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Classical and quantum extended superconformal algebra
Nuclear Physics B359 ( 1991) 168-200 North-Holland
CLASSICAL AND QUANTUM EXTENDED SUPERCONFORMAL ALGEBRA Shiro KOMATA, Kenji M O H R I and Hiroshi N O H A R A
h~stitute of Physics, Unit'ersityof Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Received 16 July 1990 (Revised 12 February 1991)
The integrable supersymmetric Toda field theories enable us to construct several infinite series of the extended N = 1 and N = 2 superconformal algebras. In particular, in the cases of two series of the extended superconformal algebras associated with sl(n + 1 In) and osp(2n + l l2n), we find the supersymmetric Miura transformations. The classical extended superconformal algebras are realized as the Poisson bracket structure among the coefficients of a superdifferential operator. We also obtain the exotic minimal discrete series of the osp(312) quantum extended N = 1 superconformai algebra. This minimal series is different from those known so far in the sense that the corresponding central charge has no upper bound.
1. Introduction
We are still far away from establishing the classification of the superconformal theories in spite of much effort for many years. The key to this problem is to construct the extended superconformal algebra containing higher spin generators in addition to the ordinary stress-energy tensor. From our experience in the bosonic case [1] we expect that there is a beautiful correspondence between this algebra and some physical theory, for example the soluble lattice model with superconformal symmetry or the one coupled with supergravity. At the present stage however, we do not have any method for the systematic approach to superconformal field theory such as the KP theory [2] in the bosonic case. Therefore it is important to discover any examples which have the extended superconformal symmetry. Our motivation to investigate the supersymmetric Toda field theory lies in a fuller understanding of integrable systems with superconformal symmetry. It is probable that the supersymmetric Toda field theory [6,9] has an extended superconformal symmetry and a close relation to the hierarchy of the integrable non-linear supersymmetric differential equations because it is a natural super version of the Toda field theory. Indeed in ref. [3] we investigated the classical supersymmetric Toda field theory in order to construct a classical extended 0550-3213/91/$03.50© 1991 - Elsevier Science Publishers B.V. (North-Holland)
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169
superconformal algebra which can be considered to characterize the theory at the classical level. It was shown that, as in the bosonic case, there are several series of Lie superalgebras which lead to the integrable supersymmetric Toda field theories with as many independent conserved currents as the degrees of freedom. (By super Toda field theory, we mean Toda field theory with supersymmetry in two-dimensional space-time.) These conserved currents constitute the closed algebra with respect to the chiral Poisson bracket which can be thought to be the classical limit of the operator product expansion (OPE) between superfields. This algebra is quite new, because it is different from the ones which are derived from a previous analysis [4] in that the Jacobi identities hold for arbitrary values of the central charge, and that in general the generator with spin higher than 3 / 2 does not become a primary field in contrast with the bosonic W-algebra [5]. In this paper, we discuss the relation between super Toda field theory and super KP theory. We also construct the quantum analogue of several classical extended superconformal algebras constructed in ref. [3], and discuss possible physical applications of this algebra. In sect. 2 we briefly review the result of ref. [3] in the case of the classical super Toda field theory and show some new results about the existence of infinite series of sets of conserved currents, i.e. classical extended superconformal algebras, and we show an interesting correspondence between the N = 1 case and the N = 2 case. In sect. 3 we establish the manifestly supersymmetric Miura transformation starting from the supersymmetric zero curvature form of the equation of motion of the sl(n + 1 In) and osp(2n + 112n) N = 1 super Toda field theory. There exists an infinite series of superdifferential operators with the structure of the classical superconformal algebra. In sect. 4 we construct, for some simple examples, the quantum version of the classical extended superconformal algebra in the sense of sect. 3 and investigate the algebraic structure of the quantum extended superconformal algebra. In sect. 5 we examine the representation theory in the case of the osp(312) N = 1 super Toda field theory. We will see that in this case there is no upper bound of the central charge of the minimal series in contrast with the bosonic minimal series, that is the central charge of this minimal model takes the form
9q
c=---3,
(1.1)
where p and q are relatively prime positive integers. Finally in sect. 6 we discuss some relevant issues of the quantum extended super-conformal algebra and future problems.
170
S. Komata et al. / Extended superconfomml algebra 2. Classical theory
First we briefly summarize the results in ref. [3] for the reader's convenience. To begin with, we explain the integrable super Toda field theory with N = 1 space-time super-symmetry in two dimensions. The action is given by
s = f d2z d20
1
r r 1 r Y'~l exp /3 E Kij~g}j ~ KoD_q)iD +4)j + -~ i= j=l i,j=l
Z+ = XO -I"/X! ,
Z-=Xo--/X!
,
,
(2.1)
(2.2)
where Kij is a symmetric and invertible matrix of dimension r and ¢~i are N = 1 superfields. We regard z ÷ as the space coordinate and z - as the time coordinate in two-dimensional euclidean field theory so that we may pick up only the chiral part. The chiral Poisson bracket is defined by
( K-l)ijOl2 {D+*,(Z,),
%(Z2)} = -
(2.3)
ZI2
where z,:
=
(2.4)
- o?o;,
and D+_ are the superderivatives defined by 0
0
D+=O0 + + 0 + Oz +-
(2.5)
-
The classical equations of motion are r
D_D + qbi = exp /3 Y'~ Kij~ j
(2.6)
j=l
and are soluble if we can construct r independent conserved currents T ts,) which satisfy the equation D_ T ts,) = 0
( 1 ~< i ~
(2.7)
where s i denotes the spin of the conserved current. In general, one of the conserved currents is inevitably the stress-energy tensor
TO/2'=
r 1 r E KijD 2+~io + % - -~ ~ O 3 ~ , . i,j=l
i=
(2.8)
S. Komata et al. / Extended superconformal algebra
171
(By spin of a generator we mean the number of superderivatives each of which is assigned spin 1/2. Note that in general we can not choose all generators to be primary fields in contrast with the case of the bosonic W-algebra [5].) It depends on the choice of K o whether we get full conserved currents. In ref. [3] we demonstrated the derivation of conserved currents by adopting for Kii the Caftan matrices of osp(ll2), si(211), osp(312) and s1(312). In refs. [3, 6] it was shown that there are several classes of Lie superalgebras the Cartan matrices of which lead to the integrable theories, that is the ones in which we can choose all simple roots to be odd elements so that eq. (2.6) is rewritten in the form of the super zero curvature condition (see eq. (3.16) and below). The Lie superalgebras which satisfy this condition are sl(n + l l n ) ,
osp(2n + l12n),
osp(2nl2n),
osp(2n + 212n), (2.9)
D(211;a).
(If readers would like to know the Cartan matrices of these Lie superalgebras, see appendix D.) We also comment on super Toda lattice theories. They are associated with affine Lie superalgebras for which all of the simple roots are generated by odd elements. In refs. [6] and [7] such types of affine Lie superalgebras are also classified as follows: psl(n In) tl~ ,
sq(2n + 1) t2) ,
osp(2n 12n) t2~,
osp(2n + 212n) tl~,
psl(2n + l12n + 1) t2),
psl(2n 12n) t2~,
D(2I 1; a ) tl~.
(2.10)
Use of a Cartan matrix of the above Lie superalgebras leads to an integrable Toda lattice theory. The conformal symmetry is lost in the Toda lattice theories. We next review the classical N = 2 super Toda field theory. The action is
s =f
gij(I)i¢~j +
dEz dE0
i,j=l
r
{f
dEz dE0 - exp fl i=1
)
13 E Kij~j + c.c.
(2.11 )
j=l
where Kij is a hermitian and invertible matrix, ~i and ~ are the chiral superfields which are complex and satisfy the following conditions, m
D
D +_.cI)i=D _+qbi - 0 .
(2.12)
172
£ &~m,am cl ai. / E~tendt~g ~u,l~,rconfonnal algebra
~ e chiral Poison brackets are defined by 2( K-*)#012
{D
Z~2
2( K= '),0,, Z~,
(2.14)
where
01,_ = 0 i - 0 +,., +
0~, = ,a-
-0~_ , +
+
Zl,.=z~( -z,_ -0~0,_ - 0 ~ 02 .
(2.15)
The equations of motion are D_D +q~i = exp
(2.16)
(r)
D_D+ q~ = exp /3 E K#q~ .
(2.17)
j=!
In this case, the integrability means that there are as many independent conserved currents as the pairs of chiral superfields. The equations which are satisfied by the consem'ed currents become
D _ T {~'}= D _ T {~'~= O.
(2.18)
We always have the stress-energy tensor as a conserved current T'"=
E K#D+q~iD+~i + -~ ~ D+D+q~i- -- E D+D+~i. i,i=l i= /3 i=l
(2.19)
In ref. [3], we derived integrable theories by using the Cartan matrices of osp(1J2), sl(211) and osp(312). Among them, osp(312) is the first, rather simple but non-trivial example. By using the Cartan matrix of osp(3[2), we obtained two conserved currents T ~l} and T {2~ (see appendix A for the explicit expression of T{2~). It was shown that T ~2~does not become a primary field with respect to T t~ even if we are allowed to add a term (D +D + - D+D +)T ~l~ in contrast with the N = 1 super Toda field theory. As for the chiral Poisson bracket {T ~2~,T~2q we mention that there is a term (T(l~) 3 besides ( T ~ ) 2 and T ~ T ~2~ reflecting the fact that there is a spin-3
S. Komata et aL / Extended superconfonnd algebra
t 73
TABLE 1 A proposal for the complete lisl of spins of d~ conserved currents in N = | super Toda t-mid ~ e o t i e s Lie superalgebra
Spin of conserved currents
sKn + 1Bn) oslg | ~2)
1 / 2 x ~2.3,4. . . . . 2 n , 2 n + | ) 3/2 3/2,2 1 / 2 x ~3.4, 7,8~ | L 12, . . . . 4n - 5,4n - 4,4n - l) 1 / 2 × (3,4,7,8, I1,12 . . . . . 4n - 1,4n)
osIg2n - 1~2n) (n >t 2) osFg2n + l~2n)(n >i 2)
os~2t 2) ( ---sK2Di D oslg2n a2n) (n >I2) oslg2n + 2U2n)
1.3/2 1 / 2 x (3,4,7,8,11,12, . . . . 4n - 5,4n - 4,4n - | L n i / 2 x {3,G7,8,11, | 2 . . . . . 4n - 1 , 4 n L n + I / 2 3/2,3/2,2
[~2] 1;a)
generator in the N = 2 multiplet T (2~. ( S e e
appendix A for the expression
{T (2~, T(2~}.)
In consequence of further study along the line of ref. [3], up to n = 6, in the N = 1 super Toda field theory we have succeeded in solving eq. (2.7) in. all the cases (2.9) and in the N = 2 super Toda field theory, in the cases of sl(n + 1 In) and osp(2n +_ l U2n). From the observation for Lie superalgebras of lower rank,, we propose a complete list of the conserved currents in the N = 1 super Toda field theory in table 1. (In appendix B we only give the explicit form of the conserved current T (2) for reference since those of higher spins are inessentially complicated.) In table 1, osp(2n +_ ll2n), osp(2nl2n) (n >/2), osp(2n + 212n) and D(211;a) are associated with the extended N = 1 superconformal algebra since the spins of the conserved currents are more than or equal to 3 / 2 . However sl(n + l ln) is associated with the extended N = 2 superconformal algebra since T ~1~ and T ~3'2~ constitute the classical N = 2 superconformal algebra although we started from the N = 1 super Toda field theory. We can identify the components of T (3/2~ and T c2~ with the generators of the classical N = 2 superconformal algebra up to normalization as follows. T ('~ = J + 0( G - G ) ,
(2.20) (2.21)
T (3/2~ -- G + G + O T .
This enhancement of supersymmetry originates from the symmetry of the Dynkin diagram under the reflection with respect to the center of the diagram (see fig. 1).
n
n
®
®
®
.....
Fig. 1. The Dynkin diagram of sl(n + 1 in).
®
174
S. Komata et al. / Extended superconformal algebra TABLE 2 A proposal for the list of spins of the spins of the conserved currents in N = 2 super Toda field theories Lie superalgebra
Spin of conserved currents
sl(n + 1 In) osp(2n - 112n) osp(2n + l l2n) osp(212) ( = sl(211))
1,1,2,2 . . . . . n - l , n - l , n , n 1,2,3 . . . . . 2 n - 2 , 2 n - 1 1 , 2 , 3 , . . . , 2 n - 1,2n 1,1
We remark that when we take the limit /3 ~ oo the conserved currents are transformed into D +~i. In this case if we have N superfields, N-extended supersymmetry is induced in the theory and as the extended superconformal algebra we obtain the so(N) superconformal algebra rather than the algebra with higher spin generators. Finally we propose in table 2 the list of the spins of the conserved currents in the N = 2 super Toda field theory. (See appendix B for the expression of Tt2).) In the case of sl(n + 1 In) there are two conserved currents of spin 1 which are decoupled from each other if we take certain forms of them as can be seen easily from the discussion in ref. [3]. In terms of components, the extended N = 2 superconformal algebra associated with o s p ( 2 n - ll2n) (resp. osp(2n + l12n)) has generators with the same spins as the generators of the sl(2nl2n - 1)(resp. sl(2n + 112n)) extended N = 2 superconformal algebra. In particular, we immediately see that the generators associated with osp(ll2) and those associated with sl(211) have equivalent chiral Poisson brackets as well. Therefore we conjecture that for general n the series of the extended superconformal algebra are equivalent to each other also at the level of the algebraic structure. As for the case of osp(2n + 212n) and osp(2nl2n) (n >f 2), we could not derive any conserved currents of spin higher than 1. We only checked that there is no conserved current of spin 2 in the Toda field theories associated with these Lie superalgebras.
3. Supersymmetric Miura transformation Here we derive a supersymmetric version of the Miura transformation. The supersymmetric differential operator we will construct is expected to play a central role in formulating the hierarchy of supersymmetric non-linear soliton equations which has a hamiltonian structure associated with superconformal symmetry and is a candidate for a powerful tool to study the extended superconformal algebra. In this section we set/3 - 1 for simplicity.
175
S. homata et al. / Extended superconformal algebra
It is instructive to review the case of the bosonic Toda field theory before going to the supersymmetric one. The equation of motion of the Toda field theory based on the Lie algebra g of rank r is rewritten in the form of a zero curvature condition [8] as [O++A+,O_+A_]=O (3.1) r
,~
E r,Aj
O+O_d& = exp
(3.2)
j=l r
r
A += ~ O+dpihi + Y'~ ei, i=l i=l
exp
A_=
d~s
,
(3.3)
i=l
where hi, e i and fi are the Chevalley generators of g, and they satisfy the following commutation relations [ei, fs = S i j h j ,
(3.4) [hi,fj] = -gijfj.
We remark that eq. (3.1) is invariant under the gauge transformation (3.5)
A +_~ g-lO +g + g-~A +_g,
where g is an element of the group generated by g. We will see that a matrix-valued differential operator 0+ + A + leads to the well-known Miura transformation [11, 12]. To see it explicitly, we take g = sl(n). The generators are given by e,=E,.,+ l ,
hi---Ei.i-Ei+,.i+,,
f~ = E,+ i. , ,
(3.6)
where E~s is a matrix with a one at position (i, j) and zeros elsewhere. Let us regard 0 + + A + as operator acting on an n-dimensional vector v and consider the following equation (O++A+)v=O. (3.7) Writing eel. (3.7) in components as 0++0+¢pl
1 0++0+~ 2
O++A+=
1 , (3.8)
D D
0+ + O+~ n_ 1
1 ¢~+ J¢- 0+q~n
where q~, = 6 1 ,
~2--~2--~!,''',
~n--I --¢~n--! - - ~ n - - 2 '
q~,,=-~bn_~,
(3.9)
S. Komata et al. / Ertended superconfonnal algebra
176
and
(3.10)
V -"
we obtain the differential equation for v~ (3.11)
Lc n =0,
where
+o+(d,,,_,-4,,_:))...(0+
L = (o+-o
+ o+
- 6,))(o+ +o +4,,).
(3.12) On the other hand, we can get the following forms of gauge fields, A + and A_, by adopting the unique gauge transformation generated by negative roots, 0
1 0
A+=
A_=0.
o
(3.13)
1 Wn
Wn- 1
W2
0
Note that eq. (3.11) is invariant under the gauge transformation generated by the lower triangular matrix with ones above and alongside the diagonal. In this way we get the well-known Miura transformation L
=
0"+ + (
-
1)" w 2 o..,-2 +
= (0+-- 0+¢]},,_!)(0++
+ ... -w,,_lO++w,,
0 + ( t ~ , , _ i -- ¢ ~ , , - 2 ) ) ' ' "
(0+-I- 0 + ( t ~ 2 -- ~}1))(0+-'1- 0 + t ~ l ) .
-
(3.14) In eq. (3.14), L is called Lax operator in the (n-reduced) KP hierarchy. We remark that in the Drinfeld-Sokolov gauge (3.13) [11], the zero curvature condition (3.1) means the conservation of the currents wi. Hence the coefficients wi constitute a closed algebra with respect to the chiral Poisson bracket in the Toda field theory (K-t)ij
{O + ¢~i( Z ) , f~j( W ) } "- __
Z
+
--W
+ .
(3.15)
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
S. Komata et aL / Extended superconformal algebra
t 77
N = 1 super Toda field theory based on the Lie superalgebra g is also written in the supersymmetric zero curvature condition,
{D++A+,D_+A_}
=0,
(3.16)
where
A + = Y'~D+ rP~hi + Y',e,, i
(3.17)
i
Here h i, e i and f~ are the Chevalley generators of g. They satisfy relations similar to (3.4) except that the commutator [ , ] is replaced by the graded commutator [ , }. Kij is the Cartan matrix of g (see appendix D). We remark that the coordinate in the spacetime (z-+, 0 -+) is embedded in that of the Lie superalgebra. Therefore e i and fi must be odd because the I.h.s. of eqs. (3.17) and (3.18) are. As a concrete example, let us first consider the case g = sl(n + 1 In). The explicit forms of h i, e i and fi are given by
hi=El,! + E,+2,,+2,
el=El,n+2, fl=En+2,!,
h2i = - g n + 2 - i , n + 2 - i - E n + 1 + i , n + i + i ' )
(1 < i < n )
e2i ~ g n + ! + i , n + 2 - i , f2i = -gn+2-i,n+! h2i+!
+i,
= En+2_i,n+2_ i + En+2+i,n+2+i, ~
(1 < i ~ n -
e 2 i + ! -- E n + 2 _ i , n + 2 + i , f2i+!
1)
(3.19)
= En+2+i,n+2-i.
We regard the operator D ++ A +
(3.20)
as the differential operator acting on the super vector space, the elements of which are b. 1
t' n + ]
1
(3.21)
Un + i
U2n+
where lfli (1 ~1 n + 2) are Grassmann-odd
S. Komata et aL / E~tended superconfonnal algebra
178
n u m b e r s . F r o m (3.17),
-- ~ 2n
Q
-q~4 + q~5
A+=
Q
- q 2 + q~3
q'l - ~°2 q~3 - ~4 I Q
0
1
q~2n-
1 -- q~2n
(3.22) where (3.23)
qi = O +~I~i .
L e t us c o n s i d e r t h e e q u a t i o n
(D++A+)v=O.
(3.24)
As b e f o r e , if w e write eq. (3.24) in c o m p o n e n t s o f t h e v e c t o r v, t h e n (3.24) r e d u c e s to t h e s u p e r d i f f e r e n t i a l e q u a t i o n with r e s p e c t to v~,
Lv i = 0 ,
(3.25)
where L -- ( D + - D + qb2.) ( D + + D + ( ~ 2 , , _ , - ~ 2 , , ) ) ( D + - D + ( q~2,,- 2 - q ~ 2 . - , ) ) - - -
x(D+-D+(~2-~3))(D++D+(~-~z))(D++D+~i).
(3.26)
By e x p a n d i n g L as L = D2n + + ! + ( - + 1 ) 2 " - l w l /~ )+ 2 " - 1
W3/2L'+/'~2n - 2 --
• • • +
Wn + !/2
,
(3.27)
we discover t h e s u p e r s y m m e t r i c M i u r a t r a n s f o r m a t i o n ( D + - D + ~ 2 , ) ( D + + D + ( q~z,,-, - q ~ 2 , ) ) ( D + - O + ( ~ z , , - z - ~ 2 , , - , ) ) - . -
x ( D + - D +( ~ 2 - ~ 3 ) ) ( D ++ D +( q~' - ~ 2 ) ) ( D + + D + q J I ) = D 2+ n + l - wID+2n-! +
W3/2~+
r)Zn-2
--
"
.. + W ,
+ 1/2
.
(3.28)
=D;-
%D++
W3/2
3.
l
and w3/2 in (3.30) constitute the respect to the chiral Poisson bracket
q
ssical
. (33
-(K-')ijlnZ,2-
(@i(z,)9@j(z2))=
In general, Wi in eq. (3.27) generates the classical extended algebra with respect to the chiral Poisson bracket (3.31) Cartan matrix of sl(n + 1 In). We suspect that there is the supersymmetric Drinfeld-Sokolov A + and A _ take the following forms, 0 W-1/2
... 0
gauge in w
0 w3/2
*. . . . l
wn -3/2
w,z-
1/2
. . . . .
0
.
1
A_=O. although we have not yet found the explicit form of t which connects (3.22) and (3.32) and leaves eq. (3.25) i Let us consider the case osp(2n + 112n). Firstly we symmetric Miura transformation for osp(l12) and osp(3
3.32)
I SO
S. &~nmta et aL / E2~-tendedsul~rconformal algebra
and f, are given by
h=
0 1 0
0 0 0
0 0 - 1
0
1
1
0 0
e=
0
,
f--
(i'!) "
0
,
(3.33)
0
for o sp(1,2) and hi = E44 - E55,
h2 =
EI~
e I = E ~ + E42,
e 2 = E~ - Esi,
f l = E,.4 - Es_,,
f2 = - E i s
- E33 - E44 +
E55,
(3.34)
- E43,
for osp(3,2). The corresponding A+ in (3.17) becomes 1 0 -D+~
0 D+~
A+=
0 D@ 2
A+-
0
0
0 0
0 0
0 -D~
0
1
0
0
0
-1
(3.35)
0 0 1
2
D+q~ i - D + ~ 0
0 1 0 2
(3.36)
0 -D+q~
i +D+q~
2
for osp(!, 2) and osp(3, 2), respectively. By rewriting (3.24) as before, we obtain the supersymmetric Miura transformations for osp(112) L=(D+-D+q~)D+(D++D+q~)
= D +3- ( D Z + q ~ D + q ~ - D 3 q g ) ,
(3.37)
and for osp(312) L -- ( D + + D + q~2)( D + - D + ( ~
- q~z))D + ( D + + D + (q~, - q~2))( D + - O + ~ 2 )
= D 5 + T { 3 / 2 ) D 2 + ( - 2T t2~ - D T t3/2~) D + - D + T (2) ,
(3.38)
where T {3/2) is given in eq. (2.8) and T (2) is given in eq. (B.5). For general n, the super Lax operator L for osp(2n + i , 2 n ) is given by L = ( D + + D +q~2,,) ( D + - D + ( q~2,,-, - ~ z , , ) ) ( D + + D + ( ~2,,- 2 - q~2,,- ,)) • - • × (D+-D +(q~= ~+f)4n+l+
q~2))D+ ( D + + D + ( ~ , -
T ( 3 / 2 ) o 4 n - 2 + ...,
q~2))-.- ( D + - D + q~2n) (3.39)
S. Komata et al. / Extended superconformal algebra
181
and for o s p ( 2 n - 112n) L = ( D+-D
+ q ~ 2 n - , ) ( D + + D + (q~2#_2 - @ 2 # _ , ) ) ( D + - D + (q~2#_3 - @ 2 . _ 2 ) ) . . .
× ( D + - D + ( g ) , - @2))D + ( D + + D + ( @ , - q~2))... (D+ + D +q~2n_ ,) _ i ) 4 n - i - T ~ 3 / 2 } D +4 n - 4 + . . . .
Corresponding Chevalley generators are given in the appendix E. Finally we remark that L in (3.27) does not admit the time flow of the super KP type in ref. [20], while L in (3.39) and (3.40) do. In general we can uniquely define L l / ( 2 m + !} a s L l / ( 2 ' n + l} = D + + u o + u i D + ~+ . . . ,
(3.41)
where L is a superdifferential operator of order 2m + 1. Calculating the (2m + l)th power of (3.41), we see that, in (3.27) u o = 0 and u~ ~: 0, while in (3.39) and (3.40) u o = u~ = 0. Recall [20] that the super KP hierarchy is a system of the equations among the Lax operators defined by A = D + + r o + clD~_l+ . . . ,
(3.42)
D +c o + 2c I = 0.
(3.43)
with
Therefore we conclude that, in the case of osp(2n + 112n), L in (3.39) and (3.40) are derived from the ½(4n + 1)- and ½ ( 4 n - 1)-fold reduction respectively of the super KP hierarchy, while in the case of sl(n + 1 In) we cannot obtain (3.27) by ½(2n + 1)-fold reduction. In mathematical literature [20,21], the n-fold reduction of the super KP hierarchy is defined by (A2")_=0,
(3.44)
where n is a positive integer. Although it would be interesting to construct another hierarchy which produces (3.27) by reduction, it is beyond our aim in this paper. 4. Quantum extended superconformai algebra
In this section, we construct an operator version of the classical extended superconformal algebra for further studies in later sections. The direct quantization of the super Toda field theory with the action (2.1) is a very difficult problem [3]. Therefore we will take the following route. First, we construct the conserved currents of the super Toda field theory at the quantum level using the free field normal ordering regularization [I0]. These
182
S. Komataet al. / Extendedsuperconformalalgebra
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 difficulties in quantization. Generally speaking, the classical conformal symmetry is easily seen by non-linear soliton equations, while for the realization of the conformal symmetry at the quantum level, it is convenient to adopt the free field 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 [5, 12, 13]. classical level: quantum level:
WZW model current algebra Feigin-Frenkel realization
Toda field theory W-algebra Fateev-Lukyanov realization
We will briefly review the quantization h la Mansfield in the bosonic case. The free field quantization is applied to the Toda field theory, i.e. the chiral Poisson bracket
(g-l)iy
{O+d)'(z')'6J(z2)}= z~-z~
(4.1)
is replaced by the OPE given by
(~i( Zl)~j(Z2) ------( K-l)ij In( z~-
+ "~i( Zl)~j( z2)',
- z~-)
(4.2)
where : : denotes the normal ordering. Therefore the equation of motion is given by
:exp(flY' Kijdpj)'.
a+a_4,i =
•
j
(4.3)
Now let us turn to the super Toda field theory. In this case, we replace the chiral Poisson brackets (2.3), (2.13) by the following OPE
(~i(Zl) (~j(/2) -- - ( K-l)ij ln( Z!2 ) + :Oi( Z1)Oi(Z2):
(4.4)
(N = 1 super Toda field theory), D
012012 > + ~i(ZI)~j(Z2) = - ( K-l)ij(ln( Z12) + ZI2
(4.5)
(N = 2 super Toda field theory), and have equations of motion
00+
,
(46 J
S. Komata et al. / Extended superconformal algebra
183
( N = 1 super Toda field theory),
exp / ~ E
, (4.7)
( N = 2 super Toda field theory). We make use of these equations to derive conserved currents which are conserved at the quantum level. (To be rigorous, we need some justification which re-inforces the free field quantization. It seems to us that a key to this problem lies in establishing the correspondence between the super Toda field theory and the supersymmetric W e s s - Z u m i n o - W i t t e n (WZW) model on some Lie supergroup with certain constraints. See refs. [5, 12, 13] in the bosonic case and sect. 6 for the discussion.) 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 N = 1 super Toda field theory. (i) osp(212)(= sl(211)). We can easily see that T {l~ and T ~3/2~ defined by
T{I~ = :DclgiDcI92. +
T{3/2)=
1
/3
D2~l
--
1 --D2t~b2,
/3
2 1 2 ~ K i j : D 2 ~ i D ~ j : - -- Y'~ D3clgi,
(4.8)
/3 i=1
i,j=l
are conserved. Note that T t~ and T (3/2) 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. (ii) osp(312). We have the following conserved currents
T'3/2~=
2
E
Kij:D2dPiDtI~y " -
(1)
1
-~ +/3 D3Cb, - --D3@2 '
i,j=l
(4.9)
/3
Tt2~-- - - D 2 ( ~ I - q b 2 ) D t ~ b l O ~ 2 " +
1
"l- ~/3 "a2t~202t~2"+
(1 t
(~1 )-[- 1
fl +/3 : D 3 ~ i D ~ 2 :
D4(~)2"
We see the quantum correction in the terms of order/3 o and/31.
(4.10)
S. Komataet al. / Extazdedsuperconformalalgebra
184
Next we show the conserved currents of the N = 2 super Toda field theory associated with osp(ll2), osp(212) and osp(312). (i)
osp(!
12) 1_
1
T tl)= :D~D~: + =DD~
- --DD~.
IJ
(ii)
(4.11)
IJ
osp(212) ( = sl(211))
Ts i) =
2 1 2 1 2 E :gijocI)iO~)J'-F -~ i=~lOOCI)i- -~ i=IE OO(I)i,
i,j=l
1
1_
T( 1 ) = :D~2D@,'+ : D @ 2 D ~ , ' - - ~ D D ( ~ t -
(iii)
~2) - -~DD(@I
-'/'2).
(4.12)
osp(312) 2 T (1) =
1 2
1
2
E "rijO@iD~j: + E D D ~ i - -- ~_. D D ~ i , i,j=l -~ i=l fli=l
[ = ~fl~:D~D~2D~D~ ~]
2+
a ijk :fl DDdPi DclgjDclgk :
i,j,k=l
2
2
+-= ~_, xijDD~iDDclJy + ~_, Yij:DDclgiDD~j" i~ i , j = l
i,j=i
2
1
)
+ ~., zij:DDD~iDclgj'+ - ~ D D D D ~ l + c.c.,
(4.13)
i,j=l
where
aijk--XijKjk , x~j =
(01) 1
0
,
Yij =
(4.14)
(1 0 ) ( 1 0
--~1
,
zij =
0
1) 0
.
(4.15)
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 s1(3]2) in the N = 1 super Toda field theory, it can be shown that by taking appropriate forms of T (2) and T (5/2), none of the generators suffer from quantum correction. For general n in the sl(n + 1In) case, we expect that :L: becomes the quantum Lax operator in the sense that :wi: (1 ~
S. Komata et al. / Extended superconformal algebra
185
form the closed algebra, where L and w~ are given in eq. (3.27). On the other hand, it is also probable that there is no quantum correction to the conserved current associated with the esp(2n +_ 1 [2n) N = 2 super Toda field theory as well as the sl(n + l ln) N = 1 case. Therefore we think that our conjecture in sect. 2 is also valid at the quantum level. In the case of osp(2n +_ 112n), the Lax operators (3.39) and (3.40) suffer from non-trivial quantum correction, which is characteristic of the non-simply-laced Lie (super) algebra, as we have already seen in the case of osp(312) eqs. (4.9), (4.10). Now we discuss the operator algebra of the osp(312) extended N = 1 superconformal algebra in detail because it is a simple and non-trivial example. We show the OPE of the generators in appendix C*. The central charge is given by c=
/32
3.
(4.16)
The fusion rule becomes [f(2)][ f(2)] = a[1] + b[ ~,2)1,
(4.17)
according to appendix C and in general b 4:0 for all values of /3. In ref. [4], assuming the simple fusion rule between the primary fields ~t2) of spin 2 [ ~,2)][ 7~(2)] = a[ 1],
(4.18)
and imposing the associativity condition of four-point functions it was shown that the central charge c must be fixed to a special negative value, c = - 6 / 5 . Indeed, if we set /32= - 5 in eq. (4.16), then c = - 6 / 5 , and in the OPE /~(2)~(2) the coefficients of ]~(2), D~(2) and DEw (2) vanish, that is, the self-coupling becomes zero. We conclude that the supersymmetric W-algebra discussed in ref. [4] is realized in the extended superconformal algebra we have invented using the super Toda field theory.
5. The representation theory Let us consider the representation of the osp(312) extended N = 1 superconformal algebra, since it is the most simple, instructive and non-trivial example. In the previous section, we treated superfields ~i as if they were free fields although there is an interaction term in the action. In this section we regard ~i as "free fields" and consider a conformal field theory containing the local spin-2 field T(2)(Z) as well as the stress-energy t e n s o r Tt3/2)(Z). In our notation Tt2)(Z) is the same as in sect. 4 and Tt3/2)(Z) is minus half the Tt3/2)(Z) in eq. (4.9) *The same algebra has independently been obtained by Figueroa-O'Farrill and Schrans [22] via bootstrap analysis.
186
S. Komata et al. / E~tended superconfonnal algebra
To begin with, we consider the Neveu-Schwarz sector of the osp(312) extended N = 1 superconformal field theory. The generators can be expanded in operator Fourier modes as follows T~3/2'(Z) =
~ GrT. - r - 3 / 2 + 0 ~.~ L , , z - n - 2 r~Y+ 1/2 he2
r~'-'(z) = ~2 U,,z-"-" + o n~Z
~.,
W,z -'-s/'-.
(5.1)
r ~ 2 + i/2
Now let .~ be a space of local fields which itself is incorporated with an operator algebra and involves T~Z~(Z) a n d T~3/2~(Z). T h e total symmetry of the theory is generated by { Q , L,,, U,,, W~}. 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 A~ which satisfies the conditions G r A i = L , , A i = U,,A i = W r A i = 0, L o A i = AiAi,
for r, n > 0,
UoA i = uiA i ,
(5.2) (5.3)
where the conformal dimension Ai > 0 and the U-charge u i are real numbers. Motivated by the fact that, in the case of the bosonic W,,-algebra, there are many solutions to the quantum conformal field theory corresponding to the degenerate representations, we will now construct the degenerate representation of the osp(312) extended N = 1 superconformal algebra. It is reasonable to expect that a similar situation can occur in the quantum superconformal theory. In fact, the 19-vertex model has a solution corresponding to the minimal series of N = ! superconformal algebra [15]. The degenerate representation is, by definition, characterized by the existence of the null field x ( A u) which is a secondary field associated with a primary field A(A,u), for r, n > 0,
Grx=L,,x=U,,X=WrX=O,
LoX(a,u)
= (a
+
t)x(a,u),
(5.4)
where (5.4) shows that X appears at level l. Now we construct the degenerate representation using the technique known as Coulomb gas formalism [16]. The primary field A(A, u) can be expressed in terms of free fields q~i as A.(A,u)
= O ** ,
(5.5)
S. Komata et al. / Extended superconformal algebra
|87
where
(5.6)
A--gq~
ql
~
~
+_,q2 2ql +
,
)
(5.7)
A and u have the following 7/2 symmetry
A(q) = ~ ( ~ - q),
.(q) = . ( - - q ) ,
(5.8)
where
1
/3
(5.9)
ql~ =
From eq. (5.8), we see that A(ot)= u ( a ) = 0 , so that there is an external charge denoted by at in the theory. In order to find a non-trivial null field, we introduce screening operators V
(vo ±., = e"--*,,* V.+. " ÷= *e* _ } ,
V=
a _+=/3 +-l ,
(5.10)
which satisfy the equations
96dz 2d02 T(s/2)(ZI)V(Z2) = ~ d z 2d02 T(2)(ZI)V(22) = 0.
(5.11)
Using definition (5.10), null fields have the following expressions
~dz, dO, V. ±.,(Z,) ...~dz,, dO,,V,~±.,(Z.)V,,_q_,,.±.m(O) , ~dz, dO, V.+e2( Zl) ...~dz. dO. V.+e2( Z.)V._q_..+e:(O) ,
(5.12)
where q is given by
q
=
qlmm
=
l-1 2fl
() (m, 1 + 1
2
fl+~
n ,)(0) 2/3
(5.13) '
188
S. Komata et al. / Extended superconformal algebra
satisfying the charge neutrality conditions [16] ( l - 1) ~ 27/,
(m - 1) + (n - 1) ~ 27/,
l,m,n
> 0.
(5.14)
Although the last term in (5.13) is very similar to what appeared in the expression for null fields of the ordinary N = 1 superconformal algebra, the first term is not, in accordance with the fact that the Cartan matrix is not symmetric under the exchange of the fields ~1 ~ ~2- A(l,m,n) and u(l,m,n) can be calculated from eq. (5.7) for each l, m and n. As for the Ramond sector, we only show the result that the central charge (4.16) and the expression of the spectrum corresponding to the degenerate representation (5.13) are the same as those for the NS sector except that the condition on l, rn and n yields in this case ( / - 1) ~ 2 7 / + 1,
(m-
1) + ( n -
!) ~ 2 Z + l ,
l,m,n >0,
(5.15)
and that we must add 1/8 to the dimension of the primary field, coming from the spin operators which change the boundary condition of fermion fields. Now let us observe an operator algebra of primary fields given by (5.13). It is immediately seen that the primary fields
{V,,,, - V(l= l,m,n) =eq,,,,,,¢'*},
(5.16)
constitute a closed algebra and have no singular part in their OPE
E
C~,i~]i'},,,,,,lZ{-2J('""'-a('"'""+'a('"'""[ V,n,,,,,,(Z2) ] ,
(5.17)
tn"/'l" A(m"n") >/A( m n ) + A( m 'n')
because t/)~(Zl)f/)~(Z2)-- :(~:(Zl)t~:(Z2): by definition. This fact can be seen also from the expression of A in eq. (5.7) in which there are only linear terms with respect to q2, indicating that a simple rule for dimension counting is valid A(:V,.,Ym,.,:) = a ( V m . )
+ A(Vm,,,,) •
(5.18)
This OPE structure is reminiscent of the chiral ring of the N = 2 superconformal algebra [17]. It will be interesting to investigate the relation between the N = 2 superconformal algebra and the algebra (5.17) characterized by the rule (5.18). We will now concentrate on a particular case, the minimal theory in which /3 takes some discrete ~alues 1
q
- -p,
(5.19)
S. Komata et al. / Extended superconformal algebra
189
where p and q are relatively_prime positive integers. Under thi~ condition on/3, the space of primary fields =
~) 1 ~m
V,,,,,
~
(5.20)
1
l <~n
is endowed with a closed operator algebra. Moreover we can write (5.20) as a union of ~¢+ and s¢'_ which are defined by ~¢' = d + U s ¢ ' _ ,
~¢'+=
~ m-
Vm,,
aV'_=
~)
Vm,.
1
q
m-
1
q
n -- 1
p
n - 1
p
(5.21)
We see that in each of ~¢+ and ~ ¢ , 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 ~e'+. The equations (5.20) and (5.21) will impose further restrictions on the coefficients of the operator algebra (5.17). If we include e ('-q,,,,,,)'~'*, which has the same values of A and u as e q''--¢'* in (5.16), then the elements of ~'+ do not constitute a subalgebra. To exclude e ('~-q,,,,,,)'~* is crucial to the realization of the unitary representation. Let us pay attention to the central charge, corresponding to the minimal theory, which is given by 9q c = -- - 3 > -3, (5.22) P which is analogous to minus the central charge of the N = 2 superconformal discrete series. We have found a novel central charge without upper bound for minimal series. Though a 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. The theory we have discussed here is the first minimal model associated with this type of central charge. 6. Discussion
We discuss in this section some unsolved problems and possible applications of the extended superconformal algebra we have constructed. Firstly we consider the possibility of the realization of the super Toda field theory by certain reduction of an appropriate superconformal theory. We remind readers of the fact that, as mentioned in sect. 3, in the bosonic Toda field theory the equation of motion coincides with that of the constrained WZW model. Therefore it is reasonable to expect that the super Toda field theory can also be interpreted as the constrained supersymmetric WZW model defined on the Lie supergroup. Below we show some evidences which support our speculation. (1) The Cartan matrix of the Lie superalgebra appears in the potential term just as in the bosonic Toda field theory.
1~)
S. Komata et aL / Ewended SUl~,rconfomzal algebra
(2) The equation of motion can be written in the form of the supersymmetric zero curvature condition (3.16). (3) In the cases of sl{n + 1In) and osp(2n + l]2n), there are natural super versions of the Miura transformations. We suppose that it is not straightforward to construct the action corresponding to the supersymmetric WZW model since we must invent a theory in which the internal symmetry is generated by the elements of the Lie superalgebra and the equations of motion become (3.16) by imposing appropriate constraints. If we overcome this problem, it will become easier to construct the generators and in particular the unitary representation in the general case. As for the former, the existence of the supersymmetric Miura transformations for sl(n + l ln) and osp(2n + 112n) extended superconformal algebras implies that for these algebras any Casimir invariant can be defined by the supertrace of an appropriate power of vector representation matrices [18]. In the case of sl(n + lln) we discover the interesting relation between the degree of the Casimir invariant and the spin of the conserved current of the super Toda field theory. From ref. [18] we know the degree takes following values, 2 , 3 , . . . , 2 n , 2 n + 1,
(6.1)
which are twice the spins listed in table 1 in sect. 2. From this fact we expect to derive the generators by contracting s p i n - l / 2 supercurrents with the Casimir invariant. In the case of osp(2n + 1 [2n), we have not yet found a simple relation between the spins of conserved currents and the degrees of Casimir invariants. As for the latter, it may be possible to construct the unitary representation by using some technique based on Lie supergroup theory which will be analogous to the coset construction in the bosonic case. Next we propose possible physical applications of the extended superconformal algebra. It is reasonable to expect that our construction of the extended superconformal algebras could shed new light on the program for the classification of the supersymmetric integrable lattice models. Finally we comment on the application to the supergravity. It may happen that the partition function of the non-perturbative superstring theory satisfies the "extended superconformal algebra condition". This is a natural super version of the Virasoro condition discussed in ref. [19]. We suspect that the non-perturbative superstring equation can be expressed using the super Lax operator L constructed in sect. 3 as
{( Lk/'2"+l))_,
L} = const.,
(6.2)
inspired by the fact that there is a chiral Poisson bracket structure among the coefficients of L and that according to ref. [20], we can define L k/~2,+ ~ uniquely for general n, where k is a positive integer and L is given by eq. (3.26). It needs
S. Komata et al. / Exto~ded superconfonnal algebra
19t
constructing the matrix model with supersymmetry in space-time to justify, eq. (6.2). We hope to report some results on this issue in a future publication. We thank Prof. T. Yoneya for useful discussions and carefully reading our manuscript.
Appendix A We show the explicit form of {T ~2~,T t2~} in the case of the osp(312) N = 2 super Toda field theory. We omit the subscript +. T ~2~ is given by 2
T'2,-
2
~_. aijkDDq'iDq'iO*k+
[3
i,j,k= 1
Y'~ i.j= i
2
[3Xi j
= DDq, iDDq' ~ [3
2
1
+ Y'~ y#DD'I',DD'I'~ + Y'~ ziiDDD'I),D¢' j + --fiDDDD'I~,I + c.c., (A.I) i.j=l
i,j=! X l l - - X 2 2 -- __ ~' ,
auk = 2 x u K i , , !
Yli =
2,
Zi! =
--Z21 =
Y12 =Y21 = 1, -- 1,
(,)4 (,)3 (,)3 o,~,~or., (1)2 (1):
A = 96 ~
+ 32 ~
0,20,2 ( T ' ' , ) - 16 ~
+8 ~
=0
,
y:,2 = 0,
(A.3)
B + C,
(A.4)
( O , 2 D T " ' - O,2DT"' )
(aT"-'- ( ~ o - D~)T"')
+4 ~
(OnD(3T ~2~- 2~DT"') +~nD(3r a' + 2DDr~"))
+2 ~
0,2~,2(3~0T ~ ' - 3O~r ~2'+ (DDDO + O~Db)r"')
+2 ~
(A.2)
Zi2 = Z 2 2 -- 0 .
{ T (2) , T (2)} = A +
where
X12 = X 2 1
12
+4012012 -Z12 ( DD - DD)OT ~2~,
(A.5)
S, ~+++++++++,+++ c+ ++I / Ett+:'mfi+d ~+:~'+m:fim+++d a!m++:l,~ra
++2
C~++++°,+,`( ¢ ~ x ) + 2[i+l ~, o- + I+l o,~o l-~++x
e= -t:
++
- 8
ii+} +,,.
O,,,0,.++(.~X
+
+
+
(
++ +-<",+')
X--- T'+'T +(:.)+
} +
+
C. + = !6
+
(
++
6
(A.6)
<
2
(++ +i'~r~t'r.~+,+ 2
O++:D+ 0+,+
,
= ~+ ~
- ~
O,,0+,( DD - DD){T~"T
2~
a(
~(0,,D + O,,DlalCgr,',r ,~,) + ~(O,,+D + (3,,~)++~r"'-ar ''))
(' }0,.+,.{' -. + +
-
_
(A.7)
~n~ix
B
|n this appendix we show the exp|icR forms of the spin-2 conserved currents of the imegrable N = 1 and N = 2 super Toda field theories. |n the fo|lowing we denote by r the rank of the Lie superalgebra. We set 3 = 1 and drop the sub~fipt + for simplicity. |. T H E N = | S U P E R T O D A F I E L D T H E O R I E S r
E
r
T'2)= E a+Da*+- E a,K+jD"*jD*, + ½ E b,,D%P+D2(Pj ~i=|
i.~=l
i.}=l
( K+,jb+:k + ( a+ + a k ) K+jK+~)D2~k DcbiD~bj .
(B.~)
i,j,k=|
(i) sR~+ + fin): osp(2n + 112n). In expression (B.I), the a+ are given by 0 ~
1 (for even r ) ,
a+ =
-1 2 -2
q q
PI
I?l
(for odd r ) ,
(B.2)
X Komata et a L / E,ctended superconformai algebra
and the b# in expression (B.I) are for s|(n + lln): 1
-1
-1
1 1
-1
-1
bij =
1
(B.3)
0 0
-I
1
1
-I
for o s l ~ 2 n - l12n): '0 l
-|
|
-1
1 bij =
-1
(B.4)
1
-1
Q 0 D
1
-1 1
-1
and for osp(2n + ll2n):
1 -1
-1 1
(8.5)
b/j =
1 -1
(ii) osp(2nB2n), osp(2n + 212n). The r
ai =
O' 0 1 -1 2 -2 °
ai
-1
and b# are given by
!] (for even r ) ,
1
ai =
2! -2 ,m
(for odd r ) .
(8.6)
194
S. Komata et al. / Extended superconfonnal algebra
For osp(2nl2n):
1
-1
-1
1 1
-1
-1
bij --
1
(B.7)
I I
1
-1
-1
1
For osp(414) there is one more solution Tt2)= 2 D ~ i D ~ 2 D ~ 3 D ~ 4
+ D2( - ~ !
+ D2( - q b ! - (Jb2 -Jr d P 3 ) D d P 2 D ~
+ ~2) D~3D~4 + D2(2~23+
~4) D ~ I D ~ 3
2D2(-- (J]~3"[- qb4) DqblDqb2
- D2qbiD2qb3 + O 2 ~ l D 2 ~ 4 + D2~2D2qO3 - D2qb2O2qb 4
+ D3(2q~2- q~3)Oq~,-D3( 2q~, -q~3)Dq~2- D4( q~, - q~2) •
(B.8)
For osp(2n + 212n): 1
-1
-1
1
(B.9)
bij --
1
-1
-1 (iii) D(2I 1; a) ai --
1
o)
--2
(B.10)
,
0 bij --
l+a 1-a -1-a
1-a l+a -l+a
-l-a) -l+a. l+a
(B.11)
B.2. THE N = 2 SUPER TODA FIELD THEORIES
m
r
2flcijKjkDDCPiD~jD~ k + E i,j,k=l
i,j=l
~
_
~
r
l
flciy DDq2iDDq)Y + -~ E bijDD~iD-D-~i [3
i,j=l
(B.12) i,j=l
i=l
S. K o m a t a et aL / E x t e n d e d superconfonnai algebra
195
(i) sl(n + lln), osp(2n + l12n). In expression (B.12), a i and bij are given by '
0' 1
0 ~ 1
-1 2
(for odd r ) ,
-2
ai =
-1 2 -2
ai =
m
~m
k w m ,
(for even r ) .
(B.13)
j
For sl(n + l ln): 0 0
0 0 1
1 0
bij =
(B.14)
9
0 1
Cii -~-
l
2 ,
1
0 0
0 0 (B.15)
(i:¢=j) .
Cij = 0
There is another solution in this case as follows. r m
2flcijKjkDD~iD~jD~k +
Y'. flcij _ D -- D ~ i D_D ~ j + v, i,j=l
i,j,k=l
r
•
1 r
}
Y'~ axKijDDD@iD@y + -fl ~ l a i D D D D * i i,j=l i=
'0
--C.C.,
(B.16)
i
0
1 1 a i --
bijED~iDD~ j i,j=l
1 1
2 2
(for odd r ) ,
ai =
2 2
(for even r ) .
(B.17)
:l m
~m
0 0
0 0 -1
1 0
(B.18)
Q
bij =
0
0
-1 cii
(-I) i+l'
Cij '-- 0
(i~j).
(B.19)
t%
S° K~:~4natact aL / EL~tendcd~u~'rc~m~nat al~,bra
For osp(2 n - I J2 n): -1 0
0 0 -1
-1 0 0
0 0
(B.20) Q ¢.
0 0
1
Ci i
Cii ~" -~
-
-
0 0 -1
-1 0
(iCj).
0
(B.21)
And for osp(2n + 112n): 0 0 -1
-1 0
-1 0 0
0 0
bij =
(B.22)
Q Q Q
0 -1
Cii -- ~,_
¢ij ----0
-1 0 0
0 0
(B.23)
( i 4=j ) .
Appendix C We show the OPE among the generators of the osp(3[2) extended N = I superconformal algebra.
T°/Z'(Z,)T'S/2'(Z2) = -
() 6
-2~
- - "
-
-
1
- ~ + 2 Z-----~l,_-
012 D2T(3/2)( 2
3 012 T(3/2)(Z,) -Z22
Z2)
-
1 Z,2
DT (3/2)( Z 2)
(C.1)
1 0]2 D2~(2) ( 4 Z--~2012 ~(21( Z2 ) - -~12 D7~(21(Z2) - 2 Z!2 Z2)' (C.2)
S. Kgnnataet a£ / Ertendedsuperconformalalgebra 2
25
1
~+g-~ + 6
3-fl + 5 -
{() 5
+Z-'~2 -
+~12
0|2 (-+ Z, 2
)
/3 D~2~(Z2) +
V+3
-
7 3
3 z~ + ~ + 2/3 ~a}(Z2) +
+-UT2 Z~2 -
0~2
1(4
/32
+
2fl 2) 01_, r o / : } t 7 3 Z~2 ',"2)
('7F + 97
2~2)DT,3/2,(Z2)} 9
(8,,+2 +
9-~
9
9
"~+
18
9
Z2)+
1 /32 ) /3 :{ T O/2), ¢,2)}. (Z2) + 6 + T :{ T°/2}'
-
(,)
t97
D2T~3/2)(Z2) ) D3TO/2}(Z 2))
DT°/2)}'(z2)
(, D4TO/2)(Z2)}, (C.3) ,(2 + [3)DTt3/2), (C.4)
fl 03/~(2)(Z2)+
where
~ + 4 +
6
~(2)____T,2)+ 3 fl" and
{A,a} :AB +aA.
(c.5)
Appendix D
Below we give the Cartan matrices of Lie superalgebras which lead to the integrable super Toda field theories• (i) osp(1 ]2):
(,).
(DA)
(ii) sl(n + 1 In), r = 2n:
0 1
1 0
-1
-1 0
(D.2) 0 -1
-1 0
1
1
0
l g:S
So K~n,am e~ aL / E~-tended sut~,wonfo~al algebra
(iii) o s ~ 2 n + 112n) (r = 2 n , n > 0): 1
-1
-1
0 1
1 0 (D.3)
.8 Q
1 0 -1
-1 0
(iv) o s ~ 2 n - 112n) (r = 2 n - 1, n > 1): 1
-1
-1
0 1
1 0
(D.4)
Q
-1 0 1
0 -1
(v) osl~2nl2n) ( r = 2n, n > 1): 0 2 -1
2 0 -1
-! -1 0 1
(D.5)
Q Q
0 -1
-1 0 1
(vi) osp(2n + 212n) (r = 2n + 1): 0 2 -1
2 0 -1
-1 -1 0 1
1 0
(D.6) 0 1
1 0 -1
(vii) D ( a ) (r = 3): [/
0 11 --
--~
1
-1-a
0
a
O/
0
(D.7)
S. Komala el a[. / E~tended superc~mformal atgebra
|~
E W e s h o w the C h e v a l | e y generators o f o s ~ 2 n - I t 2 n ) and o s ~ 2 n + 1 i 2 n ) corresponding to the Cartan matrices given in appendLx D. (i) o s p ( 2 n - l12n):
e2~- ! = E,,+ l-i.3,,-l +i +
l
(t
/ 2 i - i = E n - l + i . 3 n - i -- E 3 , - ! +i.n+ 1 - i , h2~- l = E 3 n - i . 3 n - i e2i
=
En +i, 3n - i -- E3n - 1+i,. - i ,
f 2 i = - E . -i.3n h2i
=
- E 3 n - i +~.3n- 11+i -- E n + 1t- L n + II-i -b E n _ [ +~.n- i +i,
-
i +i -- E 3 , - ~ . n +~,
En-i.,,-i - En +i.n+~ -- E3n-i.3,,-i + E3n-l
+~.3n-| +i,
)
(I ~ < i < n -
I),
(E.I)
(ii) o s p ( 2 n + l [ 2 n ) : e 2 i _ I = E n + 2 _ i , 3 n + i + i 4- E 3 n + 2 _ i . n + i, f 2 i - ! = E n +i. 3n + 2 - i - E3n + ! +i. n + 2--i,
h 2 i - I = E 3 n + 2 - i . 3 n + 2 - i - E 3 n + l +i.3n+ ! +i - E n + 2 - i . n + 2 - i
4- E n + i . n + i ,
i
e2i = E n + ! + i , 3 n + 2 - i -- E3n+! +i,n+ ! - i , f 2i = - E n + i - i . 3 n + | +i - E3n+ 2 - i , n + l +i,
hzi = E,,+ i-i.,, + I-i - E,,+ t +i.,+ i +i - E3,,+ 2-i.3,, + 2-i + E3,+ ! +i.3, ÷ I+i,
(1 ~ < i < n ) ,
)
(1 ~
(E.2)
References [1] V. Pasquier, Nucl. Phys. B295[FS21] (1988) 491; M. Jimbo, T. Miwa and M. Okado, Nucl. Phys. B300[FS22] (1988) 74; I. Kostov, Nucl. Phys. B300[FS22] (1988) 559 [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, ill Proc. RIMS Syrup. on Non-linear integrable systems-Classical and quantum theory, Kyoto, Japan; ed. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983)p. 1 [3] H. Nohara and K. Mohri, Nucl. Phys. B349 (1991) 253 [4] T. Inami, Y. Matsuo and I. Yamanaka, Phys. Lett. B215 (1988) 701 [5] P. Forgfics, A. Wipf, J. Balog, L. Feh6r and L. O'Raifeartaigh, Phys. Lea. B227 (1989) 214; preprint ETH-TH/90-2 (1990) [6] D.A. Leites, M.V. Saveliev and V.V. Serganova, Serpukhav prepfint IHEP-85-81 (1985) [7] L. Frappat, A. Sciarrino and P. Sorba, Commun. Math. Phys. 121 (1989) 457
2~}
Is] [91 [1o1 [11] [12] [131 [14] [151 I161 [171 [181 [191
S. Komata et aL / Eatended superconformal algebra
A.N. Leznov and M.V. Seveliev, Commun. Math. Phys. 74 (1980) 111 M.V. SaveIiev, Commun. Math. Ph~,s. 95 (19,84) 199 P. Mansfield, Nucl. Phys. B208 (19,82) 277 V.G. Dtinfeld and V.V. Sokolov, L Soy. Math. 30 (19.85) 1975 M. BershadskT and H. Ooguri, Commun. Math. Phons. 126 (1989) 277 Q-H. Park, Nucl. Phys. B333 {1990} 267 A.B. Zamolodchikov, Theor. Math. Phys. 63 (1985) 1205 P. Di Francesco, H. Saleur and J.-B. Zuber, Nuc|. Phys. B300{FS22] (1988) 393 VI.S. Dotsenko and V.A. Fateev, Natl. Phys. B240IFSI2] (1984) 312 W. Lerche, C. Vafa and N.P. Warner, Nucl. Phys. B324 (1989) 427 P.D. Jar,is and H.S. Green, J. Math. P h i . 20 (1979) 2115 M. Fuk-ama, H. Kawai and R. Nakayama, Tokyo preprint Lrr-562 (1990); E. Verlinde and H. Verlinde, preprint IASSNS-HEP-90/40 (1990) [20] Yu.l. Manin and A.O. Radul, Commun. Math. Phys. 98 0985) 65 I21] K. Ueno and H. Yamada, Adv. Stud. Pure Math. 16 (1988) 373 i22] J.M. Figueroa-O'Farril| and S. Schrans, [}reprint KUL-TF-90/16 (1990)
CLASSICAL AND QUANTUM EXTENDED SUPERCONFORMAL ALGEBRA Shiro KOMATA, Kenji M O H R I and Hiroshi N O H A R A
h~stitute of Physics, Unit'ersityof Tokyo, Komaba, Meguro-ku, Tokyo 153, Japan Received 16 July 1990 (Revised 12 February 1991)
The integrable supersymmetric Toda field theories enable us to construct several infinite series of the extended N = 1 and N = 2 superconformal algebras. In particular, in the cases of two series of the extended superconformal algebras associated with sl(n + 1 In) and osp(2n + l l2n), we find the supersymmetric Miura transformations. The classical extended superconformal algebras are realized as the Poisson bracket structure among the coefficients of a superdifferential operator. We also obtain the exotic minimal discrete series of the osp(312) quantum extended N = 1 superconformai algebra. This minimal series is different from those known so far in the sense that the corresponding central charge has no upper bound.
1. Introduction
We are still far away from establishing the classification of the superconformal theories in spite of much effort for many years. The key to this problem is to construct the extended superconformal algebra containing higher spin generators in addition to the ordinary stress-energy tensor. From our experience in the bosonic case [1] we expect that there is a beautiful correspondence between this algebra and some physical theory, for example the soluble lattice model with superconformal symmetry or the one coupled with supergravity. At the present stage however, we do not have any method for the systematic approach to superconformal field theory such as the KP theory [2] in the bosonic case. Therefore it is important to discover any examples which have the extended superconformal symmetry. Our motivation to investigate the supersymmetric Toda field theory lies in a fuller understanding of integrable systems with superconformal symmetry. It is probable that the supersymmetric Toda field theory [6,9] has an extended superconformal symmetry and a close relation to the hierarchy of the integrable non-linear supersymmetric differential equations because it is a natural super version of the Toda field theory. Indeed in ref. [3] we investigated the classical supersymmetric Toda field theory in order to construct a classical extended 0550-3213/91/$03.50© 1991 - Elsevier Science Publishers B.V. (North-Holland)
S. Komata et al. / Extended superconformal algebra
169
superconformal algebra which can be considered to characterize the theory at the classical level. It was shown that, as in the bosonic case, there are several series of Lie superalgebras which lead to the integrable supersymmetric Toda field theories with as many independent conserved currents as the degrees of freedom. (By super Toda field theory, we mean Toda field theory with supersymmetry in two-dimensional space-time.) These conserved currents constitute the closed algebra with respect to the chiral Poisson bracket which can be thought to be the classical limit of the operator product expansion (OPE) between superfields. This algebra is quite new, because it is different from the ones which are derived from a previous analysis [4] in that the Jacobi identities hold for arbitrary values of the central charge, and that in general the generator with spin higher than 3 / 2 does not become a primary field in contrast with the bosonic W-algebra [5]. In this paper, we discuss the relation between super Toda field theory and super KP theory. We also construct the quantum analogue of several classical extended superconformal algebras constructed in ref. [3], and discuss possible physical applications of this algebra. In sect. 2 we briefly review the result of ref. [3] in the case of the classical super Toda field theory and show some new results about the existence of infinite series of sets of conserved currents, i.e. classical extended superconformal algebras, and we show an interesting correspondence between the N = 1 case and the N = 2 case. In sect. 3 we establish the manifestly supersymmetric Miura transformation starting from the supersymmetric zero curvature form of the equation of motion of the sl(n + 1 In) and osp(2n + 112n) N = 1 super Toda field theory. There exists an infinite series of superdifferential operators with the structure of the classical superconformal algebra. In sect. 4 we construct, for some simple examples, the quantum version of the classical extended superconformal algebra in the sense of sect. 3 and investigate the algebraic structure of the quantum extended superconformal algebra. In sect. 5 we examine the representation theory in the case of the osp(312) N = 1 super Toda field theory. We will see that in this case there is no upper bound of the central charge of the minimal series in contrast with the bosonic minimal series, that is the central charge of this minimal model takes the form
9q
c=---3,
(1.1)
where p and q are relatively prime positive integers. Finally in sect. 6 we discuss some relevant issues of the quantum extended super-conformal algebra and future problems.
170
S. Komata et al. / Extended superconfomml algebra 2. Classical theory
First we briefly summarize the results in ref. [3] for the reader's convenience. To begin with, we explain the integrable super Toda field theory with N = 1 space-time super-symmetry in two dimensions. The action is given by
s = f d2z d20
1
r r 1 r Y'~l exp /3 E Kij~g}j ~ KoD_q)iD +4)j + -~ i= j=l i,j=l
Z+ = XO -I"/X! ,
Z-=Xo--/X!
,
,
(2.1)
(2.2)
where Kij is a symmetric and invertible matrix of dimension r and ¢~i are N = 1 superfields. We regard z ÷ as the space coordinate and z - as the time coordinate in two-dimensional euclidean field theory so that we may pick up only the chiral part. The chiral Poisson bracket is defined by
( K-l)ijOl2 {D+*,(Z,),
%(Z2)} = -
(2.3)
ZI2
where z,:
=
(2.4)
- o?o;,
and D+_ are the superderivatives defined by 0
0
D+=O0 + + 0 + Oz +-
(2.5)
-
The classical equations of motion are r
D_D + qbi = exp /3 Y'~ Kij~ j
(2.6)
j=l
and are soluble if we can construct r independent conserved currents T ts,) which satisfy the equation D_ T ts,) = 0
( 1 ~< i ~
(2.7)
where s i denotes the spin of the conserved current. In general, one of the conserved currents is inevitably the stress-energy tensor
TO/2'=
r 1 r E KijD 2+~io + % - -~ ~ O 3 ~ , . i,j=l
i=
(2.8)
S. Komata et al. / Extended superconformal algebra
171
(By spin of a generator we mean the number of superderivatives each of which is assigned spin 1/2. Note that in general we can not choose all generators to be primary fields in contrast with the case of the bosonic W-algebra [5].) It depends on the choice of K o whether we get full conserved currents. In ref. [3] we demonstrated the derivation of conserved currents by adopting for Kii the Caftan matrices of osp(ll2), si(211), osp(312) and s1(312). In refs. [3, 6] it was shown that there are several classes of Lie superalgebras the Cartan matrices of which lead to the integrable theories, that is the ones in which we can choose all simple roots to be odd elements so that eq. (2.6) is rewritten in the form of the super zero curvature condition (see eq. (3.16) and below). The Lie superalgebras which satisfy this condition are sl(n + l l n ) ,
osp(2n + l12n),
osp(2nl2n),
osp(2n + 212n), (2.9)
D(211;a).
(If readers would like to know the Cartan matrices of these Lie superalgebras, see appendix D.) We also comment on super Toda lattice theories. They are associated with affine Lie superalgebras for which all of the simple roots are generated by odd elements. In refs. [6] and [7] such types of affine Lie superalgebras are also classified as follows: psl(n In) tl~ ,
sq(2n + 1) t2) ,
osp(2n 12n) t2~,
osp(2n + 212n) tl~,
psl(2n + l12n + 1) t2),
psl(2n 12n) t2~,
D(2I 1; a ) tl~.
(2.10)
Use of a Cartan matrix of the above Lie superalgebras leads to an integrable Toda lattice theory. The conformal symmetry is lost in the Toda lattice theories. We next review the classical N = 2 super Toda field theory. The action is
s =f
gij(I)i¢~j +
dEz dE0
i,j=l
r
{f
dEz dE0 - exp fl i=1
)
13 E Kij~j + c.c.
(2.11 )
j=l
where Kij is a hermitian and invertible matrix, ~i and ~ are the chiral superfields which are complex and satisfy the following conditions, m
D
D +_.cI)i=D _+qbi - 0 .
(2.12)
172
£ &~m,am cl ai. / E~tendt~g ~u,l~,rconfonnal algebra
~ e chiral Poison brackets are defined by 2( K-*)#012
{D
Z~2
2( K= '),0,, Z~,
(2.14)
where
01,_ = 0 i - 0 +,., +
0~, = ,a-
-0~_ , +
+
Zl,.=z~( -z,_ -0~0,_ - 0 ~ 02 .
(2.15)
The equations of motion are D_D +q~i = exp
(2.16)
(r)
D_D+ q~ = exp /3 E K#q~ .
(2.17)
j=!
In this case, the integrability means that there are as many independent conserved currents as the pairs of chiral superfields. The equations which are satisfied by the consem'ed currents become
D _ T {~'}= D _ T {~'~= O.
(2.18)
We always have the stress-energy tensor as a conserved current T'"=
E K#D+q~iD+~i + -~ ~ D+D+q~i- -- E D+D+~i. i,i=l i= /3 i=l
(2.19)
In ref. [3], we derived integrable theories by using the Cartan matrices of osp(1J2), sl(211) and osp(312). Among them, osp(312) is the first, rather simple but non-trivial example. By using the Cartan matrix of osp(3[2), we obtained two conserved currents T ~l} and T {2~ (see appendix A for the explicit expression of T{2~). It was shown that T ~2~does not become a primary field with respect to T t~ even if we are allowed to add a term (D +D + - D+D +)T ~l~ in contrast with the N = 1 super Toda field theory. As for the chiral Poisson bracket {T ~2~,T~2q we mention that there is a term (T(l~) 3 besides ( T ~ ) 2 and T ~ T ~2~ reflecting the fact that there is a spin-3
S. Komata et aL / Extended superconfonnd algebra
t 73
TABLE 1 A proposal for the complete lisl of spins of d~ conserved currents in N = | super Toda t-mid ~ e o t i e s Lie superalgebra
Spin of conserved currents
sKn + 1Bn) oslg | ~2)
1 / 2 x ~2.3,4. . . . . 2 n , 2 n + | ) 3/2 3/2,2 1 / 2 x ~3.4, 7,8~ | L 12, . . . . 4n - 5,4n - 4,4n - l) 1 / 2 × (3,4,7,8, I1,12 . . . . . 4n - 1,4n)
osIg2n - 1~2n) (n >t 2) osFg2n + l~2n)(n >i 2)
os~2t 2) ( ---sK2Di D oslg2n a2n) (n >I2) oslg2n + 2U2n)
1.3/2 1 / 2 x (3,4,7,8,11,12, . . . . 4n - 5,4n - 4,4n - | L n i / 2 x {3,G7,8,11, | 2 . . . . . 4n - 1 , 4 n L n + I / 2 3/2,3/2,2
[~2] 1;a)
generator in the N = 2 multiplet T (2~. ( S e e
appendix A for the expression
{T (2~, T(2~}.)
In consequence of further study along the line of ref. [3], up to n = 6, in the N = 1 super Toda field theory we have succeeded in solving eq. (2.7) in. all the cases (2.9) and in the N = 2 super Toda field theory, in the cases of sl(n + 1 In) and osp(2n +_ l U2n). From the observation for Lie superalgebras of lower rank,, we propose a complete list of the conserved currents in the N = 1 super Toda field theory in table 1. (In appendix B we only give the explicit form of the conserved current T (2) for reference since those of higher spins are inessentially complicated.) In table 1, osp(2n +_ ll2n), osp(2nl2n) (n >/2), osp(2n + 212n) and D(211;a) are associated with the extended N = 1 superconformal algebra since the spins of the conserved currents are more than or equal to 3 / 2 . However sl(n + l ln) is associated with the extended N = 2 superconformal algebra since T ~1~ and T ~3'2~ constitute the classical N = 2 superconformal algebra although we started from the N = 1 super Toda field theory. We can identify the components of T (3/2~ and T c2~ with the generators of the classical N = 2 superconformal algebra up to normalization as follows. T ('~ = J + 0( G - G ) ,
(2.20) (2.21)
T (3/2~ -- G + G + O T .
This enhancement of supersymmetry originates from the symmetry of the Dynkin diagram under the reflection with respect to the center of the diagram (see fig. 1).
n
n
®
®
®
.....
Fig. 1. The Dynkin diagram of sl(n + 1 in).
®
174
S. Komata et al. / Extended superconformal algebra TABLE 2 A proposal for the list of spins of the spins of the conserved currents in N = 2 super Toda field theories Lie superalgebra
Spin of conserved currents
sl(n + 1 In) osp(2n - 112n) osp(2n + l l2n) osp(212) ( = sl(211))
1,1,2,2 . . . . . n - l , n - l , n , n 1,2,3 . . . . . 2 n - 2 , 2 n - 1 1 , 2 , 3 , . . . , 2 n - 1,2n 1,1
We remark that when we take the limit /3 ~ oo the conserved currents are transformed into D +~i. In this case if we have N superfields, N-extended supersymmetry is induced in the theory and as the extended superconformal algebra we obtain the so(N) superconformal algebra rather than the algebra with higher spin generators. Finally we propose in table 2 the list of the spins of the conserved currents in the N = 2 super Toda field theory. (See appendix B for the expression of Tt2).) In the case of sl(n + 1 In) there are two conserved currents of spin 1 which are decoupled from each other if we take certain forms of them as can be seen easily from the discussion in ref. [3]. In terms of components, the extended N = 2 superconformal algebra associated with o s p ( 2 n - ll2n) (resp. osp(2n + l12n)) has generators with the same spins as the generators of the sl(2nl2n - 1)(resp. sl(2n + 112n)) extended N = 2 superconformal algebra. In particular, we immediately see that the generators associated with osp(ll2) and those associated with sl(211) have equivalent chiral Poisson brackets as well. Therefore we conjecture that for general n the series of the extended superconformal algebra are equivalent to each other also at the level of the algebraic structure. As for the case of osp(2n + 212n) and osp(2nl2n) (n >f 2), we could not derive any conserved currents of spin higher than 1. We only checked that there is no conserved current of spin 2 in the Toda field theories associated with these Lie superalgebras.
3. Supersymmetric Miura transformation Here we derive a supersymmetric version of the Miura transformation. The supersymmetric differential operator we will construct is expected to play a central role in formulating the hierarchy of supersymmetric non-linear soliton equations which has a hamiltonian structure associated with superconformal symmetry and is a candidate for a powerful tool to study the extended superconformal algebra. In this section we set/3 - 1 for simplicity.
175
S. homata et al. / Extended superconformal algebra
It is instructive to review the case of the bosonic Toda field theory before going to the supersymmetric one. The equation of motion of the Toda field theory based on the Lie algebra g of rank r is rewritten in the form of a zero curvature condition [8] as [O++A+,O_+A_]=O (3.1) r
,~
E r,Aj
O+O_d& = exp
(3.2)
j=l r
r
A += ~ O+dpihi + Y'~ ei, i=l i=l
exp
A_=
d~s
,
(3.3)
i=l
where hi, e i and fi are the Chevalley generators of g, and they satisfy the following commutation relations [ei, fs = S i j h j ,
(3.4) [hi,fj] = -gijfj.
We remark that eq. (3.1) is invariant under the gauge transformation (3.5)
A +_~ g-lO +g + g-~A +_g,
where g is an element of the group generated by g. We will see that a matrix-valued differential operator 0+ + A + leads to the well-known Miura transformation [11, 12]. To see it explicitly, we take g = sl(n). The generators are given by e,=E,.,+ l ,
hi---Ei.i-Ei+,.i+,,
f~ = E,+ i. , ,
(3.6)
where E~s is a matrix with a one at position (i, j) and zeros elsewhere. Let us regard 0 + + A + as operator acting on an n-dimensional vector v and consider the following equation (O++A+)v=O. (3.7) Writing eel. (3.7) in components as 0++0+¢pl
1 0++0+~ 2
O++A+=
1 , (3.8)
D D
0+ + O+~ n_ 1
1 ¢~+ J¢- 0+q~n
where q~, = 6 1 ,
~2--~2--~!,''',
~n--I --¢~n--! - - ~ n - - 2 '
q~,,=-~bn_~,
(3.9)
S. Komata et al. / Ertended superconfonnal algebra
176
and
(3.10)
V -"
we obtain the differential equation for v~ (3.11)
Lc n =0,
where
+o+(d,,,_,-4,,_:))...(0+
L = (o+-o
+ o+
- 6,))(o+ +o +4,,).
(3.12) On the other hand, we can get the following forms of gauge fields, A + and A_, by adopting the unique gauge transformation generated by negative roots, 0
1 0
A+=
A_=0.
o
(3.13)
1 Wn
Wn- 1
W2
0
Note that eq. (3.11) is invariant under the gauge transformation generated by the lower triangular matrix with ones above and alongside the diagonal. In this way we get the well-known Miura transformation L
=
0"+ + (
-
1)" w 2 o..,-2 +
= (0+-- 0+¢]},,_!)(0++
+ ... -w,,_lO++w,,
0 + ( t ~ , , _ i -- ¢ ~ , , - 2 ) ) ' ' "
(0+-I- 0 + ( t ~ 2 -- ~}1))(0+-'1- 0 + t ~ l ) .
-
(3.14) In eq. (3.14), L is called Lax operator in the (n-reduced) KP hierarchy. We remark that in the Drinfeld-Sokolov gauge (3.13) [11], the zero curvature condition (3.1) means the conservation of the currents wi. Hence the coefficients wi constitute a closed algebra with respect to the chiral Poisson bracket in the Toda field theory (K-t)ij
{O + ¢~i( Z ) , f~j( W ) } "- __
Z
+
--W
+ .
(3.15)
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
S. Komata et aL / Extended superconformal algebra
t 77
N = 1 super Toda field theory based on the Lie superalgebra g is also written in the supersymmetric zero curvature condition,
{D++A+,D_+A_}
=0,
(3.16)
where
A + = Y'~D+ rP~hi + Y',e,, i
(3.17)
i
Here h i, e i and f~ are the Chevalley generators of g. They satisfy relations similar to (3.4) except that the commutator [ , ] is replaced by the graded commutator [ , }. Kij is the Cartan matrix of g (see appendix D). We remark that the coordinate in the spacetime (z-+, 0 -+) is embedded in that of the Lie superalgebra. Therefore e i and fi must be odd because the I.h.s. of eqs. (3.17) and (3.18) are. As a concrete example, let us first consider the case g = sl(n + 1 In). The explicit forms of h i, e i and fi are given by
hi=El,! + E,+2,,+2,
el=El,n+2, fl=En+2,!,
h2i = - g n + 2 - i , n + 2 - i - E n + 1 + i , n + i + i ' )
(1 < i < n )
e2i ~ g n + ! + i , n + 2 - i , f2i = -gn+2-i,n+! h2i+!
+i,
= En+2_i,n+2_ i + En+2+i,n+2+i, ~
(1 < i ~ n -
e 2 i + ! -- E n + 2 _ i , n + 2 + i , f2i+!
1)
(3.19)
= En+2+i,n+2-i.
We regard the operator D ++ A +
(3.20)
as the differential operator acting on the super vector space, the elements of which are b. 1
t' n + ]
1
(3.21)
Un + i
U2n+
where lfli (1 ~1 n + 2) are Grassmann-odd
S. Komata et aL / E~tended superconfonnal algebra
178
n u m b e r s . F r o m (3.17),
-- ~ 2n
Q
-q~4 + q~5
A+=
Q
- q 2 + q~3
q'l - ~°2 q~3 - ~4 I Q
0
1
q~2n-
1 -- q~2n
(3.22) where (3.23)
qi = O +~I~i .
L e t us c o n s i d e r t h e e q u a t i o n
(D++A+)v=O.
(3.24)
As b e f o r e , if w e write eq. (3.24) in c o m p o n e n t s o f t h e v e c t o r v, t h e n (3.24) r e d u c e s to t h e s u p e r d i f f e r e n t i a l e q u a t i o n with r e s p e c t to v~,
Lv i = 0 ,
(3.25)
where L -- ( D + - D + qb2.) ( D + + D + ( ~ 2 , , _ , - ~ 2 , , ) ) ( D + - D + ( q~2,,- 2 - q ~ 2 . - , ) ) - - -
x(D+-D+(~2-~3))(D++D+(~-~z))(D++D+~i).
(3.26)
By e x p a n d i n g L as L = D2n + + ! + ( - + 1 ) 2 " - l w l /~ )+ 2 " - 1
W3/2L'+/'~2n - 2 --
• • • +
Wn + !/2
,
(3.27)
we discover t h e s u p e r s y m m e t r i c M i u r a t r a n s f o r m a t i o n ( D + - D + ~ 2 , ) ( D + + D + ( q~z,,-, - q ~ 2 , ) ) ( D + - O + ( ~ z , , - z - ~ 2 , , - , ) ) - . -
x ( D + - D +( ~ 2 - ~ 3 ) ) ( D ++ D +( q~' - ~ 2 ) ) ( D + + D + q J I ) = D 2+ n + l - wID+2n-! +
W3/2~+
r)Zn-2
--
"
.. + W ,
+ 1/2
.
(3.28)
=D;-
%D++
W3/2
3.
l
and w3/2 in (3.30) constitute the respect to the chiral Poisson bracket
q
ssical
. (33
-(K-')ijlnZ,2-
(@i(z,)9@j(z2))=
In general, Wi in eq. (3.27) generates the classical extended algebra with respect to the chiral Poisson bracket (3.31) Cartan matrix of sl(n + 1 In). We suspect that there is the supersymmetric Drinfeld-Sokolov A + and A _ take the following forms, 0 W-1/2
... 0
gauge in w
0 w3/2
*. . . . l
wn -3/2
w,z-
1/2
. . . . .
0
.
1
A_=O. although we have not yet found the explicit form of t which connects (3.22) and (3.32) and leaves eq. (3.25) i Let us consider the case osp(2n + 112n). Firstly we symmetric Miura transformation for osp(l12) and osp(3
3.32)
I SO
S. &~nmta et aL / E2~-tendedsul~rconformal algebra
and f, are given by
h=
0 1 0
0 0 0
0 0 - 1
0
1
1
0 0
e=
0
,
f--
(i'!) "
0
,
(3.33)
0
for o sp(1,2) and hi = E44 - E55,
h2 =
EI~
e I = E ~ + E42,
e 2 = E~ - Esi,
f l = E,.4 - Es_,,
f2 = - E i s
- E33 - E44 +
E55,
(3.34)
- E43,
for osp(3,2). The corresponding A+ in (3.17) becomes 1 0 -D+~
0 D+~
A+=
0 D@ 2
A+-
0
0
0 0
0 0
0 -D~
0
1
0
0
0
-1
(3.35)
0 0 1
2
D+q~ i - D + ~ 0
0 1 0 2
(3.36)
0 -D+q~
i +D+q~
2
for osp(!, 2) and osp(3, 2), respectively. By rewriting (3.24) as before, we obtain the supersymmetric Miura transformations for osp(112) L=(D+-D+q~)D+(D++D+q~)
= D +3- ( D Z + q ~ D + q ~ - D 3 q g ) ,
(3.37)
and for osp(312) L -- ( D + + D + q~2)( D + - D + ( ~
- q~z))D + ( D + + D + (q~, - q~2))( D + - O + ~ 2 )
= D 5 + T { 3 / 2 ) D 2 + ( - 2T t2~ - D T t3/2~) D + - D + T (2) ,
(3.38)
where T {3/2) is given in eq. (2.8) and T (2) is given in eq. (B.5). For general n, the super Lax operator L for osp(2n + i , 2 n ) is given by L = ( D + + D +q~2,,) ( D + - D + ( q~2,,-, - ~ z , , ) ) ( D + + D + ( ~2,,- 2 - q~2,,- ,)) • - • × (D+-D +(q~= ~+f)4n+l+
q~2))D+ ( D + + D + ( ~ , -
T ( 3 / 2 ) o 4 n - 2 + ...,
q~2))-.- ( D + - D + q~2n) (3.39)
S. Komata et al. / Extended superconformal algebra
181
and for o s p ( 2 n - 112n) L = ( D+-D
+ q ~ 2 n - , ) ( D + + D + (q~2#_2 - @ 2 # _ , ) ) ( D + - D + (q~2#_3 - @ 2 . _ 2 ) ) . . .
× ( D + - D + ( g ) , - @2))D + ( D + + D + ( @ , - q~2))... (D+ + D +q~2n_ ,) _ i ) 4 n - i - T ~ 3 / 2 } D +4 n - 4 + . . . .
Corresponding Chevalley generators are given in the appendix E. Finally we remark that L in (3.27) does not admit the time flow of the super KP type in ref. [20], while L in (3.39) and (3.40) do. In general we can uniquely define L l / ( 2 m + !} a s L l / ( 2 ' n + l} = D + + u o + u i D + ~+ . . . ,
(3.41)
where L is a superdifferential operator of order 2m + 1. Calculating the (2m + l)th power of (3.41), we see that, in (3.27) u o = 0 and u~ ~: 0, while in (3.39) and (3.40) u o = u~ = 0. Recall [20] that the super KP hierarchy is a system of the equations among the Lax operators defined by A = D + + r o + clD~_l+ . . . ,
(3.42)
D +c o + 2c I = 0.
(3.43)
with
Therefore we conclude that, in the case of osp(2n + 112n), L in (3.39) and (3.40) are derived from the ½(4n + 1)- and ½ ( 4 n - 1)-fold reduction respectively of the super KP hierarchy, while in the case of sl(n + 1 In) we cannot obtain (3.27) by ½(2n + 1)-fold reduction. In mathematical literature [20,21], the n-fold reduction of the super KP hierarchy is defined by (A2")_=0,
(3.44)
where n is a positive integer. Although it would be interesting to construct another hierarchy which produces (3.27) by reduction, it is beyond our aim in this paper. 4. Quantum extended superconformai algebra
In this section, we construct an operator version of the classical extended superconformal algebra for further studies in later sections. The direct quantization of the super Toda field theory with the action (2.1) is a very difficult problem [3]. Therefore we will take the following route. First, we construct the conserved currents of the super Toda field theory at the quantum level using the free field normal ordering regularization [I0]. These
182
S. Komataet al. / Extendedsuperconformalalgebra
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 difficulties in quantization. Generally speaking, the classical conformal symmetry is easily seen by non-linear soliton equations, while for the realization of the conformal symmetry at the quantum level, it is convenient to adopt the free field 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 [5, 12, 13]. classical level: quantum level:
WZW model current algebra Feigin-Frenkel realization
Toda field theory W-algebra Fateev-Lukyanov realization
We will briefly review the quantization h la Mansfield in the bosonic case. The free field quantization is applied to the Toda field theory, i.e. the chiral Poisson bracket
(g-l)iy
{O+d)'(z')'6J(z2)}= z~-z~
(4.1)
is replaced by the OPE given by
(~i( Zl)~j(Z2) ------( K-l)ij In( z~-
+ "~i( Zl)~j( z2)',
- z~-)
(4.2)
where : : denotes the normal ordering. Therefore the equation of motion is given by
:exp(flY' Kijdpj)'.
a+a_4,i =
•
j
(4.3)
Now let us turn to the super Toda field theory. In this case, we replace the chiral Poisson brackets (2.3), (2.13) by the following OPE
(~i(Zl) (~j(/2) -- - ( K-l)ij ln( Z!2 ) + :Oi( Z1)Oi(Z2):
(4.4)
(N = 1 super Toda field theory), D
012012 > + ~i(ZI)~j(Z2) = - ( K-l)ij(ln( Z12) + ZI2
(4.5)
(N = 2 super Toda field theory), and have equations of motion
00+
,
(46 J
S. Komata et al. / Extended superconformal algebra
183
( N = 1 super Toda field theory),
exp / ~ E
, (4.7)
( N = 2 super Toda field theory). We make use of these equations to derive conserved currents which are conserved at the quantum level. (To be rigorous, we need some justification which re-inforces the free field quantization. It seems to us that a key to this problem lies in establishing the correspondence between the super Toda field theory and the supersymmetric W e s s - Z u m i n o - W i t t e n (WZW) model on some Lie supergroup with certain constraints. See refs. [5, 12, 13] in the bosonic case and sect. 6 for the discussion.) 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 N = 1 super Toda field theory. (i) osp(212)(= sl(211)). We can easily see that T {l~ and T ~3/2~ defined by
T{I~ = :DclgiDcI92. +
T{3/2)=
1
/3
D2~l
--
1 --D2t~b2,
/3
2 1 2 ~ K i j : D 2 ~ i D ~ j : - -- Y'~ D3clgi,
(4.8)
/3 i=1
i,j=l
are conserved. Note that T t~ and T (3/2) 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. (ii) osp(312). We have the following conserved currents
T'3/2~=
2
E
Kij:D2dPiDtI~y " -
(1)
1
-~ +/3 D3Cb, - --D3@2 '
i,j=l
(4.9)
/3
Tt2~-- - - D 2 ( ~ I - q b 2 ) D t ~ b l O ~ 2 " +
1
"l- ~/3 "a2t~202t~2"+
(1 t
(~1 )-[- 1
fl +/3 : D 3 ~ i D ~ 2 :
D4(~)2"
We see the quantum correction in the terms of order/3 o and/31.
(4.10)
S. Komataet al. / Extazdedsuperconformalalgebra
184
Next we show the conserved currents of the N = 2 super Toda field theory associated with osp(ll2), osp(212) and osp(312). (i)
osp(!
12) 1_
1
T tl)= :D~D~: + =DD~
- --DD~.
IJ
(ii)
(4.11)
IJ
osp(212) ( = sl(211))
Ts i) =
2 1 2 1 2 E :gijocI)iO~)J'-F -~ i=~lOOCI)i- -~ i=IE OO(I)i,
i,j=l
1
1_
T( 1 ) = :D~2D@,'+ : D @ 2 D ~ , ' - - ~ D D ( ~ t -
(iii)
~2) - -~DD(@I
-'/'2).
(4.12)
osp(312) 2 T (1) =
1 2
1
2
E "rijO@iD~j: + E D D ~ i - -- ~_. D D ~ i , i,j=l -~ i=l fli=l
[ = ~fl~:D~D~2D~D~ ~]
2+
a ijk :fl DDdPi DclgjDclgk :
i,j,k=l
2
2
+-= ~_, xijDD~iDDclJy + ~_, Yij:DDclgiDD~j" i~ i , j = l
i,j=i
2
1
)
+ ~., zij:DDD~iDclgj'+ - ~ D D D D ~ l + c.c.,
(4.13)
i,j=l
where
aijk--XijKjk , x~j =
(01) 1
0
,
Yij =
(4.14)
(1 0 ) ( 1 0
--~1
,
zij =
0
1) 0
.
(4.15)
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 s1(3]2) in the N = 1 super Toda field theory, it can be shown that by taking appropriate forms of T (2) and T (5/2), none of the generators suffer from quantum correction. For general n in the sl(n + 1In) case, we expect that :L: becomes the quantum Lax operator in the sense that :wi: (1 ~
S. Komata et al. / Extended superconformal algebra
185
form the closed algebra, where L and w~ are given in eq. (3.27). On the other hand, it is also probable that there is no quantum correction to the conserved current associated with the esp(2n +_ 1 [2n) N = 2 super Toda field theory as well as the sl(n + l ln) N = 1 case. Therefore we think that our conjecture in sect. 2 is also valid at the quantum level. In the case of osp(2n +_ 112n), the Lax operators (3.39) and (3.40) suffer from non-trivial quantum correction, which is characteristic of the non-simply-laced Lie (super) algebra, as we have already seen in the case of osp(312) eqs. (4.9), (4.10). Now we discuss the operator algebra of the osp(312) extended N = 1 superconformal algebra in detail because it is a simple and non-trivial example. We show the OPE of the generators in appendix C*. The central charge is given by c=
/32
3.
(4.16)
The fusion rule becomes [f(2)][ f(2)] = a[1] + b[ ~,2)1,
(4.17)
according to appendix C and in general b 4:0 for all values of /3. In ref. [4], assuming the simple fusion rule between the primary fields ~t2) of spin 2 [ ~,2)][ 7~(2)] = a[ 1],
(4.18)
and imposing the associativity condition of four-point functions it was shown that the central charge c must be fixed to a special negative value, c = - 6 / 5 . Indeed, if we set /32= - 5 in eq. (4.16), then c = - 6 / 5 , and in the OPE /~(2)~(2) the coefficients of ]~(2), D~(2) and DEw (2) vanish, that is, the self-coupling becomes zero. We conclude that the supersymmetric W-algebra discussed in ref. [4] is realized in the extended superconformal algebra we have invented using the super Toda field theory.
5. The representation theory Let us consider the representation of the osp(312) extended N = 1 superconformal algebra, since it is the most simple, instructive and non-trivial example. In the previous section, we treated superfields ~i as if they were free fields although there is an interaction term in the action. In this section we regard ~i as "free fields" and consider a conformal field theory containing the local spin-2 field T(2)(Z) as well as the stress-energy t e n s o r Tt3/2)(Z). In our notation Tt2)(Z) is the same as in sect. 4 and Tt3/2)(Z) is minus half the Tt3/2)(Z) in eq. (4.9) *The same algebra has independently been obtained by Figueroa-O'Farrill and Schrans [22] via bootstrap analysis.
186
S. Komata et al. / E~tended superconfonnal algebra
To begin with, we consider the Neveu-Schwarz sector of the osp(312) extended N = 1 superconformal field theory. The generators can be expanded in operator Fourier modes as follows T~3/2'(Z) =
~ GrT. - r - 3 / 2 + 0 ~.~ L , , z - n - 2 r~Y+ 1/2 he2
r~'-'(z) = ~2 U,,z-"-" + o n~Z
~.,
W,z -'-s/'-.
(5.1)
r ~ 2 + i/2
Now let .~ be a space of local fields which itself is incorporated with an operator algebra and involves T~Z~(Z) a n d T~3/2~(Z). T h e total symmetry of the theory is generated by { Q , L,,, U,,, W~}. 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 A~ which satisfies the conditions G r A i = L , , A i = U,,A i = W r A i = 0, L o A i = AiAi,
for r, n > 0,
UoA i = uiA i ,
(5.2) (5.3)
where the conformal dimension Ai > 0 and the U-charge u i are real numbers. Motivated by the fact that, in the case of the bosonic W,,-algebra, there are many solutions to the quantum conformal field theory corresponding to the degenerate representations, we will now construct the degenerate representation of the osp(312) extended N = 1 superconformal algebra. It is reasonable to expect that a similar situation can occur in the quantum superconformal theory. In fact, the 19-vertex model has a solution corresponding to the minimal series of N = ! superconformal algebra [15]. The degenerate representation is, by definition, characterized by the existence of the null field x ( A u) which is a secondary field associated with a primary field A(A,u), for r, n > 0,
Grx=L,,x=U,,X=WrX=O,
LoX(a,u)
= (a
+
t)x(a,u),
(5.4)
where (5.4) shows that X appears at level l. Now we construct the degenerate representation using the technique known as Coulomb gas formalism [16]. The primary field A(A, u) can be expressed in terms of free fields q~i as A.(A,u)
= O ** ,
(5.5)
S. Komata et al. / Extended superconformal algebra
|87
where
(5.6)
A--gq~
ql
~
~
+_,q2 2ql +
,
)
(5.7)
A and u have the following 7/2 symmetry
A(q) = ~ ( ~ - q),
.(q) = . ( - - q ) ,
(5.8)
where
1
/3
(5.9)
ql~ =
From eq. (5.8), we see that A(ot)= u ( a ) = 0 , so that there is an external charge denoted by at in the theory. In order to find a non-trivial null field, we introduce screening operators V
(vo ±., = e"--*,,* V.+. " ÷= *e* _ } ,
V=
a _+=/3 +-l ,
(5.10)
which satisfy the equations
96dz 2d02 T(s/2)(ZI)V(Z2) = ~ d z 2d02 T(2)(ZI)V(22) = 0.
(5.11)
Using definition (5.10), null fields have the following expressions
~dz, dO, V. ±.,(Z,) ...~dz,, dO,,V,~±.,(Z.)V,,_q_,,.±.m(O) , ~dz, dO, V.+e2( Zl) ...~dz. dO. V.+e2( Z.)V._q_..+e:(O) ,
(5.12)
where q is given by
q
=
qlmm
=
l-1 2fl
() (m, 1 + 1
2
fl+~
n ,)(0) 2/3
(5.13) '
188
S. Komata et al. / Extended superconformal algebra
satisfying the charge neutrality conditions [16] ( l - 1) ~ 27/,
(m - 1) + (n - 1) ~ 27/,
l,m,n
> 0.
(5.14)
Although the last term in (5.13) is very similar to what appeared in the expression for null fields of the ordinary N = 1 superconformal algebra, the first term is not, in accordance with the fact that the Cartan matrix is not symmetric under the exchange of the fields ~1 ~ ~2- A(l,m,n) and u(l,m,n) can be calculated from eq. (5.7) for each l, m and n. As for the Ramond sector, we only show the result that the central charge (4.16) and the expression of the spectrum corresponding to the degenerate representation (5.13) are the same as those for the NS sector except that the condition on l, rn and n yields in this case ( / - 1) ~ 2 7 / + 1,
(m-
1) + ( n -
!) ~ 2 Z + l ,
l,m,n >0,
(5.15)
and that we must add 1/8 to the dimension of the primary field, coming from the spin operators which change the boundary condition of fermion fields. Now let us observe an operator algebra of primary fields given by (5.13). It is immediately seen that the primary fields
{V,,,, - V(l= l,m,n) =eq,,,,,,¢'*},
(5.16)
constitute a closed algebra and have no singular part in their OPE
E
C~,i~]i'},,,,,,lZ{-2J('""'-a('"'""+'a('"'""[ V,n,,,,,,(Z2) ] ,
(5.17)
tn"/'l" A(m"n") >/A( m n ) + A( m 'n')
because t/)~(Zl)f/)~(Z2)-- :(~:(Zl)t~:(Z2): by definition. This fact can be seen also from the expression of A in eq. (5.7) in which there are only linear terms with respect to q2, indicating that a simple rule for dimension counting is valid A(:V,.,Ym,.,:) = a ( V m . )
+ A(Vm,,,,) •
(5.18)
This OPE structure is reminiscent of the chiral ring of the N = 2 superconformal algebra [17]. It will be interesting to investigate the relation between the N = 2 superconformal algebra and the algebra (5.17) characterized by the rule (5.18). We will now concentrate on a particular case, the minimal theory in which /3 takes some discrete ~alues 1
q
- -p,
(5.19)
S. Komata et al. / Extended superconformal algebra
189
where p and q are relatively_prime positive integers. Under thi~ condition on/3, the space of primary fields =
~) 1 ~m
V,,,,,
~
(5.20)
1
l <~n
is endowed with a closed operator algebra. Moreover we can write (5.20) as a union of ~¢+ and s¢'_ which are defined by ~¢' = d + U s ¢ ' _ ,
~¢'+=
~ m-
Vm,,
aV'_=
~)
Vm,.
1
q
m-
1
q
n -- 1
p
n - 1
p
(5.21)
We see that in each of ~¢+ and ~ ¢ , 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 ~e'+. The equations (5.20) and (5.21) will impose further restrictions on the coefficients of the operator algebra (5.17). If we include e ('-q,,,,,,)'~'*, which has the same values of A and u as e q''--¢'* in (5.16), then the elements of ~'+ do not constitute a subalgebra. To exclude e ('~-q,,,,,,)'~* is crucial to the realization of the unitary representation. Let us pay attention to the central charge, corresponding to the minimal theory, which is given by 9q c = -- - 3 > -3, (5.22) P which is analogous to minus the central charge of the N = 2 superconformal discrete series. We have found a novel central charge without upper bound for minimal series. Though a 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. The theory we have discussed here is the first minimal model associated with this type of central charge. 6. Discussion
We discuss in this section some unsolved problems and possible applications of the extended superconformal algebra we have constructed. Firstly we consider the possibility of the realization of the super Toda field theory by certain reduction of an appropriate superconformal theory. We remind readers of the fact that, as mentioned in sect. 3, in the bosonic Toda field theory the equation of motion coincides with that of the constrained WZW model. Therefore it is reasonable to expect that the super Toda field theory can also be interpreted as the constrained supersymmetric WZW model defined on the Lie supergroup. Below we show some evidences which support our speculation. (1) The Cartan matrix of the Lie superalgebra appears in the potential term just as in the bosonic Toda field theory.
1~)
S. Komata et aL / Ewended SUl~,rconfomzal algebra
(2) The equation of motion can be written in the form of the supersymmetric zero curvature condition (3.16). (3) In the cases of sl{n + 1In) and osp(2n + l]2n), there are natural super versions of the Miura transformations. We suppose that it is not straightforward to construct the action corresponding to the supersymmetric WZW model since we must invent a theory in which the internal symmetry is generated by the elements of the Lie superalgebra and the equations of motion become (3.16) by imposing appropriate constraints. If we overcome this problem, it will become easier to construct the generators and in particular the unitary representation in the general case. As for the former, the existence of the supersymmetric Miura transformations for sl(n + l ln) and osp(2n + 112n) extended superconformal algebras implies that for these algebras any Casimir invariant can be defined by the supertrace of an appropriate power of vector representation matrices [18]. In the case of sl(n + lln) we discover the interesting relation between the degree of the Casimir invariant and the spin of the conserved current of the super Toda field theory. From ref. [18] we know the degree takes following values, 2 , 3 , . . . , 2 n , 2 n + 1,
(6.1)
which are twice the spins listed in table 1 in sect. 2. From this fact we expect to derive the generators by contracting s p i n - l / 2 supercurrents with the Casimir invariant. In the case of osp(2n + 1 [2n), we have not yet found a simple relation between the spins of conserved currents and the degrees of Casimir invariants. As for the latter, it may be possible to construct the unitary representation by using some technique based on Lie supergroup theory which will be analogous to the coset construction in the bosonic case. Next we propose possible physical applications of the extended superconformal algebra. It is reasonable to expect that our construction of the extended superconformal algebras could shed new light on the program for the classification of the supersymmetric integrable lattice models. Finally we comment on the application to the supergravity. It may happen that the partition function of the non-perturbative superstring theory satisfies the "extended superconformal algebra condition". This is a natural super version of the Virasoro condition discussed in ref. [19]. We suspect that the non-perturbative superstring equation can be expressed using the super Lax operator L constructed in sect. 3 as
{( Lk/'2"+l))_,
L} = const.,
(6.2)
inspired by the fact that there is a chiral Poisson bracket structure among the coefficients of L and that according to ref. [20], we can define L k/~2,+ ~ uniquely for general n, where k is a positive integer and L is given by eq. (3.26). It needs
S. Komata et al. / Exto~ded superconfonnal algebra
19t
constructing the matrix model with supersymmetry in space-time to justify, eq. (6.2). We hope to report some results on this issue in a future publication. We thank Prof. T. Yoneya for useful discussions and carefully reading our manuscript.
Appendix A We show the explicit form of {T ~2~,T t2~} in the case of the osp(312) N = 2 super Toda field theory. We omit the subscript +. T ~2~ is given by 2
T'2,-
2
~_. aijkDDq'iDq'iO*k+
[3
i,j,k= 1
Y'~ i.j= i
2
[3Xi j
= DDq, iDDq' ~ [3
2
1
+ Y'~ y#DD'I',DD'I'~ + Y'~ ziiDDD'I),D¢' j + --fiDDDD'I~,I + c.c., (A.I) i.j=l
i,j=! X l l - - X 2 2 -- __ ~' ,
auk = 2 x u K i , , !
Yli =
2,
Zi! =
--Z21 =
Y12 =Y21 = 1, -- 1,
(,)4 (,)3 (,)3 o,~,~or., (1)2 (1):
A = 96 ~
+ 32 ~
0,20,2 ( T ' ' , ) - 16 ~
+8 ~
=0
,
y:,2 = 0,
(A.3)
B + C,
(A.4)
( O , 2 D T " ' - O,2DT"' )
(aT"-'- ( ~ o - D~)T"')
+4 ~
(OnD(3T ~2~- 2~DT"') +~nD(3r a' + 2DDr~"))
+2 ~
0,2~,2(3~0T ~ ' - 3O~r ~2'+ (DDDO + O~Db)r"')
+2 ~
(A.2)
Zi2 = Z 2 2 -- 0 .
{ T (2) , T (2)} = A +
where
X12 = X 2 1
12
+4012012 -Z12 ( DD - DD)OT ~2~,
(A.5)
S, ~+++++++++,+++ c+ ++I / Ett+:'mfi+d ~+:~'+m:fim+++d a!m++:l,~ra
++2
C~++++°,+,`( ¢ ~ x ) + 2[i+l ~, o- + I+l o,~o l-~++x
e= -t:
++
- 8
ii+} +,,.
O,,,0,.++(.~X
+
+
+
(
++ +-<",+')
X--- T'+'T +(:.)+
} +
+
C. + = !6
+
(
++
6
(A.6)
<
2
(++ +i'~r~t'r.~+,+ 2
O++:D+ 0+,+
,
= ~+ ~
- ~
O,,0+,( DD - DD){T~"T
2~
a(
~(0,,D + O,,DlalCgr,',r ,~,) + ~(O,,+D + (3,,~)++~r"'-ar ''))
(' }0,.+,.{' -. + +
-
_
(A.7)
~n~ix
B
|n this appendix we show the exp|icR forms of the spin-2 conserved currents of the imegrable N = 1 and N = 2 super Toda field theories. |n the fo|lowing we denote by r the rank of the Lie superalgebra. We set 3 = 1 and drop the sub~fipt + for simplicity. |. T H E N = | S U P E R T O D A F I E L D T H E O R I E S r
E
r
T'2)= E a+Da*+- E a,K+jD"*jD*, + ½ E b,,D%P+D2(Pj ~i=|
i.~=l
i.}=l
( K+,jb+:k + ( a+ + a k ) K+jK+~)D2~k DcbiD~bj .
(B.~)
i,j,k=|
(i) sR~+ + fin): osp(2n + 112n). In expression (B.I), the a+ are given by 0 ~
1 (for even r ) ,
a+ =
-1 2 -2
q q
PI
I?l
(for odd r ) ,
(B.2)
X Komata et a L / E,ctended superconformai algebra
and the b# in expression (B.I) are for s|(n + lln): 1
-1
-1
1 1
-1
-1
bij =
1
(B.3)
0 0
-I
1
1
-I
for o s l ~ 2 n - l12n): '0 l
-|
|
-1
1 bij =
-1
(B.4)
1
-1
Q 0 D
1
-1 1
-1
and for osp(2n + ll2n):
1 -1
-1 1
(8.5)
b/j =
1 -1
(ii) osp(2nB2n), osp(2n + 212n). The r
ai =
O' 0 1 -1 2 -2 °
ai
-1
and b# are given by
!] (for even r ) ,
1
ai =
2! -2 ,m
(for odd r ) .
(8.6)
194
S. Komata et al. / Extended superconfonnal algebra
For osp(2nl2n):
1
-1
-1
1 1
-1
-1
bij --
1
(B.7)
I I
1
-1
-1
1
For osp(414) there is one more solution Tt2)= 2 D ~ i D ~ 2 D ~ 3 D ~ 4
+ D2( - ~ !
+ D2( - q b ! - (Jb2 -Jr d P 3 ) D d P 2 D ~
+ ~2) D~3D~4 + D2(2~23+
~4) D ~ I D ~ 3
2D2(-- (J]~3"[- qb4) DqblDqb2
- D2qbiD2qb3 + O 2 ~ l D 2 ~ 4 + D2~2D2qO3 - D2qb2O2qb 4
+ D3(2q~2- q~3)Oq~,-D3( 2q~, -q~3)Dq~2- D4( q~, - q~2) •
(B.8)
For osp(2n + 212n): 1
-1
-1
1
(B.9)
bij --
1
-1
-1 (iii) D(2I 1; a) ai --
1
o)
--2
(B.10)
,
0 bij --
l+a 1-a -1-a
1-a l+a -l+a
-l-a) -l+a. l+a
(B.11)
B.2. THE N = 2 SUPER TODA FIELD THEORIES
m
r
2flcijKjkDDCPiD~jD~ k + E i,j,k=l
i,j=l
~
_
~
r
l
flciy DDq2iDDq)Y + -~ E bijDD~iD-D-~i [3
i,j=l
(B.12) i,j=l
i=l
S. K o m a t a et aL / E x t e n d e d superconfonnai algebra
195
(i) sl(n + lln), osp(2n + l12n). In expression (B.12), a i and bij are given by '
0' 1
0 ~ 1
-1 2
(for odd r ) ,
-2
ai =
-1 2 -2
ai =
m
~m
k w m ,
(for even r ) .
(B.13)
j
For sl(n + l ln): 0 0
0 0 1
1 0
bij =
(B.14)
9
0 1
Cii -~-
l
2 ,
1
0 0
0 0 (B.15)
(i:¢=j) .
Cij = 0
There is another solution in this case as follows. r m
2flcijKjkDD~iD~jD~k +
Y'. flcij _ D -- D ~ i D_D ~ j + v, i,j=l
i,j,k=l
r
•
1 r
}
Y'~ axKijDDD@iD@y + -fl ~ l a i D D D D * i i,j=l i=
'0
--C.C.,
(B.16)
i
0
1 1 a i --
bijED~iDD~ j i,j=l
1 1
2 2
(for odd r ) ,
ai =
2 2
(for even r ) .
(B.17)
:l m
~m
0 0
0 0 -1
1 0
(B.18)
Q
bij =
0
0
-1 cii
(-I) i+l'
Cij '-- 0
(i~j).
(B.19)
t%
S° K~:~4natact aL / EL~tendcd~u~'rc~m~nat al~,bra
For osp(2 n - I J2 n): -1 0
0 0 -1
-1 0 0
0 0
(B.20) Q ¢.
0 0
1
Ci i
Cii ~" -~
-
-
0 0 -1
-1 0
(iCj).
0
(B.21)
And for osp(2n + 112n): 0 0 -1
-1 0
-1 0 0
0 0
bij =
(B.22)
Q Q Q
0 -1
Cii -- ~,_
¢ij ----0
-1 0 0
0 0
(B.23)
( i 4=j ) .
Appendix C We show the OPE among the generators of the osp(3[2) extended N = I superconformal algebra.
T°/Z'(Z,)T'S/2'(Z2) = -
() 6
-2~
- - "
-
-
1
- ~ + 2 Z-----~l,_-
012 D2T(3/2)( 2
3 012 T(3/2)(Z,) -Z22
Z2)
-
1 Z,2
DT (3/2)( Z 2)
(C.1)
1 0]2 D2~(2) ( 4 Z--~2012 ~(21( Z2 ) - -~12 D7~(21(Z2) - 2 Z!2 Z2)' (C.2)
S. Kgnnataet a£ / Ertendedsuperconformalalgebra 2
25
1
~+g-~ + 6
3-fl + 5 -
{() 5
+Z-'~2 -
+~12
0|2 (-+ Z, 2
)
/3 D~2~(Z2) +
V+3
-
7 3
3 z~ + ~ + 2/3 ~a}(Z2) +
+-UT2 Z~2 -
0~2
1(4
/32
+
2fl 2) 01_, r o / : } t 7 3 Z~2 ',"2)
('7F + 97
2~2)DT,3/2,(Z2)} 9
(8,,+2 +
9-~
9
9
"~+
18
9
Z2)+
1 /32 ) /3 :{ T O/2), ¢,2)}. (Z2) + 6 + T :{ T°/2}'
-
(,)
t97
D2T~3/2)(Z2) ) D3TO/2}(Z 2))
DT°/2)}'(z2)
(, D4TO/2)(Z2)}, (C.3) ,(2 + [3)DTt3/2), (C.4)
fl 03/~(2)(Z2)+
where
~ + 4 +
6
~(2)____T,2)+ 3 fl" and
{A,a} :AB +aA.
(c.5)
Appendix D
Below we give the Cartan matrices of Lie superalgebras which lead to the integrable super Toda field theories• (i) osp(1 ]2):
(,).
(DA)
(ii) sl(n + 1 In), r = 2n:
0 1
1 0
-1
-1 0
(D.2) 0 -1
-1 0
1
1
0
l g:S
So K~n,am e~ aL / E~-tended sut~,wonfo~al algebra
(iii) o s ~ 2 n + 112n) (r = 2 n , n > 0): 1
-1
-1
0 1
1 0 (D.3)
.8 Q
1 0 -1
-1 0
(iv) o s ~ 2 n - 112n) (r = 2 n - 1, n > 1): 1
-1
-1
0 1
1 0
(D.4)
Q
-1 0 1
0 -1
(v) osl~2nl2n) ( r = 2n, n > 1): 0 2 -1
2 0 -1
-! -1 0 1
(D.5)
Q Q
0 -1
-1 0 1
(vi) osp(2n + 212n) (r = 2n + 1): 0 2 -1
2 0 -1
-1 -1 0 1
1 0
(D.6) 0 1
1 0 -1
(vii) D ( a ) (r = 3): [/
0 11 --
--~
1
-1-a
0
a
O/
0
(D.7)
S. Komala el a[. / E~tended superc~mformal atgebra
|~
E W e s h o w the C h e v a l | e y generators o f o s ~ 2 n - I t 2 n ) and o s ~ 2 n + 1 i 2 n ) corresponding to the Cartan matrices given in appendLx D. (i) o s p ( 2 n - l12n):
e2~- ! = E,,+ l-i.3,,-l +i +
l
(t
/ 2 i - i = E n - l + i . 3 n - i -- E 3 , - ! +i.n+ 1 - i , h2~- l = E 3 n - i . 3 n - i e2i
=
En +i, 3n - i -- E3n - 1+i,. - i ,
f 2 i = - E . -i.3n h2i
=
- E 3 n - i +~.3n- 11+i -- E n + 1t- L n + II-i -b E n _ [ +~.n- i +i,
-
i +i -- E 3 , - ~ . n +~,
En-i.,,-i - En +i.n+~ -- E3n-i.3,,-i + E3n-l
+~.3n-| +i,
)
(I ~ < i < n -
I),
(E.I)
(ii) o s p ( 2 n + l [ 2 n ) : e 2 i _ I = E n + 2 _ i , 3 n + i + i 4- E 3 n + 2 _ i . n + i, f 2 i - ! = E n +i. 3n + 2 - i - E3n + ! +i. n + 2--i,
h 2 i - I = E 3 n + 2 - i . 3 n + 2 - i - E 3 n + l +i.3n+ ! +i - E n + 2 - i . n + 2 - i
4- E n + i . n + i ,
i
e2i = E n + ! + i , 3 n + 2 - i -- E3n+! +i,n+ ! - i , f 2i = - E n + i - i . 3 n + | +i - E3n+ 2 - i , n + l +i,
hzi = E,,+ i-i.,, + I-i - E,,+ t +i.,+ i +i - E3,,+ 2-i.3,, + 2-i + E3,+ ! +i.3, ÷ I+i,
(1 ~ < i < n ) ,
)
(1 ~
(E.2)
References [1] V. Pasquier, Nucl. Phys. B295[FS21] (1988) 491; M. Jimbo, T. Miwa and M. Okado, Nucl. Phys. B300[FS22] (1988) 74; I. Kostov, Nucl. Phys. B300[FS22] (1988) 559 [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, ill Proc. RIMS Syrup. on Non-linear integrable systems-Classical and quantum theory, Kyoto, Japan; ed. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983)p. 1 [3] H. Nohara and K. Mohri, Nucl. Phys. B349 (1991) 253 [4] T. Inami, Y. Matsuo and I. Yamanaka, Phys. Lett. B215 (1988) 701 [5] P. Forgfics, A. Wipf, J. Balog, L. Feh6r and L. O'Raifeartaigh, Phys. Lea. B227 (1989) 214; preprint ETH-TH/90-2 (1990) [6] D.A. Leites, M.V. Saveliev and V.V. Serganova, Serpukhav prepfint IHEP-85-81 (1985) [7] L. Frappat, A. Sciarrino and P. Sorba, Commun. Math. Phys. 121 (1989) 457
2~}
Is] [91 [1o1 [11] [12] [131 [14] [151 I161 [171 [181 [191
S. Komata et aL / Eatended superconformal algebra
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