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Free field realizations of WZNW models. The BRST complex and its quantum group structure
Volume 234. number 3
PHYSICS LETTERS B
I I January 1990
FREE FIEI,D REALIZATIONS OF WZNW MOI)ELS. T I l E B R S T C O M P L E X A N D I T S Q U A N T U M GROUP S T R U C T U R E Peter B O U W K N E G T Ccnter lbr Theoretwal l'hysic~ and Department ol Physics, Massachusetts Instttute ql'Technoh~g.v. (amhrtdge. M..I 02139. l :S..t Jim McCARTHY e Department <~[l'hj'~tcs. Brandeis L'ntversity. It altham. .iL,t 02254. USA and Krzysztof PILCH 3 D~Tartment (#Mathematics. Masyachusetts Institute ol'Technology. Cambrtdge. ..'tl.,1 02 I39. /. "S,.1 attd Department of Ph)wics. Universitj" ol'Southern ('alilornta. Los..Ingeh,s. ('4 90089-0484 USA 4 Received 17 October 1989
We derive the BRST complex for the Wakimoto type free field realization of ,~(3 ). using as a guide the analogous complex which arises in the study of the finite dimensional algebra. We show that a quantum group structure underlies the algebra of screening operators. Some remarks on the vemex operators are made.
Perhaps the s i m p l e s t c o n f o r m a l field t h e o r i e s are the m i n i m a l m o d e l s o f the Virasoro algebra [ 1 ], but e v e n here the c o n f o r m a l b o o t s t r a p presents f o r m i d a b l e difficulties. W i t h an ingenious a p p l i c a t i o n o f the results in ref. [ 2 ], the c o m p l e t e solution for these m o d e l s on the sphere was given in ref. [ 3 ]. All correlators o f p r i m a r y fields could be r e p r e s e n t e d by c o r r e l a t o r s o f free fields. Recently, F e l d e r [4] has f o u n d that u n d e r l y i n g these results is a " ' B R S T - I y p e " structure: the i r r e d u c i b l e r e p r e s e n t a t i o n s o f the m o d e l , o v e r which one must sum to c o m p u t e c o n f o r m a l blocks, are e m b e d d e d in the free field Fock space as the kernel m o d u l o the i m a g e o f a nilpotcnt operator. Hc showed that screened vertex operators - those representing the primary fields - arc "'BRST i n v a r i a n t " , and that the trace o f such objects o v e r an i r r e d u c i b l e r e p r e s e n t a t i o n could be c o m p u t e d as an alternating sum o v e r Fock space traces. T h i s allowed him to give the free field r e p r e s e n t a t i o n on the torus. For the s a m e reason the e x t e n s i o n to higher genus R i e m a n n surfaces is possible [ 5 ]. T h e s a m e s t r u c t u r e a p p e a r s in the sl ( 2 ) W Z N W m o d e l [ 6 - 9 ] ( see also refs. [ 10,11 ] tbr related p a p e r s ) , and its is i m p o r t a n t to e x t e n d its d o m a i n o f applicability, c o n c e i v a b l y to all rational c o n f o r m a l field theories. In this p a p e r we p r o p o s e h o w it arises in all s l ( N ) W Z N W m o d e l s ( a r b i t r a r y integer level k > 0 ) . T o illustrate the v a r i o u s c o m p l i c a t i o n s that arise in g e n e r a l i z i n g the work o f ref. [9] to the higher rank case, we will explain in s o m e detail how the a n a l o g o u s ( a n d q u i t e well u n d e r s t o o d ) structure arises in the discussion o f the finite dim e n s i o n a l algebra. T h u s , we c o n c e n t r a t e first on this finite d i m e n s i o n a l e x a m p l e , and the results o b t a i n e d there arc then e x t e n d e d to the W Z N W m o d e l in a rather s t r a i g h t f o r w a r d way. In a lot o f respects it turns out that the Supported b} the US Department of Energy under Contract # DE-A('02-76ER03069. -' Supported b,. the NSF grant # PHY-88-04561. 3 Supported in part by the NSF grant # PHY-87-08447. 4 Present address. 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
297
Volume 234, number 3
PHYSICS LETTERS B
11 January 1990
affine extension can be viewed as the "'q-analogue". i.e. as a q u a n t u m group version, o f the finite dimensional example. For simplicity we will restrict our attention mainly to s l ( 2 ) and s l ( 3 ) , as this already illustrates the main features. The complete generalization to s l ( N ) , more details and proofs, are contained elsewhere [ 12 ]. Consider. as a toy model, any quantal system whose spectrum decomposes into irreps, o f a simple Lie algebra, (:,, and whose d y n a m i c s is algebraic, i.e. the h a m i l t o n i a n is built from the generators o f ':~. An example is the free particle on a group manifold, where the h a m i l t o n i a n is just the quadratic Casimir of the algebra. A complete solution to the problem is obtained if we find a free field representation o f the algebra, and give a way to project onto the irreducible subspace. The free field representation o f ':6 is easily m o t i v a t e d from the well known representation o f sl(2 ) on the space o f polynomials in one complex variable, which for s p i n j is E+ = a - a~ .
ll=-2z-
a +~/,
az
E
--_~
0 +2jz.
(1)
O:
This is just the action in the "generalized coherent state" representation [13]. and all finite-dimensional irreducible representations (irreps.) o f simple Lie algebras can bc found in this way [14]. Geometrically, this realization describes the right action of the group on sections o f a holomorphic line bundle over (a maximal Schubert cell ot") the flag manitbld G , . / B , d e t e r m i n e d by a characterL~ of B . Here Gc is the complexification of tile group and B_ is a Borel subgroups (generated by the negative root operators and Cartan subalgebra ). Introduce the set of free oscillators, rt[~I, ,,+~,, one pair for each positive root, [[I ~, ,";] = - ~;j. Also introduce a m o m e n t u m and position pair ',p'. q'}, i = 1 ..... rank G, [p', q'] = ~,/. Then we may build a set o f Fock spaces. F j, labelled by the vector ofeigenvalues o f the m o m e n t a p', generated by the products of 7;'s acting on the vacuum I , l ) , p ' l , l ) = , , I ' 1 , t ) , such that [ 3 + I A ) = 0 . On F.~ we can find a rcpresentation o f any simple Lie algebra. In particular for sl (2) one must just replace z . . . . ,', ~J/O.z • [3 and 2J ' v/2 P in ( 1 ). For sl ( 3 ) the realization, in the C a r t a n - W e y l basis, is given by /:,,.=ill.
E,~:=f12_71f13"
l:,.,~=f13.
11'= ~ o~)(;'+[fl)+p ' , +
I; .... = -- "/ ' ( "/ ' fl ' - "/ 2f12 + "/ 3[13 ) + "/ 3f12 - "/ ' ( oq , p ) , E ..... : __~,3(~,1/]1 _]_ ;,2fl2 ..]_,,3/]3 ) __],3(~3.
E
....
= _ .fl 2 ~, 2 f l 2
p) __ ~,1;.2 (£~2 " /)) __ ~,1:,2:,2D~2 .
.fl g f l , _ .fl 2 ( OL2 . p ) .
(2)
Expressions for the simple root operators of sl ( N ) are easily obtained by invoking the explicit formulae for the Gauss d e c o m p o s i t i o n G = N _ HN + = B N +, where for example N _ is the group generated by the negative root operators. The Fock space gives a reducible representation. In F,j the irrep, with highest weight ,'1..,~. is clearly embedded as an invariant subspace. For example the sl(2) spin d irrep, is spanned by the set of 2/-t- 1 monornials. ~'/;= ',1...- ..... --2; I , and so is embedded in F, as IJ). E - I J) ..... (1:'-)2/Ij ). The BRST structure which picks out the irrep, from F.t is known in the m a t h e m a t i c s literature as the B e r n s t e i n - G c l ' f a n d - G e l ' f a n d resolution [ 15 ]. We will now describe this resolution for sl(2 ) and s l ( 3 ) . The case o f sl ( 2 ) is particularly simple. There is one "screening o p e r a t o r " S = [3 cxp ( - ix,/2q). Geometrically this represents the left action of N+ on the flag manifold G J B _ , and thus obviously c o m m u t e s with the right action of N +, which wc have identified with the positive root generators in our representation [ 16]. From the set of powers o f S w e can construct one invariant operator, Q2:~ ~=$2'+ ~, acting on F,. Since Q2,+ ~ ~ (O/Oz) 2'+ ~, its kernel is clearly given by .Y/,. llowever, for our purposes it is more convenient to extend the above considerations to a complex o f Fock spaces for a nilpotent operator, d, in such a way that .~ is found as the natural cohomology o f this complex. In particular we can then obtain the trace o f invariant operators over the irrep, as an alternating sum o f traces over the Fock spaces in the complex. For sl ( 2 ) "'nilpotency'" translates into the complex 298
Volume 234. number 3 d(-I)
0
PHYSICS LETTERS B
d(O)
1I January 1990
d{I)
, F j . . . . . F_,_I
- ~0,
where d (- ~> is the trivial inclusion, d (°> = (2, and d (~) is the operator that annihilates all the states. Clearly, d~')d (i- ~ = 0 . and there is only nontrivial cohomology in this complex at Fj ((2 is onto, being a derivative operator). We have already seen that the cohomology there is isomorphic to .~. Thus if d(' = (' d then, by Lefschetz's principle. T r , , ( = Try, (: - Trr;_,_, ('. In particular this reproduces the Weyl character formula. The case of sl (3) is more complicated. Corresponding to the simple roots there are two screening operators S, and $2 Sl = (fll_;.2fl3) e x p ( - i o t l "q),
S2=fl2exp(-iot2.q)
(3)
,
which have the same geometric interpretation as before. From the set of powers o f & and $2 we may build two algebra invariants acting on F.~ (A is a dominant integral weight, i.e., (A, o¢)~7/+, i = I, 2) Q),'~=(.S',)I': F.t--,F.~_/ .....
I,=(A, o6)+I,
i=1,2.
One can prove that the intersection of the kernels of these two operators acting on FA is precisely .*A [ 14]. Notice that A - I , oc= r,.A. where w.A = w(A + p ) - p for Weyl group elements w(r, is the reflection in the simple root o~,). This extends (with Q~'~ acting in thc corresponding way. if (or, w.A ) in 77+ ) to a set of Fock spaces F,.A. one for every Wcyl group element w. However, unlikc thc familiar (0. 0) complcx, the two %1(2 ) dircctions" corresponding to Q C,~ and Q CZ~do not give rise to a double complex because the operators Q~J~ and Q(2~ do not (anti-) commutc! The solution is to introduce additional operators Q~3~ and (~3~ satisfying the rclations (13=1~+12), (2},~Q}.'--,=Q}~3,QI,,,
(4)
Q}._,,Q~2,=QI,Z,Q}~',
(and similar relations for (2 ~3~by interchanging 1~ 2 ), so that the following diagram is commutative:
c2~,"
Q~," F,,..~ ~ 0 .... FA
F,-,,_..I O~,"~
~, ,,,
Fr, r2r, *A ~ 0
c2 ~,~ '
O~,:' F.=., ~ "
QI?'
.
/
F, 2.... , O~,"
We will denote this complex by Corn [A ]. Note, that Q},3) (and (~3)) will contain an equal number, n. both S~ and $2 screening operators, and thus lowers the weight by not> Then, by defining F"~'[A] =Fa, F~t~[A] = F,,..~@F,~.A. F~Z~[A]=F,.,,.~..~®F,.~,.A and F ~ 3 ) [ A ] = F . . . . . . . ~. a nilpotent operator d°~: F °) -,F °+ can be obtained, as follows: d ~ ° ) ( v ) = (Q~')t,, Q<2h,),
d<')(t'~,/'2) = ( - Q <3>l', +Q<'~/:2,
d(2)(/.l./.'2 ) = Q ( 2 ) / I +Q<'~v2 .
Q<2)l'~ -(.~(3)v2), (5)
II is known that the cohomology of this one-dimensional complex vanishes except at F ~°~ [A], where ~t is isomorphic to ~ [ 15]. Explicit expressions for Q~X~ and O~ 3~ can be obtained from the defining relations (4), e.g., Q/:3)=
y"
O.<.J<~12
(1~ +12)!12!
.
)!i-i2L~ii-/~-~-);-~ (S2)'~--'(S3)'(St) t'-',
(6)
where $3= [S~, $2]. It follows from (4) that Q~,3~ commutes with the algebra, and in fact it is not difficult to verify this explicitly. 299
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Thus we have found the structure which allows the solution o f our toy models in terms of a flcc field representation. We may go further and introduce vertex operators, i.e. a multiplet of operators I~., ~.,. ).,c ~'G:, m a p p i n g ~f, to Y/,h- The significance o f these operators per se will depend on the particular toy model, but hcrc we simply introduce them as a direct and instructive analogue of the crucial chiral reflex operators in the W Z N W model. Using them wc can c o m p u t e the analogues of conformal blocks - for one loop this is just a trace over such operators, suitably chosen so that the internal representations are fixed. Wc may introduce a free field representation for the vertices acting on the Fock space, but then we must require " B R S T invariance" to ensure that only states corresponding to the irreducible representation propagate internally, itence for each Fock space F,,.,,, ~ Corn [,1, ], we introduce the vertex I ~2,~._.[ w , A ~, w , A 3 ] which maps it to F,..,t ~ in ( ' o m [A3]. These combine into opcrators I" ~'].).~[,I~..~l 3 ]" F~'~ [,,ll ] ~ F'~[,13 ] via the "'diagonal sum", e.g. 1'(11.'.~;,2["11 '''13] ( l ' l . /'2) = (I~1:.,;.2[1"1 * i l l , rl *,'13]/'1.
l'.l:z.;,2[r2*A
I.
t'2..13]l'2)
.
(7)
Then "'BRST invariancc'" is more precisely the requirement that I ('~ are cquivariant with respect to d ~') within the complex, i.e.
d ( ' ) l ' l " = I'('+l~d~'~
(8)
which by ( 5 ) and ( 7 ) reduces to the set of eight equations, one for each Q-mapping in the complex. The explicit form o f the vertices is easy to find. Note that e x p ( i A : . q ) creates the highest weight state o f Y,',f: from 10). However it is not quite suitable, since it maps F.,, into F~,+,j: which is not equal to F,j~ in general. Tiffs difficulty may be resolved iI'A~=A~ + A 2 - n , o e ~ - n 2 o e , by an insertion o f screening operators. Then the highest weight c o m p o n e n t of this vertex is c x p ( i A 2 . q ) × [all independent terms with n. S'~s and n2 S ' s ] . The conditions ( 8 ) may nov,, bc solved, and all coefficients in the expansion into independent screening terms are fixed. The procedure is the same for all other vertices ( m a p p i n g between the other Fock spaces in Corn [,I~ 1 and Com[A~] ) and they have the samc form. Wc have verificd explicitly this program for s l ( 2 ) , and in some examples for sl (3). Let us just point out the interesting observation that ( 8 ) already encodes the fusion rule algebra (here this means the usual tensor product rule). For instance in sl ( 2 ) ( 8 ) reduces to
(_),_j,+, ~ ~: ..... [J,. i~ ]
=
) ;~.,,,.. [ , ' * j , . r . / ~ ] Q , , , +, .
The vertex on the LHS needs.j~ + J 2 - J 3 insertions o f the screening o p e r a t o r S. while that on the right needs J 2 + / 3 - h such. This clearly requires -.j2<~j,-j3<~j2. Moreover we have checked that there is more than one sol ution to ( 8 ) when they are required, as in the case o f sl( 3 ) with 8 × 8--.8 + 8 + .... We now turn to the analysis o f the W Z N W model, in light o f what wc have learned from the toy models. The free field representation can be obtained directly from (2). by interpreting the/3. 7.... as the zero modes of a set o f free fields. Introduce a set o f spin ( 1 . 0 ) conjugate pairs o f first order bosonic fields [ 3 ~ ( z ) . / ( z ) . as many as the n u m b e r o f positive roots oe~. with
/~'(z)=
Y~/~(,z ..... '
:,'(z)= y. :,',,--,,
n~ 7
;,,(z)/~Jiw)=-/~Jiz)/(,,.)=,~,~-]-
n~ 2
_ --
Also introduce a set of scalars rp,, i = 1..... rank G such that
~ o ' ( : ) = q ' - i p ' l o g z + i ~ a,', . . . . n :# 1) J1
A
Then wc have for sl ( 3 )~ [ 7.8.17.11 ] 300
~p'(-)~'(w)=-g,,log(z-w).
W
Volume 234, number 3 ]~'~,, =/(] I
PHYSICS LETTERS B
/:'~y2=[]2__;,I]]3
11 January 1990
1(c~'¢ ____,(] 3
Y oG " ; , ~ / 3 ; + v ( i @ ~ ' ) ,
/f'=
I
E
..1(7'131-7,2/]2+,'3fl3 } -kOTl+;'3[32-v;'l(~l.ic~q~).
"'=-
1:..... = l:" o:3 =--
";'272/32 " - ( k + I )a;'2--)'3/f,,/1- V72(Oz2. i&p) , . ;'l(;'l/]l-f-;'2f12"f''/31t]3
)
: - k O ; ' 3 - ( k + 1 );')~);,2- u73(~'s,- iOq~) .
v;' i;,_(~2, i0q~)--"
: , ' I,,_;,. _ , ,/,,, . .
(9)
whcrc u2=/, + 3. The Sugawara e n e r g y - m o m e n t u m tensor can be computed 7 = ! ' (ia~. i a ~ ) • - c ~ , , @ . i a ' q ~ ) - 7. •/fla;'; • .
(lO)
I
where C~o= 1/v and p = ~ x- oq = 0~3. We see that fl and 7 have d i m e n s i o n s I and 0. as advertised above, while the ~p-systcm has background charge ~o. The Fock spaces F.t are labeled by the eigenvalues of the m o m e n t u m p' of e' and are generated by the modcs o f f l ~, ;'; and ~' acting on the vacuum I,'1 ), p ' I A ) =,4'IA ). satisfying fl~,l,l)=0
|'orn>~0.
"'; I tl I.l) =0,
a'. , 1 ) = 0
forn>_- 1
The screening operators are
S1(-)=(/31-;'2/;
3) :exp(-io:oo~l'q~)
(z.),
S2(z)=fl2:exp(-iozoo~2.~)
• (z).
(11)
The only nontrivial c o m m u t a t o r is with the negative root operators, for which we find ( i . l = 1.2 )
(,
?
E .... ( z ) S ; ( w ) = d " 0 ~ - '
z-w'exp[-i~°°~Jq~(w)]
)
.
(12)
We now want to build the analogue of the complex in the finite dimensional example, so that we may project the irreducible representation of (9) out of F.v First consider for a given simple root ~,. i = 1, 2, the operator
@,',. (")~"> = 1- d:, ...d:,,S,(:l
.... %',(:,,) .
(13)
I
where the contour F is g~ven by a set o f nested contours from I to I around the origin, such that I--~ I > ... > Iz,, I fi~r _-,¢- 1. The integral is defined by analytic c o n t i n u a t i o n from the ordering 0 < : , , < . . . < z ~ on the real axis. ~cting on the Fock space F,~. the value o f n is determined by requiring that the operator 0~, '~ commutes w i t h all the currents. This happens precisely when n =/, = (,1, ~, ) + 1. The same condition determines that when acting on F~_;,,,., we must put n = L , = k + 3-1,. This continues in both directions and wc get a line of Fock spaces and mappings between them that commute w i t h the algebra o f currents. One can show that (_)}',~ QI, '~ = w.()u~/,~zO~'£,~= O. i = 1.2. by reducing the contours in ( 13 ) to standardly ordered contours with all variables on a unit circle. The result is an iterated integral times a factor which vanishes. This construction thus yields two one-dimensional complexes in the direction of the two simple roots. Wc can put them together to obtain a grid o f s l ( 2 ) lines, a s before, since Q<~> and Q~2~ do not commute, we must introduce a third operator to make it into a complex. The effect of the third operator must bc that all the "squares" in the diagram arc either commutative or anti-commutative. The result is a complex of Fock spaces and mappings between them. part of which is depicted in fig. I . W e m a y c o n s t r u c t a n i l p o t e n t o p e r a t o r d u ) F ~ , F ~+ t ~from this complex by s u m m i n g over the Fock spaces in vertical directions. The rigorous statements, as well as their proofs, appear in ret: [ 12 ]. To find this third operator is, at a first glance, much rnorc complicated than before. Howcvcr. there is a very nice property of the S,'s which makes the solution a simple extension of the result from the finite-dimensional model. Recall that in the finite dimensional case the screening ,operators ',,,'ere the generators of the positive root 301
Volume 234, number 3
PHYSICS LETTERS B
L,
F2(0'l)
/~ Fa~(1,1)
\
\
Fl~m 1) '
Ii
/'
z.._.~ F(0,o)
~
,,,, Li
\
L,
Ls
~L, 12
~L2
/
117'21(1,1)
F21(°'1) " L2
h
a,
~
L,
FI(o,o)
~
~
11 January 1990
F2(-ao)
Ls
/"
F12(o,o)
Is Is
'
L3
/
F~(o' o) ---* F~l(o,o)
F21(-lo) '
\
L,
L~
x/~ L~
It
%
F121(o,0)
I~
L~
---'
/
11
)<:La
~
q
/'
F(-L-O
"x,12
Lt L~
",,,
><] Lt
Fig. 1. BRST complex, C o m ( A ) , for ~ ( 3 ) . F,.j(,,,.,2) is the Fock space corresponding to the sl(3) weight × (k + 3 ); the labels on the arrows indicate powers of the screening charges.
L:
/
2=r,...r,*A+ (n~e,+n,(~,)
algebra n +. In the infinite-dimensional case this is of course no longer true, because the two screening currents are mutually nonlocal. However, one can show that inside the contour integrals, e.g. with all contours running from 1 to I, the S, satisfy the relations of the positive generators of the quantum groups 4/q(n+ ),
SISIS2-(q"kq-I)SIS2SI-kS2SISI=O,
S2S2SI-(q-kq-I)S2SIS2-kSIS2S2=O,
(14)
where q = exp [ i n / ( k + 3) ]. The proof is again by phase arguments, and is given in ref. [ 12 ]. We can expand Q~n 3) , which again always has an equal number of both S~ and $2, in the set of nested contour integrals
f dZl ...dz, S,,(zl
)... S,,(z,) ,
(15)
1.
with all possible ordering of S, and $2. We then find: (i) There are n + 1 independent terms (the number of independent elements of the quantum group at this level, which is in turn just the number of elements in the enveloping algebra '//(n + ) at this level ), (ii) The solution for Q(3) which can be inferred from ref. [ 18], is a q-analogue of the solution in the finitedimensional case. e.g. 0}23)=
f,
~ qj+(z.+;2-,)(;2-j) [ll +12]q.[12]q. ($2);2--;($3)j(S,);2-, ' +12, - j ] , ! (,-<,-~;2 [j]~! [12 - j ] , ! [l, '
(16)
where
$3=q-IS132-$2SI,
[/7]q-
q-q-
i,
[n]q!=[1]q...[n]q.
This solution commutes with the currents o f the entire algebra; the proof is once more similar to the finite dimensional case. up to the phase subtleties. (iii) There exists a 1-1 correspondence between invariant operators of the type ( 15 ) and singular vcctors in the Verma module of the quantum group #q(sl(3 ) ) [ 12 ]. Thus we have explicitly found the complex whose cohomology should yield the irreducible representation '~,t. The structure of the vertices is exactly the same as in the finite-dimensional model and in fact even somewhat simpler. That is because, from ( 12 ), contour integrals of the screenings commute with E - " (z) (after appropriate analytic continuation ) and so it suffices to solve the equations dI,'= Vd for the highest weight vertex. These can be read off from the complex exactly as in the finite dimensional case - of course there are more since the complex is extended - and can be solved using the quantum group relations (14). Once again these constraints already know about the fusion rules, as we found in the finite dimensional example. In particular for k>_-3 the 302
Volume 234, number 3
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vertex in the representation 8 which maps 8 ~ 8 has a two parameter solution. However, now it is possible to find vertices which go outside the fusion rules, since they are again naturally limited by the usual tensor product rule whilst the fusion rule algebra is in general a truncation of this. Thus (as in the minimal models [4] ) it is necessary to show that these vertices actually vanish on physical states. We will return to this problem in ref. [12]. In this paper wc discussed the emergence of a quantum group structure in the algebra of screening operators. 11 has been observed by several groups that the structure of a two-dimensional rational contbrmal field theory ( R ( ' F T ) is very similar to that of a quantum group by comparing various "derived quantities" like 6j-symbols. To our knowledge this is the first occasion that the quantum group itself has been observed in a RCFT. We believe that this structure will ultimately lead to a better understanding of the relation between quantum groups and RCFT's. Wc would like to thank M. Frau, A. Lerda and S. Sciuto for discussions. Note added. While this manuscript was being typed, we received a paper by Fcigin and Frenkel [ 19 ], who discuss the complex and its cohomology from a geometrical point of view. In our approach one obtains explicit formulac for thc BRST operators, which arc needed for the derivation of the screencd vertex operators and ultimately for the computation of correlation functions. Wc would like to thank Ed Frcnkcl for an intercsting discussion.
References [ 1 ] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] B.L Feigin and D.B. Fuchs, in: Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York. 1989 ), to appear. [3] VI.S. Dotscnko and V.A. Fateev, Nucl. Phys. B 240 [FSI2] (1984) 312: B 251 [FSI3] (1985) 691. [4] G. Felder. Nucl. Phys. B 317 (1989) 215. [5] M. Frau, A. Lerda, J.G. McCarthy and S. Sciuto, Phys. Lett. B 228 (1989) 205: J. Bagger and M. Goulian, Harvard preprint HUTP-89/A019: G. Felder and R. Silvotti, Zurich preprint ETH-TH/89-38. [6] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605. [7] B.L. Feigin and E.V. Frenkel, Usp. Mat. Nauk. 43 (1988) 227 [Russian Math. Surv. 43 (1988) 221 ]. [ 8 ] B.I,. Feigin and E.V. Frcnkcl. Moscow preprint ( March 1989 ), in: V. Knizhnik memorial volume, to appear. 19] D. Bcrnard and G. Felder, Zurich prcprint ETH-TH/89-26. [ 10] D. Ncmcschansky, Phys. Left. B 224 (1989) 121" preprint USC-89-012: J. Distler and Z. Qiu, prcprint CLNS 89/911 : A. Bilal, Phys. Lett. B 226 (1989) 272. [ I 1 ] A. Gerasimov. A. Marshakov, A. Morozow, M. Olshanetsky and S. Shatashvili, Moscow preprint. [ 12 ] P. Bouwknegt, J. McCarthy and K. Pilch, MIT preprint CTt' # 1797: and in preparation. [ 13 ] A.M. Pcrelomov, Generaliled coherent states and their applications (Springer, Berlin. 1986 ). [ 141 D.B. Zelobcnko, Compact Lie groups and their representations (American Mathematical Society, Providence, RI, 1973 ). [ 15 ] I.N. Bernstein, I.M. Gel'fand and S.I. Gel'fand, in: Summer school of the Bolyai J~inos Mathematical Society, ed. I.M. GeFfand (New York, 1975). [ 16] B. Kostant. Lecture Notes in Mathematics. Vol. 466 (Springer, Berlin, 1974) p. 101. [ 17] A.B. Zamolodchikov. unpublished. [ 18] G. Lusztig. MIT preprint (1989). [ 19] B i . Feigin and E.V. Frenkel. Moscow preprint (April 1989).
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FREE FIEI,D REALIZATIONS OF WZNW MOI)ELS. T I l E B R S T C O M P L E X A N D I T S Q U A N T U M GROUP S T R U C T U R E Peter B O U W K N E G T Ccnter lbr Theoretwal l'hysic~ and Department ol Physics, Massachusetts Instttute ql'Technoh~g.v. (amhrtdge. M..I 02139. l :S..t Jim McCARTHY e Department <~[l'hj'~tcs. Brandeis L'ntversity. It altham. .iL,t 02254. USA and Krzysztof PILCH 3 D~Tartment (#Mathematics. Masyachusetts Institute ol'Technology. Cambrtdge. ..'tl.,1 02 I39. /. "S,.1 attd Department of Ph)wics. Universitj" ol'Southern ('alilornta. Los..Ingeh,s. ('4 90089-0484 USA 4 Received 17 October 1989
We derive the BRST complex for the Wakimoto type free field realization of ,~(3 ). using as a guide the analogous complex which arises in the study of the finite dimensional algebra. We show that a quantum group structure underlies the algebra of screening operators. Some remarks on the vemex operators are made.
Perhaps the s i m p l e s t c o n f o r m a l field t h e o r i e s are the m i n i m a l m o d e l s o f the Virasoro algebra [ 1 ], but e v e n here the c o n f o r m a l b o o t s t r a p presents f o r m i d a b l e difficulties. W i t h an ingenious a p p l i c a t i o n o f the results in ref. [ 2 ], the c o m p l e t e solution for these m o d e l s on the sphere was given in ref. [ 3 ]. All correlators o f p r i m a r y fields could be r e p r e s e n t e d by c o r r e l a t o r s o f free fields. Recently, F e l d e r [4] has f o u n d that u n d e r l y i n g these results is a " ' B R S T - I y p e " structure: the i r r e d u c i b l e r e p r e s e n t a t i o n s o f the m o d e l , o v e r which one must sum to c o m p u t e c o n f o r m a l blocks, are e m b e d d e d in the free field Fock space as the kernel m o d u l o the i m a g e o f a nilpotcnt operator. Hc showed that screened vertex operators - those representing the primary fields - arc "'BRST i n v a r i a n t " , and that the trace o f such objects o v e r an i r r e d u c i b l e r e p r e s e n t a t i o n could be c o m p u t e d as an alternating sum o v e r Fock space traces. T h i s allowed him to give the free field r e p r e s e n t a t i o n on the torus. For the s a m e reason the e x t e n s i o n to higher genus R i e m a n n surfaces is possible [ 5 ]. T h e s a m e s t r u c t u r e a p p e a r s in the sl ( 2 ) W Z N W m o d e l [ 6 - 9 ] ( see also refs. [ 10,11 ] tbr related p a p e r s ) , and its is i m p o r t a n t to e x t e n d its d o m a i n o f applicability, c o n c e i v a b l y to all rational c o n f o r m a l field theories. In this p a p e r we p r o p o s e h o w it arises in all s l ( N ) W Z N W m o d e l s ( a r b i t r a r y integer level k > 0 ) . T o illustrate the v a r i o u s c o m p l i c a t i o n s that arise in g e n e r a l i z i n g the work o f ref. [9] to the higher rank case, we will explain in s o m e detail how the a n a l o g o u s ( a n d q u i t e well u n d e r s t o o d ) structure arises in the discussion o f the finite dim e n s i o n a l algebra. T h u s , we c o n c e n t r a t e first on this finite d i m e n s i o n a l e x a m p l e , and the results o b t a i n e d there arc then e x t e n d e d to the W Z N W m o d e l in a rather s t r a i g h t f o r w a r d way. In a lot o f respects it turns out that the Supported b} the US Department of Energy under Contract # DE-A('02-76ER03069. -' Supported b,. the NSF grant # PHY-88-04561. 3 Supported in part by the NSF grant # PHY-87-08447. 4 Present address. 0370-2693/90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland )
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affine extension can be viewed as the "'q-analogue". i.e. as a q u a n t u m group version, o f the finite dimensional example. For simplicity we will restrict our attention mainly to s l ( 2 ) and s l ( 3 ) , as this already illustrates the main features. The complete generalization to s l ( N ) , more details and proofs, are contained elsewhere [ 12 ]. Consider. as a toy model, any quantal system whose spectrum decomposes into irreps, o f a simple Lie algebra, (:,, and whose d y n a m i c s is algebraic, i.e. the h a m i l t o n i a n is built from the generators o f ':~. An example is the free particle on a group manifold, where the h a m i l t o n i a n is just the quadratic Casimir of the algebra. A complete solution to the problem is obtained if we find a free field representation o f the algebra, and give a way to project onto the irreducible subspace. The free field representation o f ':6 is easily m o t i v a t e d from the well known representation o f sl(2 ) on the space o f polynomials in one complex variable, which for s p i n j is E+ = a - a~ .
ll=-2z-
a +~/,
az
E
--_~
0 +2jz.
(1)
O:
This is just the action in the "generalized coherent state" representation [13]. and all finite-dimensional irreducible representations (irreps.) o f simple Lie algebras can bc found in this way [14]. Geometrically, this realization describes the right action of the group on sections o f a holomorphic line bundle over (a maximal Schubert cell ot") the flag manitbld G , . / B , d e t e r m i n e d by a characterL~ of B . Here Gc is the complexification of tile group and B_ is a Borel subgroups (generated by the negative root operators and Cartan subalgebra ). Introduce the set of free oscillators, rt[~I, ,,+~,, one pair for each positive root, [[I ~, ,";] = - ~;j. Also introduce a m o m e n t u m and position pair ',p'. q'}, i = 1 ..... rank G, [p', q'] = ~,/. Then we may build a set o f Fock spaces. F j, labelled by the vector ofeigenvalues o f the m o m e n t a p', generated by the products of 7;'s acting on the vacuum I , l ) , p ' l , l ) = , , I ' 1 , t ) , such that [ 3 + I A ) = 0 . On F.~ we can find a rcpresentation o f any simple Lie algebra. In particular for sl (2) one must just replace z . . . . ,', ~J/O.z • [3 and 2J ' v/2 P in ( 1 ). For sl ( 3 ) the realization, in the C a r t a n - W e y l basis, is given by /:,,.=ill.
E,~:=f12_71f13"
l:,.,~=f13.
11'= ~ o~)(;'+[fl)+p ' , +
I; .... = -- "/ ' ( "/ ' fl ' - "/ 2f12 + "/ 3[13 ) + "/ 3f12 - "/ ' ( oq , p ) , E ..... : __~,3(~,1/]1 _]_ ;,2fl2 ..]_,,3/]3 ) __],3(~3.
E
....
= _ .fl 2 ~, 2 f l 2
p) __ ~,1;.2 (£~2 " /)) __ ~,1:,2:,2D~2 .
.fl g f l , _ .fl 2 ( OL2 . p ) .
(2)
Expressions for the simple root operators of sl ( N ) are easily obtained by invoking the explicit formulae for the Gauss d e c o m p o s i t i o n G = N _ HN + = B N +, where for example N _ is the group generated by the negative root operators. The Fock space gives a reducible representation. In F,j the irrep, with highest weight ,'1..,~. is clearly embedded as an invariant subspace. For example the sl(2) spin d irrep, is spanned by the set of 2/-t- 1 monornials. ~'/;= ',1...- ..... --2; I , and so is embedded in F, as IJ). E - I J) ..... (1:'-)2/Ij ). The BRST structure which picks out the irrep, from F.t is known in the m a t h e m a t i c s literature as the B e r n s t e i n - G c l ' f a n d - G e l ' f a n d resolution [ 15 ]. We will now describe this resolution for sl(2 ) and s l ( 3 ) . The case o f sl ( 2 ) is particularly simple. There is one "screening o p e r a t o r " S = [3 cxp ( - ix,/2q). Geometrically this represents the left action of N+ on the flag manifold G J B _ , and thus obviously c o m m u t e s with the right action of N +, which wc have identified with the positive root generators in our representation [ 16]. From the set of powers o f S w e can construct one invariant operator, Q2:~ ~=$2'+ ~, acting on F,. Since Q2,+ ~ ~ (O/Oz) 2'+ ~, its kernel is clearly given by .Y/,. llowever, for our purposes it is more convenient to extend the above considerations to a complex o f Fock spaces for a nilpotent operator, d, in such a way that .~ is found as the natural cohomology o f this complex. In particular we can then obtain the trace o f invariant operators over the irrep, as an alternating sum o f traces over the Fock spaces in the complex. For sl ( 2 ) "'nilpotency'" translates into the complex 298
Volume 234. number 3 d(-I)
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d(O)
1I January 1990
d{I)
, F j . . . . . F_,_I
- ~0,
where d (- ~> is the trivial inclusion, d (°> = (2, and d (~) is the operator that annihilates all the states. Clearly, d~')d (i- ~ = 0 . and there is only nontrivial cohomology in this complex at Fj ((2 is onto, being a derivative operator). We have already seen that the cohomology there is isomorphic to .~. Thus if d(' = (' d then, by Lefschetz's principle. T r , , ( = Try, (: - Trr;_,_, ('. In particular this reproduces the Weyl character formula. The case of sl (3) is more complicated. Corresponding to the simple roots there are two screening operators S, and $2 Sl = (fll_;.2fl3) e x p ( - i o t l "q),
S2=fl2exp(-iot2.q)
(3)
,
which have the same geometric interpretation as before. From the set of powers o f & and $2 we may build two algebra invariants acting on F.~ (A is a dominant integral weight, i.e., (A, o¢)~7/+, i = I, 2) Q),'~=(.S',)I': F.t--,F.~_/ .....
I,=(A, o6)+I,
i=1,2.
One can prove that the intersection of the kernels of these two operators acting on FA is precisely .*A [ 14]. Notice that A - I , oc= r,.A. where w.A = w(A + p ) - p for Weyl group elements w(r, is the reflection in the simple root o~,). This extends (with Q~'~ acting in thc corresponding way. if (or, w.A ) in 77+ ) to a set of Fock spaces F,.A. one for every Wcyl group element w. However, unlikc thc familiar (0. 0) complcx, the two %1(2 ) dircctions" corresponding to Q C,~ and Q CZ~do not give rise to a double complex because the operators Q~J~ and Q(2~ do not (anti-) commutc! The solution is to introduce additional operators Q~3~ and (~3~ satisfying the rclations (13=1~+12), (2},~Q}.'--,=Q}~3,QI,,,
(4)
Q}._,,Q~2,=QI,Z,Q}~',
(and similar relations for (2 ~3~by interchanging 1~ 2 ), so that the following diagram is commutative:
c2~,"
Q~," F,,..~ ~ 0 .... FA
F,-,,_..I O~,"~
~, ,,,
Fr, r2r, *A ~ 0
c2 ~,~ '
O~,:' F.=., ~ "
QI?'
.
/
F, 2.... , O~,"
We will denote this complex by Corn [A ]. Note, that Q},3) (and (~3)) will contain an equal number, n. both S~ and $2 screening operators, and thus lowers the weight by not> Then, by defining F"~'[A] =Fa, F~t~[A] = F,,..~@F,~.A. F~Z~[A]=F,.,,.~..~®F,.~,.A and F ~ 3 ) [ A ] = F . . . . . . . ~. a nilpotent operator d°~: F °) -,F °+ can be obtained, as follows: d ~ ° ) ( v ) = (Q~')t,, Q<2h,),
d<')(t'~,/'2) = ( - Q <3>l', +Q<'~/:2,
d(2)(/.l./.'2 ) = Q ( 2 ) / I +Q<'~v2 .
Q<2)l'~ -(.~(3)v2), (5)
II is known that the cohomology of this one-dimensional complex vanishes except at F ~°~ [A], where ~t is isomorphic to ~ [ 15]. Explicit expressions for Q~X~ and O~ 3~ can be obtained from the defining relations (4), e.g., Q/:3)=
y"
O.<.J<~12
(1~ +12)!12!
.
)!i-i2L~ii-/~-~-);-~ (S2)'~--'(S3)'(St) t'-',
(6)
where $3= [S~, $2]. It follows from (4) that Q~,3~ commutes with the algebra, and in fact it is not difficult to verify this explicitly. 299
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Thus we have found the structure which allows the solution o f our toy models in terms of a flcc field representation. We may go further and introduce vertex operators, i.e. a multiplet of operators I~., ~.,. ).,c ~'G:, m a p p i n g ~f, to Y/,h- The significance o f these operators per se will depend on the particular toy model, but hcrc we simply introduce them as a direct and instructive analogue of the crucial chiral reflex operators in the W Z N W model. Using them wc can c o m p u t e the analogues of conformal blocks - for one loop this is just a trace over such operators, suitably chosen so that the internal representations are fixed. Wc may introduce a free field representation for the vertices acting on the Fock space, but then we must require " B R S T invariance" to ensure that only states corresponding to the irreducible representation propagate internally, itence for each Fock space F,,.,,, ~ Corn [,1, ], we introduce the vertex I ~2,~._.[ w , A ~, w , A 3 ] which maps it to F,..,t ~ in ( ' o m [A3]. These combine into opcrators I" ~'].).~[,I~..~l 3 ]" F~'~ [,,ll ] ~ F'~[,13 ] via the "'diagonal sum", e.g. 1'(11.'.~;,2["11 '''13] ( l ' l . /'2) = (I~1:.,;.2[1"1 * i l l , rl *,'13]/'1.
l'.l:z.;,2[r2*A
I.
t'2..13]l'2)
.
(7)
Then "'BRST invariancc'" is more precisely the requirement that I ('~ are cquivariant with respect to d ~') within the complex, i.e.
d ( ' ) l ' l " = I'('+l~d~'~
(8)
which by ( 5 ) and ( 7 ) reduces to the set of eight equations, one for each Q-mapping in the complex. The explicit form o f the vertices is easy to find. Note that e x p ( i A : . q ) creates the highest weight state o f Y,',f: from 10). However it is not quite suitable, since it maps F.,, into F~,+,j: which is not equal to F,j~ in general. Tiffs difficulty may be resolved iI'A~=A~ + A 2 - n , o e ~ - n 2 o e , by an insertion o f screening operators. Then the highest weight c o m p o n e n t of this vertex is c x p ( i A 2 . q ) × [all independent terms with n. S'~s and n2 S ' s ] . The conditions ( 8 ) may nov,, bc solved, and all coefficients in the expansion into independent screening terms are fixed. The procedure is the same for all other vertices ( m a p p i n g between the other Fock spaces in Corn [,I~ 1 and Com[A~] ) and they have the samc form. Wc have verificd explicitly this program for s l ( 2 ) , and in some examples for sl (3). Let us just point out the interesting observation that ( 8 ) already encodes the fusion rule algebra (here this means the usual tensor product rule). For instance in sl ( 2 ) ( 8 ) reduces to
(_),_j,+, ~ ~: ..... [J,. i~ ]
=
) ;~.,,,.. [ , ' * j , . r . / ~ ] Q , , , +, .
The vertex on the LHS needs.j~ + J 2 - J 3 insertions o f the screening o p e r a t o r S. while that on the right needs J 2 + / 3 - h such. This clearly requires -.j2<~j,-j3<~j2. Moreover we have checked that there is more than one sol ution to ( 8 ) when they are required, as in the case o f sl( 3 ) with 8 × 8--.8 + 8 + .... We now turn to the analysis o f the W Z N W model, in light o f what wc have learned from the toy models. The free field representation can be obtained directly from (2). by interpreting the/3. 7.... as the zero modes of a set o f free fields. Introduce a set o f spin ( 1 . 0 ) conjugate pairs o f first order bosonic fields [ 3 ~ ( z ) . / ( z ) . as many as the n u m b e r o f positive roots oe~. with
/~'(z)=
Y~/~(,z ..... '
:,'(z)= y. :,',,--,,
n~ 7
;,,(z)/~Jiw)=-/~Jiz)/(,,.)=,~,~-]-
n~ 2
_ --
Also introduce a set of scalars rp,, i = 1..... rank G such that
~ o ' ( : ) = q ' - i p ' l o g z + i ~ a,', . . . . n :# 1) J1
A
Then wc have for sl ( 3 )~ [ 7.8.17.11 ] 300
~p'(-)~'(w)=-g,,log(z-w).
W
Volume 234, number 3 ]~'~,, =/(] I
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/:'~y2=[]2__;,I]]3
11 January 1990
1(c~'¢ ____,(] 3
Y oG " ; , ~ / 3 ; + v ( i @ ~ ' ) ,
/f'=
I
E
..1(7'131-7,2/]2+,'3fl3 } -kOTl+;'3[32-v;'l(~l.ic~q~).
"'=-
1:..... = l:" o:3 =--
";'272/32 " - ( k + I )a;'2--)'3/f,,/1- V72(Oz2. i&p) , . ;'l(;'l/]l-f-;'2f12"f''/31t]3
)
: - k O ; ' 3 - ( k + 1 );')~);,2- u73(~'s,- iOq~) .
v;' i;,_(~2, i0q~)--"
: , ' I,,_;,. _ , ,/,,, . .
(9)
whcrc u2=/, + 3. The Sugawara e n e r g y - m o m e n t u m tensor can be computed 7 = ! ' (ia~. i a ~ ) • - c ~ , , @ . i a ' q ~ ) - 7. •/fla;'; • .
(lO)
I
where C~o= 1/v and p = ~ x- oq = 0~3. We see that fl and 7 have d i m e n s i o n s I and 0. as advertised above, while the ~p-systcm has background charge ~o. The Fock spaces F.t are labeled by the eigenvalues of the m o m e n t u m p' of e' and are generated by the modcs o f f l ~, ;'; and ~' acting on the vacuum I,'1 ), p ' I A ) =,4'IA ). satisfying fl~,l,l)=0
|'orn>~0.
"'; I tl I.l) =0,
a'. , 1 ) = 0
forn>_- 1
The screening operators are
S1(-)=(/31-;'2/;
3) :exp(-io:oo~l'q~)
(z.),
S2(z)=fl2:exp(-iozoo~2.~)
• (z).
(11)
The only nontrivial c o m m u t a t o r is with the negative root operators, for which we find ( i . l = 1.2 )
(,
?
E .... ( z ) S ; ( w ) = d " 0 ~ - '
z-w'exp[-i~°°~Jq~(w)]
)
.
(12)
We now want to build the analogue of the complex in the finite dimensional example, so that we may project the irreducible representation of (9) out of F.v First consider for a given simple root ~,. i = 1, 2, the operator
@,',. (")~"> = 1- d:, ...d:,,S,(:l
.... %',(:,,) .
(13)
I
where the contour F is g~ven by a set o f nested contours from I to I around the origin, such that I--~ I > ... > Iz,, I fi~r _-,¢- 1. The integral is defined by analytic c o n t i n u a t i o n from the ordering 0 < : , , < . . . < z ~ on the real axis. ~cting on the Fock space F,~. the value o f n is determined by requiring that the operator 0~, '~ commutes w i t h all the currents. This happens precisely when n =/, = (,1, ~, ) + 1. The same condition determines that when acting on F~_;,,,., we must put n = L , = k + 3-1,. This continues in both directions and wc get a line of Fock spaces and mappings between them that commute w i t h the algebra o f currents. One can show that (_)}',~ QI, '~ = w.()u~/,~zO~'£,~= O. i = 1.2. by reducing the contours in ( 13 ) to standardly ordered contours with all variables on a unit circle. The result is an iterated integral times a factor which vanishes. This construction thus yields two one-dimensional complexes in the direction of the two simple roots. Wc can put them together to obtain a grid o f s l ( 2 ) lines, a s before, since Q<~> and Q~2~ do not commute, we must introduce a third operator to make it into a complex. The effect of the third operator must bc that all the "squares" in the diagram arc either commutative or anti-commutative. The result is a complex of Fock spaces and mappings between them. part of which is depicted in fig. I . W e m a y c o n s t r u c t a n i l p o t e n t o p e r a t o r d u ) F ~ , F ~+ t ~from this complex by s u m m i n g over the Fock spaces in vertical directions. The rigorous statements, as well as their proofs, appear in ret: [ 12 ]. To find this third operator is, at a first glance, much rnorc complicated than before. Howcvcr. there is a very nice property of the S,'s which makes the solution a simple extension of the result from the finite-dimensional model. Recall that in the finite dimensional case the screening ,operators ',,,'ere the generators of the positive root 301
Volume 234, number 3
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L,
F2(0'l)
/~ Fa~(1,1)
\
\
Fl~m 1) '
Ii
/'
z.._.~ F(0,o)
~
,,,, Li
\
L,
Ls
~L, 12
~L2
/
117'21(1,1)
F21(°'1) " L2
h
a,
~
L,
FI(o,o)
~
~
11 January 1990
F2(-ao)
Ls
/"
F12(o,o)
Is Is
'
L3
/
F~(o' o) ---* F~l(o,o)
F21(-lo) '
\
L,
L~
x/~ L~
It
%
F121(o,0)
I~
L~
---'
/
11
)<:La
~
q
/'
F(-L-O
"x,12
Lt L~
",,,
><] Lt
Fig. 1. BRST complex, C o m ( A ) , for ~ ( 3 ) . F,.j(,,,.,2) is the Fock space corresponding to the sl(3) weight × (k + 3 ); the labels on the arrows indicate powers of the screening charges.
L:
/
2=r,...r,*A+ (n~e,+n,(~,)
algebra n +. In the infinite-dimensional case this is of course no longer true, because the two screening currents are mutually nonlocal. However, one can show that inside the contour integrals, e.g. with all contours running from 1 to I, the S, satisfy the relations of the positive generators of the quantum groups 4/q(n+ ),
SISIS2-(q"kq-I)SIS2SI-kS2SISI=O,
S2S2SI-(q-kq-I)S2SIS2-kSIS2S2=O,
(14)
where q = exp [ i n / ( k + 3) ]. The proof is again by phase arguments, and is given in ref. [ 12 ]. We can expand Q~n 3) , which again always has an equal number of both S~ and $2, in the set of nested contour integrals
f dZl ...dz, S,,(zl
)... S,,(z,) ,
(15)
1.
with all possible ordering of S, and $2. We then find: (i) There are n + 1 independent terms (the number of independent elements of the quantum group at this level, which is in turn just the number of elements in the enveloping algebra '//(n + ) at this level ), (ii) The solution for Q(3) which can be inferred from ref. [ 18], is a q-analogue of the solution in the finitedimensional case. e.g. 0}23)=
f,
~ qj+(z.+;2-,)(;2-j) [ll +12]q.[12]q. ($2);2--;($3)j(S,);2-, ' +12, - j ] , ! (,-<,-~;2 [j]~! [12 - j ] , ! [l, '
(16)
where
$3=q-IS132-$2SI,
[/7]q-
q-q-
i,
[n]q!=[1]q...[n]q.
This solution commutes with the currents o f the entire algebra; the proof is once more similar to the finite dimensional case. up to the phase subtleties. (iii) There exists a 1-1 correspondence between invariant operators of the type ( 15 ) and singular vcctors in the Verma module of the quantum group #q(sl(3 ) ) [ 12 ]. Thus we have explicitly found the complex whose cohomology should yield the irreducible representation '~,t. The structure of the vertices is exactly the same as in the finite-dimensional model and in fact even somewhat simpler. That is because, from ( 12 ), contour integrals of the screenings commute with E - " (z) (after appropriate analytic continuation ) and so it suffices to solve the equations dI,'= Vd for the highest weight vertex. These can be read off from the complex exactly as in the finite dimensional case - of course there are more since the complex is extended - and can be solved using the quantum group relations (14). Once again these constraints already know about the fusion rules, as we found in the finite dimensional example. In particular for k>_-3 the 302
Volume 234, number 3
PHYSICS LETTERS B
11 January, 1990
vertex in the representation 8 which maps 8 ~ 8 has a two parameter solution. However, now it is possible to find vertices which go outside the fusion rules, since they are again naturally limited by the usual tensor product rule whilst the fusion rule algebra is in general a truncation of this. Thus (as in the minimal models [4] ) it is necessary to show that these vertices actually vanish on physical states. We will return to this problem in ref. [12]. In this paper wc discussed the emergence of a quantum group structure in the algebra of screening operators. 11 has been observed by several groups that the structure of a two-dimensional rational contbrmal field theory ( R ( ' F T ) is very similar to that of a quantum group by comparing various "derived quantities" like 6j-symbols. To our knowledge this is the first occasion that the quantum group itself has been observed in a RCFT. We believe that this structure will ultimately lead to a better understanding of the relation between quantum groups and RCFT's. Wc would like to thank M. Frau, A. Lerda and S. Sciuto for discussions. Note added. While this manuscript was being typed, we received a paper by Fcigin and Frenkel [ 19 ], who discuss the complex and its cohomology from a geometrical point of view. In our approach one obtains explicit formulac for thc BRST operators, which arc needed for the derivation of the screencd vertex operators and ultimately for the computation of correlation functions. Wc would like to thank Ed Frcnkcl for an intercsting discussion.
References [ 1 ] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [2] B.L Feigin and D.B. Fuchs, in: Representations of infinite-dimensional Lie groups and Lie algebras (Gordon and Breach, New York. 1989 ), to appear. [3] VI.S. Dotscnko and V.A. Fateev, Nucl. Phys. B 240 [FSI2] (1984) 312: B 251 [FSI3] (1985) 691. [4] G. Felder. Nucl. Phys. B 317 (1989) 215. [5] M. Frau, A. Lerda, J.G. McCarthy and S. Sciuto, Phys. Lett. B 228 (1989) 205: J. Bagger and M. Goulian, Harvard preprint HUTP-89/A019: G. Felder and R. Silvotti, Zurich preprint ETH-TH/89-38. [6] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605. [7] B.L. Feigin and E.V. Frenkel, Usp. Mat. Nauk. 43 (1988) 227 [Russian Math. Surv. 43 (1988) 221 ]. [ 8 ] B.I,. Feigin and E.V. Frcnkcl. Moscow preprint ( March 1989 ), in: V. Knizhnik memorial volume, to appear. 19] D. Bcrnard and G. Felder, Zurich prcprint ETH-TH/89-26. [ 10] D. Ncmcschansky, Phys. Left. B 224 (1989) 121" preprint USC-89-012: J. Distler and Z. Qiu, prcprint CLNS 89/911 : A. Bilal, Phys. Lett. B 226 (1989) 272. [ I 1 ] A. Gerasimov. A. Marshakov, A. Morozow, M. Olshanetsky and S. Shatashvili, Moscow preprint. [ 12 ] P. Bouwknegt, J. McCarthy and K. Pilch, MIT preprint CTt' # 1797: and in preparation. [ 13 ] A.M. Pcrelomov, Generaliled coherent states and their applications (Springer, Berlin. 1986 ). [ 141 D.B. Zelobcnko, Compact Lie groups and their representations (American Mathematical Society, Providence, RI, 1973 ). [ 15 ] I.N. Bernstein, I.M. Gel'fand and S.I. Gel'fand, in: Summer school of the Bolyai J~inos Mathematical Society, ed. I.M. GeFfand (New York, 1975). [ 16] B. Kostant. Lecture Notes in Mathematics. Vol. 466 (Springer, Berlin, 1974) p. 101. [ 17] A.B. Zamolodchikov. unpublished. [ 18] G. Lusztig. MIT preprint (1989). [ 19] B i . Feigin and E.V. Frenkel. Moscow preprint (April 1989).
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