Extended superconformal algebra as symmetry of super Toda field theory

ANNALS

OF PHYSICS

214, 1-52 (1992)

Extended Superconformal Algebra as Symmetry Super Toda Field Theory*

of

HIROSHI NOHARA’ Institute

of Physics, Meguro-ku,

Received

February

University Tokyo 22, 1991;

of Tokyo, 153, Japan Revised

Komaba,

July

3, 1991

The extended superconformal algebra is studied extensively based on the super Toda field theory. At the classical level, it is realized as the Poisson bracket structure in the superdifferential operator which we call the super Miura transformation and the generators can be interpreted as the conserved currents of the integrable super Toda field theory associated with the particular class of Lie superalgebra. The quantization is performed by free field regularization and the superfield of the Toda field theory embodies the Coulomb gas representation. The unitary representations are obtained using the supercoset construction in the cases of simply laced Lie superalgebras. ‘0 1992 Academic Press, Inc

1. INTRODUCTION Since the discovery of the conformal field theories, much attention has been paid to the classification of the conformal field theory. This problem is relevant in studying the statistical system because in the presence of the conformal symmetry, the theory is invariant under the scale transformation and describes the critical phenomena of the second-order phase transition. The systematic study in this direction was initiated by Belavin, Polyakov, and Zamolodchikov, in [ 11, for the case of the Virasoro algebra and provides rich information on the critical exponent and the scaling dimension of the primary field. Many models which correspond to the solution of the conformal field theory have extended symmetry as well as conformal symmetry. For example, the 3-state Potts model has the spin-3 conserved current which together with the energymomentum tensor constitutes the W, algebra [2]. The systematic study of the extended conformal algebra was initiated in [3, 161. In [4], using the technique of the soliton theory, the W,, algebra which is the generalization of W, algebra was derived and it was shown that the degenerate representation of it can be expressed in terms of the highest weight of the finite dimensional representation of the s/(n) current algebra. * Revised version December, 1990. ’ Present address:

of Ph.D. Yukawa

Thesis Institute

submitted

to Department

for Theoretical

Physics,

of Physics Kyoto

University,

of University Kyoto

of Tokyo, 606, Japan.

1 0003-4916192

$7.50

Copyright 0 1992 by Academic Press, Inc. All rights of reproductmn in any form reserved.

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The main issue in this paper is to generalize the conformal algebra to the supersymmetric one in the case where there exist generators of higher spin. From our experience in the bosonic case, this type of algebra may correspond to the two dimensional physical model which has the extended superconformal symmetry and hence is very interesting. At the present stage, one of the promising methods for studying superconformal algebra seems to be an approach based on supersymmetric soliton theory such as the super KdV hierarchy [S] or the super Toda field theory [6]. In this paper, we consider the super Toda field theory since its Poisson bracket structure can be seen easily at the classical level and we can quantize the theory in a certain manner as we will see in Section 6. We slightly digress to a short explanation of the (super) Toda field theory. The Toda field theory is one of the integrable system associated with the Lie algebra and has conformal symmetry. If we affinize the Lie algebra, the resulting system is not conformally invariant but is still integrable. In this paper, we call it the affine Toda field theory. The super (afline) Toda field theory is the natural super version of the bosonic (afftne) Toda field theory and is obtained by including fermions in a supersymmetric fashion. The systematic study of the super Toda field theory was initiated in [6] from the view of Lie superalgebra. In [6], the supersymmetric integrable system was constructed by imposing the zero curvature condition on the superspace into which the coordinate of the Lie superalgebra is embedded. One of the interesting results is that this procedure can lead to the various types of the integrable system by choosing the way of embedding. Also, by assuming the action of the natural superversion of the Toda field theory and requiring its integrability, some of the results obtained in [6] were derived in [7]. Recently, there has been remarkable progress in the study of superconformal algebra from the approach based on integrable super Toda field theory. We recall that one can construct as many conserved currents as the number of degrees of freedom. In [7-91, they are identified with the classical extended superconformal algebra since they constitute the closed algebra with respect to the Poisson bracket induced from the action. We mean by the extended superconformal algebra, that the algebra contains generators of higher spin besides the energy-momentum tensor and in their operator product expansions, unlike the ordinary Lie superalgebra, there are non-linear terms. The quantization of this algebra is performed by free field regularization although the theory is not free and does not have a well defined classical vacuum. We will see that the fields of the super Toda field theory give us the explicit form of the generator in the free field representation. This paper is the extended version of [7, 81. It is organized as follows. In Section 2, we briefly review the bosonic Toda field theory. In Section 3, the integrable super Toda field theories are derived from two different points of view. First, we directly solve the conservation law using the equation of motion, and second, the same results are derived in the Lax formulation in which the equation of motion is expressed in the form of the zero curvature condition. It will be shown that the structure of the Lie superalgebra enters the theory instead of the ordinary

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3

Lie algebra. We also calculate the spins of the conserved currents in several simple cases. In Section 4, we show the super version of the Miura transformation. The Miura transformation [31] is the celebrated formula which expresses the conserved currents in terms of the super held of the super Toda held theory [32] and we see that as in the bosonic case, they appear in the coefticient of a certain differential operator. In Section 5, we slightly digress to the super Wess-Zumino-Witten model delined on the non compact Lie supergroup. It is argued that by imposing appropriate constraints on the super Kac-Moody currents, the equation of motion of this model reduces to that of the super Toda field theory. We discuss the possibility of interpreting the gauged currents as the generators of the gauge symmetry. In Section 6, we construct the conserved currents at the quantum level in the light cone quantization. We comment on the operator algebra of the N = 1 extended superconformal algebra associated with ~~~(3~2). In Section 7 the representation theory is investigated. In the case of 0~~(312), we will see the exotic minimal series in which the central charge has no upper bound and the operator algebra has a structure similar to that of the chiral ring in the N = 2 minimal conformal model. As for sl(n + 1 in), the ZH symmetry arises in the sector of the chiral primary held. This symmetry is related to the invariance under the permutation of the lields in the formula of the Miura transformation. We also mention the unitary representations of the algebras associated with the simply laced Lie superalgebras, 0~~(2ml2n)(m = n, H + 1) and S/(H+ lin). In Section 8, we discuss some unsolved problems and relevant topics.

1. REVIEW OF THE BOSONIC TODA FIELD THEORY In this section we review the classical Toda held theory. It is defined by the action [lo]

where Kg is the Cartan matrix of some Lie algebra in a symmetrized form, p is the dimensionless coupling constant, and 4i are the i-component scalar lields. The equation of motion are derived from (2.1)

The Poisson bracket in two dimensional Euclidean field theory is (2.4)

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where we consider z+ to be the space coordinate and z- to be the time coordinate [ll, 121. We remark that l/(.z+ -w+) is the delta function with respect to the space integral + ~$2+ . This theory is invariant under the conformal transformation. z* -+f*(z+),

where f * are the two independent conformal transformations. Indeed the action is invariant under this transformation up to boundary terms and hence the equation of motion is inva~ant. The Noether current which generates the conformal transformation is the energy-momentum tensor

The appearance of the linear term, although it causes the central term in the classical Poisson bracket of two T(*%, guarantees the correct transformation properties of the potential terms exp (/I Ej KV#j) with the conformal dimension 1. Put in another way, the fact that this term does not have a definite mass scale since it can be shifted by a translation of 4i is the reason why the conformal symmetry in this theory is anomaIous. The integrability of the Toda field theory is immediately seen in the Lax formulation [13]. We introduce the gauge potentials AZ+ and AZ- which are the connections in pure gauge theory and can be expressed in terms of elements of the Lie group g:

Let us consider the zero curvature condition which is simply

in two dimensions. From the general argument, it is guaranteed that the Lax equation (2.8) is integrable and hence, to see the integrability of (2.3), we only have to rewrite it in the Lax form (2.8). It can be shown that the equation of motion is obtained by taking the specific gauge for A=& and AZ- as

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SUPERCONFORMAL

ALGEBRA

3

where ej, fj and & are the Chevalley generators which satisfy the foliowing commutation relations:

Using Eq. (27), we obtain the solution [lo]

where Aj are the fundamental which satisfy the conditions

weights and jAj) are the highest weight vectors

eijAj}=o,

(Aj~~.=o.

(2.12)

xk are the arbitrary functions of z* which are related to the initial conditions on the forward light cone through a certain point (we choose it to be the origin),

(2.14 and ran satisfy the equations

~3+m+(z+)=m+(z+) x x:(z+)ei,

(2.15)

i= 1

db?-(z-)=m-(z-)

i

g(z-)A..

(2.16)

i=l

In (2.11) the space and time coordinates the conformal symmetry. The fact that implies the existence of the extended expression, in the case of d(r + i), Bilal tion of order r + 1 [ 151,

are separated from each other thanks to there are r arbitrary functions in (2.11) conformal symmetry. Indeed, using this and Gervais derived the differential equa-

6

HIROSHINOHARA

where I,Gis expressed as

(2.18)

(2.19) i=l

where IAi) is the highest weight vector in the fundamental representation, ai are arbitrary constants. wi is the differential polynomial of c?: X+/C?+x+ and can be interpreted as the conserved current a -wi = 0 if we identify 8: x+/a +x+ with the a +q5. They constitute the closed algebra, that is, classical Wr+, algebra delined by Fateev and Zamolodchikov 1161 with respect to the Poisson bracket (2.4). In general, we can construct as many independent conserved currents as the rank of Lie algebra g and consider them to be the generators of the classical Wg algebra. 3. CLASSICAL SUPERTODA FIELD THEORY

3.1. Derivation of the Integrable Super Toda Field Theory I We discuss the construction of the integrable Toda held theory with supersymmetry in two dimensions in this section. There are several ways to establish the integrable super Toda field theory. As one of them, we lirst introduce the method in [7]. To begin with, we briefly explain the procedure in the case of N= 1 supersymmetry. First, we assume the natural super version of the action (2.1) to be the one which is obtained by replacing di with the N= 1 superfield Qi and a* with the super-derivatives D &. We remark that at this stage KU is a parameter and assumed to be invertible and symmetric. Then Kq is determined so that there exist as many independent conserved currents as the number of degrees of freedom. It can be shown that the action is invariant under the super conformal transformation and the conserved currents constitute a closed algebra with respect to the Poisson bracket. From this fact they can be identified with the generators of the classical extended super conformal algebra. As in the bosonic case, we deline the spin of the conserved current to be one half of the number of the superderivative to which spin $ is assigned. We derive the integral super Toda held theory following the above strategy in the simple cases explicitly. We assume the action [ 171

.

.z+ =XO+ix,,

Z-

= x0 - ixl,

(3.21

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SUPERCONFORMAL

7

ALGEBRA

where Kg is a symmet~c and invertible matrix of dimension r and Qi are N= 1 superfields, cPi=q5i+e+l)+i+e-$wi+e+e-Fi,

(3.3)

where #i are the scalar fields, $J+~are the right (left) handed spinor Iields, and Fi are the residual fields. The chiral Poisson bracket is defined by

(3.4) where

(3.5) and Dk are the superderivatives detined by

The classical equations of motion are

and we see that the action (3.1) and the equation under the superconformal transformation

which is generated by the energy-momentum

of motion

(3.7) are invariant

tensor

(3.9) Requiring the integrabiIity of (3.7), we have succeeded in obtaining several solutions for Z$ each of which coincides with the Cartan matrix of a certain Lie superalgebra [7]. It is not yet clear why Ku should be identilied with the Cartan matrices of Lie superalgebra in this method, but will be clarified after explaining another one in subsequent subsection. Let us discuss the detail of each case. (See Appendix B for the expression of &.) 1. ~~(112)

(the super Liouville theory).

This is already well known as the super LiouvilIe

theory and has been studied

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NOHARA

extensively from various points of view [18]. energy-momentum tensor

The conserved current is only the

There are two conserved currents with spin 1 and ; which generate the classical ZV= 2 conformal algebra,

Their Poisson brackets are {Tti)(Zl)

Tc1)(Z2)] =$

Tc3~2+$+ 12

{Tf3’2yzl)

Tyz2)j

12

T(i)(Z2++

= -2 g

T9Z2)-2+; 12

12

T(‘)(Z*), 12

(3.12)

This enhancement of supersymmetry invariant under exchange @t w Q2.

comes from the fact that the action is

3. o&3/2). This case is the first non-trivial

one because there are spin-2 conserved currents, (3.13)

(3.14)

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ALGEBRA

We show the Poison brackets { Tf3iz)T(2)j and {T(2)T(2)j,

We remark that f2 = Tcz) - (2/3fi) D+ T31’ becomes a primary fieid, that is,

This typ of algebra was first proposed in [2Q] by bootstrap analysis on the assumption of simple fusion rule such as [T(2j][T(‘1] = @[l-j. It can exist oniy for the special value of central charge e= - $?*The classical version of this fusion rule coincides with (3.14) by the reason which will be explained in later section.

In the finag example there arc two spin $.generators and a spin-2 one,

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It should be noted that T(*) does not become the primary field by adding the total divergence of TC3’*) and Ty’*’ to it while TC3’*) is a primary held. (We summarize the Poisson brackets of these three conserved currents in Appendix A.) This situation does not happen in the bosonic W algebra as discussed in [ 191. If we identify @, with 03, T!j’*) vanishes and T(*) and TC3”) constitute the algebra which is equivalent to the one in the case of o.s~$3[2) since the equation of motion of the o.rp(412) super Toda held theory reduces to (3.7) with Kq given by

cJ -2

1 2 0

(3.19)

which is equivalent to the Cartan matrix of 0.~~(312). We remark that the above solution is not in fact unique and actually the general solution is

Kg=(

-;-b

i

(3.20)

-ib),

where ub(a + b) # 0. The Lie superalgebra corresponding to this matrix is D(2, 1;a) which is the deformation of o.s~(4[2) parameterized by a = b/a. We have not succeeded in finding other solutions for Kg which provide more natural super versions of Zamolodchikov’s W3 algebra, generated by the spin $ and 5 conserved currents. In [20], such an algebra was lirst proposed and in [21], it was derived explicitly by the coset construction (SU(3)3@%r(3),/sU(3)d) at the specific value of the central charge c = y. We next treat the Toda field theory with the N= 2 supersymmetry in the same way as before. Since we already encountered such an example in the case of the Toda held theory associated with sQ211), our aim is to lind the examples which are not reduced to the one obtained from (3.7). To begin with, we deline the N= 2 super Toda field theory. The action is given

by WI (3.21)

S=~d2zd4tl~K~~i@j+~d2zd20~~exp(/?~Kti@j)+c.c.,

ij

j

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11

ALGEBRA

where Kg is an invertible and hermitian matrix, Qi and Gj are the chiral super fields which are complex and satisfy the following conditions:

Pi = 0,

(3.22)

{D+,Bp}=&

{D-,D+}=(l,

(3.23)

Dp,D-}=O.

(3.24)

The YV= 2 super chiral Poisson brackets are .detined by

{D+ @J.Z,), Gj(Z2)} = -2(K-$

2, 12

{D-&(Zl),

(3.25)

@,(Z2)} = -2(Kp1)g 2, 12

where

The equations of motion are

(3.21) and (3.27) are invariant under the Iv’ = 2 superconformal

transformation

cDi(z+, e+, /9+, z-, o-, e-)

(3.28) which is generated by the energy-momentum U( 1) charge

tensor with conformal spin 1 and zero

As before, Kg is determined so that there are as many independent conserved currents as the number of pairs of the chiral super-field. Now we explain two solutions below.

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1. OSP(112). The lirst is the N = 2 super Liouville theory which can be identilied with the super Toda field theory associated with ~Z(211). The conserved currents in (3.11) and the Poisson bracket (3.12) take the following compact forms: T1=-D+@D+dD+

D&D+ D

D+CE

(3.30)

P

Ts(z*)+$D+T~(z*)-g%+

Ts(ZJ

12

12

(3.31)

In the second example, there are conserved currents of spin 2 F2) as well as the energy-momentum tensor:

2

+ x

z&

D+

ii+GiD+Qj+b+

agk

y11=

-i,

Yl2

=

2.-ruKjk,

=y21=

19

x11=x22=

Y22 = (4

D+

b

V=l

-;,

z11=

-z21=

&

(3.33)

D+Ql+c.c.,

x12

=

-

1,

x2,

=

(3.34)

0,

Z12= z22 = 0.

(3.35)

Their chiral Poisson bracket is {%W~

T2@‘2J}

=

T1(Z2)+$2D+

+s

T1(Z2)+$+ 12

T1(Z2) 12

8012’J12 +

12

-G2

96

4~12~12 a+ +

~2V25j7

12 012(712 z;2

.

(3.36)

z12

In contrast with the case of N = 1 super Toda held theory, T2 does not become primary by adding terms (D + D + - D + D + ) T1 or (T(‘))*. Since there is no restriction on the central charge for the algebra to be closed, this type of the N=2

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13

extended super conformal algebra is different from the N = 2 super W algebra discussed in [23]. In the cases of ~I(21 1) and ~Z(3\2), we found that in both cases the generators can be divided into the two sets which are decoupled with each other and that each of them constitutes the same algebra as classical N = 2 superconformal algebra and the N = 2 extended superconformal algebra, respectively, From this fact, since we have not succeeded in linding any non-trivial solutions for Kq which are not reduced to the former examples, up to rank 2, 3, or 4, we suppose that there is only one N = 2 extended super conformal algebra in our approach. 3.2. Derivation

of the Integrable Super Toda Field Theory II

We illustrate another method which is called the super Lax formulation and enables us to construct the general supersymmetric integrable systems more systematically [6] than by the previous method. To begin with, we briefly review the formulation of the super Yang-Mills gauge theory [24]. We introduce the gauge potentials AM = (AO+, ,40-, A=+, A=-) which take values in some Lie superalgebra in the flat superspace x”” = (0 +, I!-, z +, z - ). We deIine the covariant derivative to be VM=D.,,+AM,

(3.37)

where DM=(DO+,

DO-, a+, a-).

(3.38)

The torsion TLN and the held strength FMN in this superspace are expressed in terms of D,+, and V,,,,,

where [ , } is the graded commutator [A, B} = AB - ( - )‘(‘)‘@‘BA, A: grassmann even, A: grassmann odd.

(3.40)

The Bianchi identities or the Jacobi identities read

rvhf,rv,v,VP}} =a

(3.41)

These equations are just the identities and do not impose any constraint on the gauge potentials. We turn to the super Lax formulation of the integrable super Toda held theory.

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In this approach, the equation of motion of the theory we consider is translated into the zero curvature condition which is expressed simply as FMN = 0.

(3.42)

In this situation (3.42), the independent components of the field strength are only Fo+o+ 3 F@+@-,Fo-@-) since the vector potentials A+ can be expressed in terms of the spinor potentials &. Let us consider a pure gauge theory, i.e., A,,, takes the form AM=G-‘.DMG,

(3.43)

where G is the superfield which takes value in some Lie supergroup expressed in terms of the generators of the Lie superalgebra ti as G = exp x aiti (Ii

.

and is

(3.44)

J

In the cases of Lie super algebras, there are two types of roots, bosonic and fermionic, and in (3.44), the two dimensional coordinates z”” are embedded in those of the Lie superalgebra ai. For example, superderivatives Dk anticommute with fermionic roots in ti. Thanks to (3.43), there is only one independent condition in (3.42), F8+@m= 0.

(3.45)

Now we discuss the Lax formulation of the integrable super Toda lield theory. In [6,26, 371, it was shown that the Lie superalgebras which admit the superprincipal embedding of 0~~(112) lead to the integrable supersymmetric system and that they are only

(For the Cartan matrices of these Lie superalgebras and their Dynkin diagrams, see Appendix B and Fig. 1.) In cases of such Lie superalgebras, all the simple roots can be chosen to be fermionic ones and the Cartan matrices are regular. Indeed, by setting G in (3.43) an element of the Lie supergroup generated by these Lie superalgebras, we can impose the gauge conditions on A@+ and Ao-,

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sl(n+i

ALGEBRA

15

In)

osp(2nill2n)

osp(2nl2n) osp(2n+2

FIG.

j2n)

1. Dynkin diagrams of Lie superalgebras which admit the super principal embeddings.

where hi, ei and fi are the Chevalley generators of the Lie superalgebra which satisfy the commutation relations

(See Appendix B for detail of the notation.) The equation of motion (3.7) is obtained from (3.45) with Kg the Cartan matrix of the Lie superalgebra as easily seen by a direct calculation using the commutation relations (3.48). If we adopt a Lie superalgebra other than (3.46), we cannot impose the gauge condition (3.47) but the one given by [25],

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The appearance of O* in (3.2) breaks the super-symmetry and leads to integrable equations

In this paper, we are mainly interested in the supersymmetric cases and do not pay much attention to non-supersymmetric cases. For the reader who want to know the detail studies of them, we refer to the reference [25]. From the above discussion, we understand the reason why the previous examples associated with 0.~~(112), ~~~(312) and s/(312) ~~(412) Q2, 1;~) lead to the integrable and supersymmetric Toda field theory, from the Lie superalgebraic point of view. As a result offurther study along the lines of Ref. [7], up to n = 6, in the N= 1 super Toda held theory we have succeeded in obtaining all the conserved currents in (3.46). From the observations in the cases of the low rank of Lie superalgebras we propose a complete list of the conserved current in the N= 1 super Toda lield theory below, given in Table I. In Table I, 0~&2n k 112#), o.~~$2n[2n)(n > q2), 0z9(2n + 212n) and @2/ 1;~) are associated with the extended N = 1 superconformal algebra since the spins of the conserved currents are greater than or equal to 2. However .Y~(PZ + 11~~)is associated with the extended N = 2 superconfo~al algebra, because T(i) and YI’C3’zjconstitute the classical N = 2 superconformal algebra, although we started from the N= 1 super Toda field theory. We can identify the components of T(l) and TC3’2) with the generators of the classical N = 2 superconformal algebra up to normalization as follows: T(‘)=J+O(G-G)

(3.51)

9

P3j2) = G + G + @-.

(3.52)

This enhancement of super-symmetry is originated from the symmetry of the Dynkin diagram such that the diagram is invariant under the reflection with respect to the center of the diagram. (See Fig. 2) TABLE Lie superalgebra

Spin of conserved

s&z+

ix(2,3,4

l[n)

w~~l~~ w(3 121 osp(2n - 1[2fl)(n > 2) osp(2?l+ 112)(n > 2)

wJPl~~~=N2l~~~ usp(2n ]2Fr)(n > 2) osp(Zn + 2 1Zn) WI k a)

I current

,..., 2n,2n+l)

s

$22 ix (3,4,7,8, 11, 12, .. .. ix (3, 4, 7, 8, 11, 12, .. .. l>$ +x(3,4,7,8,11,12 ,..., $x (3,4, 7, 8, 11, 12, .. .. $3 g a?

4n--5,4n-4,4n4n- 1,4n) 4n-5,4n-4,4n-l),n 4n - 1,4n), n -I- 4

1)

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SUPERCONFORMAL

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17

2. Symmetry of Dynkin diagram of s1(n + 11n) under reflection,

We remark that o.sP(~~& 1[2n),which motion associated sZ(2n[2n - 1) (~Z(~PI+ (see Fig. 3):

there is an interesting relation between sI(n + 1in) and we have seen in the simple cases before. The equation of with o.sP(~~- 1~2~)(0~~(2~ + 1/2n)) is obtained from 112n)) by the following identification which we call Zz folding

1. OSJI(2PI- 112Iz). Qji=CD 2n+

1 -i?

(3.53)

2. 9?(2n + 112Iz). @jz@2n+2-i.

(3.54)

This is possible because of the symmetry of the Dynkin diagram mentioned above. According to this fact, we see that the conserved currents in the former cases are also derived from the latter ones by .Zz folding and have even parities with respect to this operation. Note that this situation also exists between 0~~(2ml2n). (See the previous subsection for example and Fig. 4.)

o++----w

FIG.

3. Zz folding of sl(n + 1 1H).

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FIG.

4.

Z2 folding

NOHARA

of mp(2m

We remark that when we take the limit formed into D+ Qi. In this case if we symmetry is induced in the theory and we obtain so(N) superconformal algebra generators.

+ 1 1n)(m

= n, n + 1).

/I + OZ, the conserved currents are transhave N superfields, N-extended superas the extended superconformal algebra rather than the algebra with higher spin

3.3. IVfan$estly N = 2 Supersymmetric Formulation

It is possible to formulate the N = 1 super Toda field theory associated with d(n + 1 in) which was shown to have the N = 2 supersymmetry in the N = 2 super formalism by embedding osp(212) into d(n + l[n) [26]. For studies in later Sections, we review the construction in this example. To begin with, we show that the simple roots of osp(212) can be embedded into those of d(n + 1 In). We divide the simple roots of d(n + 1 In) {hi, ei, fi} (1 < i < 2n) into two subsets {& eli,fzij, {/rziP1, eziP1,jzi-,} and deline {Ai*, ei*,L+} to be

hn-zi=hGT

It is easy to see that {/z;, e:,

enf

l+2ize2i9

fn-

F } satisfy the commutation [e:, [Al*,

f:}

e,+ }

=

Svh,~,

=

Kuel*,

1+2i=fG,

relation

(3.56)

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TABLE

ALGEBRA

19

II

Lie superalgebra

Spin of constants of motion

d(rl+lln) q(2r.z - I [2??) Oq42n+l~2~) OS~(2~2)(=d(2~1))

1, 1, 2, 2, .. .. n - 1, ll- 1. tl, tI 1, 2, 3, .... 2il-2,2n-1 1, 2* 3, .... 2?I- 1, 2fi i, 1

where Kg becomes the Cartan matrix of o&& + 112~) and O.YJJ(~~ - 112n) for ~Z(2n+ 112n) and ~l(2n/2n - 1) respectively and in particular, for i= 1, (3.56) coincides with the same commutation relations as in the case of 0~~(212). Next we consider the N= 2 supersymmetric pure gauge theory in which the held strength vanishes: pz**, fP, 0%. (3.57) F.tm = 0, If we choose the gauge potential (A$, Ai)

to be

we obtain from (3.57) the equation of motion of the integrable N= 2 super Toda held theory associated with OS~(~E& 112n). We have seen that when Kti is the Cartan matrix of sl(2[1) or ~2(312), the conserved currents are divided into two sets which are decoupled with each other. The reason for this is that if Kg in (3.56) is the Cartan matrix of ~/(n + 1 in), then {A:, e$, f F ] span the root system of sZ(~-t lhz) @sZ(~ + lhr). We also remark that there is only one super Toda field theory with N= 2 su~rsymmetry since the nontrivial embedding of o&2/2) is possible in the case of S/(B + 1 hr) and not in other cases. Finally we propose the list of the spins of the conserved currents in the N = 2 super Toda held theory based on the same calculation in the case of N = 1 Toda held, theory; see Table II. 4. THE MURA TRANSFORMATION

4.1. Bosonic

Case

In a previous subsection, we have constructed the integrable super Toda held theories associated with various Lie superalgebras which admit the super principal embedding. The integrability comes from the extended superconformal symmetry generated by the conserved currents. In this section, we show that the conserved

20

HIROSHI

NOHARA

currents can be expressed simply using the superdifferential operator which provides us with the supersymmetric version of the Miura transformation as in the case of the bosonic Toda held theory. The superdifferential operator is expected to play a central role in formulating the hierarchy of the supersymmetric non-linear soliton equations which has the Hamiltonian structure associated with the extended superconformal symmetry and is a candidate for the powerful study of the extended superconformal algebra. In this section, we set /? = 1 for simplicity. It is instructive to review the case of the bosonic Toda field theory before going to the supersymmetric one. We recall that the equation of motion of the Toda held theory based on the Lie algebra g of rank r is equivalent to the zero curvature condition (2.8). It is obvious that (2.8) is invariant under the gauge transformation A k +c1U+g-‘4g>

(4-l 1

where g is an element of the group generated by g. We will see that the matrix valued differential operator d + A+ leads to the well known Miura transformation. To see it explicitly, we take g = s/(n). The generators are given by hicEiiBEi+l,i+17

eicEi,i+12

.ficEi+l,i3

(4.2)

where Ev is a matrix with a one at position (i, j) and zeros elsewhere. Let us regard a+ + A+ as the operator acting on the n-dimensional vector 5 and consider the following equation: (a+ + A+)C=O. Writing

(4.3) in components as a+ +a+%

c?++A+=

(4.3)

1 a++a+q2

1 ..

..

(4.4)

a+ +a+vn-1 where (4.5)

(4.4)

EXTENDED

SUPERCONFORMAL

ALGEBRA

21

we obtain the differential equation for ui LVI = 0,

(4.7)

where

On the other hand, we can get the forms of gauge lields A + and A - by adopting the unique gauge transformation generated by negative roots,

l * IT 0

A+=

1 0

W” W”-,

.. . . ...

A- =O.

. .. 1 0 w2

(4.9)

Note that (4.7) is invariant under the gauge transformation generated by the lower triangular matrix with ones along the diagonal. In this way we get the well-known Miura transformation

In (4.10), L is called the Lax operator in the (n-reduced) KP hierarchy. We remark that in the Drinfeld-Sokolov gauge (4.9) [32], the zero curvature condition (2.8) means the conservation of the currents wj. Hence the coefficients wi constitute a closed algebra with respect to the chiral Poisson bracket in the Toda field theory, Id+ 4itzL

#j(wJ}

= pz’f~)~.

(4.11)

In the same way discussed above, we can derive the Miura transformations in the cases of ~0(2n + 1) and ~0(2n) and corresponding gauge fields AZ+ in an analogous way. For later convenience, we give the Miura transformations also in the cases of ~0(2n) and ~~(2n + 1). 1. s0(2?r + 1).

(4.12)

HIROSHl

NOHARA

,

2. m(2n). 2

-1

-1

2 -1

-1 **. 2 -1 -1

-1

(4.16)

-1 2 0

0 2<

A+= -1

(4.17)

....

%-2=L2-L3 .

(4.18)

EXTENDED

SUPERCONFORMAL

ALGEBRA

23

Finaliy, we comment on the spin of the conserved currents from the group-theoretic point of view. Since the differential equation L~I= 0 is invariant under the gauge transformation L+GLG-', (4.20) which can be identi~~d with the group action on L in the adjoint representation, it is reasonable to expect that the spins are related to the group invariants, for example, the Casimir invariants. The Casimir invariant is detined by

where the symmetric summation is necessary to extract the independent part which cannot be expressed in terms of elements of lower degree. Indeed, from the l-dimensional Toda molecule, it is proved that the spins coincide with the degree of the non trivial Casimir elements by an argument similar to the ones in [33, 341. We show the list of the degree of the independent Casimir elements of S/(U), ~0(2n -t- l), and ~0(2n), in Table III. One can verify that the degrees listed above coincide with the spins using L in the cases of low rank. For example, let us examine the case of SO(~)z #o(2) @ SO(~). By expanding (4.19) we obtain

(4.22) where T and F are the conserved currents of spin 2 as indicated in Table III.

Now we turn to treat a supersymmetric case in the context of the above discussion. As mentioned in the previous section, the equation of motion of the N= 1 super Toda field theory based on the Lie superalgebra g is aIso written in the supersymmetric zero curvature condition, {D+ +A+,D-

+A-}=O,

(4.23)

(4.25) TABLE

III

Lie aIgebra

Degree

d(n)

2, 3,4, .. .. it - 1, n 2, 4, 6, .. .. 2n - 2, 2n 2,4,6 ,,.., 2n-4,2n-2,n

so(2n + 1) so(2n)

of Casimir

invariant

24

HIROSHI

NOHARA

(4.23) is invariant under the gauge transformation (4.26)

A+-+G -‘A+G+G-‘D*G.

Here we treat three types of Lie superalgebras, sZ(~+ l\n), O.Y~(~PZ k 1\21~), and &2m[2n) (m = n + 1, n), which will be shown to produce the natural versions of the Miura transformation sZ(n), so(2n + 1) and so(2n), respectively. As a concrete example, let us first consider the case of g = sZ(~+ 1 in), The explicit forms of !zi, ei, andh are given by el =&,n+~T

AI =E,I +-%+z,n+z>

fl=K+z,~,

hzi= -E~+2~i,~+2-i-E~+l+i,~+l+i, ex=En+l+i,n+z-iy i fIi= -En+2-i,n+l+iy

(1
We regard the operator (4.28)

D+ +A+

as the differential operator acting on the super vector space, the elements of which are Vl V fI+1 -

SE

vII+2

II

(4.29)

’

v2lt+1

where vi (1 < i < PI+ 1) are Grassmann-even numbers. From (4.24)

and vi (i > n + 2) are Grassmann-odd 1 0

0 . ..

.

0 1

A+=

EXTENDED

SUPERCONFORMAL

25

ALGEBRA

where q;= D+Qi.

(4.31)

(D+ +A+)C=O.

(4.32)

Let us consider the equation

As before, if we write (4.32) in components of the vector 5, then (4.32) reduces to

L=D~+‘+(-l)~~~l~,D~~l+~,~,~D~~*-

we discover the supersymmetric

. . +wH+I,2,

(4.35)

Miura transformation

U’+-D+@znW+ +~+~@~~-I-@BJN~+ -~+~@~n-~-%~-~~~-~~ tD+-D+C@z-WW+ +D+t@,-@2lND+

+D+@l)

=D2n+1-w,D2~-~+w~,2D2~-2+

. . . +w

n+

(4.36)

112.

We see that each coefficient of L becomes the generator of the classical extended superconformal algebra. We note that there is a supersymmetric Drinfeld-Sokolov gauge in which A + and FI ~ take the forms 0 w n+k?

0 0

w3/2

..'

K'n - 3,2

fi'n - I,2

A+=

0

1

,

A-=0,

0

0

(4.37)

and correspond to the last equation in (4.36).

26

HIROSHI

NOHARA

Let us consider the above statements in the case of n = 1 for simplicity, The corresponding A+ and L are

If we perform the gauge transformation

(4.26) generated by negative roots

then the A + and A- take the forms

3

A- =O,

which correspond to the super Drinfeld-Sokolov gauge. the classical N= 2 superconformal wl ad w3J2 in (4.39) constitute respect to the chiral Poisson bracket [7] {@iCzlL

@jtz21)

= - tK-‘lij

h Z12.

(4.41)

algebra with (4.42)

In general, We in (4.35) generate the classical extended N= 2 superconformal algebra with respect to the chiral Poisson bracket (4.42) with Kq given by the Cartan matrix of d(n + 1 in). Let us next consider the case of mp(2n k 112n). First we explicitly derive the supersymmetric Miura transformation for O~P(112) and O~P(312). In each case, Ai, ei, and fj are given by

EXTENDED

and the corresponding

SUPERCONFORMAL

II + in (4.24) becomes 0

A+=

D& 0 0

A+=

0 -1

0 0

1

I 1

0 0 0 0 0 -DQI 1 0

27

ALGEBRA

0 0 1

D+@I-D+% 0

0 1 0

m(312)).

0 -D+Ql+D+& (4.46)

By rewriting (4.32) as before, we obtain the supersymmetric Miura transformations w~~P~*

L=(D+-D+@)D+(D++D+@)=D;-(D:@D+@-D;@),

CD;

+ l-(3jz)D; + (-27934)7+9D+

eD+ T(2).

Here Tc3'2)is given in (3.9) and Tc2)is given in (3.14). For general n, the super Lax operator L is given by

osp(2rz+ lj2?I).

(4.47)

(4.48)

28

HIROSHI

NOHARA

Finally we treat the case of 0~~(2mi2n) (m = n + 1, n). We show the general form of the super Miura transformation, mp(2n[2n). n=l

L=v)+

+D+@*n+lW+

-~+~%"-@%n+l~w+

+~+~@2n-~-@*rJ~~-~

(D++D+(~~-~~))(D+-D+(~,+~~-~~))(D++D+(~~-~*))D~l P+ -~+~@,-@d~~~~~~+

(4.53)

-D+%n+lh

(See Appendix B for corresponding Chevalley generators.) Let us pause to compare the super Lax operators with the bosonic ones, It is easily seen that the Lax operators L for sl(n + l[n), 0~~(2n& 1[2n), and 0~~(2n[2n) (m = n, n + 1) can be thought to be the super-version of those of s/(n), ~0(2n + l), and ~0(2n). This is because each Lie superalgebra of the former contains the latter as its subalgebra. It would be convenient to rewrite the Miura transformation of s/(n + 1 In) using the chiral superlields for the study of the representation theory in later section. To this end, we reduce the N= 2 superdifferential equation for multi variables (D + A@++ &+)u = 0,

D=D++D-,

to one for a single variable as before. The result is

(4.54)

29

EXTENDEDSUPERCONFORMALALGEBRA

2. SZ(2?r+ 112n). Lv=O, L=P+

-~@*J~+

+m@2n-l-~2nw+

CD+ -W@2-~3lHD+ XV)+ - D(@, -&))(D+

-~~~2n-2-@*n-l~~~~~

+W*1-@2)W+ + D(d&

P+ +~~%n-~-@~n-~~W+

+~~~I-@,~~ Q$))...

-W=%-I-hJW+

(4.56)

+Dhl.

Using the super Lax operators constructed here, we checked that the list of the conserved currents which was proposed in the previous section is correct up to n = 10. Although we have not yet justilied the result in a more rigorous way based on the mathematical point of view, we believe that it gives the right answer. It is worthwhile to note that in the case of S/(H + 1 in) (4.35), the degrees of the Casimir invariants are exactly twice the spins of the conserved currents given in Table I. 5. DIGRESSION TO THE SUPER WESS-ZUMIN~WITTEN

MODEL

We show that in the case of ~SP(112) the equation of motion of the super Toda held theory can be derived from the constrained Wess-Zumino-Witten model and the conserved currents are gauge equivalent to that of the super Toda held theory. This result is a generalization of the bosonic case to the supersymmetric one and possibly gives us a hint on constructing a new physical model which has a gauge symmetry generated by the extended superconformal algebra. We deline the action of the super Wess-Zumino-Witten model by [27]

where G is the superlield which takes value in ~SJ$112) and the last term means the integral over the three dimensional manifold with the boundary V. The equations of motion are given by Dp(D+G.G-‘)=O,

D+(Gp’%G)=O.

If we expand .J* in terms of the generators r’ E {/z, e, e2, A f2}

then the Poisson brackets of J’i read

(5.2)

30

HIROSHI

NOHARA

where fUk is the structure constant of ~~~(112). We decompose G into three parts (Gauss decomposition), G= F-H. E=exp(fxz+f2y2)

.exp( --ah) .exp(exi +e2y1).

(5.5)

In terms of (xi, yi), the J* can be expressed as J+=(-D+@+xzD+xiexp(-@)+yzwl)h +(-D+xrexp(-@)+w1x2)e - w,e2 -(D+x2+x2D+~+D+xly2exp(-~)+x2y2wl)f (5.6)

+(-2~~~+~+~+~2+x2~+x~+2x*~+x~~*exp(-~)-~~w~)~*,

wl=exp(-2~)(-D+yl-xlD+xl),

(5.7)

J-=(D-@+xlD+xzexp(-@)-y1w2)/r +(D+x*exp(-@)-wlxl)e +%f2 +(D-x,+xlD-@+D-xzylexp(-@)+xryiwz)~ (5.8)

+(2~~~+~+~-~~+x~~+x~+2x~~+x~~~exp(-~)-~~w~)~2, w2 = exp ( - 2@)( - D-y2 + x2DPx2).

(5.9)

If we impose the constraint on (5.9) -STr(J+j)=STr(J-e)=

1,

S Tr(J+ f2) = S Tr(J- e*) = 0,

(5.10)

that is D+xl

=w

(@I,

D+x2=ev

(@I,

w,=w2=o,

(5.11)

then the equations of motion (5.2) are reduced to D-D+@=exp(c?),

(5.12)

which is the super Liouville equation. From the above observation, there should be the gauge transformation J++&=

12J+Q-1+D+Q.QP1,

(5.13)

under which J+ also satisfies the same constraint (5.10) and takes the form J+ c -e + T~‘~*~*,

(5.14)

31

EXTENDED SUPERCONFORMALALGEBRA

where P3’*) is the conserved current of the super Liouville equation. Indeed, if we choose Q to be

Q=evtft +f2d <=D+@-x2,

(5.15)

v=D+%+x~D+@+y2+D+{,

then we obtain (5.14) where Tc3j2'= -D;@D+@+D;@.

(5.16)

We remark that in this case there is no potential term when we rewrite the action (5.1) in term of D + @ by substituting (5.11) into it. The analogous phenomenon is already observed in the bosonic case [28,29]. In the case of the Wess-Zumino-Witten model associated with G, it can be shown that by imposing the constraint J;

z

p 0

i E positive simple roots, i E other positive roots,

J[

c

v 0

i E negative simple roots i l other negative roots,

(5.17)

the equation of motion coincides with that of the g-super Toda held theory although we do not prove it in this paper [30]. In both super Toda held theory and the constrained Wess-Zumino-Witten model, it is obvious that at least the extended superconformal algebra generates the symmetry of the equation of motion. The interesting question is whether it is related to the gauge symmetry of a certain model or not. When we think of the constrained Wess-Zumino-Witten model, it is easy to see that in the case of s/(2), CXS~( 112), and s/(211), the conserved currents generate the symmetry of action since they are the generators of diffeomorphisms, while in other cases we have not yet arrived at a delmite conclusion. The difficulties in the latter cases come from the fact that the product of the two group elements which satisfy the constraint in general does not obey the same constraint [29]. It seems to us that a key to this problem lies in the geometric interpretation of the constraint.

6. QUANTUM

EXTENDED

SUPERCONFORMAL

ALGEBRA

In this section, we construct an operator version of the classical extended conformal algebra for further study in later sections. The direct quantization super Toda field theory with the action (3.1) is a very difficult problem [7]. fore we take the following route. First, we construct the conserved currents of the super Toda held theory quantum level using the free field normal ordering regularization [lo].

superof the Thereat the These

32

HIROSHI NOHARA

currents close by themselves under OPE. Then we regard these expressions of currents as the Coulomb gas representation of the extended superconformal algebra. At this stage, there are no difliculties in quantization. Generally speaking, the classical conformal symmetry is easily seen by nonlinear soliton equations, while for the realization of the conformal symmetry at the quantum level, it is convenient to adopt the free held realization and BRS cohomology. In particular, in the bosonic case, the constrained WZW model on the maximally noncompact Lie group reduces to the Toda field theory [19,29, 351. Indeed the equation of motion of the former is gauge equivalent to that of the latter and ,4 + can be identified with the gauge current: classical level: quantum level:

WZW model

Toda tield theory

Current algebra Feigin-Frenkel realization

IV-algebra Fateev-Lukyanov realization

We briefly review the quantization a la Mansfield in the bosonic case [lo]. The free field quantization is applied to the Toda held theory, i.e., the chiral Poisson bracket

is replaced by the OPE given by a+~i(zt)~j(z2)~

A!cL zt+ -z;

+ ... .

(6.2)

Therefore the equation of motion is given by

where : : denotes the normal ordering. Because of the normal ordering in the LHS of (6.3), there are the quantum corrections to the conserved currents. For example, let us consider the quantization of the classical WJ algebra associated with 43). In the classical case, we have the conserved currents

EXTENDED

SUPERCONFORMAL

ALGEBRA

33

On the other hand, in the quantum case, they are modified to

(6.5) which are obtained by simply replacing l/b with (l//I + fl). T(') and T13'generate W3 algebra and their OPE are

1 +(z-w)*

A J2 7-c*)+ 10 + (6.6)

In the case of s/(n), the quantum conserved currents are derived from the classical ones by the modification of the coupling constant. From this fact, we see that the latter are lifted to the quantum ones which constitute the closed algebra with respect to OPE (6.2) without any corrections. In other cases, that is, the non-simply laced Lie algebra, there is a non-trivial correction since for example the correction to T(*) is given by

34

HIROSHI NOHARA

Now let us turn to the super Toda field theory. In this case, we replace the chiral Poisson brackets (3.4), (3.25) by the OPE

D+@i(z~)@j(z~)= -(K-y++

.. ..

(Ar= 1 super Toda field theory)

(6.9)

...

(Ar=2 super Toda held theory)

(6.10)

12

D+@i(zl)6j(z2)=

-2(K-I)&+ 12

and have the equations of motion D-D+4#ji=:exp(/3FQDj): D-D+@i=eXp

(py

(A’ = 1 super Toda held theory),

(6.11)

(A = 2 super Toda held theory).

(6.12)

KgSj)

i D-D+Si=exp(/?FKV@j) It is necessary to justify this procedure by showing the integrability of (6.11) and (6.12), but in this paper we assume it to be valid without giving a rigorous proof and only recall that the supersymmetric Wess-Zumino-Witten model with certain constraints leads to the super Toda held theory and A + is gauge equivalent to the gauge current: classical level:

SWZW model

Super Toda tield theory

quantum level:

Current algebra

Super w-algebra

One may think it strange that there are bosonic operators on the 1.h.s. of the above Eqs. (6.11) and (6.12) although there are fermionic ones on the r.h.s. In spite of this paradox, we manage to make use of these equations to derive currents which are conserved at the quantum level. From now on, we demonstrate the quantization of the classical extended superconformal algebra in several cases. Hereafter we drop the subscript +. To begin with, we treat the Ar= 1 super Toda held theory. I.

&212)

(Xl(2[

1)).

We can easily see that T(‘) and Tc3'*'delined by

2 ~(312)

4

(6.13)

2

x Kti:D2QiDQj: -;,;

= i,j=

I

D3ai L

1

EXTENDED

SUPERCONFORMAL

35

ALGEBRA

are conserved. Note that T(” and 7’C3’2) are exactly the same as those of the classical Toda field theory, i.e., that there is no quantum correction to the classical conserved currents.

We have the conserved currents 77(3/V

;

=

Kg:D2QiDcPj:-

(6.14)

i. j = I

(6.15) We see the quantum correction in the terms of order /I0 and /?‘. Next we show the conserved currents of the Ar= 2 super Toda field theory associated with 0~~(112), 0~~7(212),and 0~~(312). I.

OSP(li2).

Tcl'=:D@D@:+bD@-iD~. fl B II.

(6.16)

OSJJ(212) (ZSl(2/ 1)).

T;'=

t

:KgD@iD@j:+A i DD@+

i,j=l

fli=l

DDQi, z

I

(6.17)

--

T$)=:D@2D@l:+:D@2D@,:-;DD(+62)-bD(@,-@2). B

+;.i

x$DQiijDGj+ LJ

+~DDD~ B

i i,j=

1

I

+c.c.

yg:ijDcljiDD@jj:+ 1

i i,j=

z$DD@~D@~: 1

(6.18)

36

HIROSHI

NOHARA

where auk = xgKjk,

(6.19)

+=(Y A), &=(-A $),

%=(A--A).

(6:20)

In the above examples, there is no quantum correction to the conserved currents, in contrast with the N= 1 supersymmetric case. Although we did not treat s/(312) in the N = 1 super Toda held theory, it can be shown that if appropriate forms of T(2) and Tc512)are taken, all generators do not suffer from quantum corrections. For general n in the sZ(~+ 1 ln) case, we expect that :L: becomes the quantum Lax operator in the sense that :wi: (1~ i < n + i) form the closed Ugebra, where L and wi are given in (4.35), and hence that this situation occurs to the N= 2 Lax operator associated with ~SP(~H& 1/2n). In the case of 0~~(2n k 1[2n), the Lax operators (4.49) and (4.50) suffer from non-trivial quantum corrections, which are characteristic of non-simply laced Lie (super) algebra, as we have already seen in the case of ~SP(3 12) (6.14), (6.15). Now we comment on the operator algebra of the 0~~(312) extended N = 1 superconformal algebra in detail because it is a simple and nontrivial example. We show the OPE of the generators in the Appendix C. The central charge is given by

The fusion rule becomes (6.22) according to Appendix C, and the self coupling b is a function of /I and vanishes at fi2 = -5 (c = - g). In Ref. [36], I‘t was shown that the same algebra is derived through the bootstrap analysis and that at c = - $ the algebra is reduced to the one proposed in [20]. 7.

REPRESENTATION

THEORY

Now that we have quantized the classical extended superconformal algebra, we discuss its representation theory in this section. In quantization of the supertields Qjj, we adopted the light cone quantization of the “free field” in spite of the fact that there are interacting terms in the theory. In order to study the representation, we consider the chiral part of the theory and replace Qi with the free Iields which are simply bi + O$i and assume that the resulting generators also constitute the same algebra. Before we consider concrete examples, we comment on the signs of the eigenvalues of the Cartan matrix of Lie superalgebras. In [14], it was shown that the

EXTENDED

SUPERCONFORMAL

ALGEBRA

37

Cartan matrix is positive delinite only in the case of osp( 112) among the Lie superalgebras which admit the super principal embedding. This is because the indelinite inner product is introduced in the super root systems in general except o.rp( 112). From this fact, the kinetic term of the action does not become positive definite and the theories are not unitary at the quantum level unless we project out the negative norm states as in the case of the covariant quantization of the string theory. As we will see, for the simply-laced Lie superalgebras, S/(H + 1 in) and 0.sp(2~ 1 2m)(m = n, n + l), we can construct the unitary representation and hence actually exclude such unphysical states. First, let us consider the representation of the 0sp(3 12) extended N= 1 superconformal algebra, since it is the most simple, instructive, and non-trivial example. In our notation Tc2)(Z) is the same as in the previous section and T’3’2)(Z) is half the minus of Tc312)(Z) in (6.14). We consider the Neveu-Schwarz sector of the 03~~(312) extended N= 1 superconformal field theory. The generators can be expanded in operator Fourier modes as p3121(~)= 1 ~~~-~-312+0 1 ~~~~-2, rsz+ 112 nL?z (7.1) T(2)(Z)z

x unz-*-2+tI PIeiT

z rezi

Wryr-sj2. 112

Now let & be a space of local lields which itself is incorporated with an operator algebra and involves Tc2)(Z) and T (3’2)(Z). The total symmetry of the theory is generated by {Gr, L,,, U,,, Wr}. The space & is a direct sum of the subspaces &., each corresponding to an irreducible representation of the generators. For each i, there is a primary field ,4i which satisfies the conditions GrAi= L,,Ai= U,,Ai= WrAi=O, for r, n >O, (7.2) LoAi=AiAi, (7.3)

UoAi = uiA;,

where the conformal dimension Ai > 0 and the U-charge ui are real numbers. Now we construct a degenerate representation using the technique known as Coulomb gas formalism [39]. Our motivation lies in the fact that in the case of the bosonic Wn-algebra, there are many solutions to the quantum conformal held theory corresponding to the degenerate representations and also, the 19 vertex model has a solution corresponding to the minimal series of N = 1 superconformal algebra [38-J. The degenerate representation is, by detinition, characterized by the existence of the null field x(A, U) which is a secondary held associated with a primary lield AtA> u), Grx=L,,x= b&d

U”x=

Wrx=O,

ul= (A + Q x(A, ~1,

where (7.4) shows that x appears at level 1.

r, n > 0, (7.4)

38

HIROSHI

NOHARA

The primary field ,4(A, u) can be expressed in terms of free fields Qi as /$(A,

u) = P*,

(7.5)

where @F =x

KqQj,

(7.6)

(7.7)

A and ~4have a Z2 symmetry A(q) = A(Z- ij),

u(q) = u(Z- q),

(7.8)

where

(7.9)

From (7.8), we see that A(Z) = ~(2) = 0, so that there is an external charge denoted by 2 in the theory. In order to find a nontrivial null field, we introduce screening operators I’, vc &@Y,

vm+e*c &++a?},

a* = p,

(7.10)

which satisfy the equations + dq d& Tc3’z$ZJ J’(Z2) = f dz2 do2 Tc2)(Z,) V(Z2) = 0.

(7.11)

Using (7.10), null fields have the expressions j&

de1 ~~~e,~W+k4,

J’a~e,VJ ~~-q+,atzl@h (7.12)

where q is given by (7.13)

EXTENDED

satisfying the charge neutrality (Z- l)Cz2Z,

SUPERCONFORMAL

conditions

39

ALGEBRA

[39] I, m, n > 0.

(m-l)+(?r-l)E2Z,

(7.14)

We remark that VX+e,and Va+ezhave structures of the OPE similar to those of the screening operators in the ordinary JJ= 1 and IV= 2 superconformal algebra, respectively, From this observation, it is natural to obtain the spectrum of the degenerate representation (7.13 ). A (/, m, ?z) and ~(f, m, n) can be calculated from (7.7) for each of 2, m, and B, 1 -+/n-l 4fi A(Z,rn,n)=~ 1 Z(Z-2 1) u(Lm,n)=-

I

+-

1 l+?l-2 p p+T(rn-1) 4G i 2/?

lm- 1 4

H

/+a+2 -+i(m+l)

(7.15) p

2P

I

.

(7.16)

Now we turn to observe the operator algebra which generated by the primary fields {~~~~~(~=l,rn,~)=~~/~.~*~.

(7.17)

From (7.13), A is linear with respect to ql and the following simple rule for dimension counting i valid: A(:~~~~~,~,:)=A(~~~)+A(~~.~,).

(7.18)

That is, the primary tields have no singular part in their OPE

~~nm~~~~lC~d”~G31

This OPE structure is reminiscent of the chiral ring of the L’V= 2 superconformal algebra [40]. Formally, we see the correspondence given by A + U( 1) charge, u + conformal dimension. It is an interesting question whether there exists a model which realizes such identification. We will now concentrate on the particular case, the minimal theory in which fl takes some discrete values 1 --s= -b iy

(7.20)

40

HIROSHI

NOHARA

where p and q are relatively prime positive integers. Under this condition on /I, the space of primary lields

is endowed with a closed operator algebra. Moreover union of &‘+ and &- which are defined by

d=d+ tJ&,

d+ =

Q

vmn, Jc =

cm - 1 vcn - 1 J a dP

we can write (7.21) as a

Q

vmn.

(7.22)

cm - 1 J/Cn - 1 J G q/p

In each of z&‘+ and J&, the elements also constitute a closed algebra. Since in the former A is non-negative and in the latter non-positive, physical states should be constructed from the primary fields in &‘+. The central charge corresponding to this minimal theory is given by (7.23) which is analogous to the minus of the central charge of the Ar= 2 superconformal discrete series. This is a novel type of central charge without upper bound for minimal series. Though the central charge which can take an arbitrary large value has already appeared in the case of the N= 3 or N= 4 superconformal unitary series, it is impossible that these superconformal theories have any minimal models. Second, we construct the degenerate representation of the 0~~(2n k 1 12n) extended Ar= 2 superconformal algebra by applying the technique used in [4] and treat the NS sector for simplicity. To lind the screening operators, we consider the equation :L: (Z~)exp(~.~*+~.~)(Z*)=~~~(Z,,

Zz),

(7.24)

where L is given in (4.55), (4.56), on the assumption that there is no quantum correction to L as we conjectured in the previous section. We see easily that the equation (7.24) is equivalent to i~~(Zi)exp(d.$*

+$.$*)(Zz)=D,R(Z1,

Z1),

(7.25)

since there is only one argument Z, in :L:. To begin with, we treat the case of O~P(112) for illustration. :L: becomes L = :(D - /I Dcq(D + D(pD - pD))(D + /I II@):

(7.26)

and the solutions to the equation (7.24) with respect to z are (7.27)

EXTENDED

SUPERCONFORMAL

41

ALGEBRA

The degenerate representation of the N=2 superconformal algebra is studied extensively in [41-441, we show only the result. The degenerate primary tields take the form exp(m@ + fi$),

(7.28)

where m and 51 are labeled by two non-negative charge q,

integers r, s and the U( 1)

(7.29)

and the conformal dimension corresponding to each spectrum becomes r2 s A= -qB-j(r+l)-

f2 -4 cl2@.

The generalization to the case of arbitrary rr is straightforward observation. The vectors

from the following

(7.31)

where Zi (2i) has a one in the ith component and zeros elsewhere, are the solutions to (7.24) because in :L:, only three successive parts, L=

-c(D+D~~+

-.)(D+D(~~cD-@~))(D+~~D&+

. ..)....,

(7.32)

contribute to the singular parts in the OPE. Note that ei and Fi coincide with the fundamental weights of the Lie superalgebra q1(2rr k 112rr). Now that we have derived the screening operators which commute with the generators wi, we can construct the degenerate primary fields VG,ti in the same ways as in the previous example. The spectrum of m and fi which corresponds to the complete degenerate representation of the qn(2rr k 11211) N = 2 extended superconformal algebra is given by

where ri, s, are positive integers and qi are arbitrary constants. From this expression of m and fi, we know the conformal dimension -!%

‘irj

4l36

Let us consider the representation

2

‘i’jiqiqj

(7.34)

whose basis vectors are the chiral primary

42

HIROSHI

NOHARA

lields {exp(*$)} or {exp(&$)}. In this sector, there is the Zn symmetry which has a feature similar to the one in the case of sl(n) [4]. This symmetry comes from the fact that when we express the super Lax operator in terms of the ZV= 1 superlields as

L=H (D+ +&.D+!s),

D+&qZ,)

Xj&(ZJ =

-xi.

l

K-lxje12

z

+

. . .,

(7.35)

12

it is invariant under permutation

among { @2i} and also among { OznP 1} since ti+d (i= j=2i) (i=j=2i+

(7.36) 1).

Thanks to this symmetry, we can identify the representation VA with I’*, when G and Gi’ are transformed into each other. Finally, we comment on the unitary representations of the algebras associated with the simply laced Lie superalgebras sl(n + 11n) and CX~(~CV [2rz)(m = n + 1, n). In these cases, by choosing /I appropriately, the energy-momentum tensors can be identiIied with those of certain models obtained by the supercoset construction [45] since there are no quantum corrections. Below we show the central charge and the coset space corresponding to the unitary representation in each case: 1. d(n + 1 1n), 2. osp(2t1+ 2 /2n),

3. mp( 2n /2n), (n>Zl

3nk ‘=k+n+l’

W(n+

3(2n + 1)k Cc 2(k+2n) ’

SO(2n + 2)/SO(2n + l),

3nk ‘=k+2n-l’

SO(2n + l)/SO(2n).

l)/sU(n)x

U(l), (7.37)

The unitary discrete series associated with d(n + 1 1n) was obtained by Hamiltonian reduction in [46] and is called the CP” model; it Iist appeared in the study of the ordinary ZV= 2 superconformal algebra [47].

8.

DISCUSSION

As we comment in Section 6, it remains to show that free field regularization is applicable to the equation of motion in a more rigorous way. In the bosonic case, this type of regularization is justified since the operator valued equation of motion has the well defined finite solution [48]. We think that by analogous argument, our

EXTENDED

SUPERCONFORMAL

ALGEBRA

43

results will be shown to be correct and indeed in the case of osp( 1[2), it is easily seen that the equation of motion is solvable on the quantum level [ 181. We expect that the extended superconformal algebras we have proposed in this paper give a new framework to classify the statistical models which have the supersymmetry at the critical point. In application to these systems, it is important to find the models which correspond to the unitary representation of the algebra. For the cases of the N= 2 extended superconformal algebras, it does not seem to be totally impossible to lind other unitary series than the one discussed in Section 7 in the light of the Landau-Ginzburg theory. The N= 2 superconformal field theory can be viewed as the lixed point which is determined by the form of the superpotential [40-J. Although we have not yet found the relation between the N = 2 Landau-Ginzburg theory and the N= 2 super Toda field theory, we are inclined to suppose that such a situation can really exist. For example, if we have two chiral superlields, the extended superconformal algebra associated with s/( 3 12) has unitary representations which are given by the lixed points of the D,,, E6, E,, and Es types of the superpotential [50]. In the cases of the non-simply laced algebra 0~~(2n & 112n), we cannot directly apply the coset construction to lind the unitary representations since there are non-trivial quantum corrections to the energy-momentum tensors which cannot be canceled out by redetining the coupling constant. From the fact that the structure of the algebra essentially depends on the Lie algebra, the key to this problem may be given by the study based on the Lie superalgebra, such as the generalization of works in [49]. The important point is whether or not the current algebra has a natural interpretation in terms of the Lie superalgebra. We have not succeeded here in finding the algebra which contains the bosonic W algebra as its subalgebra and is considered to be the more natural supersymmetric generalization. This type of algebra may be obtained from the super Wess-ZuminoWitten model delined on the ordinary Lie algebra. The strategy we have is to impose certain constraints on the gauge currents and reduce the number of degree of freedom to that of rank of the Lie algebra. Perhaps the previously mentioned super W5,* in [21] will be derived from the so Wess-Zumino-Witten model by following this approach. We would like to comment on the afIine super Toda held theory [6] which is associated with certain affine Lie superalgebra, This theory is obtained by the supersymmetric integrable deformation of the super Toda held theory and not conformally invariant since the Cartan matrix is singular and one cannot define the conformal transformation. It is an interesting problem to study the perturbation theory throughs the renormalization group analysis in [52]. One of the relevant issues is the supersymmetric soliton equation which is the evolution equation defined on the space of the classical solution. For the systematic derivation of this equation, the super Toda lattice hierarchy in [5 I ] is a powerful tool in some cases. For example, in the case of s/(n 1n)(l), we obtain the one with the N = 2 extended supersymmetry. Recent studies of two dimensional gravity based on the matrix model revealed

44

HIROSHI

NOHARA

the new relation between the conformal algebra and the conformally invariant field theory. For example, in pure gravity the Virasoro algebra is interpreted as the operator which acts on the function of the coupling constant [53] and hence is related to symmetry [54] of the theory space. The associated energy-momentum tensor is not the conserved current in the held theory defined on the Riemann surface. We hope that there exist some physical models which give the novel grasp of the superconformal algebra.

APPENDIX

A

We report the chiral Poisson brackets of Ty2, Ty2, and T* in this appendix:

64.2)

1 T?2Vl

15 T2P2J

=$$-;$

12

-$D;T2(Z2),

1

12

Ty2(Z2)-%

12

T2(Z2)-$D+T2(Z2) 12 (A.3)

12

(A.4)

EXTENDED

SUPERCONFORMAL

APPENDIX

ALGEBRA

45

B

We explain Lie superalgebra in some detail here [55]. Lie superalgebra is defined by the graded commutator, [ },

E(A)EZ~.

P3.21

e(A) = 0,

(B.3)

.$A) = 1.

(B.4)

We call A even, if

or odd, if

Usually we write the element of Lie superalgebra as

where A (D) is an n x n (PHx m) matrix and B (C) is a n x m (m x n) matrix. The diagonal elements A D are even and the off diagonal ones B C are odd. Super trace (St A) and super transpose (Ast) are defined by =TrA-TrD,

46

HIROSHI

NOHARA

The commutation relations among the elements of Lie super algebra are conveniently expressed in the basis of the Chevalley generators Ai, ei, fi which satisfy [hi, ei} = Ktiej

(B.7)

[hi, j’i} zz -Kgjj

u3.8)

Lei,

(B.9)

j}

=

6ijhi

and the positive (negative) root is generated by ei (YJ. In contrast with the bosonic Lie algebra, we do not have the unique Cartan matrix in Lie superalgebra. In many references (for example, see [55]), the basis is chosen so that there is only one zero in the diagonal elements. We give the definition of the several Lie superalgebras below. 1. S&r 1??z)* {‘4csZ(n~m) 1 St(A)=O,}.

cIn01

(B.10)

When n = m, there is the center which is generated by

0

1”’

(B.11)

and we can project it out without any diffkulties and denote the resulting Lie superalgebra as @(n 1n). Let us consider the case ~l(2 1l)(n = 2, m = 1) here in

(B.12)

(B.13)

(B.14) (B.15) (B.16)

EXTENDED

SUPERCONFORMAL

47

ALGEBRA

In the above basis, the Cartan matrix Kg is given by (B.17)

Ku=

If we choose the other basis (B.18)

hl,h2~h2,h~(=hl+h2), the Cartan matrix becomes

(B.19)

Kg=

We next treat q~(k 12l)(n = k, PPZ = 2/) which is delined by

A E mp(k 124 1AI + l.Ust = 0,

.

(B.20)

Below we write the Cartan matrix of Lie superalgebra which leads to the integrable super Toda lield theory and its odd. simple roots. We omit the case of D(2, 1; E) here because it is too complicated. (r denotes the rank.) For the Dynkin d&gram,. see Fig. 1. 1. d(rl + 1 1n), r = 2n. 0 1

1 O-l -1

0 *

(B.21)

* 0 -1 -1

0 1

1 0

(B.22)

48

HIROSHI

NOHARA

1 -1 -1

0 1 1 0 *

Osp(2rn + 1[2m)(r = 2m, m > 0)

(B.23)

* 0 1

1 O-l -1

0

(B.24) 1 -1 -1

Osp(2m - 112m)(r = 2m - 1)

0 1 1 0 *

.

* 0 -1 -1

0 1 1 0

(B.25)

EXTENDED

3.

SUPERCONFORMAL

49

ALGEBRA

Osp(2n 12m), (n = m + 1, n = m).
2-l O-l

2 -1 Osp(2m 12m)(r = 2m, m B 11

-1

0 1

1 0 -1 -1 . . .

. (B.27) 0 -1 -1

0 1 1 0

L

(B.28)

Osp(2m + 2 12m)(r = 2m + 1)

2-l 0 2 O-l -1 -1 0 1

1 0 -1 -1 . . .

(B.29) 0 1

1 0 -1 -1

0

C

APPENDIX

We show the OPE among the generators of the 0.sp(3 12) extended IV = 1 superconformal algebra. To begin with, we replace TC312)and T(2) by ~(312)

+

fc312)

= I-! 2

After some calculation,

fc2)(Zl)

733

~(312)

y

Ti2) + (1/3)(2//l + fi) DTc312)

+ f’(2) =

J(4132/9)(l/4

we obtain

c/2 6!1l,~ A pc2)(Z2) = + Tc3’2)(Z2) z2

G2

+ -&

{aP2)(Zz) 12

+ 2 DF3’2’(Z2)}

- l/~%M2

+ 2)’

EXTENDED SUPERCONFORMALALGEBRA

where 2(5c + 6)

G?=

-9 c=--3. b2

&4c+21)(15-c)’

ACKNOWLEDGMENTS I thank Professors M. Kate, T. Yonea, K. Kawarabayashi, Y. Fujii, T. Eguchi, Y. Kazama, and H. Kawai for reading the manuscript and useful comments, K. Mohri and S. Komata for collaboration with me, and I. Ichinose and K. Hori for encouraging me at the most difftcult stage of this work.

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