Volume 235, number 3,4
PHYSICS LETTERSB
1 February 1990
C H E R N - S I M O N S t l O L O N O M I E S AND T I l E A P P E A R A N C E O F Q U A N T U M G R O U P S E. G U A D A G N I N I ~,b, M. M A R T E L L I N I c,,u and M. M I N T C H E V b,~ " Dipartimento di Fisica dell'Universit& di Pisa, L56100 Pisa, Italy b lstituto Nazionale di Fisica Nucleare, Sezione di Pisa, 1-56010 Pisa, Italy CERN, CH-1211 Geneva 23, Switzerland d Department of Mathematics, University of Geneva, C11-1211 Geneva, Switzerland Received 4 September 1989
The monodromy representation defined by the quantum holonomies and realized on the physical state space of the ChernSimons model is investigated. It is shown that this representation naturally extends to a representation of the braid group related to a particular solution of the quantum Yang-Baxter equation. The presence of a hidden quantum group symmetry is established.
The three-dimensional C h e r n - S i m o n s (CS) model is extensively studied nowadays in connection with conformal field theory ( C F T ) [ 1-3] and link invariants [ 1,4-6 ]. In this paper we investigate a related subject, namely, the representation of the braid group B. realized on the physical state space of the quantized CS theory. We first apply the canonical formalism to construct the wave functions of the states associated with non-abelian point-like sources (punctures) on a fixed time plane. Afterwards we report the expressions of the m o n o d r o m y matrices {B}, defined by the transport of a puncture along a closed path in the punctured plane R~_ ~. We show that the B-matrices give rise to a representation ~ of the fundamental groupx~ ( R 7,~ ~) which naturally extends to a representation q~ of B~. Recently, K o h n o [7] has proven that ¢0 in turn is equivalcnt to a well known B,-representation 0 related to a particular solution of the q u a n t u m Yang-Baxter equation. Therefore, the representation 0 provides a bridge between the CS model and q u a n t u m groups ( Q G ) . It turns out that the physical states associated with the fixed time cuts of Wilson line operators are QG-singlets. In this sense, the q u a n t u m group appears as a sort of hidden sym-
On leave from Dipartimento di Fisica dell'Universit'adi Milano, 1-20133Milan, Italy,and INFN, Sezioncdi Pavia, 1-27100 Pavia, Italy.
metry of the three-dimensional generally covariant CS field theory. We start by briefly recalling the construction of the physical states of the CS model in the presence of Wilson lines. The CS action in a ( 1 + 2)-decomposition o f R 3 reads ~
kf
S o s = 4~ .a
•
d3x(A~A~+A~)F'{2)'
(l)
a
wherc F 12 - 0~A 2 - 02A ~ - f °b"A~A ~ a n d f ~" arc the structure constants of a simple group G. The timec o m p o n e n t s Ag appear in ( 1 ) without time-derivatives and can be viewed as Lagrange multipliers enforcing the Gauss law, which in the absence of sources reads 1:~2 = 0 .
(2)
The quantization is performed in the usual way: wc impose the canonical c o m m u t a t i o n relations [A ~(x), A ~ ( y ) ] = it~ab ~ c)2 ( x - y ) ,
(3)
and select the physical states by means of the Gauss constraint. As is well known, ~ In our notations x"= (x °, x) = (x °, M) with i= 1, 2. We shall use also the two-dimensionalantisymmetric tensors % and ('J defined by (12=~2~= 1, (21= ~ 1 2 = - - 1.
0370-2693/90/$ 03.50 © ElsevierSciencePublishers B.V. ( North-Holland )
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G(A)=-~
PHYSICS LETTERS B
kf d2xF'~2(x)Aa(x),
(4)
1 February 1990
(iG(A)tP[y, R] )zj = i~V[y, R],,KR~A"(y).
(8)
where A~(x) has compact support, generates the residual local gauge transformations. In what follows we use the standard SchrBdinger representation of (3). It is convenient [8] to introduce the field variable U(x) defined by
Eq. (8) shows that ~V[y, R ] ts describes a state associated with a single puncture located in y and in the R-representation of G. The different vectors of the Rrepresentation on which G"(y) act, are labelled by thc second index J of ~U[y, R]w. Analogously, the wave function
T'~Ag(x) = - i U - ~(x)0, U(x),
~V[y, R]IJ= U-'(y, R),s~Po
(5)
where T ~ are the generators of the defining represcntation of G and their normalization is given by Tr T~Th= ½gab. Thc wave function 5%of the vacuum takes a simple form in terms of U(x) [ 8 ], namely To=cxp
-il~ n
d3y~ a~ D
× Tr( U- t O,~UU- I O#UU- IOrU )
)
--i-4-~ d2xTr(U-lOiUf-102 U) ,
(6)
where the integral of the Wess-Zumino three-form is performed on a three-dimensional manifold D whose boundary is R 2. The requirement of single valuedness of ~vo implies that k takes integer values (our normalization refers to the case G = SU (N)). One can easily veri~; that ~o satisfies the Gauss law (2), i.e. G(A) ~Vo=0. We would like to describe now the states corresponding to the case when there are some distinguished points (punctures) on the plane R 2 with a definite non-trivial reprcsentation of the local gauge group assigned to each puncture. Such states appear as intermediate states at fixed time t=to in computing the expectation values ( W ( L ) ) of Wilson line operators. The punctures arise as intersection points of the link L with the plane t=to [ 1 ]. The corresponding wave functions can be constructed as follows. Since the matrix valued variable U(x) takes values in G, we can define U(x, R) to be the representative of U(x) in the R- representation of G. Consider then the wave function tp[y, R]ts= U(y, R)wtPo,
(7)
where the indices I and Jlabel the matrix elements of U(y, R). By making use of eqs. (3), (4) one finds 276
(9)
also describes a puncture in y associated with the Rrepresentation of the group G. Differently from the wave function (7), however, the vectors of the Rrepresentation associated with G'~(y) are labelled in this case by the first index I of ~V[y, R]H. Clearly, multiplying 5% by several matrix elements of the variable U (and/or of its inverse U -* ) in different points and in different representations, one obtains a wave function of the type ~r't[Yl, RI
]IJ[Y2, R2 ]KL...
= U(yl, Rt )wU-l(y2, R2)KL...~0,
(10)
which describes several punctures (located in Yl, Y2, ...), associated with the representations R~, R2, ... of G respectively. Observe that the first lower index and the second upper index of the wave function (10) are free; thcy label the states of the charges "at infinity" and G(A ) does not act on them. We call these indices q-indices. As we shall see later, the q-indices give rise to the degrees of freedom on which the "quantum deformation" Gq of G is acting. In the wave functions corresponding to Wilson lines defined in a finite region, all q-indices are contracted. Let us consider indeed the wave function 7-'(F),
¢VH(F)=W,j(F)~Jo, obtained operator
(11)
by applying the quantum
holonomy
W(F)=Pexp(ifAT(x)R~cL'd)
(12)
F
on the vacuum. The path-ordered integral in eq. (12) is performed along an oriented non-intersecting smooth path F in R 2 connecting the point.v~ with Y2.
Volume 235, number 3,4
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1 February 1990 2
Since G"(x) generate gauge transformations, it follows that
( iG a(x) ~( F ) )tJ = - i~(x-yl )RC]K~KI(I') +ifi(x-y2 ) ltlIK( F ) R ~ .
( 13 )
Therefore, ~'(F) describes two punctures in y~ and
Y2 associated with the same representation R of G. Eq. ( 13 ) is exactly the operator Gauss law constraint in the present of (in this case two) punctures. As usual, we call physical the states satisfying the operator Gauss law. In deriving the explicit form of W(F), some care is needed. In order to preserve general covariance, a framing procedure has to be introduced [ 1,6]. For Wilson lines defined on a plane, one can adopt the so-called constant framing [9]. In constant framing one finds (see for details ref. [9] ) t/t/j ( F ) = ~ [ y , ,
R]IK[y2, R]Kj,
(14)
which shows that in wave functions corresponding to Wilson lines, the q-indices are indeed contracted. We want to study now the m o n o d r o m y properties of the physical states. A similar problem has been investigated by different methods in refs. [ 10,1 1 ]. Let us consider for example the wave function ~ associated with a four-puncture configuration, 7 t= 7J[yl, Ri
]IM[y2, RI ]MJ
X [Y3, R2]KN[y4, R2]NL"
(15)
The parallel transport, say, of the puncture Y2 along a loop C defines a m o n o d r o m y on the tbur-puncture physical states. This operation gives the state
, ~ = U-I (yl , Rl )IMU(y2, RI )MQ X [ P e x p ( i f da'iA~(x)R~)] C
-I ~2J
X U - 1(Y3, R2 ) KN U(y4, R2 )NI. tIlo-
( 16 )
For a clockwisc oriented C enclosing the puncture Y4 only, see fig. l, one gets [9] (in constant framing) ;A7J= B~,,% tP[yl, RI ]1M[y2, Rl ]ss X [Y3, R2]KN[Y4, R2](2L,
where B~,,q,, are the matrix clcments of
(17)
Fig. 1. Parallel transport of the puncture 2 along the loop C.
B=exp(ikR~®R~).
(18)
The matrix B acts on the tensor product of the two representations R~ and R2 o f t h c group G, whose vectors are labelled by the q-indices S and Q in the expression U(y2, R, )sjU(y4, R 2)QL. Moreover, all the indices of B are contracted with q-indices. Eqs. ( 17 ), (18) can easily be generalized to the case of several punctures. Apart from some phase ambiguities related to the framing, i.e. to the "self-interaction" of the quantum holonomy operator [8,9], the results (17), (18) are invariant under smooth deformations of the contour C on the punctured plane. For a loop enclosing several punctures the associated monodromy is the ordered product of B-matrices of the type (18). Summarising, the m o n o d r o m y properties of the physical states of the CS model are described by a set of matrices {B} of the type (18), which act on the qindices of the physical states. In order to study the representation of the braid group B,, it is sufficient to concentrate on {B}. It is useful to introduce now a few definitions. Let R ] = R 2 \ { Y t , Y2 . . . . . Yn} be the punctured plane, Xn = {y~,Y2..... y.: yp~yq for p ¢ q} and Y,, = X./S~, S,, being the permutation group of n elements. The fundamental groups P ~ = n , ( X . ) and B . = ~ t ( Y . ) are called the pure braid group and the Artin braid group respectively. Our strategy in deriving the representation of B~ is the following. First of all we show that an appropriate subset o # c {B} gcncrates a representation Q of 2 ~). Afterwards we introduce a flat connection ~, ( R ~_ A on R n--l, 2 whose holonomies define a rcpresentation Z o f n j (R 2~_ ~) which is equivalent to 0. The advantage of introducing A is that the representation Z can naturally be extended to a representation of P., 277
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PHYSICS LETTERS B
which in the case of cquivalcnt punctures gives rise to a representation q~of B,. We apply at this point a theorem of Kohno [7], which provides a precise relation between ~0and a quantum deformation of the Lie algebra of the group G. This final step cstablishes the relation between the CS model and quantum groups. In order to construct 0, we consider the state
T[xj, R.
]/IKI [Yl,
1 FebruaD' 1990
The relativc curvature reads 4n
R~,R qd(x-ys, ),
which shows that A(q) is a flat connection on R 2r t - - l . Therefore the holonomy
z(C)=Pexp(i~,4(q)idx ~)
Rl ]K,J,
(24)
(25)
C
X...iXn, Rn]I"g"[yn, Rn]K,,j,,
(19)
and concentrate on the set ,,E of B-matrices obtained by moving, say yq, along loops which do not enclose any of the punctures {x~, ..., x,}. ,,E gives a representation of x! (R2,_~). Denotc by {ep: p= 1.... , q - l , q + 1.... , n} the generators of n! (R~,_ ~); ep is associated with a clockwise oriented loop winding once around yp and is represented by
defines a representation Z of nq (R ,2- t ) . The point now is that 0 and Z are equivalent. Consider indeed the connection
A(q)l(X)=O, /~)2(x)=
4~z
T- ~
ha ; a 2 .2 ,! l RpRq~(X -yu)O(y,,-x ).
(26)
o(ep)=exp (i k R~,®R~) .
(20)
Let us turn now to the construction o f the representation X. We consider the punctured plane R ~,; to each puncture yp there is associated a finite dimensional linear space Hp on which a representation Rp of G is acting. The "induced" Gauss constraint reads [ 1 ] l:~,(x)=-
4a" ~
-
l~],iS(x-yp),
(21)
k p=l
where /~, acts on H =Ht®...®H, as I®...®R~,®...® 1. Eq. (21) can be interpreted as a constraint induced on H by the Gauss law (13). Clearly (21) has no c-number solutions, but one can easily show [ 9 ] that 2 /i,~(x)= ~ ( % + i d o) ~ /~g 0 / I n l x - y p l
is a matrix-valued solution of (21). When the punct u r c y q is moved along a closed contour C c R~_ ~, the connection m(q) which is relevant in computing the associated holonomy is obtained from eq. (22) by eliminating the self-interaction, i.e. ha
A(q)i(x ) - A (q),(x)Rq
278
2 p¢q
RpRq
0 ( C ) =ZO"(C) = £26- t (yu)z(C)-(2o (yq),
which establishes thc equivalence between Z and 0. Wc would like to compute now the holonomics associated with the simultaneous displacements of several punctures. In this case all the "interactions" between the point-like sources have to b c t a k c n into account; each puncture is moving in the potential (23) produced by the remaining n - l punctures. In complex coordinates cq. (23) can bc rewritten in the form 2 A(q)(z) d z = ~< ~
(23)
ff~pq
d In(z-z!,),
(28)
P¢ q
where.Qpq = Rp "~ R"~q. Moreover it is easy to vcrify that
[ff2pq,~2pr -t- ~qr ] = [ff~pq"3F~2pr,~2qr ] = 0~ p
ffln I x - y p I
(27)
(22)
p=l
-- ^a
It is easily seen that the curvature o f / l ~ ) equals l"~q) and that the relative h o l o n o m y z ~ ' ( C ) coincides with 0 ( C ) . One can show also that therc exists a "gauge transformation" -(20acting in H and interpolating between A~q) and A~q).hQ°Therefore
for distinct p, q, r a n d s ,
(29)
which are known as infinitesimal pure braid relations. Let us consider now the superposition
Volume 235, number 3,4
1 February 1990
PHYSICS LETTERS B
(ii) 0 commutes with the diagonal action of o n H and defines actually a Hecke algebra representation of B,, known as P i m s n e r - P o p a Temperley-Lieb representation. From point (i) of the theorem it follows that there exists an invertible linear mapping V: H ~ H such that SU(N)q
2 - k
~
g?oq d ln(zp-zq),
(30)
I~
which is precisely the Kohno connection. This connection appears also in the Knizhnik-Zamolodchikov equation [ 12] for the Wess-Zumino-Witten (WZW) model, The relation between the WZW level l and k is given by k = l+ c~, where c, is the quadratic Casimir operator in the adjoint representation. The one-form (30) represents a connection of the trivial vector bundle over X , = { ( z , .... , z,)eC2: ZpCZq if p~q} with fibre H. As a consequence ofeq. (29), co is a flat connection. Let :~'be an element of the fundamental group rt~ ( X , ) and let Z(?.) = p exp ( -
~ o9)
(31)
Y
be the associated holonomy. Eq. (31 ) generalizes eq. (25) to the case in which several punctures are moving simultaneously. By construction Z(7) acts on H and defines a linear representation of P~ in H. In the particular case when all the {Rp}-representations coincide, o9 is symmctric under permutations of the coordinatcs {zp} and induccs a connection over Y~. Therefore in this case 7.. extends actually to a representation ~0of the braid group B. on H. This representation, called the monodromy reprcsentation of B~, has been investigated by Kohno [ 7 ] who has shown that ~0 is related to the B,-representation defined by the universal R-matrix [ 13 ] associated with the Lie algebra of G. More precisely, for the fundamental representation ofSU (N) the following theorcm holds: (i) Considering 1/k as a parameter, the braid group representation ~0is equivalent to
O(a,)=q"[l®...®T(q)®...®i ],
~o= V-'OV.
(34)
Since ~pis an extension of Z, eq. (34) relates Z with 0 as well. The diagonal action of SU(N)q on H is defined by the standard co-product in thc quantum algebra. From eq. (34) it is clear that ~0commutes with the V-conjugate action of the SU (N)q representation commuting with 0. For generic q, one can show [ 14 ] that O(ai) generate actually the centralizer of the quantum group. Finally, the above theorem holds [ 7,14 ] also for the fundamental rcpresentations of all non-exceptional simple Lie groups and, with slight modifications, for an arbitrary irreducible finite dimensional representation of SU (2). The theorem of Kohno establishcs the role of quantum groups in the CS theory. As is well known any link can be obtained as the closure of a suitable braid element. The trefoil T is, for example, the closure o f f , B 3 shown in fig. 2. The intermediate physical states contributing to ( W ( T ) ) are determined by intersecting T with the planes t =const. At t = 0 onc can choose a standard physical state "h 3J2~J3 12 I3 ~[xl ]KIJI[x2]K2J2[x3]K3J3 q~o=6s,
× [y, ],,L, [Y2 b~2[Y3 ]i~.3-
(35)
I,: 1
(32)
where ~ = ( I - N ) / 2 N , T(q) acts on the ith and ( i + 1 )th factors of H by
t=0
J
(a) +(l-q)
Z E,~®E#/s,
ct>[I
and q = exp ( - i2~z/k);
(33)
(b) Fig. 2. The trefoil as a closure of the braid aeB3.
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PHYSICS LETTERS B
We arc i n t e r e s t e d in the state ~ at t = 1. In order to use the 0 representation, it is necessary to p e r f o r m a change o f basis in the q-indices. T w o steps are involved in this operation. First, one has to t-elate the B-matrices with the Z r e p r e s e n t a t i o n ( 3 1 ) . In this case, the i n t e r t w i n i n g opcrator.Q is the n a t u r a l extension in X , o f the operator -(2odefined previously. T h e n we have to use cq. ( 3 4 ) to c o n n e c t cpwith 0. T h e comb i n e d i n t e r t w i n i n g o p e r a t o r '~7/=~. V only d e p e n d s on the d e f o r m a t i o n p a r a m e t e r q. We i n t r o d u c e then the wave f u n c t i o n s ~.IKIK2Ks;QIQ2Q3 ( j]/-- I ) OlO2O?~ ,,I/Ilt213 LI L2L3;SIS2S3 ~ JIJ2J3 t¢ SI,S'233
x ~[x, ]~'J' [x2 ] ~2J: [x3 ] ~ 3 X [Yl ]fiLl [Y2 ]hi.2 [Y3 ]131.3-
(36)
The indices Q~ Q2Q3 a n d S~$2S3, which originate from the q-indices, represent q u a n t u m group ( Q G ) indices in the s t a n d a r d basis a n d eq. ( 3 5 ) can be rewritten as (i)o _ .~SI .~32 )~S3 ILIKIK2K3; ~I(22Q3 - - V Q I 0(,_92 ' " a 3 - - I , I1.2£ 3;,S ISz,";3 "
(37
)
T h e physical state ~ l at t = 1 is o b t a i n e d from 4o by replacing the d-factors in ( 3 7 ) by thc m o n o d r o m y
O(a), i.e. ~t
~), ~. ~s,s~s~ tuK, x~x3:0,0~o~
U~ I.X ]QIQ2() 3 X L I L 2 L 3 ; S I S 2 S 3 .
(38)
O b s e r v e now that 0 acts on the Q G - i n d i c c s a n d that these indices a p p e a r c o n t r a c t e d in the physical states. Thereforc, any i n v c r t i b l e linear t r a n s f o r m a t i o n 5 acting on the Q G - i n d i c e s (the lower a n d u p p e r Q G indices t r a n s f o r m by ~ a n d Y - 1 rcspcctively) a n d c o m m u t i n g with 0 represents a h i d d e n symmetr3; o f the q u a n t i z e d CS theory. The theorem o f K o h n o states that Gum {.~}. T h e Q G - i n d i c c s a p p e a r quitc naturally in the above construction a n d are associated with the group indices o f the charges "at i n f i n i t y " . T h e q u a n t u m group s y m m e t r y has a peculiar realization: any physical state o b t a i n e d by c u t t i n g a W i l s o n line ( d e f i n c d in a finite r e g i o n ) is a Gq-singlet. Q u a n t u m group symmetries play an i m p o r t a n t role in integrablc systems [ 13,15 ] a n d arc expected to cmcrge also in certain c o n f o r m a l field theories [ 16 ]. The CS theory represents a r e m a r k a b l e e x a m p l e o f a c o n t i n u o u s t h r e e - d i m e n s i o n a l generally c o v a r i a n t field theorY, a d m i t t i n g a h i d d e n q u a n t u m group symmetry.
280
1 Februa~' 1990
It is a pleasure to t h a n k A.P. B a l a c h a n d r a n , D. Gross, T. K o h n o a n d R. Stora for discussions.
References [1] E. Witten, Commun. Math. Phys. 121 (1989) 351; Nucl. Phys. B 322 (1989) 629. [2] G. Moore and N. Seiberg, Phys. Left. B 220 (1989) 422; S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the ChernSimons-Witten theory, preprint IASSNS-HEP-89/20. [3]J.M.F. Labastida and A.V. Ramallo, Phys. Lett. B 227 (1989) 92; J. FriShlieh and C. King, Two-dimensional conformal field theory and three-dimensional topology, preprint ETH-TH/ 89-9; The Chern-Simons theory and knot polynomials, p reprint ETH-TH / 89-10; M. Bos and V.P. Nair, Coherent state quantization of the Chern-Simons theory, preprint CU-TP-432 ( 1989 ). [4] M.F. Atiyah, New invariants of three- and four-dimensional manifolds, in: The mathematical heritage of Hermann Weyl, Proc. Symp. Pure Math. Vol. 48, ed. R. Wells (American Mathematical Society, Providence, RI, 1988 ). [5] P. Cona-Ramusino, E. Guadagnini, M. Martellini and M. Mintchev, Quantum field theory and link invariants, preprint CERN-TH.5277/89, Nucl. Phys. B, to appear. [6] E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett. B 228 (1989) 489; Wilson lines in Chern-Simons theory, and link invariants, preprint CERN-TH.5420/89, Nucl. Phys. B, to appear. [ 7 ] T. Kohno, Ann. Inst. Fourier, Grenoble 37 (1987) 139. [ 8 ] G.V. Dunne, R. Jackiw and C.A. Trugenberger, Ann. Phys. 149(1989) 197. [ 9 ] E. Guadagnini, M. Martellini and M. Mintchev, Braids and quantum group symmet~' in Chern-Simons theor2:',preprint CERN-TH-5573/89. [ 10] A.P. Balachandran, M. Bourdeau and S. Jo, The non-abelian Chern-Simons term with sources and exotic source statistics, Syracuse preprinl SU-4228-402 ( 1989 ); J.M.E Labastida and A.V. Ramallo, Phys. Lett. B 228 (1989) 214. [ 11 ] J.E. Nelson and T. Rcgge, Homotopy groups and 2+ 1 dimensional quantum gravity, prepritn DFTT 10/89; S. Carlip, Exact quantum scattering in 2+1 dimensional gravity, preprint IASSNS-HEP-89/4. [ 12 ] V.G. Knizhnik and A.B. Zamolochikov, Nucl. Phys. B 247 (1984) 83. [ 13] V.G. Drinfel'd, Soviet Math. Dokl. 32 (1985) 254; L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan, Quantization of Lie groups and Lie algebras, preprint LOMI E- 14-87; M. Kimbo, Commun. Math. Phys. 102 (1986) 537. [14] N.Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I and II, preprints LOMI E-4-87 and E-I 7-87.
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[1510. Babelon, Phys. Left. B 215 (i988) 523; V. Pasquier and H. Saleur, Common structures between finite systems and conformal field theories through quantum groups, preprint SPhT/89-031 ; H.J. De Vega, Yang-Baxter algebras, conformal invariant models and quantum groups, prcprint PAR LPTHE 88-46. [ 16 ] L. Alvarez-Gaum6, C. Gomez and G. Sierra, Phys. Left. B 220 (1989) 142; Duality and quantum groups, preprint CERN-TH.5369/89;
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G. Moore and N. Rcshetikhin, a Comment on quantum group symmetry in conformal field theory, preprint IASSNSHEP-78/18; J-L-Gcrvais, thc Quantum group structure of 2D gravity and minimal models, preprint LPTENS 89/14.
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