Superconformal affine Liouville theory

Physics Letters B 292 (1992) 67-76 North-Holland

PHYSICS LETTERS B

Superconformal affine Liouville theory Francesco Toppan ~and Yao-Zhong Zhang

2

Physikalisches Institut, Universitiit Bonn, Nussallee 12, W-5300 Bonn I, FRG

Received 27 May 1992; revised manuscript received 23 July 1992

W.~epresent a supereonformally invariant and integrable model based on the twisted affine Kac-Moody superalgebra osp (212)(2) which is the supersymmetrization of the purely Ix)sonic conformal affine Liouville theory recently proposed by Babelon and Bonora. Our model reduces to the super-Liouville or to the super-sinh-Gordon theories under certain limit conditions and can be obtained, via hamiltonian reduction, from a superspace WZNW model with values in the corresponding affine KM supergroup. The reconstruction formulae for classical solutions are given. The classical r-matrices in the homogeneous grading and the exchange algebras are worked out.

1. Introduction

Associated to a given bosonic Lie algebra, three kinds o f models can be introduced in terms of a Lax pair formalism: they are related to the algebra itself, to its loop extension and to its affine K a c - M o o d y ( K M ) extension, respectively. For instance in the case o f sl (2) the three models are the following: (i) the Liouville theory, which is conformally invariant and integrable; (ii) the sinh-Gordon (SG) theory [based on the loop algebra ] which is still integrable but no longer conformally invariant; (iii) the conformal affine Liouville ( C A L ) model proposed by Babelon and Bonora [ 1 ] [based on the affine K M algebra ~ ], which is like the Liouville theory conformally invariant and integrable; moreover the Liouville and the SG theories can be recovered from it with two different limit procedures. Similar steps can be performed in correspondence with any Toda and W Z N W theory [ 2 ]. It is natural to look for supersymmetric generalizations o f the above construction. The supersymmetric extensions o f the Liouville and SG theories have indeed been extensively studied [3-5 ]. The super-Liouville theory, which is superconformally invariant and integrable, is found to correspond to an obvious superalgebra generalization - osp ( 1 ] 2 ) - o f the bosonic algebra sl (2). However, as explicit computations show, the counterparts of the bosonic construction based on the loop superalgebra osp~-~2) and the affine K M superalgebra o s p ~ 2 ) do not work. Rather it turns out that the superalgebra underlying the integrable but no longer superconformally invariant super-SG theory is the twisted loop superalgebra osp~ (212 ) (2) . Therefore one expects that a supersymmetric counterpart o f the bosonic CAL-theory should arise from the construction based on the twisted affine K M superalgebra osp--"~2x (21 J (2). This is indeed the case. In this paper we generalizethe Babelon-Bonora construction to the N = I supersymmetric case and provide a new supcrconformally invariant and intcgrable model (letus callitsuper-CAL theory). Our supcr-CAL model realizesan interpolationbetween super-Liouville and super-SG theories which are recovered as special limits.Moreover its bosonic sector corresponds to the C A L theory. As in the bosonic case [6-8 ], the supcr-CAL theory can be obtained, via hamiltonian reduction, Address after 1 October 1992: Laboratoire de Physique Th~orique ENSLAPP, Ecole Normale Sup~rieure de Lyon, 46 All~ d'Italie, F-69007 Lyon, France. 2 Addressafter 28 October 1992: Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia. 0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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from the superspace WZNW model with values in the corresponding affine KM supergroup. We will work at any step with a manifestly supersymmetric formalism.

2. Notation: twisted affine KM superalgebra o s p ~ 2 ) t2) This section is devoted to a brief account for the twisted affine KM superalgebra osp(-~2)(2) [ 9 ] and to introduce some useful notations. The superalgebra osp (212) is a Z2 graded algebra, osp(212) =osp(212) .... ~osp (2[2)°dd, with osp(212)even=u( 1 ) ~SI(2) = {H'}(3{H, E~,, E_~,},

osp (212)°ad= {F#, F_#, Fg, F_a}.

( 1)

These generators satisfy the following (anti-)commutation relations: {F_+#,F+#}=0= {F+a, F_+a}, {F+#,F+a} = _+E+~, {F#,F_#) = - ½( H - H ' ) ,

{F~,F_a}= - ½(H+H'),

[E+_c~,F~=#]=F+_H, [E+c, F~=/~]=F+_#, [E+_c,,F+#] =0= [E+_c~,F+.a] , [H, F+#] = +F+_#, [H, F+~] = +F+_a, [H', E+-~] =0,

[H, E+_~,]= +2E+,~, [E,~,E_,~] = H ,

[H', F+-/~] = +F_+/~, [H', F+a] = q=F+-a.

(2)

The fundamental representation of osp (212 ) is three-dimensional,

H=el,--e21,

E~,=el2, E_,~=ez~,

H'=e,, +e22 +2e33,

F#=e32, F_a=e23, Fa=e,3, F _ a = - e ~ , ,

(3)

with eij being a 3 X 3 supermatrix and (e o)/a = c~i~:t. The nondegenerate, invariant and supersymmetric bilinear form on osp (212) is simply the usual supertrace "str" which is defined as

strX=tra-trb,

forX=(d

b)"

(4)

The superalgebra osp (212 ) has an automorphism z of order two and with respect to the eigenvectors of r we have the decomposition osp(212) =osp(212)o~OSp(212) 1, where osp(212)0- {Xeosp(212), z(X)=X}={H, E,, E_~, i(F#+Fa), i(F_# +F_a) }, osp(212), - {Xeosp (212), ,(X) = - X } = {H', (F# - F a ) , ( F _ # - F _ a ) } .

(5)

The twisted loop superalgebra osp (21"---'2) (2) is defined as [ 5 ] 0sp(21~2)(2)= ~ )n osp(212)nmod2 x ~2n+,H, , A2n+l(F~p-F~a), n~Z}, = {A2nH,~.2nE+_o~,i~2nl'F ~ +-~+ F +-at,

(6)

where the spectral parameter A represents the homogeneous grading. We will always work with the homogeneous grading throughout this paper. The twisted affine KM superalgebra osp(-'~2)(2 ) is obtained by adding to the twisted loop superalgebra OSp-'~2) (2) the center ~ and the derivative a = 2 ( d / d g ) . Namely, osp-'~2)(2)=OSp--~2)(2)~Cc~ca. The graded bracket on o s p ~ 2 ) (2) reads [.(', I7"}^=[2, 68

~"}~ +e~-~m ~ dA Str{[0~.~(A) ] Y(A)},

(7)

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where "Str" stands for the invariant, nondegenerate and supersymmetric bilinear form on o s p - ~ 2 ) t2) which is defined as S t r ( ~ L , ~ j ) =dm+.,o str(X,~ X_.j),

Str(~m~r n) =dm+.,o str(HH),

Str(Jq 'm/q") =&.+~,o str(H'H' ), Str(~a) = 1, the rest=0,

(8)

where the quantities ~m-2mT, m = 2Z or 2Z + 1. We perform the Cartan decomposition osp-'~T2) <2)= ~< ~ ~= ~ ~ . ,

(9)

where the Caftan subalgebra ~= is spanned by {H, ~, a}, ~> is the nilpotent subalgebra generated by the positive root vectors

,~.2nH,E., 2 2hE+ a, F.t, 2 2nF+_ al, F,~2,2 2nF.2 , ,~.2nF_a2,

2/-/', ,~2n+ IH, '

n > 0,

(10)

and ~< is the nilpotent subalgebra generated by the negative root vectors ~. - 2 n H ,

E_., 2 - 2 h E + . , F_,~j, 2 -2nF+_al,

~. - 2 h E . 2 , F _ a 2, ,~. - 2 n f - a z, ). - 1H', ,~. - (2n+ 1)H',

n > 0.

( 11 )

In the above F+.,-- i(F+a+F+H),

F+_.2=2+-~(F~a-F~H)

(12)

are odd simple root vectors of the twisted affine KM superalgebra osp-'~T2 ) t2).

3. Super-CAL. Zero-curvature formulation

The super-CAL is a super-Toda system associated to the twisted affine KM superalgebra o s p ~ 2 ) (2) It is the generalization of the purely bosonic CAL [ 1 ] to the N= 1 supersymmetric case. Let x +, 0 +, x - and 0- denote the light-cone coordinates of the world-sheet ( 1,1 ) superspace. The superderivatives are 0 D + - 00 + ~-i0+0+,

0 D _ - 00- +i0-0_ ,

(13)

which satisfy (D_+)2=i0_+ and {D+, D_} =0. We introduce the scalar superfield ~(x +, 0 +, x - , 0 - ) with values in the Cartan subalgebra of o s p ~ 2 ) <2):

~=½~H+a.½Aa+½~,

(14)

where ~, A and 3 are scalar superfields and a is a real parameter. The Ramond or Neveu-Schwarz boundary conditions will be assumed for fermionic fields. We define two Lax-pair superpotentials, Aa+=D+~+e"d~8+,

Aa_=--D_~+e-ad~'8_ ,

(15)

where 8+ and 6'_ are respectively the sum of odd positive and negative simple root vectors, 8+ =F,~, +F.2,

g_=F_.,+F_~,

(16)

and write the linear system (D_+ +Ae_+)3"=0,

(17)

where a-belongs to the affine KM supergroup whose Lie algebra is o s p - ~ 2 ) <2) The zero-curvature condition, which is the compatibility condition of the above linear system, 69

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D + S ¢_+D_S+ +{S+, S_}=O

8 October 1992

(18)

can be worked out using only the Lie superalgebra structure of o s p - ~ 2 ) (2). We get D+D_@=e~-e ~-~,

D+D_A=0,

D+D_~=e ~-~,

(19)

which define our super-CAL.

3. I. Remarks (i) The super-CAL is superconformally invariant; this is seen as follows. Let Z + = (x +, 0 + ) and Z - = (x-, 0- ). Then the superconformal symmetry of the super-CAL is realized as

Z÷~g+(Z+)=(Y,+(Z+),g+(Z+)),

Z - - . ~ - ( Z - ) = ( ~ - ( Z - ) , g- ( Z - ) ) ,

@(Z +, Z - ) - , ~ ( 2 +, Z - ) = @(Z +, ~ - ) +In(D+ g+ D_ g - ) , A ( Z +, Z - ) ~ 7/(g+, g - ) = A ( g + , g - ) +

2 ln(D+ g+ D _ g - )

S ( Z +, Z - ) - - , g ( Z +, ~ - ) = 3 ( Z +, 5 - ) -

1B ln(D+ g÷ D_ g - ) ,

tl

(20)

where B is an arbitrary constant. It is easy to show that ( 19 ) is indeed invariant under the transformations (20) thanks to the covariant transformation property of D ± under Z +---, Z ± (Z ± ) [ 10 ]: D+ =(D_+g±)b±,

I~± = ~

0

0 + i g ± 02 ± ,

(21)

which is subject to the constraints D±2+-=g ± D ± g ± .

(22)

(ii) In (19) the equation for the superfield @ reduces to the equations of motions for the super-Liouville or for the super-SG theories under the limits a--,oo and a-,0, respectively; therefore the super-CAL provides an interpolation between the super-Liouville and the super-SG. From now on we work with the fixed normalization a=l. We now expand the scalar superfields in terms of their components, @ = ~ - i 0 + ~ + + i 0 - ~ / _ + i 0 + 0 - f , A = t / - i0 + a + + i0- or_ + i0 + 0 - g and S = ~ - i0 + fl+ + i0- fl_ + i0 + 0- h. Eliminating the auxiliary fields f, g and h in the super-CAL (19) [ f = i ( e ~ - C - ~ ) , h = i C -~, g = 0 ] , we are left with, in components, 0+ 0 _ ~ = e 2~ --e2~t-2°q- ~+ {//_ e o - (t~+ --{//+ ) (t~_ --{//_ ) e"-*,

0+ 0_~/=0,

0+a_ = 0 = 0 _ a + ,

0+ ~_ = - i [ ~ + e*+ ( a + - ~ , + ) e"-~],

0_ ¥+ = i [ ~ _ e~+ ( a _ - ~ / _ ) e ~-~1 ,

0+ 0 _ ~ = e 2 " - 2 * - e " + (a+ -~/+) ( a _ -~/_) e ~-* , 0+fl_ =i(ct+ -~,+) e ¢-*, 0_fl+ = - i ( a _ - g _ ) e "-~ .

(23)

Setting the fermionic components ~/+_,a_+ and fl± to be zero, we get the purely bosonic limit of the super-CAL, 0 + 0 _ ~ = e 2 ° - - e 2't-2~,

0+0_~/=0,

0+0_~=e2'l-2~--e" .

We thus recognize the purely bosonic CAL [ 1 ], by the following redefinition of the ~ field: 70

(24)

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X~

(25) It follows that the super-CAL is the supersymmetrization of the purely bosonic CAL. Notice that the super-CAL can be obtained from the action S-- f d x + d x - dO + dO- ( D + ~ D _ ~ + D + S D _ A + D + A D _ S + 2 e ~ + 2 e

A-~) .

(26)

4. Super-CAL as constrained super-WZNW

In this section we show that the super-CAL can be obtained from superspace WZNW reduction [ 11 ] by implementing certain constraints in the left and right KM supercurrents simultaneously. A remarkable difference between the superspace subgroup-valued WZNW reduction studied here and the non-superspace supergroup-valued WZNW reduction considered in ref. [ 12 ] is that in the former case one has a whole set of first-class constraints and in the latter case a whole system of first-class constraints can be chosen only if auxiliary fields are introduced into the constraints. Let G(x +, 0 +, x - , 0 - ) be a superfield with values in the affine KM supergroup whose Lie algebra is osp-~T 2 )(2), formally written as exp [osp~'~2)<2)]. We start with the manifestly supersymmetric WZNW action with values in the affine KM supergroup exp [ osp( - ~ 2 ) (2)],

r.SwzNw(G)=½x f d2xd2O(Str(G-1D+GG-1D_G)+ f dtStr(G-' OtG{G-ZD+G,G-XD_G})). (27) The corresponding equations of motion read D+ J = 0 ,

D_~=0,

(28)

where J=G-I

D_G,

~=-D+GG

-1

(29)

are left and right KM supercurrents taking values in osp(-'~'T2) (2). With respect to the Cartan decomposition formula (9) we can write a general supergroup element in the following form:

G=G ,

(30)

where G= =exp[ - (ChH+Aa+_-~) ] and G< ~exp( ~< ), G> ~exp( ~> ). Correspondingly, the supercurrents J and ~ are separated into three parts, respectively, J = J < + J = + J>,

~/'=3< + ~= + ~ > .

(31)

Inserting (30) into the WZNW equations of motion (28), we can show that (28 ) become the "zero-curvature conditions"

D+(G-1j + ( J > , G-XJ G --1 + D + G= G= ~) + D + J < + {G=,/> G -x + D + G= G -x , J< ) = 0 ,

(32)

where J< = G ~ 1 D_ G< and J> = - D+ G> G ~ x, respectively belonging to ~< and ~>. We introduce constant elements M< e ~<, ~r> e ~>, which are Grassmann odd and have non-zero components along the simple roots only. Namely, 71

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M < =pu:F-a,+p:F_,,,

8 October 1992

M> =pLFLy,+&Faz,

(33)

where ,u:, p%, p I and p’> are non-zero real constants. Let us impose the constraints J <=G=M
J;=-G;‘a>G,,

(34)

then some algebraic works show that eq. (32) takes the form D+D_@= -&p>

e”+&:

e”-*,

D+D_n=O,

D+D_3=

-&p$

en-@,

(35)

which are nothing but the super-CAL equations ( 19 ) if we set p\ fl: = - 1 = ,uU: p%. It can be shown that the constraints (34) are equivalent to ~<=(G;‘M=M,,

.$,=-(G
(36)

thanks to the fact that M, and a, consist of only odd simple root vectors. Eq. (36) implies that (34) can be expressed as linear constraints on the left and right affine KM supercurrents. We now present a lagrangian realization of the above WZNW reduction. For this purpose we need to impose the constraints of the type (36) by means of Lagrange multipliers. This can be implemented by gauging the superspace WZNW action (27) as follows. Introduce the fermionic gauge superfields A, and A, which take values in the nilpotent subalgebra @< and @>, respectively. The gauge invariant action turns out to be KSgauged(G,&,-&) =Ics wZNw(G)+~

s

d2xd28Str[A,(G-1D_G-M,)+A,(D+GG-1-M,)+A
(37)

One can easily check the invariance of this action under the following super gauge transformations: G+g,Gg;‘,

A> +g,A>g;‘+D+g,

g;‘,

A, -+g
g;’ ,

(38)

where the superfield g, E exp ( @<) and the superfield g, f exp ( @, ). In the gauge A, =O=A,, the Euler-Lagrange equations derived from the gauged action reproduce the equations of motion together with the constraint (36). Therefore the gauged WZNW model is entirely equivalent to the constrained WZNW theory. On the other hand, there exists the diagonal gauge, A, = G z ‘i%f, G= and A, = G= M, G = ’ . In this gauge the constraints set G in the maximal torus generated by the Cartan subalgebra &, that is, G= G=. Taking these facts into account and we have, from the action ( 37 ) ,

Jd2xd2r3Str(G,M,G=*J?,),

&r(G=)=&,,(G=)-

which coincides with the action (26) of the super-CAL if we set p :fl\ = - 1= ,$.pt

(39) .

5. The classical solutions

In this section, we give a Leznov-Saveliev reconstruction theorem [ 13 ] for the classical solution to the superCAL. Instead of working with the transfer matrix Y it is convenient to introduce the matrices Q, Q belonging to the afftne KM supergroup, through the relations Q=eupY, 72

&S-’

eey.

(40)

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Consider now any representation of the superalgebra with highest weight vector 2 (r~)x), namely 8+ 1 2 ~ ) = O, (2 g)ax [ ~- = 0. Then, once the superfields ~ (r) = ( • (mr)x [ Q and ((r) = Q I2 (mra)x) are introduced, they turn out to be superchiral: D + ((o = D_ ~ (') = 0, namely ((r) = (tr) (X-, 0 - ) and ~(r) (X +, 0 + ). The whole dynamics of the super-CAL is contained in the fields ~(~) and (~o. The reconstruction formula expressing the original superfields • , A and S i n terms of~ (r) and (~) is exp[ -- 93 (r)--max ~,l"~/) ] = ~(r)~-(r) ,

(41)

with 2g~)x being the highest weight corresponding to the highest weight state 2 (m~). We can perform as in the ordinary case the Gauss decomposition of Q and ~. let us write Q=N, 0=A3< e e= A3>, where K=,/~= live in the Caftan subalgebra, N>, 37> e exp( ~> ) and N<, A3< e exp( ~< ) are respectively strictly upper and lower triangular supermatrices. As a consequence of the linear systems satisfied by Q and (~, the following equations hold: D_K= =D_N> =0,

D+N> N ; 1 =--e-adX=8+,

D+/~= =D+A3< =0,

A3~l D_A3< =eaatr=8_ ,

(42)

which tell us that iV>, A3<, K= and/~= are chiral superfields and N>, A3< can be solved in terms of K= and/~= respectively. The reconstruction formula can be expressed as

~(r)~(r)__~/~(r)__ \-.max

let= N>A3< e e= --m~x~(r)) •

(43)

This reconstruction formula can also be obtained via hamiltonian reduction from the classical solutions of the super-WZNW model. The classical equation of motion (28) implies

G(x +, 0 +, x - , 0- ) = U(x +, 0 + ) V(x-, 0- ),

(44)

where the chiral superfields U(x +, 0 + ) and V(x-, 0-) belong to the affine KM supergroup. Applying the decomposition formula for G, U and V, (44) takes the form G<(x +, O+,x -, O- )G=(x +, O+,x -, O- )G> (x +, O+,x -, 0-) = U < ( x +, 0 +) e x p [ K + ( x +,/9+)] U>(x +, O+)V<(x -, 0 - ) e x p [ K _ ( x - , 0 - ) ] V>(x-, 0 - ) ,

(45)

with K+ = O+ H - A + a + 1~+ ~, K_ = O_ H - A_ a + 1"_ ~ (here O+, A_+ and ~"+ are all chiral scalar superfields); U< and V< are chiral superfields belong to exp ( ~ < ) , and U> and V> are chiral superfields belonging to exp ( ~> ). Then the constraints (36) tell us that V~ l D_ V< = e ~dK- M<,

D+ U> U> 1 = e -adK+/~> .

(46)

We saw in the previous section that we were able to gain the super-CAL from the superspace WZNW reduction if we set/t 1<# ~>= - 1 =/z2. Without loss of generality we can assume/z 1<= 1 =/z 2< and # ~>= - 1 =/z2. Then (46) becomes V~ID_ V< = e adK- 8_,

D+ U> U;, 1 = - e -adK+ #+ .

(47)

Moreover we have D+ V< = 0 = D_ U>, D_K+ = 0 = D+K_. Now calculating the quantity (2m~x it) IGlAmax (r) ), we obtain, by (45), exp [ - 22
H~,-(D+-D_)~

(here ~ - ~, A, Z). As a consequence of the relations (49), (50) we get that the superfields and their supermomenta satisfy the following Poisson brackets at equal "supertime": (52)

{H~,(X, T), ~(Y, T)}=iy6(X, Y) ,

which look the same as for the ordinary CAL theory with the replacement of the prefix super in front of everything. The integrability property of our theory is implied by the relation for the connection A°= A°++ AC:

(53)

{L~(X, T),.Y2(Y, T)}f[r~2,~ll(X, T)+.Y2(Y, T) IJ(X, Y).

By definition .Y~-.Y®I, -Y2---I®.Y. The r-matrices r e are defined in terms of the generato~ of the osp~T2) (2) algebra and are given by 22 r + = 2 L½H®H+ (e®a+a®~,:)+ ~ 22 H ® H - -2 (1 + ~_T:_~.~_22)H,®H, # 22

22 22 _ (1 + ~_7.__.~_A2)F,t® F _ a . (1-~-~'-~_):~)Fa2®F-,~2

22 22 22 -I +2 ~-T-~-~_22E_ ® e + + -~----~_22F_ctl®Fa1-4- -~-~---~--~,~2F_ot2®Fa2],

(54)

for 12[ < 112[and

r-=-Z-~zr[½H®H+(e®a+a®3)+ ~u2

H ® H - ~ ( 1 + 2212~2122)H'®H'

122

+(1

2

122

A2-@122)F_a,®F,x, + (1--~-~_122)F_,~2®F,~2

(55) for 1121< I~I.

The r -+matrices satisfy the classical Yang-Baxter equations [r~2, r~31+ [r~2, r23] + [r13, r23] = 0 . 74

(56)

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The exchange algebra which summarizes all the algebraic structure underlying the theory can be computed with the method explained in ref. [ 14 ]. The Poisson brackets of the vectors ~(r), (~o are {~tr)(X) , ~ (r') ( Y)} = ~(r)(x) ®~¢rO (Y) [O(X, Y)r + +0( Y, X ) r - ], {(~) (X) ,~(~') ( Y)} = [O(X, Y ) r - +0( Y, X)r + ] ((~) (X) ® ( ~ ' ) ( Y ) ,

{ ~ ) (X) ~, ~ ' ) (Y) } = - ~r) (X) ® 1.r-. 1® ~ ' ) ( Y ) , {(tr) (X) ~, ~t~') (Y) } = - 1 ® ~ r , ) ( y ) .r +.(t~) (X) ® 1 .

(57)

These Poisson brackets look formally similar as the Poisson brackets of the CAL theory and even of the Liouville theory. We stress the fact that however the r-matrices are different and that now we made use of a superfield notation. If expressed in terms of c o m p o n e n t fields the above exchange algebra is more singular than the corresponding algebra for the purely bosonic theory since the Poisson bracket between fermionic fields develops a delta function [ 3 ].

7. Concluding remarks We have proposed an integrable and superconformally invariant model based on the twisted affine K M superalgebra o s p - ~ 2 ) ~ 2 ) that we call the super-CAL model. This model is a manifestly N = 1 supersymmetric generalization of the purely bosonic CAL theory recently proposed by BB [ 1 ]. The generalization to the twisted (or untwisted) affine K M superalgebras which give rise to superconformal affine Toda theories is possible but the analysis is expected to be more involved. The generalization of the above manifest superspace formalism to the case with arbitrary (p, q) world-sheet supersymmetry is very interesting but will not be straightforward. The Coulomb gas realization of the exchange algebra (57) can be provided with the methods developed in ref. [ 15 ]. In forthcoming papers we will analyze the quantization of our model.

Acknowledgement The authors are grateful to O. Babelon, L. Bonora, J. Distler, R. Flume, G. von Gehlen, W. Riihl, P. Schaller and M. Scheunert for discussions. The authors wish also to thank the referee for interesting suggestions and remarks which led the paper to its present form. F.T. has been financially supported by the Deutsche Forschungsgemeinschaft ( D F G ) and Y.Z.Z. by the Alexander von H u m b o l d t Foundation ( A v H ) .

References [ 1] O. Babelon and L. Bonora, Phys. Lett. B 244 (1990) 220. [ 2 ] L. Bonora, M. Martellini and Y.-Z. Zhang, Intern. J. Mod. Phys. A 6 ( 1991 ) 1617; Y.-Z. Zhang, Mod. Phys. Lett. A 6 ( 1991 ) 2023. [3] J.F. Arvis, Nucl. Phys. B 212 (1983) 151; B 218 (1983) 309; O. Babelon, Nucl. Phys. B 258 (1985) 680; F. Toppan, Phys. Lett. B 260 ( 1991 ) 346. [4] A.N. Leznov, M.V. Saveliev and D.A. Leites, Phys. Lett. B 96 (1980) 93; M. Chaichan and P.P. Kulish, Phys. Lett. B 78 (1978) 413. [5 ] M.A. Olshanesky, Commun. Math. Phys. 88 (1983) 63; O. Babelon and F. Langouche, Nucl. Phys. B 290 (1987) 603. 75

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PHYSICS LETTERS B

8 October 1992

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