Quantum group approach to affine Toda field theory

o, E Weyl group of su (3) ,

ao

1 2P( P + 1)

p=e 1 +e2 ,

,

k 1 , k2 E Z ,

a ± - 2(ao±

w;-e,=S;j

0 +2),

(i, j= 1 ., 2),

(1 .4)

and p is a positive integer satisfying p > 3. The coboundary operators (d,} are composite operators constructed from the screening charges . The cohomology of this BRST complex is Hi := (2)

Ker di

lm di _ 1

for j * 0 ;

H o is non-zero and isomorphic to an irreducible W3 module .

Especially, elements of H ° exist only in Fß , where (3 = a(,P + (l,w l + 1 2 w 2 )a + + (liw l + 12w2 )a_

(1 .5)

with 11 ,12 > 1, 11 + 12 < p - 1 and l 1,12 > 1, 11' + 12 < P . We can easily verify that the BRST complex (1 .2) survives after the perturbation (1 .1) is turned on . So according to the idea of ref. [7] this BRST complex is the hidden BRST structure of ;u,-(3) Toda field theory at p 2 = p/2(p + 1). So su(3) Toda field theory at ß 2 =p/2(p + 1) is the effective theory of the Z3 symmetry preserving deformed W, minimal unitary model labelled by the same p. This is confirmed by the renormalization group analysis in ref. [19].

T. Nctkatstt / Afjne Tocfct ,fielcl theoty

Si)2

Although the integrability of quantized su(3) Toda field theory with a pure imaginary coupling constant (at least, with special coupling values) appears to hold from the discussion in ref. [6], the corresponding S-matrix theory is little known. In this paper we try to construct the factorizable S-matrix of this theory by imposing the quantum group su(l(3) symmetry. The assumption of this symmetry is natural because su(3) Toda field theory with a pure imaginary coupling constant is the natural extension of the sine-Gordon theory which has suq(2) symmetry. In sect. 2 we discuss our approach in the case of suq(2) symmetry (i.e. the sine-Gordon theory). In sect. 3 we construct classical soliton solutions (we call them "soliton triplet" and "anti-soliton triplet") of su(3) Toda field theory with an imaginary coupling constant . Based on the analysis of the spectrum of these classical solutions we construct a vertex model type S-matrix making use of the quantum R-matrix with suq(3) symmetry. We also construct an R-matrix relevant for the description of the crossed channel scattering by means of the fusion procedure . In sect. 4 we construct the S-matrix from the "fused R-matrices" in sect. 3. Here we switch to the face model type description of S-matrix, which is related to the vertex model type description by vertex-face correspondence . We assume the unitary property of the constructed S-matrix even at generic values of coupling constant . And then we discuss the consistency of this S-matrix . In sect. 5 we study this S-matrix at special values of the coupling constant . At these values of the coupling constant the S-matrix is consistent and moreover the Hilbert space of the theory must be restricted. We describe an explicit correspondence between this restricted Hilbert space and the corresponding BRST cohomology classes. In sect. 6 we give some discussions. 2. From suq(2) to sine-Gordon S-matrix In order to describe the strategy of our analysis, it is very useful to review the known result of the sine-Gordon theory which has the quantum group symmetry. In the quantum sine-Gordon theory a soliton and an anti-soliton form an su q(2) doublet, i .e . a soliton is spin-up and an anti-soliton is spin-down. The S-matrix of these elementary particles is given by [14,15] SsC,(e) = ( -1)s(8)P X

(

e",7 /4

e -i, /4

~

qu/4

X

9-u/4 x

X( e -irr/4

e«/4

®

~

e -ir~ /4

R(u : q)

®

e irr/-~

eiTr/4 q-

q-i~/4

)

q u/4

~~ /4

e-i7r/4

g11/4

®

9

q-u/4

T. Nakatstt / Affine Toda field theory

503

where

it = io/'r , q = e - ""' 2' 1 y o (rapidity difference)

(2.2)

and P= is a permutation matrix. R(tt : q) in eq. (2.1) is the well-known quantum R-matrix, which commutes with the quantum group suq(2). So the Su q(2) symmetry is hidden in this R-matrix. AO) in eq. (2.1) is the scalar function which determines the bound-state structure of the theory and is given by x ôl ôl Sln(ô?r 2 /y ) )( s(o) = r 1 + -o r 1 + -(iir -o) n R k (o)R k (isr -o), y

RJO) -

y

k=

r(8i(-2k7ri + o)ly)r(1 + 8i( -2 k7, i + o)/y) . r(8i(-(2k + 1)7ri + o)/y)r(1 + 8i(-(2k - 1)7ri + o)ly)

(2.3)

The renormalized coupling constant y is related to the sine-Gordon coupling constant p2SG : y

~SG

1

(2.4)

1

- 3sG/8 Tr

The construction and justification of the S-matrix (2.1) was, in some sense, the corner-stone in two-dimensional integrable quantum field theory. But it is interesting to see to what extent the S-matrix (2.1) can be determined by its symmetry. The manifest symmetry in (2.1) is the quantum group su,(2), which is hidden in the R-matrix . The R-matrix is given by [211

R(u :q) = r[1 i

+u]Eii®Eii+

+ F [u]E ij 0 Eji . i*j

rq "Eii ®Ejj +

i
~ q -uEii 0E

i>j

jj

(2.5)

Here 1 < i, j < 2 and Eij is the 2 x 2 matrix whose matrix element is (Eij )k, = 5ik Sjt . . The symbol [ x ] denotes [ q a - q--v]/[q - q - ' ].

T. Nakatsii /

504

Affine Toda field theory

For the notational convenience we write the matrix elements of eq. (2.5) as RWkl (1 < i, .i, k, l < 2) : R(u :q)ei 0ej = 1: R(u)i,'1ek0e k,1

with e1 =(1,O), te 2 =(0,1). (R(u)k1) satisfy the "crossing symmetry" relation:

where t = 3 - i and $ 1, $, = q' /4, q- 1/4 . The signs on the r.h.s. of eq. (2.6) are easily fixed by one of the following two methods : (a) Modification of the gauge transformation, 92 =

t (b)

e i :r/4

(2 .7)

92

Introduction of the new variables {fil, f =e

/4

f2 =

e ir. /4

(2.8)

Then eq. (2.6) is turned into R(u)l.,

ifi f = R( -1 _

fjfk

j,1)

j k f fk fifl

g'gi gigk

(2.9)

On the other hand, the R-matrix satisfies the "first inversion" relation : F

in, n

R(u)k'1n R(_u)in~T = [1 + u][1 - u]Skis1j .

(2.10)

At this stage we define the S-matrix as S(u)ki=f(u)R(u)kl

fifl fjfk

(

gigk

j

-a

gig/

(2 .11)

Here f(u) is the scalar function which is determined in the following. First we note that S(u)k'i satisfy the relations S(u)iki = S( - 1 - u)~"

ij ES( u)km1 n S( - u)inn = skiS1j .

m, n

(2.12)

These are the crossing symmetry and the unitarity property of the S-matrix (2 .11).

T. Nakatsu / Affine Toda field theory

505

From eqs. (2.9) and (2 .10) the conditions (2.12) reduce to the following equations for f(u): f(U)f(-U)[1

+u][1 -u] = 1,

f(u) =f(-1 -u) .

(2 .l3)

Comparing (2 .11) with (2 .1) we can see easily that s(0) [eq. (2.3)] satisfies the above equations if we change the variable u for the rapidity difference 8 by (2.2). At this stage we want to discuss our criteria in selecting acceptable solutions to the functional equations (2.13). The bound-state spectra of the theory (2.11) are determined by the physical poles of f(u) which satisfies (2.13). We consider it to be very natural that the bound states form some irreducible representations of su9(2) because we assume the suq(2) symmetry in the theory . So we must choose the solution of eq. (2.13) whose physical poles correspond to some irreducible representations of suq(2). Such a correspondence can be checked by comparing the physical poles of f(u) with the following spectral decomposition of the R-matrix (2.5): R(u : q) _ [ 1 + U ] Ptriplet -

+ [1

-

U ]Psinglet

sin 87r2 (8/i7r- 1) /y P -

tnplet triplet

Sin 8îr 2/,y

+ sin 87r2 (8/i7r+ 1) /Y P sin 87x 2/y

_

singles

_ (2.14)

Here Ptriplet 9 Psinglet are the projectors onto the spin triplet and the spin singlet representations . Of course s(e) [eq. (2.3)] is the consistent solution from this point of view. To end this introductory section we give some comments. First, as we have above, the sine-Gordon S-matrix can be constructed from the quantum group data by solving some simple functional equations. Second, in the case of 87r/y being rational, the physical Hilbert space of sine-Gordon theory is restricted because of their hidden BRST structure [7,12]. To describe this feature of the sine-Gordon S-matrix theory it is convenient to change the vertex model type description of the S-matrix to the face model type S-matrix [8]. 3. Vertex model type S-matrix In this section we try to construct the S-matrix of su(3) Toda field theory with an imaginary coupling constant . Let us begin by constructing classical solutions of su(3) Toda field theory. The starting point is the lagrangian û(3>(`P) `P)

1

= 2 d,,4) - P+ +

m2

-~

ß2 i-1,2,3

e

(3 .1)

_5(6

T. Nakatsu / Affine Toda field theory

Here 4 = (01, 02) . e 1, e, are simple roots of su(3), and e3 = - (e 1 + e 2 ). Eq. (3.1) has two classical symmetries. The first is the Z3 discrete symmetry which corresponds to the cyclic permutation of {e l , e,, e3}, and the second is the invariance under + H + + (2-,T/ß)(11o)1 + l,w,) (11.) 1., E Z). Here {(OA .1,1,2 are fundamental weights with ej - wj = Sij. The second translation symmetry signals the existence of classical (anti-)soliton solutions of this theory. If we set +(1,(1) = w 1 0 (0 is a real scalar field) and insert +(1, (,) into eq. (3.1), we obtain the following sine-Gordon lagrangian : ~) ( ( )) _ ~(~``~) (

m2 2 cosße . +

(3 .2)

It is well known that the sine-Gordon theory has a soliton OSG and anti-soliton OsG , So if we set 4(1,()) = w 1OsG, this is a solution of eq. (3.1). Moreover, by the Z3 symmetry we can obtain other solutions which are equivalent to +(1 , 0); +(1,(1) = w10SG +(-1,1) = ( +(0, -1)

(6 l + w2)OSG w20SG

(3 .3)

These solutions form a vector representation p of su(3). We call them the soliton triplet . In the same way we can construct an anti-soliton triplet : +(- 1,0) = w10SG +(1, -1) _ ( -(0l + w2)4)SG +(0,1) -

w2(')SG

(3 .4)

These form a representation 0Here we remark that in the case of classical Z(n) Toda field theory, the Z symmetry of the lagrangian can be interpreted as an su(n) symmetry at the level of the classical solutions. For example, in the case of n = 4, we can construct four types of classical solutions by using the Z4 symmetry. Two of them are interesting. The first type forms a four-dimensional representation of Z4 and the second forms a two-dimensional representation of Z4. These two types of solutions have the same mass at the classical level. So they can be combined to form a representation 8 of su(4). In the light of these classical results, we try to construct the S-matrix of these particles assuming their existence in the quantum theory of (3.1). We also assume

T. Nakatste / Afftne Toda field theory

507

that our case also has the same feature as in the case of the sine-Gordon theory, i .e . the classical su(3) symmetry is enhanced to the quantum group suq(3) symmetry, which is hidden in the corresponding R-matrix. So we propose the following form for the S-matrix of the soliton-soliton scattering in the quantum theory of (3.1): (3 .5) Ssoliton -soliton( 8 ) - f (e) P - RO0(tt : q) , where P is the 3 x 3 permutation matrix. Here " - " means that the l.h.s. of (3.5) is equal to the r.h.s. of (3.5) up to some suitable gauge transformation . Furthermore we set 3i q=e -'r.-i/y (3 .6) where 8 denotes the rapidity difference . The coupling constant y can be expressed in terms of ß'- in (3.l ). 16-ß-' -' . (3 .7) y= I -2ß Eqs. (3.6) and (3.7) are the natural generalizations of the sine-Gordon case. The scalar factor f(8) in (3.5) will be discussed in sect. 4. In this section we discuss the quantum group nature of (3 .5). R 00( u : q) in (3.5) is the quantum R-matrix which acts on the space V0 ®V0. Here V E3 - C3 is a vector representation space of su~l(3) (in our language the one-soliton Hilbert space) . Its explicit form is given by Rpp(tt : q) = y [ 1 + tt ] Eii ® Eii + i

Y q"Eii 0 Ejj +

i
y, q -"Eii ® Eü

i>j

+ E [u]Ei! 0 Eji i :A j

Vp ® V[:] -4 VE] ® V[:) .

(3 .8)

In the above i, j run from 1 to 3 and [x] denotes [q -` - q --' ]/[q - q_'] . Eij is the 3 x 3 matrix whose matrix element is (Eij )k l = 5ik Sjl . Here k, l denote the basis of V0. R 0(it : q) obeys the Yang-Baxter relation (1 ® ROO(v : q))(RC:O(u + v : q) ® 1)(1 ® R[=(it : q)) =(R®(ct :q) ® 1)(10 ROO(u+v :q))(ROO(t' :q) (9 1), (3 Vp ® Vp ® Vp -i Vp ® VC:] ® VC] .

.9)

And moreover R 111-:1 (u : q) is a commutant of the quantum group su,,(3) algebra ~'', [21]. The quantum group su(,(3) is the associative algebra generated by (q ±'t,

508

T. Nakatsu /

Affine Toda field theory

Ej , F;}i =1, 2 obeying the following relations: gtt,/2E_q-H,/2 .l

q",,1-)E j,

gHj l2Fq -H;/2 = q-a;,/2F H, - - H, q E;F -FE;=Sij -1 , q -q q

E;Ej -[2]E;EjEj +E~E;=0

if a;j = -1

.= 0 F; - F - [2] F; F Fi + F-'

if a lj = -1,

(aij) _

2

_1

-1

(3 .10)

2

suq(3) algebra has the following coproduct 4 : ±H,/2~ ;/2) =q ±H;/2 4(q±H ®q 4(E,) =q+H,/2®Ei+EI®q-H;/2, 4(Fi)

=q+ t1, / 2 ®Fi +Fi 0q -Ii, /2 .

(3 .11)

Here we note that an irreducible representation of su q(3) is isomorphic to its counterpart of classical su(3) and thad the tensor product of two representations of su,(3) is defined by the coproduct 4 (3.11). Under the above definitions R00(u : q) has the property [RE][3(u : q),4(A)1 = 0,

A

E

su,(3) .

(3.l2)

From the property (3.12) R 00(W q) has the following spectral decomposition [211: R00(u :q) = [1 +u]PCO+ [1 (3 .l3)

-u]Pe .

Here Pm, P8 are the projectors from V 0 ®V0 onto V ., V . VM, V 8 denote the irreducible components (with respect to su q(3)) of V0 O V0. How can we construct other S-matrices from the point of view of su q(3) symmetry? According to our assumption, the classical antisoliton triplet (3.4) of su(3) should survive in the quantum theory . So we must obtain, at least, the R-matrices which correspond to antisoliton-soliton and antisoliton-antisoliton scatterings . These R-matrices can be obtained by the fusion procedure of the

e

our can ®VB spectral ordinary ucase we R8G(u = we easily (V0 -1, as note which is lattice set are well we ®V0) decomposition prove eq choice q) R0 arbitrary will q)is models (1 defined = as 0(-1 developed ®Pq(RM(u ®V0 see (3ofin becomes -numbers u0, q)vEF](_e®,0)[u Here uo by sect 1/2] q) to (3 u, and a only V mainly we P8 4,8u, is (~(u can we using u0 briefly +% / because ®V0, Amine may _ specify be by -1, E] the +derived take Vp®V® explain Toda the 0 i+231 u,_+2 z®1)(10 Yang-Baxter u, them of q) [u®1)(1®Rm(u field Kyoto any R00(u v=® (30from this theory values as 1)(10 3]v RM(u Under school fusion the From q)Ive®vp ~(u as relation general dNO) the +u, far method [21,22) eq +u, as choice -(3 (3 rule u0 iVB and by -[22] q)), (3 u, If O studying we apply we can VD--> set we it

T. Nakatscs

R-matrix, integrable to

509

.

. First

RO®00(u :q) =(RC][3(u+u, :q)01)(10Rm(u+uo :q)), (VE] ®VO) 0VO~VO ®(VO®VO), Here

() , u,

. R

00

(3 .14)

(u : q) satisfies

RO®00(u :q)( R[]E](uo-u, :q) ® 1) =

(1 ® i?b[:3(uo-u,))(Rm(u +u o, :q)

We u 0 u,

:q)) .

.15) . (3 .15)

(3 .l5)

.9).

RO ,& OO( u : 9) ( pe 0 1) =

:q)

Here

:q)) .

:

.13).

restrict

.e.

VO

. :

(3 .16)

.16) :

.

However,

. > UO=-2,

>

3.17 ()

.

.17)

.)

(The define

: RBO(u :

(3 .18)

.

VB®Vp--> The

:

:

u

.18) RBO(u : RBO(u :q)v

(-0®ße), -eeo,

=

(3 .19) ~(-0

10

T. Nakatsu /

Tocki fïelcl theon,

(_ (s ,) is the highest weight vector of the irreducible component 8 ® V Q. The subscript ® ®, o denotes adding a box to the third row of ,

For example, V

Affine

in V

t,

~

e

® from which we can read off the ordering of the tensor product. Next we define R (it : q) and calculate its spectral decomposition . First we set

80

Roo=(1 ®Rp[3(tt+ l :q)0 1)(1 ® 1 ®Rpp(u :q)) X(R®[3(it :q) 0 1 0 1)(1 ®RpO(it - I :q) 0 1),

(vo ® VD) 0 (Vo ® Vo)

~

(vo ® vo) ® (Vo ® vo)

(3 .20)

Along the same line as (3.16) we can prove that we can restrict (V 0 ®V 0) (V® O V®) to V 0 O V . Then we define R (it : q) : V ® V V ® V as e R®8(u :q)

e

00

1

= [u ] [tt - 1 ][tt +2]

e

e

8

(10 R®(zt + 1 : q) ® 1) (1 ® 1 0Rm(1t :q))

x(ROp(u :q) 0 1 0 1)(10 R®(tt- l : q) ® 1),

VB8

®

Vee V88 ® VBB .

(3 .21)

-11

In eq. (3.21) the following equality holds:

1 R88(it : q) = (1 ®REP(u + '-, : q))(REJ(u - '- : q) ® 1) . [it [ + 2]

(3 .22)

The spectral decomposition of (3.21) reads R88(u : q) v ®(= (0oßo)®,0) = R88(u : q) ~ [4]

- [u +

1]c'

~[2] c'O~

®(_0®,(p®,8))

=(8®~p)®~O)1 (3 .23)

To derive eq . (3.23) we used [22] L'(o ®; [rrt ]) ® Io = "o®, ( [rrr]®,~) +

~[ bij -

1] [bij + 1]

[bij ]

L

'o® ; q[m] ®lo) .

(3 .24)

T. Nukatsru /

Affine

Toda field theory

511

In eq. (3.24), i =#j and [m] = [1,,12, l;] denotes the Young diagram, which has l; boxes at the ith row (i = 1, 2,3). bij = bi - bj with b; = l; - i. Therefore,

Ree(u :

q) r( e® . O)®,O

-L2 : q ) 0 1 )"(00,O)OD

[ u + 2]

(10 R~(u + 2 : q))v(p®,8)®,p

[u + 1] (10 RRD(u + -! [u + 2] (3.25) In eq. (3.25) we used eqs. (3.19) and (3.24). The second line in eq. (3.23) can be derived similarly. The R-matrices in this section enjoy several Yang-Baxter relations. For example, (10 RRD( v : q))(RE]o(u + 1 , : q) 0 1)(10 I?bo(u : q)) (ROD(u : q) 0 1)(10 REJO(u + v : q))(R[]0(1- : q) 0 1), VO 0 V[:] 0 VC]

V[:] 0 VO 0 VO,

(10 Rn(t, : q))(RE]o(u + v : q) 0 1)(10 RE]D(u : q)) (R[]D(u : q) 0 1) (10 RE]D(u + v : q)) (Ro](1' : q) 0 1), VO 0 VO 0 V[:] -3. VED 0 VO 0 VO, (10 RM(t, : q)) (RE]B(u + v : q) 0 1) (10 REJ](u : q)) (R[]B(u : q) 0 1) (10 REJ](u

,

+ 1 : q))

(REI](l , : q) 0 1),

VO 0 VB 0 VO -~ VO 0 VO 0 VB .

(3.26)

T. Nakatsu / Affine Toda field theory

512

4. Face model type S-matrix In sect. 3 we have studied the S-matrix of the quantum theory of eq. (3.1) from the view of sU Q(3) symmetry and we have used the vertex model type description of the S-matrix [i.e. eq. (3.5)]. However, to proceed further, it is convenient to change the vertex model type description to the face model type one. This is done by the vertex-face correspondence [21,231 . First we decompose the N-body solitons Hilbert space (V )®N into the direct sum of its irreducible components (with respect to suq(3)), i.e.WV )ON= ® m [m]V[]-Here V[,,,, is an iireducible component of (V0) ®N and [m] denotes the Young diagram which corresponds to the highest weight of this irreducible component . The sum with respect to [m] denotes the irreducible decomposition of (V 0) ®N. With these notations we can write the irreducible decomposition of the (N + 2)body solitons Hilbert space as (V )®(N+2) = ® [1121( ®-ijV ). 0 ([ni]®j0)®j0 The direct sum with respect to i and j denotes the irreducible decomposition of V[, n] ® V o V13. We can study the following operator on this space :

1 ® R00(u : q) : V[,n1® (Vo ® VD)

-4

V[ ,n ] ® (VE] ® VD) .

(4.1)

From (4.1), because of eq. (3.12), we can define the set of numbers W0®0(u) = WOOD

a

b

d

c

u

a, b, c, d : integral weight of su q (3)

(4.2)

as follows: 1 ® R00(u : q) V([,n]®,D)oj0 = Woo[]

la+i

a +i +j

a

a+i

a+j

a+i+j

u

V'([tn] (& ,0)®j0

for i$j

1 ® R00(u : q)UC[m]®~o)® ;o = u WOOD

a a+i

a +i

 a+2i

v([in ]® ;0)®j0 . (4 .3)

T. Nakatsu /

Affine

Toda field theory

513

In eq. (4 .3) c.([ ,]®;0)®jD denotes the highest weight vector of V([m1® :3)®A0 which is an irreducible component of V[,, ® V11 ® V 0. For notational convenience we define an integral weight a by a = w + p, where w is the highest weight corresponding to the Young diagram [ m] and p is half the sum. ofAthe positive rests . a, j run over 1, 2 and 3: the weights of V,:,. Explicitly, we set 1 =eel, 2 = - w, + w2 and 3 = - w2(_ - (1 + 2)). The other elements of WD D(u) which do not appear in (4.3) are all equal to zero. WD D(u) is called a face operator for p ® p, whose explicit form is given by

WOOD

u

=[I +u],

a+i

a+2i

a

a +i

[aij - u ]

a+i

a+i+j

[ai,]

a

a+i

+j

for i

* j,

[aij +1l [a j

for i * j, (4.4)

[a ii~ la

a +i +j

where a;j = ai - a, and a; = (a + p) - i (i = 1, 2,3). The Yang-Baxter relation (3.9) can be rewritten in terms of the face operator WO®O(u),

WOOD g

-

u )Wool] f e

b

~Woo cl

a 18

b d I u ,

WOO

d I u ,W ~

a

~1f

g I u

IWOOD

Eq. (4.5) can be derived from eqs. (3.9) and (4.3). Other face operators W o D(u), WDO and W

(u)

f

1e

g

(4 .5) d I u,'

8 e 8® eqs. (3.18) and (3.21) can be obtained in the same way as in the case of Woo 0, i.e.

8(u) which correspond to

eq. (4.1). But here we use the fusion procedure of face operators which is developed in ref. [24]. First for convenience we perform the following gauge transformation for

T. Akrkatstu / Affüre Toda field theory

'5 14

W00 0(u) which preserves the Yang-Baxter relation (4.5): Woo[]

a

b

d

c

11

-)1

WO®0

= WO®0

Here s(a, a +1) _ are

a

b

d

c

a

b

d

c

11

Il

s(a,b) s(b,c)

s(a,d)s(d,c)

.

(4 .6)

.0(u) VI- I,; # ;[ a; j ] [ a;j + 1] . The explicit matrix elements of W-,

a

a

+t

a+2i

a+i

[a;j - ct] [aij]

r W0,00

for i *J,

a

R+i

[a ij _ 1]

a+j

a+i+j

au

In this gauge we apply the fusion procedure of ref. [24], which is parallel to the ,(u), case of R-matrices, and we obtain W,,, . (cr) and W W,,, Subsequently, we perform the gauge transformation WIR d

b1 Il ) cl

I

-~

-

Wla kd Wi

bI c

a

b

d

c

0

. B11B(u)

u

tc

~js(a,b)s(b,c)

VH j , j [a jj ][a j.j + 1] and s(a, . _ Vr[a_ jk ][a jk, + 1][a ;_~( u )

Here

s(a, a + î) _

(4 .8)

s(R,d)s(d,c) a +

î + .1)

satisfy the Yang-Baxter relations which correspond to eq . (3.26), and that the gauge transformation (4.8) preserves these Yang-Baxter relations . So W(ct ) also satisfy these Yang-Baxter relations .

Affine Toda field theory

T. Nakaisu /

515

The explicit form of W(u) reads a+i WD®p

®

W8 0

wo o

W0®8

Q+i

a+i+j

a

a+i

a+J

a+i+j

a+j

a+i+2j

a

a +î +Î

a +k

a

a

a+i+j

a

it

II

a+l+j

a+2t+j

e

lt

.

[a ;j -it]

LaijJ [a,j+ll[a,j-11

[ltJ

2

[ail]

jk + 11

+ 1J[ajk- - 11 [a i ]2 [a [al-

" a +i

WO®

=

[a ;k - 1][a ;k

a

WOOR

tt

+ 11],

=

a

+t

I_ [I+ lt J ,

"

a+1 +j

woo[:]

a+2l " a+~

a

a

o

tt

it

[aki

[a;j + 11 [aik+ 11 + 2 I

[ ajk ] [aij ] [aik ]2

= [ 'i +u],

[a;k - 1 ] [a; k + 1 ] [ Qjk - 11 fajj, + lj

"

a

a+k

a +i +j

a

a

a+i

a+i+j

a

[ ik ] [ ik J

tt

= [Qki + ~ +"I

[a lj + 1] [aj~ + 11 , , [ aj~ ][a~j ] [Q ~~ ]~ (4 .10)

We®e

a

" " a +l +j

WB®8

WB®e

a+i+j a+2i+2j

a

a+i+j

a +i +j

a +î

a

a+~+ j

a +î +k

a +î

ct

=[1+u],

u

=

ct

= [tt]

[ a jk _u ] [aj k ]

,

[aik + 1 1 [aj k [ajk]~

-lJ

(4 .11)

T. 1Vakatsit / Affine Toda field

516

theory

In eqs. (4.4) and (4.9)-(4.11) i, j and k are all distinct. Other matrix elements of W(u) which do not appear in eqs. (4 .4) and (4.9)-(4.11) are all equal to zero. The face operators W(u) satisfy the following "crossing symmetry" relations: (l

b

W[30°I d

c

a W000

W8 ® 0

b

c

i ~a

d

b

b

c

d

c

a

d

a

b

b

c

d

c

a

d

a

b

b

c

d

c

(- I)W[],Do

)

=

u

I) WOO[]

a

d

gbgd

3

a e

g g

;

gb gd gag`

;

gb gd

919C

-2 -u

gbgd

919C

(4.l5)

Here we set g a = {II i j aij]}' /2. We can prove the above "crossing symmetry" relations by direct calculations . For example,

wo 00

a+i

a+21

a

a+t

r

i

2 ga + i ga ga + 2i

r -1 [aki ] [aji ] [a ki - ][ aji - zj [aji - 1 1 2[aki - 1]2

1]2 aik + [a ik ] [ aij ] [aik + 2 ] [aij -}- 21 [ aij ;

112[

+u _ - WD ® 0

a

a+i

a+i

a+2i

u

(4 .16)

We give some comments concerning the above "crossing symmetry" relations. First we can multiply W ® ,,(u) and W,,,,, (u ) by ( -1) without violating the 8 8 Yang-Baxter relations. So we can change the minus signs in eqs. (4.12)-(4.15) to plus signs. A second comment concerns the signs in eqs. (4.14) and (4.15). We can

T. Nakatsu / Affine Toda field theory

517

not find (g.) which unify the signs of (4.14) and (4.15) as in the case of the suq(2) R-matrix . It seems to require a more complicated gauge transformation . On the other hand, these face operators satisfy the following "first inversions" relations :

E Wp®p x

We®8 W13

x

a

g

d

b

a

g

d

b

a

g

d

b

u

u

u

Wooll

a

c

(g

b

a

c

W8®8

b

Wp®8

a

c

g

b

-u

= [1 + u][1 - ul Scd'9

(4.17)

-u

= [1 +u][1 -u]8,d ,

(4.18)

-u

= [ + u] [ 2 - u] S,d . (4 .19)

The above relations are easily proved by using the equality (4.20)

Rpp(u :q)R[]0(-u :q) = [1 +u][1 -u]ll . At this stage we define the following face model type S-matrix; Sa ®0

SB®e S8,&[]

SC],&

0

a

b

d

c

a

b

d

c

a

b

d

c

a

b

d

c

=F(u)Wp ® []

8

= F(u)

8

a

b

d

C

a

b

Woo0 d

u

u

C

e

_ u ) W 8 o~

B

- u)WO ®

8

ga gc

(

gbgd

ga g c

(4 .2l)

9

)" 6

(4.22)

( gbgd a

b

d

c

a

b

d

c

u

u

e/

g ° g`

(

gbgd

, (4.32)

)

g a g~ 8 ( gbgd )

7r

/

'

. (4 .24)

Here 8 and u are related by eq. (3.6) and F(u) is the scalar function discussed below. The meaning of the above face model type S-matrix is as follows. For an b

describes the scattering amplitude between soliton example S 001:1 triplets. The incoming two solitons with the rapidity difference 0 have b - a and 11

T. Nakatstt /

if te

Toda field theory

c - b as the weight of su q(3). The outgoing two solitons have d - a and c - d as the weight of su q(3). (Fig. 1). Suppose that F(u) satisfies the following equations: F(tt)F(-t( )[1 +tt ] [1 -u]= 1, F(-

2

- it)F(- _ + u)[tt +

2

] [ -u + 2 ] = 1 .

(4 .25)

From the "crossing symmetry" relations (4.12)-(4.15) and the "first inversion" relations (4.17)-(4.19), the S-matrix (4.21)-(4.24) satisfies F Sa ®p

ESOOG( 9

a

g

Id

bI

a

g

d

b

a

Yse®o ' d Sco[3

S8®8

E]O S

0

a

b

d

c

a

b

d

c

a

b

fd

~

d d

c

g b

e

1

Sci ® o (

a

c

a

So® 8

9

= Sp®8

8

= S8®[3

,

c

0 ) IV 0 0 0( g e

e 3t-it 1 - ~ =(

b

a

c

g

b

b

c

a

d

b

c

a

d b

( ) o e1= ± l se efa a

-e

=

i7r -

8

i7r -

8 ,

,

c dl «-B~

d

(4.26)

.

(4.27)

The equalities (4.26) and (4.27) represent the unitarity and the crossing symmetry of the theory. And moreover, the factorizability of this theory is derived from the Yang-Baxter relations. At this stage it is instructive to construct the vertex type S-matrix which corresponds to the face model type description (4.21)-(4.24).

T. JVakatsit /

ra

b

~d

c I

S~0OI

S8®8

S8®]

Sp®8

Affine Toda field theory

a

b

d

c

a

b

d

c

a

b

d

c

-}-- :

I

5)9

, \~/ e I = a~l~c d

0

= a

",4 b "/ /

c

`~b 8

=a

~ c d `

8

= a ~ '

soliton triplet[]

b~

c

d

- --> -- : antisoliton triplet

8

Fig. 1 . The diagrammatical representations of the face model type S-matrix, in which a, b_ c and d denote integral weights of su(3).

First for notational convenience we write the matrix elements of R-matrices as

REU(ii : q)f- i

of-j

k,l

REU(u)

j

Of

(4 .28)

Here {e;} , are the canonical basis of V,:,, i.e. e, _ `(1, 0, 0), e, _ `(0,1, 0) and e3 ='(0, 0,1) . And also { f- ;} , are the canonical basis of V , which are represented by ej:

8

1 [2~ 1

V=[-2]

1/2 -q

(-q

1/2 e3 0 e, + q- e 2 0 e3),

1/2 e o e, + q- 1/2e, Oe 3) , 3

T. Nakatstt / Affine Toda field theory

5,20

TABLE 1

The matrix elements of the fused R-matrices. Other matrix elements which do not appear in this table are all equal to zero. [1 +u] if i =j 1313 [il] if i *j q" if i j [1+11] if i =j R~--(tt) =; '. = [u] if i * j q" if i >j if q -" i
[ ; +tt]

R

if i =j

[ + tt]

[]Ei

«,

2

=

if i *j 1-1

Illi

R (u);! ;= (R,-L(u)i;;)-l = E113

11

if i =j

+111

R,-,p(u)!-2 = (R

(U)2

-q- Ot- 1/2)

i i)-' =q`112

R[]0(u )2-2 -3 ) -1 -33 = (R[]B(u)3-22 3 3 R [30(u)'-33 - 1 = (R[j](u) , j ) - I

= qII+ 1/2 = _ q u- 1/2

The explicit forms of the above matrix elements can be obtained via eqs. (3 .18), (3.21) by using eq. (3 .8). These are listed in table 1 . With respect to the R-matrices the following "crossing symmetry" relations hold: j 1 gjg- i -i k 9-19k

3

Rpp(u)k1 = -RE:] 0( - 2

j1 g -jgi R00( u) -k -1- -R0(:](-' - u )i -k g1g-k Re (3 (

j1 gj g i u)k -I - ± ROO( - 2 - u)ik

Rp8(u)`

g1gk

k1 = ±R

ee

(-

2 -

u)

~-k

g -19 g _1g _-rk

(4 .29)

T. Nakatsat /

Affine

Toda field theory

521

Here gj = ei7rj'P /2 g j 'Pl2. Eqs. (4.29), which are the counterpart ofeqs. (4.12)-(4.15), can be checked by direct calculations . We note that, as in the case of the su4 R-matrix, it is possible to unify the signs of the r.h.s. of eqs. (4.29). Introducing the new variables { fj): fj = e'Trj'P/2, eq. (4.29) can be written as R0o(u)kl f'

fi

R[30(-

fjfk

_j f-'f-1 R88(u) -ik _1 Tif-k

o(u) k-' j f-if-1 .if

R

k

i _j fifl R8(u) _kl f_jf_k

A A u )j-1 fjfk gig-i A A a ik f-if-1 g-lgk

U

-R -

A A

3 ) -j1 f-if- l g-j g- k 2 - i -k A A f;fk gigi

A A _ 3 - U)j1 fjfk gjgi -RE: 0( 2 )i k fif. ' e 919k

_R

(

ee

A A g; U ) -j -1 f-if-k ;9A A 1 i k .i -if-1 g-19-k

2

(4.30)

where gj = e -1Tr;'P /2 gj. As in the case of the face model type description, we introduce the vertex type S-matrix :ri 0/ S0[3(e)k1 =f(u)R~~(u)kl ij

grgk

( 9j9j .

S

(e)-k-jl=f(u)R

(u)_k-j1

g-lg-k

( g-jg_1 )

Sep(e)k'j = - f( - i - u)ROE](u)k ii Soe(e)' kl =

-f(2

u)RE]

8

Ai-. :

(u)r

kl

g-igk ( 9j9 -t

glg_

k

9/Tri

( g-jgl

(4.31)

The scalar function f(u) is determined below. Because of eq. (4.29) the S-matrix (4 .31) satisfies the "crossing symmetry". On the other hand, there exist the "first inversion" relations, which correspond to (4.17)-(4.19); ROO(u : q)RO [:] ( - u : q) = [1 - u][1 + u] 11,

RBB( u : q) Ree( -u : q) = [1 -u][1 +u]11,

RBE](u :q)Rpe(-u :q) = [i -u ] [z +u] ll .

(4 .32)

522

T. Nakatsu

/ fl fete

Todei field theory

From eq. (4 .32) the S-matrix (4.31) satisfies the unitarity if f(II) is the solution of eq. (4.25). Can we solve the functional equations (4.25) imposing the unitarity condition? As described in the suq(2) case, we must choose solutions whose physical poles correspond to some irreducible representations of su q(3). There are the following two solutions Fl(0) and F,(0):

F,(8) - ±

sin in

87r22

v

8~r - ( 30

87r

30

8a

30

~ y ~

27ri Y l2ai 11 1 11 s~ 3e s~ 3e r(-( +3k~)ry+-(+3k-y) 27ri s 7r

X

F2(0) -

3e

11T

87r

se

l'' y (-+3k-2(1+ Y (- 2 ~l + 3k-all l

8m

\Z~i 38 8m 30 +3k-211T(1+-( +3k-311 27ri Y 12ai 11 1

sin

sin

87r

y 8 7- 2 ( 30 y

Brr

2 7ri

30

87r ( + 3k - 2 )Jr~i + y ( 27ri y x Brr 30 Brr ( F 1 + +3k-2 F 1 + y 27ri y T i +

30 + 3k - 3 27ri 30

27ri

+3k-3

Although F,(0) and F2 (0) appear to be rather complicated, we can easily check

T. Nakatsu / Affine Toda field thcoty

that eqs. (4.33) and (4.34) satisfy (4 .25): F,( 8 + 2,ri) F,( - 0)sin

87r2

38 8rr-' 38 sin +3 Y (27,i Y ( 27H'

2 87r 2 87r2 38 87r2 ( 38 sin sin )sin +3 Y y (2r, i y 27rî 87r2 38 87r2 38 sin -+2 sin -+I y ( 27rî y ( 27ri 87r

38 ») (1 8T 38 + , +3(k- i +3(k + i y 27rt y 27ri ) 87r 38 r(1 8T - 38 + F + . + 3(k + 1) 3( k - 1 ) y 27r l y 2 .-. i I'

x

F

-

8,r ( 38 + 8T ( - 38 3k + 1 F 1 + -+3(k-I)-3)) y 27ri y 2-,ri 8-,r 38 8 '~ 38 . + F . + 3(k- 1) - 2 T 1 + 3k 27rt y 2~t ( y ( 8,r ( 38

+3k

y 27ri 8,7r 38 + 3k y 27ri

F

8,r (

38 --+3k-1)) 27ri Y 8,r ( 38 F 1+ +3k-1 y 2,ri

( _( ))

F

F (I +

( 87r 38 8 ,r ( 38 + 3k-2 2))F(1 +3k-3 y 27ri y 27ri 87r 38 87r 38 F +3k-2 F 1 + + 3k-3 y 27ri 27ri )~ ( y F

8-r 8-7r2 2 38 87r2 ( 38 sine sin sin +3 y ( 2,ri y 27rî Y 87r 2 39 87r 2 38 sin + 2 sin +1 y ( 2,ri y ( 27rî (

(

I

7r

v

30

8y 2~~30 )) r ( - 8 12~~ +3 11 3e +Z1~~ls~l 3B +I r(r+ 8 ) y ~ 27ri ~ l y ~ 2ori I

523

T. Nak-aistt

524 r ( a7r l 3e

?r, 8v 3®

+3))r(1+

r ~

T I-y ( 27ri +

30

87r2

F,(8)F,( -)sin

( 27ri

y

/ Affine Toda field theory

1

))

F

3e 

y ~ 27ri 8,zr 3®

- y

- 1 sin

8~r (

a~(

87r2 y

3e

11

2 2Tri +

38 ( 27ri

Y

+ 1

( 8~r ( 38 T 1 + + 3k - 1 87r, 2 7ri y 2,rri 8 sin` 8v (87r( 3 e ( 3e T + 3k T i + --+3k-1)) y 2ri y 27ri .

T

-

+ 3k

))

s7r se +3k-211!'(1+-( +3k-3,1 Y l2 7ri Y t tai I1 t 11 87r 30 8Tr 30 l'(-(+3k-2,11(1+-f +3k-3ll ~ y ~ 27ri Y 12 7ri 11 1 11 s 7r

T(-(

3e

(87r( 38 87r 38 l' -+3k1,T(1+-(1--+3k-111 y 2ai 1 tai 1 11 1 Y 1 a7r 3e s7r se T(-(+3k11Tf 1+-( +3k-111 ~ y ~ 27ri Y 12 7ri 11 1 l 87r

30

8a 30 11 +3k-211I'~1+-~ +3k-31' ~~ y 27ri Y 1 2~i +3k-2)r1+ 8a 30 +3k-31, y ~ 2ai / ~ y ~ 27ri I1 (4 .35) These two relations reduce to (4 .25) by using eq. (3 .6). Similarly we can check that F2 (e) satisfies eq. (4.25).

T. Nakatsei /

Affne Toda field theory

525

The simple poles of F,(0) which satisfy the conditions Re 0 = 0 and 0 < Im 8 < Tr, are 0= 0=

27ri Y 8Tr 3 1k with 1 < k < : integer, ( 8Tr Y

(4.36a)

27ri

12r Y k with 1 < k < : integer . 3 (8v y

(4.36b)

The simple poles of F(i v - 0) which satisfy the above conditions are 0= 0=

Tri

8Tr (1 + Y k with 1 < k < :integer, 3 4Tr y

(4.37a)

2Tri

3 127r Y - k with 1,
(4.37b)

On the other hand, the simple poles of F2(0) which satisfy these conditions are 0=

27ri y 8Ts k with 0 < k < 3 1:integer, ( 8Tr ) Y

0=

27ri 3

y

(4.38a)

12Tr

:integer, k with 1 < k < ( 8Tr ) y

(4.38b)

whereas the corresponding simple poles of F2(Tri - 0) are 0= 0=

7rt

(I + Y k with 0 < k <

27ri 3

8Tr

:°integer

(4 .39a)

3 127r Y - k with 1 < k<- :integer . (2 8Tr ) y

(4.39b)

Do these poles of F, (F2) give the bound states which form some irreducible representation of suq(3)? First we study the soliton-soliton scattering . The corresponding S-matrix is given in eq. (4.31). From the spectra! decomposition (3.13) the poles in (4.36a) [(4 .38a)] must correspond to the representation of su,(3). On the other hand, the poles in (4 .36b) [(4.38b)] should not be physical in this channel. Secondly we study the antisoliton-soliton scattering . The corresponding S-matrix is given in eq . (4.31). From the spectral decomposition (3.19) the poles in (4.37b) [(4.39b)] which are equivalent to the poles in (4.36b) [(4.38b)] by crossing symmetry, must give the bound states whose representation of su q(3) is a singlet . The poles in (4.37a) [(4.39a)], which correspond to the poles in (4.36a) [(4.380], should not be

8

326

T. 1Vukatstt /

Atttie Tcxica field theopy

~

physical in this channel. Thirdly we study the antisoiiion-antisoliton scattering. The corresponding S-matrix is given in eq. (4.31). From the spectral decomposition (3 .23) the poles in (4.36a) [(4.38a)] must correspond to 13 of su q (3). And the poles in (4.36b) RUM] should not be physical in this channel . By the above heuristic arguments F,(9) [eq. (4.33)] and F,(0) [eq. (4.34)] are the good candidates for the scalar function of the S-matrix (4.31). Concluding this section we give some comments . Can we apply our face model type S-matrix (4.21)-(4 .24) (or, equivalently, the vertex type S-matrix (4.31)) to the quantum theory of (3.1) at irrational values of ß'. Our S-matrix, based on the classical solutions (3 .3) and (3.4), is constructed under the assumption that there exists suq(3) symmetry in the theory, such that the unitarily and the crossing symmetry is satisfied. Especially the scalar functions F,(®) and F,(8) are derived from the unitarity condition (4.25). On the other hand, it is quite conceivable that the quantum theory of (3 .1) at generic values of ßneed not be unitary. Then our S-matrix (4.31) obtained by imposing the unitarity condition may have some internal inconsistency at generic values of ß - . As an example we study the soliton-soliton scattering in eq. (4.31). At a generic 13 - there exists a pole with non-zero k in (4.36a) [(4.38a)]. This pole must correspond to a bound state with mass 2 M cos T(1 - (y/87r )k ). Here we denote the mass of (anti)soliton by M. On the other hand, this particle forms the representation of su (a(3). So this particle has the same irreducible representation of su,(3) as the antisoliton triplet with a different mass. Hence the bootstrap is not closed . As we will describe in sect. 5 we are mainly interested in the region ,-' < ß- < In this coupling region F, has no simple poles satisfying the conditions Re 0 = 0 and 0 < Im 0 < Tr. So if we choose F, to be the scalar function of the S-matrix (4.31), there does not exist a bound state in this region . On the other hand, F,(0) has the simple pole at 0 = 27,-i/3 in this region . This corresponds to an antisoliton triplet in the soliton-soliton scattering. Here we note that the bootstrap equation with respect to this simple pole can be proved from the definition (3.18) and the following equality:

8

8

(0+ Tri ) sin(8Tr'/y F, (4 F 0 - --- _ ± ri

d

)

(

F,( ,rr i - 0) .

.40)

Here the ± signs in the above equality correspond to the ± signs in F2 , eq. (4.34). 5. Restricted model In this section we study the case when p2 is rational . We are especially interested in the case p2=

p 2(p+1)

p > 3, integer .

(5 .1)

T. Nakatsu /

Affine Tmla field theory

527

In this case qn = ± 1 . When q is a root of unity, representations of the quantum group su q(3) are classified as "type 1" or "type 2" in the terminology of ref. [161. So the possibility that the physical Hilbert space is restricted to "type 2" representations will appear at rational values of the coupling constant as in the case of the sine-Gordon theory [7,81. First we consider the q-deformed Casimir operators C, and C, of su,(3), which are given by [161 (q +q-1)C1 = (q --q-1)C2 -

~

(9

-

+ (q

q -l )

2g

(411,+211:)/3 +

-q-1)_

2q (q _ g-')

q-(2h,-2H,)/3 +

-(211,+411,)/3

q 1 +(11,+211_)/3F I El

+q -1 - (211,+11,)/3FE,) -q(1i, - H,)l3(F,F,, - gF,F I)( E I E2 - q -l E2E1)

- {q

(5 .2)

H q-1) .

For a given integral weight w =11 6)1 + 12 w2, these Casimir operators are invariant under the following affine Weyl transformations : wHw'=o*w+p(n 1 e,+n 2e2 ),

(5.3)

where n,, n 2 E=- 7, (r E Weyl group of su(3) and or * w = or(w + p) - p. From the physical point of view it is very natural to define a "type 2" representation as the one whose highest weight w =1 1(0 1 + 1 2 (02 belongs to "type 2" when 1, + 12 < p - 3 and 1,,12 > 0. Here we note that this convenient definition does not contradict the definition in terms of the q-dimension [161. Our next problem is whether we can restrict the physical space to "type 2" representations . This can be read off if we choose the face model type description in sect. 4. The face operators in eqs. (4 .4) and (4.9)-(4 .11) become singular unless we restrict a = w + p = (1 1 + 1)6), + (12 + 1)6)2 to within 1, + 1, < p - 3. .82 = p/2(p + 1) must be From this observation the physical Hilbert space at restricted to a "type 2" representation, which satisfies 1 1 +12
11 ,12 >0 .

(5 .4)

We note that we can make the explicit correspondence between the physical

52 8

T. Nakatsit / Affine Toda field theory

Hilbert space (5.4) and the BRST cohomology classes (1.5). The former is determined from the S-matrix view-point, and the latter is determined by perturbation theory from w3 minimal models. As described in sect. 1 these cohomology classes p2 2determine the physical Hilbert space of the quantum theory of (3.1) at p/2(p + 1). On the other hand, the "crossing symmetry" (4.12)-(4 .15) and the "first inversion" relations (4 .17)-(4.19) are satisfied even when we restrict the physical Hilbert space to (5.4). This can be proved as in ref. [3]. And as described at the end of sect. 4, 87r/y = 1/p (i.e. p2 =p/2(p + 1)) belongs to the coupling region It is useful to compare the above results with the corresponding features of the sine-Gordon theory, which has the quantum group su g(2) symmetry. In the sine-Gordon S-matrix theory, the N-body Hilbert space is factorized to the direct sum with respect to irreducible representations of su,(2). Especially at ps~/8 -rr = p/(p + 1) (p > 3, integer), there exists no bound state in the theory and moreover we must restrict the allowed spins of irreducible representations to j < (p - 2)/2 [8]. On the other hand, this restriction results from the hidden BRST symmetry of the sine-Gordon field theory at ps~/8 -rr = p/(p + 1) [7]. It is possible to explicitly show the correspondence between these allowed spins and the BRST cohomology classes. From these discussions it is clear that our restricted S-matrix (4.21)-(4.24) with p2 = the allowed weights (5.4) describe the quantum Z(3) Toda field theory at p/2(p + 1), whose short-distance limit corresponds to the W3 unitary minimal CFT labelled by the same p. 6. Conclusion We have studied the su(3) Toda field theory with a pure imaginary coupling constant . Based on the analysis of classical soliton solutions and suq(3) symmetry in the quantized theory, we constructed two kinds of factorizable S-matrices . At a special value of the coupling constant, 132 = p/2(p + 1) (p > 3, integer), these S-matrices, which satisfy the unitarity condition, are consistent and their N-body Hilbert space is restricted. _ su(3) Toda field theory at the hidden BRST symmetry of the On the other hand, p2 = p/2(p + 1), which is the remnant of the BRST structure in the corresponding W3 minimal unitary CFTs, determines the physical Hilbert space of the theory as its BRST cohomology classes . There exists an explicit correspondence between these two spaces . w Although our treatment of the su(3) Toda field theory with a pure imaginary coupling constant may not be complete, our results very strongly suggest that quantized sû(3) Toda field theory at p2 = p/2( p + 1) is integrable and its unitarity is recovered by the reduced Hilbert space of the theory.

T. Nakatsu / Afne Toda field theory

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We would like to express our gratitude especially to Professor T. Eguchi for suggesting the problem addressed in this paper and for many fruitful discussions and continuous encouragement. We could not have performed this work without him. We are also grateful to S. Mizoguchi for useful discussions and critical comments, and to K. Ogawa, A. Kato, and S. Ishihara for enjoyable discussions. References [11 C. Itzykson, H. Saleur and J.B. Zuber, eds. Conformal invariance and applications to statistical mechanics (World Scientific, Singapore, 1988) [21 G.E. Andrews. R.J. Baxte r and P.J. Forrester, J. Stat. Phys. 35 (1984) 193 [31 M. Jimbo, T. Miwa and M. Okado, Nucl. Phys. B300 [FS221(1988) 74 [41 D.A. Huse, Phys. Rev. B30 (1984) 3908 [51 A.B. Zamolodchikov in Advanced Studies in Pure Mathematics 19 (1989) 641 [61 T. Eguchi and S.K. Yang, Phys. Lett. B224 (1989) 373 [71 T. Eguchi and S.K. Yang, Phys. Lett. B235 (1990) 282 [81 A. LeClair, Phys. Lett. B230 (1989) 103 [91 N. Reshetikhin and F. Smirnov, Commun . Math . Phys. 131 (1990) 157 [101 F.A. Smirnov, Int. J. Mod. Phys. A4 (1989) 4231 [III T.J. Hollowood and P . Mansfield, Phys. Lett. B226 (1989) 73 [121 M. Henkel and S. Saleur, J. Phys. A: Math. Gen. 23 (1990) 791 [131 V.A. Fateev and A.B. Zamolodchikov, Int. J. Mod. Phys. A5 (1990) 1025 [141 A. Luther, Phys. Rev . B14 (1976) 2153 [151 M. Karowski and H.J. Thun, Nucl. Phys. B130 (1977) 295 [161 A.B. Zamolodchikov and A.B. Zamolodchikov, Ann. Phys. (N.Y.) 120 (1979) 253 [171 V. Pasquier and H. Saleur, Nucl. Phys. B330 (1990) 523 [181 R. Sasaki and I. Yamanaka in Advanced Studies in Pure Mathematics 16 (1988) 271 [191 M.T. Grisaru, A. Lerda, S. Penati and D. Zanon, Nucl. Phys. B342 (1990) 564 [201 M.T. Grisaru, A. Lerda, S. Penati and D. Zanon, Nucl. Phys. B346 (1990) 264 [211 S. Mizoguchi and T. Nakatsu, Tokyo preprint UT-566 (1990) [221 M. Jimbo, Int. J. Mod . Phys. A4 (1989) 3759 [231 E. Date, M. Jimbo, and T. Miwa, in Physics and mathematics of strings, ed. L. Brink, D. Friedan and A.M. Polyakov (World Scientific, Singapore, 1990) [241 V. Pasquier, Commun . Math. Phys. 118 (1988) 355 [251 M. Jimbo, A. Kuniba, T. Miwa and M. Okado, Commun . Math. Phys. 119 (1988) 543