo of the cycles s rather subtle ad we refer to [12] for the detals. Itutvely, ca be thought of as cocetrc crcles about 0 that are pched together at 1. Let us try ad uderstad the plcatos of Theore 2 a lttle better. Sce α,k s a Vrasoro hghest weght vector, the so s Q [] [] α,k by vrtue of Q beg a Vrasoro hooorphs. Thus, wheever k 0, the vectors Q [] [] α,k ad Q αk, geerate Vrasoro subrepresetatos Fα,k ad F αk,. Such Vrasoro hghest weght vectors are called sgular vectors. For µ = α,k, µ = α k,, 1 j j κ ± α ±µ ± = 1 1 j j κ± k 1, 2.38 where we have used the defg forulae 2.14 for α,k,α k, ad 2.29 for κ ±. For later use we defe G ;κ± 1 = 1 κ± j j Ths seegly odd choce of κ 1 ± the defto of G s to ake our otato Secto 3 cofor wth the stadard covetos the syetrc polyoals lterature [13]. The costat c κ ± 2.31 oralses the cycles 2.32 ad 2.34 such that 1 c κ ± G : κ 1 ± d 1 d j 1 j = Heceforth, we wll therefore deote by [ ] the hoology class of the cycle that has bee rescaled by c κ ± 1, the κ ± -depedece beg left plct, that s, [ ] = c κ ± The codtos o κ ± at the begg of Theore 2 esure that c κ ± 0. These codtos are et for the applcatos to the al odels ths ote. For a systeatc dscusso of how to regularse [ ] whe these codtos are ot et, see [5, Sectos ]. By expadg the forulae for the Vrasoro hooorphss Q [] ± o F α,k ad F αk,, oe sees that Q [] ± = e α ±â [ ] G ;κ 1 ± k a exp α ± p exp a α ± p d 1 d Apart fro the ultvalued fucto G, the tegrad cossts of a fte su of ooals UH UH where the coeffcets are products of polyoals ether postve or egatve powers of the. It turs out that for syetrc polyoals f,g, the parg f,g κ 1 ± = G ;κ± 1 f 1,..., g 1,..., d 1 d, [ ]

7 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 7 where g 1,..., = g 1 1,..., 1, defes a o-degeerate syetrc blear for o the rg of syetrc polyoals varables. Evaluatg the Vrasoro hooorphs Q [] ± 2.42 therefore reduces to evaluatg er products of syetrc polyoals. As we shall see, ths s a well kow proble wth a elegat soluto. 3. SYMMETRIC POLYNOMIALS For a coprehesve study of syetrc polyoals, we recoed the book by Macdoald [13]. Let Λ be the rg of syetrc polyoals wth coplex coeffcets. As a coutatve rg, Λ s geerated by a uber of terestg sets of polyoals cludg the eleetary syetrc polyoals e = 1 j 1 < < j j1 j, = 1,..., 3.1 ad the power sus p = j=1 j, = 1,...,. 3.2 These polyoals are algebracally depedet ad geerate Λ freely, that s, Λ = C[e 1,...,e ] = C[p 1,...,p ]. 3.3 The rg Λ s clearly also a coplex vector space ad t s atural to look for coveet bases. Oe such bass s costructed fro the power sus. Let = 1,..., k, k 0, be a partto of a teger wth largest part 1 we follow the coveto of lstg the parts weakly descedg order. The, for all such, the p = p1 pk are learly depedet ad for a bass of Λ. Aother coveet bass of Λ s gve by the ooal syetrc polyoals. Let µ = µ 1,..., µ be a partto of legth lµ at ost f the partto s shorter tha pad t wth 0s at the ed utl t s legth. The, the ooal syetrc polyoals are defed as 3.4 µ = τ 1 1 τ, τ {all dstct perutatos of µ}. 3.5 τ We shall refer to these polyoals as the syetrc ooals for brevty. As we ca see fro the power sus ad the syetrc ooals, the set of parttos that label bass eleets ust be trucated oce the weght = s greater tha the uber of varables. Specfcally, there exst parttos wth 1 >, whch are ot allowed for the power sus, or l >, whch are ot allowed for the syetrc ooals. Ths s why t s coveet to work the lt of ftely ay varables: Oe ca the easly recover the fte varable case by the projecto Λ = l Λ. 3.6 γ : Λ Λ 3.7 j 1 j j 0 j > that sets to 0 all but the frst varables. The power sus ftely ay varables ow geerate Λ as ther fte-varable versos dd Λ. We cotue to use 3.4 to defe p the fte-varable case. Λ = C[p 1,p 2,p 3,...]. 3.8

8 8 D RIDOUT AND S WOOD The power su ad syetrc ooal bases of Λ are ow labelled by all parttos of tegers wthout restrctos o parts or legths: Λ = Cp = C. 3.9 The projecto to the fte varable case s partcularly easy the syetrc ooal bass: γ : Λ Λ 3.10 µ 1,..., lµ µ. 0 else Recall that, by 2.7, the uversal evelopg algebra UH s also a polyoal algebra ftely ay geerators. Idetfyg these two algebras wll be portat below. Proposto 3. For f,g Λ ad κ C such that d/κ / Z for all tegers satsfyg 1 d, the blear for f,g κ = G ;κ f 1,..., g 1,..., d 1 d s 1 syetrc, 2 o-degeerate, [ ] 3 graded: f,g κ = 0 f deg f degg. Proposto 3 leads us to the bass of Λ that s ost portat for our purposes, the Jack polyoals P κ. These polyoals are charactersed by two propertes [13]: 1 The Jack polyoals have upper tragular expasos the bass of syetrc ooals wth respect to the doace orderg of parttos 3, that s, where the u,µ κ C ad u, κ = 1. P κ = u,µ κ µ, 3.12 µ 2 The Jack polyoals are utually orthogoal: P κ,p κ κ µ = 0, f µ Sce the doace orderg of parttos s oly a partal orderg, tryg to detere the Jack polyoals by eas of Gra-Schdt orthogoalsato s a overdetered proble. Showg that they exst s therefore o-trval, see [13]. We prepare soe otato regardg parttos. For a partto, let s =, j be a box the Youg tableau of, so that = 1,...,l ad j = 1,...,. The, the ar legth, coar legth, leg legth ad coleg legth are defed to be as = j, a s = j 1, ls = j, l s = 1, 3.14 respectvely, where s the cojugate partto of, that s, the partto for whch the colus ad rows of the Youg tableau have bee exchaged. Proposto 4. 1 Jack polyoals exst for all ad the fte varable lt. 3 The doace orderg s a partal orderg of parttos of equal weght defed by k k µ µ, k 1.

9 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 9 2 The Jack polyoals satsfy the sae projecto forulae as the syetrc ooals: γ P κ = P κ 1,..., l else 3 For ether a fte or fte uber of varables,y j, 1 y j 1/κ 1 p p y = exp κ, j 1 asκ ls 1 b κ = as 1κ ls. s 4 The or of the utually orthogoal Jack polyoals s P κ,p κ κ = s as 1κ ls asκ ls 1 = b κp κ P κ y, 3.16 a sκ l s a s 1κ l s For X C, let Ξ X : Λ C be the algebra hooorphs defed, the power su bass, by The ap Ξ X s called the specalsato ap. The, ad 1 1 X/κ = exp Ξ X p y = X l Ξ X P κ X a y = sκ l s asκ ls 1 s X κ p 3.19 = b κp κ ΞX P κ y We stress that whle ths hooorphs apples to syetrc polyoals ay varables, we wll oly be applyg t to those the y varables. 6 Let =,..., be the partto cosstg of copes of. The, γ P κ = P κ 1,..., = 1,..., = See [13] for proofs. Ared wth ths kowledge of Jack polyoals, we ca ow explctly evaluate the acto of screeg operators o Fock odules. Recall fro equatos 2.7 ad 3.8 that both UH ad Λ are polyoal algebras a fte uber of varables, C[a 1,a 2,...] = UH = Λ = C[p 1,p 2,...], 3.22 ad are therefore soorphc. For δ C, we defe the algebra soorphs ρ δ : Λ UH, 3.23 p y δa. As wth the specalsato ap, we wll oly be applyg δ to polyoals the y varables. Theore 5. For k 0, let k = k,...,k be the partto cosstg of copes of k. The, the Vrasoro hooorphss Q [] : F α, k F α, k, Q [] : F α k, F α k, 3.24

10 10 D RIDOUT AND S WOOD ap the vectors α, k ad α k, to the o-ero sgular vectors Q [] α, k = bk κ 1 ρ 2 α k y α, k, Q [] α k, = bk κ 1 ρ 2 α P κ 1 k y α k, Proof. We prove the forula for Q. The oe for Q follows slarly. The proof follows by drect evaluato usg the theory of Jack polyoals: Q [] α, k = G ;κ 1 [ ] 1 = = 2 = µ k exp p k, exp α k,ρ 2 α b µ κ 1 3 = b k κ 1 4 = b k κ 1 ρ 2 α k k exp,p κ 1 µ,p κ 1 k α p a κ 1 κ p p y κ 1 κ 1 Here we have used te 6 of Proposto 4 for 1 = to detfy ρ 2 α ρ 2 α a d 1 d α, k 1 α, k κ 1 k α, k y α, k y α, k k y α, k k = k ; 3.27 te 3 of Proposto 4 for 2 = reeberg that the tegrato the er product s over the varables; the orthogoalty of Jack polyoals for 3 =; ad te 6 of Proposto 4 to see that k P κ 1 k = 1 k κ 1 κ 1,P k = 1, 3.28 whch justfes 4 =. By drect evaluato of forula for b k κ 1 te 3 of Proposto 4, oe sees that b k κ 1 s a product of quotets of postve ratoal ubers ad s therefore o-ero. Ths theore s orgally due to Mach ad Yaada [3], though the uch spler ad strealed proof that we have preseted here frst appeared [5, Proposto 3.24]. 4. THE MINIMAL MODELS AND THEIR REPRESENTATIONS I Secto 2, we costructed the uversal Vrasoro vertex operator algebra at cetral charge c = 1 3α0 2, α 0 C, 4.1 as a vertex operator subalgebra of the Heseberg vertex operator algebra. At geerc values of the cetral charge c or equvaletly, at geerc values of α 0, the uversal Vrasoro vertex operator algebra s sple ad cotas o o-trval deals. However, there s a dscrete set of cetral charges at whch the uversal Vrasoro vertex operator algebra s ot sple. The al odel vertex operator algebras are the sple vertex operator algebras obtaed, for these cetral charges, by takg the quotets of the uversal vertex operator algebras by ther axal deals.

11 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 11 Thus, the al odel vertex operator algebras ca be realsed as subquotets of Heseberg vertex operator algebras. The al odel cetral charges, that s, the cetral charges at whch the uversal Vrasoro vertex operator algebras are o-sple, are precsely c p,p = c p,p = 1 6 p p 2 p p, 4.2 where p, p 2 are copre tegers [9]. We deote the al odel vertex operator algebra of cetral charge c p,p by M p, p. To obta these al odel cetral charges for the Heseberg algebra, we set 2p 2p α 0 = α α, α =, α =. 4.3 p p The paraeters α ± are precsely the Heseberg weghts whch, by forula 2.12, correspod to coforal weght 1. The κ ± paraeters troduced Secto 2 are thus, κ = α2 2 = p p, κ = α2 2 = p p. 4.4 The deal I p, p of the uversal Vrasoro vertex operator algebra of cetral charge cp,p s geerated by a sgular vector of coforal weght p 1p 1 [9]. By usg the screeg operator forals of Secto 2, partcular Theore 5, we ca realse ths sgular vector usg the screeg operator Q = V α or Q = V α. Wrtg Q [] ± = V α± 1 V α± d 1 d, 4.5 [ ] we deduce that the sgular vector F 0 whch geerates the deal of the uversal Vrasoro vertex operator algebra, sttg sde the Heseberg vertex operator algebra, s gve by Q [p 1] 1 p α = bp 1 p 1 κ 1 ρ 2 α Q [p 1] 1 p α = bp 1 p 1 κ 1 ρ 2 α p 1 p 1 P κ 1 p 1 p 1 y 0, y 0. The above equatos are obtaed drectly fro Theore 5, the frst by choosg = p 1, k = 1 p ad the secod by choosg = p 1, k = 1 p. For a gve coforal weght, the Vrasoro sgular vectors of a Fock odule are uque up to rescalg [14], f they exst, so the two vectors 4.6 are proportoal to each other. As a fal deostrato of the power of cobg the screeg operator ad syetrc polyoal foralss, we wll classfy the represetatos of the al odel vertex operator algebras. Sce the uversal Vrasoro vertex operator algebras are subalgebras of the Heseberg vertex operator algebras, the Fock odules F µ are represetatos of the uversal Vrasoro vertex operator algebras for ay µ C. However, the Vrasoro represetato geerated fro µ ca oly be a represetato of M p, p f each feld correspodg to a vector the deal I p, p acts trvally. Moreover, ay rreducble hghest weght represetato of M p, p ust be realsable as a subquotet of a Fock odule as, for ay coforal weght, there exsts a Fock odule whose geeratg vector has that coforal weght, by Theore 6. Let h r,s = rp sp 2 p p 2 4p p. 4.7 Up to soorphs, there are exactly 1 2 p 1p 1 equvalet rreducble M p, p represetatos. They are gve by the rreducble represetatos of the Vrasoro algebra geerated by hghest weght vectors of

12 12 D RIDOUT AND S WOOD coforal weght h r,s, 1 r p 1, 1 s p 1, rp sp p p. 4.8 Proof. We oly prove that the above lst of rreducble represetatos of the Vrasoro algebra s a upper boud o the set of equvalet rreducble M p, p represetatos. I order to show that the lst s saturated, oe ca the ether costruct all these represetatos by, for exaple, the coset costructo [15, 16], by quatu haltoa reducto [17], or use Zhu s algebra [4, 8], ths beg the assocatve algebra of ero odes of the felds of the vertex operator algebra actg o hghest weght vectors. Cosder the sgular vector [p χ = Q 1] 1 p α F0 4.9 of equato 4.6. The correspodg feld s obtaed by tegratg p 1 vertex operators V α over [ p 1] about the vertex operator V 1 p α, wth [ p 1] cetred about the arguet of V 1 p α : χw = Q 1 w Q p 1 wv 1 p α wd 1 d p [ p 1] The vector χ s a eleet of the deal I p, p, so the feld χw ust therefore act trvally o ay M p, p represetato. Cosequetly, µ χw µ = 0, 4.11 where µ s the hghest weght vector of F µ ad µ ts dual whch satsfes µ µ = 1, µ a = µ δ,0 µ, We ca evaluate µ χw µ usg the theory of Jack polyoals. Applyg forula 2.26 to splfy the coposto of vertex operators the defto of χw, we see that Q 1 w Q p 1 wv 1 p α w = G p 1;κ 1 p 1 a exp α exp α a w α a0 w 1 p α a 0 p 1 w,..., p 1 w 1 p w p 1 w,..., p 1 w 1 p w The expoetals of Heseberg geerators a, 0, 4.13 ahlate µ ad µ. Thus, µ χw µ = µ Q 1 w Q p 1 wv 1 p α w µ d1 d p 1 [ p 1] = G p 1;κ 1 p 1 [ p 1] = G p 1;κ 1 p 1 [ p 1] = p 1 p 1 p 1, 1 p p 1 p 1 1 w α µ w α µ w 1 p α µ d 1 p 1 1 p 1 p 1 κ 1 p 1 = w p 1p 1 b p 1 p 1 κ 1 Ξ α µ = w p 1p 1 s p 1 p 1 1 α µ d 1 d p 1 w 1 p 1 p 1 p 1 α µ a s/κ l s as 1/κ ls y

13 FROM JACK POLYNOMIALS TO MINIMAL MODEL SPECTRA 13 = w p 1p 1 p 1 p 1 α µ j 1/κ 1 j=1 p j/κ p 1, 4.14 where we have evaluated the er product usg te 5 of Proposto 4. Clearly, the deoator of the above product s o-sgular, sce κ = p /p s a postve ratoal uber. Therefore, µ χw µ = 0 wheever 0 = p 1 p 1 α µ j 1/κ 1 = C j=1 p 1 where C s a o-ero costat. We group the, j-factor wth the p, p j-factor: Thus, p 1 µ α, j, 4.15 j=1 µ α, j µ α p,p j = 2hµ 2h, j µ χw µ = 0 h µ h r,s = 0, 4.17 r,s where the dex r,s rus over all 1 r p 1 ad 1 s p 1, wth rp sp < p p. The above costrats ply that the coforal hghest weght of a M p, p represetato ust be a root of the polyoal f h = h h r,s, 4.18 r,s that s, t ust be equal to h r,s for soe 1 r p 1 ad 1 s p 1, wth rp sp < p p. Showg that the represetato theory of M p, p s copletely reducble ad that t ca be used to costruct ratoal coforal feld theores requres oly a lttle ore work. The Vrasoro Vera odule of coforal weght h r,s, where 1 r p 1 ad 1 s p 1, cotas a axal subrepresetato geerated by two depedet sgular vectors of coforal weghts h = h r,s rs ad h = h r,s p rp s [9]. However, ether h or h are roots of So the M p, p represetato of coforal weght hr,s ust be soorphc to the rreducble quotet of thevrasoro Vera odule of coforal weght h r,s. Ths also ples that there exsts o o-trval extesos betwee rreducble represetatos wth dstct coforal weghts. I [18, Prop. 7.5], t was show that the rreducble Vrasoro represetato of coforal weght h r,s adts o self extesos as represetatos of the Vrasoro algebra. Thus, ether do the rreducble M p, p represetatos. Ths proves that rreducble M p, p represetatos do ot adt ay o-trval extesos ad that therefore the represetato theory of M p, p s copletely reducble. Corollary 7. The Vrasoro al odel vertex operator algebras are ratoal, that s, they adt oly a fte uber of equvalet rreducble represetatos ad all represetatos are copletely reducble. REFERENCES [1] L Alday, D Gaotto ad Y Tachkawa. Louvlle correlato fuctos fro four-desoal gauge theores. Lett. Math. Phys., 91: , [2] A Moroov ad A Srov. Falg the proof of AGT relatos wth the help of the geeraled Jack polyoals. Lett. Math. Phys., 104: , [3] K Mach ad Y Yaada. Sgular vectors of the Vrasoro algebra ters of Jack syetrc polyoals. Co. Math. Phys., 174: , [4] W Wag. Ratoalty of Vrasoro vertex operator algebras. It. Math. Res. Not., 7: , [5] A Tsuchya ad S Wood. O the exteded W-algebra of type sl 2 at postve ratoal level. It. Math. Res. Not. to appear, arxv: [hep-th]. [6] D Rdout ad S Wood. I preparato. [7] B Feg, T Nakash, ad H Oogur. The ahlatg deals of al odels. It. J. Mod. Phys., A7S1A: , [8] Y Zhu. Modular varace of characters of vertex operator algebras. J. Aer. Math. Soc., 9: , [9] B Feg ad D Fuchs. Represetatos of the Vrasoro algebra. Adv. Stud. Cotep. Math., 7: , [10] K Iohara ad Y Koga. Represetato theory of the Vrasoro algebra. Sprger Moographs Matheatcs. Sprger-Verlag, Lodo, 2011.

14 14 D RIDOUT AND S WOOD [11] V Dotseko ad V Fateev. Coforal algebra ad ultpot correlato fuctos 2D statstcal odels. Nucl. Phys. B, 240: , [12] A Tsuchya ad Y Kae. Fock space represetatos of the Vrasoro algebra tertwg operators. Publ. Res. Ist. Math. Sc, 22: , [13] I Macdoald. Syetrc fuctos ad Hall polyoals. Oxford Uversty Press, New York, [14] A Astashkevch. O the structure of Vera odules over Vrasoro ad Neveu-Schwar algebras. Co. Math. Phys., 186: , arxv:hep-th/ [15] P Goddard, A Ket, ad D Olve. Vrasoro algebras ad coset space odels. Phys. Lett. B, 152:88 92, [16] V Kac ad M Wakoto. Modular varat represetatos of fte-desoal Le algebras ad superalgebras. Proc. Natl. Acad. Sc. USA, 85: , [17] B Feg ad E Frekel. Quatato of the Drfeld-Sokolov reducto. Phys. Lett. B, 246:75 81, [18] K Kytölä ad D Rdout. O staggered decoposable Vrasoro odules. J. Math. Phys., 50:123503, arxv: [ath-ph]. Davd Rdout DEPARTMENT OF THEORETICAL PHYSICS, RESEARCH SCHOOL OF PHYSICS AND ENGINEERING; AND MATHEMATI- CAL SCIENCES INSTITUTE; AUSTRALIAN NATIONAL UNIVERSITY, ACTON, ACT 2600, AUSTRALIA E-al address: davd.rdout@au.edu.au So Wood DEPARTMENT OF THEORETICAL PHYSICS, RESEARCH SCHOOL OF PHYSICS AND ENGINEERING; AND MATHEMATI- CAL SCIENCES INSTITUTE; AUSTRALIAN NATIONAL UNIVERSITY, ACTON, ACT 2600, AUSTRALIA E-al address: so.wood@au.edu.au