UNIVERSITE DE PARIS-SUD CEN?TRE D'ORSAY
THESE présentée pour obtenir
Le GRADE de DOCTEUR EN SCIENCES DE L'UNIVERSITE PARIS XI ORSAY
par
NIR SOCHEN
sujet: CONTRIBUTION A L'ETUDE DES THEORIES CONFORMES ET DES MODELES INTEGRABLES
Soutenue le 19 Mai 1992 devant la Commision d'examen MM. E. BREZIN président P. BINETRUY K. GAWEDZKI C. ITZYKSON V. PASQUIER J.-B. ZUBER UNIVERSITE DE PARIS-SUD CENTRE D'ORSAY
THESE présentée pour obtenir
Le GRADE de DOCTEUR EN SCIENCES DE L'UNIVERSITE PARIS XI ORSAY
par
NIR SOCHEN
sujet: CONTRIBUTION A L'ETUDE DES THEORIES CONFORMES ET DES MODELES INTEGRABLES
Soutenue le 19 Mai 1992 devant la Commision d'examen MM. E. BREZIN président P. BINETRUY K. GAWEDZKI C. ITZYKSON V. PASQUIER J.-B. ZUBER
i.r 1
A nies parents et à Lori et Carmel
T REMERCIEMENT
Deux personnes ont joué un rôle déterminant dans ma formation et m'ont montré la voie de la recherche. Je tiens à remercier Shimon Yankielowicz qui m'a enseigné la théorie des champs et a guidé mes premiers pas dans la physique. J'ai eu la chance de travailler avec Jean-Bernard Zuber, mon directeur de thèse, qui est aussi devenu un ami et m'a enseigné le métier de chercheur avec enthousiasme et bonne humeur. Je le remercie tout particulièrement de l'infinie patience dont il a fait preuve et de la gentillesse qu'il a bien voulu me témoigner. Je voudrais remercier Edouard Brézin qui m'a accueilli en France. Il m'a aidé à trouver le début de la piste qui m'a amené jusqu'ici et il a accepté d'être président du jurj'. Je remercie tout particulièrement Vincent Pasquîer et Krzysztof Gawedzki qui ont accepté dans des conditions difficiles la lourde tâche de rapporteur. Je remercie également Claude Itzykson qui s'est toujours intéressé à mon travail et qui a accepté de faire partie du jury. Je remercie Pierre Binétruy d'avoir eu la gentillesse de faire partie du jury. Je remercie Michel Bauer pour son amitié et sa collaboration, et tous les membres du Service de Physique Théorique qui m'ont offert un accueil chaleureux et un environnement stimulant. Je tiens à remercier André Morel qui m'a permis de finir ma thèse dans de bonnes conditions. Je veux aussi remercier Jean-Yves Ollitrault qui a eu le double courage de partager avec moi son bureau et d'essayer de corriger mon français. J'aimerais remercier mes parents qui m'ont toujours soutenu et encouragé. Ils ont toujours été présents même quand une grande distance nous séparait. Je remercie Lori pour la force qu'elle me donne et pour sa patience. Sans elle cet ouvrage n'aurait pas \'u le jour. Je remercie enfin Carmel qui, malgré son très jeune âge, sait grâce à ses sourires me donner espoir. TABLE DES MATIERES
INTRODUCTION 1 1. GENERALITES SUR LES THEORIES COFORMES .... 3 1.1 Le grdbpe conforme 3 1.2 Théorie de champs conforme 4 1.3 Théorie de champs conforme à d=2 5 1.4 L'algèbre de Virasoro 8 1.5 Théorie des représentations de l'algèbre de Virasoro ... 10 1.6 Caractères 13 1.7 Fonctions de corrélation 15 2. MODELES DE WESS-ZUMINO WITTEN, ALGEBRES DE KAC-MOODY ET LEURS VECTEURS SINGULIERS . . 19 2.1 Le modèle WZW 19 2.2 La théorie des représentations des algèbres de Kac-Moody 21 2.3 Les vecteurs singuliers et les règles de fusion 29 3. LES ALGEBRES W 40 3.1 Les algèbres W quantiques 40 3.2 Les équations différentielles linéaires covariantes 43 3.3 Le formalisme matriciel 44 3.4 Les différentielles dans les algèbres w 46 4. BOSONISATION 51 4.1 La bosonisation à la Feigin et Fuchs 51 4.2 La cohomologie de Felder et la généralisation à SU(2) . . 53 4.3 La bosonisation des fonctions de partition 57 5. MODELES INTEGRABLES SUR RESEAU ET REPRESENTATIONS DE L'ALGEBRE DE HECKE .... 63 5.1 Motivation 63 5.2 Les modèles de face 64 5.3 Les modèles ADE 67 5.4 Entrelaceurs et conditions aux limites 69 5.5 Les modèles quotients 72 REFERENCES 75 PUBLICATIONS 77
- Vale u'il faut INTRODUCTION
La dernière décennie a été témoin d'une explosion d'activité dans le domaine de la physique bidimensionnelle. La source de cette activité a été la théorie des cordes qui a ses racines dans les modèles duaux d'il y a plus de vingt ans. La théorie des cordes a reçu une nouvelle impulsion avec la reformulation de Polyakov en 1981. et elle est devenue une sérieuse candidate à être une théorie cohérente de la gravité quantique et à décrire dans une théorie unifiée toutes les forces connues. En tous cas elle est quasiment la seule qui peut prétendre à l'être. L'avenir dira si la théorie des cordes est la bonne manière d'attaquer ces problèmes difficiles. Nous ne pouvons qu'affirmer à présent qu'en poursuiv- ant le chemin de recherche indiqué par la théorie des cordes nous en avons tiré profit dans de nombreux domaines qui a priori n'ont rien à voir avec cette théorie. Notamment les théories conformes qui sont les "vides classiques" du point de vue de la théorie des cordes sont les clefs pour comprendre le comportement à grande échelle des modèles de mécanique statistique bidimensionnels ayant les symétries de translation, rotation et dilatation à leur point critique. Les théories»conformes sont liées naturellement aux modèles intégrables de la mécanique statistique. D'autres domaines qui sont inspirés par la théorie des cordes sont la gravité quantique bidimensionnelle, les théories topologiques, les équations différentielles intégrables et les algèbres w classique et quantique. Les idées et méthodes de la théorie conforme jouent un rôle important dans tous les domaines mentionnés. Nous présentons une revue de la théorie conforme au chapitre 1. Nous poursuivons au chapitre 2 (et dans les articles 3 et 4) l'étude de la théorie des représentations des algèbres de Kac-Moody. On se sert de ces algèbres pour construire des exemples de théories conformes, et en effet toutes les théories conformes rationnelles connues peuvent être obtenues de cette manière. Les vecteurs singuliers sont des objets importants du point de vue de la théorie des représentations et ils sont très importants aussi du point de vue de la physique puisqu'ils engendrent des équations différentielles pour les fonctions de corrélation. Leur forme qui n'était connue qu'implicitement est donnée explicitement au chapitre 2 pour le cas de l'algèbre A\^. La limite classique d'une classe de vecteurs singuliers est liée au formalisme de Drinfeld-Sokolov et aux algèbres w classiques. Nous présentons brièvement ces formalismes au chapitre 3 et nous calculons les nombres de différentielles d'un degré donné dans une algèbre w classique. La méthode de bosonisation est présentée au chapitre 4 et utilisée dans l'article 5 pour identifier le spectre de la théorie topologique sl(2)/sl(2). La forme des vecteurs singuliers présentée au chapitre 2 est utilisée
1 aussi dans l'article 5 pour construire les états physiques de la théorie topologique. La bosonisation des fonctions de partition de différents modèles avec diverses conditions aux limites est présentée au chapitre 4 (et dans l'article 1) également. Cette méthode fait partie d'un vaste programme de classification des théories conformes rationnelles en faisant un lien entre elles et des modèles de spin intégrables. Nous décrivons ce programme au chapitre 5 et dans l'article 2 où nous avons montré l'existence de relations intéressantes entre les poids de Boltzmann de différents modèles. Ces relations nous permettent de prouver l'intégrabilité de certains modèles par un calcul direct de leurs poids de Boltzmann.
D'abord nous démontrons qu'une fonction de corrélation des descendants est uniquement 1. GENERALITES SUR LES THEORIES CONFORMES
1.1. Le groupe conforme
Le groupe conforme dans Rd est défini par l'ensemble des changements de coordonnées pour lesquels la métrique se transforme de la manière suivante [1]
(1-1)
C'est une transformation qui conserve localement des angles. Sous une transformation infinitésimale X14 —> a-,, + e/( l'équation (1.1) s'écrit
Up/5 cfP /V(LV)f> \C\ *••*•) 1}
où on a développé autour de ,,„ et O11 = ^f- . Quand la dimension est plus grande que 2 il s'ensuit que
2 fii = an + Xx11 + u>nvxv + 2(e • X)X11 — C11X (1.3)
est la solution générale de (1.2) où U11^ est antisymétrique et U1111, a,,, C11, A 6 R. On reconnaît en (1.3) les quatre types de transformations conformes:
a) Translation : aM d paramètres
b) Dilatation : AxM 1 paramètre c) Rotation : u^x,, ^d(d — 1) paramètres 2 d) Transformation conforme spéciale : 2(c • X)XP — cMx d paramètres soit au total ^(d + 1 )(d -f- 2) paramètres , c'est-à-dire autant de paramètres que dans le groupe SO(l,d+l) et en effet les deux groupes sont isomorphes. Nous remarquons aussi que (1.3) est défini également globalement, c'est-à-dire que les transformations a)-d) sont des applications bijectives. En d=2 par contre on tire de l'éq.(1.2)
(1-4)
\
et celle de 3 champs primaires à Nous définissons
S = Xj + ZX2
C = X1 - ZX2
" 9d i (L5) d = d; = -%: = -(di + iÔ2 ) dz 2 e = C] 4- z«2
ê = ei — ze2
On traite X] et .T2 comme des nombres complexes et par conséquent z et 5 sont indépen- dants. Nous rétablirons la condition de réalité de x, plus tard en imposant z = r* ( z" est le conjugué complexe de z). L'équation (1.4) s'écrit maintenant
de = de = O (1.6)
ce qui montre que en d=2 le groupe conforme est un produit direct du groupe de trans- formations holomorphes de z par Ie groupe de transformations holomorphes de z. Notons que dans ce cas il existe des transformations conformes locales (qui ne définissent pas des applications bijectives sur la sphère de Riemann).
1.2. Théorie de champs conforme
Considérons un système décrit par l'action
S = Jddx£{
Le tenseur énergie-impulsion est donné par la réponse de l'action à un changement de
coordonnées x(, —» X1, + e,, éS = -^- IddXd^T * (1.8) 2* J 11 Le facteur 2?r est inséré par commodité. Une fonction de corrélation, dans l'approche Lagrangienne, est définie par
(<^j l^n) = I Î^C &1 • • • $n (1--9)
C!.!.] où on a normalisé / î>0e = 1.
grâce à l'invariance conforme on écrit Sous un changement infinitésimal de coordonnées ( pas nécessairement conforme) les champs se transforment o -+ . L'intégrale fonctionnelle est insensible à ce change- ment et on obtient donc l'identité suivante:
O = £(0, ...... On) = J](P, . . . 64, . . . -An) - -^ j tfxd^yp^oi ...*„) (1.10)
On suppose la variation des champs locale
1 où Bj^1 ^ „ sont des champs locaux et la somme est finie. Une differentiation fonctionnelle ^^7 de (1.10) donne
n N
i= i *=o
d ce qui prouve que le tenseur énergie-impulsion est conservé pour x € fi = R — {z"i,..., Xn }. L'action d'une théorie conforme est invariante par changements de coordonnées con- formes. Il est facile donc de déduire de (1.3) et (1.8) que le tenseur énergie-impulsion est symétrique et de trace nulle à des dérivées totales près. On démontre qu'en espace plat on peut toujours ''améliorer" le tenseur énergie-impulsion de telle sorte que les propriétés de symétrie et de trace nulle soient vérifiées.
1.3. Théorie de champs conforme à d=2 Notre point de départ est l'équation (1.10) pour un changement infinitésimal arbitraire de coordonnées xp —» X11 + g ^
i ... (j>n) = - dxt\ dyd^g^T^i ...„) (1.13) /T J
II ne reste en d=2 que deux éléments independents de
On définit
T=J(Tn-ITi,) 2 (1.15)
2. MODELES DE WESS-ZUMINO-WITTEN, ALGEBRES DE KAC- Dans la base introduite en (1.3) on a (Lr A dy = d: A r/c/2/ et l'éq. (1.12) s'écrit
0(T(c.c)o,Ui.c, 1...O11(C11. c,,)) =0 c <= iî = C - {c, C11} (1.16) d(T( c. c >o,( c,..-,)... OnU11-C11 )) = O c € SÎ = C - {c, C11} pai' consequent T (T) est hulomorplu* (antiholomorplu») sur fi (U). L'équation (1.13) devient
iU,. C1)... (1-17) c, C)(T(C)O1 (C1, c, (...C)n(C111C11) + 00( :. c)(T(c )o,(r ,,-,)... On (c,,. =„))) Nous définissons un champ primaire comme une différentielle de poids (h, h): (1.18) soit sous forme infinitésimale (1-19) Ces deux façons d'écrire doivent être comprises dans le sens d'insertion dans les fonctions de corrélation. On peut choisir g(~.:) de telle façon qu'il soit holomorphe au voisinage de Zi et que g soit antiholomorphe au voisinage de c,. On dénote e; = g(z,z) (ëj = g(z,z)) au voisinage de c; (z,) et on demande que Remarquez que F C îî. En utilisant le théorème de Stokes nous déduisons (pour les 6 qui satisfont 0J = O et 0J = O. Nous écrivons donc J = J(c) et J = J(z). Les Ra sont les matrices des générateurs infinitésimaux dans une représentation de l'algèbre de Lie finie dite o ' ,o ta e champs primaires ) nICn.Cn)) /• d- = 4 5A«,- Jc, -1*' ZI , Z\ 'liri c,(c,) sont des contours autour des points Zi(Si) dans les regions où g(z,z) (g(z.z)) est holomorphe (antiholomorphe). Il s'ensuit par le théorème de Cauchy et la condition que T — » O à l'infini que St, i ...<~-n,~n ; (Z ~ zil Z-Zi h d (1-21) h ^r- X*i(*i,*i)" •*«(*.,*«)) Ce sont là les identités de Ward pour les théories de? champs conformes. On écrit parfois dans un abus de langage T(s)0A(u>, u-} = - - —(f>h(w,ûi) H --- d(t>h(w,w) + termes réguliers (1.22) (r - w)2 z — w II faut comprendre cette équation au sens d'insertion dans les fonctions de corrélation. Cette forme d'écriture émerge naturellement dans une autre approche de la théorie quan- tique des champs, l'approche de l'algèbre des champs locaux. L'adjectif locaux veut dire que la fonction de corrélation < .4](;rj }. . . An(Xn)) € C de champs locaux Ai(xi) est an- alytique partout sauf ayx points x, = Xj . Cette propriété est invariante par rapport aux opérat:rns linéaires. On peut donc considérer les champs locaux comme des éléments d'un espace vectoriel *4. Cet espace est en général de dimension infinie. On suppose qu'il est engendré par une base dénombrable Ai i = l,2,... ^1-(X1 )Aj(Xj) = , Cfai, Xj)At(Xj) (1.23) h où C*j(xi, Xj) sont des fonctions appelées fonctions de structure. A muni de cette multiplication est une algèbre. On caractérise maintemoit une théorie conforme comme Le produit des opérateurs de _,2"^-» _.^(---) <2-17) une algèbre de champs locaux qui contient l'identité et le tenseur énergie impulsion. A contient un sous-ensemble non trivial P de champs primaires qui sont définis par leur produit avec T. Dans une théorie conforme A est engendré par [T] U P, Les champs dans A - P (y compris T) s'appellent descendants. 1.4. L'algèbre de Virasoro Une question intéressante est comment se transforme le tenseur d'énergie impulsion lui-même sous un changement de coordonnées? L'équation (1.8) nous indique que la dimension d'échelle de T est 2. L'expression la plus générale qui prenne en compte la dimension d'échelle de T et la symétrie conforme est T(z)T(y) = *° >4 + , ~ T(V) + -^—&T(y) + termes réguliers (1.24) (.r - yY (x -yY x -y c est une constante qui s'appelle la charge centrale. Elle est normalisée de telle façon que la charge centrale du boson libre soit 1. L'équation (1.24) est une identité dans une fonction de corrélation ).--T(zn) i=2 *• ' (1.25) II est commode, à ce stade, de changer le point de vue et d'utiliser le formalisme Hamiltonien. On sait que chaque symétrie est engendrée par un courant conservé J*1 O11 J" = O (1.26) On associe à chaque courant conservé une charge conservée Q= /V-1ZJV) (1-27) où on intègre sur une surface à temps fixé. Les transformations infinitésimales sont engen- drées par le commutateur à temps égaux 6(A = e(Q,A]T.E. (1-28) 8 où e est infinitésimal et A est un champ quelconque de la théorie. Dans le cas qui nous intéresse, la théorie conforme, le courant conservé est J11 = Tliuev où fv est un changement conforme infinitésimal de coordonnées. Le changement de coordonnées (1.5) transforme l'intégration sur une surface à temps fixé en intégration dans le plan complexe sur un cercle de rayon constant. La charge correspondante est Q = T(z)£(s) + et la variation d'un champ primaire 6(i Le changement de base (a-j, ^2) — » (-,.?) fait que la quantification est par ordre radial A B W W N>H /i,n B(w)A(z) \z\<\w\ (1'31) Ceci nous permet d'écrire pour la variation d'un champ primaire eq.(1.22) -» = * TT- :e(s) ( ; - rz4>h(u>, û) + - d = (ed + h(de)) StT(w) = (ed + 2(Oe)) + ~e"' (1.33) d'où pour une transformation finie T(z)^^\(w)+^c{w,z} (1.34) où { u), z} est la dérivée schwarzienne \ 9 J - 2 "1 1 Nous développons T(r) en série de Laurent 2 n^"- (1-36) II s'ensuit de l'éq.(1.22) que [In, Ms)] = s*((n + \)h + zd)d>h(z) (1.37) La même analyse à partir de l'éq.(1.24) donne l'algèbre de Virasoro 2 [l«,Lm] = (n - m)Ln+m + ^n(n - l)6n+m (1.38) qui joue un rôle essentiel en théorie des champs conforme. Avant de me plonger dans la théorie des représentations de l'algèbre de Virasoro, j'aimerais faire un détour en carac- térisant les propriétés des descendants. On définit les descendants par récurrence comme les termes dans le membre de droite du produit d'opérateurs U-wr'-r^'Sjf*1 ..... -*"+l)(«M5) (1.39) 0 4>h ' " («>,«>) sont construits en termes de ^i. ° (w,w) = (j>k(v>,w) le champ pri- maire. Une autre manière de voir les choses est d'intégrer l'éq.(1.39) "(U>,«')= et, donc, par récurrence tl n) *JT ~" ( «>,«') = L-kn(w)L.kn_t(w)... Lk1(W)Mw^w) (1.41) On appelle l'ensemble de 1.5. Théorie des représentations de l'algèbre de Virasoro Une représentation de plus haut poids est une représentation cyclique munie d'un vecteur de plus haut poids caractérisé par les conditions (Ln - h6n,o)\h) = 0 n > O (1.42) 10 Li» _ Une base pour le module de Verma de l'algèbre de Virasoro est donnée par les vecteurs I_tlI_t,...I.tJ/i> (1'43) Ces vecteurs sont des vecteurs propres de LO associés aux valeurs propres h + J^JL i Jk1-. Le module de Verma est donc naturellement gradué. Chaque espace propre associé à la valeur propre h + n est dit espace homogène au niveau n et il est de dimension finie P(n)- Ie nombre de partitions de l'entier n. Le "vide" d'une théorie conforme est invariant par rapport aux transformations conformes globales et il est vecteur de plus haut poids. Il satisfait, donc, les conditions In)O) = O n>-l (1.44) II est facile de vérifier, à l'aide de l'éq. (1.37) , que Hm <£fc(z)|îï) (1-45) 2—>0 obéit aux conditions (1.42) et nous l'identifions, donc, à \h). Il s'ensuit que le même genre de relation s'applique aux descendants _„.„ , ,. .=L.kl...L.kn\h) (1.46) Ces relations sont en effet des conséquences directes de la discussion à la fin de la section dernière et de l'éq. (1.45). La famille [<£&] est en relation bijective avec une représentation de plus haut poids de l'algèbre de Virasoro. Ces représentations sont caractérisées par deux paramètres, la charge centrale c et le plus haut poids h. Deux questions importantes qui intéressent naturellement les physiciens sont les questions de la réductibilité et de l'unitarité de ces représentations. Nous citons ici les résultats Théorème:(Kac, Feigin et Fuchs [2]) Pour une charge centrale donnée p€C (1.47) P(P +1) les représentations sont réductibles si et seulement si h est une des valeurs suivantes *--""^"+Tf"1 0 InU)=O 17 > 1 (1.49) ou n l\) ^ 1^)- O décompose |\) en ses parties homogènes par rapport à £0- U est clair que chacune des parties satisfait l'éq. (1.49) parce que Ln\\m) est homogène de niveau m — n si \\m) est homogène de niveau KI. On traite donc le vecteur singulier, sans atteinte à la généralité, comme un vecteur homogène. Il est évident qu'une représentation est réductible si et seulement sielle contient un vecteur singulier. La présence de vecteurs singuliers et leur forme explicite sont des questions intéressantes et importantes. Nous les aborderons abondamment à la section 6 et au chapitre 2. Nous passons au problème d'unitarité. Un champ hermitique vérifie . i 1 [MOr = Mi)]^r ^1-50) La conséquence sur les coefficients de Laurent de T(Z) est Ll = £-„ (1.51) On introduit une forme contragradiente F : Vh 0 Vh -» C .... d-52) La représentation en question est unitaire si et seulement si F est une forme bilinéaire définie positive. Théorème: (Friedan, Qiu et Shenker [3]) Soient c,h réelles. Pour que la représentation soit unitaire il est nécessaire et suffisant que 1) c>l h >0 ou 2) O < c < 1 c et h dans la forme du théorème de Kac, Feigin et Fuchs où p e IN -{0,1} l 12 On voit qu'une théorie unitaire avec UIK? charge centrale positive inférieure à 1 a un nombre fini !! clo champs primaires. Ces théories sont dites théories minimales. 1.6. Caractère* Le caractère d'une représentation de plus haut poids R/, est par définition (1.54) Le caractère d'un module de Verma I '/, est L \h(q) = Tr\'h(j °~'^ = Vj dim( espace homogène au niveau n)ç" n=0 (1.55) -- _ ,1=0 où J? est la fonction eta de Dedekind. Si /! ^ /irj le module de Verma est irréductible et (1.55) est le caractère de la iepresentation de plus haut poids Rh = Vi1. Si par contre h = hr3 le module de Verma l/, est réductible Ri, ^ V/,. Il est très important de savoir, dans ce cas, la structure exacte du plongement des sous-modules. Ce problème a été résolu par Feigin et Fuchs [4]. Le résultat est le suivant: on dénote, pour les modèles minimaux, par (r,s) le module Vi,rr. Le diagramme du plongement est Fig. 1.1 Le diagramme d'inchision des sous modules de Virasoro où une flèche de x à y veut dire que y est un sous-module de x. Pour avoir une représentation irréductible de plus haut poids Rhri à partir de (r,s) il faut soustraire de (r,s) les sous-modules (2p - r,s) et (r + p.p + 1 — s). Mais, l'intersection de ces sous- modules n'est pas triviale et elle est soustraite deux fois au lieu d'une seule. En ajoutant 13 l'intersection on ajoute deux sous-modules avec une intersection non-triviale qu'il faut soustraire etc. Le résultat est une somme alternée de caractères des sous-modules (HV+uroM2 -5 ^ (1-56) où A = r(p + l) — .sp, N = 2p(p -(- 1) et WQ A = r(p + 1) + sp. On construit la fonction suivante par la conjugaison de la partie holomorphe avec la partie antiholomorphe Z(q) = TTW"-'- = ;xâ; (1.57) »,ï La deuxième égalité vient du fait que H = ®Ri ® R\ et JV4 tj compte donc le nombre de champs primaires de type «', / dans la théorie et il est par conséquent un nombre entier non- négatif. L'unicUé du "vide" impose en plus que A/ô.o = 1. Nous verrons au chapitre 3 que Z(T) = Z(q = e2*IT) est la fonction de partition du système sur le tore, où r caractérise le tore. Cette caractérisation n'est pas unique et en effet tous les points dans une orbite du groupe modulaire PSL(2,Z) caractérisent le même tore. Ce groupe agit sur r de la manière suivante T^^ + b ad -cb= I a,b,c,d£Z (1.58) CT + a transformations qui sont engendrées par deux générateurs T-.T-4T+1 1 (1-59) S : T -> — T Evidemment, la fonction de partition Z(T) doit être invariante modulaire [5]. Ceci pose des contraintes fortes sur le contenu opératoriel de la théorie. Pour les modèles minimaux la somme en (1.57) est finie. Les caractères, dans ce cas, forment une représentation de dimension finie du groupe modulaire T\\(T) = A OÙ SAV = (1.61) 14 Table I: Une liste des invariants modulaires de modèle quotient Str(2)t_i x SU(2)i/SU(2)k, avec n' = n ± 1, k + 2 = max(n,?i'). n'-l n-1 77 >2 i V 2 2 2-é 5^I\A,,| M-l) A=I /1== 1 "'-» 2p-l n=4p + 2 2 4Z / _. |X A /i ' | + %A2+ll4 (IV») A=I ( n'-l 4p-l 2p-2 77=4p 1 >2 Z l\A/i|'J + |\A2p|2 + E (X'A»«XA4p-/. + c-<,} ,^+1, A=I (/i odd = 1 /i even =2 n'-l 77 = 12 »£ {|XA1 + XA7I2-+• |XA4 + XAsI2 + IXAS + XAiil2} (fie) A=I n'-J 1 V 2 2 2 n = 18 2 Z-*, {1XA I+ \A17| + |\A5 + \Al3| + |XA7 + XAll| (fir) A=I |2 I \ A 9 1 T K X A3 + U15 XA' 9 +C-C. } n'-l 1 V~* n = 30 " Y \ 1 Q ~fr~ Y \ 9Q I ~t* 1 Y )k 7 "^ V \ 1 1 "4" \ \ 1 7 " 2 2 Z-* {1|\A1 + \All4 /\ jA 1 » i A « *" I ' (A"' ' A " A •> i AAlI I-XA23I } (Et) A=I La condition d'invariance modulaire se traduit par les conditions a) [^1T]=O = [JV1S] b) JV"; € JV c) .V00 = 1 qui permettent une classification exhaustive [6](voir la Table I). 1.7. Fonctions de corrélation Nous allons voir dans cette section comment l'invariance conforme fixe les fonctions de corrélation de 2 et 3 champs primaires et comment la forme explicite d'un vecteur singulier peut aider à calculer une fonction de corrélation de n > 4 champs primaires. Nous illustrons cette technique par le calcul de la fonction de corrélation de 4 spins dans le modèle d'Ising. 15 «r D'abord nous démontrons qu'une fonction de corrélation des descendants est uniquement déterminée par la fonction de corrélation des champs primaires correspondants. Considérons une fonction de corrélation qui contient un descendant t =2 )...*»(*»)) parréq.(1.41) =^(L-k(:t ) (1.63) o ^-S(S^*^) La généralisation est immédiate (1.65) Les cas avec plusieurs descendants sont analysés de la même manière en utilisant l'éq. (1.24) au lieu de l'éq. (1.22). Dans tous les cas, la fonction de corrélation qui contient les descendants s'exprime comme un opérateur différentiel linéaire agissant sur la fonction de corrélation des champs primaires correspondants. La forme de la première fonction est donc fixée par celle de la deuxième. On tire de l'équation (1.21) que ««**( = ) = (fd + M0e)tofcU,f ) = O (1.66) pour e quadratique en z. Cette condition fixe la forme de la fonction de corrélation de 2 champs primaires à la valeur 16 T et celle de 3 champs primaires à hi+k..hk t (1.68) -12 *13 -23 ou 2jj = z, — z} et les C',jfc s'appelle les constantes de structure. L'invariance conforme globale ne suffit pas à déterminer complètement la forme des fonctions de corrélation pour n > 4. Néanmoins elle réduit la fonction de corrélation à 4 points à la forme où h = £V ft; et x est le birapport x = (z\ La présence d'un vecteur singulier et sa forme nous permettent d'aller plus loin. Re- gardons l'exemple du modèle d'Ising. Ce modèle est identifié avec la théorie minimale c = i où les poids conformes des champs primaires sont O, ^ et 5. Le champ de spin est identifié avec le champ primaire -^ parce que le propagateur du champ de spin x c.c (1.70) (Z - est identique à la fonction à 2 points pour un champ primaire de poids (^, ^)- Nous considérons dans l'analyse qui suit la partie holomorphe. Dans le module de Verma Vh= \ il y a un vecteur singulier (1(L1 )' - L2) |i) (1.71) Ce vecteur est identifié à zéro dans la représentation irréductible, et par conséquent on a la relation suivante pour une fonction de corrélation qui contient le champ de spin (1-72) Pour la fonction de corrélation de 4 spins on a Il est plus commode, pour la définition de la fusion, de récrire grace à l'invariance conforme on écrit (1.74) et l'équation différentielle devient (1.75) On effectue le changement de variable a- = sin3 6 et on obtient (1.76) la solution est n(0) = (1.77) Une raisonnement analogue pour la partie antiholomorphe donne, en conjuguant avec la partie holomorphe. u(0,0) = flu cos-flcos-0 + ai2cos-0sin-0 (1.78) + 02] sm-0cos-0 + a22sin-0sin-0 On requiert que la fonction de corrélation soit univalente sur le plan complexe quand x = x*. Cette condition impose que a2i = aj2 = O et au = 022- Dans la bonne normalisation on trouve le résultat final ^l 3-24 {*,(*!)... (T4(S4 )) = (ii + v/r^i+11- vT (1.79) -12^23-34241 Une comparaison avec le même calcul fait microscopiquement [7] montre, d'abord que notre méthode donne de bons résultats, et aussi qu'elle est beaucoup plus efficace. 18 i.» ou nous prenons g = a = 1 et on dénote j = jr,*,±. On obtient que r = 2j + 1 et 2. MODELES DE WESS-ZUMINO-WITTEN, ALGEBRES DE KAC- MOODY ET LEURS VECTEURS SINGULIERS 2.1. Le modèle WZW Un modèle de Wess Zumino Wit ten (WZW) [8] est un modèle tridimensionnel où les champs sont les applications d'une surface de Riemann dans un groupe de Lie. L'action est (2.1) ou £ = (€i>&) sont des coordonnées locales sur la surface de Riemann S et T(S) = ^- / d3«'"'1(sr1a»0)(9-IÔ/tf Xr'a,») (2.2) -4îr JB où B est une variété de dimension 3 qui a S comme bord. Nous prenons d'abord un groupe de Lie serai -simple et compact. Pour deux choix différents de B, T varie par un élément H3(G) = Z. Pour que l'intégrale fonctionnelle ait un sens, k doit donc être un entier. Une étude détaillée du groupe de renormalisation de cette théorie montre l'existence d'un point fixe dans la limite infra-rouge . La constante de couplage A y est liée à k par A2 = k/4it. La théorie effective, à ce point là, est évidemment de masse nulle et l'action est donnée par a 1 Sk(g) = kW(g) = H d €Tr(ô^- ôl,s) + T(g)} (2.3) On peut démontrer à l'aide de l'identité de Polyakov et Wiegmann W(gh~1) = W(g) + W(h) + -i- f d2 ^!(g-1 Bgh-ldh) (2.4) !DÎT J que l'action (2.3) est invariante par (=) (2.5) o -<•** ,2.6) = = & - ifc Cette symétrie est engendrée par les courants a 1 0 l J = J R" = -Ik(Bg)9- J = J R" = -\kg- (dg) (2.7) 19 I.I l'histoire finirait là, mais pour l'algèbre affine on a des vecteurs singuliers supplémentaires. On répète le calcul pour eux qui satisfont d J = O et d J = O. Nous écrivons donc J = J(z) et J = J(z). Les Ra sont les matrices des générateurs infinitésimaux dans une représentation de l'algèbre de Lie finie dite horizontale [Ra,Iîb} = fïhRc (2.8) Une transformation de jauge infinitésimale est engendrée par les courants conservés (2.9) Cette théorie est aussi une théorie conforme et les transformations conformes s'écrivent (2.10) Par (2.5) on voit que pour une transformation infinitésimale îî(r) = / + u>(z) (2.11) La partie holomorphe est donc a a 6wg(z) = w(z)g(z) =w (z)R g(z) (2.12) Nous utilisons (2.9) et (2.12) pour obtenir wa(z)Rag(z,z) = _. R" g c _ ,gUi ) + termes réguliers (2.14) D'une manière générale J3a Ja(£) Les champs 4>3(~) sont dans la représentation de spin j de l'algèbre de Lie horizontale et ils sont aussi des champs primaires par rapport à l'algèbre de Virasoro. Le tenseur d'énergie- impulsion est une forme bilinéaire des courants conservés (construction de Sugawara [9]) = EI - 1-2 (2.16) 20 dans un module j montre que Ic- noyau n'est pas trivial et. à cr moment là. il y a une contrainte sur les valeurs possibles clos j.ju et ^i pour que la fusion soit possible. Ces contraintes sont les règles de fusion. Le produit des opérateurs de (2.17) Les courants conservés sont exprimés en termes de g(z) (ou plus généralement par les fi) . Leur loi de transformation est donc calculable et le résultat est qu'ils sont des différentielles de poids conforme (1,0) (2.18) et une transformation infinitésimale de jauge donne (2.19) On développe J en série de Laurent (2.20) Les relations de commutation sont (2.21) 2 [Ln, Lm] = (n - rn)Ln+m + rnfa - l)6n+m.o où c = 3k/(k + 2). Les modes zéro JQ sont 'es générateurs de l'algèbre horizontale. Nous ajoutons pour une utilisation ultérieure la relation de commutation des courants avec les champs primaires [J«, V(z, S)] = zTWz, S) (2.22) où R" est une représentation de l'algèbre horizontale, ainsi que la relation de commutation 2.2. La théorie des représentations des algèbres de Kac-àfoody Nous allons donner les théorèmes fondamentaux de la théorie des représentations des algèbres de Kac-Moody. Il est plus commode à cette fin d'utiliser la base de Serre-Chevalley. 21 est Hm déterminé pour toits les «^ qui sont entourés dans la Fig. 2.2. Ces *•> g sont plus petits que i-/,.m par Tordre partiel que nous avons défini. Dans le membre de gauche de (2.80) on voit le coefficient Nous suivrons largement l'exposition de Malikov, Feigin et Fuchs (MFF) [1O]. A la fin de cette section nous donnerons la transformation que nous permet de passer de la base de Serre-Chevallcy à la base usuelle de l'algèbre des courants. Soit A une matrice symétrique telle que AU = 2 Vz et At] < O entier si i ^j. Une algèbre de Kac- Moody g(A) est définie comme une algèbre de Lie complexe engendrée par les générateurs h,, /j, e; ? = 1. . . . , n et les relations [/>„/>>]= O [hi Jj] = -Ai,}fj (2.24) (ade,rAi-'+lej = 0 1"'+1/, = 0 On peut décomposer g(A) = N- ^ H 6 A'+ où les sous-algèbres N-, H et TV+ sont engendrées par les /,, hi et c, correspondants. La sous-algèbre de Cartan est H, et on introduit un produit scalaire sur H par < hi,hj >= Ai1J. L'algèbre enveloppante universelle U(g(A)) est l'algèbre tensorielle quotientée de droite comme de gauche par les relations de commutation. Elle se décompose naturellement en U(g(A)) = M(N-)®U(H)® U(N+). Nous introduisons un espace linéaire T de dimension n avec une base ai,...,an. F est le réseau engendré par les Oj, l'octant positif est le sous-ensemble {a € TJa = 53,-XiO,- t.q. Xi > O Vi } et F+ est l'intersection de F avec l'octant positif. Soit a = 53 mto/i où m, € Z, et ga le sous-espace de g engendré par les monômes de ci et fi pour lesquels la différence entre le nombre de e,- et le nombre de /f est m,-. Nous dirons que Q est une racine de g(A) si dim Jf0 ^ O et qu'elle est une racine positive si elle est une racine dans l'octant positif. On appelle A+ l'ensemble des racines positives. Pour x a chaque point Q = 53 i > € T on désigne le point correspondant dans H par ha = J^Xi/i,-. Il est clair que T et H sont isomorphes. Néanmoins, on identifie traditionellement T avec H* l'espace des fonctionnelles linéaires sur H par la relation a(h) =< A0, h >. On a par héritage de cet isomorphisme T ~ H* un produit scalaire sur T: < 0, f >=< fe/j, h^ >. Nous définissons pour chaque racine < Q, a >^ O une réflexion par rapport au plan orthogonal à Q passant par l'origine. S0 : H' -* H' 22 f— I.I _ 2X(ho) a (2.25) < Q1Q > où Q est une abréviation à la fonctionelle Q(-) =< H0,- >. D'une manière analogue nous définissons une réflexion sur T S0 : T -» T (2.26) < Q,Û > S0, sont les générateurs du groupe de Weyl. On peut définir des réflexions par rapport à des points différents de zéro. Il est utile pour la suite de considérer par exemple des réflexions autour du point p £ H* qui est défini par p(h\ ) = p( /J2 ) = ... = p(hn ) = 1. On dénote par S£ la réflexion S£\ = S0(X + p) -p (2.27) Une représentation de plus haut poids est construite de la manière suivante. Nous choisissons une représentation unidimensionnelle |A > de l'algèbre JV+ (&H sur laquelle N+ agit trivialement N+[X >= O et (A > est un vecteur propre de la sous algèbre de Cartan: h\X >= X(h)\X > VA € H. |A) s'appelle le vecteur de plus haut poids. L'action de l'algèbre enveloppante de l'algèbre de Kac-Moody en question sur (A > est isomorphe à l'action de IA(N- ) sur |A >. On dénote par M(A) le module U(N-)\\ >. M(A) s'appelle aussi le module de Verma et on le dénote parfois par VA. Le module de Verma est naturellement gradué par le réseau F. On donne à un vecteur du sous espace > ç 9i3i • • • 9^k I^ • l degré A-J^ /?,. Nous appelions M(A)0 le sous espace homogène de degré Q. On a la décomposition M(A) = eo€r+M(A)0 (2.28) et le caractère du module de Verma est A ch M(A) = 9 P(Ij)5-' (2.29) où P(T; ) est la dimension de l'espace M( A)1, et s'appelle la fonction de partition de Kostant. Cette fonction de partition est le nombre des partitions {fca}, où ka sont des entiers non négatifs tels que »; = 53o€A+ kQa. Deux concepts clefs qui sont liés doivent être introduits dans l'étude de la question de réductibilité. 23 l.r I.» 1) Un vecteur singulier est un vecteur \ € M(A),, pour »; ^ O, qui est lui même un vecteur de plus haut poids e,|\>=0 f = l,....n (2.30) Ce vecteur engendre un sous-module de Verma. L'existence, donc, d'un vecteur sin- gulier dans un module de Verma signifie que ce module est réductible. Il est facile de se convaincre aussi que dans chaque module réductible il existe un vecteur singulier. 2) Une forme contragradiente est une forme F : M(A) ® M(A) —» C. Pour la définir on commence par la définition d'un espace unidimensionnel E\ engendré par |A >. Il existe une involution a sur l'algèbre de Kac-Moody / \ e i F \ ^ /1 _ \ ^ i /e\ Q^ \ On désigne par PrA(a-) la projection de a-|A > pour x € H(g) sur E\. Nous appelons I t'isomorpmsnie de EA et Ç. La forme contragradiente est Si x et y ne sont pas des membres du même sous espace M(A)17 la forme contragradiente s'annule trivialement. Il n'est pas difficile de démontrer que det M(A)17 s'annule si et seulement si il existe un vecteur singulier dans ce sous espace. L'annulation du déterminant est équivalente à l'existence d'un vecteur singulier dans ce sous espace et à la réductibilité du module de Verma. Théorème: (Kac-Kazhdan [U]) OO p( rQ) det M(X)^= (const) JJ J] (X(ha) + p(ha) - ±r < a,a >) '- (2.33) r=l En conséquence de ce théorème il s'ensuit que M(A) est réductible si et seulement si il existe une racine positive telle que 2(A + p)(/iû) = r Si a est une racine réelle < Q,Q >^ O, on peut écrire cette condition d'une façon plus suggestive S£A - A = -m (2.35) 24 La signification géométrique de cette formule est transparente: SJA-A est proportionnelle à Q. Le théorème de Kac et Kazhdan affirme que la constante de proportionnalité est un entier négatif. Nous allons voir dans la suite que le groupe de Weyl joue un rôle important dans la construction de vecteurs singuliers. Prenons l'exemple de s/(2) affine: La matrice / 2 -''\ de Cartan de cette algèbre est I J^ 0" 1 et les racines positives sont dans une des trois classes 1) SQi +(S - 2) (S-I)QI + 3) SQj + SQ2 où s est un entier positif. Dans la première classe: A(A0) = «A, +(S P(Jt0) = 2* -I (2.36) < Q, Q > = 2 La condition de réductibilité est donc i) sAj + (s - 1 JA2 + 2s - 1 - r = O et d'une manière analogue on obtient de la deuxième classe U) (s - I)Ai + sA2 + 2s - 1 - r = O et de la troisième classe m) Ai +A2 + 2 = O (2.37) où on utilise s ^ O. Théorème: (Malikov, Feigin et Fuchs) 1 ) si A = ( AI , A2 ) satisfait la condition i ) et seulement i) pour un couple r, s, le module de Verma est réductible et le sous module propre est engendré par un vecteur singulier unique à un facteur global près: Fl \\ fr+(»-l)t fr+(J-2)t fr+(a-3)t fr- (»-2)< ,r-(»-D«| \ \ ^ It 1R\ r (AI = h h h ---h h |Ai,A2> (4.Jo) où A, =r-l -(s-l)t (2.39) A2 = -r - 1 + st 25 2) si Ia condition ii) et seulement U) est satisfaite on change /i en /3 Remarques: 1. On a utilisé une représentation paramétrique de A par t, où t = Aj + A2 +2. Le cas iii) correspond à t = O. Ce cas est particulier et plus compliqué et nous ne le traiterons pas. 2. Comment faut-il comprendre l'expression (2.38) ? Quel est le sens des exposants négatifs ou complexes? La réponse est que cette expression doit être comprise comme un symbole et non comme une description operative. Malikov, Feigin et Fuchs ont trouvé un ensemble infini de valeurs de t telle que cette expression a un sens et, pour ces valeurs là on peut l'écrire dans la manière suivante r( 1) | F(A)= £]Pp.,(/,,/:,,t)/a '~ - 7r~'|A> (2.40) JM=O où Pp1, = O si p > r(s — I) ou q > rs. Les polynômes Pp,q appartiennent à la sous-algèbre 1 ^([fl ! 01) C U(g) et ils sont définis par cette équation. La dépendance de ces Pp Il est difficile en général de trouver la forme explicite de ces Pp (2 41) qui sont faciles à vérifier pour tout les x € IN, et les prolonger analytiquement pour les valeurs complexes a- € C. Cette méthode n'est pas très commode et nous allons voir par la suite qu'il existe un moyen plus rapide et plus facile de trouver l'expression explicite des vecteurs singuliers. 3) Nous allons voir le sens géométrique de la forme que MFF ont donnée aux vecteurs 1+1 singuliers. Par les relations de commutation (2.41) on peut vérifier que /x |Ai, AZ > 1+1 est un "vecteur singulier" c'est à dire que formellement Cj^ IAj1A2 >= O . Ce vecteur est le vecteur de plus haut poids | — 2 — A1, A2 + 2A1 + 2 >. De façon analogue on obtient pour A2 Aa+1 /2 |A, , A2 >~ |A, + 2A2 + 2, -2 - A2 > (2.42) 26 Nous dénotons .-i(A,. A2) = I -2- A1. A2 +2A, +2 > (2.43) -.'(A1. A, ) = |A, + 2A2 + 2. -2 - A., > 1 + 1 Remarquer quo ••>i(A,.Aj) = S% A c'est à dire que les opérateurs S1 = /, sont les réflexions de Weyl sur (A,. A2 > et ils transforment un vecteur de plus haut poids en un autre vecteur do plus haut poids. Il va de soi que cette opération a un sens seulement pour A]1A2 € IX et dans ce cas on a une représentation graphique de l'action de ces opérateurs Fig. 2.1 Les réflexions de Weyl laissent la droite A, + A2 + 2 = t invariante. La réflexion si est la réflexion de cette droite autour du point a, et la réflexion «2 est la réflexion de celle-ci autour du point b. Par exemple dans la Fig. 2.1 $2 transporte le point 1 en 2 et si transporte le point 2 en 3 etc. Les orbites du groupe de Weyl sont caractérisées par les points sur la ligne située entre a et b (dans le quart positif). Les réflexions successives autour de a et b donnent des vecteurs singuliers de niveau de plus en plus élevé. Les vecteurs singuliers dans ce langage sont les réfléchis Si... SiS2S]S? • • • 5>|Aj, X^ >. Lorsque les A< sont complexes il faut être prudent parce qu'en général ces expressions n'ont pas de sens. Néanmoins xtne de ces réflexions a un sens et c'est précisément l'expression que MFF ont donnée. On peut généraliser cette construction pour une algèbre de Kac-Moody quelconque. Soit g une algèbre de Kac-Moody de rang N et W son groupe de Wcyl. Nous représentons un élément w de W par les réflexions associées aux racines simples «• = Sin Sin _, ...Si1. Soit A G H*, on introduit une série A0, AI , . . . , An par A = A0 ..... Aj ; = Sf. \j.i ..... An = SfnAn-, (2.44) Le vecteur Aj_i — A^ est proportionnel à la fonctionnelle < a,,, • >. On appelle 7^ la constante de proportionnalité. Théorème:(Malikov, Feigin et Fuchs) Le vecteur singulier dans M(A) (pour les valeurs de A données par le théorème de Kac et Kazhdan) est /""'-"/iiA> (2-45) Remarque: Comme dans le cas de sl(2), il faut comprendre cette expression d'une manière symbolique, le vrai sens étant = ; Ph. . .,„ /f'-" .../#" '"!A > (2.46) j.=o où Qj = — St, =ï 7s sont des en*iers pour les valeurs de A données par le théorème de Kac et Kazhdan. Pour finir nous présentons le "dictionnaire" entre la base de Serre- Chevalley et la base de l'algèbre des courants qui est la plus utilisée par les physiciens. Nous prenons une fois de plus notre exemple favori sl(2) affine et définissons = Jo- fci- 2J0° (247) + /2 = J ! fe2 = A--2J,O° On dénote |A > par | J, f). Les conditions sur un vecteur de plus haut poids sont J1" | J, t) = O pour 2n + a > O et (2.48) <*-2)| J1*) 28 Nous obtenons en conséquence les relations suivantes A, = 2J (2.49) A2 = * - 2 - 2J Tous les autres modes sont les commutateurs des modes que nous avons donnés. Le théorème de Kac-Kazhdan se lit maintenant avec les valeurs de J suivantes 2) 2Jril,,_ + 1 = -r + et 3) t = 0 Le théorème de MFF donne, dans ce langage, le vecteur singulier correspondant au cas 1) par )r-("-l)t\J,t) (2.50) Un changement de base similaire existe pour une algèbre de Kac- Moody quelconque. 2.3. Lts vecteurs singuliers et les règles de fusion Nous allons définir dans cette section une fusion entre deux champs primaires de la théorie conforme de type sl(2) affine, et nous utiliserons des méthodes standard de la théorie des champs conformes pour trouver les vecteurs singuliers et les règles de fusion. Il est commode d'organiser les champs primaires de l'algèbre sl(2) affine sous une forme un peu différente afin d'avoir une réalisation de l'algèbre horizontale par des opérateurs différentiels au lieu de la représentation matricielle. Les champs primaires sont définis, par exemple, par leurs relations de commutation avec les coefficients de Laurent des courants conservés. Nous avons vu dans la section 1 que celles-ci sont où R" est une représentation matricielle de l'algèbre horizontale. Remarquez que pour une valeur générique de j la représentation est de dimension infinie. Nous suivons Fateev et Zamolodchikov [12] et introduisons une variable auxiliaire x qui nous permet d'utiliser des opérateurs différentiels au lieu des matrices infinies. Nous définissons n=0 29 ^'^X- .*F• " ces al 'èbres. Nous e o s ' ' t nou assons directement à la discussion I et les relations de commutation sont (2.52) Les R" forment une réalisation différentielle de l'algèbre sl(2) horizontale. La relation entre les opérateurs et les états de cette théorie est donnée comme d'habitude par HmUmo^(3,*)|n,«) = M (2.53) où |fi,t) est le "vide" invariant de sl(2,R) (c'est à dire que c'est le plus haut poids du module FO,J quotienté par le module engendré par le vecteur singulier J0" |0, /)). La relation entre l'algèbre sl(2) affine et l'algèbre de Virasoro est donnée par la con- struction de Sugawara \K-mJ+ =) (2-54) - / Le charge centrale est c = 3(< - 2)// et (2.55) ïl est naturel de prendre L-\ comme le générateur des translations de direction z et J0" comme celui de direction x. On peut, donc, écrire (2.56) Par un calcul direct on obtient (2.57a) •fa.r' (2.576) (2.57c) et pour Virasoro d_ (2.58) ~'d: On normalise les champs primaires selon (2.59) 30 érivée schwarzienne II est plus commode, pour la définition de la fusion, de récrire )j.-Jt;-Lo (2.60) Cette forme est tout à fait cohérente avec les relations de commutation (2.57a, b, c) . En effet (2.60) est la solution globale de (2.576) et la forme (2.56) est la solution globale de (2.57o). La fusion s'écrit maintenant de la manière suivante /f\ /»T \ } Nous utilisons l'hypothèse de base de la théorie des champs conformes qui nous permet, au point critique, d'écrire tous les opérateurs comme des combinaisons linéaires des champs d'échelle, et en particulier o>i x> oo 0>°d, D^-(O, o)|n,o = £q ]T £ J où »/>o,o(0, 0) = 0 (0,0) et les autres «/>n,m sont ses descendants WJ- -(* + „*, Jî*j.. = (>-•" Wi1- Cj°}l est une constante qui s'appelle la constante de structure. Nous allons voir dans la suite que la comme sur j est contrainte par l'existence des vecteurs singuliers. Il faut se rappeler aussi que le symbole de sommation doit être compris comme une intégrale pour les cas où Cj°'Jl ne s'annule pas pour des intervalles continus de j. Le résultat de cette discussion est m(0,0) (2.64) j n=0m=— n Nous prenons un exemple pour illustrer comment les vecteurs singuliers contraignent les valeurs de j à un ensemble discret et fini. Nous allons reproduire les résultats bien connus sur les modèles intégrables de WZW. Dans ces modèles 2 9 t > — 2 et 2j < t — 2. Deux des trois conditions présentes dans le théorème de Kac et Kazhdan sont satisfaites. Les conditions sont r 2J r.s,+ + l=r-(5-l)t (2.65) 2 Jr> iS.,- + l = -r'+s'< 31 Drinfeld et Sokolov ont remar ué t vie cette forme .*>~ où nous prenons s = a' = 1 et on dénote j = jr,»,±- On obtient que r — 2j + 1 et r' = t — 2j — 1. On a par le théorème de Malikov Feigin et Fuchs que les vecteurs 2j+1 2 ( JJ" ) |j, t) et ( Jl1 )'~ J~' |j. /) sont les vecteurs singuliers. Il est facile aussi de le vérifier directement. Si jo et ji e Z/2, on a l'identité 0>°(r,.T)(Jo")2>l+Vl(0,0)|n,f) = O (2.66) On commute JJ" à gauche à l'aide de la relation de commutation (2.57a) et on obtient 2>1+ 0 JI (J0" - ^) V (2,z) Ou change a- —> — a- et ; —» — z et on multiplie à gauche par elj* +ïi-' dT (2.68) x (e:l-l+lJ° eiL_,+rJ0-( j- + * ^,+Ig-ïL., -xJ- _ (^.)2>t+l (2.69) dx dx et 8L t+xJ ïL 1 J ( e - û"^*>(_z,_T)^>«(0,0)e- - "* «"|ft,*> = ^ (z^)^°(0,0)|n,t) (2.70) On a donc o = () et par la fusion ,9 -.,% OO OO n=0m= — n On sait que ^'0,o est différent de O. On a, donc, par l'expression (2.71) la contrainte (-3 + Jo + Ji X-J + Jo + Ji - 1) ... (~j + Jo - Ji )q°Jl = O (2.72) d'où on déduit que j € Z/2 et Ij0 - Ji | < j < Jo + Ji • On a mis la valeur absolue à gauche parce que dans |ji) il y a un vecteur singulier du même type. Pour l'algèbre sl(2) finie 32 Une autre observation remarquable [18] est que "la quantification" de Qn donne une l'histoire finirait là, mais pour l'algèbre affine on a des vecteurs singuliers supplémentaires. On répète le calcul pour eux 10 2 On commute à gauche - (2 74) La même analyse que ci-dessus donne la contrainte ( -J ~ Jo + Ji )( - j - Jo + Ji + 1 ) • • • ( -j - Jo + Ji + < - 2Ji - 2)Cf J'' = O (2.75) on dénote k = t — 2 et on a les deux contraintes IJi - JoI < J < min(j0 + Ji , & ~ Jo - Ji ) (2.76) qui est le résultat connu[l2] [13] Jusqu'à maintenant nous avons vu comment les vecteurs singuliers fixent les règles de fusion. Nous allons voir qu'on peut aussi trouver les vecteurs singuliers à l'aide de la fusion. On fixe d'abord la forme des t/'nm P8* 1& covariance par rapport au sl(2) affine. Le principe de covariance dit que les deux membres de (2.71) doivent se transformer de la même façon sous l'action des générateurs de sl(2) affine. En particulier = O (n,a) = 2n+a>0 (2.77) La même stratégie que ci-dessus donne les équations ( ~ J + Jo + Ji + m + 1 )_p,m+1 P > (2.78a) (J - Jo - m W,-,,m P > 1 (2.786) (J - Jo + Ji - m + 1 )V'Lp,m-i P > (2.78c) qui s'appellent les équations de descente. En effet seulement deux de ces équations sont + indépendantes, celles de J0 et de Jf. On obtient toutes les autres par les relations de commutation. Ces équations fixent uniquement les 0^n,, pour les valeurs arbitraires des + J> Jo et ji, tant que le noyau des J0 et J-Ij est trivial. L'existence d'un vecteur singulier 33 b -» b + 6b tel que s = &' - \b2 reste fixé. Nous traitons d'abord, pour des raisons de simplicité, le cas d'un seul «v La forme la plus énérale d'une différentielle de de é R dans un module j montre que !<• noyau n'est pas trivial et. à ce moment là. il y a une contrainte sur les valeurs possibles dos j.ju et jj pour que la fusion soit possible. Ces contraintes .sont !«•> règles tic fusion. Nous utilisons l'équation df Knizhnik-Zamolodchikov [14] pour donner à ces équations une forme triangulaire et pour obtenir des informations supplémentaires sur les conditions de solvabilité de ces équations. Oi; a. par la construction de Sue;a\vara. 4»(z.x) I_, - y L n€Z (2.79) Nous utilisons les relations de commutation des champs primaires avec les coefficients de Laurent de l'algèbre sl(2) affine et de l'algèbre de Virasoro pour les commuter à gauche et après un calcul long mais direct nous obtenons (nt 1 ) 11+1 +2(J-J -m) 0 (2.80) +(./• - jo+ji - m+ •TA-, *+<*"> Ceci est notre résultat principal et nous allons voir qu'il nous donne beaucoup d'information sur les vecteurs singuliers et. sur les règles de fusion. Nous introduisons, d'abord, un ordre partiel sur les couples (7?,???) tel que (n.m) -< (n',m') si les deux conditions n > n' et n + m > n' + n' sont satisfaites. Tous les ?/'£,, dans le membre de droite de (2.80) sont plus petits que ti^nm dans le membre de gauche. On a. donc, une équation de récurrence, et d'une manière graphique On voit immédiatement que .Y(S. T) - N(S, T - 1 ) > O (pour S > O), et que si > S2 si et seulement si 5r(.~i ) > K(SI). On fait maintenant un changement de variables en (3.41) I est ^'im déterminé pour tous les «>jjj? qui sont entourés dans la Fig. 2.2. Ces «/>£ 9 sont plus petits que <;'/,,,„ par l'ordre partiel que nous avons défini. Dans le membre de gauche de (2.80) on voit le coefficient Aj(n,m) = ni + m(2j + 1 - m) (2.81) L'annulation de A}(n,m) est une condition nécessaire (mais elle n'est pas suffisante) pour la réductibilité du module Vjit. Pour le voir, imaginez qu'il y a un vecteur singulier |\n,m) dans le sous espace homogène de type (n,m). On a, par la construction de Sugawara. = (U - JS(JS+ l))\Xnm) 0 (2.82) = (t(h + n)-(j- m)(j - m + l))|vnm> d'où ni - 77?(2j + 1 -m) + th- j2 - j = O (2.83) mais h = yj(j + 1), et on obtient Aj(n,m) = O. Le théorème de Kac et Kazhdan nous dit qu'il faut en plus ajouter la condition que m divise n pour avoir des conditions nécessaires et suffisantes. Nous avons donc trois possibilités, et nous allons voir que l'équation (2.80) a quelque chose à nous dire dans chacune d'elles. • Si .4j(77,m) ^ O pour tous les couples (n,m) alors Vj,t est irréductible et l'équation (2.80) nous permet de calculer pouj chaque (p, q) le descendant 4'pq- On peut, dans ce cas, diviser l'équation (2.80) par A'nm et vérifier d'une manière recursive que ^n, est une solution des équations de descente. Il n'y a aucune contrainte sur la valeur de j,jo et ji ( sauf si jo ou jj sont réductibles, nous reviendrons sur cette possibilité dans la suite). • Si Aj(n,m) = O pour un couple (77,177) mais JT? ne divise pas n alors Vj,t n'est pas réductible. Nous agissons avec les opérateurs j£ et J1" sur le membre à droite. Nous utilisons les équations de descente et les relations de commutation pour trouver que le membre de droite de l'équation (2.80) est dans le noyau des ces deux opérater/s. Il est, donc, égal à zéro parce qu'il n'y a pas un vecteur singulier dans un module irréductible. • Si Aj(n, JTJ) = O et m divise ??, on est dans le cas où le module Vj,t est réductible et nous avons un candidat sérieux pour être le vecteur singulier: le membre de droite de l'équation (2.80) . Des vérifications sur les valeurs basses de (n,m) confirment que 35 Prenons enfin le cas S = O. Il est clair que pour T > O, .V(O, T) = O et que N(O, O) = 1. Nous compensons alors le coefficient négatif dans (3.50) en ajoutant à cette expression q. En prenant x égal à qh (parce que R — liS + T) nous obtenons finalement 1.1 • le membre de droite ne s'annule pas identiquement et il est proportionnel en effet au vecteur singulier. Plus précisément Le de droite = Pnm(j, Jo- Ji )|\»»i) (2.84) où Pnm(jjoiji) sont des polynômes des j,j0 et ji et |\,,,m) est le vecteur singulier. Pour les valeurs basses que nous avons calculées il apparaît que ces polynômes sont précisément les règles de fusion!! C'est à dire que = Pnm(j,jo,jï) (2.85) L'annulation de ce polynôme nous donne les règles de fusion et garantit l'annulation du membre de droite dans l'équation (2.80) . On est dans la situation absurde où pour trouver le vecteur singulier et les règles de fusion on commence avec une fusion interdite qui fait que (2.80) n'a pas de sens parce que le membre de gauche est nul et le membre de droite ne l'est pas. Néanmoins cette absurdité nous donne l'information qui nous intéresse: le vecteur singulier et les règles de fusion. Deux exemples simples illustrent le mécanisme: dans le module VQ,/ il existe un vecteur singulier au niveau (n, m) = (0,1). Il est facile de voir que c'est |\oi) = J0~|0,f). On a par l'équation (2.80) (membre de droite)0] = (-J0 +Ji )J0~I On a obtenu le résultat bien connu que seule la fusion d'un champ primaire avec lui même contient l'identité. La deuxième exemple est celui du module V_ i f>f. Le vecteur singulier est l\ii) = -i - *2 J-i I - *' (2.87) et un calcul direct par l'équation (2.80) donne (membre de droite)n = -(j - J0 + ji )(j + J0 - j\ )(j + J0 + Ji - l)lxn) (2.88) où j = -^t. On peut vérifier facilement que le coefficient de \\u) est précisément la règle de fusion qui vient du vecteur singulier \\u). Il est commode parfois de choisir 36 u» Fig. 2.3 un J0 et/ou un j'j particulier(s) qui contrai(gne)nt une partie des il'i,m à s'annuler identiquement. Par exemple si on prend j\ — ^ on a la contraint e( voir (2.71)): (} ~ Jo " - »' KJ - Jo - - '» + !)(/'/,„, = O (2.89) Si on prend par exemple le canal j = J0 + 5 alors forcément =0 (2.90) qui nous indique que ^j111 = O V»7? > 1. L'équation de récurrence (2.80) est, par conséquent, tronquée (voir Fig. 2.3) II est intéressant enfin de remarquer qu'on peut donner aux vecteurs singuliers une forme matricielle. On organise les ojL dans un vecteur colouue de la bonne dimensionnalité (2.91) selon un ordre approprié. On définit aussi F = (\nm.O O)' (2.92) de la même dimensionnalité. et on peut écrire F = .V// (2.93) où les éléments de la matrice J\/ sont les opérateurs dans l'algèbre sl(2) affine. 37 Nous illustrons la méthode par les exemples j = —(n — l)/2 et J0 = 1/2, Ji = —1/2 — ((n - l)/2)f. Notez que j\ n'est pas de la forme du théorème de Kac-Kazhdan et est donc irréductible. Nous prenons aussi j = jo + j\ et nous définissons f]+n — JU)_^"'0)^ (2.94) / est un vecteur 2n — 1-dimensionnel / = (fj+n-l,9j+n-l,fj+n-2, où fj = V^0 = LM)'et F = (-0>+n, 0,...,O)' (2.96) où 1 F= [J- + Y1 R3+^h"]/ (2.97) \ »=o / ou /T~\.. c.. •; 10 9« (2.98) où [x] est la partie entière de a: et ^2, fl ê=jmod2 (29£) IJ IO sinon La somme en (2.97) est finie parce que A"2"-1 ~ O. On peut aussi démontrer directement que _. i f ., = = r ^O fj+r O •/! /j+r ~ «7j+r ... ._ (2-100) Hr = O c'est à dire que g}+n est le vecteur singulier que nous cherchons. Il est différent de O parce ll+1 que le coefficient du terme ( Jl^ )"( J0" ) est 1. Un exemple clarifiera la structure de cette 38 i.» matrice. Pour j = —2 on a 3 4 5 6 J -3J-. J -6J Ji 12J JI, 12J Ji -36J JI X ^ /l\4.l)\ ( O -3/'Jr1 2 3 3 /f /7r>y° 2 3 4 5 O J+ JJ-ii -2J Jl2 -4J Ji2 4J Jl3 12J Ji3 0 2 3 4 9j+2 O O 1 J- 2JJ , -4J Jr1 -4J Ji2 12J JI2 /J+2 O = O 1 2 3 O 1 Jl1 2JJi1 -2J Jl2 -6J Ji2 2 O fj+l O O O 1 J0- 'Jî, -3J JI1 O 0 O O O O 1 Jl1 3JJ , 9} V o J V V 0 O O O O 1 J0~ > fi ' (2.101) On a en général sur la diagonale j - i + 1 = N les opérateurs de degré In + a = N multipliés par ÎN. Sur la diagonale N=O on a le nombre 1, et en dessous il y a des zéros. Quand N est impair on a les opérateurs J° N+l et pour les N pairs il y a deux possibilités + J~JV-1 et J JV+,. Les deux apparaissent dans une façon alternée et le nombre des J~ est plus grand que celui des J+. Les constantes numériques sont données par la matrice K et ses puissances où /O (n-D O O \ O O 1 O O O O O (n-2) O ... O O O O O 2 O ... O A' = (2.102) O O 1 O O O (" - D Vo , . , O / Cette forme des vecteurs singuliers est particulièrement intéressante à cause de la ressemblance d'un côté avec les vecteurs singuliers de l'algèbre de Virasoro et de l'autre côté avec les algèbres w classiques. Les algèbres w classiques sont précisément notre préoc- cupation principale dans le chapitre 3. 39 fS 3. LES ALGEBRES W 3.1. Le.f algèbre» W quantiquex Nous avons considéré jusqu'ici des théories conformes comprenant le tenseur d'énergie- impulsion (qui est de dimension d'échelle 2) soit seul et il s'agissait d'une théorie conforme stricte, soit en compagnie de courants de spin 1, et il s'agissait des théories conformes éten- dues de type Kac-Moody. Il est naturel de se demander quelles sont les théories engendrées par les systèmes d'opérateurs de spins entiers plus grands que 1 (T(z), W3(z), . . . , H'n(z)}. Ces systèmes ont été introduits par Zamolodchikov [15] et on les appelle algèbres W. Il- lustrons par un exemple comment la structure de l'algèbre est fixée par les spins et la condition de l'invariance conforme. L'algèbre Wy a deux générateurs {T(z), W3(z)}. En termes de produits d'opérateurs on a c/%' •' d T(:)T(w) = - -^ - + - - ^-T T(W) + - T(W) + termes réguliers (3.1) (; - H')4 (: - w)2 qui est simplement le constat qu'il s'agit d'une théorie conforme, et . 3 .,) + . Iy (U.) + ' ' ' (3-2) (z-w)3 (z-u<)22 3 qui vient du fait que TV3(r) a un spin 3. Il faut enfin calculer le produit de W3 avec lui-même. Il est clair que 4 1 4 + (= - W) Q3U T(W) + (z- w) /34A.(w) + (z- wf fa(?T(w) + (z- w)5fad\(w)} + • (3-3) 2 où A(Z) =: T(z)T(z) : — -^d T(Z ). /iu dépend du choix de normalisation de W3. On prend par convention /?o = c/3. Les autres coefficients /^ sont fixés en comparant entre elles les identités de Ward suivantes (T(Z1)- (3.4) 40 où un chapeau sur un opérateur signale que cet opérateur est omis de la fonction de corrélation. La deuxième façon de calculer la même fonction de corrélation est par l'identité ~ / 3 - *•**! i ff>m > v ^TT/ \ T1/ \¥ir/ v \TLT I \\ _ _ v( ;,-)••-T(Cn)Ir (W2/-•• Ir (j»m)> •=i VV"M ""'* '"" ~" ' m 6 , _,)g"- t=2 ff=0 * ' "'' ' (3.5) II suffit de prendre m = 2,3 pour fixer les six paramètres /Jj. Le résultat est 11 1 ft + 7 - (TP^T(U') + ^A(u')} + termes réguliers C — 1C IO — — T" vC En termes de coefficients de Laurent 2 T(Z) = VLn.'"- 3 (3.7) i.n--) = £w_n--"- l'algèbre obtenue est [Ln, Ln] - (n - m)Ln+m + ^n(n* - l)6n+m,0 (3.8a) [Ln, Wn] = (2n - m)Wn+m (3.86) [W , W ] = 0(n - ?77)A + (n - m){~(n + m + 2)(n + m + 3) n n I1+m lo 2 - i(n + 2)(m + 2)}In+m + ~(n* - 4)(n - l)n*n+m,0 (3.8c) où 0= 16/(22 4- 5c) et An = .rnln + : LmLn-m : (3.9) OÙ *2m = i(l ~ »»2) *2m-i = ^d + m)(2 - m) (3.10) O O Cette algèbre n'est pas une algèbre de Lie parce que dans le membre de droite de (3.8c) il y a un terme quadratique dans les générateurs Ln. Le module de Verma est construit sur 41 le vecteur de plus haut poids. Nous définissons d'abord le "vide" invariant de la théorie par les conditions T(c)|Q> = régulier (3.11) W( c)|Q) = régulier qui se traduisent par les conditions (3.12) wn IQ) = O 7i >-2 Un vecteur de plus haut poids est défini par l'action d'un champ primaire sur le "vide" dans la limite z —» O |/(, «>) = Hm 4>h,u'(-)|Q) (3.13) Ce vecteur est défini aussi par les conditions Io|/i, te) = h\k, «') In|/i,10) =0 n > O (3.14) H'o|/i, «') = u'|/i,u') Wn\h,w) = 0 n > O Les vecteurs de base du module de Verma sont de la forme W-kt ... W-krL.mi ...L.m.\h,w) (3.15) Pour des valeurs génériques de la charge centrale et de (h,w) le module de Verma est irréductible et son caractère est h /1 0 ti \,,.3(ç) = Tr1x..,-/ = -FfOcT-JT nj (3-16) ll/=o(l ~ 9 ) Pour le module du "vide" il faut soustraire les contributions qui viennent des L-i,W-j et \V'-2- On a, par conséquent, ( }, La théorie des représentations pour les algèbres W a été beaucoup étudiée ces dernières années [16]. Néanmoins, on a obtenu plusieurs résultats par analogie avec le cas de l'algèbre de Virasoro mais pas ou peu de preuves rigoureuses. On manque aussi d'une compréhension géométrique de ces algèbres. Mais, en admettant ces résultats on sait calculer les fonctions de corrélation des modèles unitaires et identifier des modèles de spin qui correspondent à 42 I ces algèbres. Nous ne donnerons pas de détails et nous passons directement à la discussion des algèbres W classiques. 3.2. Les équation* différentielle» linéaires covariantes On dénote par f\ l'ensemble des différentielles de degré A [17]. Ceci signifie que ce sont des objets représentés dans deux systèmes de coordonnées .T et à1 par les fonctions / rt / telles que = f(x)dx* (3.18) Soit Dn un opérateur différentiel qui s'écrit dans le système de coordonnées x comme 2 3 Dn = d" + a2(:r K/"- + a3(x)d"~ + •••+ an(.r) (3.19) où d = djdx et le coefficient a\ est pris égal à zéro. On peut toujours le faire par la transformation Dn -* g~*(x)Dng(x) où g(x) = exp{£ f f/f«i(£)}. On démontre que Dn est un opérateur covariant Dn : F_«çi ~* ^n_+l (3.20) On a par conséquent des lois de transformation assez compliquées pour les coefficients a,. Pour les trouver il faut d'abord écrire explicitement la loi de transformation de Dn. Puisque Dn f € -Fn+i on peut écrire = Dn(x)f(x) = où nous avons utilisé dans la deuxième égalité le fait que / € T_«=±- Cette loi de trans- formation se traduit par des lois de transformation pour les coefficients a;. On prend par exemple n — 2: (3.22) La transformation a- —» * entraîne la relation 3 = d] ' -dxjY (3.23) =rf? 43 où {f.x} est la dérivée schwarzienne fit /,Iv3 Z /tfltlflr'*\'2\ (3.24) et on obtient rt et pour n générique un calcul similaire donne + "t"'"1'{f..r} (3.26) Cette loi de transformation n'est autre que la loi de transformation du tenseur d'énergie- impulsion de la théorie conforme! (eq. (1.34) ) Les lois de transformation pour les autres a, sont beaucoup plus compliquées. Il est avantageux de réexprimer Dn par des objets qui ont des lois de transformation simples. Ce sont en général des polynômes différentiels des coefficients a;. Par un polynôme dif- férentiel nous entendons un polynôme des variables et de leur dérivées. On obtient une telle représentation par le formalisme matriciel. 3.3. Le formalisme matriciel Nous représentons, dans ce formalisme, l'opérateur Dn comme une matrice Ji x n (3.27) II est clair que les noyaux des Dn et Dn sont isomorphes parce que pour chaque vecteur / € KerDn le dernier élément /„ € KerZ?n Dn = : = O =» Dn fn = O (3.28) et inversement pour chaque / € KerD,, il existe un vecteur unique / avec un dernier élément / tel que / € KerPn. 44 l.r Drinfeld et Sokolov ont remarqué que cette forme (3.27) n'est pas la seule a préserver -1 7 cet isomorphisme. Tous les Q1, = .V D11A , où N est la somme d'une matrice triangu- laire supérieure avec la matrice d'unité, le préservent. Ces transformations peuvent être considérées comme des transformations de jauge. On dénote /O 0\ 1 (3.29) Vo i Q) Dn prend la forme D11 = - (3.30) et 1 1 Qn = A- D11A' = -J + d + (N- U0N + N-* (dN + [N, J])J (3.31) où a est toujours une matrice triangulaire supérieure. Inversement toute matrice triangu- laire supérieure a est équivalente de jauge à une matrice Ci0. Nous changeons la jauge de telle façon que a prenne une forme particulièrement commode: Nous définissons la matrice ( J+)ij = i(n — i)6j.i+i et en termes de cette matrice n-l 1 Qn = N- DnN = - (3.32) k=i On peut démontrer que la covariance de Qn est équivalente aux propriétés (3.33) On a obtenu, donc, une représentation de Qn par des objets géométriques-lés différentielles «'it ((k > 2), «'2 ayant l'anomalie de la dérivée schwarzienne). On introduit aussi une structure de crochet de Poisson naturelle sur l'espace des fonctionnelles linéaires sur Qn et on calcule les crochets de Poisson {u>n,u>t}. L'ensemble {u'z, • • •, «'„ } plus le produit entre les «', défini par le crochet de Poisson engendre l'algèbre w classique. Les éléments de cette algèbre sont des polynômes différentiels des «>*. D se trouve que l'algèbre obtenue est identique (à une normalisation de coefficients près) à celle des produits d'opérateurs Wn(X)Wk(U)*- 45 racines de l'unité. Nous considérons le tore comme un parallélogramme dont on identifie Une autre observation remarquable [18] est que "la quantification" de Qn donne une classe spéciale de vecteurs singuliers de l'algèbre de Virasoro. On remplace les fonctions w k par les opérateurs tk~lL-k où 1 est lié à la charge centrale et à la dimension conforme par c^l3 + 6r + r') (3.34) et j est déterminé par n — '2j + I. On remplace aussi la dérivée par L-\. Les vecteurs singuliers dans ces modules de l'algèbre de Virasoro sont donnés par l'équation n-1 \ ( -J+£!_*_, «J+)M/ (3.35) k=0 / 1 où F = (F, O, ••• .0) et '/ = (/,, /;_i,---,/_;). F est, alors, le vecteur singulier dans le module où f-j est le plus haut poids. La forme matricielle pour les vecteurs singuliers de l'algèbre A\l* (voir chapitre 2 eq. (2.97)) est liée aussi à (3.32) par le changement Jo' «>„ !»0 Jtn -» àlM n > O (3.36) J°n -» O 77 > O La question que nous analyserons, dans ce sujet vaste et fascinant, est la suivante: Y a t-il des différentielles de degré R > n dans l'algèbre «',, classique? et si oui combien ? S. 4- Le3 différentielles dan» les algèbre» w II est commode, afin de trouver le nombre des différentielles indépendantes de degré R dans une algèbre wn, de travailler dans un système de coordonnées spécial u où Wy(U) = O. Cela signifie que « est la solution de M.2(a-) = cn{u,j.-} (3.37) ou autrement dit que la dérivée logarithmique du Jacobien b = j^(\ogdu/dx) est telle que = w2(x) = V(x) - (3.38) Puisque les objets que nous considérons ne dépendent que des coefficients de Dn (et ils ne dépendent pas explicitement de b), ils doivent être invariants par un changement de 46 où pour un réseau R: b —» 6 + 66 tel que s = 6' — ^t2 reste fixé. Nous traitons d'abord, pour des raisons de simplicité, le cas d'un seul M'A. La forme la plus générale d'une différentielle de degré R est ,r...<<')r (3.39) On dénote l'homogénéité en WA par S = £, «, et le nombre total des dérivées par T = ^isj. / est -^ne différentielle de degré R = h S + T. On change le système de coordonnées, et on obtient (3.40) où Dk = (d - (h + k)b)(d - (h + k - 1)6)... (d - hb) et b est comme plus haut la dérivée logarithmique du Jacobien de la transformation u —» x. On impose la condition 6f/6b = O, et on obtient, par conséquent, le système de contraintes (3.41) {».} i où Aj = Sj $Zi=0(/î + i) = Sj(J -f l)(h + ^j). Le nombre de différentielles de degré R pour des S et T donnés est, donc, le nombre des coefficients dans (3.39) moins le nombre de contraintes indépendantes. On dénote s s= {s0 , S1 , . . . , Sk } — une partition de S en k+1 entiers > O P(S, T) = {les partitions de type s de T) c'est à dire (3.42) «o fois 3i fois fois JV(S, T) = |P(S, T)| Démontrons que pour S > O le nombre de différentielles de degré R avec des S et T donnés est N(S,T) — N(S,T — I). Nous définissons un ordre: si > J-j si et seulement si si.fc > $2,k ou *i,t = S2,k et si |t_i > Jilfc-i, c'est à dire si la première différence Si1I — «2,1 non nulle est plus grande que zéro. On définit ensuite une application injective (3.43) ,O) = (S0,...,sk - 1,1) 47 Table I. Une liste des invariants modulaires connues de la théorie quotient SU(3)k-i x SU(3)\/SU($)k caractérisés par (G(n)); n = k + 3. On voit immédiatement que .Y(S. T) - N(S, T - 1) > O (pour S > O), et que J1 > S2 si et seulement si JT(S] ) > Tr(Jj ). On fait maintenant un changement de variables en (3.41) j- .Sj-J +1 J=J-I /', = < Sj - 1 J=J (3.44) I Si sinon. et on obtient les AT( 5,T-I) contraintes =0 (3.45) Chaque contrainte indexée par un r € P(S, T — 1) s'applique à des coefficients a indexés par f ^ P(S, T). Il est clair que le plus grand t qui apparaît dans la contrainte f est fmax = 7r(r). Notre système des contraintes peut être mis maintenant sous une forme triangulaire: .V(S.T) = 0 ^NiS,r-i) \ O O s. T-l),t- ,V(S T-D + 1 (3.46) où rj > f*2 > ... > r*/v(s.r-i)- Toutes les contraintes sont, donc, indépendantes et le nombre de différentielles de degré R pour Set Tdonnésest, par conséquent, N(S,T)- JV(S5T-I). Nous construisons une fonction génératrice pour ces nombres. Un terme générique dans l'expression suivante 2 (1 + x + a; + ...)(!+, + • • - + *V...)... (3.47) est Ts°(.r3 V1 K-I-1V*2 )••-=-I1 (3.48) d'où .S,.T JV(S,Î> (3.49) S1T=O et — = £ [JV(S,T)-JV(S,T-l)]ar (3.50) 2 ' s.r=o 48 Prenons enfin le cas S = O. Il est clair que poui T > O, .V(O, T) = O et que TV(O1O) = 1. Nous compensons alors le coefficient négatif dans (3.50) en ajoutant à cette expression q. En prenant x égal à qh (parce que 7? = hS + T) nous obtenons finalement q r R G( h ) = q + nœ ^~_ <+/l. = JT A (difFérentielles de degré ï\)q (3.51) La généralisation de ce résultat à plusieurs un- est immédiate. La forme la plus générale, dans ce cas là, est (3.52) ou (3.53) Ti Ii o (!)''•' (2)''.' (*!)''•*.• w,;. = «.;, wh. wl.' ...«'),." On définit un ordre fj > f2 si fn,i > f,,.j ou si fn.j = fn,> et /I|A— i > fzU-i- Nous avons aussi l'application injective I,. . . ,/„) = (*i,... ,<„_!, TT(^n)))) (3.54) On voit maintenant que la matrice des contraintes est triangulaire de la forme O (3.55) O O ... A où -4i,fc,+i sont des matrices triangulaires elles-mêmes. Ceci prouve l'indépendance des contraintes, et par le même argument que plus haut on obtient la fonction génératrice pour le nombre de différentielles de degré R g G(hi) = q + I"1 - J-J- = V ^(différentielles de degré R)9" (3.56) iii=i Ih=O^ 9 ') R=0 Nous allons voir que ces nombres, qui apparaissent dans des algèbres v classiques, sont présents aussi au niveau quantique. Prenons le cas de 1173 par exemple. Nous avons vu que le caractère du module du "vide" est (3.57) 49 5. MODELES INTEGRABLES SUR RESEAU ET REPRESENTATIONS DE L'ALGEBRE DE HECKE -\ On le décompose sur des modules de Verma VA pour un c générique et h un entier (3-58) T on constate que Cn = A ( Différentielles de degré n) parce que Co = 1 et les caractères de Virasoro sont vir = n>0 (3.59) X, oVi r" _ En utilisant ces formules on obtient G(S) = q = X, c»i" = de degré n)ç" (3.60) U=O n=0 en accord avec (3.56). Ceci montre une fois de plus qu'il y a une relation forte entre,d'un côté, les algèbres w classiques, et de l'autre, la théorie des champs conformes. Cette relation est entendue dans la "limite classique" c —» —oo, mais on vient de voir qu'elle existe aussi pour des valeurs génériques de la charge centrale. 50 1,1 Ll'- 4. BOSONISATION 4-1. La bosonisaiion à la Feigin et Fuchs Nous présentons dans cette section une description bosonique des théories conformes. Nous allons voir comment on construit des espaces de Fock d'un boson libre pour représen- ter des théories conformes avec c ^ 1 et comment les champs primaires peuvent être représentés en termes de ce boson. Pour des valeurs génériques des c et /i, l'espace de Fock est isomorphe au module de Verma de l'algèbre de Virasoro. Pour les modèles minimaux, la représentation irréductible de l'algèbre de Virasoro est plus petite que l'espace de Fock et il faut faire attention à ce que les états non physiques ne contribuent pas aux fonctions de corrélation. Commençons par construire l'espace de Fock. Soit 0 un champ bosonique (rtrWy)) = -log(.«'-(/) (4.1) Les coefficients de Laurent de • • V * 1 n / . ri\ (Jt(X) = q — /0ologa- — / y -CnX (4.2) Le "vide" est donné par les conditions 4>n (O) = O n > O (4.3) et les vecteurs de plus haut poids sont |A-)=e''**|0) (4.4) Ils sont donnés aussi par les conditions *B|*} = O n >0 L'espace de Fock F(k) est engendré par les vecteurs de base f(k) = eC<*-n, • • • Le tenseur d'énergie-impulsion est O2 (4-7) 51 i.-» L: I (en fait c'est une construction de Sugawara pour U(I)). Il est facil de voir que (4'8) Pour représenter des théories avec c plus petit que 1 on ajoute une charge à l'infini. Celle-ci change en même temps le tenseur d'énergie impulsion et la façon dont on calcule les fonctions de corrélation. On définit 2 En multipliant T par lui-même, on trouve C=I- 12Q2, (4.10) et pour les valeurs on obtient pg qui sont les valeurs connues clés modèles minimaux. Les dimensions conformes des opéra- teurs changent aussi. Par exemple (4.13) (c-w)2/ c-u; ao d'où on déduit que la dimension conforme de V0 = t' est ^Q(Q — 2ao). Chaque champ primaire $/, de la théorie conforme peut être représente par V0 ou par Vïao-Q où h = \ a(a -2a0). La charge à l'infini affecte aussi les fonctions de corrélation. La nouvelle règle de conservation de la charge est (V01 - - • V0n ) = O si at -,ï 2Q0 (4.14) d'où on pourrait déduire naïvement que = O (4.15) 52 lit' Nous savons par nu calcul direct (dans le cas d'Ising par exemple) que ce résultat est faux. La bonne prescription pour calculer les fonctions de corrélation a été proposée par Feigin et Fuchs [19] et réalisée par Dotsenko et Fateev [2O]. Elle est basée sur l'insertion d'opérateurs de dimension conforme nulle avec une charge non triviale. Les propriétés conformes de la fonction de corrélation restent, donc, les mêmes et la conservation de la charge peut être obtenue. Les opérateurs nécessaires s'appellent les opérateurs d'écran. Leur définition est (4.16) où ^e±(e± — 2a0) = 1. A l'aide de ces opérateurs, nous construisons Qnm =ldti-l II est important de noter que V011111 (on le note parfois V111n) où Qnm =5(1-11)«+ + J(I -m)e_ (4.18) est la représentation de $*„„, donnée par la table de Kac. On dénote par Fnm l'espace de Fock T{ an™)- La bonne prescription de calcul est, donc (V0nmVanmVa ..... Vi0o_aiimQn-i.m-i> (4.19) On voit que la charge est conservée £ Qi = 2oo- Le point auquel il faut faire attention est le choix des contours de Q,,-i,m-\. En effet, différents contours correspondent à différents blocs conformes de la fonction de corrélation {$/,„„, $*„„, $knm$hnm)- On peut vérifier directement sur des modèles simples, comme par exemple le modèle d'Ising, que cette méthode donne de bons résultats. C'est une justification a posteriori. Il n'est pas évident a priori que cette méthode soit la bonne parce qu'on travaille dans un espace de taille différente de celui de la représentation irréductible de l'algèbre de Virasoro. La preuve générale que cette méthode est correcte a été donnée par Felder [21]. 4-2. La cohomologie de Felder et la généralisation à SU('2) Felder a reformulé le problème comme un problème de cohomologie. Cette formulation a l'avantage de donner à la fois une preuve que la méthode de Feigin et Fuchs est correcte et une généralisation de cette méthode aux surfaces de Riemann de genre quelconque. 53 Los fonctions d<> corrélation dans ce formalism^ sont rxprinuV's en tc-rrnes d'operateurs do vertex l"'1* • T T II. Il' ' •'III. I"' •* II+ III—Jl — I ./!'-<• Ill'--'••— I (4.20) ' il.H' = ' n.n'Qr.a où les contours sont montrés sur la Fig. 4.1. Fig. 4.1 Les contours des operateurs de vertex. Felder a introduit l'opérateur nilpotent Q11, = Qo.m- II a démontré les propriétés suivantes • Qn, commute avec l'algèbre de Virasoro [Qm-Xf]=O V* (4.21) • Les opérateurs de vertex sont invariants par rapport à Q1n en ce sens que: r\ Tr'ir/ \ Vm-M-2r-l >V.n'-' = • II existe un complexe €,„',,„ p--. -, " T- "— T. ', — m+2p * -fin',m ' Jm'.- m * -r m' .m—2 (4.23) tel que Q,»Q,,~m = Qf-mQm = O (4.24) et la cohomologie (4.25) où JRh est la représentation irréductible de l'algèbre de Virasoro basé sur le plus haut poids /». 54 I On tire de ces propriétés la conclusion que les fonctions k =*) = const( VU1)) (4.26) ou 1 *'"' *"' (4.27) m'i-l = n't + 111', - 27'', - 1 ?7?i_| = 77, + 777, - 2/', - 1 sont les blocs conformes de la fonction de corrélation k "'i"i On peut démontrer que l'expression (4.2G) est équivalente à la prescription de Dotsenko et Fateev. La dernière propriété nous montre que la méthode de bosouisation ressemble aux théories de jauge. On calcule dans un espace plus grand que l'espace physique. Ce dernier est un sous-espace: l'espace de la cohomologie par rapport à la charge BRS dans les cas des théories de jauge, et par rapport à la charge Qn, dans les cas des modèles minimaux. La même méthode peut être appliquée pour l'algèbre Kac-Moody s/(2) [22]. On introduit le champ bosonique (- ~ W)2 (4.29) alors les courants définis par = 3(z) (4.30) où 2a2 = k + 2, sont les générateurs de l'algèbre Kac-Moody sl('2) [23] avec une charge centrale k. Le tenseur d'énergie-impulsion est 55 A l'aide des opérateurs d'écran r = /3e-'> (4.32) on construit l'opérateur nilpotent VH • J~n,n' * -r—n,»' , . (4.33) où fn,n' est l'espace de Fock avec un plus haut poids | Jn,«'). Ceci est le vecteur propre de JQ avec une valeur propre Jn,»' donnée par la formule de Kac-Kazhdan (voir chapitre 2): 2J,,,n> + l = 7?-??'(*•+ 2) n,n'£ Z (4.34) Les opérateurs de vertex sont • • ' -, Ll-' ~- *.*",•.«• € ^1,,,,. (4.35) /i*nn',//' = «=2 OÙ J 1 t,Vj(z) =: -) -"(z) : e'«* =' (4.36) et r = Jnni + Ju- — Jkk' • On a des propriétés analogues au cas de Virasoro • Qn commute avec l'algèbre des courants [Qn,^] = O Va,fr (4.37) • Les operateurs de vertex sont invariants par rapport à Qn : • Pour les modules réductibles, qui sont caractérisés par (4.34) on a le complexe Cnn'- "" *r> ***-n ^ "" ^ **f—n ^ "" y M OQl ' " ' * •'—n+2p,n' ' J"n.n' ' J~—n,n' ' J~n~ïp,n' * ' ' " \t.3a) tel que QnQp-n = Qp-nQn = O (4.40) 56 et la cohomologie (4.41) où Rj est la représentation irréductible de l'algèbre de Kac-Moody sl('2) basé sur le plus haut poids J. Nous avons utilisé ce résultat et l'analyse de vecteurs singuliers faite au chapitre 2 (cf. articles 2 et 3) pour étudier le modèle topologique sl(2)/sl('2). Le spectre de ce modèle a été identifié et il se trouve qu'il est identique à celui de la gravité bi-dimensionnelle couplée aux modèles minimaux (cf. article 5). 4-3. La boionisation des fonctions de partition La bosonisation des fonctions de partition signifie la décomposition de celles-ci en fonctions de partition de modèles plus simples-lés bosons libres par exemple. C'est une méthode efficace notamment pour comparer les fonctions de partition des modèles de spin sur réseau (voir chapitre 5) aux fonctions de partition des théories des champs conformes. Nous traitons les fonctions de partition diagonales de type (A,A) avec des conditions aux limites tordues, dans les modèles quotients SU(N) et des fonctions de partition dites exceptionnelles dans le modèle de quotient SU(Z). Les modèles quotients (An)i-n-ï * (•4n)i/(-4n)t_n_i sont équivalents pour I — n — 3 aux algèbres W. La fonction de partition sur le tore avec des conditions aux limites périodiques est un bilinéaire de caractères [5] lfi\i(r)x-t(T) (4.42) ou W est une représentation, JV1-; compte combien de représentations de type IVj x W-t existent dans la théorie et \, ( \ ; ) est le caractère de la partie holomorphe ( anti holomorphe) q = C2i*T IlllT > O (4-43) La somme est finie parce qu'il s'agit d'une théorie rationnelle, et les JVjj sont tels que la fonction de parti.ion soit invariante modulaire Z(T) = Zn( = ZB(-1/T) (4.44) Le caractère pour les théories quotientes unitaires que nous considérons a été trouvé par Jimbo et al [24]: U.,.= £ f«.Altil,(r) (4-45) u' € S,1+i 57 I où \L - L'\ — 1. S,,+ j est le groupe de Weyl de .4n, fw = det(u<). et (A+ 2 /VA-"'(T) = (»/)-" XI « *'-"*' *' (4.46) ij est la fonction de Dedekind. Q est le réseau des racines de An et A € P++ est un poids dans l'alcôve de Weyl P++ = {A e P\ 1 < Ai < Jl/ JT A, < :U} (4.47) * L'étape cruciale de la bosonisation est l'utilisation de la formule de Poisson pour réexprimer les différents termes de la fonction de partition d'une façon bosonique [25] ' (4.48) •»'£ .4n dans le secteur (7,7'). Il se trouve que tous les invariants modulaires connus sont de la forme Z = (conft) Y. Y. Y. AA«\AA (4-50) où R et .R' sont des sous ensembles de P++ avec peut être des multiplicités. On utilise l'identité si 7 € 0 sinon pour écrire Z(T)= 53 Z1iV(r)p(7,7') (4-52) •>.V€Q où (t ) r (V 2 ) 7 eu,'fu^ ' (7' + «V-«'7t)e" * " " ' (4.53) On généralise pour les modèles diagonaux JV1; = ô" - pour les fonctions de partition avec des conditions aux limites tordues. Les théories que nous considérons ont une symétrie Zn. Les conditions aux limites qui respectent cette symétrie sont des torsions par les 58 On peut généraliser ce résultat pour des modèles avec conditions aux bords plus com- racines de l'unité. Nous considérons le tore comme un parallélogramme dont on identifie les côtés opposés. Les conditions aux limites "spatiales" affectent l'Hamiltonien LQ + LQ. La périodicité dans la direction "temporelle" est la raison pour laquelle la somme sur les états devient une trace. Un changement des conditions aux limites dans la direction "temporelle" est équivalent à une insertion d'un opérateur dans la trace. Dans le cas qui nous intéresse il est plus commode de traiter d'abord le changement de conditions aux limites "temporelles". Les fonctions de partition tordues s'écrivent: ueP$+ (4.54) où s = O signifie des conditions aux limites "spatiales" périodiques. La charge Zn+i est Q(U, v) = LQ(u) -L'Q(v) Q(U) = w1; - I)1 (4.55) 1=1 où Ui est une composante du poids i/ dans la base des n poids fondamentaux. En effectuant la même analyse que ci-dessus et en utilisant la covariance modulaire on trouve *7 I \l \ * ' -•" "-"•' (n + 1)! u'és,r+'i (4.56) Nous définissons le tenseur anti-symétrique n-l e' ~ f| (4.57) 1=1 où ti sont définis par C1 = A1 (4.58) e»+i = -An où A1 sont les poids fondamentaux. En termes de ce tenseur on trouve finalement ry i \i \ \ / (4.59) 59 où JVfA11 sont des entiers non négatifs. La matrice d'incidence de ce graphe est ait où pour un réseau R: 4V*)= c (4.60) et .,[(# des cycles de la permutation «')— 1] //i ei\ A ut — X* . (4.Dl ) La fonction de partition périodique est explicitement pour SU('2) de type (A, G) [25] [26] os77(»7,mVm.m'(-^r) (4.62) ~ ngexpoG m, ni' où h est le nombre de Coxeter du groupe G1 expoG est l'ensemble de exposants de G et (m, m') est le plus grand diviseur commun de »7? et /??'. La fonction de partition de type (.4, .4) pour SU (3) (avec conditions aux limites périodiques) est 44"' = I [Zi1Q(O-O) + 24"(0,O) - 3^1J ,(0,O)] (4.63) où .Rpiï = Zpei + Zge2- Pour les invariants modulaires dits exceptionnels l'analyse devient de plus en plus difficile. La Table I donne les invariants modulaires connus du quotient SU(S). Pour le modèle (.4, P) où 2 AVAV = XX* V-V AePiV1' /i€QnP# (4.64) A-O on obtient en répétant les étapes (pour le cas périodique) 3 Z,D<°»>) = 1 ^(M^oj + 2ZjJJ(O1O) - SZj1 J1, (0,O)] (4.65) On a pour les autres cas les identités suivantes 2_^ COS —A • J- = expo(E») COS —A • ï = 60 "\ où P9(x) est le polynôme de Chebichev de deuxième espèce de degré s. et qui ont les mêmes Table I. Une liste des invariants modulaires connues de la théorie quotient 5I'(3)*_i x SU(3)i /SIf(S)* caractérisés par (G(n)); n = k + 3. z = J E E lE\A. si 3 divise ?? *'"*'*) ft est la rep. conjuguée de fi *=° 7 — • Z - 3 \A,(3,:J)! \A,(l.ii)l + I\A,(2,3) + XA,(6,1)I + I\A,(4,1) + \A,(1,4)|2 + I\A,(1,3) + \'A.(4,3)|2 + |\A,(3,1) + X'A,(3,4)|2 = Z 3 E I\A,(1,1) + XA1(IO1I) + XA1(I1IO) + XA1(S1S) + XA,(5.2) + X'A,(2,5)I + 2|\A,(3,3) + XA,(3,6) + XA,(6,3)I A1(I1I) + \A,(10,1) + XA1(I1IO)I + I\A,(3,3) + X'A,(3,6) + XA,(6,3) I\A,(5,S) + .\A,(5,2) + \A,(2,5)I + I\A,(4.7) + \A,(7,J) + \A,(1,4)I I\A,(7,4; + \A,(1,7) + \A,(4,1)I + 2|\A,(4.4)I (XA,(2,2) + \A,(8.2) f \A,(2,8))\À,(4,4) +c-c- 2 — \ / . VA.(lO.l) I\A,(3,3) + \A,(3,6) + X'A,{6,3)| I\A,(5,5) + \A,(5.2) + \A,(2,5)I + 2|\A,(4,4)I (\A,(4,7) + \A,(7,1) + \A,(1,4))(\A,(7,4) + \A,(1,7) + \'A,(4,1))* (\A,(2,2) + \A,(8,2) + \A,(2,8))\Â,(4,4) +C'C- 24) (£< ) A.d.l) + \A,(22,1) + \'A,(1,22) + \A,(5,5) + \A,(5,14) + XA,(14,5) .0 + \A.(11.11) + \A,(11,2) + \A,(2,11) + \A,(7,7) + \A,(7,10) + X'A,(10,7)I I\A,(7,1) + \A,(16,7) + X'A,(1,16) + X'A.(1.7) + XA,(7.10) + X'A,(16,1) + \A,(5,8) + XA,(11,5) + \A,(8,11) + XA,(8.5) + XA1(S1Il) + XA1 -\ References 1 lit cos ^A • x = 24 + 61*2 + par la méthode d'essai et d'erreur on trouve pour -f Tl —(7173 - TT TfU *V.4Q COS — -(7i72 - 7271 ) - 26-,, ,2Q COS --(7l72 - 727i) 3ir£' , ,. , - cos ~^- 2 - 727i ) ~ cos -- 2 ~ 727i) (4.66) L'exemple de SV(Z] nous montre que la généralisation de l'équation (4.62) est très difficile. Le manque d'une équation analogue à (4.62) nous empêche aussi de comparer directement ces théories conformes et certains modèles de spin, que nous allons définir dans le chapitre suivant, conjecturés être les modèles microscopiques ayant ces théories conformes comme théories critiques. 62 A. Bilal and J.-L. Gervais, Adv. Scr. in Math. Phys. 7 World Sci. 1989 483/semi B. Feigin and E. Frenkel. Phys. Lett. B24G (1990) 75; 5. MODELES INTEGRABLES SUR RESEAU ET REPRESENTATIONS DE L'ALGEBRE DE HECKE 5.1. Motivation Nous décrivons dans ce chapitre des modèles de spins intégrables [27]. Mais avant d'«r.trer dans les détails, motivons la construction de ces modèles. Pour nous, la con- struction de ces modèles fait partie d'un vaste programme destiné à classifier les théories conformes rationnelles et peut apporter un éclairage nouveau à ce problème difficile. Les théories conformes décrivent le comportement universel à longue portée au point critique. L'idée est de chercher dans chaque classe d'universalité un modèle canonique exactement soluble. La classification, ensuite, de ces modèles canoniques revient à classifier les théories conformes et à trouver donc tous les comportements à grande échelle possibles en dimen- sion deux. Bien sûr, c'est un programme ambitieux et il est très difficile en réalité de résoudre ce problème en toute généralité. Néanmoins, il a pour mérite d'être bien défini et de nous servir de phare dans la mer parfois agitée des modèles intégrables. Ce programme comporte plusieurs étapes: On commence- par une théorie conforme rationnelle . Celle-ci est caractérisée par les dimensions conformes des opérateurs, les règles de fusion et les constantes de structure. La deuxième étape consiste à construire à partir de ces données un graphe qui va jouer le rôle d'un espace- cible pour un modèle de spins sur réseau. La troisième étape est peut être la plus difficile. Il far,?. trouver pour chaque modèle des poids de Boltzmann tels que le modèle soit intégrabîe. Ensuite on réexprime la fonction de partition comme la comme des fonctions de partition d'un modèle plus simple-le modèle de vertex. On ferme la boucle dans la dernière étape en prenant la limite continue et en comparant avec la théorie conforme de départ. Dans certaines classes de théories conformes ce programme a été réalisé, notamment pour les modèles minimaux de type (.4,G), pour lesquels Pasquier [28] est parvenu à réaliser toutes les étapes. Récemment Kostov et puis Roche ont fait des progrès dans les modèles de type (G, .4). Une autre classe de modèles où ce programme a éîé réalisé est la classe des modèles de type (A, A) et (A, D) dans les modèle.-, quotients Nous allons définir les modèles et étudier les propriétés des solutions qui ont été trouvées afin de généraliser à des modèles quotients de type (A, G) et notamment pour 63 ARTICLE 1 5.2. Les modèle» de face Nous travaillons sur un résrau carré tourne clé 45 degrés pour dos raisons de commodité et orienté de gauche à droite par convention (voir Fig. 5.1). A chaque point dii réseau on attache une variable dynamique -le spin. Les spins sont des applications du réseau dans un graphe, et on demande «jiu- ces applications soient "continues" c'est à dire que si j est un voisin de / sur le réseau, a} doit être un voisin de a, sur le graphe. Un graphe est défini par un ensemble de sommets et un ensemble de liens orientés entre les sommets. Remarquez que la notion de voisinage n'est pas symétrique. On dira que j est un %-oisin de i s'il y a une flèche de / à j, et on définit d'une manière analogue le voisinage sur le graphe. Pour mieux formuler la notion de voisinage on introduit la matrice d'incidence d'un graphe. Pour un graphe à .V sommets, la matrice d'incidence G est une matrice A" par .V où G'oj =# de flèches de Q vers .i. Une configuration de spins est dite admissible si Ga,,a> ^ O pour tous les j qui sont voisins de /'. La situation où il y a plusieurs liens entre deux sommets ( Ga, ,„, > 1 ) est interprétée comme si on attachait des variables dynamiques aux liens: II faut sommer sur toutes les variables, celles de spins et celles de liens. Les poids de Boltzmann sont attachés aux faces et ils dépendent de toutes les variables autour de la face. La fonction de partition est ]T JJ lF(face|w) (5.1) faces où W(face|u) sont les poids de Boltzmann. et tt est un paramètre complexe qui s'appelle le paramètre spectral. Le paramètre spectral est important dans la mesure où il entre dans la formulation des conditions d'intégrabilité, notamment dans l'équation de Yang-Baxter[27]. 1 2 3 4 5 6 1/0 1 O O O (A 2 O O 2 O O O 3 1 O O 1 1 1 4 O 1 O O O O 5 O 1 O O O O 6 ^o 1 O O O O, Fig. 5.1 Le réseau tourné avec l'orientation. A droite, une matrice d'incidence et le graphe correspondant. 64 »0 »i l Fig .5.2 La matrice de transfert de face avec les indices des liens. Nous définissons une matrice de transfert de face :x .Y1(M) = 1°'-' ? !!>._,.(7,,(T1+1XlM) Zl " (5-2) Les variables attachées aux liens sont implicites dans l'éq. (5.2) et explicites dans la Fig. 5.2. On construit à l'aide de la matrice de transfert de face les matrices de transfert paire et impaire t-l)/2] -Y2,(«) (5.3) et (1/2J-1 0L(u)= JI -Y21+1(I/) (5.4) 1=1 On peut voir une ligne en zigzag (voir Fig. 5.2) comme un état donné à un temps fixé et on le dénote parfois par |«u. a\ ..... «/.). Les matrices de transfert propagent cet état d'une ligne de parité donnée à un autre ligne de parité opposée. La matrice de transfert complète est TL(U) = EL(U)OL(I,) (5.5) et la fonction de partition avec les conditions aux limites périodiques est M Z = n-(TL(u) ) (5.6) et on atteint la limite thermodynamique quand L et M tendent vers l'infini. Une condition suffisante pour l'intégrabilité du modèle est que la matrice de transfert de face satisfasse l'équation de Yang-Baxter .Yi(ti)Jfi+1(u + V)Xi(P) = .Y,+1(»).Yj(« + W)-YH-I(U) (5.7) 65 UI Fig. o.3 Une représentation dingrammatique de l'équation tie Yang-Baxter Nous i )IiN par mi An*;itz que la matrice de transfert de face .Y1(Ji) soit de la forme .Y1(Ii) = siii(n-(À - u))l +sin(7ru (5.8) Quand A est réel, le modèle est critique. Nous allons prendre A = I/;? pour des raisons que nous expliquerons plus loin. La condition de Yang-Baxter se transforme en les conditions suivantes pour les matrices U: (U1)- = JU1 (5.Qa) U1Uj = UjU, \i-j\>2 (5.96) [',U1+iUi-U1 = U,+iUiUi+1 - (V1+, (5.9c) où /3 = 2 cos JTA. b 2 I c.eiu A.t Fig. 5.4 Une représentation diagrammatique de l'équation (5.9c) Ce sont les relations définissant l'algèbre de Hecke. Il se trouve que ces modèles ont beaucoup à voir avec les groupes quantiques SU(X)g. La matrice de transfert des modèles quotients SU( N) connues (voir sections 3 et 5) est dans le commutant des groupes rA quantiques SU(N)9 où q = e" est une racine de l'unité. On suppose que cette propriété est générale et on requiert que À = 1/»? et que les générateurs de l'algèbre de Hecke, les Ui, appartiennent au commutant du groupe quantique SU(N)1. On a donc une condition supplémentaire [29] (- GG L't -\ où S.v + i est le nnmpe do permutation cle A" + 1 objets. On décompose a = fLe/ ri.i+i où r,.,+ ! est une transposition, et on note A",, = Hie/» -^' l>t -^- = -' um f'""-^")- La H •"•" 1OC condition pour SL'('2) est [*,[•.+,[*, = [•, (5.11) L'algèbre de Hecke devient alors re qu'on appelle l'algèbre de Temperley-Lieb. La condi- tion pour Sf(S) est plus coniplitiiiée (voir Fig. 3.5) Jf1+Ir1 + f,+i)([',+i[',+if,+1 - f,+i) = o (5.12) Fig. 5.5 Une représentation diagramme tique de l'équation (5.12) Le problème que nous avons à étudier est donc la théorie des représentations de l'algèbre de Hecke quotientée par la condition (5.10). 5.S. Le» modèles ADE Une solution générale pour les graphes symétriques et sans liens multiples (c'est à dire les graphes qui ont une matrice d'cence symétrique d'éléments au plus égaux à 1) a été trouvée par Pasquier. Soient i/'1''' les vecteurs propres orthonormés de la matrice d'incidence. D'après le théorème de Perron-Frobenius les composantes du vecteur propre associé à la plus grande valeur propre de la matrice d'incidence sont positives. On les 67 \ dénote par Vo où a appartient à l'ensemble des sommets. La solution est donnée en termes du vecteur de Perron-Frobenius[2S] n . i 6 On peut démontrer que les modèles de face de ce type ont une transition de phase du deuxième ordre si et seulement si la plus grande valeur propre est plus petite ou égale à 2. Il se trouve que les matrices de ce type ont été classifiées par Cartan, ce sont les matrices associées aux diagrammes de Dynkin des algèbres de Lie simplement lacées. On démontre, pour ces cas, que dans la limite continue ces modèles sont décrits par les théories conformes de type quotient s '^g.5 (2>1 • Ces modèles sont les modèles minimaux du chapitre 1. Ils sont caractérisées par les exposants des algèbres simplement lacées (voir Table I chapitre 1). Nous ne poursuivrons pas ici cette analyse de la relation avec la théorie conforme. Nous nous sommes intéressés davantage à la détermination des poids de Boltzmann et aux relations qui peuvent exister entre différents poids dans le même modèle ou dans des modèles différents. La propriété la plus importante de ces modèles est l'existence d'une trace de Markov. r Nous appelons AN l'algèbre engendrée par les générateurs U1 ---- , C' .v. On a naturellement r l'inclusion AN C -4/v+i VA . Une trace de Markov est définie sur A00 = U/v=i ^JV par les propriétés suivantes 1) Tr( .4B) = Ti(BA) VA, Be. 4« 2) Tr(XUN) = /(/î)Tr(A') VA' € -4/v-i où /(/3) ne dépend pas de la représentation. Cette propriété s'appelle la propriété de Markov. On définit pour les modèles de face la famille des traces W.("> Tr^ (G)(Ui1...^,)= Y) <«,«2 ..... aL\Ufl... U11^a3 ..... «i-fe .,ïr.t *« Les deux propriétés de trace de Markov sont facilement vérifiées pour /i = 1. La condition de cyclicité (pour tous les /*) vient du fait que les conditions aux limites dans la direc- tion "espace" ne peuvent pas être modifiées par l'action des générateurs de l'algèbre de Temperley-Lieb. La propriété de Markov (pour /* = 1 ) est la conséquence des identités ) = '*!ti1> (5.14) c:6 t 68 **-?***•- 6 n/»*1' = i/.J1' (5.15) ..„ O 6 où c : b veut dire que c est adjacent à 6 et /3 est la plus grande valeur propre de la matrice d'incidence. La chose la plus importante à retenir est le fait que la trace Tr(1'(G) ne dépend que de la valeur propre la plus grande de la matrice d'incidence. La trace est la même, donc, pour tous les modèles qui ont la même valeur propre maximale. On a des identités similaires pour tous les vecteurs propres ?•<>"-3 ^ >T ,#" = cV" (5.17) c.rf (5.18) b:a Ces identités vont être très utiles dans la suite. Une autre identité intéressante, dans la mesure où on va la généraliser dans la suite, est (5.19, 5-4- Entrelaceura et conditions aux limites Nous nous posons le problème suivant: soit A un graphe pour lequel nous avons une solution pour le poids de Boltzmann et soit G un graphe qui satisfait les conditions ( 1 ) G a la même coloriabilité que le graphe .4. (2) La matrice d'incidence G est diagonalisable dans une base orthonormée. (3) expo(G) C expo(.4) où expo(R) est l'ensemble des valeurs propres de R (pour les modèles quotient 517(2) elles sont caractérisées par les "exposants" des algèbres sim- plement lacés). Il peut arriver qu'un exposant de A apparaisse avec une multiplicité plus grande dans G. 69 (4) II existe une involutions -» a telle que G0* = G-b-tt. Existe-t-il des relations entre les poids de Boltzmanu du modèle .4 et ceux du modèle G? La réponse est fournie par des entrelaceurs [3O]. Nous définissons les entrelaceurs Vk par = VhG Les entrelaceurs sont de la forme (5.20) p€expo(G) où (A^' sont les vecteurs propres orthonormés de la matrice .4 et Nous choisissons C p = V';, /$i et nous prenons pour .4 le diagramme de Dynkin associé à l'algèbre An (ou plus généralement le graphe associé à l'alcôve de Weyl de l'algèbre St^(JV), voir section 5). Les matrices carrées (V)0I, = Vj*, ont des propriétés intéressantes (a) E, A,, VJl = Ec V.*eGc6 = Ec G.cV£ (b) V.'t = èak ((a) + (b)=> V/4 = G) où 1 est un point extremal du graphe .4 et / est son seul voisin. (c) VJi € TN (à) VV" = EA ^" VA où N^ sont les règles de fusion données par la formule de Verlinde 0|J,p)A(>)~(f)*< v;"« ^y^; ^1'* 1,'^ (5.2l) Nous allons voir que pour les modèles .4, P, £ les entrelaceurs lient les fonctions de partition, avec conditions aux bords fixes, des différents modèles. La fonction de partition avec conditions aux bords fixes est obtenue en développant le produit (5.6) avec (5.8) . 2 (L) A IT= 51 ^=OJi1 ..... i,} Jo2 ..... at_i} (5.22) où /i, ..... i, sont des fonctions universelles (ce sont de polynômes en sin Jf(A — u) et en sin TTU ) qui ne dépendent pas du graphe G. 70 I.» Nous définissons une fonction de partition modifiée (5.23) On peut démontrer (cf. Article 2) en utilisant les propriétés de la trace de Markov que ~-V.Z Remarquez que la modification affecte seulement le bord. Une telle modification ne change pas l'énergie libre, dans la limite thermodynamique. L'équation (5.24) nous indique que les deux modèles A et G ont le même comportement critique. Si un modèle a une transition de phase du deuxième ordre l'autre en a une aussi. Les propriétés (5.16) (5.17) et (5.18) nous permettent d'aller plus loin. En les itérant nous démontrons que ) I) £zifVl" = /.»,-Mr.«n.*i' (5-25) 6€G pour tous les G qui satisfont les propriétés (l)-(4). On démontre (cf. Article 2) à l'aide de ce résultat l'identité suivante - E WWW = *~ E W a,6€G (5.26) Nous inversons le membre de gauche en utilisant la propriété d'orthonormalité des vecteurs propres pour obtenir notre résultat principal (G) _ (5.27) Ce résultat ne dépend pas de la taille du réseau et peut être également utilisé pour de petits réseaux. Pour le réseau le plus petit possible notamment, on obtient (5.28) où à gauche nous reconnaissons les poids de Boltzmann du modèle G et à droite ceux du modèle A. L'équation (5.28) nous donne donc une relation forte entre les poids de Boltzmann dans des modèles différents. N. Sochen / Intcgrahle models On peut généraliser ce résultat pour des modèles avec conditions aux bords plus com- pliquées. Toujours pour les modèles de type .4, D, E, on définit une fonction de partition ( Z ab.c où les deux spins de gauche sont fixés aux valeurs a et b et celui de droite est fixé à la valeur c. Par les même méthodes que plus haut on arrive à démontrer (cf. Article 2) que «*"* <5-29> 1 où e06 est le point extremal dans la branche où a et 6 se trouvent, et a est le point du graphe .4 donné par 5 3 ^TI (fjt)T = %JL(M7) ( ' °) 6' est défini d'une manière analogue et a' est adjacent à b1. 5.5. Lea modèles quotients Les modèles .4, D, E correspondent aux théories conformes minimales qui sont des modèles quotients Sf (2)t x SU(2)i/Sf (2)k+i- La généralisation des modèles de face pour les qutients SU(N)t x SU(N)i/SU(N)k+i est le sujet de cette section. On a pour ces modèles l'analogue du graphe de type .4. Le graphe de ce type est l'alcôve de Weyl au niveau n = k + N définie comme suit: les sommets du graphe sont dans l'ensemble A'-l 0< A, / = n} (5.31) où P est le réseau des poids de SU(N) et A = £, A,A, où A/ sont les poids fondamentaux. Nous représentons les N vecteurs linéairement dépendants e^ par = A1 = A/ - Aj_i 1 < i < N (5.32) Les EI donnent la direction des flèches entre les sommets. Le graphe obtenu est lié aux règles de fusion [31] [32] d'une représentation R\ de l'algèbre de Kac-Moody par la représentation fondamentale R/ ~ TfXvR1, (5-33) 72 N. Sochen / Integrable models où Nf\fi sont des entiers non négatifs. La matrice d'incidence de ce graphe est .4^ = Les représentations des générateurs de l'algèbre de Hecke ont été trouvés par Wenzl [33] et Jimbo et al [34] A+1, où SIJ(T) = sin(^(e, - £j)-r) et J = 2COS(JTA) = 2cos(£). Pour St:(2) nous retrouvons les solutions de Pasquier (5.35) parce que pour le diagramme de Dynkin -4n_i les composantes du vecteur de Perron- Frobenius sont ^i/' = sm(irv/n) 1 < v < n — 1 et la valeur propre qui lui correspond est /3 = 2cos(7T/r?). Le vecteur de Perron-Frobenius pour A* > 2 est (Q • /<)) (5.36) et les valeurs propres de la matrice d'incidence sont (5.37) 1=1 On peut pousser l'analogie avec les modèles .4, D, E encore plus loin et résoudre (au moins partiellement) le problème posé dans la section 4. à savoir, trouver les graphes G qui ont les propriétés 1-4 où le graphe .4 est celui que nous venons de donner. Di Francesco et Zuber[30] ont proposé une liste de tels graphes pour le cas N — 3. L'obstacle à la généralisation est, à ce stade là, l'absence d'une description des poids de Boltzmann en termes des graphes. L'analogue de l'équation (5.35) n'existe pas pour N > 2. Par conséquent nous ne pouvons démontrer ni la propriété de la trace de Markov ni les relations entre les fonctions de partition, avec les conditions aux bords fixes, de différents modèles. Au lieu de les démontrer nous allons les supposer et les utiliser afin de contraindre et déterminer les poids de Boltzmann. Nous cherchons des modèles qui satisfont la propriété de Markov 6 ' (5.38) c:6 73 N. Sochen / Integrable models où PJ(X) est le polynôme de Chebichev de deuxième espèce de degré s. et qui ont les mêmes valeurs propres que le modèle .4 " (5.39) «WMV c:6 c Nous demandons aussi que les générateurs de l'algèbre de Hecke satisfassent l'identité ,(i) ,(i) é >e ( 1} Y, 0"' = M" ''"fc % + PN-3(J/-2)tl' c ètc6ma6kl (5.40) Cette identité assure la compatibilité entre la trace de Markov et l'algèbre de Hecke. On suppose enfin que la relation suivante est vraie où 1 est le sommet de la chambie de Weyl. La relation (5.29) est plus difficilement appli- cable parce qu'il n'y a pas toujours de solution à l'équation (5.30) (cf. Article 2). L'expérience montre que ces relations sont suffisantes pour décider si un graphe donné peut accommoder une telle représentation de l'algèbre de Hecke. Nous avons constaté empiriquement que l'algèbre de Hecke plus l'identité (5.41) suffisent pour trouver une solution unique (si elle existe). Dans tous les cas étudiés la solution qui a été trouvée satisfait toutes les autres contraintes. Il est intéressant de remarquer que chaque fois que l'équation (5.30) a une solution, l'identité (5.29) est satisfaite aussi. Cette méthode nous a permis de déterminer les poids de Boltzmann de plusieurs mod- èles qutients SU(Z), pour lesquels la solution était inconnue, et de vérifier, par conséquent, leur intégrabilité complète. 74 N. Sochen / Integrable models References [1] Le nombre d'articles sur le sujet est clénombrable. Nous suivons l'article des fondateurs du sujet: A.Belavin, A.M. Polyakov and A.B. Zamolodchikov, N'ucl. Phys. B241 (1983) 333. Nous avons trouvé les revues suivantes extrêmement utiles: P. Ginsparg, "Applied Conformai field theory" in Les Houches, Session XLIX, 1988 éd. by E. Brézin and J. Zinn-Justin; A.B. Zamolodchikov and Al.B Zamolodchikov, Sov. Sci. Rev. A. Phys. Vol. 10 (1989) 2G9. Un livre où on peut trouver la plupart des articles importants: Conformai Invariance and applications to statistical mechanics, C. Itzykson, H. Saleur and J.-B. Zuber editors, World Scientific (1988). [2] V.G. Kac, Lecture Notes in Math. 33 (1982) 117: B.L. Feigin and D.B. Fuchs. Funct. Anal. Prilozhen IG (1982) 47. [3] D. Friedan, Z. Qiu and S. Shenker. Phys. Rev. Lett. 52 (1984) 1575; Comm. Math. Phys. 107(1986)535. [4] B.L. Feigin and D. B. Fuchs. Lecture Notes in Math. 1000 (1984) 230. [5] J.L. Cai-dy, Nucl. Phys. B270 [FSlG] (1986) 18G. [6] A. Cappelli, C. Itzykson and J.-B. Zuber. Comm. Math, Phys. 113 (1987) 1. [7] A. Luther and I. Peschel, Phys. Rev. B12 (1975) 3908 [8] J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95; S. P. Novikov, Usp. Mat. Nauk. 37 (1982) 3; A.M Polyakov and P.B. Wiegmann, Phys. Lett. B131 (1983) 121; E. Witten, Comm. Math. Phys. 92 (1984) 455. [9] H. Sugawara, Phys. Rev. 170 (1968) 1659; R. Dashen and Y. Frishman, Phys.Rev. DlI (1975) 2781. [10] F.G. Malikov, B.L. Feigin and D.B. Fuks. Funkt. Anal. Prilozhen 20 No. 2 (1987) 25. [11] V. G. Kac and D. A. Kazhdan Adv. Math 34 (1979) 97; V. G. Kac, "Infinite dimensional Lie algebras". Cambridge University Press 1985. [12] A. B. Zamolodchikov and V. A. Fateev, Sov. J. Nucl. Phys. 43 (1986) 657 [13] D. Gepner and E. Witten, Nucl. Phys B27S (1986) 493. [14] V. G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83. [15] A.B, Zamolodchikov, Theor. Math. Phys. 63 (1985) 1205. [16] V. A. Fateev and S. L. Lykyanov, Int.. J. Mod. Phys. A3 (1988) 507; S. L. Lykyanov, Funct. Anal. Appl. 22 (1988) 1; F. A. Bais, P. Bouwknegt, K. Schoutens and M. Surridge, Nucl. Phys. B304 (1988) 348 et 371; 75 N. Sochen / Inlegrable models were found by Wenz! [12] and Jimbo et al. (13] and read A. Bilal and J.-L. Gervais, Adv. Ser. in Math. Phys. 7 World Soi. 1989 483/semi B. Feigin and E. Frenkel. Phys. Lett. B24C (1990) 75: et beaucoup d'autres. [17] V. G. Drinfeld and VA' Sokolov, J. Sov. Math. 30 (19SÔ) 1975: P. Di Francesco, C. Itzykson and J.-B. Zuber, Comm. Math. Phys. 140 (1991) 543. [18] M. Bauer, P. Di Francesco. C. Itzykson and J.-B. Zuber, Nucl. Phys. B362 (1991) 515. [19] B. L. Feigin and D. B. Fuchs Moscow Preprint (1983). [20] V. S. Dotsenko and V. A. Fateev. Nucl. Phys. B240 [FS12] (1984) 312; Nucl. Phys. B251 [FS13] (1985)691. [21] G. Felder, Nucl. Phys. B317 (1989) 215. [22] D. Bernard and G. Felder, Comm. Math. Phys. 127 (1990) 145. [23] M. Wakimoto, Comm. Math. Phys. 104 (1986) 605. [24] M. Jimbo, T. Miwa and M. Okado, Lett. Math. Phys. 12 (1987) 123 [25] I. Kostov, Nucl. Phys. B300 [FS22] (1988) 559. [26] P. Di Francesco, H. Saleur and J.-B. Zuber, Nucl. Phys. B2S5 [FS19] (1987) 454; J. Stat. Phys. 49(1987) 57. [27] R. J. Baxter, Exactly solved models in statistical mechanics. Academic Press,New York, 1982. [28] V. Pasquier, Nucl. Phys. B.300[FS19]( 1987)162; J. Phys. A 20(1987)5707. [29] N. Yu. Reshitikhin, "Quantized Universal Enveloping Algebras, the Yang-Baxter Equation and Invariants of Links", I and II, Leningrad preprint (1988). [30] P. Di Francesco and J.-B. Zuber, Nucl. Phys. 6338(1990)602; P. Di Francesco and J.-B. Zuber in "Recent developments in conformai field theory, Trieste Oct. 1989, éd. S. Randjbar-Daemi et al. World Scientific, Singapore. [31] D. Gepner and E. Witten, Nucl.Phys. B278 (1986) 493. [32] E. Verlinde, Nucl. Phys. 300 [FS22] (1988) 360. [33] H. Wenzl, Inv. Math. 92 (1988) 349. [34j M. Jimbo, T. Miwa and M. Okado, Lett.Math.Phys 14 ( 19S7) 123; Comm.Math.Phys. 116 (1988) 507. 76 620 N. Sochen / IntfKrable model* 3, Properties of the known models ARTICLE 1 Volume 231. number 4 PHN SICS LETTERS B Ib November I BOSONIZATION OF TWISTED PARAFERMIONS AND APPLICATION Nir SOCHEN ' Sen-ice de Physique Théorique :. CE\-Sacla\: Wl IVI (iit-sur-YreneCede.\. l-'rancf Received IO July 1989 The panilion functions for Z, statistical models are considered through conformai field theories based on the coset iheon. (SU(/)i®SU(/), J/SU(/)j. The bosonization of the associate parafermions for the sectors with twisted spin is carried out. An application to a bosonic description of the coset theory [SU( 2 )m_28SU< 2 )w| /SU ( 2 )»*„.} is given. A lot of efforts were aimed recently in conformai Kostov [6], who represented the untwisted parafer- field theory (CFT) toward the problem of bosoniz- mions as a combination of n compaciified bosons. ing the parafermions [I]. Parafermions were intro- Other methods use either nor 2 bosons [ 6 ]. We will duced by Zamolodchikovand Fateev [2] >odescribe use the result to give a bosonic description of the Z, models in statistical mechanics, and were general- models based on the coset [SU(2)m_2®SU(2)n,]/ ized afterwards by Gepner [ 3 ]. The will to bosonize SU(2)/v+m..2. them stems from several sources. First, in the Eight of The partition function of a CFT in the untwisted the spin-statistics theorem, we would like to describe sector is a bilinear form in the characters x things in terms of fields that obey Fermi or Bose sta- tistics only. In two dimensions there is also equiva- lence between fermions and bosons. A bosonization <7=exp(2i;rr), Imr>0, (I) of parafermions will enable us to describe everything in a bosonic language. Second, and maybe more im- where W is the representation space and L0 is the ze- portantly, parafermions appear frequently, them- roth coefficient in the Laurent expansion of the en- selves or with other fields, as ingredients in CFT based ergy-momentum tensor T. The explicit form of x for on Kac-Moody algebras and coset constructions [4]. the general coset theory G/H is not known. Yet for Bosonizing them may deepen our undemanding of several cases they have been worked out. For the the- the structure of these CFT. Another interesting fea- ory [(An)t_n_2®(AH),]/(AB)t_n_, Jimbo et al. ture of CFT is the possibility to expose different sec- [7] found tors of the same theory by twisting the boundary con- Xu.n— £ 4U^Lt-Z-Ij(T) ' (2) ditions. These sectors are usually related to one wcSit-f I another by modular covariance [S]. where \L-L'\ = l, S , is the Weyl group of A . In this letter we will describe the bosonization of H+ n tw=det(K-),and the coset theory [(A1I)t_l,_20(AB)11/(An)^.,,., with twisted boundary conditions. This theory coin- -" I <,u+ (3) cides with the parafermions of Zamolodchikov and yeQ Fateev for L=M+3. Ve will follow the approach of t\ is the Dedekind function, Q is the root lattice OfAn and AeP^+ is a weight in the Weyl chamber 1 Supported by bourse "CHATEAUBRIAND" du CIES Bureau des Affaires Etrangères, France. P^ =JAeP (4) 1 Laboratoire de l'Institut de Recherche Fondamentale du + Commissariat a l'Energie Atomique. 0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. 425 ( North-Holland Physics Publishing Division ) Volume 231. number 4 PHVSICS LETTERS B l6No\cmbcr l Denote Zn(r. .S)(T) the partition function for the f,, is the mh fundamental weight. One sum o\cr the twisted Z,,*, spin We\ I group is equal to the volume of the We> I group Xexp(i2KG/(«+l)]). (7) (13) The Zn+, charge is [9] and by modular covariance Q(U, v) = LQ(u)-L'Q(V) , Q(U)= Î (»,-l)j, (8) r.r'«Q where the u, are the components of the weight u in the basis of the n fundamental weights. The fusion rules are invariant under 0(r)-exp[i2ji(?/(n+1 ) ] XexpliJt(L/L')(y'-2i;)-y]. (14) X 0( z ) [ 91. Reorganizing ( 7 ) a little we get Defining the anti-symmetric tensor (-1)" n-l Zn(OJ)(T) = On= I te-M®*, (15) w-H /»l we c^n show that (9) 426 I-.* Volume 231. number 4 PHVSlCS LETTERS B 16 November I4.S1- ( - I )" ii>. The second is the central charge of the coset the- ;?„(/-, A -, I e,,.V,, («+!)!.,,t., or> [(A\_, ),®(A\_, ),]/(A\_, ): which we con- sidered above. The theory can be written in terms ot •. T) one bosonic field and one ( .V— I (-vector boson field. The energy-momentum tensor is (16) where + i£0/>'o:0'. QH=LQ@ (I ->v)s. (17) seP/i P with the normalization The Weyl group can be viewed as the group of per- mutations of an orthonormal basis h, for («+1 )-di- (24) mensional space. The root space is an n-dimensional hyperplane in that space, and e,=h,-hl+l, i= 1 n. p is the sum over fundamental weights of Av_,. In these terms The primary fields are y _£'((*of cycles of permutation w)-l (18) :exp IiOn-1^U) J (25) The exponent factor that contains B can be obtained n where from an action by adding an antisymmetric tensor to it [10,6]. t=\n-kmod2N\ , Notice that all the elements in the same conjugacy ( N \ class of Sn+, contribute the same. Explicitly for the Oa = ( I twisted Ising model we get \4m(m+N)J — Zy_3r»i. y + 31,1(3, T)] , (.9) and for the twisted Potts model: (26) The conformai dimension of these vertex operators is t(N-t) ~3 (27) x ***= 4m(m+N) where and the screening operators are -• (21) dz q± The e* are the simple roots. Next we consider the coset theory G= [SU ( 2 )m_2 exp[ -2i£ <'0!(z) ] . (28 ) ®SU(2)jv]/SU(2)w+m_2. Following Ravanini and ± Bagger, Nemeschansky and Yankielowicz [11] we decompose Di Francesco, Saleurand Zuber [12] showed that a class of partition functions for this theory labeled (22) by a pair of simply laced Lie algebras ( A, G ' ) ( where m(m+N) yv+2 G' CPJI be A, D or E type ) can be written as The first piece can be understood as coming from a free boson field with an appropriate charge at infin- 427 I:» \ Volume 231. number4 PHYSlCSLETTERSB Ib November I References 7-m.m (P/-V/''. T) r t = () W= (r 1\ m « un (I]P Griffin and D Nemeschansk\. Bosonuanon ot paratcrmions. SLAC report SL \C-PliB-4ftfift. — 27T(/M. /H' )fl D. Nemeschansk>. Fvigm-Fuchs representation of -,!ring X £ COS - ; — , (29) functions, preprint USC-89/012. ne oiponcnti of G P J. Distler and Z. Qm. BRS cohomolog> and Feigm-Fuchs where p and p' are the Coxeter numbers of A and G representation of Kac-Moody and parafcrminnic theories, respectively, (m, m ) means the greatest common preprint CLNS 89/911: divisor of/M, m', and (r) means rmod A'. With our A. Gerasimov. A. Marshakov. A Morozov. M. Olshaneiskx N and S. Shatashvili. ITEP preprints nos. 64. 70. 72.74 ( April previous result for Zn(r, S)(T) we can express now 1989): ZG ( T ) in terms of bosonic partition functions A Gtrasimov. A. Marshakov and A. Morozov. Free field representations of parafermions and related coset models. ZG(T)= ITEP preprint no. 73 ( 1989); Nl m.m'cZ A. Marshakov. Dotsenko-Fateev representation for WZW ^e S/v models, Lebedev preprint no. 76 ( !989): ,n,m')n A. BiIaI, BosonizationofZ,parxfermionsandsu(2)\ Kac- X 1 cos—*—j—- Moody algebra. CERN preprint CERN-TH.5370/89 (April nt exponents of G' P 1989); G. Dunne. I. Halliday and P. Suranyi. MIT preprint CTP , T) 1663 (December 1988). (2) A.B. Zamolodchikovand V.A. Faieev. Sov. Phys. JETP 62 y-0-y], (30) (1985)215. [3)D.Gepner.Nucl.Phys.B290[FS20] (1987) 10. That concludes OLT boson izat ion of the coset theory [4] P. Goddard, A. Kent and D. Olive, Phys. Lett. B 152 ( 1985 ) G= [SU(2)m_2®SU(2)A,]/SU(2)A,+m_2. 88. Another application is to the A, vertex model [ 1 2 ]: [5) J.-B. Zuber. Phys. Lett. B 176 ( 1986) 127. (6) I. Rostov, Nucl. Phys. B 300 (FS22) ( 1988) SS9. ZTV(T)= f' Zf,., (r, s) (T) J Zm.m.(frT), F. Bais. P. Bauwkntft, M. Surridfeand K. Schoutens. Nucl. r.J-0 OT-(T)(V Phys.B304 (1988) 371; ".«(DA- (31) A. BiIaI and J.L. Gervais. Phys. Leu. B 206 (1988) 412: and for g= I /N we get the partition function for the Nucl. Phys. B 314 (1989) 646;B 318 ( 1989) S79. (7] M. Jimbo. T. Miwa and M. Okado. Lett. Math. Phys. 12 SU (2 )w WZW model. Again with our expression for (1987)123. ZN(r, s) we get a bosonic description: [8 J d. Gepner, Phys. Lett. B 27.2 (1989) 207. [9] V.A. Fateev and S.L. Lykyanov. Intern, i. Mod. Phys. A 3 (1988)537. Nl [1O]S. Elitzur, E. Gross. E. Rabinovici and N. Seiberg. Nucl. Phys. B 283 (1987) 413. [U]F. Ravanini. Mod. Phys. Leu. A 3 ( 1988 ) 397: y.x'eQ, J. Bagger, D. Nemeschansfcy and S. Yankielowicz. Phys. Rev. Lett. 60 (1988) 389. (32) [ 12 ] P. Di Francesco. H. Saleur and J.-B. Zuber. Nucl. Phys. B 300 [FS22]( 1988) 393. It is a pleasure to thank J.-B. Zuber for guiding me through this work. 428 ARTICLE 2 Nuclear Physics B360 ( 1991 ) 613-640 North-Holland !NTEGRABLE MODELS THROUGH REPRESENTATIONS OF THE HECKE ALGEBRA Nir SOCHEN Service de Physique Théorique de Saclay, Direction de Science de la Matière du C.E.A., F-91191 Gif-sur-Yvette cedex, France Received 13 September 1990 (Revised 18 March 1991) We present new solutions to the Yang-Baxter equation through representations of the Hecke algebra. The generators of the Hecke algebra are considered as Boltzmann weights for face models. The heights live in a graph. These models were conjectured to be integrable by Di Francesco and Zuber. We prove integrability for some of the suggested models by building explicitly the Boltzmann weights. Some relations that the Boltzmann weights satisfy and the consequence on partition functions with various boundary conditions are also discussed. 1. Introduction The Yang-Baxter equation (YBE) plays a major role in 2D exactly solvable modeîs in statistical mechanics where it appears as a condition on Boltzmann weights in order that the model be integrable [I]. In conformai field theories the analogous hexagon equation appears as a condition on braiding matrices for the theory to be symmetric under duality transformations [2]. An important class of solutions to the YBE can be obtained by representations of the Fscke algebra. In this paper we shall present some new representations of the Hecke algebra where the generators of the algebra appear as Boltzmann weights for a system of "spins" that take their values in s graph. These models were constructed by Di Francesco and Zuber [3], and were conjectured to be integrable and to correspond in the continuum limit to non-diagonal modular invariants of the coset models SU(S)/ x SU(3),/SU(3)/+1. The dream is of course to find a canonical representative integrable model in the stabilizer of each of the conformai field theories to be thought here i-s fixed points. The classification of all representatives will amount then to the classifica- tion of all conformai field theories. The link between a given CFT and its canonical representative is not totally clear. In some of the known examples the connection is made by a "fusion rule"-like algebra or more precisely a c-algebra [4]. For certain models this c-algebra is identical to the fusion r jles of the KM 0550-3213/91/$03.50 €>1991 - Elsevier Science Publishers B.V. (North-Holland) u» 614 N. Sochen / Integrable models theory which is in the denominator of the coset (and not to the fusion rules of the coset!). For others the c-algebra contains the fusion algebra of the extended chiral algebra as a subalgebra, yet for other models there ij no clear relation. We think of the generators of this c-algebra as operators and distinguish one operator as the "fundamental". From this c-algebra we build our statistical mechanics representa- tive as follows. We take the labels of the operators of the theory as the vertices of a graph and the fusion of an operator with the "fundamental" one gives the links between this vertex and the others. Our model is thus a spin system where the spins take their values in this set of operator labels and admissible configurations will be governed by the adjacency matrix of the graph, that is, by the fusion rules. In order to close the circle we must show that this model is indeed integrable and is described in the continuum limit and in the critical point by the above CFT. This program was realized* by Pasquier [S] in the special case of the minimal models which correspond to the coset theories SU(2), x SUG)1XSUQ)/+,, see also in this context refs. [6,7]. There are several steps to take in the course of generalizing Pasquier's analysis. First we have to find a set of requirements to be obeyed by the graph-fusion rule, then to find a classification of all graphs, to prove the integrability of the spin systems based on these graphs, and finally take the continuum limit and show that it coincides with the expected CFT. Several steps in this direction have been taken by Di Francesco and Zuber [3] for the more general coset theories SU(AT)7X SU(AO,/SU(AO/+1. They gave a set of empiric rules for this generalized set of graphs and a list of graphs that fulfill all requirements for the SU(3) coset models. They showed that there is a great similarity between this problem and the problem of non-negative integral representation theory of the fusion algebra of SU(N)n Kac-Moody algebra. They also investigated the connection with the continuum limit (modular invariant partition functions) in an algebraic approach [8]. In this paper we will prove the integrability of some of the above mentioned graphs by building explicitly the Boltzmann weights that satisfy the Yang-Baxter condition. The main technical tools are the Markov trace over the Hecke algebra and the relation between partition functions with fixed boundaries for the various graphs. These relations hold even for small lattices providing thus strong relations between the unknown Boltzmann weights and the known ones. This paper is organized as follows: in sect. 2 we describe in general the spin systems, discuss their relation to the Hecke algebra, and present two families of known solutions. In sect. 3 some properties of these solutions are shown. They, or their natural generalization, are listed in sect. 4. Sect. 4 presents also the important concept of intertwiner between graphs and some relations between partition functions with fixed boundaries for the various graphs. These relations are the *More carefully put, he found the statistical mechanics representatives for the CFT models (A1G) for G - A, D, E. The representatives for the models (D1A) and (E1A) await such a construction. N. Soclien / !ntegrahle models 615 basic tools in finding the explicit solutions presented in sect. 5. Sect. 6 is a summary and discussion. 2. Generalities and known solutions Let us describe the models. Our system consists of a square lattice with orientation which we fix once and for all as, say, from left to right (up and down) when we rotate the lattice by 45 degrees for later convenience (see fig. 1). The spins live at the nodes of the lattice and take their values in a graph. What we mean by graph is a set of vertices and a set of directed links between these vertices (fig. 1). We can summarize all the information about the graph in its incidence matrix Gab = (number of arrows that point from a to b) (fig 1). The admissible configurations are those where for every two sites i and j on the lattice that are connected by an arrow from / to ;', the spins cr, and oy are vertices in the graph which are connected by an arrow from at to o; (G0. a * O). Moreover the case where several links connect two vertices is interpreted as a situation where additional degrees of freedom are attached to links: one has to sum over all spins and over all link variables. This will be illustrated below. We attach to each face of the lattice a family of Boltzmann weights W( Z = £ H W(face|u). (2.1) J0.) all faces Let us define a face transfer matrix by X1(U) = !•'-' 6 ir(oi_|l0i,oi+I,ii/|B)< (2.2) 1 2 a 4 5 6 1 (O 1 O O O 0\ 2 O O 2 O O O 3 1 O O 1 1 1 4 O 1 O O O O 5 O 1 O O O O 6 VO 1 O O O O/ Spxc Fig. 1. The square (rotated) lattice with orientation. Furthermore an example of adjacency matrix and corresponding graph. 1 616 N. Sochen / Integrable models »o »i SL Fig. 2. The explicit link indices used in defining the face transfer matrix. where in fig. 2 the link indices are explicit. We define also even and odd transfer matrices by [(L-D/21 £*•(«)- H X2i(u) (2.3) i-l and 0L(u)~ H *2,+i(«) (2-4) i-l We think of a zigzag line as of a state in a given time. The odd or even transfer matrix propagates this state from a zigzag line with one parity to a zigzag line with another. The full transfer matrix is TL(U) = EL(u)OL(u) (2.5) A sufficient condition for the integrability of the model is that the AXu)'s will satisfy the Yang-Baxter equation + V)X1(U) -X^Wu + vW+M (2.6) We take the following ansatz for the AX M) = SUI(U(A - «))1 + sin( XU)U1 (2.7) then, by inserting (2.7) in eq. (2.6), we find vhat the l^'s are the generators of the e Fig. 3. A diagrammatic representation of the Yang-Baxter equation (2.6). A'. Sochen / Iniegrable models 617 E M E Fig. 4. A diagrammatic representation of eq. (2.8c>. Hecke algebra: (2.8a) \i-j\>2, (2.8b) (2.8c) where /3 = 2cosirÂ. Another constraint comes from the requirement that the generators U should be in the commutant of the quantum group SU(N)4 (for a good introduction to quantum group ideas in connection to integrable lattice models see reference [9]). The commutant is obtained by imposing the vanishing of the rank N + 1 antisym- metrizer. . £ (-<7)"-X = 0, (2.9) where a e SN+1 the permutation group of N + 1 objects. It can be decomposed as o- = n,ei/,.i+1 where ru+l is a transposition of neighbors, ATff "FI1-Si4AT,- and Xi = 2i limu'_ 1M6""ATXM).' For SU(2) it reads (2.10) so the Hecke algel ,a reduces to the Temperley-Lieb algèbre Tor SU(3) it is (see fig. 5): ~ Ui+2Ui+1Ut + Ui+l)(Ui+lUi+2Ut+l - U1+1) = O, (2.11) we are thus led to study the representation theory of quotients of the Hecke algebra by this constraint. Several classes of solutions are known, first for the graph that describes the fusion of a representation of SU(N) Kac-Moody algebra ai level k = n-N with the fundamental representation. We remind the reader that the integrable repre- sentations of SU(N) level k Kac-Moody algebra asre indexed by the finite set 618 N. Sochen / Integrable models b< /» , C Fig. S. A diagrammatic representation of eq. (2.11). (Weyl aïcove) N-I \ AeP| (XA1 I = I N-I £A, where P is the weight lattice of SU(N) and A = L1A1A1 where /I, are the weights of the fundamental representation. We take these points as vertices of our graph. We represent the N linearly dependent vectors et by KKN-I, N -AN_v (2.13) The bonds between the above vertices are directed along the ef's. This graph describes the fusion rules [10,11] of representation /?A of KM algebra by the fundamental representation R1 x ^A (2.14) where NfAM are non-negative integers. We take A^11 - /VfAM to be the adjacency matrix of our graph. The generators for the Hscke algebra in this representation /V. Sochen / Iniegrahle models were found by Wenzl [12] and Jimbo et al. [13] and read k,( O„,..,, -(I -B11)L:* where s,/*) = sinWir/wXe, -e,)*). The number that characterizes this represen- tation of the Hecke algebra is /3 = 2cos(irA) = 2cos(ir/«). Notice that for SU(2) it reduces to I1/2 where ^*,1' = sindr^/n) 1 « v « n - 1 is the vth component of the eigenvector of A that corresponds to the largest eigenvalue /3 = 2cos(ir/n). This class of solutions will be referred to in the following as the "regular" representation, a term which will be justified later on. The second class of solutions corresponds in the continuum limit to the coset models SU(2)7x SU(2)j/SU(2),+ ]. The requirements on the graphs are to be unoriented (thus G is symmetric) and to have a largest eigenvalue 0 < 2. The reason for these requirements will be more clear below. This problem is closely connected with the problem of classifying simple Lie algebras. The solutions are the ADE Dynkin diagrams that correspond to simply laced Lie algebras. The partition functions correspond in the continuum limit to the ADE classification [14] of modular invariant partition functions of the minimal conformai field theories. The Boltzmann weights are given by A. f ^(i>0(i)|l/z U1 = „< > -£r Sad (2.17) where i/»(1) is the eigenvector of G that corresponds to t largest eigenvalue. Another class of solutions is known. This is the class Oi arbifolds of the class of regular graphs, that is, the quotient of a regular graph by its discrete symmetry. Details can be found in refs. [15,16]. Further solutions for spins that take only two values were presented in réf. [17j. Having explicit solutions (for these classes) we were able to check our conjectures and have more confidence on them. A remark concerning notation. We denote by A, Greek letters and <£ the regular graph (or equivalently its adjacency matrix), its vertices and its eigenvectors respectively, while G.. Roman letters and «A denote the non-regular graph (or equivalently its adjacency matrix) (orbifold or exceptional), its vertices and eigen- vectors. 620 /V. Sochen / Inregrable models 3. Properties of the known models In this section we discuss the properties of the above mentioned models. We will concentrate on relations which do not depend on representation (for fixed rank and level) in order to impose these relations or their direct generalizations as constraints on new representations. We start with the class of "regular" graphs. The fusion coefficients for the SU(AO Kac-Moody algebra at level k - E N^VRV (3.1) are non-negative integers and, by the Perron-Frobenius theorem, there exists for any such uatrix an eigenvector <£(1) of the largest eigenvalue y(1>, such that all its components ^0 are real and non-negative. In this case however we can say more since it is well known that the fusion coefficients are diagonalized by the represen- tation of the modular transformation T-* - 1/r over the space of characters (T) . (3.2) This is summarized in a beautiful formula due to E. Verlinde [11] A^- E 5(P where 1 refers to the apex of the Weyl alcove, and Sf is given by the Kac- Peterson formula wp -\, (3.4) where eu = det(o>) and C is a normalisation constant fixed by the requirement that (p> {S } is an orthonormal system. Our graph A111, is then given by 5(P)5(P>5 * A — V _ V f M " ^M" ~fjfia>- L, (3.5) N. Sochen / Iniegrahle models 621 so we see that (S"1') are the eigenvectors of the graph A, and £ / 27T (3.6) are the eigenvalues. The largest eigenvalue corresponds to the apex of the Weyl alcove y, =PA,_2(/3/2), where PN(x) is the Chebichev polynomial of the second kind of degree N, and the Perron-Frobenius eigenvector that corresponds to it is (up to normalization) (3.7) where the product is over the positive roots. The largest eigenvalue and its corresponding eigenvector play an important role in what follows. Next we consider the partition function with fixed boundaries and draw some results from the underlying algebraic structure. We take the partition to be periodic in the temporal direction and fix the boundaries in the spatial one, then it reads t M *£' = E M = E E /i,.... .,,(*•" ) E (a,a2,...,aL_t,b\Ui>Uiî...Uir\a,a2,...,aL.1,b). '-O 111 ..... i,» (a2 ..... aL_,} (3.8) where ff ir are universal functions, that is, they are (for given N and n) independent of the representation G. Define 3 9 E <«,«2,...,fl|.-i,*Hlfl,fl2. — . «L-i.*). ( - ) («2 ..... 0/.-|) then since the U's do not affect a, and aL tr( AB)=Ir(BA). (3.10) Moreover, using the cyclicity of the trace and the defining relations of the Hecke algebra, we can write (see proof in appendix A) (3 n 1I/,,... I/,,) - E é::;;;(ÂW...^), - > '-o (A A where again g, ..,,(A) are universal. Define now M to get finally M E E htl ,,tr(i/(i...i(;). (3.13) /-o {,-,<...<,•,} We are now in a position to introduce the modified partition function, or a Markov trace in mathematical language. Doing so we will reveal some nice properties of the weights and, at the same time, come out with some new physical understanding. We define and TY^(G)(I/,,...l/,,)- E <«,a2,..., by the same reasoning as above, we can write Zffid(G)-E E A1, ,, TYiH(G)(LJ1... U1,), (3-15) 1-0 (/,<...<,,} with the same A1- l; as before. We will prove now that (3.16) The proof follows by recursion from the identities 1 1 E M>A ^A > = FA,_2(/3/2)^ >, (3.17) x v 1 0 E <#>- 3X11^- V-IOXZK , (3.18) where v.fi means >> which is adjacent to /i- The second equation follows from the simple fact that 4>(1) is the Perron-Frobenius eigenvector of A. The first can be N. Sochen / IntegraHe models 623 verified directly from the explicit expression (2.15). We impose now the condition thai the largest eigenvalue of the graph C will be the same as that of A (3.19) c :b c and assume further that v-2(0X2)fi". (3.20) holds, then eq. (3.16) is universal and so is ZJ^(G): (3.21) The modification of the partition function is only through the boundaries, an effect that cannot change, in the therrnodynamic limit, the free bulk-energy. As a consequence the two models share the same critical behavior i.e. if one has a second-order phase transition so does the other. From the definition of the central change as a universal correction in finite-size effects, we conclude further that they both have the same central charge. The two theories are different of course due to differences in their operator content. All this can be checked explicitly for the minimal models where the largest eigenvalue is the same for any two theories with the same Coxeter number and eq. (3.20) can be verified directly using eq. (2.17). We impose then the conditions (3.20) and (3.19) on our Boltzmann weights. By inspection, we notice further that for the minimal models the set of eigenval- ues of an exceptional graph is included in the set of eigenvalues of the regular one with the same Coxeter number. It is thus possible to find an intertwiner between the two adjacency matrices AV=VG. (3.22) We shall return to it, and to some important consequences, in the next section. Another useful relations are the unitarity relations discovered by Deguchi et al. [18] for the A graphs. The first unitarity relation (from here until the end of the section we use roman letters also for the A model in order to be compatible with fig. 4) ,e,c,b\u)~smir(\-u))sm(w(X+u))Slle, (3.23) -\ 624 Af. Sochen / Iniegrable models follows directly from U2 = pU. The second reads (3.24) where p(u) = SHI(TT(A - M)/sin(iru) and y = N\/2. Writing explicitly the Boltz- mann weights and with some trigonometric manipulations we get (3.25) This relation can be understood as the simplest one that guarantees the compatibility between the commutation relation of the generators of the Hecke algebra and the Markov trace properties. To see that let us multiply relation (2.8c) by <#/' and sum over d (with the same notations as fig. 4). Using the projection property (2.8a) of the weights, the Markov trace property (3.17), and (3.18), we can carry out the summation over three terms in (2.8c), getting finally «>. (3.26) Relation (3.25) is obviously the simplest one that leads to the above relation. Since we imposed the Markov trace property on the exceptional models, it is natural to impose this second unitary condition as well + PW_3(0/2)* The last remarks are the simple observations that, for SU(2), the adjacency matrices are diagonalizable, and that the graphs are 2-colorable. We make the ansatz that for the SU(N) cosets the adjacency matrices are diagonalizable as well and that the graphs are Af-colorable. The Hecke generators for the two classes that N. Soclien / Intenable model!, 625 we presented are real, for the orbifold series they are hermitean. We assume then that they are hermitean for the other models as well. 4. More about graphs and intertwiners In the last section we detailed some properties of graphs on one hand and of Boltzmann weights that correspond to these graph models on the other. We concentrate in this section on graphs and the intertwiners between them. We list the expected properties for graphs (or equivalently for adjacency matri- ces) that are conjectured to be related to SU(N) coset models (1) The graph G is N-colorable (2) The adjacency matrix G is diagonalizable (3) Denote by exp(G) the set of eigenvalues of the graph G, then exp(G) c expM) (possibly up to multiplicities) for a graph A that has the same largest eigenvalue as G. (4) There exist an involution: a -» a such that Gab = G63. eads to tne same (5) G is not a quotient of another graph (i.e A2n/ 12 ' statistical mechanics model as A2n [3]). These conditions and especially condition (3) are very restrictive. Di Francesco and Zuber found a list of graphs satisfying (I)-(S) for the cosets of SU(3). It is important to note, however, that this list is not restrictive enough in the sense that their list contains graphs that cannot give rise to a representation of the Hecke algebra. We will be interested mostly in the graphs they called E(,I2) and E<24) (see figs. 6,7). But before giving the explicit form of the Boltzmann weights that correspond to these models we would like to talk more about intertwiners. For any G that has properties (I)-(S) we can find the following intertwiners V"= E C*|0<">><0<">| (4.1) p eexp(C) } } We choose C* = $tf /4ff . Then denoting (K")a6 = Vf0 we have the special set of square matrices <*<*> *X- L T5o*ipVip)* <4-2) p e exp(G> ^' These matrices have the following properties (a) LrA119VJ, = F.,K£Grt = ZeG.eV& (b) Vab = *,b «a> + (W imply V^ = G), (c) KfcelM, (d) V*V = LWV\ where Nf are the fusion coefficients of the "regular" representation of the fusion i.i 1 fi 626 N. Sochen / Integrable models KM algebra. That is the intertwiners form a m = IG I dimensional representation of the Kac-Moody fusion algebra over the non-negative integers. The fact that these are integer valued matrices follows from (a). Using the explicit form of A111. we have a recursion relation for the ^""''s with J"", Vtf } and V^ ' as coefficients. Thus the K 's can be written as polynomials of V\ Vf and V{. In particular. taking for example the SU(2) case A i/""'= V{v+ "+ V(v~ !)= V(v}G = V(v)V{f) , (4-3) and for SU(3) by similar considerations y(i>l + 2.l) _ 1/(V1 + 1.I)(J _ I/<"|.1)(J« _|_ ^("1-1-I)1 (4-4) and 1/(V1, V2) _ 1/(^ i-i- 1. V2-I)Q _ ^(VI +2.1-2-1) _ J/OM + Lf2-Z)1 (4.5) The fact that they are non-negative is more difficult to prove and the best we can say is that it was checked to be so in all known cases. The problem of finding all the representations of the SU(2) Kac-Moody fusion algebra over the non-nega- tive integers leads again to the ADE classification of lattice minimal models. For higher-rank algebras the equivalence, between the representation theory of Kac- Moody fusion algebra and graphs with the properties (D-(S) above, remains a (very appealing) conjecture. In order to understand better the physical meaning of intertwiners and to give a partial explanation of their non-negativity, we discuss some relations of partition functions with fixed boundaries. We restrict ourselves to the case of SU(2) where we have an explicit expression for the Boltzmann weights of all the models (regular and exceptional), and postpone a discussion of more general relations to later on. Let us start by generalizing the Markov trace property*. By using (2.17) we have ^i1W*1* O -e»=^*• , and * These arguments have been developed in collaboration with J.-B. Zuber [9]. I ] £ ft N. Sochen / Integrable models 627 Since iA(M) is an eigenvector of G we have also Z>i*> = y<*>*i">. (4.8) b:a where the sum is over all the spins which are adjacent to a. We are now in a position to prove = E +^ZttW-t^mtW/W *« «r.^ a, beG A,i Notice that it is sufficient to prove » ^ ail G. (4.10) Using this result we see that (4.9) follows since all the expressions are equal to ^/universal- T° PrOVe 6I- W-IO) WC WHtC the l.h.S. 3S i.h.s.= ELE VTrW-IO, <4-n> where >, (4.12) («2 ..... OL) we prove it now by induction. For / = O repeatedly using eq. (4.8) we get For / = 1 using eqs. (4.6) and (4.8) we get T^(LJ1) = (y^^-V'tyr. (4.14) Suppose now that ) Tri">(^i...^,)-/univerMie (4-15) then for p > 1 Ll 628 /V. Sochen / lntegrahle models and for p = 1, by eqs. (4.7) and (4.8) which completes the proof. Inverting now the left hand side using the orthonormality of the i/»'s we have = E Er>eVr* Ztf> = £(K*UZtf». (4.18) The generators £/ of the Hecke algebra act on paths in the graph G. It has been shown by Pasquier and Saleur [19] that the only irreducible representation of this algebra are those that start in an extremal point in the graph A and end in a generic point A in that graph. All the other representations associated with other paths in graphs A or G are reducible. The above formula shows how traces over these representations decompose with coefficients which are our intertwiners. It is thus natural that those intertwiners be non-negative integers. M. Bauer and H. Saleur [20] studied various systems with more complicated boundary conditions. Although in a different context, their results can be used in our framework. We are interested particularly in the results for lattices with two spins fixed on one side and one on the other. We denote the partition function for ( this system Z $c. They calculated these objects for the series of Dynkin diagrams A and D. We show now that the results have an algebraic explanation, that is, they follow from the relation where we denote eab the extremal point in the branch where a and b are found, and a' is the point in the graph A given by , (4.20) i, j (•*-»' M. Sochen / Integrable modela 629 b' is defined in an analogous way and we require a and b' to be adjacent. The proof is given in appendix B. What happens now when we try to generalize these relations to higher-rank algebras? Our proofs were based on the knowledge of the form of the generators of the TL algebra. Since we don't have analogous expressions for the higher-rank algebras we cannot prove such kind of relations directly. Nevertheless relation (4.18) can be taken as it stands as a conjecture for the more complicated graphs. It was checked on small lattices to hold on all known cases, i.e all regular graphs, their orbifolds (generalized D series) and exceptional graphs like E(1!) of the SU(3) coset. It is also naturally related to the analysis of Pasquier and Saleur [19] as was mentioned above. All that strongly suggests that it holds in full generality. The second relation (4.19) is more difficult to generalize and the reason is twofold. First the notion of extremal point is not always clear since there are graphs where no vertex has only one neighbor and is a neighbor of only one vertex. Second the generalization of the concept of branch is an hyperplane (a plane for SUO)), but taking for example SU(3) it is not clear how to map a triplet from the G graph to the A one when the three vertices are not in a definite plane see for example points 4, 5 and 6 in the graph E\12\ Yet in the situations where the map is unambiguous and the extremal point is identified this formula was checked to hold on small lattices (see for example 5,2,4 in the same E(,I2)) (see fig. 7). 5. New solutions The systems for which we present the Boltzmann weights are E(,12) and E(24). The method of calculation was to use the Markov trace property (3.17), and relation (4.18), together with the defining properties of the Hecke algebra. Notice that relation (4.18) holds locally for any size of system and in particular for the smallest ones we have b v- L >> and (5.2) where 1 is the apex of the Weyl alcove. These relations, the Markov trace property and the commutation relations of the Hecke algebra are strong enough to make the calculation feasible. These are the only relations we used in order to find the 630 N. Sochen / Integrable models 1, Fig. 6. The special graph E(24). Fig. 7. The special graph E(,12). 1 N. Sochen / Integrable models 631 Boltzmann weights. All other relations were checked to hold a posteriori in particular the vanishing of the quantum anti-symmetrizer (2.9). ]2 We start by listing the weights for the model E\ \ This model has obvious S3 symmetry that permutes the three planes from which the graph is built. Our notation is as follows «,> £/<•*> ... t/J!*' >b {a, a,} IJ(ab) Ij(ab) ^a.a, ' • • '-'a.a. we will give only the weights that are different from those of the regular graph where the mapping from G to A was given in eq. (4.20). They are 2 -x) c* ' ft -x where the star means complex conjugation, the indices 1 and 2 refers to the links between the vertices 4 and 5 for l/<4-2'> and 5,6 for t/(3-6), jS = 2cos(ir/12), x = 0.58600, and ot = e2iri/3 (an example of a calculation is given in appendix C). (y//32)* 2 -(U23)*, ' 1/0 J where y = j82 — 1, the non-diagonal elements are factorized U^ = yU^*' and Ua = (Ua)*. (1,1) (1,2)(2,1) (2,2) a OO -1/1/2"' (1,2) O OO O (2,1) O OO O (2,2) -l/v/2 OO d L.' 632 N. Soclten / Integrable models where a + d = /3 , ad = {. and we take a > d. A O T ro» rta O B -S -s -s —s re" rta ~~s te.'1 re10 -s te" re here 1 = 1.18882, 5 = 0.22539. r = 0.54262, s = 0.35806, t = 0.49398, 0 = 0.44917. All these numbers are ratios of trigonometric functions of tractions of TT, hence algebraic numbers. We present now the solution for the model E<24). Again we give only the weights that involve relatively more work. Parts of thi graph which are close to an extremal point can be mapped safely to the A graph a.id the corresponding weights follow. Other weights can be found easily using only relation (4.18). The other ones read as foliot- ai,, 12,) s(5)/s(4)-s(4)/s(5) + j_ cf4\ /v( *!^ — ' \ // \ / t 5(3)A(4) <10r4.,,8,+1,8fV * ^y-I)77-,/ _/ + 12r (7r.9f.9p+l) (10,4.,,8,,.1,S,), + -i s(3)/s(4) T N. Soclien / Integrahlc motlch 633 where d = 0.89128. (12,,12,. ,,5,) s(6)/2s(S) *(6)/2s(5) (12,,12,.,.5,) -I -i (12,.12,,,.5,) -z -i i i z- (12,,12,,.,,S,) -i - 1//3 where z = 0.35660. 0,,,.8r) 12"1 " "-O7' (+ 7,* ,.9,) where y = 0.26105. 6. Summary and conclusions In this paper we presented new solutions to the Yang-Baxter equation. The main motivation in finding explicitly the Boltzmann weights, which correspond to a given graph and solve the Yb equation, besides its value as new solvable models, is to verify a number of relations between the Boltzmann weights in a given model and between Boltzmann weights in different models. We see these relations not as further constraints of the Hecke algebra, but as relations that come from the existence of a representation of the Hecke generators in terms of properties of the graph. Although we were not able to find the generalization of (2.17) for SU(W), N > 2 models, we feel that the number of relations that we presented is close to some "critical number" of relations from which it would be possible to find such kind of a realization. Some problems and directions for further research are (i) continue this program by finding an explicit realization for the Boltzmann weights, and calculating the partition function for these models, (ii) take the continuum limit and make contact with known modular invariants that were calculated from the CFT point of view, and (iii) find an extension to solvable models off criticality and investigate the phase space. It is a pleasure to thank J.-B. Zuber for numerous discussions, suggestions and for his constant encouragement. 634 Af. Sochen / Integrable models Appendix À To prove eq. (3.11) we notice first that we can always write (A.I) /-J m-L + l where tr is defined in (3.9), «/+, > t, V/, /„, > /, V/, /w and denoting M = minm{/m}, UM appears at least twice (L may be O, in that case UM = U2). This is so since if UM appears only once then trr, n i/-trni/^ni/ \/-l m-L + l j \'~1 WI m2 (A-2) m, where we used eq. (2.8b) and /m| -i/> 1 Vm1,/ for the second equality and the cyclicity of the trace for the third. This procedure can be repeated till the first index that appears more then once. We will call R the length of the word, and L the length of the ordered part of the word. We proceed by induction on the length of a word, and on the length of the ordered part of a word. We suppose that it is possible to order any word of length R-I, and any word of length R with ordered part of length L + 1. It is straightforward to see that any word of length R = 1,2,3 can be ordered, and that any word of an arbitrary length R can be ordered if L = R, R - 1, R — 2. We write rm, n /-1 m-L + l /-l m, where the two l/M's that are shown explicitly are the two leftmost of in total k0 UM's. There are three possibilities: 2 (i) A/+l<£{m,}, in this case [UM, nmit/m,] O and using (UM) we reduce R by one. N. Sochen / Intenable models 635 (ii) UM+] appears in Yln, Um] exactly once. In this case we use eq. (2.8c) to write / L \ I L K tr H FI ",,"M II U, UM U «/,., = Fl V,fJMU^ ,t/M H U, U = I m, ' m, -/ W-I m-L+4 / L R \ -T m^t/M+I^M^II+l FI ",,,} W=I m = /.+4 / I L R-2 ^ i L K~2 \ -tr FlW+, H 1/,1+tr ni/,,1/* U U1]. W-I m-L + 2 I W-I m=L + 2 / (A.4) The two rightmost terms can be ordered by the induction hypothesis. In the first term R and L were not changed, but k0 (the number of t/M's in the initial word) was reduced by one. If k0 = 2 then in the new word k0= 1 and by the procedure that we described in the beginning of this appendix we increase L by one. If k0 > 2 then we take again the two leftmost UM's and repeat the analysis. The above argument (together with point GiD) shows by induction that we can always reduce k0 to 1 which completes the proof. (iii) UM+i appears in YlmfJim &,*»2 times. Then we take the two leftmost s aR UM + i d repeat the above analysis for UM+2. The length of the word is finite and does not increase in the procedure, so at some level p, Um+p+l must appear once or not at all between the two leftmost Un +P s. In that case we can reduce kp by one. By induction kp can be reduced to 1, and by induction on p, k0 can be reduced to 1, so the proof is completed. Appendix B The way to prove (4.19) is analogous to the way we proved relation (4.18). We first write it in an equivalent form (B.I) We will write Za h as a sum of terms each of which has the form R-In i-i mK where /m(+1 =/m. + 1 1 < j < ft - 1 and jm-+1 >jmR. This is the best we can do since there are no summations over the two leftmost spins (in contrast to the one leftmost spin-fixed case where we could have a sum over ordered monomials). The Ul 1536 Af. Sochen / Integrable models proof of this result is in the same spirit as the proof we gave in appendix A, we omit it for the sake of brevity. We prove now eq. (B.I) term by term. Let us take first the case R = PK = 1, that is, we prove it first on ordered monomials. We prove it again by induction. For / = O E ( (a j a/1 -c^ï- L (B.2) for 1 = 1 i (03 V/.-ÏI-1 )\ i~'/~I VL^ <«.fr,«/ l. 3 Ca3 o,,} )/"""1 E ( ("3 «„-!> E (a',b',a3,...,aL\Uii\a',b',a3 (B.3) (B3 O1) N. Socheii / Inifgrahle modch 637 for / = 1 i, = 1 1.(Ml E (a,b,a3 aL\Uh\a,b,aj a,) -^7 (a, a,) ™f:ih MM) , , "a\ l fc(M) (03) (a,b,a3 aL\Ub\a,b,a3,...,a (B.4) suppose now that (B.5) (a, ..... aL] then " E (O3,...,a " E (O3 ..... * " E (B-6) 638 N. Sochen / Integrable models The prove for the other kind of terms make use of the relation the proof then is completely analogous to the ordered monomials one. Appendix C We give in this appendix an example of a calculation of one of the more "unusual" matrix element of the generators of the Hecke algebra for the graph E<24). Let us first write down the elements of the Perron-Frobenius vector _ 3y + 3) , _ y - where f 5 22 3 «0>2 = E(^)42 1<()>A = (K )42 1<^>3 = (G')«/3 = G24ft= ft, 5 A 2 2 where we used on the r.h.s. our knowledge of the A Boltzmann weights and the Ul Ll N. Sochen / Integrable models 639 intertwiners. The numérotation principle for the A graphs can be understood from fig. Cl. (b) , *, A so by (a) In the same way (c) (d) ii 4 = 1 =* 7 11 11 Using now the Markov trace property ii ii ii ii n ii we finally get ii 1.54982. ii -\ 640 N. Sochen / Imegrable models References [1] RJ. Baxter, Exactly solved models in statistical mechanics (Academic Press. New York. 1982) UJ G. Moore and N. Seiberg, Phys. Lett. B212 (1988) 451; Nucl. Phys. B313 (1989) 16: Commun. Math. Phys. 123(1989)177 [3] P. Di Francesco and J.-B. Zuber, Nucl. Phys. B338 (1990) 602 [4] E. Bannai and T. Ito, Algebraic Combinatorics I: Association schemes (Benjamin/Cummings. Menlo Park, CA, 1984) [5] V. Pasquier, Nuc!. Phys. B285 [FS19] (1987) 162; J. Phys. A20 (1987) 5707 [6] V.V. Bazhanov, N. Yu. Reshetikhin, Int. J. Mod. Phys. A4 (1989) 115 [7] H.J. de Vega, Int. J. Mod. Phys. A4 (1989) 2371 [8] P. Di Francesco and J.-B. Zuber, in Recent developments in conformai field theory, Trieste Oct. 1989, éd. S. Randjbar-Daemi et al. (World Scientific, Singapore, to appear) [9] H. Saleur and J.-B. Zuber, Intcgrable lattice models and quantum groups, lectures at the 1990 Spring School on String theory and quantum gravity, Trieste 1990, Saclay preprint SPhT/90-071 [10] D. Gepner and E. Witten, Nucl. Phys. B278 (1986) 493 (11) E. Verlinde, Nucl. Phys. 300 [FS22] (1988) 360 [12] H. Wenzl, Inv. Math. 92 (1988) 349 [13] M. Jimbo, T. Miwa and M. Okado, Lett. Math. Phys 14 (1987) 123; Commun. Math. Phys. 116 (1988) 507 [14] A. Cappelli, C. Itzykson and J.-B. Zuber, Nucl. Phys. B280TFS18] (1987) 445: Commun. Math. Phys. 113 (1987) 1 [15] I. Rostov, Nucl. Phys. B300 [FS22] (1988) 559 [16] P. Fendley and P. Oinsparg, Nucl. Phys. B 324 (1989) 549; P. Fendley, J. Phys. A22 (1989) 4633 [17] Y. Akutsu, T. Deguchi, M. Wadati, J. Phys. Soc. Jpn. 57 (1988) 1173 [18] T. Deguchi, M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. 57 (1988) 2921 [19] V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523 [20] H. Saleur and M. Bauer, Nucl. Phys. B320 (1989) 591 ARTICLE 3 Physics Letters B 275 ( 1992 ) 82-86 PHYSICSLETTERSB North-Holland Singular vectors by fusions in Aj '} M. Bauer and N. Sochen Service de Physique Théorique CEN-Saclay '. F-91191 Gif-sur- Yvette Cedex. France Received 27 Augt-jt «991 Explicit expressions for the singular vectors in the highest weight representations of Aj" are obtained by simple recursion relation using the fusion formalism of conformai field theory. We study in this letter some aspects of the repre- proach is based on basic facts and methods in confor- sentation theory of Aj". The subject was studied mai field theory, with which the form of the singular thoroughly by mathematicians [ 1 ] and physicists *' vectors for A1," is derived by means of a recursion in the last years. It is known that a Verrna module relation. The recursive form is very efficient in ac- ( i.e. highest weight representation ) is reducible for tually calculating the singular vectors, and the differ- certain values of the highest weight. This indicates ent terms in it are interpreted as descendent^ in a fu- the existence of a singular vector in the module, thai sion process. Then singular vectors appear as an generates a submodule. An irreducible representa- obstruction to fusion. This approach is in the same tion is obtained by quotienting the submodules out, spirit of BDIZ and we have analogously a mysterious or equi valently by setting the singular vectors to zero. relation between the singular vectors and the classi- The latter point of view is the most appropriate to cal w-algebras [4]. physical applications. We present the infinite-dimensional Lie algebra a Conformai field theory Ward identities permit us Aj " as a current algebra. The generators are J n, neZ, to express any correlation function that contains des- a= + , O, — , and K. The non-vanishing commutators cendents as a differential operator that acts on the are correlation function with the descendent replaced by its primary [ 3 ]. A singular vector is a descendent that f/0 / ± 1 _ + we equate to zero. We get, therefore, a differential I»' n» «»m J — — • equation for any correlation function that includes the corresponding primary field. The explicit form of a this equation is fixed by the form of the singular vec- The generators J n are doubly graded, with respect to tor. Explicit forms for singular vectors in Verma a (generated by /g ) and to n (generated by L0). It is modules for the Virasoro algebra were obtained re- convenient to introduce a combined gradation d(n, cently by Bauer, Di Francesco, Itzykson, and Zuber a) = 2n+a. We are interested in the representations ( BDIZ ) [ 4 ]. Forms for A^ > were given by Malikov, generated by a highest weight vector. The highest Feigin and Fuks (MFF) [5]. The form that MFF weight vector is defined by found reflects the symmetries of the AV > algebra, but it is not very suitable to physical application. Our ap- 0 J 0\J,t>=J\J,t>, Laboratoire de la Direction des Science de la Matière du Commissariat à l'Energie Atomique. Fractional levels appear naturally in 2D quantum gravity [ 2 ]. Notice that J is not a half-integer as usual but takes Volume 27S. number 1.2 PHYSICS LETTERS B 23 January 1992 a value in the complex plane, and we work with com- this allows to continue analytically to the whole com- plex central charge as well. All the states are reached plex plane. Actually calculating the RHS of (3). by by applying repeatedly J"n with d( n. a ) ^ - 1 on the applying successively the analytical continued com- highest weight, and taking linear combinations. A mutation relations turns out to be tedious. It is the combMaiiop I K) = UC1U1J, [«,j )•/!!: --Cl-/. O « aim of this letter to give a short-cut for it. called a descendent. A descendent I R~> that satisfies Let us consider for the moment only the +sign in J"H I R ; - O for d( n, a ) 2 1 is called a singular vector. ( 1 ) then J=J-J' t where 2j+ 1 . Ij' + 1 eM. We denote i ii .;asy to see that singular vectors are linear com- the corresponding highest weight representation as (_/, binations of homogeneous singular vectors with re- / ). The singular vector in the class of representations spect to ui; graduation L0 and J%. We denote the ho- of type (Q, j) admits a matrix form, the Ai ' ' analog mogeneous pan as of the formulae of Benoit and Saint-Aubin [ 7 ] and BDIZ in the case of Virasoro. Let «= 2j+ 1. We define the (2n- 1 )-dimensional ,4< {a, vectors /= (fj+a- I » gj-t-n- I .-//-Ki-2. •••• './/)' » F=(-gJ+a,0 ..... O)', Theorem (Kac and Kazhdan [6]). The highest weight \J, i ?*0> generates a reducible module iffy where f,= \J, f> and the other components are de- fined by solving the triangular system takes the value JrJ. ± defined by (4) r, S€\ . (D The singular vector is in where ±r>. (J- )u=SH+t, 2n-l, Theorem (Malikov, Feigin and Fuks [5]). The /2I form of the singular vector in \Jrj.+, IJ=Q) js - « " )<5,, , , = { [n- J(I+ 1 ) ]6<ï> + Wff }0JJ+, , (2) where [x] is the integer part of x, and and ./0 *-*•/- 1 for the sign ( - ) in ( 1 ). JJj» = !, /=/mod2, =0, otherwise . This is not to be interpreted as a multiplication of operators raised to complex powers, but in the fol- In fact K2"- ' =0, so the sum in (4) is finite. One can lowing way: there is an infinite set of integer values t show that the components of/, F satisfy such that (2) is well defined, and by ordering the generators and applying the Poincaré-Birkhofï-Win theorem, this vector can be expressed as from which is clear that £/+„ is a singular vector. Take for example the representation (O, \) (that is J= -J/), then (4) reads X|7rj.+ ,O. (3) where p, q are non-negative integers, /"„., are polyno- a mials in the J n with d(n,a)^-2. The heart of the proof is to show that Pp,q are polynomials in t, and 83 IiI Volume 275. number 1,2 PHYSICS LETTERS B 23Januar\ -Sj + :\ Sugawara construction ° L = £ (:J°,.,,,J", +1 J,:_,,,J- +1J-_„, J,T,:) . O J n In this formulation L_, and J /J<7 'J% -rJ:,\ /J+, in : and in .v respectively. Thus, we can write = I J-i i/°-, (gj+i 0-7U, A-) =exp(.vj,r +:L_, ) 0'(O. O) Vo i Jn / /j Xexp(-A-J,7-rI_|). which gives the singular vector It follows that 0 gj+i = (JôJ-{Jô -tJôJ ., -t d (5) /-J- .. J <6a) ~" " dv Notice that in the "classical limit", and after "gauging" in the spirit of hamiltonian reduction, (6b) J0- -4 ./In -O, [J*, 0y(r. A-)] =:"( -x1 — +2A-Jj 0'(T. A). (6c) this matrix, that maps/) to £/+„, becomes the covari- Let us look now at the short distance operator prod- ant differential operator of Drinfeld and Sokolov [8 ] uct expansion for these chiral primary fields. For this that maps {(!-«) forms to j ( 1 +«) forms. aim it is more convenient to write We show now how the matrix form appears natu- rally from fusions of primary Fields in conformai field theory. Our main result will be a recursive formula which is a consequence of (6b). Imagine that we are for the compcnents of/from which the matrix can be interested only in the J sector in the fusion of J0 and recovered. The advantage of this approach is that it J, then gives us also the form of the singular vector in a gen- 0'0U A-) |J, , /> =0'°(::, jt)0y- (O, O) ifi, /> eral O, / ) reducible representation. It gives us im- portant information about fusion rules as well. Ji-Jg0J0J i i )0'i(0, 0) |i2, /) J We introduce a chiral primary field 84 i.t Volume 275. number 1.2 PHYSICS LETTERS B 23Jamiar\ and in the same way troduce an order (;;„./>;„)> (n. IH) if /;,,>« or if H,, = I/ and /H0 > IH. We first note that ( 1 1 ) is a recursion re- lation since ( n. m ) of the LHS is bigger than all the This fixes ^.,,, to be in the homogeneous subspace of pairs of indices on the RHS. We define (»/„. /'J11) to conformai degree n and charge - m, as expected from be the smallest solution of .)/(/). m)=0 if it exists (7), and VQ.O is proportional to \J, />. For the gen- and (oo. so ) otherwise: note that there is at most one a erators J n, d(n, a)^\ there are only two indepen- solution if / is irrational. Taking vi'.o to be propor- dent equations: tional to \J. />. all v'/,.,,, for (O. O) < (n. IH) < (U11. w,, ) are uniquely defined in terms >-Vo.o- we define ( a,,. -/O+ Vn m = (J-J +Jl -m+ I )«//,.„,- I , 0 IH0) as the solution of AJ(U, IH)=O if it exists and J Jïri.,n = (-J+Jo+Ji+m+l)v n_l.in+l. (8) (oo, oo ) otherwise. From the theorem of Kac and J Kazhdan we know that if there is in a representation These descent equations determine v/ n.m as long as Ja singular vector in the homogeneous subspace { U11. the kernel of J£ , Jf is trivial. The existence of a sin- IH0) then (i) /Mn0, in0)=0 and (ii) IHO divides n,,. gular vector in IJ, t, n0, m0> means that the kernel of We conclude that if J is such that .-!./( n. IH ) - O has no Jo ,Jr is non-trivial, and a a consequence J0, J\ and solution than the J module is irreducible and the des- J must verify a relation in order for (8) to have a cendants i//J,,,,, are easily calculated from (11). solution. This relation is the fusion rule. Take the One can check recursively that the descendents cal- vacuum sector for example, Vo.o = \Q O and Vo., = culated by ( 1 1 ) verify the descent equations. In the Ao.iJôVo.o «s a singular vector. The non-trivial de- case where Aj(m, n)=0 admits a solution («„. IH,,) scent equation gives O=Jo Vo.\ = (J\ -Jo)Vo.o and but m0 does not divide n0 the representation is irre- we see that only the fusion of a primaiy field with ducible but Vno.mo cannot be found from (11) since itself contains the vacuum sector. in this case both the LHS and the RHS vanish iden- A solution to the descent equations is obtained us- tically. For the reducible representation J where ing the Knizhnik-Zamolodchikov equation [10] 0, w0)=0 and IHO divides n0 we cannot find combined with the fusion procedure. We write from (11) since its prefactor on the LHS of (11) vanishes, but this time the RHS does not vanish (9) in general and it is proportional in fact to the singular vector we are looking for. Take for example J= - 5 / multiply on the left by 0"'°(z, x) to get then .4,(1,I)=O: no matter how we choose J0 and J, J Q= °(z,x)ltL_, -(Jl1Jo + 2JtJ°_,)] the RHS is found to be proportional to ( 5 ). We stress the fact that this recursion relation generalizes the >. (10) matrix form we presented before. To see this we ar- We commute the Aj" and Vir operators ;o the left range the descendents as column vectors using (6), plugging (7), and after some manipula- F=(s ,0 ..... O)1. tions we obtain nojno J lnt+m(2J+l-m)]v n.m The recursion relation then gives a triangular matrix = (-J+J +J +m+\) T J-kVl. i 0 t m+ form, and smjna is the singular vector wanted. Details and proofs will be given elsewhere. Quantum hamil- tonian reduction [H]*2 between the physical spaces + 2(J-J0-M) I J°_kVi k + l-n of Vir and Aj ", and the relation between the Ai " singular vectors and classical w-algebra are under (U) investigation. A.»0 We define Aj(n, m) = nt+m(2J+1 -m) and in- « For an approach akin to ours, see réf. J12J. 85 U'f Volume 275, number 1.2 PHYSICS LETTERS B 23 January References ( 5 ] F.G. Malikov. B.L. Feigin and O.B. Fuks. Fanki. tnal. PnI. 20. No. 2 (1987 )25. [ 1 ] See e.g. V. Kac. Infinite dimensional Lie algebras |6] V.G. Kac and D. A. Kazhdan, Adv. Math. 34 (1979) 97. (Cambridge U.P., Cambridge, 1985), and references [ 7 ] L. Benoît and Y. Saint-Aubin. Phys. Lett. B 215 ( 1988 ) 517. therein. [8) V.G. Drmfeld and V.V. Sokolov. J. Sov. Math. 30 ( 1985) [2 ] A.M. Polyakov. Mod. Phys. LeU. A 2 ( 1987) 893: 1975. V.G. Knizhnik. A.M. Polyakov and A.B. Zarnolodchikov. (9J A.B. Zamolodchikov and V.A. Fateev. Sov. J. Nucl. Phys. Mod. Phys. Lett. A 3 ( 1988) 819; 43(1986)657. D. Bernard and G. Felder, Commun. Math. Phys. 127 110) V.G. Knizhnik and A.B. Zamolodchikov. NuO. Phys. B 247 (1990) 145. (1984)83. [ 3 ] A. Belavin, A.M. Polyakov and A.B. Zamolodchikov. Nucl. [Il] A.A. Belavin. Adv. Stud. Pure Math. 19 ( 1989 ) 117; Phys. B 241 (1983)333. M. Bershadsky and H. Ooguri. Commun. Math. Phys. 126 14) M. Bauer. Ph. Di Francesco, C. Itzykson and J.-B. Zuber, (1989)49. Nucl. Phys. 6362(1991)515; see also M. Bauer, Singular vectors in Virasoro Verma [12] P. Furlan. A.Ch. Ganchev R. Paunov and V.B. Petkova. modules. Saclay preprint SPht/91-096, in; Trieste Workshop Phys. Lett. B267 (1991) 63. on String theory (Trieste, Italy, April 1991), to appear. 86 l.i ARTICLE 4 Fusion and singular vectors in AI highest weight cyclic modules M. BAUER and N. SOCHEN Service de physique théorique de Saclay1 F-91191 Gif-sur-Yvette cedex, France Abstract We show how the interplay between the fusion formalism of conformai field theory and the Knizhnik-Zamolodchikov equation leads to explicit formulae for the singular vectors in the highest weight representations of A, '. I Introduction Infinite dimensional Lie algebras occur everywhere in the study of 2-d conformai field theo- ries: the Virasoro algebra and the affine algebras are the most common examples. However the construction of the irreducible representations of these algebras is quite involved. Sin- gular vectort. are important because they indicate the existence of subrepresentations in a given representation. In the affine case, Kac and Kazhdan [10] gave the criterion for the reducibility or irreducibility of the Verma modules and Malikov, Feigin and Fuks [14] found a formula for the singular vectors in the A$ case. This formula looks very simple, but involves an analytic continuation to make sense, which makes it very difficult to use. Apart from the purely mathematical description, several approaches motivated by physics have been proposed, based on vertex operators (see [16] for a general reference dealing with AI ), bosonization and variants of the Feigin and Fuks construction and BRST cohomology [4]. In the physical context, the importance of singular vectors comes from Ward identities: to calculate a correlation function involving a descendent of a primary field, one simply applies a linear operator to the correlation function of the primary [2]. A singular vector is a descendent that is set to zero in an irreducible representation, with the consequence that the correlation functions of the corresponding primary satisfy closed linear relations, leading to a contour integral representation. One of the aims of this paper is to show that elementary methods of conformai field theory allow us to understand some important features of the structure of representations of theses algebras. Our inspiration comes from remarks at the end of the seminal paper of 1 Laboratoire de la Direction des Sciences de la Matière du Commissariat à l'Energie Atomique / INTRODVCTIOS Belavin, Polyakov and Zamolodchikov (see appendix B in [2]). We restricted our attention to the A\l) algebra not only for simplicity (although generalization is not straightforward, we believe that the same methods applied to other affine algebras will lead to interesting results), but also because we hoped to get a better understanding of the construction made in [1] by Bauer, Di Francesco, Itzykson and Zuber for the singular vectors in Virasoro Verma modules. The basic idea is the following : the symmetries of conformai field theories are so large that they determine "almost" completely the structure of the operator product expansion of primary fields. A remarkable homogeneous linear system, the system of descent equations (see section IV.3), encodes this structure. The singular vectors are in the kernel of the descent equations, and by duality, they also appear as an obstruction to solve the linear system. This can be used to compute them. The A* case has its own peculiarities, but is in a sense easier to deal with than the case of the Virasoro algebra, and a more complete treatment is possible. We still expect a precise connection between the two cases via Hamiltonian reduction [5], although as yet we have only been able to work out some simple examples. The organization of this paper is as follows. We begin with a short reminder of the basic notions in the representation theory of affine algebras in our particular case. We introduce Verma modules, singular vectors, and the contragredient form. This is standard material, included only for the sake of completeness. For a more detailed and pedagogical presentation, see [U]. The next section quotes (again restricting to the A[ case) the results of Kac and Kazhdan [10], and the formula for singular vectors given by Malikov, Feigin and Fuks [14]. We decided to include some of the proofs, hoping that a physicist's style could make them accessible to a larger audience. Furthermore, some features of our constructions have counterparts in these proofs, showing clearly that for the time being, our work is not a substitute for the usual representation theory, but uses it in several places. In section IV we introduce the notion of Verma primary fields and explain fusion from a naive point of view. This leads to the "descent equations", which summarize the structure of the operator product expansion. We end this section with some comments showing the relation with a more mathematical definition of fusion. In section V we derive important consequences of the descent equations, using the contragredient form as a fundamental tool. This leads to the existence of fusion rules. In section VI we recast the descent equations in triangular form, and point out the role played by the so-called Knizhnik-Zamolodchikov equation. This allows us to calculate recursively all the descendants of a primary field in a fusion process. We use this recursive form in section VII to obtain explicit recursion relations or matrix forms to calculate the singular vectors. The next section is devoted to some simple comments related to our initial motivations, i.e. the relation with the case of the Virasoro algebra via Hamiltonian reduction. Some technical details are treated in appendix. We have tried to give a self-contained and pedagogical presentation, but decided to refer systematically to [1] for the comparisons with the case of the Virasoro algebra. // BASIC DEFIMTIO.\S II Basic definitions 11.1 The A\l} algebra The Aj1' algebra (which we shall also denote simply by ,4) can be presented as a current algebra with generators Jk and J°, n € Z, a € {-, O, +} satisfying the following commutation relations: + + + + [J , Jn ] = O - = O [J' , k] = O [J° , Jn ] = J +r. [J° , Jn] = - J-+n [J+, JnI = km6n+m + 2J°m+n (I) This algebra is doubly graded if we define 0 = a 3(Jb)=O 4(Jn ) = n J(Ib) = O The so-called principal gradation d = 2d •+ d is used to define several subalgebras needed to construct the A, _Verma modules. We remark that the commutation relations with Jo simply calculate the d gradation i.e. ad(J°) is multiplication by d. It is also useful to add to A a generator called D_ with analogous properties with respect to d, that is The Jacob! identities are still true because A is graded by d. Shifting D. by a constant does not change the commutation relations. We set A = A ® CZ). In physical applications, the Sugawara construction will provide an explicit form for D. so adding it to A is not completely artificial. Up to an additive constant, — D_ will be the energy operator, which we require to be bounded below in representations. We write t€Z where & is the subspace on which d = 2d+d takes the value i and £_ (resp. £+ ) is the direct sum of the £,'s for negative (resp. positive) i's. Finally we let B = £o ® £+• The dimension of £Q is 3 and the dimension of £;, i ^ O is 1 or 2 depending on whether i is even or odd. It is easy to check that the smallest Lie subalgebra of A containing £_! (resp. £\ ) is £_ (resp. £+). Furthermore £_i @ £\ generates A. This last observation can be generalized (s-ee [9]) to give an axiomatic definition of affine algebras by generators and relations, leading to a theory very akin to the theory of finite dimensional complex semi-simple Lie algebras. We introduce now the basic tools to study a certain class of representations of A. We begin by recalling some useful concepts. For the rest of this section, we more or less follow [U]. // BASICDEFIMTIONS 4 II. 2 Verma modules Let G be a Lie algebra. We shall denote by U(Q) its universal enveloping algebra. This space can be defined abstractly as the quotient of the tensor algebra over Q by the two-sided ideal generated by the commutation relations in G. This definition is just what is needed to make representations of Q and left U(Q)-modu\es the same thing. Naively, when we make calculations in a representation of C on a space E we manipulate the representatives of elements of Q in End(E), and U(Q) is the space where we can make all the manipulations which do not really depend on the particular representation we are dealing with but only on the commutation relations in Q. We now state two results which we shall need later on. • The first one is the Poincaré-Birkhoff-Witt theorem: fix a basis 7, of Q as a vector space, where ? belongs to some ordered set /, then monomials of the form 7,, ---- ^n, where i\ < • • • < in, form a basis of U(Q) as a vector space. The hard part is of course the fact that these monomials are linearly independent. To see that they span U(Q) we simply apply the commutation relations. In the special case when the algebra we deal with is an oscillator algebra this simply tells us that Jt is possible to put the annihilation operators on the left and the creation operators on the right by applying the commutation relations (in fact we shall see in the next section that A is not too different from an oscillator algebra and use an interesting consequence of this fact). • The second one is the fact that U(S- ) does not contain zero divisors. For an elementary and lucid account on universal enveloping algebras, see [12]. Verma modules are usually defined by giving properties that characterize them. Ths starting point is a one dimensional representation of £o, a maximal Abelian subalgebra of A. In this representation, J° and k act by scalars which we denote generically by j and t — 2. By analogy with the finite dimensional Lie algebra AI, we shall sometimes call J the spin of the representation. The value of £> is immaterial, we take it to be O. We can turn this space into a one dimensional representation of B by letting £+ act as O. We denote this representation of B by C'J|t'. A Verma module V^.') for A is a representation of A with the following properties: 1. The module V^1'' contains a one dimensional subspace VQ.O carrying a representation of B isomorphic to C(j'(). 2. The smallest subspace of stable under the action of A and containing Vo.o is itself. 3. Any representation of A satisfying the first two properties is isomorphic to a quotient of These properties make it clear that two Verma modules associated with the same C'*'' are canonically isomorphic, so Verma modules if they exist are unique. Usually, representations satisfying properties one and two are called cyclic representations. "T // BASICDEFlMTlONS To prove existence we consider the induced representation U(A) 3i'(B) CM. As an U(A)-modu\e this is isomorphic to the quotient of U(A) by the left ideal generated by ^o ~ J' k ~ (< ~ 2). D, and the J^s in £+. We denote this ideal by !<*". It is easy to check properties 1, 2 and 3 for this representation. We can now order the generators according to the principal gradation and apply the Poincaré-Birkhoff-Witt theorem to see that any element in U(A) can be written as a linear combination of terms of the form X-XoX+ with x* € U(£a) for a € {-,G,+}. This implies that V^ is isomorphic to U(S-) as an U(S-)- module. If x € U(A) we denote its image in the quotient by \x). The module property is simply that x\y) = \zy), and we call |1) the highest weight vector, a terminology borrowed from the theory of semi-simple Lie algebras. Later, when we need to manipulate several Verma modules at the same time, we shall use the notation \j,t) for the highest weight vector in V(j'°. Let us finally remark that Vlj>() is a doubly graded representation. In fact the Poincare- Birkhoff-Witt theorem implies that the monomials +00 +00 (2) 1=1 f'=0 (where all but a finite number of the integers p's are zero) form a basis of the Verma module. The values of — d and — d on such a monomial are respectively n = £i,azPi.o and m = ~Yli,aaPi,a, and we see that n is always non-negative and m is never less than — n. We denote by / (see figure 1 ) the set of couples (n, m) and end up with a decomposition (n,m)€/ Explicit summation over the p's then leads to a formula for the generating function of the dimensions of the graded subspaces (which is the character, up to an overall factor) as +0° 1 Highest weight cyclic modules are quotients of Verma modules. Thus they are doubly graded, and we shall see below that their characters are alternating sums of characters of Verma modules. II.3 Singular vectors, the contragredient form and representa- tion theory The first question we have to understand, now that we have defined Verma modules, is whether V(j>l* is irreducible as an U(A)-modu\e or not. We are going to introduce two important tools that allow us to reformulate this question and that will also prove useful later on when we discuss fusion: // BASIC DEFINITIONS m (0,0) (0,1) (0,2) (0,3) (0,4) (0,5) (1,-1) (1,0) (1,1) (1,2) (1,3) (1,4) (1,5) (2,-2) (2,-l) (2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (3,-3) (3,-2) (3,-l) (3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (4,-4) (4,-3) (4,-2) (4.-1) (4,0) (4,1) (4,2) (4,3) (4,4) (4,5) Figure 1: The set / • Vectors lying in V(jit) but not in VQ.O and annihilated by £+, called singular vectors, • A bilinear symmetric form on V(ji() called the contragredient form. We begin by recalling an elementary lemma in linear algebra. Lemma II. 1 If a linear map I on a vector space E is diagonalizable, and if F is a subspace such that /(F) C F, then the restriction of I to F is also diagonalizable. By hypothesis, E is the algebraic direct sum of invariant subspaces of /, that is E ~ with /|£A = A/rf. Any vector in E, hence in F is a finite linear combination of eigenvectors with distinct eigenvalues, say / = £"=1 /, with /< € E\t. By assumption, /(F) C F, hence /> '(/)• ' ' -i /'""1H/) belong to F, and by inverting the linear system with non-vanishing (Vandermonde) determinant £JL, A}/, = /(j)(/) for j = O, • • • ,n - 1 we see that /i € F for i= l,---,nand F = ®i As a consequence of this lemma, using the operators D_ and JQ, we check that any submodule M of V^1'' can be decomposed as M= ® Mn, (n,m)e/ with A/n,m = Vn,m (") M. We see here that enlarging A m A is very useful. If we define, for p > O, Mp = ®2n+m=P Mi,mt we end up with a decomposition of M according to the principal gradation. By the definition of a Verma module if M0 is non-trivial then M coincides with '. If M is a proper submodule we choose p minimal among those for which Mr £ {0}. Then Mp is annihilated by S+, i.e. consists of singular vectors. Hence any proper submodule // BASICDEFlMTlONS 7 of V'J>*> contains a singular vector. The converse is also true. In fact let 5' be the subspace of V'(jit) annihilated by £+. As ad(D_) and ad(J°) act diagonally on A they map 5' into S' and we can apply the lemma to get (n,m)€/ with Sn,m = Vn,m fl-S'. VVe call 5 the direct sum of the Sn.m's with S0.o omitted. If S ^ {0} let A/s be the smallest submodule containing it. By definition it is U(A)S but because 5 consists of singular vectors, by applying the Poincare-Birkhoff-Witt theorem, this is the same as U(£-)S. Hence Ms has no intersection with VQ.O and is proper. But as we saw any proper submodule contains a singular vector, so is contained in MS. This proves that lA»-') either contains no singular vector and is irreducible, or contains a unique maximal proper submodule, generated by the space of singular vectors. We remarked at the beginning of this section that U(S-) does not contain divisors of zero. As a consequence we see that a non-zero vector in 5n,m, if there is one, generates under the action of U(A) (which acts non- trivially only through (/(£_)) a submodule of V'J|<' isomorphic to V'J~m>''. This proves that the characters of cyclic highest weight representations are alternating sums of characters of Verma modules, as claimed at the end of section II.2. We are now going to recover MS from another object, the contragredient bilinear form on V'->'(>. We endow the algebra A with the linear anti-automorphism a of order two defined by 0 0 Ut // BASIC DEFINITIONS 8 Theorem II. 2 The following properties are equivalent: 1. The module V'(jil) is irreducible. 2. The module V1*'' contains no singular vector. 3. The contragredient form on KtJ>() is non degenerate. UA The Sugawara construction The idea that in some quantum field theories, the energy-momentum tensor is a suitably renormalized bilinear combination of the currents proved to have many applications in the representation theory of affine algebras (see for instance [9]). We shall see several examples in the rest of this paper. Let us define elements Cn for integral n by the following formulae : 0 0 Co = + 2J0 J0 ) + + JI + 2J-° m=l A priori these operators live in some completion (to allow infinite sums) of U(A). The expression for Co is some normal ordered version of the generic expression. It is easy to see 1 that acting on a state in V^ '' all but a finite number of terms in the expression for Cn give 7 O. Thus the Cn 's are well-defined linear operators on V'- ''*. As such it is well known that they satisfy the following commutation relations 0 [Cm, Jn ] = -in J +n (D, Cn] = -nCn 3 [Cn, Cn] = t(m - n)Cm+n + -t(t - 2)(m - So, for t ji O, V(j-') carries automatically a representation of the Virasoro algebra with central charge c = 3(t - 2)/t and conformai weight h, = j(j + l)/t. We set Ln = Cn/*. This leads to [D_, Ln] = -nLn (3) TTT — m [L ,I ] = (m - n)L m n m+n ~Û~( (4) As a byproduct, we remark that the enlargement of A in A is also automatic in the class of representations we are studying. We simply use L0 instead of D_. In the next sections we shall need the following expressions for Co and C-1 which are direct consequences of the definition, and show clearly their action on the space 5. +00 i = J°(Jo i.'l " Fractional levels appear naturally in 2D quantum gravity [2]. Notice that J K not a half-integer as usual but takes /// FlSDAMESTAL RESl 'LTS • ( Figure 2: The subset /<""»> of /. +00 m-, m=l III Fundamental results We introduce some notations. The set of couples (n, m) € / such that m ^ O and n is a multiple of m is denoted by /'"""I (see figure 2). The elements in /<"'"»> are in one to one correspondence with the elements of the set j(*ln*) of couples of integers (a, 0) surk that « ^ O, /9 > O, and Q + |a|/3 > O, by the map (Q, /3) -»• (|a|/?,a). We shall often use this ( ns parametrization of /'""«». For (Q, /3) € J *' ', we define ja,0(t) to be the solution of The first theorem, due to Kac and Kazhdan, localizes the singular vectors in certain subspaces Vn,m. Theorem III.1 (Kac-Kazhdan, [1O]) For nonzero t the Verma module V(jit' contains a singular vector at level (n,m) if and only if there is a couple of integers (a, ft) € Jt*"**' such that (n,m) = (|a|/î,a) and j = ja,0(t). Then the dimension of 5n,m is exactly one, i.e. the singular vector is unique up to an overall factor. This is an immediate consequence of the following lemma Lemma III.2 (Kac-Kazhdan, [1O]) The determinant Dn,m of the contragredient form in Ki.m (defined up to & non-vanishing basis dependent overall factor) is proportional to dim V In the basis ('J), the matrix elements of the contragredient form are polynomials in j and t. because they are calculated by repeated use of the commutation relations (1). It follows that Dn,m is a polynomial in t and j. To compute this polynomial, we restrict / and j to be real. Then there is a basis in which the matrix of the contragredient form is real, and of course symmetric, so that it is possible to diagonalize it. As noted before, states with different n or m are orthogonal to each other. First of all we remark that, because of their commutation relations with the generators of A, C0 and J° act diagonally on Vn We want to study the polyncmial Dn A real symmetric matrix can be diagonalized, so that when j is such that tno + m0(2j + 1 — TOO) = O, the determinant Dn,m has a zero of order the dimension of MS at this level, that is dim Sno.modim Vn_n0im_mo. So we end up wiih a factorization of Dn,m as a function of j for irrational t: dinl 5 dim Dn,m oc J] (tn0 + m0(2j + 1 - m0)) "o--o ^-'o.«—»o (no.m0)/(0,0) The product is in fact finite because dim K-no.m-mo vanishes for all but a finite number or couples (no, TOO). The proportionality factor depends on t but in a very simple way. In fact, when t £ O, if (n0, m0) ^ (O, O) in / is such that tn0+m0(2j+1 —TOO) = O then m0 ^ O, hence if t jt O any singular vector gives a factor containing j, and all these have been included. Thus, up to a basis dependent constant we have : di lS diml Dn,n = «""•- n ( where the integer We have now going to give an asymptotic estimation of Dn-m when we let j and then / go to infinity. The trick is simple: we shall change the scales of the generators, and go to a limit where the affine algebra reduces to an assembly of independent oscillators. We define modified generators: _!_ 7+ — ! /+ ~ " T " .!•' We recall that monomials of the form +00 4-00 i=0 (where all but a finite number of the integers p's are zero) form a basis of V^1''. When we substitute the modified generators for the original ones, such a monomial picks a factor .rHp>.+ +p..-yIIp".o but we still have a basis. It is easy to check that when we let a- and then y go to infinity, the commutation relations for the modified generators have a limiting form [J-, Jn-] = O [J',*] = O (J°,J-] = Thus, in this limit, the Verma module reduces to a Fock space for independent oscillators, with JQ and k as normalizations for the scalar products. The modified monomials form an orthogonal basis for this Fock space. This scaling argument shows that, if we let j and then t go to infinity, the diagonal terms dominate the determinant of the contragredient form, and contribute in this limit to a factor (j5Jp''+"l"pl--)(<5Jp>-0). Hence every monomial in our basis for the Verma module contributes additively with a factor £ p;,+ + p,,_ to rn-m and £ Pt.o to an E wnym = x(9, y )( £ E « \i'=0j=l I=Ij=I and also Comparison with (5) and (6) gives the value of dimSno,mo and 0n.mO, O « <£•>>'.<>>>dim """*- H (*IQI/* + "(2J + ! ~ a))dil" Vn-Mt-m-" (o,,/3)6J("n»> where J('in3) is the set of couples of integers (a,/3) with a ^ O, 0 > O, and a + \a\0 > O.D III FUNDAMENTAL RESULTS 12 For nonzero /. this formula allows the inclusion of Verma modules in \ 'lj-/' to be described completely, leading to explicit formulae for the characters of irreducible cyclic representations of A\l\ The case when t = O is much more complicated (see for instance the conjectures in [14], and the closely related ["]). We shall have very little to say about it in what follows. Now that we know when and where the singular vectors are to be found, it is possible to look for "explicit" expressions. This was done by Malikov, Feigin and Fuks in the A^ case. We quote their result for TV = 1. Theorem III.3 (Malikov-Feigin-Fuks, [14]) Fix a nonzero t. The vector |o|+(/3 0 (J0-) ( j+.)i«i^("-')(jo-)i»i+'(/'-2)(i/+i)N+^-3)... (j-)M-' \Jai0(t),t) (7) for positive a (resp. the vector (J+jM+«W-t)(j-)M+«W-a>(j^^^ (8) 0 for negative a) is a non-trivial element of S\a\0l0 in V^ '"'''-'' i.e. is a singular vector. These are expressions involving complex exponents of the operators JQ and Jl1, and they do not make sense a priori. Malikov, Feigin and Fuks are able to prove that they make sense by using the following trick: they prove identities relating products of integral powers of generators of £_, and observe that these identities admit an analytic continuation for complex powers. Starting from the above expression, by repeated application of these identities, they end up with a well-defined expression belonging to U(S-) and depending polynomially on t. Moreover, naive manipulations using the commutation relations as if the exponents where non-negative integers "show" that the above expressions are singular vectors. Uniqueness of the analytic continuation ensures that this is indeed the case. In the case when o is a positive integer and /3 = 0, there is no analytic continuation to 0 implement, because (7) reduces to (JJ" J Ij0^(O, t). One recovers the well-known singular vector for the Ai-subalgebra [Jo-, JQ>JQ }• The simplest non-trivial case where analytic continuation is needed is (a, /J) = (1,1). We treat this example in appendix A.I to illustrate the method. It is fair to say that explicit calculations of singular vectors remain quite complicated, but these compact formulae exhibit naturally many non-trivial properties. Among these, we quote • The singular vectors are naturally normalized. We denoted by £_ the Lie algebra of generators of degree (with respect to the principal gradation d) less than O. The gener- ators of degree less than -1 form an ideal in £_, and we can consider the quotient Lie algebra. In this quotient J /\ • PRIMARY FIELDS A.YD Ft'S/O.V 13 • Another useful property of the singular vectors is that with the above normalization they are polynomial in t. In the rest of this paper we shall give alternative formula? for the singular vectors. They a.e quite efficient and have an intuitive physical interpretation. They are connected with fusion rules. However we have neither been able to show the relation between the two approaches, nor to check directly the above properties. IV Primary fields and fusion We first give some motivation for our abstract definitions, considering for a while general properties of quantum and conformai field theories. Later we shall return to our special case. In a Euclidean quantum field theory, we know that short distance singularities in the correlation functions can be understood in terms of operator product expansions: when the spatial arguments of two local operators almost coincide, we can replace their product by some asymptotic expansion in local operators with functions as coefficients, and the need to renormalize is responsible for anomalous dimensions. In 2-d conformai field theory, the operator product expansion, also called fusion, has a much stronger, and perhaps sounder, status. It is known that its convergence is only limited by the position of the nearest operator in the correlation function under study. The symmetries of the theory are rich enough to determine almost completely the structure of the operator product expansion. This in turn leads to a purely algebraic or geometric study of the fusion. IV. 1 Motivations Any 2-d conformai field theory contains two distinguished operators T and T", which are the components of the traceless symmetric stress-energy tensor in complex coordinates. Conser- vation of stress-energy leads to BT = dT = O A field $(w, w) is called a primary field of weight (k, h) if its operator product expansion with T and T reads T(z)$(w, w) = I TT + 70«,) $(w,w) + regular terms (9) \(z-w)* (z-w) J T(z)$(w,w) = I — rr; + _ -\d«») $(w'w) + regular terms Recalling that the stress-energy tensor generates coordinate transformations, this simply means that $(w,w) is an (h,h) form in the language of complex geometry. The fields appearing in this expansion are also scaling fields. They have in general more singular terms in their short distance expansion with T and T. All the fields one gets by repeated operator /V PRIMARY FIELDS .4.YD FUSION 14 product expansions of T and T with a given primary are called its descendants and they form what is called a conformai family. For instance it is a tautology to say that the stress tensor is a descendant of the identity operator, and in fact it is not a primary because c/2 T(Z)T(U-) = + regular terms (10) c -u--)« and a similar equation for T. This shows that the insertion of T(Z) in a correlation function produces a meromorphic function of £, with known singular part. When one brings two scaling fields FI(Z, z) and F2(u>, w) close together, one expects that in some weak sense (for instance after insertion in a correlation function) there is an expansion F1(Z,c)F2(u;,U.) = 5>F,,F,(* - to,z - w)F(w,w) (11) where the sum is over all scaling fields and the coefficients cj^ ^ are functions. We can split this sum by putting together scaling fields belonging to the same conformai family. If ( 11 ) is to be true, both sides of the equality should have the same geometric properties, i.e. change in the same way under a change of coordinates. In the field theoretic language, they should have the same operator product expansion with the components of the stress-energy tensor (which generates changes of coordinates). This is only a necessary condition, but it is very powerful as we shall see. In the sequel we shall concentrate on the holomorphic part of the conformai field theory but similar statements hold for the antiholomorphic part. To go from a formalism of corre- lation functions to an operator formalism, we use radial quantization (i.e. decide that the expectation value of a sequence of operators ordered according to the radial coordinate is n 2 simply the corresponding correlation function) and write T(z) = 2I-» Lnz~ ~ . A simple application of the Cauchy residue theorem gives an operator version of (9) and (10) m m+1 [L1^a(S1S)J = (h(m + l)* + z d) $(z,z) (12) C o [Lmi Ln] = (m — n)Lm+n 4- TT(HI — m)6n+m (13) In particular L-\ generates translations and LO generates dilatations. If in addition the field n h * does not depend on z, it is possible to expand it as $(z) = £l~ ^nz~ ~ . The structure of this expansion is dictated by (12) for m = O and leads to [Lm, $„] = (m(h -I)- n) (14) Similar considerations apply in the case when holomorphic currents associated to some semisiiiiple finite dimensional Lie algebra C are present. In this case primary fields have several components. The translation into operator language of the operator product expan- sion gives the commutation relations of the untwisted aff.ne algebra associated to G for the /V PRIMARY FIELDS .4.VD FI'S/O.V 15 commutators of the currents (that is (1) in the particular case Q = A\). For the commutator of a current with a primary field, we get where the matrices R" carry a representation of Q. So we see that, apart from a minus sign, the commutator acts as the loop algebra in some representation. Now a descendant is obtained by repeated operator product expansion of the currents with a primary. It should be stressed that although the Sugawara construction leads (in a Verma module) from the Q commutation relations to those of the Virasoro algebra for suitable central charge, the commutation relations (15) do not imply that the components of $ are primary fileds for the Sugawara stress tensor. An explicit calculation shows that one has to postulate the correct commutation relations with one of the Ln 's and then the other follow. The usual choice is £_i, leading to the Knizhnik-Zamolodchikov equation, which really is a dynamical equation, and not a mere tautology. We see that descendants of a primary field can split into several conformai families. By repeated use of these commutation relations (12) and (15) we can evaluate the commutator of any product of primary fields with the components of the stress-energy tensor or of the currents, i.e. in a more geometric language the behavior of such a product under a conformai or a gauge transformation. For instance if a state \s) n l 1 is annihilated by some J% then the state 9j(z,z)\s) will be annihilated by J°6J + z (K ) i. If |fl) denotes the vacuum state (annihilated by all the Ln's with n > — 1 and all the J°'s with n > O and also their antiholomorphic partners), we can create new states by applying a primary field. The states $,(0,0))Q) carry a representation of Q and we can build on this a L 1+ £ representation of the associated left and right affine algebra. We expect e* - * -' $,(0- O))Q) to coincide with $,(*, -?)|fi). All these statements made sense in some a priori known conformai field theory, where operator products were assumed to be well-defined. Things are quite different when one looks at them from an abstract point of view. All one knows from the start is that the space of states should decompose as a direct sum of representation of the left and right Virasoro algebra or any larger symmetry algebra (A^ will be the case of interest for us). No operators are a priori defined, not mentioning their product. But, as we shall see, the naive manipulations we shall use are close enough to a more axiomatic approach. Our construction is completely "chiral" in the sense that we completely forget about antiholomorphic parts. IV.2 Verma primary fields It is time now to return to the AI case. We let t be a fixed nonzero complex number (sectors of distinct central charges are decoupled). First of all we ought to define a vacuum sector. So we look for a state annihilated by all the J£'s for n > O. This state is to be found in a cyclic module (i.e. a quotient of a Verma module) and has properties of a highest weight state. As it should be annihilated by JQ 1 /V PRIMARY FIELDS A.VD FCSION 16 (0l() the obvious candidate is the highest weight vector |0, t) in V' . It not annihilated by J0 but clearly J0"\Q.t) is a singular vector (according to the theorem III.l. if t is not a rational number, this is the only singular vector up to normalization), so we choose for the vacuum sector the resulting quotient and denote the image of |0, t) (i.e. the vacuum state) by |fi). It is readily checked that if one uses the Sugawara stress tensor the vacuum is effectively 11 annihilated by the Ln S with n > — 1. We want now to associate a primary field to an arbitrary Verma module \/(j-''. As we saw above, the components of this field should carry a representation of the finite dimensional M .4i algebra generated by JQ, a = —,0,+. The subspace ®mV(0.m) of V carries such a representation. It is infinite dimensional, spanned by the Taylor coefficients of the family of states exj°\j, t) (parametrized by a complex number x). On this family of states JJ" acts as 1 2 D~ = Q1 Ox, JQ as D° = j - xdx, and J£ as £>+ = Ijx - X Ox. Hence the natural primary field to introduce ought to depend on one extra variable x, with commutation relations 3 z IVn'*;(*' OJ ~ D3V](ZiX) (16) We call such a primary field a Verma primary field. A closely related construction was proposed in [17]. This leads to define the action of Q3 on the vacuum by the formula Then we can use repeatedly the commutation relations (16) to define the action of $}(z,x) on the whole vacuum sector. For fixed z and x, e*L->+tJ°\j,t) is not a state in V^1'' but rather in some completion (i.e. in the direct product of the subspaces Vn,m) to allow infinite linear combinations. Of course, if V(jl(' is not irreducible, we can replace it by a quotient module. Let us mention a more algebraic point of view. The differential operators J* = —znD* m m+l (resp. Cn = —hj(m + \)z — z d,) satisfy formally the commutation relations of the (non-anomalous) current (resp. Virasoro) algebra. Hence the tensor product of IA*.*) with a suitable space of functions of the variables x and z will carry a graded representation of AI and of the Virasoro algebra with the correct anomaly. Thus we can interpret $,(z,z)|n) as an element of this tensor product having the properties of the vacuum (i.e. it is annihilated by the same left ideal of U(A)). We shall see a similar phenomenon when we analyze fusion. IV.3 Fusion and descent equations We shall now try to understand the structure of the operator product expansion of our Verma primary fields. Suppose that we bring $,, and $„, close together and look for their operator product expansion. For our purpose it is sufficient to consider the following state /V PRIMARY FIELDS A.VD FCS/O.Y We postulate the following expansion, which is the analogue in the operator formalism of the short distance expansion (11) h o = (18) where \j,t,z,x) is a (:,x) dependent state in V'J'(*. Covariance (with respect to the symmetries generated by the current algebra) implieies non trivial constraints for the right hand side of this expansion. This leads to the following theorem, which is crucial for the rest of our discussion. Theorem IV. 1 The covariance of the operator product expansion has the following conse- quences: 1. It fixes the (~,x) dependence of\j,t,z,x) to be (n,m)6/ with \n,m}} € Vn,m. 2. It leads to relations among the coefficients |n,m), JfIn1 m), = (-j + J0 + J1 + m + l)|n - 1, m + 1), (19) Jf\n.m), = -(-J + Jo -Ji + m - l)|n,m - I), (20) To find these constraints, we use the following trick: the left ideal in U(A) generated by JQ — Jo, k — (t - 2), L0 - /to, and the J£'s in E+ annihilates U), O- Then by using the commutators ( 16) we get relations that the right-hand side of ( 18) has to satisfy. For instance (Jo - Jo)|jo> t) =Q implies $,, (c, .T)( J£ — Jo)U), t) =Q and after commutation we get 0 ( J0 -D° - S1Z)Ui1O = O (21) In the same way we obtain also (L0 - It0 - zdz - *,)#„ (z, (22) and = O VJ" € As we noticed before, the corresponding constraints on the right-hand side of (18) do not mix different values of j, and they apply to each term in the sum separately. So we fix j and decompose |j, <, r , x) = £n,m |j, t,z,x, n, m) according to the eigenvalues of LO and J§. Then equations (21) and (22) imply that = 0 /V PRIMARY FIELDS .4.YD FfS/O.V 18 and (h + ii - I)0 - :dt - /ij)|j,<,2,x,n,m) = O so they determine completely the a- and z dependence. We write with |n, m)} € Vn,m • Then we obtain for the other constraints ^P~|n, ">), = (-j+ jb+ Ji+m+ I)Jn-RWi + !), for p > I (23) for p > (24) = -(-J+ Jo- for p > O (25) This will be the starting point of the definition of fusion. We expect that these equations, called the "descent equations", are compatible. A formal proof of this leads to the definition of a family (parametrized by J0, Ji and j) of graded representations of B. The vector space V on which they act is a direct sum of copies of C indexed by couples (n,m) € /, that is V = ®C(nimj. We denote by *„,„, the vector with component 1 in C(n,mj and O elsewhere. The action of B on V is as follows . The vectors "&n,m are eigenvectors Lo and JQ w'th eigenvalue ft + n and j — m respectively. Moreover JP *»,m = ( -J + Jo + Ji + m + 1 )*n.p,m+i for p > 1 ^p°*n,m = -(-J + Jo + m)*B_plTO for p > 1 Jf *n.m = -(-3 + Jo - Ji + m - 1 )*n_p,m_i for p > O Note that we did just mimic the descent equations. It is easy to check that we indeed get a representation whatever the parameters JQ, Ji and j are. We denote these representations } by R JlM. The representation property implies that the equations (23,24,25) are compati- ble. Then they are consequences of (19,20), because Jf and Jf generate E+ by repeated commutations. O We introduce the notation na(j — jo,Ji,"i) for the scalar factors on the right hand side of the descent equations, that is Jp*n,m = /*°(j — JOi Jli*" J^n-p.»»-.! V(n,»7l) € /, VJ" € £f a The striking fact is that fi (j - jo,ji,m) does not depend on the L0 degree. In the formalism of correlation functions, mutually local fields commute. If they are not mutually local, they do not commute, but after fusion in a given sector, they commute up to a phase. Thus, in the spirit of radial quantization we expect that $„(«, i)|jo, t) has exactly the same covariance properties as (notice the change in the operator ordering) 1 /V PAJMAAV FIELDS .4.VD Fl 'SION 19 We give the proof in appendix B. This property allows these two states to be identified, as far as covariance is concerned. According to this discussion, we propose the following definition of fusion. Fusion of the Verma modules \ '(jl -() and V(j3lt) in V(ji() is possible if and only if the descent equations (23,24,25) have a non-trivial solution. The dimension of the vector space EjlM of solutions of the set of linear equations (23,24,25) for the family of vectors |n,'«)j € Vn.m is called the multiplicity of the fusion. A solution of the descent equations is said to be proper if |0,0)^0. This deserves some comments. • The first point is that we could look for analogous definitions involving quotient mod- ules of non irreducible Verma modules. 1. The equations (23,24,25) still make sense in any quotient module of V'*1' and we can look for solutions in this smaller space, modifying the definition of EjltK accordingly. We shall use this generalized definition freely in the following. 2. The case when we consider a quotient module of K'Jl1'' or I/I-*»1'' is more compli- cated. We have to introduce new constraints because the ideal annihilating the highest weight state is bigger. We shall see examples of this in appendix C. • The second point is concerned with the relation between our construction and the existence of intertwiners between representations. As we saw above in the defini- n tion of Verma primary fields, the differential operators J* = ~z D*l (resp. Cn = m m+} —hn(m + \)z — z dz) satisfy formally the commutation relations of the (non- anomalous) current (resp. Virasoro) algebra. Hence the tensor product (denoted by V(j>t)[z, x}) of V'*'' with a suitable space of functions of the variables x and z will carry a graded representation of A1 and of the Virasoro algebra with the correct anomaly. h h lll n >+11 } m The covariance constraints ensure that the state £„,«. z ~ °~ + x* - + \n,m)1 as- sociated to a non-trivial element of R is a highest weight state with highest weight J0 in this representation. As such it generates a highest weight module. Hence there is an intertwiner between V(K^ and V(j|{)[i,i]. In the same way one can construct an intertwiner between V'J1''* and V'*'' [2,1]. Admittedly this is very formal. We do not attempt to define what we mean by "suitable space of functions" and this prevents us from elucidating the structure the tensor product representation. But this suggests that our definition of fusion is reasonably close in spirit to what is usually done. Let us also observe that solving the descent equations i.e. finding Ej1 M, is also an intertwiner } 7 problem, because it amounts to find graded linear maps from R }1 ijo to V' ''' commuting with the action of B. • The third point is that we do not impose the absence of short distance singularities in x-space, that is we do not restrict to the case when j\ + jo - J is a nonnegative integer. This is quite unconventional but well suited to our purposes. As we shall see V FIRST REFORMULATIOX OF THE DESCENT EQl'ATIOSS 20 m (0,0) Figure 3: The couples (n,;n) satisfying (?i,m) < (4,2) in appendix C, when j\ or J0 are positive integers or half-integers, the singularities in z-space disappear. This is related to the existence of singular vectors (see the first remark above). Bearing all this in mind, we can now proceed with the consequences of our definitions. Let us first explain the content of the descent equations. V First reformulation of the descent equations As they stand, the descent equations are not very tractable. For given j\, jo, and j, it is not at all clear whether or not they do have non-trivial solutions. However, we have the following simple bound. Lemma V.I The vector space of solutions of the descent equations in an irreducible highest weight cyclic module has dimension at most one. We introduce a partial ordering on the couples (n, m) by the rule (n, m) < (n', m') if and only if n < n' and n + m < n' + m' (see example on figure 3). With respect to this ordering, £+ decreases the degree. This implies that if the descent equations do have a non-trivial solution in a highest weight cyclic module, then the nonzero |n,m)j with minimal (n,m) have to be annihilated by £+. If the module is irreducible, the vectors annihilated by £+ form a one dimensional subspace generated by the highest weight state. Hence any two solutions of the descent equations are proportional. D We are going to see that if V J Using the representations R Jt-JO, we shall derive consequences of the descent equations which are much easier to deal with. We define a family of linear forms on U(S+). Let X+ be in U(S+). If we denote by TT the projection on C(0,o) in /2J1J0* the composition TTX+ defines a !inear map from C(tltm) into C(0,o| i.e. (we identify the endomorphisms of C with C itself) a cor.iplex number un,m(x+). clearly linear in X+. Then un,m o a defines a linear form on (J (/(£_), thus on V' '''. We denote this form by «n,m and observe that it acts non-trivialy only (l V.I Preliminaries To use the full strength of (27), we need to know some properties of the linear forms un,m- The action of un>m on V'-*''' is simple. We begin with Lemma V.3 If x- is a homorjeneous element of U(S-) of degree (n,m), un,m(x-\j,t)) con- tains a factor UT=\(j- Jo+ h - m + i) ifm>Q and U7=i(-J + Jo + Ji +m + i) ifm To go one step further in the calculation, we use the particular basis (2). Consider the monomial +•x +00 +00 1=1 1=1 I=O antl m tnen and set m_ = Z1P1.-- ™o = E, Pi .0- "'+ = Ei P..+I £i,a*P'.<> = " "'- ~ '"+ ~ ( .r_|j,<) belongs to V'n.m). Define polynomials m _.m0,m+(«,t> = (« - HJ- t) ~ - I=I 1=1 Then we have Lemma V.4 The linear form un>,m> takes the value 6n,n>6mtm'Pm_ Lemma V.5 For fixed m, the family of polynomials Pm_,m0,m_-m indexed by ni_ and m0 is linearly independent. Suppose 53m_,mo Am_imoPm_im0im__m is some vanishing linear combination of these polyno- mials. We can group terms to get (EA«.-.".o(«-m- mo 1=1 i=l The degree of the polynomials fn—— TÏÏ m— JJ (y - u + m + i) JJ(v + u - m_ + i) 1=1 .=1 in v is 2m_ — m, thus they are linearly independent as polynomials in v. This implies that mo Zm0 ^m_,m0(« - W-) = O Vm_. This in turn implies that the initial linear combination was trivial, i.e. that the A's were all zero.D V.2 Fusion in irreducible Verma modules Lemma V.6 // K^-'' is irreducible, the vector space of solutions of the descent equations is exactly one dimensional. Equivalently, fusion ofV^'^ and V^1-') in an irreducible always possible and unique. V FIRST REFORML'LAT/O.V OF THE DESCENT EQC.4T/O.YS 23 In the case when V'*1' is irreducible, the contragredient form is non-degenerate. Hence the equations (27) have a unique solution if we fix the value of (j, t\Q. O)1. This solution is also a solution of the descent equations. The check is easy. It is enough to check scalar products. Let .r_ belong to {'(£_) and J° belong to S+. We have = Un,m(o( Jf)X-) = Un,m(ff(X-)J*) = n''(j-Jo,Ji,m)(x.\n-p.m-a)J (28) In the sequel, we shall normalize the solution by taking JO, O)7 = \j, t). V.3 Fusion in reducible Verma modules In the case when I/**1' is reducible, the contragredient form is degenerate on Ms which is a submodule, i.e. is stable under the action of U(E-). This implies that the direct sum of the subspaces Vn,m on which the contragredient form is non-degenerate (we call /' the set of couples (n,m) such that this is true, and although /' depends on j and t, we shall not mention this dependence explicitly) is a U(£+)-modu\e. Hence the descent equations make sense when restricted to this subspace, and by the former reasoning, the vectors Jn, m}3 for ( Ji, m) € /' are completely determined once the value of (j, t\Q, Q), has been fixed, and satisfy the descent equations restricted to this subspace. However, this solution cannot always be extended to define the states |n, m)j for (n, m) € I \ /'. This means that fusion rules have made their appearance. We shall examine them shortly. They have interest in themselves, but they will also be of use later on when we shall give formulas for the singular vectors. A word of caution is needed here. For generic values of t, there is no hope of building a respectable conformai field theory, and the word fusion we use here is an extension of what is usually meant. Lemma V.7 // K(ji'' is reducible, fusion is not always possible. The descent equation have no proper (i.e. such that |0,0)j ^ Q) solution in general. A necessary condition for fusion to be possible is that JQ ana* j\ satisfy non-trivial polynomial relations. We mentioned in section III a crucial property of singular vectors, called normalization. We can rephrase it by saying that if V'Ji') contains a singular vector at level (n, m) (there is no need at this point to be more precise, but we recall that (n, m) cannot be arbitrary in 7) and if we expand it in the basis (2) the coefficient of (J-\)n(Jô)n+m >s nonzero and can be rescaled to one (this is the normalization we find if we use the Malikov, Feigin and Fuks expressions). The result on linear independence (lemma V.5) proved in the preliminaries shows that the value of un,m on this singular vector is a non zero polynomial in J0 and Ji. V FIRST REFORMiLATIO* OF THE DESCENT EQC.AT/O.V.S 24 Hence (27) implies that fusion is not possible unless either J0 and j\ satisfy a non-trivial relation containing t as a parameter, or (j,t\0,Q)} is taken to be zero.D These are a priori only necessary conditions. The second one means that the operator product expansion, if possible, is less singular than expected. Of course, if V'(ji() contains several singular vectors, each one contributes a (possibly redundant) constraint on fusion. If V V.4 Truncation of the descent equations We shall now see that the descent equation can be truncated in several ways. Lemma V.8 //— j + jo+Ji is a nonnegative integer i+, it is possible to restrict the descent - equations to the subspaces V'n,m suck that m > — i+. If —J + Jo Ji '"* a nonpositive integer i_. it is possible to restrict the descent equations to the subspaces Vnm such that —i- >m. In the first case, tha descent equations connecting the domain m > —i+ with the rest of / state that Jp \n, —i+)} = O. In the second case, the descent equations connecting the domain -i_ > m with the rest of / state that J~\n, —i-)j = O. Hence the announced truncation is possible. O In fact, we have a more precise result, stating that in the rest of /', the solution of the descent equations is identically O. Lemma V.9 // —j+jo+ Ji is a nonnegative integer i+ and if (n,m) € /' is such that m < —i+ then \n,m)j = O. //— J + JQ — ]\ is a nonpositive integer i, and if(n,m) C. I' is such that —i_ < m then \n, m)j = O. This is a simple application of lemma V.3 and the fact that the contragredient form is nondegenerate on Vn,m, (n,?n) € /'.D If both the above conditions are satisfied, (in which case j\ = l/2(z+ —z_) is a nonnegative integer or half-integer) this truncation is related to the fusion of quotients of Verma modules. This is shown in appendix C, where a derivation, using our technique, of the (well known) fusion rules for the unitary models is also given. V.5 Algebraic structure of the solutions of the descent equations To close this section, we make some comments on the behavior of the solutions of the descent equation as functions of the parameters j, JQ, Ji and t. We already remarked that all Verma modules are isomorphic to V(S- ) as £/(£_ )-modules. This allows us to consider them in a uniform way. VJ SECOND REFORMULATION OF THE DESCENT EQUATIONS 25 Lemma V. 10 The action of A (hence of U(A)) on V If V^-'' is irreducible, we have seen that the descent equations have exactly one normalized solution and we can interpret the normalized sequence |n,m)j as a sequence x*1™ in U(E-). Lemma V. 11 Each x"'m is rational in j and t and polynomial in JQ and ji. The poles in j can occur only at zeroes J0 ,0(1) of the determinant of the contragredient form. According to theorem HU, for fixed (n,m) and t ^ O, there is only a finite number of values 7 of j such that the contragredient form is degenerate on Vn,m in V^ '''. The determinant of the contragredient form is polynomial in j and t and the linear forms an.m evaluated at members of the basis (2) depend on j, JQ and j\ polynomially. Hence the solution of the system (27), whose determinant is the determinant of the contragredient form at level (n,m), has the announced properties. O m The singularities of x"' as a function of j and t may depend on the value of J0 and j\. The two above lemmas lead to the following VI Second reformulation of the descent equations VVe are now going to derive the most useful consequences of the descent equations. Then, we shall give a geometric interpretation to our computations. VI. 1 Triangular form of the descent equations The fundamental result is Lemma VI. 1 Any solution of the descent equations satisfies n (*n + m(2j + l- Wi))Km), = (-j + J0 + ji + m + l)£-/*p|n -p,m + 1), P=I (29) P=I -(-} + Jo - Ji + m - 1) 51 Jlp\n - P, m - 1), Ll *&&** V7 SECOND REFORMULATION OF THE DESCENT EQUATIONS 26 /or (11,TO)9E(O. O) Multiply the descent equations (23), (24) and (25) by J+p, J^1, and J~v respectively. Then the sum Ep=, •/+,,( 23) +1 Ep=i -/£,,(24) 4- EJU «/rp(25) gives on the right hand side of the equality the right hand side of (29). On the left hand side, one recognizes the definition of (C0 - J°(Jo + U) \n,m)}, which is nothing but the left hand side of (29).O For (n,m) = (0,0) it is natural to interpret (29) as the empty relation O = O. It will be useful later on to separate the equation (29) to get a system In1Hi), = (-J + Jo-rJi+m + l)^=lJÎP\n-p,m + l), n m -2( — J + Jo+ Tn) Ep=i J-P\ — Pi )j -(-J + Jo - Jt + rn - 1 ) Ep=O J-p\n ~ P' ™ ~ l)i The important property of equation (29) is its triangular structure. The appearance of the prefactor tn + m(2j + 1 — JTI) should not come as a surprise. If this prefactor dees not vanish, the state |n, m), is expressed in terms of states of lower degree (we still use the same ordering in /). Hence, if j and i are such that tn + m(2j + 1 — m) vanishes for no non-trivial value of (n,m) (this is more restrictive than demanding that j is not a ja,0(t)), (29) has a unique proper normalized solution, whatever the values of jo and Ji are. By unicity, this solution has to be a solution of the descent equations. However, we can show a little more. For fixed values of j and t, we call 7" the subset of / containing the set of pairs (n',m') such that tn + m(2j + 1 - m) ^ O for any (n, m) € / \ (O, O) such that (n,TO ) < (n', m'). The set /"contains (0,0). Lemma VI.2 The equation (29) restricted to I" has a unique normalized solution, and this solution satisfies the descent equations. By the definition of /", the direct sum ®(n,m)e/"^n,m is a U(£+ )-module. Hence the equation (29) and the descent equations make sense when restricted to this subspace of V^. It is clear from the triangular structure of (29) that the restricted equation has a unique solution. As /" is included in /', we know that the descent equations also have a unique normalized solution for (n, m) € /". These solutions have to coinc'de.D We also have a weaker result when (n,m) is "as close as possible" to /". Lemma VI.3 Let (n,m) C I be such that (n',m') < (n,m) implies (w',m') 6 /". Then Jp0Km), = (tn + m(2j + 1 -TO))// 0O - Jo,Ji,m)\n - p,m - a), VJ^ € £+ VI SECOND REFORMt'LATIOX OF THE DESCEKT EQVATlOXS 27 Let us first note that, with the hypotheses of the lemma, either (H. w) belongs to /" or /ii + m(2j + 1 - »» ) = O. According to lemma VI.2, (n.m)^ which is expressed only in terms of vectors |n',w')j with (n', m') ç /", is well-defined. To prove the lemma, it is i lough to check the cases J° = JQ" and J* = J1". We do the calculation in detail for JQ", and leave the other verification to the motivated reader. Using the commutation relations (1) we obtain J^|n, m)j = (-J + Jo+Ji + m + l)2^JIpJf\n-p,m+ l)} P=I + -(-J + Jo - Ji + m - 1 ) £( J_-pJ0 + J£,)|/i - p, m - 1), (30) P=O On the right hand side, the descent equations are valid, because we can simply invoke lemma VI.2 (Notice that we might also argue by induction as follows. The vector |0,0), always satisfies the descent equations. We assume that the descent equations are valid for the predecessors of (n, m) and we follow the rest of the proof of lemma VI.3. Then if (n,m) belongs to /", tn + ?n(2j + 1 — TO)doe s not vanish and we infer that |n,m)j is well-defined and satisfies the descent equations, completing the induction step and giving an alternative proof of VI.2). Using the descent equations we get P=I m)(-j + J0 - J, + m-l)E J^ln-p,™-!), P=I n • JJl) 2. J_p|Tl — Pi m)) P=I n +(-J +Jo-Ji+m- l)(-j+jo-Ji +m-2)J3 Jlp|n-p, m-2), P=O -(-J + Jo-Ji+m- 1) £ J?p|n - p,m - 1), (31) We recognize many terms of the right hand side of (29) for the couple (n,m — 1). We obtain -2(-J + Jo + m) Y, -/-pi" - P>m/j P=I +2(-V + Jo - Ji + m - 1 ) £ J°_p\n -p,m~l)} V7 SECOND REFORML'LATlON OF THE DESCENT EQL'AT10.\'S 28 P=I -2(-j + Jo-Ji -DE- (32) P=O There are many cancellations on the right hand side, and except for the first line and the term p = O in the last line, everything disappears. But J^ acts on |n - ;>,») - l)j as multiplication by j — m + 1, and we finally obtain + J0 Jn,ITi)3 = -(-j + J0 -Ji +m-l)(*n + m(2j+ 1 -7?i))|ii,m~ 1), We deduce the following result, which is reminiscent of corollary V. 12. For fixed nonzero i, we can consider the solution of the equation (29) as a function of j, JQ and J1. A given couple (n,m) belongs to /" for all but a finite number of values of j, and the form of equation (29) gives another proof that the vectors .r2'm 6 U(B-), introduced in section V.5, are rational in j and t and polynomial in jo and J1. However the prefactor in + m(2j + 1 —TO )i n (29) leads to consider "spurious" poles for xn'm. We know that the true poles are the zeroes of the determinant of the contragredient form. Hence, the only couples (n,TO )tha t contribute to the poles are of the form (|o|/9,a) for (Q,/3) € J(t'na). Corollary VI.4 Let (n,m) € / be suck that for j = -t^ + **=*•, (n',m') < (nrm) implies m (n',m') € /". If\n,m)_t^.+!Sfi - O, then x"' has a limit when j -» -t^ + 2=1. The image of this limit in 1/'"'5TS+12Z-1'' satisfies the descent equations at jtegree (n,m). m We are interested in the behavior of |n, m)} near j = -t^ + -^. The vector x2' € U(£.) (j>£) (corresponding to \n,m)} € V ) is well-defined and analytic in j in a neighborhood of 1 m -<^ + =Ci. Hence the vanishing of |n, m)_tJ!_+m-i implies that (in + m(2j-H-m))~ z"' has a limit when j -» -t-^ + ^. We take this limit to be x"'m at the point j = -<^ +3 ^1. The proof that this limit satisfies the descent equations at degree (n,m) is the same as the proof of corollary V. 12. D VI.2 The Knizhnik-Zamolodchikov equation Although the derivation of (29) is simple, its physical meaning is not clear. We shall now show that (29) is a consequence of the Knizhnik-Zamolodchikov equation, illuminating the geometrical origin of the descent equations and their associated triangular form. Lemma VI.5 Equation (29) is the constraint on o, t) coming from the fact that tL-i - Jt1Jo - 2J°jJ5 annihilates the state Ij1, f). \7 SECOND REFORM rM770.Y OF THE DESCENT EQl'ATIOSS 29 The proof is a straightforward but tedious computation. Remark that (tL-\ — J-\Jo — 2J^1J0)IjI1O = O comes from the definition of L_i by the Sugawara construction. On \]\,t), J$ acts as multiplication by Ji. so we start with L + J and multiply on the left by e* -* * ° 4>JO(— z, — x). We use the commutation relations (12) and (16) to get l +rJ 1 0 1 e* "' Mt(£_1 + dt) - (7+, - Z- DJ)(J0- + D-) - 2J1(J ., + C- ^))C"*- "^ We have checked in lemma B.I that, as far as covariance is concerned, it is not possible to L zJ distinguish e* ->+ o $K(—;,—x)\ji,t) and $,,(2,;r)|jo, t). Hence we have to compute 1 7 , 1Je-* -- '' (33) This is done by repeated use of the commutation relations (1) and (3). We compute P=I P=I 1 J1Te-'*" = J0- + 1 1 We define J (^) = £~, --"-'J+p, J°(z) = ££, z"' J£p and j~(z) = Z^0 z»' J+p. We can interpret these expressions as the "negative part" of the currents, the part which acts non- trivially on the highest weight state. The p = O part of J~(z) appears in the computation of ezL.t +ZJ~ jrjO e-,L_, -xJ- _ /7++xJ- It is now a simple matter of regrouping terms to check that (33) is equal to + + (to, + 3-'(D D; - 2J0DPn )) - ( J (Z)D- + 2 J°(Z)D» + J~(z)D+ ) (34) The exchange of J0 and J1 is somewhat unexpected, but in fact D+ D" - 2J0Dj1 = D£ 2JiDJ1. If we apply (34) to the short distance expansion projected on the j-sector \7J SINGULAR VECTORS 30 we know that we obtain zero. Term by term identification of the powers of r and .r leads to (29).D By abuse of language, we call (29) the fused Knizhnik-Zamolodchikov equation. The fundamental role played by the Knizhnik-Zamolodchikov equation, or its fused ver- sion (29)), is not really a surprise. It is well known that this equation is related to the existence of integral representations (i.e. quite explicit forms) for the correlation functions of minimal -4',1' Wess-Zumino-VVitten models (see for instance [15]). This shows that it is related to the fusion, but also to the structure of singular vectors. We shall see shortly that this is indeed true. VII Singular vectors We are finally in position to propose an effective way to compute singular vectors. VII. 1 General construction We fix a nonzero t. Lemma VILl Let (»i,m) 6 / be such that for j = -t^ + ^f1, (n',m') < (n,m) implies (n',m') € /". Then \n,m) , „_ , m^i is annihilated by U(S+) If (n, m) satisfies the hypotheses, |n, m)_, ^^m^ is well-defined. The lemma is then a direct consequence of lemma VI. 3. D Corollary VII.2 Under the same hypotheses, if \n,m)_tJ!_+m=i does not vanish, it is a singular vector. Clear from the definition of the singular vector.D Corollary VII.3 Under the some hypotheses, if(n,m) is not of the form (\a\/3,a) for some (a,/?) € J("ns>), \n.m) „ m-i does vanish. 2m 2 Clear because in this case V(~'5^+~î~''' contains no singular vector.D Lemma VII.4 Let (\a\/3,a) € / be such that for j = ja,0(t), (n',m') < (|a|/?,a) implies (n',m') € /". As a polynomial in J0 and ji, |/3|a|,a)Ja ^ cannot vanish identically. As we have seen in the proof of lemma V.7, if V(ji*' contains a singular vector at level (n,m), the equation |n,m); = 0,t|0,0),Un,m (35) cannot have a solution, unless jo and 71 satisfy non-trivial relations. But corollary VI.4 shows that whenever |/?|a|,a)Ja (() vanishes (for a particular value of jo and j\), it is possible to VIl SINGULAR VECTORS 31 define a solution of the descent equations at level (|a|/î,a) by analytic continuation. This solution is automatically a solution of (35).O This leads to the important Theorem VII.5 Ie/ t be irrational. Unless J0 and Jj satisfy non-trivial fusion rules, the t vector \0\a\,a} (() is a non-vanishing singular vector in V'l-><».<'<>''> at level (\a\f3.a). We demand that t be irrational to be sure that the condition (»', w') < (|a|/?, o) implies (n',m') € /" is satisfied. The values of JQ and j\ leading to a vanishing vector are restricted by polynomial equa- tions. Hence, we can choose J0 and ji almost arbitrarily to get the singular vector. We shall illustrate this point below. VII.2 Some matrix forms for singular vectors In equation (29), it is possible to put the vectors \n,m)3aff(t) for (n,»n) < (|a|/3,a) together to build a column vector with ((\a\ft + a + I)(|a|j3 + I)-I) components. We have to choose a total ordering for the couples (n,m) < (\a\/3,a). We can even arrange things to make this total ordering compatible with the partial ordering we had before (but there is no canonical way to do this). We write for instance / = (|/?|a|,a — 1 );<,,„(«)> • • • » I^ O)^0 „(*))*' r and F = (|/0|a|, a)]a fl((),0, • • • 10)' . Equation (29) is then recast in a matrix form F = Mf. The matrix elements of M are of course operators. We shall also use the notation | )£"*)) for the state |/?|a|,a);o/)((). The matrix M is triangular. In certain circumstances, a simpler matrix form is available. This is based on the trun- cation of the descent equations (see section V.4 and appendix C). If a is positive, we choose jo and Ji such that jo — Ji = Ja,0(t) ~ Q and jo + Ji = J, i.e. 2j0 = — 1/3 — 1, 2ji = a. In this case, we know that the couples (n,m) with m < O or m > Q do not contribute. This leads to a matrix form for the singular vector, involving only the states \n, m)Ja p^ with O < m < a and O < n < a/3. The number of components of the vectors is reduced to ((Q/? + I)(Q +I)-I) A similar construction is also possible if a is negative. To be sure that we obtain the singular vector, we ought to prove that the values of jo and Ji do not satisfy the fusion rules. We conjecture that this is true. The case, when Q = 1 is interesting. We remark that Ji,0(t) = -^. The family of equations (29) can be restricted to 0 P=I -/.pi" -P< >- P=I p=o VW SOME COMMENTS O.V HAM/LTCXV/AN REDUCTION 32 VVe recall that the singular vector is given by the right hand side of the degenerate equation corresponding to the singular level (n,m) = (/3,1). The associated matrix form can be written explicitly. VVe give an example in section A. 2. These expressions play the same role for A'" as do the matrix expressions (see [I]) of the Benoit-Saint Aubin formula» (see [3]) for the Virasoro algebra. We shall comment on this in the next section. In this case, we have computed the overlap function (see appendix D) P/j.i for (jo,ji) — (^i|^i,i) and ( JO, Ji) for small values of /?. This leads to Conjecture VII.6 When j = ji,a(<). & necessary condition for fusion from and in V^J'^ to be possible is the vanishing of the polynomial 13 II D + ij) -J«) i=-<3+2 (where in these products, i is restricted to have the same parity as j3). It t is irrational, the vanishing of this polynomial is also a sufficient condition. VII. 3 Projection of the recursion relations The family of equation (29) involves only U(E-). We have already emphasized several times that £_, which consists of generators of degree !ess than O with respect to the principal gradation, contains the generators of degree less than —1 as an ideal. The quotient is a commutative Lie algebra with JQ and J+, as generators. Its universal enveloping algebra is still graded by n and m, and there is a single generator at level (n,»n), (Jîi)"(Jô)n*m- n n+m We can write equation (29) in the quotient, replacing \n,m)} by Cn,m( Jît) ( Jo ) . The coefficients Cn,m are complex numbers satisfying (in + n?(2j + 1 - m))CB.m = ( -J + Jo + Ji +m + l)Cn_i.m+i -(-J + Jo -Ji +m- I)Cn17n-I The initial condition ior a proper solution is Co.o = 1. It follows from the previous consider- ations that, as a function of j for fixed <, Cn,m is rational, with poles only at the zeroes of the contragredient form. The residues at the poles give the fusion rules (this is a consequence of the normalization property of the singular vectors). The non-appearance of the spurious poles is highly non-obvious. Hence, this innocent-looking recursion relation contains a lot of information, and it would be of great value to be able to study it independently. We have not been able to do so, and leave it as an open problem. This is an appropriate point to close this section. VIII Some comments on Hamiltonian reduction We make some comments related to our initial motivations. There is a close connection between the structure of the representations of the A1 algebra and the Virasoro algebra. It uses quantum Hamiltonian reduction (see for instance [5) for SOAfE COMMENTS OS HAMILTONIAN REDUCTIOS 33 the quantum case and [6] for the classical one). We recall the basic steps of the construction. The idea is to introduce on Yerma modules for A\}) a modified Virasoro algebra. From now on, we denote by Z.^1 the Yirasoro generators obtained by the Sugawara construction. We set ij^' = L^ — (m + I)Jm- VVe observe that there is no modification for m = -1. It is easy to check that = (m - n)Ln + (I* -6t- 6T ' With respect to this new Virasoro algebra, we obtain n l Hence, according to (12) and (14), J+(s) = ^î~ J^ and J~(c) = E-S Jn=~ ~ are primary fields of respective weights O and 2. However, leading to Hence J°(z) is a scaling field of weight 1 but not a primary field. If we replace the above commutators by Poisson brackets, the system becomes classical. If we take J*(z) as a dynamical variable, the fact that it has conformai weight O makes it possible to reduce the phase space by the constraint J+(z) = 1 without loosing conformai invariance. The correct way to treat this problem in quantum field theory is to introduce ghosts. To the be system with commutation relations {cm,bn} = 6m+n = {bm,bn} = O we associate a graded Pock space. There are two states at level O, | T) and I i) such that C0I 1) = I T) and &o| T) = I !}• The states | T) and | J) are annihilated by 6n and Cn for positive n. By definition, the Fock space is the representation obtained by acting on the states at level O with any combination of the generators. The Fock space can be turned into a representation of the Virasoro algebra by choosing an arbitrary parameter s and taking +00 m O n=l V[II SOME COMMENTS OA' HAMILTONIAN REDUCTION 34 Then one can check that = (m - - s) --2)6m+n - 1) -»i)6n+m - « - D - Then according to (12) and (14). the fields +00 -> and _-n-l+s are primary fields of weight s and 1 - s respectively. Remark that the regularity of 6(z)| T) and c(z)| T) at z = O is equivalent to the defining properties of | Î) if and only if s = O. We shall see below another reason to fix s to be zero. We define the ghost number to be 1 for c(z) and -1 for b(z). This leads to define the ghost number operator U by U = En=i(c-nbn - b.ncn) - We can now study the tensor product of this Fock space with a highest weight cyclic A1 - 01 module. The generator J$ commutes with the Virasoro algebra (with generators Ljn ' =• L^' + L^))) and can still be diagonalized in the tensor product. To impose a quantum - analog of the constraint J+(s) = 1, we define Q = En SL00 Cn(Jtn-I ~ *».<>)• Then [C/,Q] = Q i.e. Q has ghost number 1. It is easy to check that Q2 = O. The operator Q commutes with the Virasoro algebra (with generators L^ = Ljn^ + Lffl) if and only if s = O. We assume that s = O in the following. Then Q is proportional to fc(z)(J+(z) — I) which is geometrically well-defined, showing clearly the relation with the appropriate constraint. Moreover, the representation of the Virasoro algebra in the tensor product has central charge -1 s c = 13 - 6t - 6< , and the eigenvalue of L0 acting on |j,<) ® | T) « A = ^f^ -J = (2j+i-t^-(i-«) This state is clearly annihilated by the Ln's for positive n. The fundamental remark is that if we take J = Ja,a(t) with Q positive, we get (a -*(/?+ I))2 -(I -O2 4< and these are just the weights for which the Virasoro Verma module is not irreducible and contains a singular vector at level a(/3 + I). The cohomology of Q is graded by C/, and at a given degree, the cohomology space carries a representation of the Virasoro algebra. Clearly, the state |j, t) ® | *) has ghost number O and is in the kernel of Q. It is never C-exact. This is because the only states at level O for the Virasoro algebra are obtained by repeated action of J+, and6oon|j,<)®l T)- But Q commutes with Jt1 and Qbo\j,t)®\ T) = (J^-l)\J,t)®\ T)- Hence, no finite linear combination can lead to |j,<) ® | T) by application of Q (we note however that the ill-defined - Eo"(«/-i)"6o|j)<) ® I Î) would formally do the job). Hence the cohomology at ghost number O is non-trivial. We believe that there is no cohomology at non-zero ghost number and that if the -4j module is a Verma module, the cohomology at ghost number zero is a Verma module for the /.y co:vo.r.s70.vs .A.VD REMARKS 35 Virasoro algebra. This result probably exists already in the literature, but we have neither been able to find it written in an accessible language for us, nor »o build a proof, although we think there should be some elementary argument. 1 It is easy to check that a singular vector in an ^1 '-module tensored with | f ) is annihilated by Q. Our hope was then to piwe that the singular vectors for ,4*,1' with Q positive could be easily rewritten as polynomial as in the generators of the Virasoro algebra modulo a Q-exact term. Remark that the operator -C0 has a trivial cohomology and that Q is the sum of — C0 and a term decreasing the eigenvalue of Jg by one. This ensures that a state annihilated by Q which is a finite linear combination of eigenstates of J° w'tn eigenvalues greater than j is always equivalent to an eigenstate of J^ with eigenvalue j modulo a C-exact term. Hence the situation is not hopeless. But we have not been able to proceed further except in very special examples. For instance, if j = O, lT KM) = ./Vo" HM) J- l T) = o Hence T) = Jl1 Jo 10,0® I T) = J(TlO, O 01 T) + (Jt1 - l)J0-|0,0 © I T) showing that this particular singular vector for A( flows to the singular vector for the Virasoro algebra under Hamiltonian reduction. If we could do this more systematically, we would probably understand much better the construction (see [I]) of singular vectors in Virasoro Verma modules. The special case a = 1 is promising and interesting because the relation with the Benoit-Saint-Aubin formulae (see [3]), but has nevertheless eluded us. Moreover, a precise solution to these question would give an interesting shortcut for the usual proof (see [5] and the for the mathematically inclined reader [S]) that Hamiltonian re- duction relates the minimal models for the A\ and the Virasoro algebra. The usual method is quite indirect and involves bosonization in two places, with the necessity of introducing other Q operators. A direct proof would be much more illuminating. We leave this as an open problem. IX Conclusions and remarks The interplay between fusion, fusion rules and singular vectors has be used to construct these singular vectors explicitly. It is not clear for us whether these expressions can be used in other theoretical applications, but we think that the relationship between these aspects, although not unexpected, was not recognized to be so intimate. The proper interpretation of the Knizhnik-Zamolodchikov equation in our context has been of great importance. On the A THE SINGl'LAR VECTOR AT LEVEL ( 1,1 ). 36 other hand unitarity played no role in our discussion. Some fusion rules have been computed, and a general calculation should be possible. However, many questions remain open. Among these we would like to emphasize two. The generalization to other affine algebras would be interesting. There are serious tech- nical difficulties, but they should not be insuperable. Much more intricate seems to be the extension to other chiral algebras. The Virasoro algebra is an example v/hich still needs to be better understood, and we are back to Hamiltonian reduction. We have concentrated on purely algebraic aspects, but geometry certainly plays a fun- damental role. We have some hints that a geometrical interpretation of the formulae (36) exists, and is related to the analogous geometrical interpretation of the Benoit-Saint-Aubin formulae in terms of covariant differential equations given in [1], inspired by [6j. We observe that the two cases are related by Hamiltonian reduction. We hope that these questions will motivate further work. Acknowledgements We benefited from many fruitful discussions with Denis Bernard and Giovanni Felder. They helped us to understand the links between our approach of fusion and the more math- ematical one. They also suggested improvements and raised many questions, concerning for instance the fusion rules. It is a pleasure to thank them warmly. We also thank Claude Itzykson for a careful reading of the manuscript. A The singular vector at level (1,1). The singular vector for (a, /3) = (1, 1) and j = — | is the simplest non-trivial singular vector. We compute it in two different ways. A.I The method of Malikov, Feigin and Fuks To illustrate the technique of analytic continuation, we do the calculation in detail for (Q, /3) = (1, 1). So, we are trying to make sense of The fact that J+, already appears raised to an integral power (in fact 1) makes the situation comparatively easy. However, the general computation follows analogous patterns. The starting point is the identity which is proved for instance by differentiation. Then we expand + C^-J , = (J+, - 2x7°, - XVr1)C-**" A THE SINGULAR VECTOR AT LEVEL (1,1). 37 in powers of x to get 1 (J0-)" Jl1 = -C1W -2PjO1(J0T -P(P P = 0.1. • We observe that the coefficients are polynomial in p, and we extend these identities for complex p. Both sides are ill-defined. We take p = l+t and multiply the identity by (J(T)1"' on the right. This leads to H 1 (J0-) 'Jl1(J0-) -' = (Jl1 -2(1 + OJS1(Jo")' -*(!+: If we assume that the usual rules for multiplication of powers of J0" can be extended to complex powers, we end up with 1 2 (J0-) -' = Jl1(J0-) - 2(1 + <)J-, J0" - <(l + ')J_-, The right hand side gives a definition of the left hand side. We remark that the left hand side was already well-defined for < € {-1,0, 1}. It is easy to check that at these special values, the two definitions coincide. Of course, we could have started with an identity for Jli(Jo")p- We do not prove that the result is the same. This is a consequence of the general theory of Malikov, Feigin and Fuks [U]. It is clear that even when Q = 1, if /3 > 1 formula (7) contains more factors, making the computation more and more complicated. This is to be contrasted with the form given in equation (36). A.2 The matrix form In the case when (Q, 0) = (1, 1), our method leads to the following computation. The family of equations (36) reduces to -<|0,l)_i = J0-|0,0).i «|l,0)_ = Jl1)0, 1)_.+J?,|0,0)_. It is easy to recast this in a matrix form. We write /J0- -JS1 Jr1 |i,o).j = t -Jl1 -JS1 |0,l)-e VO t Jn- |0,0)_f We solve this triangular system and obtain ( s) I > _t = -i (^Jl1J0- - <(J0-JS1 + J0-JS1) + Jr1) |o,o).f B FURTHER CO\'ARIA.YfE COKSTRAINTS 38 Using the commutation relations to rewrite the right hand side of this equation in the basis (2), it is easy to check that the different expressions for the singular vector are proportional to each other. The analogous computations for /? > 1 become more and more tedious, but they are much simpler that the ones involved in the computation by analytic continuation. There is some intuitive explanation for this: our recursion formulae define the singular vector, without specifying a basis of £'(£_ ), with the consequence that in a sense "the singular vector itself chooses the way it wants to be expressed". B Further covariance constraints We are going to study the covariance properties of the state (26) of section IV.3 with respect to the current algebra. So we apply our method to the state e'*-+'-'.- *„(-*, _z)|j,, I) (37) The left ideal annihilating \j\,t) is generated by JQ — Ji, k — (t — 2), LO — h\, and the J°'s zt xJ in E+. We use once more the commutators (16) and then conjugate with e -'+ o to get ,(-*,-*)!*,*) = O L 1+rJ e' - «~(Lo - ft, - zd, - ^(-z, -x)\3l,t) = O a ^-,, -*)!,„*) = O VJp We can now prove Lemma B.I The covariance constraints on (17) and (26) coincide. We use the commutation relations between the stress-energy tensor and the currents to check that L +zJ 0 * -> ° (J» - = J0 - Dl - J0 L l+zJ L xJ e' ~ «(L0 -h0- zd, - h.)e-* ->- = L0 - h0 - zdz - A, It is quite tedious to show directly that the operators for 7° in £+ are ((s,x) dependent) linear combinations of the operators appearing in the constraints for $3l(z,x)\jo,t). Happily, as we emphasized above, two particular constraints generate them all. So we are left with two simple computations and t + L + 0 2 + e' -«^M + D+)e-' -'-'*" = J0 -2« J0 + 2J0X-X O1 = (J0 -^) -2Z(JJ-DJ -Jo) This concludes the proof that the covariance constraints on (17) and (26) are the same. D It is in this sense that we can identify these two states. C FUSION OF QUOTIENTS OF VERMA MODULES 39 C Fusion of quotients of Verma modules We give an interpretation of lemma V.S. This will also lead to some illustrations of the comments we made after the definition of fusion. This section is very close in spirit to the computation of the fusion rules in [17]. We note that if j\ is a nonnegative integer or 2jl l I s a ( () half-integer, («/(r) ' " |ji'0 ' singular vector in V -"' . This singular vector generates a submodule, and we can take the quotient. In this quotient the left ideal of U(A) annihilating \ji , t) contains (Jo )2-" +1 . So if we try to implement fusion of this quotient module with V ,, (*,*)!*,<)= O where this time Q31 stands for the primary field associated to the quotient module. ll+l L xJ We multiply the relation (Jo)* \J\,t) = O on the left by e* -*+ ïQK(-z,-x) Then a simple application of the commutation relations (16) leads to But, as far as covariance is concerned, we have shown that it is possible to identify " From this we deduce that in the j-sector Hence for any € / -j + m- (38) In particular, either |0,0)7 = O or j € {jo + Ji, Jo H- Ji — 1, • • •, Jo — Ji} • This is a fusion rule. It looks quite familiar. If we define Z+ and i_ by ji = 1/2(1+ — z'_), J0 — j = l/2(i+ + z_) then the content of (38) is equivalent to the truncations of the descent equations obtained in V.8. We observe that the use of the quotient module of V'J1>1' to define fusion imposes that the operator product expansion has no singularity in x-space. What if jo is a nonnegative integer or half-integer? We can guess that the consequence of the existence of the singular vector in V(jo'(' is a fusion rule j € {ji H- Jo, Ji H- Jo -1, • • • > Ji — Jo} • This is indeed the case. Lemma C.2 In the fusion o/V'"-'' with the quotient module of V(j0i(), the new constraint is »*(*,< C FUSION OF QCOTIESTS OF VERMA MODULES 40 ijo+l Starting from (J0 ) |jo,') = O, which holds in the quotient module of \'l-»-'>. this is a simple application of the commutation relations (16). D We deduce that in the j-sector ij +] h h hl n + } m (Jo ~ dr) ° '£: - °- + x*> »- + \n,m)J = O n.m There is only one term belonging to V0-0, namely From this we infer (Ji +Jo- J)O1 + Jo - J -'.)••• Oi + Jo - 3 - This leads to the expected fusion rule. We turn now to the case when /li)'~2-'0~1|jOi') is a singular vector in V'*1'1. If we use the quotient module for fusion, we get a new constraint. Lemma C.3 In the fusion ofV(jl-t} with the quotient module ofV(K We deduce that in the j-sector )J = O There is only one term belonging to Vb,o, namely From this we infer O -J and this leads to a fusion rule. The spin j has to belong to {t — 2 — jo — Ji, t — 2 — jo — Ji — 1, ••• ,Jo - Ji}- This is of course also familiar. We can guess the analogous fusion rule when Putting all these results together, we obtain the usual conditions for fusion. D THE OVERLAP FC.VCT/O.V 41 • If both jo and Ji are nonnegative integers or half-integers, we simply recover i.he law of composition of roins. The spin j has to belong to {ji +JQ,Ji +Jo~ 1- ' • • < |ji —Jo|}- Thus it too is an integer or half-integer, and V(jit| contains a singular vector. \Ve have only obtained a necessary condition for fusion to be possible. But it is not difficult to show that the fusion involving the three quotient is possible and unique if t is irrational. We do not give the proof here. • If moreover t - 2 is a positive integer, we recover the full set of fusion rules for the unitary models. Note that unitarity played no direct role in the discussion. This is common in the representation theory of finite dimensional semi-simple Lie algebras, where the requirement of finite dimensionality of a representation implies its unitarity. Let us stress once more that, although our definitions did not prevent short distance singularities in .r-space, these singularities disappear when we consider fusion of quotients of appropriate Verma modules. D The overlap function In the definition of fusion, jo and j\ play the role of parameters, and it is interesting to have some kind of measure of how much the solutions of the descent equations differ at level (n,m) when jo and ji vary. The contragredient form gives such a "measure". We define the "overlap" between two solution of the descent equations, corresponding to distinct couples (Jo1Ji), and (jo'/i) but of course with the sar,e value of j and t to be r,'..m(jo,Ji,JO,JnJ,0 =} (n,m\n,m)'} lining the method of sections V and VI, it is easy to show that the overlap satisfies recursion relations. Lemma D.I The overlap Fn.m satisfies ( P=» n + (-J + Jo-Ji+m-l)(-J + Jo-j'i+'»-l)Ern_p,m_i p=0 To prove this, we first use (29) for |n, n?)J, and then we use the descent equations for j(n,m|. This procedure in not symmetric, but the final formula treats (Jo, Ji) and (j0,j'i) in a sym- metric way. D These relations are quite complicated as they stand, but by using truncation, it is possible to use them to compute for instance fusion rules. REFERENCES 42 Theorem VII.ô allows us to say something about the structure of the overlap r\a\0,0 when j = Joi,0(t). In fact, for these very special values of the indices, the right hand side of (D.I) has to split as a product of the fusion rules for (}o,j\) and OJ1,/J. This is because |,3|ar|,a) (t) and |J|a|,a)'Ja j(() are both proportional to the singular vector, and the right hand side of (D.I) computes the obstruction to the solving of the descent equation at level (N0,«). Hence, the overlap equation provides a method to compute the fusion rules. When a = 1 and j = Ji.0(t) = -0% for instance, it is possible to compute F^.i for small values of 3, taking Uo, Ji) = (^f-11.5) (these are the values leading to the truncation of the descent equations) and 00,Ji). This leads to conjecture VII.6. References [1] M. Bauer, Ph. Di Francesco, C. Itzykson and J.B. Zuber, Nucl. Phys. B362 (1991) 515. [2] A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B241 (1983) 333. [3] L. Benoit and Y. Saint-Aubin, Phys. Letters. 215B (1988) 517. (4] D. Bernard and G. Felder, Comm. Math. Phys. 127 (1990) 145. [5] M. Bershadsky and H. Ooguri, Comm. Math. Phys. 126 (1989) 49. [6] V.G. Drinfeld and V. V. Sokolov, J. Sov. Math. 30 (1985) 1975. [7] B. Feigin and E. Frenkel, Berkeley preprint MSRI04029-91 (1991). [8] B. Feigin and E. Frenkel, Phys. Letters. 246B (1990) 75. [9] V.G. Kac, Infinite dimensional Lie algebras. Cambridge: Cambridge University Press 1985, and references therein. [10] V.G. Kac and D.A. Kazhdan, Adv. Math 34 (1979) 97. [11] V.G. KUC and A.K. Raina, Highest weight representations of infinite dimensional Lie algebras. World Scientific, 1988. [12] A.W. Knapp, Lie groups, Lie algebras and cohomology. Mathematical Notes 34. Prince- ton, 1988. [13] V.G.Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B247 (1984) 83. [14] F.G. Malikov, B.L. Feigin and D.B Fuks, Funkt.Anal. Prilozhen, 20 No. 2 (1987) 25. [15] V.V. Schechtman and A.N. Varchenko, Max Planck Inst. fur Math, preprint, MPI89-51 (1989). REFERENCES 43 [16] A. Tsuchiya and V. Kanie. AcIv. Stud. Pur. Math. 16 (1988) 297. [17] A.B. Zamolodchikov and V.A. Fateev, Sov. J. Nucl. Phys. 43 (1986) 657. ARTICLE 5 April 1992 TAIT- 1961-92 Physical States in £j Models and 2d Gravity O. AHARONY, O. GANOR . J. SONNENSCHEIN AND S. VANKIELOVVICZ t School of Physics and Astronomy Beverly and Raymond Sackler Department of Exact Sciences Ramat Aviv Tel-Aviv, 69987, Israel N. SOCHEN Service de Physique Théorique Cen Saclay 91191 Gif-sur Yvette Cedex, France ABSTRACT An analysis of the BRST cohomology of the ^ topological models is performed for the case of .4 j . Invoking a special free field parametrization of the various currents, the co- homology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irre- ducible representations. Using the physical .states we calculate the characters and partition function, and verify the index interpretation. We twist the energy-momentum tensor to establish an intriguing correspondence between the ^A,! model \vith level k = * _ 2 and (p, q) models coupled to gravity. t Work supported in part by the US-Israel Binational Science Foundation and the Israel Academy of Sciences. .1. INTRODUCTION Topological Quantum field t hones ' have been a center of much interest during recent years. At the same time a great emphasis was devoted to the research of non critical string theory for c < 1. This topic was explored via matrix models ' as well as continuum Liouville theory.'3' In several cases a strong relationship between some topological models and the string models was revealed. For instance recursion relations for amplitudes in the one matrix model were found to be identical to those derived in the topological theory of gravity.wThe exploration of connections between another class of interesting topological models, the so called § models, and theories of c < 1 matter coupled to 2d gravity is the subject of the present work. The £ topological theories'5'6' are theories based on G current algebra. Formally they are obtained by gauging the G WZW model. Following the well known ^ construction1"1 the § models are derived by gauging the anomaly free group G of the Gt WZW model. This recasts, using a complexified Gc algebra, the § action into a sum of three terms: a G WZW at level fc, a (^)I+JCG WZW model which can be viewed as a G WZW model at level -(A; + 2Cc) and a (1,0) odd ghost system in the adjoint representation of the group. The ^ models have a significant overlap with other TQFT's. For one they are related to the 2+1 dimensional Chern-Simons theory . In particular the amplitudes, which are given in terms of N1Jt fusion rule coefficients, ' coincide with the scalar product of wave functions of the Chern-Simons theory. Other related TQFT are the topological flat connection models. They share with the ^ models the correspondence to the moduli space of flat gauge connections. More interesting is the relation with models of c < 1 coupled to gravity. This is the main topic of the present work. In réf. [5] the structure of the ^ theories was investigated and some striking resemblance to the 2 dimensional gravity models was revealed. One would expect that the Gt \V ZW model plays the role of the matter system, while the (^r)t+2Cc WZW model is affiliated with the Liouville degrees of freedom. Both in the gravitational models " as well as in the ^ models the full character gets a contribution only from the primary states of the matter (Gt) sector whereas the matter and Liouville (G_(t+2Cc)) oscillators are canceled out exactly by the ghost contribution. The character can be interpreted as an index which encodes the information about the BRST invariant states of the theory. This index is 7>[(-)Vl «»d Tr[(-)cqN(eme)J*~ï\ for the gravity and A(I) cases respectively, where G stands here for the ghost number, N is the level and J$ is the charge, appropriately defined ". The partition function is obtained by integrating over the moduli. In the gravity case this leaves only matter primary states while in the § model the partition function equals the number of conformai blocks. For the Sl '('-) case with an integer level k the result is k + 1. The information from the index interpretation of the partition function was used in réf. [5] to disentangle the spectrum. In view of possible cancelations, an index cannot a-priori give full information about the spectrum. A complete account of the physical states can be achieved only by working out the cohomology of the relevant BRST operator. This type of analysis for c < 1 matter coupled to gravity was worked out originally in réf. [12] and latter in réf. [13]. An extremely rich spectrum of discrete states was revealed. They were found to correspond to previous matrix model calculations ' In the case of c = 1 a " ground ring" ""of discrete states of zero ghost number and dimension was exposed with an underlying W00 symmetry. The corresponding Ward identities were implemented in the computations of correlators.15 A different method for those calculations was suggested in réf. [16] together with an extension to the c < 1 models. The counterpart of these results in the context of the ^ models is still missing. One of the main tasks of the present work is to take the first step in this direction. It is devoted to a BRST analysis of the gauged SU(2) WZW model of level k defined on a sphere. We establish rigorously the result of réf. [5] that for an integer k there are k + 1 zero ghost number primary states corresponding to the matter primaries. On each of these primaries there is a whole tower of descendant states, one at each ghost number. The outcome of the BRST analysis follows closely that of c < 1 models coupled to 2d gravity!"'"1 To be more precise (p,q) minimal matter systems coupled to gravity correspond to k = ^ -1 level of the SL(I) case. This is related and indeed has a lot in common with the observation that SL('2) WZW model is connected to the (p,q) system via the Hamiltonian reduction." Moreover, it was proven that the irreducible representation of the Virasoro algebra can be extracted from the irreducible representation of the SL(2) current algebra by putting a constraint. The quantum gravity coupled to Cp The paper is organized as follows. In section 2 the § topological theory is briefly re- viewed. This serves mainly to establish notation and put forward important topics to what follows. In section 3 a BRST analysis similar to the one employed in ref.[13] is performed for G — .4, . The main idea is to use a free field parametrization for the currents of the k and —(k + 4) levels. In fact it turns out that to implement the procedure of ref.[13] it is t Throughout the paper we shall freely move between SU('2) and 5L(2) cases depending on the values of the level fc. convenient to invoke "conjugate" parametrizations for the two sectors. The physical states of the theory correspond to the colioinology on another space Th ^ /"_(A-+4) £ Fyhost, where Fl: denotes the irreducible representation of ,4, level k. F_^.+i) and Fghost the Fock space of the —(k + 4) sector and the ghosts respectively. In section 4 we discuss aspects of the representation theory of Kac-Moocly algebra ,4j for an arbitrary complex level k. Special attention is devoted to singular vectors associated with null states. The existence of these vectors in a given Verma module indicates that it is reducible. 'This information is crucial for the determination of the BRST cohomology on the space of irreducible representations. There is a similarity between the representations of the current algebra and those of the Virasoro algebra. The duality between representations of the latter which are characterized by |fc,c > and |1 — /i,26 — c > is shown to correspond to a duality between \J, k > and I — 1 - J,—(k -)- 4) > in the /i',1' algebra. The singular vectors are used in the appendix to determine the fusion rules. The passage from the Fock space BRST cohomologies to an irreducible representation is presented in section I . It follows the work of Bernard and Felder . In section 6 we demonstrate the correspondence between (/>, The § TQFT is constructed by g.mging the anomaly 1'iw diagonal G group of the G — IV Z U 'model. The classical action takes the fonn 2 l Sk(g,A") = Sk(g) - j; J d :Tr( + gdg~ A, - .4c where g € G and Sk(g) is the WZW action at level k. In case that S is topologically l l trivial the gauge field can be parametrized as follows A, = ih~ d:li,A-, = ih*dzh*~ where M-) € Gc. The action then'7*1 reads St(g,A) = Sk(g)-St(hh') (2) The Jacobian of the change of variables introduces a dimension (1.0) system of anticom- muting ghosts \ and p in the adjoint representation of the group. The quantum action thus takes the form of Sk(g, h,p,X) = St(sr) - S4+2C0(A**) - * / d-=Tr(pdx + c.c] (3) where CG is the second Casimir of the adjoint representation. The second term can be viewed 1 as S-(Jt+JCc)C )- Since the Hilbert space of the model decomposes into holomorphic and anti- holomorphic sectors we restrict our discussion only to the former. There are three sets of holomorphic G transformations which leave (3) invariant 6jg — i\t(z),g\ 6jh = Z[E(Z), /i] and c a a c £./(•») \° = «/fce'^ ; 6j(ii»P = -z/t reV with (. in the algebra of G. The corresponding a c currents J°, /° and jW = z/6 cxV satisfy the G Kac-Moody algebra with the levels k,— (k + 2cc) and 2cc respectively. We define now J''0'' a a j(tot) = j(gh) = ja + ja which obeys a Kac-Moody algebra of level (5) The energy-momentum tensor T(z) is a sum of Sugawara terms of the 7° and I" currents and the usual contribution of a (1,0) ghost system, namely ''" /ru \ *• . /mo fa/a . •*• -wn vn . a f\ a * \ ** I I •«/*/• . (6) k + C0 fc + c( The corresponding Virasoro central charge vanishes - -2dG = O (7) k + CG -(k + 2cG) + < This last property is a first indication that the § model is a TCFT. In fact it is easy to realize that the basic algebraic structure of a TCFT is obeyed by the model. This is expressed in -\ terms of two boson ic and two fermionic operators. The former arc T(:) and the "ghost number current" J& = \,,p" • The iermioiiic currents are a dimension one current which is the BRST current J(BRST) an(j a (|jmensjon two operator G'. These liolomorphic symmetry generators are given by The TCFT algebraic structure now reads (9) j(BRST} ={(?J#(J:) where Q = / j(BRST>(z) is the BRST charge. In addition to T(Z) and J(BRST>(z), the total ol) curent J^ a is also BRST exact, (10) The extraction of physical states as elements of the BRST cohomology will be the subject of the next section. We summarize here, following réf. [5], the picture emerging from the investigation of the torus partition function. The latter is expressed asin r =CT2- f (U) where du is the measure over the flat gauge connections on the torus and r is the rank of G. 29(r, u) is the torus partition function of the Gk WZW model (12) where q = e2lTr, A^, XL denote the Gk highest weights, and for the diagonal invariant -^AA = ^ARA The character can be written as r, u) (13) A/O,O(T,M)' with Mt1A defined explicitly for the SU(I) case below, Z**'(T,U) in eqn. (11) is the con- tribution of ft € ^r at level k + 2Cc or equivalently ft G G at level -(k + 2cc). This was calculated in réf. [7] using the iterative Gaussian path integration technique. The outcome 6 is that 2*'''(r. ») ~ |.Wo.n(~. ")| " indicating that ^- contains just one conformai block. It is straightforward to calculate Z'jll(r. H). the ghost contribution to the partiton function gh 4 Z (r,u] ~ IAyO-O(T1K)I . The cancelation of the l-Uo.o^-")| factors in eqn.(ll) is similar to the cancelation of the r/ factors in the torus partition function of c < 1 models coupled to 'Id gravity."'1In both cases the resulting character is given by the numerator of the character of the "matter" sector. In the ^ model it is Mk,\- This cancelation property leads to an index interpretation for Mt.\- For G = $t'('2) this amounts to expressing 2)1 (14) /=-00 as (15) where 6 is the holonomy in the T? direction, G is the ghost number, Lo is the excitation level and J?tot\ is the J?tot\ eigenvalue of the excitation. The prefactor in front of the trace was chosen to agree with the definition of the vacuum we will use in section 3 where the cohomology is worked out and eqn. (15) is verified. Note that MJ-J(T, 6) is obtained from the torus A/j. J(T, u) by restricting to just one angle. This amounts to consider the propagation along a cylinder rather then around the torus. As long as we are interested in the spectrum it is sufficient to consider MKJ(T, 9). This index interpretation enables us to read important information about the physical spectrum from eqn. ( 14). For a positive integer k, Ij = O, ...A;. Hence there are A;+1 zero ghost number primary states which correspond to the first term in the q expansion of the different A/tj's i.e the term corresponding to / = O with LQ = t+t • On each of these states there is a whole tower of states correponding to the higher terms in the q expansion, we will refer to these states as descendants. In réf. [5] it was argued that those states apear for all ghost number, and for a given j and a given ghost number there is precisely one physical state. The ^ characters are orthonormal in the du measure (16) where j and j1 are Sf/fc(2) multiplets. Thus eqn. (11) yields for G = St/jt(2) Za = fc + 1. "5 For integer k this is precisely the number of conformai blocks. .3. A1,1' LEVEL A- BRST COIIOMOLOGY ON THE FOCK SPACE We now proceed to extract the physical states of the ^ theory for G = A1 . We use the BRST procedure to quantize the system and thus the physical states are in the cohomology of Q, the BRST charge , \pluj* >€ H'(Q). Expanding the currents J", /" and the (1,0) ghost fields p", \" in modes and inserting them into eqn. (S) we obtain the following BRST charge Q = (17) where : : denotes normal ordering namely putting modes with negative subscripts to the ( left of those with positive ones and pjj to the right of \jj. Since both J '°''n and Ln are Q exact namely (18) it follows that Lolphys >= O J(tot^\phys >= O (19). ( For non-vanishing eigenvalues of LO and J '°''0 it is easy to see that \phys > is in the image of Q which cannot be true for a non-trivial \phys >€ H*(Q). Let us now select a sub-space F(i.I) of the space of physical states on which />0 = O in ( ot) addition to J ' 0 = LQ =0. On this sub-space Q which may be written as Q = \8-/('°')0o + + Q (20) M = -4 = XoXo : equals Q. W° thus start by deducing H*(Q) the cohomology of Q. The states which correspond to the latter are built on a vaccum |J, / > obeying the following relations + 0 |J, / >=0 .'0 |J, I >= O J0 IJ,/ >= J|J,/ > >= » 'Q0I-A 7 >= 7I-7'I > (21) A convenient way to handle the J" and /" currents is to invoke the following "bosonization"|26) • -Jm-i -ni—m • a"I—. 7nv" (22) k. nt where a2 = t + 2. The fields 3 and -, form a bosonic (1,0) system with [•>,„, Ai] = Q = -nf/?» - «(^m - V>m )în-n,- • 7m7A-m-/ = "(* /,m,n +2 (23) it,m,n c - V'/ ^ i2-' f ai'.c '• -V-nA-mrm+v" v* /J n • >7i,n where An assignment that obeys this requirement is the following deg(x) = deg(t) = deg(i) - deg(+) =1 (24) deg(p) = deg(/3) = deg(P) = deg(d>-) = -1 The decomposition of Q to different degrees now reads Q = Q(U =« ~ " / _ \ -.,,O111") H -III — / j 2 (25) m.n (A- + 4)n)in] n /r, m ../Jn-m-fc = +(«(0"? + «>0) - ^ n A-, m From the fact that terms of different degree in (Q)' vanish separately it follows|13'that on ) Q1 obey the following relations = (Q<3>)2 = O {Q<°>,Q('>} = {Q<2',Q<3'} = O Q(1)2 + {Q(0),Q(2)} = (Q(2))2 {Q(0',Q(3>} = O In fact (Qt0*)2 = O holds on the entire Fock space. We want now to apply the results of réf. [13] which hold only provided that there is a finite number of degrees for each ghost number. Recall that states in J-(J, I) are annihilated by both L0=L0 + -r[J (J + I)- 1(1+ I)] (27) [: 0-«7n : + : /J-n7n : +9ab and J<"*>0 = .7 (28) It is clear from the expression of LQ that for a given | J, / > the amount of excitations and to thus the degree they carry is limited. The restriction of vanishing J^ ^0 further limits the contribution of the zero modes -yo and /?o- This proves that on .F(J,!) and in particular for each ghost number the degree carried by any state is bounded from both sides. Hence we can 10 proceed and use the lemmas proven in réf. [13]. The next step is to find the cohomology of (0) Q on ^(J1I) . It is not difficult to realize that Lo. the contribution to Lo of the exitations, is Q'0' exact G0 = - The consequence of the last relation is that LQ annihilates the slates in the cohomology of Q*0' on F(J,l) and thus there are no excitations in H*(Qa). Moreover, since LO = O, states in the latter must have either / = J or / = -(J + 1). Let us now extract the zero modes contributions to the cohomology . The general structure of these states is ,ii_ >= (30) are where obviously n+,n_ =0,1 and »•»,"/} non-negative integers. Using the following commutation relations (0 (O) + (0) [Q U] = -X0- J0] = { = (31) and the relation Q^\I, J >= \$0o\I,J >« one finds that the result of operating with on (30) is - >= (-l (32) where states with n± > 1 are identified as zero. It is now straightforward to deduce the Kernel and the Image of Q^0'. The former takes the form (33) and the latter has the same terms apport from th |0,0,1,0 > term . Therefore the only possible state in the relative cohomology of Q'0' is vjl/, J >,and from the condition j''°% = 11 UIt O we find a state only provided that / = - J - 1 and then H"'(QM) = {\t\-(J + \),J>}. (34) The passage from the relative cohomology to the absolute one is thon given by Hal"(Qm) * // \Xf'(Q(0)) (35) in the same way as in réf. [13]. Since the derived cohomology includes a single degree, //*(Q(°>) ~ H*(Q) in a complete analogy to the Liouville theory discussed in réf. [13]. We conclude that the cohomology of Q on the full Fock space includes states of arbitrary J with a corresponding / = -(J + 1) and with ghost number G = 1,2, one state at each ghost number. We shift from here on the definition of the ghost-number so that the state XjJ-I/, J > is at G = O. We should note here that since there is a symmetry between the / and J sectors we could also have used an "inverse bosonization" in the J sector and the ordinary one in the / sector with the same resulting cohomology. So far we have analyzed the cohomology on the whole Fock Space. Now one has to pass to the space of irreducible representation» of the J sector. For that we need certain results related to the'singular vectors of arbitrary level k of the A\ ' algebra. These results are presented in the next section. The reader who is not intereseted in the details can skip it and move directly to section 5. .4. REPRESENTATION THEORY OF A\l) We give in this section a short review of the representation theory of A1 and of its singular vectors. The results presented in this section will be heavily used in what follows. We follow the presentation of Malikov, Feigin and Fuks but we use the language of current algebra. We concentrate on these results which will be relevant for what follows. Let g be the Aj algebra defined by the commutation relations r o ±i _ . ± (36) The universal enveloping algebra U(g) can be written U(g) = AL © tf ® N+ (37) where N+ is the subalgebra generated by Jp" and J1" , H is the Cartan subalgebra generated by JQ and fc, .\nd N- is the subalgebra generated by J0" and J^1 . 12 The highest weight vector is characterized by two parameters: the spin J and the central charge k. It is convenient, though, to work with the variable / = k + '2. We denote the highest weight vector by \J.t). The conditions it satisfies are (38) K\J.t) = (t-2)\J,t) We recall that the Sugawara construction gives us another diagonal operator LoM, O = h(J)\J,t) = \J(J+ 1)1 J, O- (39) On this highest weight U(g)\J*t) ~ N-\J,t). We call this module the Verma module and denote it Vj j . The Verma module is an infinite dimensional representation of A1 . The Verma module Vjj is naturally graded with respect to LQ (the level) and to JQ (the spin). The homogeneous subspaces are finite dimensional. Kac and Kazhdan studied the question of reducibility of these modules and gave the following conditions for reducibility: Vjtt is reducible if and only if there exist two positive integers r and s such that the value of J is at least one of the following 2Jr,a,+ + 1 = r - (s - l)t (40) 2J,.4._ + l=-r + st or if t = O. A singular vector is a. state |\) € Vj,i such that |\) ^ \J, t) and such that \x) is a highest weight vector. It is clear that a Verma module is reducible if and only if it contains a singular vector. An interesting question is what are the level and spin of a singular vector in the module J = Jr>3,± and what is its explicit form. These questions were first studied in réf. [2Oj. It was found that if J = Jr,s,+ for a couple (r,s) then (41) The subindices of \\) keep track of the level and spin of the singular vector What does this expression mean algebraically and geometrically? 13 t;' Algebraically. MFF found an infinite sol of positive integer / such that this expression makes sense and it can be written as r(a — 1 ) — u\ j f\ I jo v Jt/r.s.-f • '/ \ »v/ P-V 1 where Pp,q depend polynomial!} on t and, thus, can be continued analytically to the whole complex plane. Take for example /• = 1, * = 2 then we use analytically continued commuta- tion relations )'-1 - *(* - (44) in order to solve 2 M) = (J0"J-i Jo' ~ 'Jo'J-i - 'J-i Jo~ - < Jli)|Ji.2,+,*> (45) If J = Jr,«,- then J0 «-» J^1 and |\) = |\rj,_r). We remark that in practice there is another way to find the singular vector which is more efficient and is briefly described in the appendix In order to see the geometrical meaning of (41) we denote AI = IJ and k — IJ. Using (44) we see that formally (46) J,t) ~ \k - J + l,t) are singular vectors for any value of J. The geometrical meaning of these operators is clear. In the A plane (see Fig. 1) they conserve the line AI + Ao + 2 = t. Fig. 1- The A plane. XI+I 2 1 1 In fact the operators ( JQ) and (J^1)* " " are reflections of this line on itself by the points (—M ~ 1) and C ~ 1» — 1) respectively. They are the generators of the Weyl group. MFF ""assert that for the values of J that were given by Kac and Kazhdan"91we get after finite number of reflections a meaningful expression. 14 Al+1 Aj+l If AI 4- 1 (A.. + 1) is positive then (J0~) |^0 ((-/±i) l-AO) is a formal singular vector in the Venna module \ j/ (I 'jj). It' on the other hand AI + 1 ( A > + 1 ) is negative then A 2 1 is a or Vy., ~ (J0~)- '-'| -J-\.t) (Vj., ~ (-/^1T* ' I*' ~ J + 1-0) l' mal singular vector in the Verma module V'_y_j , (U_j+i,/). The only reducible modules which can not be generated by (formal) singular vectors are those that satisfy \\, A-> > —1. These conditions for the case t = pjq read O < rq-p(s- \) For such Jr.» we can calculate the inclusion diagram Fig.-2 J inclusion diagram Where r — 1 s — I p 15 Fig. 3-1 inclusion diagram It is even more interesting to notice the striking similarity of this structure to the Virasoro representations. In the Virasoro case there exists also a "mirror symmetry" between \h,c) and |1 - A,26 - c)!"'\Ve will s«> in the following chapters that this similarity is the reason for the similarity between the spectra of ^'."' and that of the minimal models coupled to Liouville. We refer to this mirror symmetry as "duality". Note also that for r = p eqn.(48) gives a(l) = 6(/-r 1). Hence, in this case, we have just one set of singular vectors and the two branches of the embedding diagram of Figs. 2,3 degenerate into a single branch diagram. .5. IRREDUCIBLE REPRESENTATION AND THE BRST COHOMOLOGY The next step in the extraction of the physical states is to pass from the cohomology on the Fock space to the irreducible representations of the level A- A\ Kac- Moody algebra. In general a representation L is reducible iff 21 + 1 = r - (j> — I)(A; + 2) where r and s are 1 integers with either r,s > 1 or r < 0,a < 0.''" In the 3 model with G = A1 ' we therefore have =r-(s- I)(It + 2) - I)(A: + 2) (49) Note that JT>S = —I-r,s — 1. Completely irreducible representations, which have infinitely many null vectors, appear provided that fc + 2 = *• for p and q positive integers which can be chosen with no common divisor. In this ca.se /r.5 = IT+p,s-q and JT<3 = Jr+p,i+g. It is, thus, enough to analyze the domain 1 < * < line one can construct the irreducible representation which is contained in FT,s , the Fock space built on \JT T,3 . Qj) x Ti x f°, Q] (50) where J^ is the ghosts' Fock space built on the new vaccum |0 >c- Since Q^ acts only an 16 the J sector we can rewrite H "J as ]. (51) Moreover, since {Q.Qj} = O one can use theorems about double cohomologies and write this as isomorphic to ""0UC(^ x^xfC^Q^Qj}. (52) The theorems '"'apply only provided that each cohomology separately is different from zero only for one single degree. In the present case this was shown in section 3. In fact we have c already calculated //^(/V.s x .F/ x f .Q) since J>,s is the union of free Fock spaces. Hence the result is that the latter has one state if the Fock space of J = —I — 1 is in fr,a i and it is empty otherwise. For each Jr,s we get states at 7 = - Jr+2ij,,i - 1 and / = —J-r+2iP,s — 1 where their ghost number is equal to the corresponding degree in the complex of réf. [22] . For each such JTi3 there is an infinite set of states with 1 = I-r-Mp,t G = —21 and / = A— 2/p,» G= 1—2/ for every integer /. An example is the case of k = O which is discussed in detail in section 7. There JT,, = O since the only possible values of r and s are r = s = 1. Thus the posible states are at /_r_2/f>,s = —2/ — 1 and /,._ iif,i = —2/ which implies that the possible values of the ghost number are G — I + I for every integer /. Though the general derivation of the cohomology does not give an explicit construction of the physical states, it is clear that they involve null states. (Recall that originally there is only one physical state in the Fock space. ) In our construction an irreducible representation of the J sector was used namely null states were eliminated, while the / sector was left as a Fock space. In the latter case, it can be shown that half of the null states vanish identically on the Fock space. For instance in the k = O case, for negative ghost number one has Q\phys >»- \null >, where by |;?w// > we mean a null state or it's descendant. Hence, these states are in the cohomology provided the corresponding null states vanish, which indeed is the case. For positive ghost number there exist states |V> > for which Q\>l> >= \phys > +{null > and therefore those states are in the cohomology only when the •fol o corresponding null states are non-zero. The next step is to compute the level LO and ./( )0 for the excitations on all states. The results are summarized as follows '!I J — JT,S" ' — •— T — J//>.-;* J —• "T, s* ' — *r — 2lj),s* G = -2/ G = I- 2/ - _ ., (53) —(lol) u where IQ is given in eqn. (27) and J u = -I - J - 1 is the total charge of the excitations since the total Jo vanishes and our ghost vacuum has JQ = 1- Again we should note that had we used the reversed parametrization, the results would have been analogous to those just derived. Since the direction of the cohomology in the complex of réf.[22] would have been reversed, we would have obtained the same states but with opposite ghost numbers, namely, G = 2/ and G = 2/ - 1 instead of G = -2/ and G=I- 2/ respectively. The partition function that is computed below is not affected by those changes. The last step in the deduction of the space of physical states is the passage to the absolute cohomology which is the same as in réf. [13] since, as we have just shown, there is a single state for each / and J, Let us now examine whether we can verify the index interpretation of the torus partition fucntion which was discussed in section 2. For integer Jt j = O,.., *. The partition function in terms of the characters A'/tj was given in eqn. (14). We want to check now whether it can be rewritten as a trace over the space of the physical states. One has to insert the values of I/o and J,°jo/. of eqn. (53) into eqn. (15), with Jffot\ and G «shifted to the values defined for an SL(2) invariant vacuum. Inserting these values for every / one gets exactly the expression of eqn.(15). We can obviously add the values of the level and the eigenvalue 01 O - - a ;_i_i \ 0 ° toi ° of Jl' ' of the IJ > vacuum namely LO -* L0 + iVV and J<' ') -» J( > + J to derive a simpler expression ' '2i 18 .6. THE CORRESPONDENTF: TO <• < 1 MODELS COUPLED TO GHAN-ITY Let us now raise the question of whether one can map the JJTTTJ model to minimal models coupled to gravity. Or differently to what extent are the two types of TCFT equivalent. The minimal models can be formulated either in a Liouville approach * or in a world-sheet light-cone gauge.2* Since these two gauges are believed to be equivalent it is enough to demonstrate the correspondence of the § model with one of the two. Nevertheless, we discuss here the relations of the £j with the two 2d gravity pictures. Motivated by the comparison of the partition functions of the ^ model of level k = | — 2 and that of a (p, q) model coupled to gravity, which is discussed below, we define now a twisted energy-momentum tensor T as follows: Since both T(z) and 7"°')0(;) are BRST exact so is f(z). Hence, this twist does not introduce a Virasoro anomaly. However, both the dimensions and the contributions of the various currents to the central charge are now altered. In addition it is clear that T is not an A1 invariant operator. The contributions to c of A1 currents at level k are shifted from •$•% —» -ççï — 6k. Hence, for the case of k + 2 = | one gets the following anomalies in the J, I and J cy = 2 + cM c/= 2+ (26-cp,,) c9A = -30 (57) where Cp,, = 1 - 6(>'~ql . One can rewrite the contribution to the total central charge in the "V form they appear in the Liouville and the light-cone formulations (26 -Cp,,) -26 where K = — k — 4. Obviously so far it is only a rewriting of zero and by iteself it does not prove much. However, we want to check whether one may provide a map between the set of fields of the A1 § model and those of minimal models coupled to gravity. In particular eqn. (58) may suggest the following correspondence. The J sector contains the "matter" degrees of freedom and a bosonic (1,0) system which compensates a similar anticommuting system from the ghost sector. The / sector is the 51(2, R) gravity part in the approach of ref.[28] 19 or it is the Liou ville sector plus an additional commuting system of c = 2 which again pairs with a ghost partner. The p. \ ghosts translate into the b. <• system (or 10 the ghost system of ref [29]) plus additional aniicommutiiig (1.0) degrees of freedom. This implies that the ghost vacuum transformation |0 >— > xJlO > corresponds to the usual |0 >— » ci|0 >. Before discussing these kind of realtions let us observe another "numerological correspondence" |29 between KPZ 'and the A1 ^ model. Recall that the relation between the "undressed" and "dressed" dimensions is1"1 A + A(0) - &%$• = O. If we identify J with A and take for it J = JT,, then we find that A° = JJrrf ~ J = Tt[(ts ~ r)2 ~ (t ~ 1)J] = A- (59) for t = f . The modified T of eqn. (56) introduces the following modified dimensions: + (.-.X I- <*.-» -(/M).. (60) (\'.f*)- (1.0) «-(•»,/)) The conformai dimension of \u,/»° and Whereas it is obvious that <$> corresponds to the Fegin-Puks "free field representation of the matter part, since their central charges are the same, one cannot simply assign 4> to the Liouville degree of freedom. This is due to the fact that its background charge is 1 \(\/k + 2 — y. ) which leads to a contribution to the Virasoro central charge 2 — Cpi9 rather than 26 — cp,g. The role of d0i, where the central charge of Next we want to compare the partition function of the (p,g) model to that of § for G = SL(I) and k •- |j - 2. Comparing eqn. ( 15) to the numerator of the character of the minimal mode! it is clear that correspondence might be achieved only provided one shifts T — » r — %0 or equivalently taking e'*e = q~l. In this case the numerator of the character in the minimal 21 Ul 1 o model which is proportional to 7>[(-l )Gc/io] is mapped into Tr[( — l)cqL<>~Jl""• —}o] in the § model. That is the origin of the twisted T defined in eqn. (56). We thus need to compare the number of states at a given level and ghost number in the minimal models with the corresponding numbers at the same ghost number and "twisted level". From eqn. (53) we read / = /_,_,,,,, G = --21 I0 - J' = /V/ + /() —- . o -I*") rs / = /r-2/;>,s G=I- '11 L0 - J<""'0 = /V/ ~ '( qr which have dimension hTi3 = ~ ',L~' . The levels of the excitations are L0 = A - hr,s. For G = 21 + 1 one has A = A(I) = [(^+^+^'-(g-p)'! and for G = 21 A = B(I) = [(W-^j'-'v-;.)2]!"."! Hence, the the contribution of the • — o various levels to the partition i'unctiojj are identical to those of LQ — «7''°''o in eqn, (61) for the same ghost numbers and the respective vacua satisfy J = •J&pm and / = —• where pm and pi are the matter and Liouville momenta respectively. It is thus clear that for a given r, s we get the same number of states with the same ghost number parity in the two models and thus the two partition functions on the torus are in fact identical. The relation et n between aT,s(/) (&r,s(0) °f l - (48) and the dimensions AT,,(l) (Br,s(l)) of the null states appearing on the minimal models embedding diagram is given by 7<2(0«ÏC) + D- <ÏO = W(I) = -4?;? (62) with a simi'Ja expression relating briS(l) to BT (62) into AT in fact there are 4 different possible identifications of the § states with those of the minimal models. There are the two possible parametrizations discussed in the previous sections, and there is in each of them the possibility to assign hr 21 1 with the appropriate ghost numbers. The complete identification oi' a minimal model coupled to gravity and its ^ counterpartner requires obviously the identification of all physical states and all non-trivial correlators in both theories. This question is under current investigation. .7. PHYSICAL STATES AND COHUKLATION FUNCTIONS IN THE .S'('t=o(2) CASE'"' As an explicit demonstration of the general results obtained in the previous section we consider the simplest non-trivial case namely JT model for Sl'('2) at level k — O. Here we present the calculation of the cohomology in the free field parametrization of section 3 as well as in a scheme where both the J and the 7 sectors are parametrized according to eqn.(22) and show how the equivalence between the two methods is utilized in the determination of non-trivial correlation functions. The matter sector of the Hilbert space consists only of the \J = O > highest weight state since all other states are nulls . Therefore, in this case the choice of the bosonization in the J sector does not affect the results. The / sector is associated with the k = —4 Kac-Moody algebra and in addition there is the usual (1,0) ghost system. Following the general analysis of sections 3 and 5, the operators which furnish the cohomology of the k = O case are found to be (63) 0 + + + n l + n V_n_i = \ x P dp ...d ~ p e *o(i) + corrections for n > O. In the second scheme that we have used, the same parametrization for both the J and 7 currents is used. It is the one given in eqn.(22) where now a = \/2 and —i-\/2 for the k = O and k = -4 sectors respectively. As mentioned earlier the passage between the two prescriptions is via the automorphism 7° «-» —7° and 7* «-» 7T. The other difference is the properties of the vacuum. The operators in section 3 were constructed with respect to the 7o|0 >= O, whereas now we use the 51(2.7?) invariant vacuum which is annihilated by fa. Formally the relation between the two vacua can be written as (70 = O >= £(7)|A> = O > First it is straightforward to check that the following states are in the cohomology in this parametrization. JV0 > = I/ = O > (64) > = Next we want to extract state: with negative ghost number. The steps in the construction 22 are the following. First we search for states |\'_,, > which arc BRST closed up to a null state namely Q\V.a >= \mill >n (65) where by \null >„ we denote a null state or a descendant of a null which is built on the state I/ = n > as depicted in the embedding diagram. In addition we pick states which carry the lowest ghost number possible for the given level and J''"'1. so that they cannot be BRST exact. It is easy to verify that states of the form |Vn >= (p~lp~.î..p'2a+correctiona)\I — n > obey these conditions. For instance the tirst state which has the explicit form iv-i >= (PI, +AAT - Ki(7O-)2)!' = i > (66) leads to the null state of r = l,$ = 2 when acted with Q namely 7 7 2 7 7 67 C|V_! >= (/I1 + /Vo" - 3 *i( o") )l = 1 >= IVU >= U.il = 1 >' < ) From the general structure of singular vectors described in section 4 we can read the eigen- values of |xi,i >, ioU'1,1 >= /OIXU >= °- The next nul1 state *2,i|/ = 2 >= |X2.1 > corresponds to |V_2 > as follows Q|V_-> >= (^I1 + correction)\hat\ij\I = 2 > with the correction terms carrying LQ = 1 and J^ot\ = — 1. Notice that since |\'2,i > has LO = 1 and Jltot\ = +1, it is its descendant which is in the image of Q. It is easy to verify that the general structure of the BRST charge acting on the states K_n takes the form Q\Vn >= (^I1Pl,..p~(n_1( + correctian)\a.i\I = n> w tn It can be proven that all the states |\r(j-i),r > ' possitive r,,3 vanish upon using the bosonization eqn.(22) Since we are interested in the space which includes the Fock space of the / sector, it is obvious that |V_n >€ H*(Q). Thus there is no need to perform the procedure described in réf. [22] in this sector. Clearly restricting ourselves to the Verma modules of the current algebra gives rise to a different cohomo! y. It is presumably a general feature of the embedding diagram that half of the null stales vanish upon invoking the parametrization of eqn.(22). This is proven in réf. [22] for any k > O. The generalization to k < O seems to follow essentially the same steps. In the present case the null states which carry positive values of /, namely those which are on the right handed branch of the dual embedding diagram (Fig 3.) vanish. The situation is reversed once one uses the parametrization of section 3. Writing the state j/ = n > in terms of the operator e"* acting 23 on the SL('2.R) invariant vacuum, we can now write down the complete set of operators that span the cohomology. U=I (68) \'_,, =p~dp~...i)"~lp~e'"!> + corrtction* Note that all these operators carry zero dimension and thus they close upon a ring. The multiplication operation is just the OPE. namely V,n(--)V,,(u.-) = Un+11M + {Q,0'} (69) where O' is a dimension zero and ghost number ?) + m operator. The Q exact term does not always appear. For instance there is no such a term for n.m > O. Using OPE's we can, in principle, starting from eqn. (66) calculate all the tin- Next we proceed to the calculation of correlation functions. To calcualte the expectation vslue of products of operators which are in the BRST cohomology, one has to define the scalar product or equivalently the notion of the conjugate to a given state. Due to appearance of zero modes of the ghosts \Q", \o,,\'(T, tne zero m°de °f tne commuting field 70 and the background charge -1 of ^, it is clear that the vacuum is not self-conjugate, < 0|0 >= O. Instead , < 0| the conjugate to |0 > is given by 0| =< 0|\£\jj\oe-*«(7), <0|0>=1 (70) where we have introduced formally 6(7) to absorb the zero mode of 70- To compute correl- lators we thus define formally another operator in the cohomology V0. (71) and denote by Vn the result of the OPE of the Î with Vj, namely 1(C)Vn(U.-) = Vn(u>) + 0(Z - u>). (72) We see that the Vn are exactly the operators appearing in the second parametrization up to an interchange \'+ <-» \~ and p+ «-» />~. Using these new definitions we can write the 24 correlators of the model as follows .V 1=1 1=1 1=1 (73) In particular it is obvious that < I-,, (I',, >= 1 . This proves that the states |Vn > given above are not exact, since otherwise their correlations with closet! states would vanish . Notice that since all the operators in the correlators are of zero dimension the result is a number which is independent on r.c,, a.s it is expected for a topological model. The ghost number of Vn, Vn are n,n + 3 and the momentum P$ are — n,— (n + 1) respectively. The conditions for a non-vanishing correlator on the sphere are * = -1 (74) n which translate into a single condition for G(Z, Si^. ..:n) namely n + £3i » = O- Usually in TCFT there are in addition to the correlators of zero dimension operators also those of (1,1) forms. The general derivation of the latter from the zero forms follows the general construction of réf. [30,31] 1 Vn'- = [ The holomorphic part of the lowest operator takes the form ML1 = 2/»V/o-<* (76) The computation of non-trivial correlators, and the relation with the (2,1) model, namely pure topological gravity will appear in a future publication. .8. SUMMARY AND DISCUSSION In the present work we have worked out the space of physical states of the 3 models for the case of A\ . Strictly speaking only for integer levels we could have used 517(2) gauged WZW model. For non integer k one has to adopt the SL('2, R) counterpart. The extraction of these states was done in two stages. At first the BRST cohomology on a Fock space based on a "free fields" parametrization was derived. In order to apply a method developed in réf. [13] for the Virasoro case, we had to apply a conjugate parametrization in the J and 25 / sectors. The second stage was the translation of physical stairs from the Fock space into the irreducible representation of tlio corresponding Kac-Moody algebras. For this procedure we implemented the structure of the singular vectors of the /I, algebra for arbitrary k. A "duality" between the singular vectors associated with the Yeruia module of \J, k- > and that of I — (J + 1), -k - 4 > is expressed h the embedding diagrams (Fig 2.3). This is analogus to the "duality" in the Virasoro algebra between \6.c > and |1 - 6. '26 - c>. The physical states are built on the highest state vacua ].]. I > where J — Jr / = /r-2/p,s or / = /_r_2f,,,s for every integer / with G = 1 - 2/ and —2/ respectively. The / values are those of points in the dual embbeding diagram (Fig 3). They are related to values of the points on the J diagram (Fig 2.) by / = -(J + 1). The ghost number G is in fact the "'distance" between the point on the / diagram and the top of this diagram. Using the entire space of physical states we constructed explicitly the characters of the -jj theory and verified tiie index interpretation of those chracters. Perhaps the most intriguing and interesting observation of this work is the intimate connection of the ^jj^j models at k = *• - 2 and the (p,q) minimal models coupled to gravity. The set of primary fields correspond to the \JTi3 > in the J sector. The fusion rules of the latter are discussed in the appendix and arguments are given in favor of the conjecture " that they are closed under tlieir O.P.E.. The primaries and their / descendants at all ghost numbers carry conformai dimensions which match, after twisting, those of the minimal models. Moreover, the partition function coincides with that of the latter provided that a particular value of the moduli of the flat G connection i.e u = q~l is chosen. This amounts to shift LO —» LO — 7"H- 5J''0'' . The conforma! dimension of the various fields with respect to this modified operator are shown to correspond to those of the minimal models with an addition of two "topologicar (1,0) systems. A complete isomorphism between the theories would be established when one compares positively the correlation function in the two theories. A relation between correlators in the A,' WZW model at level k = ^9 _ 2 and those of the (p, q) models have been worked out within the Hamiltonian reduction approach. We expect similar relations to hold in the topological version which we study in this paper. Alternatively, one would like to identify the TCFT algebra of the two theories. These topics are under current investigation. Notice that there is an apprent difference in the boundaries of the Kac table between the ^ models and the (p.q) models. While in the latter there is a double line embedding diagram and hence states at every ghost number for 1 < r < p — 1, 26 t.t 1 < * < g — 1. in our case this happens also lor » = and s = q whereas in the § models they exsit only for r = p. As was discussed in section G those differences do not affect the equivalence of the torus partition functions of the two types of models. Certainly the states on the boundary of the Kac table deserve further investigation. An explicit construction of the physical states was demonstrated for the case of k = O. The zero dimension operators which correspond to the states produced can further generate (1.1) forms which are in the BRST cohomology, V(u) = / and the Wj11J21J3 are the fusion algebra coefficients. It would be instructive to establish this result once the operators are explicitly constructed. Correlators in topological 2d gravity are known to obey recursion relations!*'301 We expect, therefore, similar recursion relations involving correlators defined on higher genera to be derivable in the context of the § models. Recently, such recursion realtions were derived using Ward identities related to the W00 symmetry of the ground ring of c — 1 " as well as using "contact relations"!"it will be very interesting to recast these results in the gauged WZW framework. In particular for c < I the Ward identities approach is missing in the Liouville approach. It is still not clear what will be the fate of its § model counterpartner. A long standing problem of TCFTs is Io write down a theory which is isomorphic to the c=l model. It is not difficult to realize that the k = —1 model has the right numerologics to play the role of this theory especially when the bosonization of eqn.(22) is chosen in both sectors. This model as well as some others are under current investigation. In fact it is easy to check using eqn.(57) that the level which corresponds to a given c of the matter sector is k = — ^j(Il + c :p -\/(c— l)(c- 25)). Not suprisingly a ^ model with real k has a forbidden domain which is the familiar range of c, 1 < c < 25. This work was entirely devoted to the A1 case. It is pretty clear that a great part of the results achieved here could be extended to other An_^ cases and maybe other algebras. Generalizing the twisting of T in the form T -» T + £], 0J 27 i.i t = k + N. Here \ve have assumed a ghost system of li'.v gravity. This is identical to the c of VV;vIJ4lmodels. We note that the relationship between >'t(.V) and H',v was established also via the Hamiltonian reduction."1 Explicit discussion of ihcsr models will appear in a future publication. Another obvious generalization is to tlnj case of super Lie algebras. In particular for G = SL(\,.\ — 1) we expect to obtain the Kazama- Suzuki sUiV-ifxtfti) matter, which has been shown to have super IV'jv algebra as its chiral algebra, coupled to super WN gravity. In this respect we also recall the work of réf. [35] where the embedding of § models into topological matter theories was obtained by twisting hermitian symmetric N = 2 supersymmetric coset models. Acknowledgements: We are indebted to M. Bershadsky, D. Levy. N. Marcus, Y. Oz and M. Spigelglass for many fruitful discussions. One of us N.S would like to thank the School of Physics in the University of Tel-Aviv for its kind hospitality. He also would like to thank M. Bauer, P. Di Francesco, I. Rostov, M. Petropoulos and J.-B. Zuber for discussions. REFERENCES 1. E. Witten, Comm. Math. Phys. 117 (19SS) 353. 2. D. Gross and A. A. Migdal Pliys. Rev. Lett. 64 (1990) 127; M. Douglas and S. Shenker A'«c/. Phys. B335 (1990) 635; E. Brezin and V. A. Kazakov, Phys. Left. 236B (1990) 144 3. F. David Mod. Phys. Lett.A3 (1988) 1651; J. Distler and H. Kawai Nucl. Phys. B321 (19S9) 509. 4. E. Witten, Nuci Phys. B340 (1990) 281. 5. M. Spigelglas and S. Yankielowicz " ^ Topological Field Theories by Coseting Gt TAUP-1934 ;"Fusion Rules As Amplitudes in G/G Theories," Technion PH- 35-90 6. E. Witten, "On Holomorphic Factorization of WZW and Coset Models" IASSNS-91-25. 7. K. Gawedzki and A. Kupianen , Phys. Lett. 215B (1988) 119, Nucl. Phys. B320 (1989)649. 8. D. Karabali and H. J. Schnitzer. Nucl. Phys. B329 (1990) 625. 9. D. Montano, J. Sonnenschein , Nucl. Phys. B324 (1989) 348, J. Sonnenschein Phys. flev. D42( 1990) 2080. 28 10. E. Witten Com;». MuIh. Pluj.-. 121 (UWW) 301. 11. M. Bershadsky and 1. Klrbanov AV/. Ph y». B360 (IWl) 00!). 12. B. Lian and G. Zukennan /•%•.. Lf//. 254B (HJ1Jl) 117. 13. P. Bouwknegt. J. McCarthy and K. Pilch Cern Preprint TH-6162/91. 14. E. Witten "Ground Ring of Two Dimensional String Theory" IASSNS-HEP 91/51 15. I. R. Klebanov, Princeton preprint PUPT-1302. 16. D. Kutazov, E. Martinec and N. Seiberg Princeton preprint PUPT-1293. 17. D. J. Gross, I. R. Klebanov and M. J. Newman Nud. Phys. B350 (1991) 621. 18. M. Bershadsky and H. Ooguri COIHW. .Math. Phys. 126 (19S9) 49. 19. V. G. Kac and D. A. Kazlulan Adi: MaIh 34 (1979) 79. 20. F. G. Malinkov, B.L. Feigin and D. B. FuksFunfcf. Anal. Prilozhen 20 No. 2 (1987) 25. 21. R. Bott L. W. Tu "Differential Forms in Algebric Topology", Springer-Verlag NY 1982. 22. Bernard and G, Felder Comm. Math. Phys. (1991) 145. 23. This case was also analyzed by M.Bershadsky [private communication] 24. A. Zamolodchikov, Tear. Mat. Phys. 65 (1985) 1205 V. Fateev and S. Lukyanov, J. Mod. Phys.A.3 (1988) 507. 25. See for example K. Schoutens, A. Servin and P. van Nieuwenhuizen, "Properties of Covariant VV Gravity" in proceeding of the June 1990 Trieste Conference on "Topological Methods in Field Theory", and reference therein. 26. M. Wakimoto, Comm. Math. Phys. 104 (1989) 605. 27. N. Marcus and Y. Oz "Discrete States of 2D String Theory in Polyakov's Light-Cone Gauge". Tau preprint TAUP-1962-92. 28. A. M. Polyakov, Mod. Phys. Lett.A.2 (1987) 893. 29. V. G. Knizhnik, A. M. Polyakov and A. B. Zamolodchikov Mod. Phys. Lett.A.3 (1988) 819. 30. R. Dijkgraaf, E. Witten, Ar«c/. Phys. B342 (1990) 486. 29 tes*»*- 31. R. Dijkgraaf. E. VeiTuule. and H. Yerlindc. .Vue/. P/ij/.-. B352 (1991) 59; "Notes On Topological String THeory And 2D Quantum Gravity." Princeton preprint PUPT-1217 (1990). 32. A. Polyakov and P. B. \Vigmann Pli i/s. Lttt. 131B (1983) 121. 33. P. Furlan. A. Ch. Gamlirv. R. Pannov and V. B. Pt-tkova CVrn preprint CERN-TH- 6289/91. 34. D. Gepner Comm. Matli. Pliy*. 141 (1990) 381. 35. D. Nemeschansky and N. P. Warner "Topological Matter, Integrable Models and Fusion Rings" USC-91/031. 36. "Setting Fusion Rules in Landau-Ginzburg Space1' Teclinion preprint PH-8-91 Phys. Lett to appear. 37. D. Nemeschansky and S. Yankielowicz "N=2 W algebras, Kasama-Suzuki Models and Drinfeld-Sokolov Reduction" USC-007-91. 38. B. Feigin and E. Frenkel Phys. Lett. 246B (1990) 75. 39. A. Rocha-Caridi and N. R. Wallach, Trans. Am. Math. Soc. 277 No.l (1983)133. 40. B. L. Feigin and P. B. Fuks "Representations of the Virasoro Algebra" Seminar on Supermanifolds 5 ed. D. Leit.es. 41. M. Bauer and N. Sochen, Phys. Lttt. 2V5B (1992)82 M. Bauer and N. Sochen, "Fusion and Singular Vectors in ^1 highest weight cyclic modules" hepth/9201079 APPENDIX Fusion rules We introduce a chiral primary field Following Zamolodchikov and Fateev ' we introduce an auxiliary parameter in order to have where K* (x) is a differential operator. 30 The correspondence fields-stales is given by limj_olim-_,iOJ(r..i-)in.t >= LM >• Here |n,i > is the vacuum which U characterized as a highest weight stale that is annihilated by the whole horizontal algebra. The Virasoro algebra and .-1, are related by the Sugawara construction u L/ » ~- Y/ ^,. Tl'f-. •'/ H" -I H Jin/ •• j.' 1i) •• Jn-m/+ jm;- •• ^ J. 1.-) • •/J-n-m j/m+ -\•/ In this formulation L-\ and JQ generate translations in z and in x respectively. Thus, we can write 0J(;.,r) = f-J°~+si-' 0J(O. O)C-*J o~-si-' It follows that i) (A.I) Let us look now at the short distance operator product expansion for these chiral primary fields. For this aim it is more convenient to write which is a consequence of eq. A.I with « = O. Imagine that we are interested only in the j sector in the fusion of jo and jj then (A.2) H=OfO = - II V'n.m are fixed by the requirement that the two sides of (A.2) transform in the same manner s under the Vir and A1 algebras. It is clear also that t''o.o(0'0) « proportional to Using the Sugawara construction we can write the following recursion relation. (nt + m(2j + 1 - m))^,m = (-j + J0 -r Ji + m + 1) *+<*» *>> +'2(j — jo — rn) +U -Jo+Jl -m + 1) We define Aj(n,m) = nt + m(2j + 1 - 771) and introduce an order (no, mo) > (n,m) if 31 no > n or if no = n and mo > »1. \Ve see that (A. 3) is a recursion relation since (?i,m) of the left hand side is bigger then all the pairs of indices in the right hand side. The i/>'s are well defined as long as Aj(n.m) ^ O. In the case where AJ(H, m) = O and m divides n the L. H. S of (A. 3) vanishes and we find in the R. H. S P(J, Ju, Ji) x (singular vector). The equation P(J' Jo- Ji) = O is then a fusion rule since P should vanish for the fusion to be possible. VVe use results and notations of ref.[41] reviewed briefly above as a framework for discus- sion of the fusion rules. The constraints on the possible operator content of a given theory are a direct consequence of the fact that we work in an irreducible representation. In these representations we put the singular vectors ( if they exist) to zero. (sing, vector) |j'i,<) = O. We use the relation between states and operators and multiply from the left by <^">(2,z) to get ^0U,. r Using. vector)(/>J1(0,0)|n,/} =0 we commute the Verma module operators to the left using eq.( A.I) and we act with the 0 1 operator which is the outcome of this manipulation on the fused ^ and P . Since ^J0 is a highest weight state different from zero its coefficient in the expansion should vanish. This coefficient is a polynomial in the j,jo and j\. Its vanishing condition is the fusion rule. We demonstrate this procedure by an example: take ji = ji,i,+ = O and a generic jo. The singular vector in the Verma module Vj1 , + is Uo.l) = J(TlJl1J,+,*) We have then the following equality We change x — » — x, : — » — c and we multiply from the left by ezj° +ci-' to get U=Om=- n from which the constraint (-J+ Jo + Jl W'o.o = O follows. In our case jj = O and we have the fusion rule j = jo. Another example is jo 32 and ji = ji, 2,+. The singular vector in the Yenua module Ij1 1 + is KM) = (Jj J+iJû-t JàJ-\ ~ tJ-iJo - ''-/I, HJi..'.+ . O The same analysis as above gives the fusion rule: ( j + Jo + Ji + 1 K-J + JO - Ji )(j - Ja -Ji) = O from which we conclude jr.s,+ 3 Jl..'.+ = jr.j-1,+ + jr,s+l.+ + jr.s,- for the case s > 1. For s = 1 we can apply this method to the singular vector in VJrt to get jr.!.+ O jl.2,+ = jr.2.+ It is difficult to give the general fusion rules since we don't have an explicit formula for the singular vectors. Nevertheless we have a conjecture for a special class of modules . For ji = —%t we conjecture the condition n from which we conclude the fusion rule Jr,..+ ®Jl.-+l.+ = £ 5((-l)' + (-n")j,..+l.+ + E 5((-l)' + (-l)B /=-n " J=-n+2 " (A.4) As we mentioned above this constraint is due to the singular vector in j\. The singular vector in jo may give further restrictions. For example it is easy to show that if JQ = jr,i,+ then jr.l,+ O jl.n,+ = jr.n,+ Using this result and the associativity of the OPE we finally obtain for rational t = p/q such that jr,i = jr.*,+ = jp-r.q-3+l.- s-l r-l 5-1 T-I |=0 Let us further remark that these fusion rules can be related to the Virasoro fusion rules for the minimal models through the hamiltonian reduction procedure. Recall that in the 33 hamiltonian reduction scheme j and Ii are related by '' = 7 JU + i)-j for hT,, and t rational we find that l>oth jr.s,+ and jr.s+i,- are solutions. If we apply hamilto- nian reduction to ( A.4) (that is, we replace each j by the appropriate A) we get the minimal model fusion rule. This gives us also a clue on the way our operator should be related to the models of minimal matter coupled to gravity. Since We expect that in the sense of insertions in correlation functions. 34 Cette thèse est consacrée à l'étude de la physique bidimensionuelle. L'outil principal est la théorie des champs conformes et ses généralisations aux algèbre* de Kac-Moody et aux algèbres W. La théorie des champs conformes est la théorie qui décrit les modèles bidi- mensionnels, ayant les symétries de translation, rotation et dilatation, à leur point critique. Les théories conformes étendues décrivent des modèles ayant une symétrie plus grande que la symétrie conforme. Après une revue des méthodes de la théorie conforme nous effectuons une étude détaillée de la forme des vecteurs singuliers dans l'algèbre sl(2) affine. Cette forme est très importante puisqu'elle nous permet, en principe, de calculer les fonctions de corrélation d'une manière non perturbative et de les résoudre complètement. Nous pour- suivons par l'étude de l'algèbre w classique où nous soulignons les relations remarquables entre les algèbres w classiques et les algèbres W quantiques. La méthode de bosonisa- tion est présentée ensuite et on l'utilise avec le résultat pour les vecteurs singuliers pour étudier le modèle topologique sl('2)/sl('2). Nous identifions le spectre de ce modèle et nous trouvons qu'il est identique à celui de la gravité quantique couplée aux modèles minimaux. Nous décrivons ensuite la bosonisation des fonctions de partition de différents modèles avec diverses conditions aux limites. Cette méthode fait partie d'un vaste programme de classi- fication des théories rationnelles en faisant un lien entre les théories conformes rationnelles et des modèles de spin intégrables. Nous décrivons ce programme et nous trouvons des relations intéressantes entre les poids de Boltzmann des différents modèles. Ces relations nous permettent de prouver l'intégrabilité de certains modèles par un calcul direct de leurs poids de Boltzmann.9r+I i p/> i -
. This is achieved via the cohomology of an operator Qj which acts on ^V1J the union of the Fock spaces that correspond to JT+2ip,s and J_r+2/p,» for every |22 integer /. It turns out 'that the relevant information is encoded in H°(fT,a ,Qj) and all other levels of the cohomology vanish. The cohomology is only in the J sector and not in the / sector just as there is no use of the cohomology of the Liouville sector in models of C < 1 matter coupled to gravity ' . Thus, the space of physical states of ghost number n is given by