Modular transformations of conformal blocks in WZW models on Riemann surfaces of higher genus

Volume 241, number 4

PHYSICS LETTERS B

24 May 1990

MODULAR TRANSFORMATIONS OF CONFORMAL BLOCKS IN WZW MODELS ON RIEMANN SURFACES OF HIGHER GENUS M i a o LI 1 ICRA - International Center for Relativistic Astrophysics, Rome, Italy and Dipartimento di Fisica, Universit?t "La Sapienza % 1-00185 Rome, Italy

and Ming Y U International Center for Theoretical Physics, 1-34100 Trieste, Italy

Received 21 November 1989

We derive the modular transformations for conformal blocks in Wess-Zumino-Witten models on Riemann surfaces of higher genus. The basic ingredient consists of using the Chern-Simons theory developed by Witten. We find that the modular transformations generated by Dehn twists are linear combinations of Wilsons line operators, which can be expressed in terms of braiding matrices. It can also be shown that modular transformation matrices for g> 0 Riemann surfaces depend only on those for g~<3.

1. Introduction

There has been steady progress in u n d e r s t a n d i n g 2D conformal field theory ( C F T ) [1,2] recently. Verlinde's work on fusion rules is the first i n d i c a t i o n that one m a y classify and construct C F T by starting from some basic d a t a and relations a m o n g them [ 3 ]. Subsequently, Moore and Seiberg o b t a i n e d polynomial equations for data such as duality matrices which enable them to prove the conjecture o f Verlinde as a by-product [4,5]. They showed that classifying rational conformal theories might be equivalent to classifying all solutions to those polynomial equations. Rather surprisingly, Witten found recently that one can attack the W Z W models [6] by studying three d i m e n s i o n a l C h e r n - S i m o n s theories [ 7 ]. The idea is that in the canonical quantization of C h e r n - S i m o n s ( C S ) theory of a certain group, the Hilbert space is i s o m o r p h i c to the space o f conformal blocks in the corresponding W Z W model. In a subsequent paper, W i t t e n found a kind of basis o f conformal blocks on Address after October 1, 1989: The Niels Bohr Institute and Nordita, Blegdamsvej 17, DK-2100 Copenhagen ~, Denmark. 522

any R i e m a n n surface [ 8 ]. In this paper we work out m o d u l a r transformations acting on this basis. We found that the m o d u l a r transformations under basic Dehn twists a r o u n d h o m o t o p y cycles on genus g R i e m a n n surfaces are linear c o m b i n a t i o n s o f the hol o n o m y matrices a r o u n d the corresponding cycles. The nontrivial h o l o n o m y matrices in W i t t e n ' s basis are the ones for bj ( j = 1, ..., g) cycles which we found to be expressible in terms of braiding matrices. The fact that m o d u l a r transformations can be expressed so is not surprising, Moore and Seiberg already suggested it in their fundamental works [4,5 ], using different methods. However, we found our m e t h o d simpler and less d a t a are involved. The m e t h o d we use is similar to the one used by D i j k g r a a f and Verlinde in proving Verlinde's conjecture [ 3 ]. We find that in W Z W models one can extend the operators defined by Verlinde for genus one to higher genus, and in fact the operators can be realized by path integrals in the C h e r n - S i m o n s theory. The result is a generalization of what was obtained by Verlinde in the case of genus one, namely the hol o n o m y matrices a r o u n d b cycles can be diagonalized to the form o f h o l o n o m y matrices a r o u n d a cycles.

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The fact that our method is the generalization of the one used in ref. [3] convinces us that our result is valid for any rational conformal field theory, not only for the W Z W models we consider here for convenience. Let S(bl ), say, be the modular transformation corresponding to the Dehn twist along cycle b~. One of our main results is eq. (3.10), for the genus 2 case. This result directly generalizes to higher genus, with the help of eqs. (3.1), (3.11 ) and ( 3.12 ). The modular transformations generated by Dehn twists along a cycles, in some sense, are trivial. Since the modular groups are generated by basic Dehn twists, our result thus provides the basic blocks for constructing representations of modular groups. Moore and Seiberg conjectured in ref. [9] that all rational conformal field theories can be constructed from (2 + 1 ) C h e r n - S i m o n s theories. If this could be finally proven true, then our consideration here is essentially complete. In section 2 we define the Verlinde operators for W Z W models and describe the method we will use for higher genus in the case o f genus one. This proves Verlinde's conjectures again. We then present our main result in section 3 and discuss some applications in section 4.

2. Realization of Verlinde operators in the ChernSimons theory

As shown by Witten, three dimensional C h e r n Simons theory is closely related to the WessZ u m i n o - W i t t e n model in two dimensions. Here we shall consider the W Z W model of a given compact, simply connected and simply laced group G. Let A be a connection on a three-manifold of a bundle with structure group G. The lagrangian 5°---~

T r ( A A d A + ~ A 3)

(2.1)

M

is a general covariant one. The Chern-Simons form (2.1) is not well-defined for a group when its third h o m o t o p y group is not zero, unless the coupling constant k is quantized. Here k is an integer, corresponding to the level o f the K a c - M o o d y algebra in the W Z W model.

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To quantize the theory canonically, one starts from a three manifold of the form Z × ~, where Z is a compact Riemann surface. The Hilbert space depends only on the two dimensional surface. In fact, one can only treat the Hilbert space as a (projective) flat vector bundle over the moduli space of Riemann surface Y as follows. The action (2.1) pertains to a constrained system so one must work in the reduced phase space. The reduced phase space is the set of all flat connections modulo gauge transformations. If the Riemann surface has inherited a conformal structure, then the phase space is naturally endowed with a conformal structure and the usual symplectic form restricted on this phase space is K~ihler. By this K~ihler polarization one can quantize the theory consistently. This procedure seems to be the unique way to canonically quantize the theory for a general group with our requirements ~. But note that our theory will not depend on a specific conformal structure. We then consider all possible conformal structures, and at a point of the moduli space the Hilbert space coincides with the fiber of a flat holomorphic vector bundle. This holomorphic vector bundle is just that of the conformal blocks of the W Z W model on Riemann surfaces E [ 10-12 ]. The moduli space is the quotient of Teichmfiller space by the modular group. A flat vector bundle over it can be completely determined by its twist properties under the modular group, namely the modular transformations. Knowing all modular transformations for a given genus, one can in principle construct the flat bundle over the moduli space, hence calculate conformal blocks as holomorphic sections of the bundle. Now we recall what has been defined for a vector in the Hilbert space on a Riemann surface for the CS theory. The genus zero case is trivial since the Hilbert space is one dimensional. The first nontrivial case is of genus one. Now the Hilbert space is isomorphic to the space of characters. Let us consider the character of the primary field corresponding to the integrable representation Re. In a three manifold, given a loop C, we define the Wilson line as the holonomy around C,

~t The theory, of other kinds of groups can be quantized by proper constructions, see ref. [9].

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W/(C)=Tr Pexp ( ! A) .

PHYSICS LETTERS B

(2.2)

Consider the solid torus with boundary E, the vector in the Hilbert space corresponding to the character Zi is given by a path integral with insertion of Wilson line Wi(C) [7]. N o w C is parallel to the b cycle on the torus. As for a Riemann surface of higher genus, one considers the handlebody with boundary 2; (fig. 1 ). Instead of Wilson lines, we use graphs constructed from the basic baryon graph. Given three representations Ri, Rj and Rt, we associate the graph with two indices and ~-, each one corresponds to a chiral vertex of type (ifl) and (i*j*l*). Here we use i* to denote the conjugate of i, we reserve/-for the representation in the right moving sector coupled to i. The vectors in the Hilbert space are given by path integrals with insertion of these graphs as in fig. 1. Note that the number of choices of each vertex in the graph is given by the fusion rule N,j~, so we find that the number of independent graphs is just the number of conformal blocks. Note that in the path integrals, one must use a specific framing, here we adopt the prescription given in ref. [ 8 ]. Also the normalizations o f the baryon graphs are the same as in ref. [ 8 ]. We define the generalization of Verlinde's operators here. Consider any loop 7 on 2;. Given a representation q, we construct the operator Tq(y) a s follows. To set the problem of framing, we first consider the operators associated with the canonical basis of the homology group. The framing of a~ or b~ ~2 is given by requiring that the outward vectors are tangent to the surface (thus untwisted). Then any other loop is

Fig. 1. A state generated by the graph inside the handlebody. 524

24 May 1990

framed by smoothly connecting those generators by which the loop is presented. Now suppose there is a graph inside the handlebody which gives the vector I ~P) in the Hilbert space. We send continuously another loop 7 on Z into the handlebody, associating this loop with the Wilson line Wq(7). The new state after the action of the operator Tq(7) is given by the path integral with the presence of the original graph and the additional Wilson line. It is not too difficult to show that this definition is the realization of the holonomy operators in canonical quantization. The abelian case has been considered in ref. [ 13 ]. We show that this is the generalization of Verlinde's operators. It is sufficient to calculate the matrices o f Tq(a) and Tq(b) in the genus one case ~3. We calculate by the method developed in ref. [ 7 ] the partition functions corresponding to figs. 2 and 3, finding

(rnlTq(a)ln)= ~ ~m,,, ( ml Tq(b) ln) -Nqn . --

m

(2.3)

These formulas are just the ones obtained in ref. [ 3 ]. In fact, given an arbitrary graph F, we can also define an operator T ( F ) associated to it. The use of this general kind of operators may play a role in the fur~2 Without further noticing, by a, cycles we mean the homotopy equivalent class of 8, cycles in the notation used in ref. [ 5 ], to be more specific for later considerations. ~3 We note that these operators are denoted as ~q(a) and Oq(b) by Verlinde.

Fig. 2. Two Wilson lines resides in S2× S1,parallel to the nontrivial cycle, with an additional Wilson line linking one of them.

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Fig. 3. Three parallel Wilson lines, the partition function is not m zero only when N~ ~0. ther study o f W Z W models [ 14]. In this paper, we only make use o f these operators associated to Wilson lines. We prove the Verlinde conjecture as follows. Consider the Dehn twist along the b - ~ cycle. U n d e r this twist, we have a ~ a b and b ~ b . Note that the basis we choose for the Hilbert space d e p e n d s on the choice o f the homology basis. So in the new basis, a' = a b , b' = b , the matrix elements o f Tq(a'e), for example, will be the same as those o f Tq(a,) in the old basis. So in the case o f the Dehn twist along b - ~, we must have

Tq(a' )'ran = ( m [ ' Tq(ab)[n)'= ( m I Tq(a) I n~ = Tq(a) ......

(2.4)

where I n ) ' = ~,~ [ rn ) S,,n ( b - ~) is the new basis after performing the Dehn twist. We calculate ( m l T q ( a b ) I n ) as shown in fig. 4. We do two operations. First, ab and b b r a i d i n g twice, so there is a factor exp[2~ri(hm+hq-h,)] after resolving the braiding. Secondly, since the framing o f a b is twisted, there is an a d d i t i o n a l factor exp ( - 2 n i h q ) which contributes. We thus have

( ml Tq(ab ) ln) = e x p [ 2 ~ r i ( h , , - h~) ] ( m l T q ( b ) l n ) . Suppose the m o d u l a r t r a n s f o r m a t i o n o f the Dehn twist is S ( b - ~ ) ..... and its inverse S , 7 , ~ ( b - ~ ) = S,,,~(b), then we have from (2.4)

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Fig. 4. Three Wilson lines, among them two are braided. S ( b ) ........S ( b - ' ) .... exp[2~zi(hm, - hn, ) ] ?~ql , h i

× (m, rTq(b)lnl ) = (mlT~(a)ln)

,

(2.5)

namely, the matrix C .... = S ( b ) .... exp(2nihn) diagonalizes 7 ) ( b ) , the fusion rules, to the matrix Tq(a). This is the statement equivalent to Verlinde's conjecture, noting that S ( b ) = T - ~ S - ~ T-~. Here we use the notation that T is the m o d u l a r transformation corresponding to the Dehn twist along a, or in matrix elements, T, nn = Jm,, exp [ 2hi ( h m - ~c) ], and S is the one corresponding to z - , - 1/T, Z,,,--"Z,,, = ~,,S,,,,~Z,,. A n o t h e r condition implies that the matrix S ( b -~ ) c o m m u t e s with Tq(b). The fact that matrices Tq(b) can be simultaneously diagonalized shows that all these matrices c o m m u t e among themselves. We derive S ( b - ~) by using the surgery procedure [ 7 ]. This is one way we will use in the next section in deriving S ( b 7 ~) on higher genus. It is obvious that S ( b - I ) ..... ~--- ( m I n ) ', equals the partition function of the m a n i f o l d obtained by gluing together two solid tori with the presence o f two unknotted Wilson lines ~,, and •,,,., after performing the Dehn twist along b-~ with the first one, where m* is the conjugate o f m. To calculate this partition function, we do a surgery along a line parallel to these two lines. Gluing back again the two solid tori, we have again the manifold S~-× S ~, but now with an a d d i t i o n a l Wilson line along b. By the m e t h o d in ref. [ 7 ], we find S ( b - ~) ....

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..=E q S ( b ) 0,q( r/I To(b) I m ) . Notice that S ( b - ~) 0,q=

24 May 1990

qg-1 ( --1

q2

(TST) oq= exp (2xihq) So,q in our notation. Here we neglect the common phase depending on the central charge. We write this more compactly as S(b -1 ) = ~, exp(2xihq) So,qTq(b) .

(2.6a)

1

W 1

t~2

Wg--1

qg

q

Similarly, we have S ( b ) = ~ e x p ( - 2~iho) So2~ Tq(b).

(2.6b)

J

q

Using the matrix C defined above, we can diagonalize S ( b - ~), and hence prove that S ( b - l ) is unitary, where (ST) 3 = C has been used. C is the charge conjugation operator. Our formula (2.6) seems trivial in the case of genus one. But it is important in the case of higher genus.

3. Modular transformations in higher genus The formulas of the kind of (2.6) will play the central role in this section. In fact, let C be a cycle picked from the canonical homology basis. Again by the surgery argument and taking the central charge into account, we find exp[21ri(/~q+/~o)] So,qTq(C) ,

S(C-')=•

(3.1a)

q

q

where by/~q we m e a n h q - ~ 4 C, C is the central charge. This formula enables us to convert the problem of calculating modular transformations into the one of calculating matrices T~. The formula can be proven in another way which we will shortly demonstrate: It is a well-known fact that Dehn twists along nontrivial loops generate the modular group; that is why we only consider Dehn twists here. We denote the vector generated by the graph in fig. 5 as T(q~, ~,, w~, E~, g,). It is easy to show that this basis is orthogonal:

< T(q'~, 0~, w;, e;, ~;)1T(q~, 0~, w. e~, ~) > 1 -1 H ¢~qi,qtigOi,~,~(~wi,vv'i(~c:i,eti(~gi,~'ri" S~,o i

transformation, closed curves are mapped to closed curves. Consider a modular transformation M which maps a homotopy cycle C to another cycle C. Certainly it will change the Wilson line operator Tq(C) to Tq(C). Let S ( C ) and S ( C ) be modular transformations generated by Dehn twists along C and C, respectively. We shall consider the transformation law IT)--~ I~P) = M I T ) ,

L(c)-~L(e), s(c)-~s(e). Simply by the argument we used before, we must have

= ( c I J I M - ' T q ( C ) MI T ) ,

< q~[S(C) I T> = < ~ I S ( C ) I g'> = (q)IM-IS(C) MIT).

Tq(C) =MTq(C) M - ' , S(~)=MS(C) M-' .

(3.4)

Let us consider first a single Wilson line operator associated with the a~ homotopy cycle on a Riemann surface of genus g. Let the vector be given as in fig. 5. By some simple manipulation, we find Sq ql

(3.2)

(3.3)

Since the states l O ) and I T) are arbitrary, we deduce

Tq(a,)l T ) = ~

Modular transformations are global diffeomorphisms of the Riemann surfaces. Under a modular 526

Fig. 5. A general graph in constructing a basis of Hilbert space on lhe Riemann surface of genus g.

( ~ [ Tq(C) I T ) = (~1 Tq(C) I T )

exp[-2gi(hq+ho)] Soz~Tq(C) , (3.1b)

S(C)=~

~i 02

IT ),

(3.5)

namely, by use of this basis, Tq(ai) is diagonal. Similarly we have To(a,aF+J~)I T ) =S~,w,/So,w~[ T ) . We conclude from these facts:

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[ Tq(a,), T,,(a,) ] = 0 , [ Tq(a,), T,,(ajaG ~, )] = 0 ,

(3.6)

and some other similar relations. Now let S be the modular transformation which maps a cycles to b cycles, namely, S: a i - b i , b,-~ bi~aF~b,. Clearly, T q ( b i ) = S T q ( a , ) S -~. We then have [ Tq(b,), T,.(bj) ] = 0 , [ Tq(bi), T,,(bjbG'~ )] = 0 ,

(3.7)

which is similar to (3.6). Consider the Debn twist along a,, which keeps aj invariant, so S(aD commutes with Tq(aj). Applying the modular transformation S, we deduce that S(b,) commutes with Tq(b;). When we perform the Dehn twist along a~ or a,ai+~, the line q, or w, is twisted, so

24 May 1990

culate S(b~ ) we need to calculate T~(b, ), the given entry of the latter is the partition function of one graph in fig. 6 in the manifold S2XS ~+ S 2 X S '. We separate this manifold in two parts, each one is a manifold S 2 X S ~with boundary S 2. On each boundary there are four punctures, marked by two pairs (q,, qT) and (q',, q',*). We can close these two manifolds by gluing a ball B 3, which contains a four point block. In fact, path integrals of those graphs give a complete basis for the Hilbert space o f S 2 with such four punctures, when m varies. We then obtain two manifolds S 2 X S with graphs shown in fig. 7. We consider fig. 7a first. Cutting S 2 X S ~to result in a cylinder shows that m = q. Note that in fig. 7 we still mark the orientations of lines, even though in the case of the group S U ( 2 ) ,

S(a,) I ~> =exp(27ri/~'q,) I 7J~ , S ( a , a ~ ~,)[ 7-') = exp (2~ri/~,,,) [ ~ ) .

(3.8)

ql

rn=q

Again by the relation ( S T ) 3 = C for g = 1, we can prove the following formula: S ( a 7 ' ) = Y~ exp[2rd(hq-~C)]

So,qTq(a,).

(a)

(3.9)

q

Then by the modular transformation 5', we find that the above formula is valid also for S ( b y ~). This proves again (3.1). We now proceed to use this formula to calculate the modular transformation of the Dehn twist along by ~. Here for simplicity, we consider the S U ( 2 ) case, since the vertex ~ is unique if the fusion rule is not zero, and we need not distinguish between the representation m and its conjugate m*. As the first example, let us calculate S(b~ ) in the case of g = 2 . To cal-

e-

w'0tq

rn=q

t,

q2

(b)

Fig. 7. The graph in fig. 6 is separated into two parts.

f

w,Otq

ql

q2

\ Fig. 6. The partition function of this graph is an entry of the operator 7](bl ) when g=2.

Fig. 8. Fig. 7a can be cast into the baryon diagram. 527

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this is not necessary. After performing the cutting and gluing procedure, fig. 7a is changed to fig. 8, where the simple baryon graph resides in the manifold S 3. Now we turn to fig. 7b. Still using cutting and gluing, we obtain fig. 9. We find q~ =q2, otherwise we have the vanishing result. Separating the graph in fig. 9 into two halves, we finally reach fig. 10, with two tetrahedrons. A given tetrahedron is related to the braiding matrix, as Witten pointed out in refi [8 ]. Here we simply write down what we have obtained by the operations described above:

( q't, q'2, w'~ I Tq(b, ) lq,, q2, wl ) ( q t , q2, wl Iql, q2, wl ) = e x p [2ni(hq, +h,.a -hq~ - h , . ) ] ×

q2

q

~

.

"~ SO,qlSo,w~l ql

r,

")q2

~t

q

l/J 1

ll)

q,

qIL /

. ] w; q2

Fig. 10. Separate the graph in fig. 9 into two halves, then attach to each one a vertex, we obtain two tetrahedrons, which in turn can be written in terms of braiding matrices.

528

(q],q'2, w't IS(b?l)lq~,q2, wr ) ( q l , q2, wj I ql, q2, wl ) =

~ exp [2~zi(hu +hal +hw,t -hq, 1--hwl - - I c ) ] So,q

q

/So,~rlS~,w:

(q2 q)(~q2,q12. (3,10b)

× ~[ So.o, So,w,, B2q'w'~ wl

ql

There is another, simpler way to achieve this result. The operations involved are similar to those indicated in fig. 23 in ref. [8]. But in that way it is hard to manipulate the phases appearing in the graphs. We thus see that modular transformations generated by the Dehn twists along b cycles are expressed in terms of braiding matrices. Similarly one can calculate Tq(b2). Proceeding further, we can prove that the fundamental elements we need to know are Tu(b,) on the Riemann surfaces of genus 3. When g > 3, one can prove that all Tq(b~) can be written in the forms of those on g = 3. This in turn implies that modular transformations for genus greater than 3 can be written in terms of modular transformations on genus 3. Our result then indicates that the independent data for modular transformations are those on genus 1, 2 and 3. Modular invariance on higher genus is ensured by modular invariance on genus 1, 2 and 3. This is an interesting result. For completeness, we write down two relevant cases on genus 3. Again the group is SU (2). Firstly,

= exp [ 2zri (hq, +hw,, -hq,~ -hw, ) ]

q

w; q

Substituting eq. (3.10a) into eq. ( 3.1 ) we obtain

(qq, q~, q~, ~ , w'l, w~l T q ( b l ) [ q l , q2, q3, 02, WI, We>

Fig. 9. This is the graph obtained from the one in fig. 7b, by cutting qz and q~, and gluing them the other way. q2

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Secondly, we calculate Tq(b2), which is quartic in braiding matrices:

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= e x p [2zci(hw;

+ hw,:-["hq2 q- ho2

-h.,~ - h w 2 - h q , 2 -ho,~) ]

l /So.~ow2Soq,2aoo:n (q, s' s " q2"w'l

So.0 ~/ oo.w~oo,.:~

o.q2 o,~

q w~ q2

)

When g > 3 , To(b~) can be expressed in either

Tq(b~ ) or Tq(b2) on genus 3. For example we have ~ ( b ~ ) oc T~(b, )lg=3 and To(N) oc T~(b2)Ig=3, when j>~2. Therefore, modular transformations for g > 3 can be expressed in terms of those for g = 3.

4. Application and conclusion We discuss here only one application of our result. Further applications will be considered in ref. [ 14]. Dijkgraafand Verlinde proved in ref. [ 3 ] that if there is a nondiagonal version of the modular invariant CFT, there is a nontrivial automorphism of the fusion algebra. We can extend their result to include the braiding matrices. Suppose we have a theory in which the total Hilbert space is (4.1)

This means that the one-loop partition function can be in general written as

Z=2"H.x,

eration on genus 2 similar to that by Dijkgraaf and Verlinde. On higher genus, one can use the factorization argument to show that the partition function is the sum of products of the generalized characters in both sectors, each pair are coupled according to i~F. For genus 2, we denote the character corresponding to I ~(q~, q2, w~ ) >, which we considered in the last section. Now we have a generalized matrix H ( 2 ) such that the partition function on the Riemann surface of genus 2 can be written as Z ( 2 ) =)?.H(2 ) % Now any modular transformation must commute with H ( 2 ) , if our theory is modular invariant. Note that the following facts were used in proving the automorphism of the fusion algebra. First H is symmetric. Second, we have So/Soj=S~v/So,9 [3], this is equivalent to Su/Soj=Si,//So,;. L e t / / b e an operator such that [ ~ > = H [ ~ > . Now the above statement amounts to HTq (a)H= Tq(a). Similarly we have in the genus 2 case H(2) Tu(al ) H ( 2 ) = T~(al ). Consider the modular transformation S we discussed before. We have G(b~ ) =ST~(a, ) S +, thus we have

use~.(ql,q2,,,,,,i) to

X Bc}2,w](ql 1 0q) Bq2.w~(q,32 qq)

,Y(=®c;[¢~]® [¢~1 •

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(4.2)

where H is an integer matrix. Dijkgraafand Verlinde showed that H v i e w e d as a map, maps i to a unique F in the right sector. Then it is not hard to show that Ngjz=Nu-r. If the matrix H is nondiagonal, then this means there exists a nontrivial automorphisrn of the fusion algebra. Our consideration in this paper so far is restricted to the W Z W models. We believe that our method used here can be extended to any rational CFT. So the resuits obtained in the last section should be valid in the general case. We apply our results to some consid-

= < O I H ( 2 ) T q ( b j ) H ( 2 ) [ ~> :- ( d ) ( 2 )

IH(2)STe(a, ) S+HI ~>

= <~[Tq(b~)l W>,

(4.3)

where the fact that S commutes with H ( 2 ) is used. We note t h a t / / ( 2 ) is also symmetric, since the partition function Z ( 2 ) must be real. This is also used in the above derivation. Substitute what we have obtained for Tq(b, ) in the last section, and note that actually we can neglect the extra factors besides the square of the braiding matrix. This is because h r-hi= integer, and o f what we stated in the last paragraph. We prove more concretely that the squared terms are equal to each other. For i = 0 we find S6j/So,f=Soa/Soa= 1. So S0.#So.0 = S~,f/ So,o=Soa/So,o;this proves fSo~.rSoj _

x~ So:So,,,

?o,,So4

x~ So,So,,n

So we find that there are also nontrivial automorphisms of squares of braiding matrices. It is simply the generalization of the result obtained in ref. [3]. A similar consideration applies to the genus 3 case. In conclusion, in this paper we showed that all 529

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PHYSICS LETTERS B

m o d u l a r transformations can be expressed in terms of braiding matrices a n d m o d u l a r transformations of genus one characters. This is similar to what Moore and Seiberg suggested [4,5]. Indeed, by the use of Witten's C h e r n - S i m o n s theory, we find the relations explicitly. However, in contrast to Moore and Seiberg's work, our method shows that the S ( j ) modular transformations of the one point functions on the torus are not i n d e p e n d e n t data in reconstruction of the projective flat holomorphic vector bundles over moduli space of the R i e m a n n surfaces. Rather, we expect that they can also be expressed in terms of braiding matrices [ 14 ]. Also what we find shows that the theory is guaranteed to be m o d u l a r i n v a r i a n t for every genus provided it is m o d u l a r invariant for genus 1, 2 and 3. The polynomial equations are f u n d a m e n t a l as the basic ingredients to ensure the m o d u l a r invariance. One can in principle construct m o d u l a r transformations by starting from these equations. But things are not as clear as in the case when we used C h e r n Simons theory. For example, in proving the second equation in (2.3), one has to carry out m a n y operations [ 3-5 ]. O u r results obtained here provide m a n y relations a m o n g the braiding matrices. These relations are not easy to prove by using directly the polynomial equations. In addition, we would like to point out that the relation l~,aibia71 b y ~= 1, will provide a relation a m o n g braiding matrices. For example, the relation will be o f a quartic polynomial equation in the genus 2 case. We believe further developments along the lines in this paper are possible.

Acknowledgement M.L. is grateful to Professor R. Ruffini for his hospitality at the Physics Department, University of Rome "La Sapienza". M.Y. would like to thank Professor A. Salam and UNESCO, for their hospitality at ICTP, Trieste, and the CERN, T H division for their hospitality during the final stage of the present work.

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24 May 1990

Note added The present paper was completed in May 1989 as ICTP preprint I C / 8 9 / 1 0 6 . We thank our referee for pointing out that the papers by Bos and Nair [ 15 ] a n d by Labastida and Ramallo [ 16 ] are relevant to our work.

References

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