chern-simons theory and conformal blocks

Volume 228, number 2 PHYSICS LETTERS B 14 September 1989

CHERN-SIMONS THEORY AND CONFORMAL BLOCKS

J.M.F. LABASTIDA and A.V. RAMALLO ’ Theory Division, CERN, CH-1211 Geneva 23, Switzerland

Received 22 June 1989

We construct the operator formalism for Chern-Simons theories on the three-sphere for the case in which Wilson lines are cut.

It is shown explicitly that the states of the Hilbert space correspond to the conformal blocks of a Wess-Zumino-Witten model.

One of the most important outstanding problems of string theory is the classification of rational conformal field theories (RCFT) [ 11. Recently, it has become clear [ 2 ] that progress towards the solution to this problem falls into one of the two approaches: an algebraic approach in which the duality properties of RCFT are exploited (see for example refs. [ 3,2] ), and a geometric approach in which the starting point is the connection between Chern-Simons (CS) theory and RCFT pointed out by Witten [ 41. In this letter we will be working within this second approach. In particular, we will extend the operator formalism for CS theories which we proposed in ref. [ 5 ] to the case in which Wilson lines are cut. Explicit realizations of the connection between CS and RCFT based on the canonical quantization of CS theory have been carried out recently using both, the holomorphic representation [ 6 ] and the Schrodinger representation [ 7 1. Other works addressing some issues related to this problem within the framework of canonical quantization have also recently appeared [ 8 1. In ref. [ 5 ] we de- parted from the canonical approach and an operator formalism for CS theories was constructed in analogy to the one in string theory [ 91. This construction involved the following steps: first the three-dimensional compact oriented manifold in which the theory is defined was cut (without cutting any Wilson line) via a Heegaard splitting [ lo]; second, for each resulting g-handle body a state was defined via a Feynman path integral; finally, the resulting states were determined exploiting the symmetries of the theory. The resulting Hilbert space was identified with the space of characters of a RCFT. In addition, it was demonstrated that the insertion of non- contractible unknotted unlinked Wilson lines has the same action on the Hilbert space as the Verlinde operators [ 111 on the space of characters. As a consequence, the fusion rules of the corresponding RCFT were derived. In this letter we extend the operator formalism to the case in which Wilson lines are cut. We find by explicit construction and by deriving the Knizhnik-Zamolodchikov equations [ 12 ] that the resulting Hilbert space can be identified with the space of conformal blocks [ 13 ] of the corresponding RCFT.

Let us consider a gauge group G and a gauge connection A, on the three-sphere S3. The subsequent equations will be written for the case in which G is SU( n). Their generalization to other groups is straightforward. Inside the three-sphere we introduce some Wilson lines which may be knotted or linked. Let us split S3 into two solid balls B, and Bz in such a way that N Wilson lines have been cut. After this operation there are 2N distinguished points on the boundary of each solid ball. Let us concentrate on one of the solid balls, say B,, and let us label the distinguished points on it by P,, Q,, i= 1, . . . . N, where Pi ( QL) corresponds to the beginning (end) of the Wilson line i. Following the operator formalism construction in ref. [ 5 ] we associate to this solid ball B, with 2N distinguished points and N pieces of Wilson lines running through it a wave functional defined by means of a Feynman path integral. Fig. 1 constitutes an example of a configuration of this type for the case N=4. For each pair of

I Address after October 1, 1989: Departamento de Particulas Elementales, Universidad de Santiago, E-l 5706 Santiago de Compostela,

Spain.

214 0370-2693/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)


Fig. 1. Solid ball B~ with four Wilson lines cut. All lines are oriented from P~ to Q~.

distinguished points P~ and Q~ the insertion in the functional integral corresponding to the piece of the Wilson line i running through the solid ball B~ is the following matrix:

a i

W(~)(Pi, Q i ) , ~ , a ~ = I P e x p ( - f A ( i ) ) l Pi °ti1~i '

(1)

where A (ou = A ~ T ~) and T ~) belong to an SU ( n ) representation labeled by (i) and the integral is path-ordered through the path joining P~ and Q~ in B~. We define the wave functional associated to B~ as

; [ 1 ( ) T(Az;P,,Q, .... ,PN, QN)= [DAu]B , ~ W(~)(P~,Q~) exp ikS(Au)--~-~ d2aTr(AzAz) , 0Bt

(2)

where [ DA.]B, represents the Feynman path integral measure over gauge orbits such that Az is fixed at 0B~, and, at 8B,, A_.= ½ (Aj - iA2) , Az= ½ (A~ +iA2), Ao being in the direction perpendicular to 0B~. In (2) k is an arbitrary integer, Tr denotes the trace in the fundamental representation, a t represent real local coordinates on 0B,, and S(Au) is the Chern-Simons action

l S ( A , ) = ~ f T r (A^dA+~AAA^A) .

BI

(3)

Notice that the wave functional defined in (2) contains a tensor product denoted by ® which implies that ~u(A_,; Pl, QJ .... , PN, QN) contains 2N indices. In what follows this symbol will be used to denote products of objects belonging to different representations of the gauge group.

One of the main tools in the construction carried out in ref. [ 5 ] was the analysis of the behavior of wave functionals under gauge transformations. In this behavior the presence of the boundary term in (2) was essential. The exponential exp [ ikS(A,) ] in (2) is invariant under gauge transformations A u---, h - ~A~h + h - ~O~h for any continuous map h: B I ~ SU (n) when k is integer except for boundary terms. These boundary terms combine with the ones originated from the transformation of the last term in the exponential of (2) to give a factorized form for the transformed wave functional. In addition, it was shown in ref. [ 5 ] that the most natural choice for the measure in (2) is such that it transforms-in the same way as the rest of the functional, leading just to a shift of the integer k into k+cv where Cv is the quadratic Casimir operator in the adjoint representation. A similar pattern appears in the analysis of the transformation of (2). Let us consider a continuous map g: ÙBI~ SU (n) and the following gauge transformation: Ae~g- ~A~+g- ~0@. The map g has many continuous extensions from the boundary to the interior of B~. Let us consider one of them: g. Parametrizing Az by Az= u - ~0zu with u valued in SU (n)c and using the fact that g l0B, =g, one finds from (2) that the transformed wave functional takes the form

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7t(Az; P~, Qt ..... PN, QN)

where F(~) is the Wess-Zumino-Witten (WZW) action [ 14]

1 1 i f F ( ~ ) = ~ d2aTr(g-'Ozgg-~Ozg)+ ~ ~.UVOTr(~-lOu~-lOv~-lO¢~), (5) 0BI Bi

and

1 f d2aTr(u-lO~ugOzg-l). (6) (u,g)=-~

0B~

In (4) we have used the fact that under the gauge transformation the Wilson-line operators transform in the following way:

W(i) (Pi, Qi) ~ g ~ (P~) W(,)(Pi, Qi)g(i)(Qi). (7)

Notice that since k is integer and the function involving F(~) is an exponential, the transformed wave functional is independent of the choice of extension of g to the interior of the solid ball B I [ 14 ]. In what follows we will not make a distinction between g and ~.

As u-,.ug under gauge transformations, it is rather simple to verify by using the Polyakov-Wiegmann (PW) property [ 15], F(vW) = F(v) + F(w) + (v, w), that a solution to (4) has the form

where ~(PI, Q~, ..., PN, QN) is a function of the 2Npoints, P~, Q1, ..., PN, QN, with 2Nindices which has the same structure as in ~(A~; PI, QI ..... PN, QN). Notice that in (8) u ~ (P,) is in the representation (i) and ue.) (Qj) is in the representation (j). The infinitesimal form of the transformation property of the wave functional (4) implies that T(A~; P~, QI ..... PN, QN) satisfies the following Gauss' law with sources [4]. One finds from (4)

k+cv F~.(X)~(Az; Pt, Q, ..... PN, QN) 2~Z

N = ~., [O(2)(X-Pj)T~)~(Az;PI,QI ..... pN, QN)--~(2)(X--Qj)~(A~.;P1,Q,,...,PN, QN)Ta~)], (9)

j=~

where

F~z= O~A~- O~A~ + [A~,Ae], (10)

and [5]:

2zt a _ ( l l ) m z - k +c~ 8A~ "

From the symmetry arguments based on gauge transformations it is not possible to determine the complete form of the wave functional ~V(A~; P1, QI, ..., PN, QN). The function ~ ( P I , QI .... , P~, QN) remains arbitrary. However, it is rather simple to obtain certain equations which will help us in determining this function. In fact, it will turn out that it is precisely ~ ( P , QI, ..., PN, QN) which will be identified with the conformal blocks of RCFT. Notice that, from (8), ~(PI, QI, ..., PN, QN) = T(A~; P~, Q1 ..... PN, QN) I~= ~. First, we will show that this function is ho lomo~hic in all its variables. Let us consider the wave functional T(A~; P1, Ql . . . . . PN, QN) as defined in (2). It is rather simple to verify that

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0zp ~(Ae; P, Q, PN, "~ ~ aa T a ~(Az; P,, Q,, Pu, QN), , , , " " , ~ N ] ~ - - ~ zPi ( i ) " " ,

0~e, ~(Az; P,, Q, .... , P~., QN)=A~-e, ~(A~; P,, Q,,..., PN, QN) T~i). (12)

These equations are immediately obtained by performing a derivative in (2) and taking into account that in that functional integral Az is not integrated over. Considering now (8) one obtains

Oe,.,~ ( Pl , Q, ..... PN, QN) = Ozt,, ~(Ae; e l , Q, ..... PN, QN) lu=l

_ _ A a T a ~ t A • ..., -----~ze~ (i)~, e, PI,QI, PN, QN) Iu=I =0, (13)

and similarly,

Oeo,~ ( P, , Ql ..... Pu, QN) =0. (14)

TO obtain the promised equation for ~ ( P l , QI, ..., PN, QN) let us take an holomorphic derivative of the wave functional, say 0~p. This operation produces an insertion ofA ~e, T'~n in the path integral which defines the wave functional (2). There are two alternative ways of looking at the resulting expression. Since A~ is integrated over in the path integral we should have to perform the integration to obtain the form Of0z~, ~(Ae; Pl, Q, ..... PN, Q N ) .

This is rather complicated and we will not follow this path. Fortunately, there is a simpler way to proceed. Since A ~ lies on the boundary we may take it out of the path integral and consider it as an operator acting according zPi

to the holomorphic representation, i.e., the derivative of ~(A~; P~, Ql ..... IN, Q N ) takes the form,

0..e, ~(A_.; P, , Ql , ..., PN, QN) = --A aze, Ta(i) ~V(Ae; P, , Q, , ..., PN, QN),

m a . . . , 0zQ, ~(Az; P,, Q,, ..., PN, QN) = zQ, ~/tt(Az; PI, Q,, PN, QN) Ta(i), ( 15 )

where one has to think of A~ as the operator ( 11 ). These equations, when combined with the Gauss' law (9), are rather powerful and will lead to the promised equations, which in turn will determine ~(P, , Qt .... , PN, Q N ) .

Before exploiting (9) and ( 15 ) let us recall the form of the Green function on the sphere. It is well known that the solution to the equation OzxOexA (X) = ~(:) (X) has the form A (X) = ( 1/zr) log [p2(Zx2x) ] where/~ is an infrared cutoff. This form of A (X) is, however, singular at small distances, i.e., when X--, 0. In our analysis we will need to make sense of this Green function in those situations so we will regulate it in such a way that it is well behaved at short distances. The simplest regulator consist of the introduction of a cutoff in momentum space of the form exp ( - a [ k I ) and consider A (X) in the limit a ~ 0. The regulated Green function has the form zl(X) = (1/~) log [/z2(a2+ Z x g x ) ] . This regularization has convenient features which will become apparent in the subsequent discussion. For example, it turns out that 0~A (X) = 0 when X~0 .

Using the expression for the delta functions entering in (9) in terms of derivatives of the Green function above it follows that the Gauss' law can be written as

k+c~ 2 G ~ ( A ~ ; P,, Q, ..... PN, QN)

(~(A~;P~,QI , . . . ,Px , QN)TaU) T ~ ) ~ ( A z ; P , , Q , . . . . . PN, QN)) (16) j = 1 7 _ - - Z Q j Z - - Z p j

where the operator G~ is

Ga_A ~ _1 f --d2a' (Oz,A~z ,-fab~ab-.4~ ). (17) ~Bj

Considering ( 16 ) for the case in which u = 1 (for any of the representations involved) we finally derive the key equation of the foregoing discussion:

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k~cv+ A~ ~(Ae; Pl, Q,,..., PN, QN) Iu=~

~ ( ~ ( P I , Q , . . . . . PN, QN)TaU) T~)~(PI ,Q , . . . . ,PN, QN).) (18) j= 1 Z-- ZQ) Z-- Zpj

The first consequence of (18 ) is the global gauge invariance of the functions ~ ( P l , Q~, ..., PN, QN). Let us consider a small closed contour on the boundary of B, which does not enclose any of the points P,, Ql .... , PN, QN. Since Az~(Az; P~, QL, ..., PN, QN) is regular in the enclosed region one has

~A z ~ hV(A~; P,, Q,, ..., PN, QN) = O. (19)

On the other hand, assuming that Az~(A~ P~, Q,, ..., PN, QN) a is well defined on the whole boundary S 2 we may look at the contour as enclosing all the points P~, Q~, ..., PN, QN, being (19) still valid. Taking into account ( 18 ) this implies, after performing the contour integrations,

N

[ ~ ( P l , Ql ..... PN, QN)Tao)- T~)~(PI , Q~ ..... PN, QN)] =0, (20) j = l

which shows the global gauge invariance of the functions ~(P~, Q1, ..., PN, QN). This equation is familiar from the analysis of WZW models [ 12 ] and corresponds to the Ward identity related to global gauge invariance.

Let us now consider (18 ) for the case in which (z, ~) becomes one of the distinguished points, say P~. The right-hand side of ( 18 ) becomes singular in this limit. However, using the regularization of the Green function discussed above we may regulate this singularity in such a way that it vanishes. Remember that our regulated Green function was such that 0z~,d (X) = 0 when X-~ 0. Taking into account ( 15 ) for the left-hand side of ( 18 ) one finally obtains

Ozp,,~(P,, Ql ..... PN, QN)

_ 2 "(N~ T(,)T(j) , (P~ ~ ~ ~r ,Q, . . . . . PN, QN) ~ T~ , ) ' (PI ,Q, . . . . . PN, QN)TaU,~ (21)

k + c,: ",a~i z p , - Zpj j= l Zpi - - ZQ) J

Similarly, if the point chosen is Q~,

O~Q,~(Pt, Q, .... , PN, QN)

~ a a ~ T a ) 2 ~(PI ,Q, .... ,PN, QN)T( j )T(o_ Ta~)~(PI,Q,,'",PN, QN) (i) (22) k + c,~ zo~ - z~ v~ i j = l Z Qi -- Z pj

These last two equations correspond to the Knizhnik-Zamolodchikov differential equations [ 12] which are critical in the analysis of WZW models. Using these equations together with (20) one may derive all the Ward identities of the WZW model except for the pure KaY-Moody ones [ 16 ]. It is not clear how these last identities may be derived from CS theory. However, eqs. (20), (21) and (22) are enough to obtain the explicit form of the functions ~(P, , Q~, ..., PN, ON) up to a constant factor. First, the global gauge invariance (20) permits a gauge invariant decomposition. Then ~ ( P l , Q~, ..., PN, QN) written in such a form is plugged into (21) or (22) and the resulting differential equations are solved. This analysis is entirely parallel to the one carried out in ref. [ 12 ]. There, the resulting differential equations are solved explicitly for the case in which one has four distinguished points and both Wilson lines are in the fundamental representation.

From eqs. ( 20 ), ( 21 ) and (22 ) we can identify the Hilbert space of the CS theory with the space of conformal blocks of the corresponding WZW model. One important issue concerns the dimensionality of this space and its explicit construction. As it stands, CS theory does not give any additional insight into these problems. The

218


algebraic approach [ 3,2 ] mentioned at the beginning of this letter may be much more powerful in addressing these issues. However, there is an indirect approach based entirely on CS theory which may be of some use. For Hilbert spaces of low dimension or for higher genus it may be rather powerful. The key idea is the following. Let us consider the case in which the N pieces of Wilson lines going through B1 are unknotted and unlinked. The situation depicted for the solid ball in fig. 2a corresponds to one of these cases for N = 2. Lines 1 and 2 may be deformed to the boundary of B~ without ever touching each other while keeping the points Pl, QI, P2 and Q2 fixed. The solid ball of fig. 2b portraits one possible resulting configuration after such a deformation. Once the lines lie on the surface we may appeal to the holomorphic representation and consider the corresponding expressions as operators acting on the vacuum, i.e., the state with no Wilson lines which we constructed in ref. [ 5 ]. In explicit terms, one would be considering wave functionals of the form:

Qi

~(A~;P~,Q~,...,PN, QN)=[~Pexp(-fA(,)I~o(A~), Pi

(23)

where A ~ must be thought as the operator ( 11 ) and

To(As) =~ exp[ - (k+cv)F(u) ], (24)

being a constant. Notice that because of the path order in the exponentials of (23) there is no ordering ambi- guity from the fact that now A g must be considered as an operator. In addition, since by assumption there are no crossings of paths, the order between the different exponential factors in (23) is irrelevant. Let us assume that we are able to compute the action of those operators on the vacuum. Once this is done, by performing a gauge invariant decomposition (since (20) must hold ) of the result we may identify the states which constitute a basis of the Hilbert space. For some cases it may be simpler to solve the differential equations (21 ) and (22) than to carry out this alternative approach. For others, however, as the case of higher genus, this approach based entirely on CS theory may be much more practical. This type of analysis captures the spirit of the operator formalism in the sense that an association between states and operators is performed. To obtain the operator corresponding to a given state in the basis resulting after the gauge invariant decomposition discussed above one only needs to act with the corresponding SU (n) projector on (23).

In this letter we will explore the possibilities of this alternative approach for simple cases. First let us solve the abelian case in this framework. For the gauge group U ( 1 ), following the conventions of ref. [ 5 ] we have

if To(A~)=~exp[-2kT(u)], 7 ( u ) = ~ d2au-'Ozuu-'Oeu, 0Bl

(25)

and,

(a) (b)

Fig. 2. The solid ball (a) corresponds to a situation where the only two Wilson lines present are unknotted and unlinked. In (b) these two Wilson lines have been displaced to the surface without every crossing while keeping the points P~, Q~,/)2 and Q2 fixed.

219


7t 3 A z = - 2--k 8A~-~- " (26)

Let us consider B, with N unknotted unlinked pieces of Wilson lines with charges ri, i= l, ..., N. We displace these lines to the boundary 0B1 in such a way that there are no crossings of paths. Our aim is to evaluate the abelian version of (23):

!)l ~(Ae;PI,QI,...,PN, QN)=[@i Pexp(-r i A ~o(Az), (27)

where Az is given in (26). Notice that, though Abelian, the path ordering of the exponentials is essential since we are dealing with operators. Since for this case the Hilbert space is one-dimensional the evaluation of (27) will necessary lead to its only state modulo a normalization constant. In evaluating (27) one finds short-distance singularities which we regulate making use of the regularized Green function discussed above. This implies that we may consider as zero all the expressions of the form Ozxd(X)=0 in the limit X + 0 which appear in the calculation. For N = 1 one finds, after some algebra:

~P(Ae;P,Q)=IPexp(-riA)l~o(A~)=exp{-(r2/2k)log[(ze-za)/a]}u~l(P)u~r)(Q)~o(A,). (28) P

Comparing with (8) we may identify, up to a global constant factor,

(a7 ~ ( e , Q ) = ~ A,= 4k ' (29)

which coincides with the two-point conformal block of a gaussian model. Of course, (29) has to be understood in the limit a + 0 . The calculation of the general case is entirely analogous. After some tedious algebra one finds (dropping the a dependence into the global constant factor)

( ,rirj/2k[ N )--rirj/2k ~(',,QI,...,PN, QN)~- f i (Zpi--Zpj)(ZQi--ZQj)) ~i.j~= ' (Zp,--Z~) , (30) i<j which agrees with the standard result for gaussian models.

In general, the evaluation of expressions of the type (23) are rather intricate. Typically one has to expand the exponentials, work out the action of the operators on the vacuum wave functional, and identify from the resulting series the form of ~(PI, Ql .... , PN, QN). In the abelian case, as we have seen, this is simple but in general it may be rather tedious. Let us consider now this computation in the non-abelian case for N = 1. For this situation the space of conformal blocks is one-dimensional and so (23) will necessarily lead to its only state. First, notice that, as follows from ( 11 ) and (24),

A~7/o(A~) =u-~OzU~o(Az), (31)

where u is in the fundamental representation of SU (n). Suppose that the only unknotted Wilson line present carries a representation R. Using (31 ) one finds, after some algebra,

Q

\ze-- zQ / u rl) ( p )u(R) ( Q ) ~°( Ae)' (32) P

where CR is the quadratic Casimir operator in the representation R. From this result it follows that

220


CR a ~2da

~ ( P ' Q)"B=8"P z p - - - ~ / AR-- k+cv' (33)

which is in fact the two-point conformal block of the SU(n) WZW model. For N > 1 this approach becomes rather cumbersome as compared to the one involving the resolution of eqs. (20), (21) and (22). However, it may be very useful in other situations like higher genus where more conventional approaches are rather intricate.

In the analysis presented in this letter, either through the connection between CS theory and RCFT via eqs. (20), (21) and (22), or the purely CS-inspired method described in the previous paragraphs, the states are determined modulo a constant. This constant may be obtained via normalization conditions. However, such conditions must be imposed very carefully. As discussed in ref. [4], CS theory is severly constrained and one may choose orthonormal conditions only for the three-manifold $2× S ~. For other manifolds, at most one may choose the states in such a way that they form an orthogonal basis. For the case in which Wilson lines are not cut the analysis regarding the normalization was carried out in ref. [ 5 ]. For the case under consideration in this letter the convenient normalization for S 3 is the following. We define the inner product among the states of a basis such that it is different from zero only if the points at the cuts coincide, if at the same points the Wilson line carries the same representation and if the structure in the gauge invariant decomposition is the same. In the situations in which all these three conditions hold the inner product is defined as [ 5 ]

(~b, ~V)= f l d e t O z O z l d u d a e x p ( l ( k + c v ) f d 2 a T r ( A z A , ) ) 0B~

X ~(A~; P,, Q, .... , PN, QN) ~V(Az; Pl, QI, ..., PN, QN), (34)

where ~(Ag Pb Q~ .... , PN, QN) is the state corresponding to a solid ball with the same orientation as the one for T(A4 P~, Q~ .... , PN, QN). The first crucial check to verify that (34) is a sensible definition of the inner product is to prove that it is independent of the points Pb Q~ ..... PN, QN. This is indeed so. Consider for example the abelian case for one unknotted Wilson line in the representation of charge r. The corresponding state was com- puted in (28). Let us now evaluate the squared norm of this state. The first thing to notice is that when com- puting the corresponding functional integral in u there is still a residual gauge invariance. Factoring out the corresponding group volume and parametrizing u = exp (iz) it is simple to observe that one is left with a functional integration over Im Z. This functional integration is formally equal to the one appearing in the evaluation of the vacuum expectation value of the product of two vertex operators of charges r and - r , exp (r Im Z)exp ( - r Im Z), but with an action possessing the wrong sign for the kinetic term. Taking the same regularization as in the computation leading to (28) for the short-distance singularities one finds one over the squared modulus of (29). Therefore, all the dependence on the points P and Q disappears. Notice also that the dependence on the short-distance cutoff also drops. However, these are not the only places where the short- distance cutoff enters in the inner product, it is also present in I det 0z0_~l. In fact, the dependence on the frame of the three-manifold is hidden in such a regularization. Remember that according to the general analysis [4] the observables of CS theory are invariants of framed three-manifolds.

We would like to finish with some final remarks. The analysis presented here for S 3 in what concerns the connection between CS theory and RCFT is not complete. As we mentioned above, we have not been able to derive the pure KaY-Moody Ward identities [ 16 ] from CS theory. It would be desirable to know which feature of CS theory is connected to these identities. One way to reveal such a connection is the following. First try to construct an operator J such that the JT(Ae; PI, Qb ..., PN, Q~) I ,= ~ correspond to the same conformal block as the one originated by the insertion o fa Ka6-Moody current in RCFT. Whether or not such an operator exists is not clear at the moment. Assuming that we are able to build J, we may work backwards, i.e., we may take the null vectors leading to the Ka6-Moody Ward identities and construct the corresponding operators in CS theory. In addition, the presence would be desirable of another operator playing the role of an insertion of the energy-

221


m o m e n t u m tensor. Th is wou ld c o m p l e t e a k ind o f d i c t iona ry b e t w e e n CS theory and R C F T for S 3. In this letter, bes ides m a k i n g the c o n n e c t i o n b e t w e e n CS theory and R C F T we have p re sen ted an a p p r o a c h to

c o m p u t e c o n f o r m a l b locks based en t i re ly on CS theo ry insight. F o r S 3 this app roach does no t look ve ry advan -

tageous wi th respect to the s t anda rd ones. H o w e v e r , i t m a y be o f s o m e use in dea l ing wi th o the r s i tua t ions l ike

h igher genus in wh ich s t anda rd app roaches possess s o m e diff icul t ies . We expec t to s tudy this and o the r re la ted issues in fu ture work.

Acknowledgement

We w o u l d l ike to t hank L. A l v a r e z - G a u m 6 , C. G 6 m e z and G. Sierra for helpful discussions.

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