Bootstrap fusions and tricritical Potts model away from criticality

Volume 229, number 4

PHYSICS LETTERS B

19 October 1989

BOOTSTRAP FUSIONS AND TRICRITICAL POTYS M O D E L AWAY FROM CRITICALITY Galen SOTKOV ~and Chuan-Jie ZHU 2 International School fi)r..ldvanced Studtes. 1-34014 Trieste. Italy Rccei,,ed I 0 July 1989

We present the analysis of the tricritical 3-state Ports model perturbed by the energ:, density field r. =0¢ ~/7.~/7) and the S-matrices of the (conjectured) field theory. A general scheme for solving the minimal integrable models starting from the possible bootstrap fusions is also discussed.

The completely intcgrable 2D models obey the remarkable property of having an infinite set of commuting conserved charges P,. As a consequence their S-matrices arc purely elastic and factorizablc into products of two particle S-matrices. Although the problem of classification of all possible sets of conservation laws P, is far from a solution, the recent observation by Zamolodchikov [ 1-3 ] allows to classify a subset of integrable models: those having an action A =.4c,,n f +g,0, (0, are given fields from the conformal grid), i.e. the models with RG fixed points or certain limits (say g , - . O ) are described by the conformal minimal models. Zamolodchikov's conjecture is that for each conformal minimal model there exist a few integrable perturbations Oh2, ~2), ¢~,~, etc. (?) leading to "'minimal" integrable models with a finite number of massive particles. In other words, one should find as a consistent solution of eqs. (2), (3) and (5) below a finite set ofmeromorphic functions S,,~(O) (in the physical strip 0~
Ising model [4 ] and the 3-state Potts model [ 2 ] and the analysis for the nonunitary Yang-Lee edge singularity model [ 5 ]. In this letter we present the solution for the thermal perturbation (h, ~) = ( ~, t ~)of the tricritical 3-state Potts model ( c = -67). The main result is that the corrcsponding"minimal" intcgrable model describing the off-critical behaviour of the tricritical 3-state Ports model has the following six particle mass spectrum:

On leave absence from the Institute for Nuclear Research and Nuclear Studies. Bulgarian Academy of Sciences. Boulevard Lenin 72. Sofia 1184, Bulgaria. Permanent address: Physics Department. Graduate School. Chinese Academy of Sciences, P.O. Box 3908. P.R. China. Address after October I. 1989: International Center for Theoretical Physics, 1-34100 Trieste. Italy. Bitnet address: Zhu ,~ltssissa

0370-2693/89/$ 03.50 © Elsevier Scicncc Publishers B.V. ( North-Holland Physics Publishing Division )

m,~ =,n,~ ,

rnt,=mt~=m~(x/3+l)/v~,

m,=x//2ma,

ma=m,~(x/~+l).

(I)

Zamolodchikov's recipe [ 1-3] for describing and solving the minimal integrable models can bc summarized as follows: (a) Find the spin s ofthc conservation laws for the perturbed model P2,* t = f

./

(T2.,+ 2 d z + t 9 2 s

d~) ,

fP 2 , = J 14"z,+ ~ dz + Q2, _ ~ d~ ) , by studying thc corresponding deformations of the well known conformal conservation laws

0: 7I','+: = o = 0: w ~,?+,. (t) ~ are the quasiprimary descenHere T ~,t)+ 2 and 1472~+ dents of the stress-energy tensor T (of spin s = 2) and o f t h e s p i n s = ~ ( m - l ) ( r n - 2 ) , m = 1, 2 ( m o d 4 ) 14=

391

V o l u m e 229, n u m b e r 4

PHYSICS LE'ITERS B

currents specific for the Virasoro minimal models ~t (b) Analyse the spccific residualsymmetries of the perturbed model, i.e. describe the fundamental particlcs a,, b,, i = 1.2 ..... N a s representations o f the corresponding "group" of symmetry. This fixes the number of independent components of the two by two S-matrices S~,/(O) and the symmctry conditions for thcm. (c) Satisfy the crossing symmetry Sot,d0) = S ~ ( i n - O )

"R 5~a(O)=S~,(in-O),

,

(2)

the unitarity condition S ~T , , ( - O ) S , r ( O ) + s o ~ ( - o ) s L ( o ) = ~ . •T R R s~( -o).%.( 0) + so~( -o)sL(

o)

=0, (3)

so~(-o)s,,,,(o)=l.

and the factorization (Yang-Baxter) equations (see e.g. ref. [ 7 ] ) . The conditions (2) and (3) above are purposely written only for the very specific case of panicle-antiparticle scattering in the U( I )- or Z2,, + ~invariant models. (d) Assume the bootstrap, i.e. that all 2-particle bound states belong to the set of fundamental particles of the model; find an ansatz for the bootstrap 2particle fusions of some "'minimal subset" of particles consistent with the conservation laws and thc symmetries of the models. For example,

aa--,b+a,

bb--,c,

cc--,a.

S,-v(O) =SuM 0 - iO~c )S~o(O+ i/.-7~c) , ' U~,.+ U ~',, =2n. U,,~+

fusions are given by the tensor products rulcs for these representations. What is known in fact are only a few hints about these symmetries: ( 1 ) The relation between some o f the "minimal" intcgrable models, the affinc Toda field theories (say ,~, ,~2, f~,,, fZT, ~:s, etc. ) for specific rational values of thc squared coupling constants [8.9] and thc differcnt coset realizations of the initial conformal model. (2) Although the exact 5"-matrices for some of the simplcst models trivially satisfy the YB equations, the classical and quantum group representation method [ 10.1 I ] can play thc role of the symmctry principles at least for the integrable models bascd on the WZW minimal model. The lack of a general scheme (analogous to BPZ [ 121 for thc minimal conformal models) forces us to use "heuristic" argumcnts as the bootstrap minimal fusions consistent with {P,} and the residual symmctries remaining unbrokcn by the perturbation (for example S~ symmetry for the 3-state Ports modcls). In fact one can try to derive the fusions from the conservation laws. Hcrc wc shall present our analysis of the reversed problem: given a finite set o f massive particlesa,(p,) (or statcs la(p, ) >,,,~,,o,~ l = I. 2 ..... N) with the 2-particle fusions

a,(p, )a,(pj ) ~ ~ C,,~ak(p, +G ) •

(6a)

t,

satisfying the momenta conservation laws (P,), (4)

In other words, find some of the physical poles 0 = iL']~ of S~u(O) (.4, B=a, b, c) by using the conservation laws P, and the symmetries of the model. (e) Solve the bootstrap equations [ 3 ]

C'= n - U,

19 O c t o b e r 1989

m~ - m~ - m~ = 2m,,n, cos U~,, /-.:~+/./~a + UL = 2ft.

( 6b )

find an infinite set of conserved charges Ps consistent with the bootstrap fusions (6). such that

",a,(p,)=,',(~,)a,(p,), (5) $

following from the factorization of S,,8o, S,,,,,,( O, , 0~, 03 ) = S.,e( 0, 2 ) S . , . ( 0, 3 ) S,,,,( 02 ~ ) .

and thc bootstrap fusions discussed above. We note that points (b) and (d) in this recipe are not independent. If wc know all the symmetries of the perturbed model and the reprcsentations consistent with the conservation laws !', then the bootstrap :~

The so-called extended or H". minimal models obey, more than one such currents 16 l.

392

P~ fi a,(p,)= ~ >': (P' ) fi G(PJ). ~=l

,=l

\m,}

/=1

For N = 1 we havc only the fusion a(p~) a(p2) -, a(p,+p2). In the rapidity variables p = m c °, p = m e -°, the consistency conditions rcads I3] 2cos(~sn)=l,

s=l,5(mod6).

(7)

For N = 2 we can have one reducible fusion aa-.a, bb-.b and two new ones, aa--*b, bh--,a, or aa-.a+h, bb--.a. In the first case the consistency conditions are

Volume 229, number 4

PHYSICS LETTERS B

2y7 cos(sOgh) = y ( ' ,

2 7 ~ c o s ( s O ~ ) = 7 ,° ,

cos(sx,) cos(sx~) c o s ( s x 3 ) = ~ ,

and for }.~.h :g 0:

cos(sx,) c o s ( s x ~ ) = 1 ,

cos{sOL)=

cos(sO,%,)

1-

(8)

Two solutions ofeq. (8) are given by U-" ab= I n ,

t-,t, = ~5 R , (J~,a

(8a)

with s = 1, 4, 5, 7, 8, I1 (rood 12), i.e. the exponents ofE~, '~ and -'~

'

{~,

OL =

s = I, 3, 7, 9 (mod 10) .

(8b)

In the second case we have to solve the system of equations (7) and (8) and the corresponding solutions for the spins s arc the common sol utions of (8a), (8b) and (7). For N = 3 we have

aa ,b,

bb-,c,

cc--.a,

(9a)

with consistency condition cos(sx, ) eos(sx,) c o s ( s x ~ ) = I ,

(9b)

and solutions

x.=~n,

x2=~n,

x3=~,

s = I, 3, 5, 7, l I, 13, 15, 17 (mod 18),

xt=}n,

x2=~n,

x~=~n:,

s = l, 3, 7, 9 , 1 1 , 1 3 , 1 7 , 1 9

(10)

eos(sx~ ) cos(sx2 ) cos(sx~) = ~, cos(sx~ ) cos(sx_,) = ' , cos(sx~) cos(sx~) = ~. The corresponding solutions are given by the common solutions of (8a). (8b) and (9). The next fusions

bb-,c,

cc--.a,

lead to the following system:

cc~a+b ,

COS(SXI

) cos(sx~) c o s ( s x 3 ) = ~ ,

COS(SXl

) cos(sx~) = I ,

COS(SXa

) cos(sx~) c o s ( s x o ) = l .

COS(SX2

)cos(sx~) = ~.

COS(SX3 )

cos(sx~) = I ,

and

aa-.b+c,

bb.a,

cc--,a,

cos(sx, ) c o s ( s x 3 ) = 1 • cos(sx,) cos(sx~ ) = I • Continuing in this way one can exhaust all possible fusions of 3-particles. But we did not have a proof of exhausting all possible solutions of the corresponding consistency conditions. For N = 4 it is still possible to exhaust all the fusions. One of the simplest fusions is

bb--,c, cc--,d, dd--.a,

Xl=~rC.

obeying the system of equations

aa--,b+c,

bb ~c+a,

with solutions:

(mod 20) ,

cc--,a+h,

aa-b+c,

cos(sx, ) cos(s.<,) cos(sx3) c o s ( s x , ) = " ,

etc. The other possibility is

bb~c,

which solutions are obvious combinations of the previous ones. We give here two more nontrivial fusions:

aa--.h,

s¢0(mod7),

x,:~o,~, <~=:o:,~, ~,:~,~,

aa--,b+c,

19 October 1989

", x~=(3lr,

4//

x~=15

, x~=~x

.

s#O (rood 1 5 ) , Y 1 = ~",

X 2 = 37[. •

7 .X~ = ~(,7[,

X 4 = ~07 [ .

s=1.3,4,7,8.9.11,12,13, 16, 17, 19 (mod 2 0 ) , XI =

]'¢,

X2=

~,

s=1,2,4,6,7,8,10,

X3=~'67~,

.Y4=~R,

11.12,13,14,

16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29 (rood 30) , etc. This type of fusions can be analysed for general N: 393

Volume 229, n u m b e r 4

a,a,-.a,+,,

PHYSICS LETTERS B

a,,.a,v-.a,,

Ilcos(sx,)=2-'.

(11)

t=l

Some of the solutions of these equations can be found using the formula

,,1- 1 l-I

nj 2sin-- =P

t= I.(I.m1 = I

i f m = P ~,

t/'I

=1

otherwise.

The case when 7',=0 only for some i and some s requires a more detailed analysis. Let us assume that the initial finite set of particles contains also antiparticles and neutral particles. One can take U( I ), Zx or SU (N), etc. charged particles. We choose S~ for our discussions. The case of one charged doublet (a, d) with symmetry operations as follows: .Qa=exp(~ni)a,

Ca=a,

£2d=exp(- ]ni)d,

was considered in detail in ref. [ 2 I. It describcs thc minimal integrable model with one massive doublet rn,,=m,~ which can be identified as thcrmal perturbation of the conformal Potts model ( c = ~ ). The cxplicit construction of the lowest conservation laws shows the even spin/'~, ( s = 1, 2) chargcs arc C-odd:

CP2,=-P2,C,

i.e.l'2sd=-72,(p)2'd,

and the odd spin P2,-~, are C-even. The same phenomena also occurred in the case of the thermal perturbation of tricritical 3-state Ports model by the energy density field e=0(1/7.,/71. In fact at the critical point there exists another chain of conservation laws 0:14"(2, + ,.o1= 0 generated by the descendents of spin5 current 14'<~.o) which is C-odd. Repeating the analysis of spins and dimensions of ref. [ 2 ] for our case

0, Hi~.o)=,:.B, + . . . .

wc shall apply later Zamolodchikov's counting argument [3 ] comparing the number of quasiprimary descendents of 14"~with the number of descendents of 022/7 at level s which are not of the form L_ i ( 0 2 2 / 7 1 , _ 1" To continue the general discussion of the conservation laws corresponding to a given fusion procedure for S~ charged particles, we conjecture that the cvcn spin P21 are C-odd. The first nontrivial case is of two doublets (a, a), (b, 6):

aa--,a+fi,

[).1 = ( ~ ,

~),

one can conclude that B, ..~-i0~z~)(22/7.1/71 which is the only C-odd (spin-3) ficld wc can construct. Thcrcfore the lowest odd conservation laws is

O~W(~.o) =2P0:0c22/~.1/~) ,

bb-,a,

(12a)

with consistency condition :),,~cos(sx~) = + 7 b ,

7~cos(sx,,) = + ;'~,

2 cos([sn)= + I ,

(12b)

where the + sign is for s-odd and - sign for s-even. The solution of this system is x,,=~n,

C 2 = I =.(23,

19 O c t o b e r 1989

xh=~n,

s=1,4,5,7,8,

I1 (mod 12).

(12c)

With this sign modification one can easily repeat the previous analysis for charges particles for lhc casc of $3 particlcs also. At the end of this general discussion the natural question arises: can we find S-matrices satisfying cqs. (2), (3) and the corresponding modified bootstrap cquations for any givcn bootstrap fusions and thc corrcsponding infinite set of conscrvation laws? If yes. which are the conformal models generating these minimal integrable models? We have a dcfinite answer only for the casc givcn by eqs. (12). We shall first prove thc existence of some nontrivial conservation laws with spin s = I, 4.5, 7, 8, 1 I in the tricritical 3-state Ports model perturbed by the field e(z, ,~) =0~ 1/7.1/7). Since the field 0(1/7.,/71 havc zero Z3 charges this perturbation does not destroy the $3 symmetry of the model. Let t , +, bc the space of quasiprimary descendant fields of identity field / at level s + I: 1"s+ , = T~+ )/0:T,, and ~h.t: be the spacc of quasiprimary descendant fields of the field OJ"~-at level s. l f d i m ( ~ ' , + , ) > dim(6,~'h), it means that the mapping 0z T,+, --,20~ h should have a nonvanishing kernel. Therefore there exist ficlds T,+, (z, z:) s T , + , and 0~'_t;, ( z, f)~ O~._t;,such that

i.e./",, = j (W(~.o) dz+~oOi22z~.l/7) d#) .

0eT,, 1(z, ~) =20.0~ -~;,(z, z?) ,

To prove that the next odd conserved charge is P~,

Analogous arguments were already discussed for thc

394

( 13 )

Volume 229. number 4

PHYSICS LETTERS B

case o f the descendents o f the spin-5 current a n d spin3 field 0~22/~.,/7~. T h e g e n e r a t i n g f u n c t i o n s for the d i m e n s i o n s o f the spaces are given in t e r m s o f the characters [ 13 ]

Tablc 1 Dimensions ofthc spaces t , , , and $~,/7.,/,~ s

I

x,,,.,=

FI~=, (t-q")

X ~" (q :12m('n+l)n+°n+t)'-msl2-1t/'t''~('n+j) 1..,

--q {12''(''+')n+~'÷l)'+'n'12-':/'~m(m+t)

) ,

(14)

for the highest weight field 0h,, in the m i n i m a l model with central charge c = 1 - 6 / m ( m + 1 ). T h e tricritical 3-state Pots m o d e l have c=-67 or m = 6 . We have then (in an o b v i o u s n o t a t i o n )

~. q ' d i m ( ~ ' , ) = ( l - q ) z o ( q ) + q , ~,=0

~. q'+'/T d i m ( ~ ) l ' / ~ ' t / ' ~ ) = ( I - q ) z , / 7 ( q ) , ~ q~dim(~V~÷5)=(I-q)zs(q), A;n,~lTh(22/7.1/7)~

. . . . . ~.vs+3

dim ($,)

I

1

0

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 1 0 2 0 3 I 4 2 7 3 10 7 14 11

0 1 I I 2 2 3 4 5 6 9 I0 13 17 21 25

S

"~=11

q~+2217

dim (T,.,)

22

Table 2 Dimensions of the space~ W~÷, and ~22/, ,/,,

5=ll

S=(I

19 October 1989

dim(~/~+, }

dim(~,)

1

J=(I-q)z,_2/7(q),

2 3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17

(15) where z , ( q ) =Zh,., (q, m = 6), a n d 7.~/~(q) =Zh, 2 (q, m=6), x ~ ( q ) = z h , , ( q , m = 6 ) , Z22/7(q)=z/,,.s(q. m = 6 ) . T h e results o f the calculation for s~< 17 arc shown in tablcs I a n d 2 below. F r o m these two tables o n e discovers the C-even c o n s e r v a t i o n laws with spin s = 1, 5, 7, I 1 a n d C-odd ones with spins s = 4 , 8. We e o n j c c t u r e that there exist an infinite n u m b e r o f conservation laws with spin s = 1,4, 5, 7, 8, I 1 (rood 12 ). These spins are exactly the Coxeter e x p o n e n t s o f E 6 which was expected because o f the existing o f an alt e r n a t i v c c o n s t r u c t i o n o f the c o n f o r m a l tricritical Ports model as the first model o f the WE6-extended algcbra. As we have shown eqs. ( 1 2) give b o o t s t r a p fusions consistent with the c o n s c r v a t i o n laws a n d with the $3 s y m m e t r y o f thc p e r t u r b e d model. A c c o r d i n g to the analysis o f ref. [ 14] w h e n the anti-particle appears as b o u n d statc o f the 2-particle aa scattering thc corrcsponding rcflcction a m p l i t u d e vanishes, i.c. S ~R = 0. T h c n thc YB e q u a t i o n s are trivially satisfied by S a n d

1 0 I I 2 I 4 3 6 6 9 10 16 16

I 0 I 1 2 1 3 3 5 5 8 8 13 14

24

S v a n d wc can start solving thc b o o t s t r a p e q u a t i o n s by using the k n o w l e d g c ( 1 2 ) o f t h e three poles o f the S-matrices: "It*

=

[:; U- alT "=

Uaa =

3~z .

T h e simplest e q u a t i o n s are related with the fusion

aa -. d: 395

Volume 229, number 4

PHYSICS LETTERS B

/ I Sad(8 ) :Saa( 8 - ~in)s",~(8+ [izt),

S,,a(8 )=,Saa(8_',

3i n.l

)ST (8+~in.)

&+= Oq,,,:)2 (/;,/,)~ (~,,) ~Ct;,,3)~A,,~. S ' , = - f , , , , (11,~)'(];,,,)~(];, ,,)~ (f,,,,) ' ,

,

Sbc : , . ~ :fl/12(fl/6)2(.fl/a)2Ui/3)2fs/12

or cquivalently

S",,( 8)S"~( 0 - ]in)S,~(O+ ] i n ) + I.

(16)

The minimal solution ofeq. (16) which satisfies the unitarity condition has the form

S"~( 8) =

19 Oclober 1989

sh( ~O+ lift) sh( ~O+ ~i~r) sh( ~8+ lift) s h ( ~ 8 - lift) s h ( ~ 8 - ~ i n ) sh(~O- fin)

- f , / ~ ( 8 ) f , , , ~ ( 8 ) f , : , ( o) ,

,

S,,,,=S,;,,=/;:'.,(A,:~)3(£,,,)~(/;:2,)"(/;:8)3.~

S,, = - f , : , 2 f , , , ( / ; : , ) e f , , , A : , ~

,i=, ,

.

S,,a= - (I;,:,~)"(/;,:~)'~:,)~(/2,,)5(:.,/,,_) ~ .

S~,,=f,/~,(/;/~)2 ~:,~) 3(f,:~,)3(/3/8)2./;, ~/2,. ( 17 cont'd) The full fusion structure of the tricritical 3-state Polls model away from criticality is given as follows:

where .L(8) =sh( ~0+ ixrr)/sh (~O-ix~r). Obviously S"~(8) has two simple poles with positive residucs at 0, = ]izr and 02= ~,in which corrcspond to the physical particles O and t~ with masses ma=rnu and m a = m ~ ( v / 3 + l ) / v / - } according to eq. (6b). The simplc pole 8= ~in has negativc residue and represents a particle in the cross-channel. In fact by using of the crossing symmetry we get

a c - . a + b , ad .h ,

SJa(8) =s",,( i n - 8 ) = -.li,,~( O)fi/~( 8)f~/,:( O) ,

cc-.c+d,

which has only one simple pole with positive residue at 8= ~i~r. This polc corresponds to the neutral physical particle c, arising as an aa bound stale: aa-.c, with mass rn, = v/-} ma. The scattering amplitude S,/,(O) can be obtained from the bootstrap equation

As one can see from ( 17) the scattering S-matrices we have obtained have a rich multipole structure. The appearance of double poles was obscrvcd[ 15 ] and explaincd [ 161 in the case of scattering amplitudcs of the sine-Gordon model. This mechanism was recently generalized by Christe and Mussardo in ref. [4] to multipoles. They also argue that in the case of highcr odd poles (of order 3, 5. etc.) one can havc new bound stales if the residue of the corresponding simple pole appearing in the Laurcnt expansion of S(8) has an appropriate value. The direct chcck for the higher odd polcs in (17) shows that no new particles appeared. The only change is in the fivc of our fusions ( 18 ), namely

S,,t,(0) = Sa,,(O-- ~ in)S.,,( 8+ ~ i n ) . and the result is

Sab(O) =Jil/8(O)./;,'8(8)fl/24(O)fs:24(8) Lf7:24((.~) ]2. The pole structure ofSah lcads to the following fusions:

ab-.c+d, with r n a = m a ( v / 3 + l ) . Proceeding in this way we obtain the following full solution of the bootstrap equations with only six particles:

aa--.ti+6,

ad-~c,

ab-~(l+6,

aS-.c+d,

t~h-.gt,

bb--,no bound states,

bc-.a,

bd-~a, dd-,noboundstates,

bb-.a+(6),

bS-.(d),

bd-.a+(b),

cd-.c+(d),

cd-+c.

dd-.(c)+(d),

(18)

(19)

where parentheses denote the bound states coming through the odd multipoles.

So,, = S,,i, =.1;,:~ (I;:,~) 5 ; : ~ . A , s f , , :,.,.

References

S.: = Sa,. =.fl /~f ,/ '-,.f, / "-, f ,:* , S"a=S,a=fl/2(j;/6)2(j]l/,)~(fl/x)2f~/,~, 396

(17)

[ I ] ~..B. 7..amolodchikov, JETP l.etl. 46 (1987) 160. 121-~.B. Zamolodchikov. Intern. J. Mod. Phys. A 3 (1988) 743.

Volume 229, number 4

PHYSICS LETTERS B

[3] A.B. Zamolodchikov, Integrals of motion and S-matrix of the (scaled) T= "1~.lsing model with magnetic field, prcprint RAL-89-001 (January 1989). [4 ] P. Christe and G. Mussardo, Integrable systems away from criticality: the Toda field theory and S-matrix of the tricritical Ising model, preprint UCSBTH-89-19 (May 1989). 15 ] J.L. Cardy and G. Mussardo, Phys. Lett. B 225 ( 1989 ) 275. [61V.A. Fateev and S. Lykyanov, Intern. J. Mod. Phys. A 3 (1988) 507; lectures given in II Spring School on Contemporary problems in theoretical physics ( Kiev, 1988 ), preprints ITP-88-74-76P ( KJev, 1988 ) [in Russian 1. [ 7 ] A.B. Zamolodchikov and A1.B. Zamolodchikov, Ann. Phys. 120 (1979) 253. [ 81T. Egucht and S.-K. Yang, Deformations ofconformal field theories and sohton equations, preprint RIFP-797 (March 1989).

19 October 1989

[9 ] T.J. Hollowood and P. Mansfield. Rational conformal field theories at. and away from, cntlcality as Toda field theories, preprmt (May 1989). [10]E. Ogievetsky and P.W. Wicgmann, Phys. Left. B 168 ( 1986 ) 36O. [ I 1 ] E. Ogacvetsky, N Rcshetikhin and P.W. Wicgmann. Nucl. Phys. B 2801FSI81 (1987) 45. [ 12 ] A. Belavin, AM. Pol~akov and A.B. Zamolodchiko~,. Nucl. Ph)s. B 241 (1984) 333. [ 13] R. Richa-Caridi. Vacuum vertex representations of the Virasoro algebra, in: Vertex operators in mathematics and physics, MSRI Publication No. 3 (Springer. Heidelberg, 1984) pp. 451-473. [ 141 R. Krberle and J.A. Swicca, Phys. Lett. B 86 ( 1979 ) 209. [ 151 A.B. Zamolodchikov. Commun. Math. Phys. 55 ( 1979 ) 183. [16] S. Coleman and H.J. Thun, Commun. Math. Phys. 61 (1979) 31.

397