The quantum symmetry of rational conformal field theories

Nuclear Physics B352 (1991) 791-828 North-Holland

E

César GÔMEZ

~ ~

X

Départemer:t de Physique Tlaéorigue, Université de Geraèa~e, Ge~aera, Scvitzerlarad Germân SIERRA liastituto de Estratctt~ra de la Materia, CSIC, Serrarao 119, Madrid, Spaira Received 30 April 1990

The quantum group symmetry of the c < 1 Rational Conformal Field Theory. in its Coulomb gas version, is formulated in terms of a new type of screened vertex operators, which define the representation spaces of a quantum group Q. The conformal properties of these operators show a deep interplay between the quantum group Q and the Virasoro algebra. T):e R-matrix, the comultiplication rules and the quantum Clebsch-Gordan coefficients of Q are obtained using contour deformation techniques. Finally, the relation between the chiral vertex operators and the quantum Clebsch-Gordan coefficients is shown.

1. ntr®ciucti®n During the last two years quantum groups have become an ubiquitous mathematical concept in the framework of conformal field theories [ 1-11 ]. In spite of this fact we are still missing the very physical meaning of this mathematical structure, in particular how the quantum group is acting as a hidden, but nevertheless relevant, symmetry of conformal field theories. The quantum group in the context of conformal field theories has appeared as a by-product of the duality structure that it is encoded in the polynomial equations . Most of the work relating quantum groups and CFT deals with the centralizer of the quantum group rather than with the quantum group itself. In order to understand the quantum group as a symmetry the first thing we need to do is to characterize the objects on which the quantum group transformations Permanent address : Departamento de Fisica, Universidad de Salamanca, Spain. * * Partially supported by the Swiss National Science Foundation . OS50-3213/90/$03 .SQ ©199() - Clsevier Science Publishers ß.V. (North-Holland)

i'.

Giirrce~" G. Sic~rrcc / Qccarctccrr: s~~rrcrrcetry of rcctiorcal CFTs

are defined and next to prove that the conformal blocks and therefore the physical amplitudes are invariant `vith respect to these symmetry transformations . In a previous work 12] we have constructed an explicit realization of the quantum groups SL.1~?~~r p on the thermal subalgebras of the Coulomb gas systems describing [ 13]. This construction was done using a modified version of the the c ~ 1 C screened vertex operators of ref. 1 ]. The main difference was the choice of the contours, ®v ich are in our case defined around a cut that goes from the point ®v ere the vertex operators are inserted, to infinity. These screened vertex opera tors re ~n in some formal sense andelstam's operators of gauge theories . The first good indication that these screened vertex operators are the conformal field theoretical objects on which the quantum group is acting is their braiding commutation relations. They can be obtained using ordinary contour deformation techniques and they define a quantum -matrix. n a certain sense the quantum -n~atril appears as the effect of the infinite contours `ve attach to the vertex operators . Tl~e generators of the algebra and the co ultiplication can directly be o twined using again the same kind of techniques . In particular, the generators ~ ~ ~ of the quantum group SI~(~~~ ~ are represented as contour creation-annihilation operators, important aspect of our construction is the interesting interplay `~~e find het
C. Gômez.

Cam. Sierra / Quantum symmetry ofrational CFTs

793

2. The screened ve~iex ®pest®rs The Coulomb gas version of the c < 1 Conformal Field Theories is defined in terms of a massless scalar field ~ in the presence of a background charge 2av located at infinity . The corresponding energy-momentum tensor defines on Fock modules .t« representations of the Virasoro algebra with central extension c = 1 - 24an. The highest weight vector of .t« is given by the vertex operator V«( z ) _ :exp(i a~( z )): which has U(1) charge equal to a. For a = a ~, where 2a~ = a + + a _ and a + a _ _ -1, we obtain the screening operators J ~( z ) = V« +( z) which have conformal weight equal to one. The braiding-commutation relations satisfied by the Coulomb gas vertex operators are «, V«,(z~)V«,(z2) =e2~~ «~V«~(zz)V«,(z~)-

(2.1)

Associated with each vertex operator V«( z ) we define a family of screened vertex operators as follows :

ea .r-(z) _ ~ dtl CI

J+(t~) . . .

J

cr .

dt,+

J+(tr+)J

SD

dw 1 J_(w~) . . .

where the contours are the ones depicted in fig. 1. Notice that the integrals around these contours will vanish if the screening charges have no relative monodromy with respect to the vertex operator V«(z ). More precisely, the integrals in eq. (2.2) are defined in the analyticity region of the integand obtained by performing the OPE between the screening charges and the vertex operator V«( z ). The screened vertex operator e" , r _ is a bifocal operator which is defined at the points

Fig. 1 . Contours which define the screened vertex operator e`+ .r_(z) .

794

C. G®'t:e" G. Sioema / Qata®atatna syreanaetry of ratioaaa! CFTs

Fig. 2 . ~n the right-hand side it is shown a "lighthouse °' vertex operator é;T ,,_(z), which is proporüonal to the screened vertex c'` +. ~_( z ) .

and ~ (the dependence of the point at ~ will be considered implicit from now on). ` e shall associate to each vertex operator !~"( z ) a "representation space" 7 ~" generated by the screened vertex operators ei , r _( z ). It is our aim to show that is a representation space of a quantum group, which requires, among another things, a complete l:no`vledge of its dimension. The space ?' ~" will be finite dimensional provided there exist positive integers fiâ and n~ such that e"~ ,,1~ = 0. n this case it is easy to see from eq. (2.2) that the charge a is given by Kac's formula, _ ;(1 -~~~ )~~+ -,(1 °

-ft~ )a_=''-,(1 -fi)a ++''-,(1 - m)a_= all .lll~

(2 .3)

with ~~ _ ~a~, ~~a = j~~. The dimension of ?' ~" is therefore equal to nâ nIX = nm . n a rational theory, where a ~. = p'/p, the range of the positive integers n and ~~a, `v ich define a finite-dimensional space ~ ~" .~.»>, is restricted to the intervals 1 < j~ < p and 1 < fn < p', This is due to the path ordering of the screening operators J + and J_ in eq. (2.2). In order to see this phenomenon in more detail we will write the screened vertex operators e;~+ ,,._ in a different way* (see fig. 2), 1 - e-lr~ia"~ ~r ) where q+= ez-'`"± and (2 dti J+(t~) . . . f~ dw,._ J_(w,.-)1~"( z) . ea+ ,r_(z) _ ~~ x x

.5)

* Strictly speaking eq . (2 .4) is only true if the contributions of the integrals :n eq . (2 .2) around the point z are negligible (see appendix A for a more detailed discussion) .

C'. Gômez, G. Sierra / Quantum symmetry of rational CFTs

79 5

All the factors in eq. (2.4) come from the braiding between the screening operators J + among themselves as well as with the vertex Va( z ) . It is convenient for later discussions to path order the integrals entering the definition of é" , r _( z ) . Denoting the "path-ordering" operator by the symbol P (i.e. I t, I > I t 2 I > . . . > I z I) we obtain ( 2 .6]

where the q-numbers I [x] I + and I[ x ] I - are defined as

I+

x

1 -q+

I

x

1 -q-

We adopt the convention that I [0] I ±! = 1 . Eq. (2.6) is the q-analogue of the path ordering of an integral operator. We see from eqs. (2.4) and (2.6) that e" , r- is different from zero provided that 1 < r + < n « - 1, with 1 < n = n â < p and 1 < m = n~ < p'. 1Vloreover, in a IZCF'I' both Va = V, , and V,~~~_ a = Vp _ . p-- ,, with 1 < n < p - 1 and 1 < m < p' - 1, give rise to finite-dimensional spaces 7" "" and ~ "-"~~-~ respectively, which are not in general isomorphic. From the previous analysis on the dimensions of the spaces ~ "" we already get some indications on the desired connection with quantum groups. To simplify the discussion we will concentrate on the case of the thermal operators V_ , (or V, _ ,) which form a closed subalgebra of the c < 1 theory . For a rational theory (i.e . seen, produces the «+ = p'lp), q + is a pth root of unity which, as we have qp. = 1 and 1 < m restriction 1 < n < p (for the thermal subalgebra «,. , we have "ai .~~ would contain "null vectors", p'). For values of n greater than p, the space 7 in fact the operator ea+ =p . ~ already vanishes since I [ p] I + = 0 . This is reminiscent of the quantum group result ( X ± )p = 0 for a quantum deformation parameter a pth root of unity. This simple result strongly indicates that the quantum group generators X ± should be identified with contour creation and annihilation operators acting on the spaces of screened vertex operators . The idea that quantum group generators must be closely related to screening operators has also been discussed in refs. [22, 23]. Under this assumption the representation theory of the spaces 7" ~" reduces to the representation theory of quantum groups. For instance, the condition 1 < n < p corresponds to irreducible representations of SU(2)q (qn = 1_ ) with no null-vectors . All these irreps have positive q-dimension ( = I [ n] I ), except the irrep n = p which is excluded from the list of integrable irreps of the c < 1 theory, hence there must exist a close relationship between these two facts.

C'. G®r»e~, Cî . Sierra / Qetarttt~nt synarrretry of rational CFTs

~yi,

Indeed we know that the Virasoro characters of the representations (p, m) or o not mix under modular transformations with the remaining irreps, (t~, p') precisely because ~ [ kp ] ~ + = I [ k'p`] I - = 0 for k and k' integers. At genus zero the representations (p, nt) and (n, p') can be ruled out on the basis of unitarily . In the Coulomb gas the scalar product is defined using both the vertex operator V"( z) and its "dual" V," ( ~-"( z ). For the rase n = p the vertex operator V, «~o _ n j3. ~ z ) is equal to the operator V~, ~,. _ ,( z ), whose corresponding representation space ~ ~'"~' _""~~ ~ can be eonsistently identified with zero (recall that ar~t ). Thus if `ve want to have a positive well-defined scalar product dim ~ and no null-vectors we are forced to truncate the theory in the usual way, i.e. 1 ~'~ < p - 1 and 1 ~ ~~1 ~ p® - 1 . The results we obtain concerning the existence of null vectors and unitarity conditions are one-to-one related with the ones we know for the representation theon of quantum groups for the cases where q is a root of unity. The rest of the paper is devoted to prove that the screened vertex operators (2.2) under the restrictions (2.~) fo irreps of a quantum group which contains as subgroups t e quantum groups SU(2),~ } (q .~ = e-' °° °"} ) which are associated to the thermal subalgebras . ~n outcome of this construction will be the interpretation of X+ as contour creation and annihilation operators . To finish this section we will make two additional comments. The first one concerns Felder's definition of the RST charge [ 14]. It is easy to prove that if ~~" is finite dimensional then one can define two currents J}( z ) which acting of V"( t ) are single-valued as z goes around t. These two currents are defined as follows : .`~r=~ sa=

®

~

J±~z) =

J±(z)J±(t~) . . .J + (t jt n -~)dtl . . .dt~ -~ ~

(2 .8)

with the contours of integration the ones depicted in fig. 3. Notice that Felder's RST charge Q corresponds to the current J+(z) for a = a ,,n and nâ = n. The second comment refers to the transformation law of the screened vertex operators under the action of the Virasoro operators . Using the commutator

Fig. 3. Contours which define the BRST current J t ( z ).

C. G®lnez, G. Sierra / Quantum symmetry of rational CFTs

797

between the Virasoro generator L n and the screening operators J + ,

we get Lner ,r_( z) - er ,r_( L n v«( Z)) - lim t n t-~~

+i

J+ ( t) I ~ r + ~ +

ea~~««{ + (1 -

q+

where e ; , , _( L  V«( z )) denotes the screened vertex operator associated to the vertex L nV«(z ). The boundary terms in this equation appear as a consequence of the cut, and their presence imply that the screened vertex operators cannot in general be interpreted as conformal fields in the usual sense. VVe postpone to later chapters a more detailed discussion on this issue.

3. The quantum R-mata~ix If the vector spaces 7Z" are irreps of some quantum group Q associated to the c < 1 CFT we must be able to find, by simply using the braiding commutation relations of the operators e~ , r- ( z), the quantum R-matrix of Q as a map, independent on the coordinates, going from ~Z"~ ® ~Za-' into ~~"- ® z~;', ~-«, Z,

The braiding commutation relations of the screened vertex operators e ~, r_( z) can directly be computed by studying the effect of the contours on the braiding commutation relations satisfied by the Coulomb gas vertex operators, e2Tri«,«2 V«,( Z I) Va2( Z2) V«2(zz)V«,(zi)

(3 .2)

and they take the general form ®er2(z2) er~(Z1) 1 2

ri " rz

er?(Z2) 2

®er'~(ZI)Rr'~r'?rr 1 1 2 1 2

~

3.

where r = (r + , r-) labels the vector in the representation a associated to the vertex operator V".

C'. GÛrrtt.~®, G, Sier°rct / Qttctratttrrt swrtratctry of ratiorta! CFTs

Fig . ~ . Illa.tstration of the braiding rule (Rl ) .

or the actual computation of the `g~`a=-matrix it is more convenient to use the "lip thouse representation" ~ï o the screened vertex operators ë~ defined in eq. (~.=f). indeed, what we shall compute is a matrixx Î2`~i~= defined by the braiding of the "lighthouse'' operators ~~ as follows :

The relation between `~ ~`~-' and a a~- follows automatically from eq. (2.4). We s all denote by ttr ~tC~ ) the family of contours which enter the definition of ~,`~,(~,~zë) . he computation of `~~`~= involves the following three contour manipulations performed in consecutive order: ( 1): raiding of the operators il~ of z, ) and V~,,( z,) accompanied with a displaceent o t e contours C, attached to the vertex V~ I( z, ), which now become the contours C, of fig. 4. t R?): Splitting of the contours C t into pieces joining the family C 2 and pieces going from z, to z 1 , that we shall call C,, (see fig. 5).

R2

Fig . 5 . Illustration of the splitting rule (R 2),

C G®n :e6, G. Sierra / Quantum symmetry ofrational CFTs

799

Fig . 6. Illustration of the opening rule (R3). (R3): Opening of the contours CZt into pieces joining the families C ~ and C,

(see fig. 6). Each of these operations produces braiding factors which are all collected into the matrix R"~"-' . In this way we find a diagrammatic interpretation of quantum R-matrices . As an example let us compute the effect of a braiding on é;,®( z 1 ) é~®( z2). Following the rule (R1) we get (see fig. 7) éiô( z~) ® ëôô( z2) = J dtJ+ ( t)Va'( zi)V«~( z2) c, e2;,ia,"2J

c,

dtJ+(t)V«,(z2)Va,(z~) ~

(3 .5)

We now split C 1 into a contour C~ ending at z2 and a contour C21 going from z 2 to z~ (rule (R2), see fig. 8), J^c,

dtJ+(t)Va2(z2)V«,(z~) _ ,dtJ+(t)Va2(zz)V«,(z~) ~c2 ~~~« +«2 /' dt Va2( z2 )J +e + ( t)V«,( z~ ) ~ (3 .6) Jcz,

R1

c~ Z1

é~~ a2

Z2

Fig . 7. Application of (R1) to

ê~Y~(z,)

®é~i,~,(z~).

C. iîr~aaaez, G. Sierra / Qa~aaate~na synaa'aetry of rat~oaaal CFTs

;;C~

C1

Z1

Z2

R2

+ e2Tticx~a2

C2 Z2

Fig. 8. Application of (R2) to

Z1

C21 Z2

Zl

Jc, dtJ+(t)V~,(z,)V~1(z,).

he phase factor e-'°''" +`~= in e . (3.b) comes from the braiding of J+ and Va,( 2 necessary to perform the integral of J } from z, to t Finally, we open the contour ,~ according to ( 3) (see fig. 9), z

),

z .

tV«~(`,)J+(t)V«,(zt) _

«_

-e-'Tà«t

IdtJ~(t)Va~(z~)V«,(z') c:

tttng everything together we get

+e~eâ8 «'«,(~

-e`~TIQ' .}CY,)ei

lD(ZG)

®G1110(Z1) ~

.

We leave to appendix the calculation of Ray"= for an arbitrary number of cöntours. ~n order to make contact with more familiar R-matrices we shall restrict ourselves to the thermal operators e"+;~, where a, , = 2 (1 - n )cr + . Defining the "spin" j as j = -', (n - 1) we see that the space ?" has dimension 2j + l. We can similarly define the angular momenta ire of the state e"+;~ as m =j - r +, which satisfies m = j, . . . , -j . I-faving done these identifications it is easy to check that the matrix elements of R"=~ ~«-'~ ~ given in ey. (3.8) coincide with those of the ctuantu R-matrix of StJ(2)r+ evaluated in the representation 2 ® '-, . Denoting -an.

-~2TLicc+q2

C21 Z2

Z1

~

C2 Z2

Z1

Fig . 9. Application of (R3) to J~~~ dtV~~(z,)J+(t)Va~(zi) .

C. (8®mez, G. Sierra / Quantum symmetry of rational CFTs

8U1

e"+~ô( z ) by en(z ), eq. (3 .3) becomes e'~ m~( z 1~ ~

® e'2 m 2( z 2~

-

m~,m ;

(3.9)

e'2, , m Z ( z 2) ® ej', rn ' ( z 1) R'''2 m'm,rn~m 2~

where R .i~j2 turns out to be the quantum R-matrix of SU(2)q{ in the representation j l ® j2 (see appendix B). We observe that the effect of the contours transform the braiding-commutation relations of the Coulomb gas into a well-defined quantum R-matrix acting on the tensor product of the representation spaces ~~". One should not confuse this R-matrix with the braiding matrix for conformal blocks or chiral vertices computed for example in ref. [24]. It is interesting to compare the pictorical result obtained in fig. 6 with the explicit definition of the quantum R-matrix for SU(2)9 [25, 26), R = qH®Hl4

(1 n~o

where ~n~9 = ( q ~zl2

q-l'n

~ni9i

q

n -n(n-i)/4 nH/4 q (X~)

_ q-nl2)/(q~li _

q

-i

ll)

-nH~~(

X-)R

,

(3 .10)

and

Comparing now (3.10) with fig. 6 shows that the net effect of X~ and X- is, re~pClalVCly i+G decrease and increase the number of contours used in the definition of e t( z ). . The quantum group In this section we will proceed to the explicit construction of the generators of the quantum group Q of the c < 1 CFT. We first define contour creation and contour annihilation operators and then by using contour manipulation techniques we find the algebraic relations they satisfy . A result of this construction is that each thermal subalgebra is associated to the quantum groups SU(2)q +, which are both contained in the more general quantum group Q. This quantum group Q is acting as a symmetry of the c < 1 CFT. The quantum group transformations are defined as operators acting on the screened vertices and the conformal blocks appear as the invariants of this quantum symmetry. 4.1 . CONTOUR CREATION OPERATORS

We define two contour creation operators F+ and F- acting on the space ~~" as follows : F + ea .r_(z)

=

Jc

dtJ+(t)ea .r_(Z)

~

(4.1)

~lï?

C. Gôrreez, G. Sierra / Qeear:tear: syrnrreetry ofrational CFTs

Fig. 1~. Action of the generators F + on a generic screened vertex .

~;~' ere the contour C surrounds the whole screened vertex e~+ , r _( z ) (see fig. 10). Thus in the basis cTa . ~, of .~ ~~ we have (4-2) f we consider now the ordinary operator product e~ + , ~_( z, )e~ . ra( z,) we obtain a basis of t e space ~_~ ~ ~ _~ where we can define, similarly to (4.1), the action of the operators + which is usually called the co ultiplication, namely

+(e~~ .r_(

`t)~~

.~-" ( z,))

_

,c

dt.~ + ( t)ea ,r_( zl)eß .r~ ( zz) ,

(4-3)

`where the contour C surrounds the operator e" , r_( z t )e~ ,,. . ( z,) (see fig. 11). efo ing the contour ~C into the union C, U C,, with C; a contour surrounding

~C

+e 2~

i0L~o

Fig. 11 . Comultiplication rule of F +. The phase factor e`'~`"±" emerges from the braiding of J + and the vertex located at the point z i .

C°. Ciômez, G. Sierra / Quantum symmetry of rational CFTs

~®3

the vertex at the point z;, we obtain « ®F+(er+, r_( 21)eß ,r'_(

_

ß

Z2)) - F±(er ,r_( Z1))er~,r_( Z2) + e2si«+(« +r+«+ +r_ «_) « (4 .4)

The phase factor e2T`«±(«+r+«++r_«_~ in eq. (4.4) comes from the braiding between the screening operators J + and the vertex e~ .r_ necessary to define r_ e action of F + on e~ ,r, . The comultiplication law of F+_ follows easily from (4.4) and it takes the form 4F+ =

(4.5)

where the operators k + keep track of the braiding between the screening operators and the vertices, k±(e

r

+ " r_(Z))

= exp 2Tlia + (,TUc fi~ ea+, r-(Z) = e2 ..i«

er~ .r_ ( .Z) . ( 4.6)

Similarly the comultiplication law of k + ( = exp(iTa +~~ a~) is given by 4k + =k

(4.7)

We shall call B - the algebra generated by the operators k +_ and F +_, satisfying the relations

which follow from the definitions of F+ and k +. B - is the c < 1 analogue of the Bores 5ubalgebra Urb _ of the quantized universal enveloping algebra U~(SU(2)) with t = q -' ~a [27, 28]. The comultiplication law ® : B - ~ B - ® B -, given in eqs. (4.5) and (4.7), is co-associative and preserves the algebraic relations (4.8). 1Vloreover, the bialgebra B - becomes a Hopf algebra if, in addition to ®, we introduce an algebra homomorphism c : B - -~ C and an algebra anti-homomorphism y: B - ~ B - by the following formulae : s(k + ) =1,

E(F + ) =

0, (4 .9)

~(Z.1

C. G®))ic'z, G. Sie)°)a / Qlla))!dl)n

sy)nn)etry of ratào))a! CFTs

Jl 11

Fig, 1?, +~oyassociati~eïtS' property' of the comultiplication of an element a of the quantum group defined bs' a contour integral (i.e. a = ~c X ).

The antipode ~ is essentially a path reversing operator, while the co-unit ~ is a contour milling mapping (i.e. ~(~~ .~ ) = 0 for any X ). All these identifications give way to a geometrical interpretation of the defining relations of a opf algebra A (see figs. 1~-1 ~; id)~(a) _ (id ~~~ ( id

y)=~(~) =f'~(~ id) :~( a ) _ ( id

id)~(a) =£(a)1, ~)%1(a) =a~

(4.10)

with any element in the Hopf algebra A (see refs. [25-30] for literature on the sub]ect). To further proceed let us introduce once more the SU(2) labels for the description of the screened vertices. Recalling that a = -j +« +- j - a _ and r +_ _ j } - ~~a }, we can express eq. (4.6), or rather its squared root, as j + ,j

- ~n:± i~rjn + j + ,j

id ®Y

Fig. 13. Geometrical interpretation of )n(id ® y)®(a) _ ß(a) 1 = 0, where a = ~~ X.

G

(;~8r8ez,

G. Sierra / Quantum symmetry ofrational CFTs

805

U id E

Fig. 14. Geometrical interpretation of (e ® id)d(a) _ (id

as

~)d(a) = a, where a = ~cX.

These equations motivate the definition of the Cartan generators H + and H_ i+ ,i- = + i+,jH +e~nt,m2m-e,n+,,n- ,

(4.12)

so that k + and H + become related via

In the thermal case (a, 1 ), the operator k + reduces to q+ H -~a . Moreover, the comultiplication (4.5) implies that F + could be identified with the operator Î+q+H +~4 ~ with f+ the usual lowering operator of SU(2)q+ In fact, working on the thermal subalgebra V, 1 one finds that k + and f+ (or F+ ) generate the Borel subalgebra U, + b~+ ~ of the Hopf algebra U~ + (SU(2)) ( t + = q+' ~~); to be distinguished from the Borel subalgebra U~+ b~+ ~, which we shall consider later (see ref. [28] for definitions and conventions). In the general case we can define f+_ as F+ k + 1, which makes the comultiplication (4.5) more balanced,

as well as the antipode y(f +) q+1/2f+

(4 .15)

4.2. CONTOUR ANNIHILATION OPERATORS

To define the contour annihilation operators we shall make use of the transformation law of the screQned vertex operators under the Virasoro algebra. This law

Sll~'

C. Grintce, G. Simact j Qtrctrittrrtt sytatmetry of mttiorta! CFTs

w`~s given in eq. (2.10) and can also be written in the following form :

r~~here ~( z ) is the vector field that generates the eonformal transformation, and E ~ are tai°o Bontour destroying operators defined as -~ 1 -e

tcacr ,

1 _ e ~-8«~ _

r_ - 1

~-

- i

ne can checl: from eqs. (x.16) and (x .17) that (i) a` satisfies the Virasoro algebra; (ii) E + commutes ~~'it E _; (iii) E + commute with Virasoro . ~t a first glance eq. (x.16) seems to be telling us that the screened vertex operators are not primary fields, not even conformal fields . To clarify this issue we shall `vrite eq. (=x.16) in another form . (Jsing the fact that the vertex V r ( z ) is a primar\~ field, namely (4 .18) where .~~ is the conformal weight of V~, it can be shown that eq. (4.16) is equivalent to

(4 .19) >-Ience the field e;+,,._( z, fix) "almost" behaves as a bilocal primary field of conformal weight ®~ at the point z and conformal weight zero at infinity. The usual conformal transformation of a primary field suffers in this way a "covariantization" in the presence of internal quantum group numbers . We believe that this is a general feature of the Coulomb gas version c~f CFTs, namely that the local transformations generated by the chiral algebra operators get covariantized

C. Gômez, G. Sierra / Quantum symmetry ofrational CFTs

807

when acting on quantum group objects . Let us return to the properties of the contour annihilation operators E _+ . Their comultiplication law follows from that of the Virasoro operators, ß ~E(er + ,r_( Z1)er+,r~ ( Z2)) -

(S~e«

,r_(z~))eß ,r, (ZZ) +ea ,r_(Z~)(S~eß ,,. (z2)) .

(4.20

Combining eqs. (4.16) and (4.20) we obtain r_( Z1)eß ,r'_( Z2)) a ( +er .r_(Z1)eß er + ,r_l s EVa(Z1))eß ,r'_(Z2) .r~ (SFVß(~2)) -[(1

-q+')~(°~), +(°°) ®E t T(1 ---9-')~(°°)J-(o~) ®E-~

where

Since we have the explicit action of F+ and E + acting on the space ~ ~a, we can find out the following relations : EtF± -q±F±E ±=

1 -k 4 1 - q+

+~ -+~- F+E + =O~

(4 .23)

which are preserved by the comultiplication laws. It is an interesting fact that, as a consequence of eq. (4.23), the contour creation operators do not commute with the Virasoro algebra, (4.24) Indeed eq. (4.24) is another way to state eq . (4.23). VVe have finally arrived at an algebra Q generated by k ~_, E + and F + which summarizes the quantized symmetries of the space ~~~" . Q is a Hopf algebra which contains two Borel subalgebras : B + (generated by k + and F + ) and B - (generated by k + and F + ). If we restrict ~~~,i:selves to the thermal subalgebra a,,, then the operators k + , .~+ and F + generate the quantum group U, (SU(2)) . If, instead of k +, we consider the operator H ~. of eq. (4.12) then we find the quantum group U,, + (SU(2)), with

~()~

C: Gcirrac" G . Sierra / Qacarata`rra s~~rrtrraetrti~ of rational CFTs

dr ~. ~ 2 ;; â cx + [2~], which is usually denoted by SU(2)~ + IViutatis mutandis the tfer al Subalgebra ~~, , lead us to the quantum group SU(2)~ The Coulomb gas representation of the quantum group generators we have presented in this section is quite asymmetrical : the two Morel subalgebras play very different roles. 13 } commutes with the Virasoro algebra and it defines the lift of conformal transformations on the quantum group representation spaces. ®n the other hand, ® does not commute with Virasoro and does not participate into the ch~~nge of quantum numbers induced by conformal transformations [see eq. (4.16)]. utator between E ~. This asymmetry is, in some sense, what the non-trivial co and ~ is measuring . t seems difficult to understand how the quantum group Q, that we have found, when it is manifestly non-commuting plays the role of a symmetry of the e < I C `~=ith the Virasoro algebra and in that sense with the chiral algebra . The solution of this puzzle is quite simple once we realize that the conformal blocks of the Cl~T can be identified with the invariant tensors o t e corresponding quantum group hen it is easy to prove, using eqs. 04.16) and (4.24), that the quantum group Q commutes with the chiral algebra on the space of invariant tensors, namely ( 4.25) for any and any element in the space of i-wariants, Inv (?' ."' ® . . . ® r"~«"~"), defined relatively to the quantum group action. In Sect. ~ we shall study the identification of the conformal blocks and the invariant tensors of the quantum group.

ira ve ex o erators a qua tu C e sc coe ïcients orda 9. In a CFT the holomorphic and antiholomorphic pieces of correlations and partition functions are computed in terms of the so-called chiral vertex operators k (CVOs) [1, 31]. A chiral vertex operator z~, z~ is defined as an intertwines operator between three irreducible representations i, j and k of the chiral algebra, namely k

where ~f~, ~f~ and .~f'k are the corresponding representation spaces located at the points z,, zz and ~, respectively . The number of CVOs of the type i ~` ~ is

C. Gc~rnez, G. Serra / Quantum symmetry of rational CFTs

given by the fusion rule integer [1,31]

N~ .

809

The CVOs satisfy the intertwining equation

(5 .2) where « E ~;, ß E ~~ and pk(O) is the representation of the nth mode of an operator O(z ) of the chiral algebra acting on ~k and ( p®® p'(ß: is the corresponding operator acting on ~~ ® ~j. The CVOs satisfy also the equation c~f motion ~ . ~ :(® ))

d dz

k

(5 .3)

Eq. (5.2) simply means that the contour integral around the point at x, which defines the operator p~(O ), can be smoothly deformed through the CVO, after which it becomes a contour integral around the points z ~ and zz yielding the chiral algebra comultiplication ( p` ® p'(~~,,~,(O )). These two equations completely characterize the CVOs. For a quantum group the intertwining operator between three representations cc1 , «, and «; is given by the quantum CG matrix Kâ;"-', which is defined whenever ~ ~"; c 7 ~"~ ® ~ ~"~. If et is a basis of ~ ~" then the quantum CG coefficients are defined by the equation [26] r~

~ r, , r=

r,r,a~

r,

(5 .4)

r,

The intertwining property of Kâ;"= is reflected by the equation

with a E Q. Let us suppose now that the RCFT that we want to consider admits a Coulomb gas version, as the c < 1 theory, then each irrep of the chiral algebra gives rise to a Fock space representation « (or 2a~~ - «for c < 1). In this case the CVOs (a~ az a,) of the Coulomb gas can be built out using the screening operators . For the c < 1 models they take the form given by Felder [14], «~

«2 `', ``~ ~al

® Va, d t; J+ ( t; )

r

dsj J_(sj )va ,( z 2 ) , (5 .6)

~1t1

C.

~®Fiez,

G. Sie,°ra / Qtta®rtcurt syr~t»ietry~ ofrcitio®:al CFTs

Fig. lj. D~otsenl o® ateev choice of the cont®urs which define a CVO.

Fig. 1~. Felder's choice of the contours which define a CVO.

where the contours Cz and S~ can be chosen in various ways, yielding results which onl`= differ by proportionality factors tree figs . 15-17). The charges «,, «, and «~ in eq. t~ .b) are related to i. } and ~° _ by « t + «, + r- +« + + ,~ _« _ _ «~ . This equation has a unique solution for r° ~ and r _ whenever « + and « _ are not relatively com ensurables. ~Ve shad suppose this from now on. L.et us rata to our quantum group construction based on the Coulomb gas. We have previously shown that each representation « of the c < 1 Coulomb gas yields a representation ~ ~`~ of a quantum group Q . This question has also been posed by the authors of ref. [S], whose work has strongly motivated us; we shall obtain however different results to theirs. In this line of thought it is natural to ask about the relation between the CVOs and the quantum CG coefficients . We shall

Fig. 17. Double contour choice for the definition of a CVO .

C. G®fr~ez, G. Sierra / Quantum symmetry ofrational CFTs

8~ l

propose and demonstrate the following fundamental relation between those two objects: «~ 2 z, .~,

(V«, ® V«,) _ ~ K~~â~e~; (V«,( z' ))er; (V«,( zz)) - (5 .7) r,,r,

One goes from the left-hand side of this equation to the right-hand side by means of a contour deformation of the contours entering into the definition of the CVO. The contour conservation law implies (5 .8)

° depend of course on the choice of the The values of the coefficients K«~«-' r,r~«~ contours which define the CVO. The most natural choice is to take double contours integrals of the screening operators around the vertices located at the points z, and z 2 . In this case the Virasoro operators goes smoothly through the contours and the CVO-intertwining property is satisfied . i.e t us consider the simple case of a CVO defined with only one contour. Taking into account the phase factors produced along the double contour path one gets az

_

~_

(V«~ ~ V«,)

e4r~ia + -e4:~i«+«,)~ z, (itV«~(Z1) .J+(t)V«-(Z'2) «')(1 (1 _ z,

"

(

If we now open the contour f~~= as in eq. (3.7) we obtain al + a2 + a+ a2

(V« ~ ® V«`)

- _e2Tri«+«,(1 _ e4;" i«+«,)ei y(zl)eô ()(Z2) +(1 _ eaTi«+ «')ec%'o(Zi)ei ô(Z2)

(5 .10)

which has the form proposed in eq. (5 .7). The general formula of the coefficients Kr ~ «~ is computed in appendix D. The important feature of eq. (5.7) is that it has a quantum group interpretation . Notice that the vector (« «; «,)(V«~ ® V«= ) E .~«~ can be regarded, in the quantum group sense, as an element of `7 ~"~ ® 7 ~"=. In view of eq . (5 .4) we are lead to identify the K-coefficients of eq. (5 .7) with quantum

C, Gn»ze~, G. Sierra / Qraritzura st~r~a®~ietry of ratiof :a! CFTs

1~

C cï6

to

coefficients. Under this identification it turns out that the state `w )( VcY ' ~ Vcr, ) of .}« ; is nothing but a highest-weight vector of the quancY® space .% ~~ ~, namely

«~

«~

his explains the index ~ of the -coefficient in eq . (5 .7). To prove (5 .11) one checks the highest-weight condition. «î

j Q3,(Y~~~IY,CY :( d~r,r,(Y;

E

?

)e¢Y8

ri

(5 .12)

w 1011 can be easily checked out for the previous example (see appendix D). The highest-`weight condition (x.12) has yet another important consequence, namely that the interttivining property of the C`TOs follows from eq. (5 .12) and the equation of otion (4.21) ; indeed «;

«, __ a .

_

~ ~~ a

r,, r: -

«

(1 -~+i)~(x) .~ +(°°) L9E+ +(1 -q=')~(°°)J -(°°) 4E-leP,(Va~)e~-,'(v«,)] «;

«, _

~~(V«, ® V«=) .

(5 .13)

This equation exhibits the deep interplay between CVOs and quantum CG coefficients . After all these considerations it is justified to generalize eq. (5 .7) in the following manner: «~

r, , r,

Ka,a,r~ ( ( )) r,r,a~ e",(V r, a, Z l ))ea~(V r~ a~ Z2

(5 .14)

C. G®f~aez, G. Sierra / Quantum symmetry ofrational CFTs

r2

813

al a2r3 rl 2 a3

Fig . 18. Pictorical representation of the Clebsch-Gordan decomposition .

which displays the full content of the interplay between CV®s and quantum CGs (see fig. 18). An immediate consequence of this interplay is the close relationship between the fusion rules of the c < 1 CFT and the decomposition rules of the quantum group Q.

5.1. FUSION RULES AND CONFORMAL BLOCKS

Until now we have shown that starting with the Coulomb gas representation of the CVOs we can obtain, using contour deformation, the quantum Clebsch-Gordan coefficients which intertwine between different representations of the quantum group. In appendix D we present the explicit computation for the case of generic a + obtaining the correct quantum CG coefficients for q not a root of unity, which are not zero if the "classical" BPZ fusion rules are satisfied,

~n,-n2~+1
C. Gnaaae" G. Sierra / Quarttun: synametry ofration:al CFTs

~l~

For the rational case a + = p'/p the truncation is obtained, in the Coulomb gas approach, by taking into account the isomorphism .t"n .m = ~~"p-n . p'-!n and by requiring SL{2, ~) invariance for the three-point function . Notice that in the BRST description the previous isomorphism is crucial for deriving the fusion rules. In our approach we shall define the quantum CG coefficients, for q a root of unit, using contour deformation of the Coulomb gas CVOs operators and being consistent and with SL(2, C) invariance. The result with the isomorphism .~" = .~ we obtain, as it should be expected, are the quantum Clebsch-Gordan coefficients that define the decomposition into irreps of the modified tensor product (~~"~ «=)' that was considered in refs. [b, 22, 32~ and that it is defined as hiebest weight vectors E ( ?' ~" 1®

?' -"-' )'}

The corresponding decomposition rules reproduce the correct c < 1 RCFT

fusion algebra,

'31~ - 3t,~

+ 1 < ;t~
ft,-

1,2p-n~-n,- 1),

~'a~ - ~n,~ -~- 1
(5 .17)

It is important to stress here that the approach we have presented to the problem of quantum group symmetries in conformal field theory is a constructive one. In this sense it should be noticed that the quantum Clebsch-Gordan coefficients are defined as the intertwining operators that reproduce the contour deformation manipulations . In connection with the truncation of the fusion algebra what we obtain is that only the quantum Clebsch-Gordan coefficients defined for the restricted tensor product (5 .16) can be obtained by contour deformation of SL{2, ~)-invariant Coulomb gas vertex operators that are consistent with the isomorphism a ~ 2«~~ - a. In this sense we can say that there exists a complete correspondence between the fusion algebra of RCPT and the Brauer-V~leyl decomposition of quantum groups. In sect . 4 we identified the conformal blocks with the invariant tensors of the quantum group. Let us now see this identification in more detail . A conformal block with 3 external legs .~"~«-'«; can be computed in terms of the CV® (" "~ "`) as follows: "~"~ " ; ( Z 1 ~ Z 2 ) - ~ V2"p -"~( ~ ~

I ( CY 1

IX3 Q2

z~, z2

ir" ~ ® T~"`

,

5 .18

so that .~ "~"~a~ is not zero whenever ~~z"~~ c (~/~"~ ~ ~~" z ® ~~2"~ " -" ;)' . It is easy tû see that this condition is equivalent to looking for the identity representation id

C G~fnez, G. Sierra / Quantum symmetry ofrational CFTs

815

inside (~"~ ~ ß/'a 2 ® ~/'°`;)'. More generally, there exist a conformai block with 1V external legs ~~ a ßa2 ~ ~ - aN for each solution of tale equation (5.19) namely for each invariant tensor associated to the quantum group.

6. Conclusions In this paper we have shown that the Coulomb gas representation of the c < 1 theory provides a natural framework to define an explicit realization of the underlying quantum group structure . Our methods can be easily generalized to any RCFT having a Coulomb gas version . From a technical point of view our main result is to translate most of the quantum group objects and operations into a geometrical language which could be named as "contour algebra" . We have also clarified the non-trivial interplay between the chirai algebra and the quantum group. To get a better understanding of the quantum group transformations would require a deeper comprehension of the screened vertex operators we have introduced in order to define the quantum group representation spaces. In connection with this problem it seems natural to conjecture that the screened vertex operators are revealing the ultraviolet limit of the solitonic structure of some restricted sine-Gordon theories, the ones that flow in that limit to minimal CFT models [21, 33]. This line of thought can also explain why the Coulomb gas representation appears as the most natural framework to look for the quantum symmetries. In fact the restricted sine-Gordon models teat flow to RCFs are defined by introducing a background charge in close analogy with the way the Coulomb gas representation of minimal models is built. We expect to discuss some of these issues elsewhere . We would like to thank Luis Alvarez-Gaumé for discussions .

~ppendix A CONTOURS

The contours which enter into the definition of the screened vertex operator z ) are a multicontour version of the contour which appears in Hankel's

ea.,._(

C. GC)))IC'" ~.

SIE'l'ill ~ QIlQlltllllt Sl'/IIIilE'tly' Of l'QtlOlllll

CFTS

C

Fig. 19 . Hankers contour which defines the I=function.

formula of the 1=function, (

r) ®

dt( -t)`-t ~ -2~ sin(~-z ) c ta

e-r

dtt~ -t e -r ,

(A .1)

`~~here C is depicted in fig. 19 and z ~ . ~ 1, ± 2, . . . . Comparing eq. (A.1) with a e-r is the analogue of r_ eq. ('. ). tive observe that the integral lc dt( - t )` `while { z ) is the analogue of ~~~ , ~ = The function sin T z in eq. (A.1) like the factors in eq. (?.=1) re ect the existence of a cut. in writing eq. (2.=1) we could have considered the contribution of the integral on a small circle of radius ~ around the point z, thus e.g. (see fig. 20)

ea .

otivever these extra terms will play no role in our considerations since at the end of the computations we shall always return to the screened vertices eY, r _ ïn the representation of eq. (2.2). Another remark about the vertices e;~+ , r _ is that in order to define consistently the spaces 1 ~" we have considered that the screening operators J + commute with the screening operators J_ inside the integrals (2.2). This is made possible from the fact that J+ and J_ have no relative braiding, i.e. e 2 ~`"+"- = 1 .

Fig. 20 . Pictorical representation of eq . (A .2).

C. G®mez, G. Sierra / Quantum symmetry of rational CFTs

817

Appendix R-CALCULUS

We show in this appendix how to compute the matrix R"'"-' of eq. (3.3). As we said in sect. 3 it is easier to compute the matrix R« , a-' which gives the braiding commutation relations of the "lighthouse" operators é; , r- ( z ). Let us first attack the case where all the contours are of the J + type. To simplify our formulae we shall skip the subindex + whenever no confusion is produced. Thus e.g.. if we denote e « , o by e a, eq. (2.4) then reads e 4:~éa« t

e«( z)_

gz)éac( r z)

where q = q + and r = r+. In the sequel the charge a of the vertex T~«( z ) is not restricted to be of the thermal kind (i.e. «, 1 ). Similarly, eq. (3.3) becomes e «,( zl )

~ e «~( z2 ) _

~ e «~( z2 ) ~ e «,( z1 )R « «Tr'r_

r~, r ;

.

(B.2)

The relation between R"~"= and R"~"-' follows from eqs. (B.1) and (B.2), Ra,a, rjr~,r,r,

_ e 4;~iaia + gx,)TTr,-1 _ e4;~i«, «T g x, ) TTr,-1 l1 x 0 ( 1 x,=0 (1 i a -' ~ R ar,r~,r,r, (B.3 e4Tia,a+ e4: .ia,a+gY,)~Sz-1 _ .~ Ô(1 l lrYW q'r)_ ,,-0 (1

The application of the rules (R1)-(R3) of sect. 3 reduces the problem of finding R"~"2 to a combinatorial one. To show more clearly the combinatorial nature of this problem we shall calculate R"~"- in the most simple case where all the braiding factors are trivial (i.e. e 2 ~`"~"Z = e2~`«,«+- e2~`a -« +- e 2~`«+ = 1), It is obvious that the correct result should be

R "rjr;, ~"~ r,r, =

rjr, r ;r,

(B .4)

In the realistic case where the braiding factors are not trivial, we shall find a q-analogue version of the procedure used to derive eq. (B.4). We start by applying the rules (R1) and (R2) to éa~ ® é"z. The result is best expressed in graphical terms

S1~

C. (Jvr::ez, G. Sie'°ra / Qccacatcu~i sl~rttratetrv of rational CFTs

as

r2 r,

r

(13 .5)

The combinatorial number

~ ) in eq. (B .5) gives the number of choices of I

contours which, according to the splitting rule (IZ2), become contours from the point ~ to the point . application of the opening rule (I~3) in eq. (B.5) yields

1

~=o

(

_ 1)!_~

d

(B .6)

where the sign factor ( -1)!-`~ is caused by the change of orientation of the l -- d lines which return to the point 2 after the opening. Introducing eq. (B .6) into eq . (B.5) and changing the order of the summands we obtain

( - 1) 1-c! rl 1 ~=o t=~

1 d

(B .7)

C°. Gômez, G. Sierra / Quantum symmetry of rational CFTs

819

Finally, changing variables in eq. (B.7), we arrive at

rl n=o n

~=o

r2+ n

1) v( n (. lv)

(B .s)

The desired result (B.4) simply follows from the identity

Let us come back to the case of real interest . The object whose braiding we want to analyze is é"~( z 1 ) ® é,",-( z, ), or more explicitly dt ;J+(t~)V,(z~)

z, -

ds;J+(S;)V~~(z~)-

_ x ~-i

(8 .10)

After the braiding of Va~( z 1 ) and Va,(z2) and the corresponding deformation of the contours C, we get (see fig. 4) e2~rta,(a,+r,a +) r Z, 'x

r

~=i

dS;J+(S;)V~2(Z2)Va~(Z1) . (B .II

d t~ J+ ( t~ )

Splitting the contours one finds r,

Y1

i=o

l

e2~~i(a,+la+xa,+r,a +)

dt! J+(t~)Va~(za)J where

~'

q

z, z~

. k=1

du k

J+(uk)V~,(zi) ~

(B .12)

is the standard Gauss polynomial,

l

_ (l_

~[r~]~ ~[1~~9~[r~

t

_l]iv

(B .13)

C. Cic~aa:ez, G. Sierra / Qaearataura syaaar~aetry of rational CFTs

~~tl

laic term arises in eq. ( 12) due to braidings of the screening operators among t e selves. 1\ie~t is the opening of the integrals J_;', which gives

~t=n d~~J+(t;)f~,(z ;)

ds~J~.(s~)V«~(z,) .

(B.14)

the novel feature of (13.14), apart from the expected modifications, is the term which is caused by the inversion of the path ordering of the 1- d screening operators who fly bacl: to the point 2. sing t .14) in ( .1~) and following the same steps as in the trivial case we arrive at att_tt~t-ti-,)

ra

~_1) t9

~ e`

Ya=(~

' -~Y( z~) ~

er~+rY( z~)

~-

(B .15)

T' e identity `ve need to simplify (13.15) is (see ref. [27]) n-i

~t

- 1)t~l

t~=o

v

J

Ic~ Zt'~ tt~

(v-l)

v=o

(1 - zqv)

(B .16)

.

the final answer is given by

_

r, -rt , r: + tt ; ri , r,

_I~

n

~ ~q

e ~rri(«~ +r i « + -ntx+x«~+r,«+)

which implies, using eq. (13.3),

r~-n,r,+n:n,r, _ _

n

- I ~ Y,

I J ~

e 2~rri(a i +ri«+

-ncr+)(az+rza +)

( v=r i -rt

1 ,~ e 4Tri «+«,

q v) .

(B

18

C Gr~jr~ez, G. Sierra / Quantum symmetry of rational CFTs

821

A particular interesting case of eq. (B.18) is to consider its values for the thermal operators a 1 = -j,a + and a 2 = -j2 a + . Eq. (8.18) yields the matrix Rjtj~ of eq. (3.9), R .it .i2

-

1 - q - 1 rt q(rn'+n)rn,

J1

-ml

LJ1 -f- ri11 $Yl,f 9

1

1

which, as we shall see in sect. 4, coincides with the R-matrix of SU(2)q. The techniques we have developed so far extend easily to the situation where R"~"-' involves contours of type J+ and J-. The task greatly simplifies due to the fact that the J +contours commute with the J_ contours . The final result can be written as rt -n, rt -n~, r; +n, r~ -rt~ ; rj r; rj r~ -

e

Rrj -r:, r; +rt, rj r= ( q+ ) Rr l -n~, r, -~rt`; r' r, ( q- )

~

( ®2®)

where R"~"-'(q + ) are given by eq. (B.18), with q replaced by q +. The factor e - "' `"t"-' in eq. (B.20) takes care of a double counting of the braiding between V" and V",, already contained in R"t"'-(q We have said tiat for the thermal subalgebras (i.e. an, , or a ~, »t ) the matrices R"~"= reduce to those of SU(2)q +. We can ask now what is the situation for generic charges a, »t . Let us use again the SU(2) labels for the characterization of the vertices e"+, r , namely E

+) .

cr = - j a + - j cx,t , r + =j +_ m+

r-

m

,

( 8 .21)

with 2 j ± E 7l . In this notation eq. (B.20) becomes Riy

t ._

i, ~_

m~ +r :,mi +rt',»t ; -n,»

_

t,

-rt' ;m~ ,m~_ ,rn;rn ; (q+~

- e -2rri(m; +n)nt~ e -2rri(n:~ +n')m ; xR'~'~

_

_

__(

q_),

q-)

RJi!~

(

B .22)

where R'~+ '~ (q + ) are the SU(2)~ + matrices given in eq. (B .19). The interesting feature of eq. (B.22) is the appearance of simple but non-trivial phase factors, which mix quantum numbers of SU(2),~ + and SU(2),~_ This implies that the quantum group Q of a c < 1 CFT is not equal, in general, to the tensor

C. Gvrrtcr, G. Sier°r°a / Qtiar:ttrrrt syrrtrr:etn~ of rational CFTs

product SL3(2)$, + ~ SU(2)(~_ We postpone to sect. 4 a detailed study of the structure of this quantum group. ~ e a`'e not considered in the derivation of the R-matrix the possible extra fauns in er ~ , ,. _( ` ) coming from integrals around the small circles near the point z, as in eq. (~R.2). If one takes care of these terms it is easy to show that all the terms combine to ~=field the correct R-matrix for the braiding of screening vertex operators c' r

. ®.

°r ~ Q ~

.

~~ ~~~~~ Q

In this appendix we summarize the properties concerning the algebra Q. Q is ~`enerattd hti= k }, E } and F~ satisfying the following relations:

-~ +F+E + =

1 -k + t ,

1 -q±

E ±F -+ F+E +

(C .8), (C .9)

`vhic can also be written in more compact form as

= '«,a, , = e +~~«,«; F~k ; (C.11), k,F (C .12) E~kr , klE~ eE;E~ = E~E; , F

;F~ = F~F; ,

1 -k4 where i = +, - .

(C .13), (C .14)

C. G®inez, G. Sierra / Quantum symmetry of rational CFTs

823

Except for the minus sign in eqs. (C.3) and (C.5) the rest of the relations are those of Ut +(SU(2)) ~ Ut -(SU(2)) with t + = q +' ~4. The comultiplication ® of the generators of Q is given by

4 F; = F; and the co-unit

E

(C.18)

and the antipode are

£(k~) = 1,

£(E~) =0,

y(k~) =k~ ' ~

y(E,) _

E(F;) =0, - k~ ZE~,

(C.19)-(C.21)

__,t ~ 2Fg . (c.22)-(c.24) y(F,) _

Q with 4, y and ~ is a Hopf algebra. The representation of Q on the space ~ ~" of screened vertex operators e~ , r- is given ~y ± r+ ,r_

C.25

r~,r_ ~

F_ -er , r_- er ,

F+ea ea C + 1, r_ ~ , r_-

1 - e 4~ri"" +

r

r_ + 1

( C .26 ,

~

+ -1

q+

.27 (C.28)

r_-1 1 - e -~Tt""_ q-

Using SU(2) labels

(ea, r- = em+j,n-) eqs.

k ± ern+,rn - = q± F+en +~rn-

-

e

e

r +~ l,rn

(C.25)-(C.29) become (C.30)

ern+,rn_ -

~

F- ein ~rn -- e n +1rn - - 1

(C.31), (C .32)

E+em+,'rn -= ~[J+-m + ]~+~J + +m ++ 1 ] +en+ +l,rn -,

(C .33)

E-e n+Jrn -- ~~ .i -- m - ] ~ - ~ .i - + m - + 1]-e

( C .34 )

t+ j ,n -

+~ ~

We have kept in eqs. (C.33) and (C.34) the distinction between the quantum _ ~numbers x ]I and [x ]. Althou g h the Y are prop ortional (i .e . I~ x ]i~~ - q ~ [ x )~ _ braiding contour game . Indeed, the [x]t,-~), they remember their origin in the

C. Gdraicz, G. Sie)°ra %

~illRldtill)3 Sj'J)dl)IE'tly' of ratio®)al CFTs

quantum numbers ~ x j ~ come from braiding screening operators among themselves, while the quantum numbers [x] come from braiding screening operators and vertices I~Y . e may ask nv`v whether is a quasi-triangular Hopf algebra, i .e. if there exists an operator R such that

the per utation map (x y) ® ~' x. with ' ® cr ~ and his is essentially true, owever there are some subtleties that one must is the analogue of the opf algebra U,(SU(2)) (generated consider, ecall that y , and ~) and in fact reduces to it on each thermal subalgebra . The trouble wit ,(SU(~)) iS that the universal R-matrix which interpolates between ® and ®' does not belong to U,(SU(2)) but it belongs to Ut,(SU(2)) (generated by , E and ). t is only the to q ~~~~ in R that causes problems, although it can be i ple anted via an inner automorphism on both U~,(SU(2))®- and U~(SU(2))®-' (see ref® [~8 for more details). Taking into account the automorphism ~, it is possible to write the eq. (C.3~) inside the algebra U,(S (2))® ' as

`where

The lesson we learn from this discussion is that the universal R-matrix we are looking for does not live i but rather in an algebra Q,, generated by H+, E + and E}. t is easy to see that Qi,, aS an algebra (i .e . commutation relations and generators), is isomorphic to the tensor product U,, +(SU(2)) ® U,, -(SU(2)), but they are not isomorphic as Hopf algebras . The difference between Q,, and U,,+(SU(2)) ® U,, _(SU(2)) as Hopf algebras resides in the coproduct ® and in the antipode ; indeed for Q,, they are ®E + _ E + ® 1 + q+ :H + eITH_ ® E+

y( E+) _

-q

H+

e-iTrH_

E+

(C .37)

and this is why the R-matrices of both algebras are different . The universal R-matrix of U,,(SU(2)) is given by q ~iH~2En ® Fn

(C.38)

C°. Gômez, G. Sierra / Quantum symmetry of rational CFTs

825

It is easy to check that the matrix Rio'2, corresponding to the representation j, ®j, in the basis e;n ~ ® ein,, coincides with the one computed in appendix B using contour techniques [see eq. (B.19)]. The universal R-matrix of Q,, turns out to be

where R + +(q +) are given by _

_ ++(q+)

q+

~ n>0

~

[n]

1

n

o

~+-

q+

while the matrices R + ~ are given by R ++ =e

( C.4I )

Eq . (C.39) agrees with expression (B.22) obtained by cûïitoür tc%iïfiqücs

~lp e COMPUTATION OF K~ ;"=

The purpose of this appendix is to derive the fusion rules and the quantum CG coefficients using contour deformation arguments. «; The operator ( a )` , ~ acting on °t ® .ta, is defined as a double «~ ai z contour integral of ~r+ screening operators J+ , and r_ screening operators J_ around the vertices Va ( z, ) and V« ,( z 2 ) (see fig. 17). The state ( a~ a; a, )z~, z,(Va i ® Va, ) vanishes unless

where a; = 2(1 - n ~ )a + + i (1 - m; )a _ and n ;, m; > 1 . « and the fact that more than Eq. (D.l) follows from the definition of (a~ ; aZ) n - 1 (m - 1) contours of J+ (J_ ) type annihilate the vertex Va.,.n~~ The charge conservation condition (5 .7) implies the following relations : n~=n 1 +n 2

-2r + - 1,

;;?E

C'.

û®tttE"z,

G. Sic'r°ra l Qttarttttrr: syrrtrr:etry of rational CFTs

which together with { .1) Meld the "classical" fusion rules ~tt, -~t,g

+ 1


t +n,- 1,

If we compute now the scalar product (D.4) it is easy to see, by deforming the double contours as in fig. 21, that this scalar product never vanishes provided ~} is not rational, so that conditions (D.1) or .3) are necessary and sufficient for ~ `~' ~, to be non-zero . a n the rational case `where } = p'~p we obtain a vanishing scalar product (D.4) unless 0
e have supposed here that ~ ; is an integrable representation of the c < 1 theot-~~. From { .1), ( ,2) and {D.5) we find the fusion rules of the RCFT, tt~ -tt,~

+ 1
in(ft,

+ft,-

1,2p-ft fl -n,- 1),

~trt, - tn,~ + 1
1,2p' - m l - m 2 -1),

(D .6)

which from eq. {5.18) are also the decomposition rules of the restricted tensor product { ~ ~" I ® ~ ~`~= f .

Fig. 21 . Deformation of a double contour integral on the sphere .

Co Gômez, G. Sierra / Quantum symmetry of rational CFTs

827

Let us now compute the coefficients Kâ;«~ of eq. (5 .7). VVe shall ~~rork for simplicity with only one type of contours, say J + , and at the end we shall give the general result. If -we disentangle the double contour integral which defines ( «, «; «, ) we obtain CY 2

zl,z~

«i

xJ

Zo

. . .J

Z,

Z, Z,

dt~ . . .dtrV«o(zt)J+(t')- . .J+(tr)V«,(zz)~ _

( D.7)

where q = q +, r+ = r and '-- = 0. Opening the integrals j~; -' as in (B.14) yields é«,( r, , r, r,+r,=r

~

9

'

z )é«

,(

ZZ)

so that the final result is «,«,0

r

rl

~r,(r,-I)

2ai« + «,r,

9

X

1 - e ~ai « +«,

~x ) y = r,

with the restriction r = rl + r2 . If there are two kinds of contours the general result is r, rj , r, r, , «~

r, r, «~~ "

-r )

,

where each term of the right-hand side is given by eq. (D.9). It is a simple matter to check from eqs. (D.9) and (D.10) that 4E + = 0.

[1 ] [2] [3] [4] [5]

eferences G. Moore and N. Seiberg, Commun . Math . Phys . 123 (1989) 1 iï L. Alvarez-Gaumé, C. Gômez and G . Sierra, Nucl. Phys. B319 (1989) 155 L. Alvarez-Gaumé, C. Gômez and G. Sierra, Phys . rett . B220 (1989) 142 E. Witten, Nucl. Phys. B330 (1990) 285 G. Moore and N.Yu . Reshetikhin, Nucl. Phys. B328 (1989) 55ï

( D .9)

S2S

C. Cr'~irne" G. Sierra / Qlla®ltlllil

S)')DlI)IE'ily'

of rational CFTs

L. Al>'arzz-Gaun~é, C. Gûmez and G. Sierra, Nucl. Phys. B330 (1990) 347 ~, Ch. Ganche`e and V.B, Petkova, Phys. Lett. 8233 (1989) 374 P. Saint°knegt, J, McCarthy and K. Pitch, Phys. Lett. B234 (1990) 297 V,G, Drinfel'd, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations, preprint ITP-89-43E R. Dijkgraaf, V. Pasquier and P . Roche, Quasi-quantum groups related to orbifold models, talk presenied by V. Pasquier at the Int. Colloquium on Modern quantum field theory, Tata Institute of Fundamenial Research, 8-14 January 1990 [11 ] I,T, Todorov, Quantum groups as symmetries of chiral conformal algebras, Proc. 8th Summer `~4'orkshop on Mathematical physics, Clausthal, 1989 [1?] C, Grimez and G, Sierra, Phys. Lett. 8240 (1990) 149 [ 13] ~'.S, i?-otsenko and V,A . Fateev, Nucl, Phys. 8240 C 1984) 312 ; B251 (1985) 691 [ 14] G. Felder, Nucl, Phys. 8317 (1989) 215 [1>] J.L, Gervais and A. Neveu, Nucl, Phys. 8224 (1983) 329 [16] L,D, Faddee`~ and L,A. Takhtajan, Lecture Notes in Physics, Vol . 246 (Springer, Berlin, 1986) p, 166 [17] D, Babelon, Phys. Lett. B215 (1988) 523 [18] A, Bilai and J.L. Gervais, Phys. Lett. B206 (1988) 412 [19] T,J. Hollor~~ood and P. Mansfield, Nucl. Phys. 8330 (1990) 720 , [~t~] T,J . Hollor~~ood, P `°s. Lett . B2_ß-1(1990) 57 [21] A, LeClair, Integrable restricüons of quantum soliton theory and minimal conformal series, ila Ph`~sics, braids and links, ed, H.C. Lee, to be published [22] V, Pasquier and H, Saleur, Nucl, Phys. B330 (1990) 523 j?+] . Saleur, Phys. Rep, 184 (1989) 177 [2-1] G, Felder, J, Frdlich and G. Keller, Commun . Math. Phys, 124 (1989) 417 [?~] V,G. Drinfel'd, Quantum groups, Proc. Int. Congress of Mathematicians, Berkeley, CA, 1986. [26] A,N, Kirillov and N.~u, Reshetikhin, Representations of the algebra U9(Sl(2)), q-orthogonal pol`'nomials and invariants of links, Leningrad preprint L®MI-E-9-88 [27] M, Jim'bo, Lett. Math. Phys. 10 (1985) 63 ['8] M. Ros_Sfl, Commun. Math. Phys. 124 (1989) 307 ; Analogues de la forme de Killing et du théorème d'Harïsh-Chandra pour les groupes quantiques, Ann. Sci. Ecole Normale Superieure, to be published [29] N, Burroughs, Relating the approaches to quantized algebras and quantum groups, DAMTP/R89,~ 11 ; [ 0] V, Pasquier, Commun . Math. Phys. 118 (1988) 355 ; Thèse pour le grade de Docteur (Paris Univ. Press 1988) [;1] G. Moore and N. Seiberg, Phys. Lett. 8212 (1988) 451 ; Nucl . Phys. B313 (1989) 16 [32] G, Lusztig, Modular representations of quantum groups, MIT preprint (1988) [33] N,Yu. Reshetïkhin and F. Smirnov, Harvard preprint (1989)

[6] [7] [8] [9] [10]