- Email: [email protected]
On a new hierarchy associated with the W3(2) conformal algebra
Physics Letters B 274 (1992) 179-185 North-Holland
PHYSICS LETTERS B
On a new hierarchy associated with the Peter v a n D r i e l
W (2)
conformal algebra
1
Institute of Theoretical Physics, Valckenierstraat 65, NL- I O18 XE Amsterdam, The Netherlands
Received 17 August 1991
In a realization of the Kac-Moody algebra A~~) intermediate between the homogeneous and the principal realization, an integrable hierarchy of differential equations is constructed. The hierarchy shares features of both the AKNS and the KdV hierarchies. The same hierarchy can also be constructed using the Drinfeld-Sokolov approach in terms of zero curvature conditions, with as hamiltonian structure the Wt 1) algebra of Polyakov and Bershadsky. It provides evidence for the conjecture that there exists a general relation between hierarchies constructed in some intermediate realization and the covariantly coupled chiral algebras of Bais, Tjin and van Driel.
1. Introduction It becomes m o r e and m o r e clear that the interest in physics o f integrable hierarchies o f non-linear differential equations is not at all restricted to the description o f solitary waves in shallow canals. Nowadays, a new impetus to study integrable hierarchies stems from the observation that n o n - p e r t u r b a t i v e inform a t i o n about two-dimensional gravity might be extracted from them. In particular the p a r t i t i o n function o f 2D gravity represented by the one matrix model after the double scaling limit is known to be a z function o f the K d V hierarchy [ 1 ], The original construction o f r functions o f the KdV hierarchy is through the so-called bilinear construction [2 ]. The key point there is that z corresponds to the group orbit of the highest weight vector o f an A[ ~) integrable level one representation. The square o f the r function is in a well d e t e r m i n e d s u b m o d u l e o f the tensor product o f two such representations, such that the orbit is fixed as the o r t h o p l e m e n t o f vectors not in that submodule. Since level one representations ofA~ ~) admit a polynomial realization, this leads to differential equations bilinear in z. F o r AI ~ there are two essentially inequivalent p o l y n o m i a l realizations d e p e n d i n g on whether we take the twisted E-mail address: [email protected].
or the untwisted vertex realization. The twisted realization o f A~ ~) results in the K d V hierarchy [3], whereas the A K N S hierarchy comes out if the untwisted realization is used [ 4 ]. In general, a vertex realization exists for all simply laced K a c - M o o d y algebras at level one [ 5 ]. The n u m b e r o f inequivalent realizations grows with the n u m b e r o f conjugacy classes o f the Weyl group of the underlying finite algebra [6 ]. A unified description of the resulting r functions is however still lacking. In particular for A~N~), only the r functions in the untwisted ( h o m o g e n e o u s ) realization and in the 77N twisted ( p r i n c i p a l ) realization are studied [7], but so far the intermediate cases have not been considered. At the same time, it is well known that the KdVtype hierarchies are hamiltonian, in the sense that the time flows are generated by (mutually c o m m u t i n g ) hamiltonians. There is no direct way to derive this structure from the bilinear construction. There exists a formulation in which it is straightforward and natural to derive the ( b i ) H a m i l t o n structure and the hamiltonians, called the zero curvature formulation. In this formulation the time evolution o f an integrable system is thought o f as a zero curvature condition associated with some gauge group. The gauge group o f the A K N S hierarchy is the current algebra o f s12. By constraining this algebra by setting J _ ( x ) to a constant, one obtains the K d V hi-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V, All rights reserved.
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2
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erarchy with as gauge group the Virasoro algebra. The general setting of this reduction scheme was provided by ref. [ 8 ] and it related WN algebras with generalized KdV hierarchy, which in the bilinear picture are related to the principal realization. Recently, in ref. [ 91 the problem of finding constrained algebras starting from the current algebra group ~1,~was addressed. It was shown that to every inequivalent slZ subalgebra of ~1,~there is naturally associated a set of constraints that give rise to a conformal algebra. Since there exists a one-to-one relation of inequivalent sl, subalgebras of sl, and conjugacy classes of the Weyl group of sl, [ lo], it is natural to conjecture that the algebras of ref. [ 93 are in fact a hamiltonian structure of the hierarchies in the corresponding intermediate realization in the bilinear formulation. This relation is the subject of the present paper. To be precise, the bilinear formulation will be employed to construct the hierarchy associated with the ZZ twisted realization of Ai” which exists besides the principal and homogeneous ones. The result is a new hierarchy, with three fields w, q and u that combine features of the AKNS and the KdV hierarchies. The second hamiltonian structure coincides with the Wj2) algebra of Polyakov and Bershadsky [ 11 1, which is indeed related to the non-maximal sl, subalgebra in sl, [ 9 1.
LETTERS
B
pzm=
s
9 January
[(h,),,-(&),,,I
P2 ,n+l =(es)m+Cf3),,7+,
,
=
y&j
(3) m>O,
-_I,, I,
l(e3)
-_),I -I
satisfying the commutation IPI,
(2)
,
42m=&5~(L,-(h2)
92m+I
1992
+ti)-,,,l,
(4)
(5)
relations
411=&,k.
(61
This realization is induced by the Weyl reflection in the root -_(y, --a2, and is related to embedding su2 X u, c su3, when we twist the Heisenberg algebra associated with the su2 subalgebra. The su2 is oriented in the o3 direction, the root associated with e3. As is well known, (6) admits a polynomial irreducible representation (V, n), V=C[x,;
iEz+]
,
n(p,)=
$)
x(9,)
i
=4
(7)
The basic module L(&) is not irreducible as an smodule since the spectrum of p,, is mfi, mEB, and go is not in the Heisenberg algebra. However, if we define q,, by (6), set 7c(qO) =x,, and realize the translation operator T that generates the p0 spectrum through T= efl-“, then (8)
2. The Z2 twisted realization of L(&) over A$‘) First we will discuss the Z, twisted realization of the basic representation L (iiO) over Ai1 ). Let g be the finite dimensional simple Lie algebra sl( 3, C) with generators
We may in fact gather the Heisenberg generators in formal expansions, such that we obtain the familiar vertex realization of A$’ ) with two bosons who have a Fourier expansion
Zh.Y2k-
Q= [er, Q], f;= (P,)’ and h,= [e,, f;]. Let g=Ai’) be the associated affine Kac-Moody algebra, g=I ,,,tL g@,I’“OCkO@d, where k is the central charge and d the derivation. The algebra g contains an intinite-dimensional Heisenberg subalgebra s generated by the central charge k and 180
&r“p2h
1 - ~ L-i-‘/2pzA+, 2k+ 1
>I )
>I ,
where we have chosen o+ (z) to be the untwisted
(9)
(10) bo-
Volume 274, number 2
PHYSICS LETTERSB
son (¢ + ( z e 2m) = ¢ + ( Z ) ) and ¢2 (z) to the Z2 twisted boson (¢_ (z e 2m ) = - - ¢ _ (2")). The bosons have the following OPE exhibiting the boundary condition
¢+ (z)O+ ( z ' ) ~ l o g ( l - Z,) ,
(11)
O_ ( . ) ¢ _ ( z ) ~ 1 o g ( . ~ + V~7) .
(12)
Expressions for the other current algebra generators can now in principle be found using the expressions for the vertex operators e , ( z ) = :e("~): = ~ (e,)kz - ~ - I ,
(13)
k
where the : : refer to the normal ordering of the modes of 0.+_, and the inner product ( ' ) is fixed by defining the direction of the twisted boson (for which we took a3). The gradation d is set equal to Lo, the scaling generator of the Sugawara tensor. In particular we find ~1
d" (Xm®e nxf~x°) = ( ½m+ 3n2) (x,,, ® e "'/5~°) .
(14)
Finally, the polynomial space admits a natural and unique hermitian form ( , ) given by
( P(x) e m'fSx°, Q(x) e "',/s~° ) = P ( D ~ ) ( ~ ( x ) ]x=o&..,
(15)
where ( 0 1 0 1 0 ) Dx= 0xl'20x2'30x3'"'
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r ( x ) e LAo ~ r(X (l)) ®r(X (2)) ~ L(2Ao) c L(Ao)®L(Ao) •
If we denote the orthogonal complement of L(2Ao) with respect to the hermitian form by L (2Ao) •, then for all elements PsL(2Ao) •, (P, r ® r ) =0. It is not difficult to see that L (2Ao) • contains arbitrary polynomials in the sum variables x=x (l~ + x (2). From the inner product (15) it then follows that the condition for r ( x ~ l ~ ) N r ( x ~2~) turns into a differential equation bilinear in r ( x ) . In this way one may construct the r functions as the subset of symmetric elements in L (2Ao), but we are only interested in the polynomials that define the differential equations, the so-called Hirota polynomials. These polynomials depend only on the difference variables y=xC1)--X (2) [3]. From the symmetry properties of the bilinear equation finally one obtains that only the polynomials in the symmetric tensor product of L(Ao) can give rise to non-trivial differential equations. The number of Hirota polynomials ZHir can thus be determined from the decomposition of L(Ao)®sL(Ao) with respect to the 7/2 twisted Heisenberg algebra, which can be determined by first calculating the branching A~ l ) ~ Al I ) × U ( 1 ) and then twisting the AI l) The result of this rather lengthy calculation is )~Hir~---q3/2 -t- q7/4 q_2q2 +
+2qll/4+5q3+ (16) "
3. The bilinear formulation In this section a few of the Hirota bilinear equations are constructed. The basic input is that the z functions are elements of the orbit O.o of the highest weight vector VAo, and
q9/4 + 3qS/2
....
(18)
We are now set to calculate the Hirota polynomials of lowest degree. Apparently, the first one occurs at degree 3/2. At this degree there are two independent polynomials ~2 in y that satisfy the symmetry property: YzVland cosh (x/3 Yo). Acting with (e2) _ ~(el) _ 1 on the vacuum of L(Ao)® L (Ao) gives the polynomial .~f3y2yl + c o s h ( ~ 3 y o ) which is an element of L(2Ao). Constructing the orthoplement this uniquely fixes the one polynomial in L(2Ao) • at this particular degree, (Hit)3/2 =
~ Note that the structure of (8) is almost identical to the basic representation L(Ao) over A} ~) in the untwisted (homogeneous) realization. The only difference is the zero mode spectrum, which in our notation would be 2n, neZ. Therefore the string functions start as weight n 2.
(17)
-
l ~ y 2 f l I +cosh(x/3
yo) .
(19)
Proceeding in this way for the next few degrees we found ~2 This includes exponentsin the variables Xo and Yo.
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(Hir)7/4 = ~x/3y2 exp( ~x/3 x0) sinh( ½x/3 yo) ~Yi exP(½x/3 Xo) cosh(½x/3 Yo) ,
(20)
(Hir)2,, =Y3Y,-~Y'~-½Y, sinh(,,/3 yo),
(21)
1 sinh(xf~yo) (Hir)2,2 =1jY22 -- gYt
(22)
(Hir)9/4 = - ~Xf3 y, y2 exP(½x/3Xo) cosh(½xf3 yo) - 2 ( y 3 - ~ 2 y 3) exp(½x/3 Xo ) sinh(½xfl3 yo). (23) Let x=xj, flow
t2 =2Xf3X2,
then (19), (20) define the t2
d~' = - ~ ' l ,, + ~u~,, J
(24)
d~ dr2
(25)
dl2
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neric and in fact quite unpleasant feature of the bilinear construction. As we already see from (18), the number of Hirota polynomials grows rapidly with the degree. Nevertheless, it is always possible to eliminate all equations but one equation using the equations of lower degree. Of course, this indicates a deeper structure, which, in the bilinear formulation, is still unravelled. The other thing that points in the direction of a deeper structure is the magic substitution (27), (28). It turns out that both issues have a natural solution when they are stated in the zero curvature formulation, which will be discussed in the next section.
4. The zero curvature formulation
du
dt2
!,,,,_ !2 ,~," F, --2"F
-
,
(26)
(¢~V)',
where u= - 2 log"r .... ~'m-- I
qJ=- --, l"m
(27) ~'m+ 1
¢=--
Tm
(28)
The first two equations (24), (25) are Schr6dinger equations, which are coupled non-linearly through the third equation. We do not know whether this particular Schr6dinger equation has any physical relevance. From ( 21 ), (23 ) the equations defining the 13flow are constructed ( t 3= X3) , d~' = ~u'"- 3 u ~ ' - 43u ' ~ + 3 ~,20, dt3
( 29 )
__de =¢,,,_ ~u¢_ ]u,¢_ 3i~u~_2 ,
(30)
d/3 du _ [¼u'- ~3 u -3 - ~-(~,'¢/- ~u¢') l t dr3
Vvac(/)=exp(
k~o~>~Pk& )
,
(31)
(32)
where now the tk are genuine complex variables. Vv,c(t) is a vertex operator since Vv,c(t)r(x)= r(x+t). Therefore, we obtain the function r(t) by calculation of the "vacuum expectation value" of r(x+t), making use of the inner product (15), i.e.
~(t) = ( 1, T-~Vv,c(t)h(x) . 1 ) ,
Notice that by putting c/and ¢7to zero we get the KdV equation from (31 ). In that case we essentially forget about the untwisted U( 1 ) and reduce to the 22 twisted realization of A} ~ The equation resulting from (23) is trivially satisfied using eqs. ( 2 4 ) - ( 2 6 ) and is therefore redundant. This redundancy of Hirota polynomials is age182
The way the hierarchy of equations is reformulated in terms of zero curvature relations is generally referred to as Wilson's dressing method [ 12 ]. The zero curvature relations arise when differentiating the r function directly. Since the group orbit is a non-linear space such an operation should be treated a bit carefuly. Therefore we first replace the r ( x ) (an element of the representation space L(Ao) ) by a genuine differentiable function z (t). This comes about as follows. Following Wilson [12], we introduce the vertex operator
(33)
where h(x) is the (non-unique) group element that produced the r function. The t dependence of the group element V~=T-IVvac(t)h(x) is now quite simple:
dV~ dr,
=p,V~.
(34)
Any arbitrary group element h in the Kac-Moody group Sin (C [2, 2-1 ] ) admits a factorization
9 January1992
PHYSICS LETTERS B
Volume 274, number 2
h=h_ hoh+ ,
(35)
where h_ = 1 + 2 - IMI + 2 - 2 M 2 + .... h+ = 1 +2N~ +22N2 + ...,
(36)
where the N~, Ms are arbitrary elements of sln, and
ho =N_ wHN+ ,
(37)
where N_ ( N + ) is the exponentiation of a strictly lower (upper) matrix, w is a Weyl group element and H is the exponentiation of an element of the Cartan subalgebra. Using the Birkhoff decomposition
In (42) one can recognize a gauge equivalent form of the constrained S13 current algebra giving rise to the W~ 2) algebra as defined in ref. [ 11 ]. The U ( 1 ) current of the W~ 2) algebra is missing though. It should not be confused with the U ( 1 ) current v, as v can be gauged away. This fixes the gauge freedom in the definition of h in (33 ), transforming T ' to
T=T'+v2-v ' .
(43)
The fields in (42) can actually be related directly to the r function. For instance, by projecting (42) on the h3 subspace one finds V-----½(/].-lh3, )[-1 ( ~ / _ ) - l p l ~//_ ) = (2-1e3, g/_ ) = l o g ' z t
I7,= (Vl) _ ( Vt)o,+
while for T ' one gets
a differential equation for (Vt)o,+ can be derived, which reads
T ' = ( 2 - 'f3, 2 - ' (~d_)- 'pl q/_ )
(~--~-[(V,)=~P,(V,)_]+)(V,)o.+ =0
Di (V,)o.+ i = 1 , 2 .....
(38)
As the r function (33) is merely the vacuum expectation value of Vo.+, (38) gives rise to a hierarchy of differential equations for ft. Finally, compatibility of the equations leads to the zero curvature conditions [D,, Dj] = 0 .
(39)
To be able to appreciate (38) we consider the simplest equation that occurs for tl = x . Explicitly, the covariant derivative in (38) then takes the form
D~=[Ox-(Vt)zl pI(V,)_1+ = a x - (2f3 +e3 + [ (M,) l,f31 ) .
(40)
The matrix (Mj) ~ has arbitrary form, and for convenience is parametrized like
__--_(,~- lh3, qfl_ > = --log"r-- (1og'z) 2 .
(44)
(45)
For both calculations we only used the contravariance of the inner product and the explicit polynomial representation (3). Combined with (43), (45) and (44) yield T = - 2 log"r,
(46)
The calculation of G -+ makes explicit reference to the translation operator Twhen we use T - ~q/= V t+ ~such that G + = <2-~fl, 2 -~ ( q / _ ) - ' p , g / > = _ <,~- le2 ' eft/_ ) =-(T-1
,q/)=-
r~_~ r~
(47)
and likewise G - = rt+ ~/rt. Apparently, the substitutions ( 27 ), (28 ) are motivated by the underlying zero curvature structure.
5. Solution of the zero curvature conditions
-2a
G+
,
d
a+½r'
(41)
where we have dropped the indices l for notational convenience. This brings (40) to the form ( Dx=0x-
v G+ \T'+2
0 0 G-
_' ) O . v
(42)
In this section the starting point are the zero curvature conditions for (38). To be precise, we will focus on the curvatures involving the x derivative, i.e. the equation
dJx =[Ji, Jx]+J'i, dt,
i = 0 , 1.....
(48)
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where ,Ix is a n sl 3 valued matrix that is now constrained to the full W ~ 2 ) form [ 11 ], i.e. including the U( 1 ) current j,
J~=
G+ T+2
-2j G-
,
(49)
=\
be c,
-2r, de
.
(50)
re-at/
dj _½q+ G+_½q;_ G_ +~r,' dt,
(51)
+ I 3 G ± O+G+'+3jG±lhe,
(52)
dT dt - I -½ ~ 3 + 2 T ~ + T ' + 2 2 01te +I3G + O+~G+'-3jG+Iq? +I~G- O+½G-'+3jG-Iqi-.
(53)
These equations determine the Poisson structures [9]. Besides the W (32) algebra which is the second hamiltonian structure of this hierarchy, we now also
find the first Hamilton structure of the hierarchy {G+(x), G - ( y ) } ( , ) = - 6 ( x - y ) ,
(54)
{T(x), T ( y ) } ~ , ) = - 2 6 ' ( x - y ) .
(55)
All other brackets vanish. Clearly, the current j is a central element of the algebra, which fixes the current j to be constant under all time flows. For notational convenience and without loss of generality it is set to zero. Writing (48) like
and recursion relation j = l .... , i ,
(58)
dJ - ~ 2 R , = CJ:( ~ff t .@2)R,_2 . d6
(59)
Finally, using the bi-Hamilton structure, the hamiltonians can be found explicitly from the defining equation dF - { H i + , , F}(2)= {H,+3, F}(l) , dr,
(60)
where F is any functional of the currents and Hi has scaling dimension i. Using ( 5 1 ) - ( 5 3 ) , the initial condition (57) reads
(56)
(61)
This has two non-trivial solutions, to= 1 and ho= 1. Since the initial condition will fix the scaling behaviour of the currents rood 2 it will give rise to two essentially different and mutually commuting hierarchies. Straightforward application of the algebraic algorithm just described yields the first two (trivial) sets of equations
dG±_-7½G±, dto dG * = G +', dtl
d~T=0, dto
(62)
dT -T'. dtl
--
(63)
More interesting are however dG + - T - ~ 1G dt2
+
tt
+_½TG~,
(64)
dT -- = (G+G-) ' dt2
(65)
and dG + dt3
_ _
2~2J,+2+~,oJ,
(57)
h ; = q ~ =qff = 0 .
dG + 1 + _ + _ I - ( O + _ 3 j ) 2 + T + 2 I q T T-~G r, dte
184
~0Ro=0
reads
Due to the constraints in J~, J~ can only be of a vex3, restricted form. This fixes at, b~, c, and d, in terms of 2 and the "source" fields re, q,+, q~- and h,. This in turn fixes the flows for all currents in J~. Explicitly, they are
dJ -dte
and expanding J, = ,~~~-=o v ' ~e-ko,,k, the solution to (48) in terms of the initial condition
~oRj=~2Rj_2,
while Je is parametrized like
Z
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=
- - - ~ T3G - - 3+ Tt ' G
G +lit
+
+~G±2G
+
,
(66)
Volume 274, number 2 dT
dt~ =[¼T"-~T2+3(G+'G--G+G-')]"
PHYSICS LETTERS B
(67)
which as advertised are in complete accordance with eqs. ( 2 4 ) - ( 3 1 ) found in the bilinear formulation with u = T, q/= G + a n d ~ = G - .
6. Discussion and conclusion In the preceding sections, Hirota polynomials of A~ ~> in the 772twisted realization are constructed. The polynomials give rise to n o n - l i n e a r differential equations for r functions in a bilinear form. The r functions are related explicitly to the currents of the algebra W~ 2) , It explicitly verifies the conjectured relation between the intermediate realization associated with the m i n i m a l sl 2 subalgebra and the corresponding reduction of the s13 current algebra. A general proof of the relation for arbitrary reductions and realizations is still lacking. Though the second h a m i l t o n i a n structure of the hierarchy is identical to the one used in ref. [ 13 ], the resulting hierarchy is different. This is clearly due to the different first h a m i l t o n i a n structure. In fact, as explained in ref. [ 14 ] the resulting fractional KdV hierarchy is related to the conjugacy class of the Weyl group that one usually associates with the principal realization. The modified hierarchy in ref. [14] should be related to the one we have found by a Miura transformation. The zero curvature formulation is an entirely classical formulation in the sense that the currents satisfy Poisson algebras a n d are functions, c o m m u t i n g with respect to ordinary multiplication. It is still an open problem how to quantize these algebras in general, i.e. how to replace the Poisson algebras by c o m m u tator algebras. It is interesting to compare this with the bilinear formulation, where one is dealing with a q u a n t u m algebra, be it at fixed level. It is still unclear if this relation can be used to construct unitary representations of the q u a n t u m version of the algebra W~2)
9 January 1992
along the lines of ref. [ 15 ]. This is u n d e r current investigation.
Acknowledgement The author wishes to thank Sander Bais and Fons ten Kroode for fruitful discussions, Koos de Vos for guidance through the bilinear literature, and the "Stichting voor F u n d a m e n t e e l Onderzoek der Materie ( F O M ) " for financial support.
References [ ! ] M. Douglas, Phys. Lett. B 238 (1990) 176. [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Proc. Jpn. Acad. A 57 ( 1981 ) 342; J. Phys. Soc. Jpn. 50 ( 1981 ) 3806; Physica D 4 (1982) 343; Publ. RIMS 18 (1982) 1111; J. Phys. Soc. Jpn. 50 (1981) 3813; Publ. RIMS 18 (1982) 1077. [3]V.G. Kac, Infinite dimensional Lie algebras, 2nd Ed. (Cambridge Univ. Press, Cambridge, 1985). [4] M.J. Bergvelt and A.P.E. ten Kroode, J. Math. Phys. 29 (1988) 1308. [5] I. Frenkel and V.G. Kac, Invent. Math. 62 (1980) 23; G. Segal, Commun. Math. Phys. 80 (1981) 301. [6] V.G. Kac, Funct. Anal. Appl. 3 (1969) 252. [7] V.G. Kac and M. Wakimoto, Proc. Symp. Pure Math. 49 (1989) Part 1. [8] V. Drinfeld and V. Sokolov, J. Sov. Math. 30 (1984) 1975. [9] F.A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B 357 ( 1991 ) 632. [ 10] P. Bouwknegt, J. Math. Phys. 30 (1988) 571. [ 11 ] M. Bersbadsky, Conformal field theories via hamiltonian reduction, Princeton preprint IASSNS-HEP-90/44. [ 12 ] G. Wilson,Compt. Rend. Aead. Sci. Paris 1299 ( 1985) 587. [13] I. Bakas and D. Depireux, A fractional KdV hierarchy, Maryland preprint UMD-PP91-168. [ 14] M. de Groot, T. Hollowoodand J. Miramontes,Generalized Drinfeld-Sokolov hierarchies, Princeton preprint IASSNSHEP-91/19. [ 15 ] F.A. Baisand K. de Vos, The algebraicstructureofintegrable hierarchies in bilinear form, in: Proc. Trieste Conf. on Recent developments in conformal field theory (October 1989).
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PHYSICS LETTERS B
On a new hierarchy associated with the Peter v a n D r i e l
W (2)
conformal algebra
1
Institute of Theoretical Physics, Valckenierstraat 65, NL- I O18 XE Amsterdam, The Netherlands
Received 17 August 1991
In a realization of the Kac-Moody algebra A~~) intermediate between the homogeneous and the principal realization, an integrable hierarchy of differential equations is constructed. The hierarchy shares features of both the AKNS and the KdV hierarchies. The same hierarchy can also be constructed using the Drinfeld-Sokolov approach in terms of zero curvature conditions, with as hamiltonian structure the Wt 1) algebra of Polyakov and Bershadsky. It provides evidence for the conjecture that there exists a general relation between hierarchies constructed in some intermediate realization and the covariantly coupled chiral algebras of Bais, Tjin and van Driel.
1. Introduction It becomes m o r e and m o r e clear that the interest in physics o f integrable hierarchies o f non-linear differential equations is not at all restricted to the description o f solitary waves in shallow canals. Nowadays, a new impetus to study integrable hierarchies stems from the observation that n o n - p e r t u r b a t i v e inform a t i o n about two-dimensional gravity might be extracted from them. In particular the p a r t i t i o n function o f 2D gravity represented by the one matrix model after the double scaling limit is known to be a z function o f the K d V hierarchy [ 1 ], The original construction o f r functions o f the KdV hierarchy is through the so-called bilinear construction [2 ]. The key point there is that z corresponds to the group orbit of the highest weight vector o f an A[ ~) integrable level one representation. The square o f the r function is in a well d e t e r m i n e d s u b m o d u l e o f the tensor product o f two such representations, such that the orbit is fixed as the o r t h o p l e m e n t o f vectors not in that submodule. Since level one representations ofA~ ~) admit a polynomial realization, this leads to differential equations bilinear in z. F o r AI ~ there are two essentially inequivalent p o l y n o m i a l realizations d e p e n d i n g on whether we take the twisted E-mail address: [email protected].
or the untwisted vertex realization. The twisted realization o f A~ ~) results in the K d V hierarchy [3], whereas the A K N S hierarchy comes out if the untwisted realization is used [ 4 ]. In general, a vertex realization exists for all simply laced K a c - M o o d y algebras at level one [ 5 ]. The n u m b e r o f inequivalent realizations grows with the n u m b e r o f conjugacy classes o f the Weyl group of the underlying finite algebra [6 ]. A unified description of the resulting r functions is however still lacking. In particular for A~N~), only the r functions in the untwisted ( h o m o g e n e o u s ) realization and in the 77N twisted ( p r i n c i p a l ) realization are studied [7], but so far the intermediate cases have not been considered. At the same time, it is well known that the KdVtype hierarchies are hamiltonian, in the sense that the time flows are generated by (mutually c o m m u t i n g ) hamiltonians. There is no direct way to derive this structure from the bilinear construction. There exists a formulation in which it is straightforward and natural to derive the ( b i ) H a m i l t o n structure and the hamiltonians, called the zero curvature formulation. In this formulation the time evolution o f an integrable system is thought o f as a zero curvature condition associated with some gauge group. The gauge group o f the A K N S hierarchy is the current algebra o f s12. By constraining this algebra by setting J _ ( x ) to a constant, one obtains the K d V hi-
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V, All rights reserved.
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erarchy with as gauge group the Virasoro algebra. The general setting of this reduction scheme was provided by ref. [ 8 ] and it related WN algebras with generalized KdV hierarchy, which in the bilinear picture are related to the principal realization. Recently, in ref. [ 91 the problem of finding constrained algebras starting from the current algebra group ~1,~was addressed. It was shown that to every inequivalent slZ subalgebra of ~1,~there is naturally associated a set of constraints that give rise to a conformal algebra. Since there exists a one-to-one relation of inequivalent sl, subalgebras of sl, and conjugacy classes of the Weyl group of sl, [ lo], it is natural to conjecture that the algebras of ref. [ 93 are in fact a hamiltonian structure of the hierarchies in the corresponding intermediate realization in the bilinear formulation. This relation is the subject of the present paper. To be precise, the bilinear formulation will be employed to construct the hierarchy associated with the ZZ twisted realization of Ai” which exists besides the principal and homogeneous ones. The result is a new hierarchy, with three fields w, q and u that combine features of the AKNS and the KdV hierarchies. The second hamiltonian structure coincides with the Wj2) algebra of Polyakov and Bershadsky [ 11 1, which is indeed related to the non-maximal sl, subalgebra in sl, [ 9 1.
LETTERS
B
pzm=
s
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[(h,),,-(&),,,I
P2 ,n+l =(es)m+Cf3),,7+,
,
=
y&j
(3) m>O,
-_I,, I,
l(e3)
-_),I -I
satisfying the commutation IPI,
(2)
,
42m=&5~(L,-(h2)
92m+I
1992
+ti)-,,,l,
(4)
(5)
relations
411=&,k.
(61
This realization is induced by the Weyl reflection in the root -_(y, --a2, and is related to embedding su2 X u, c su3, when we twist the Heisenberg algebra associated with the su2 subalgebra. The su2 is oriented in the o3 direction, the root associated with e3. As is well known, (6) admits a polynomial irreducible representation (V, n), V=C[x,;
iEz+]
,
n(p,)=
$)
x(9,)
i
=4
(7)
The basic module L(&) is not irreducible as an smodule since the spectrum of p,, is mfi, mEB, and go is not in the Heisenberg algebra. However, if we define q,, by (6), set 7c(qO) =x,, and realize the translation operator T that generates the p0 spectrum through T= efl-“, then (8)
2. The Z2 twisted realization of L(&) over A$‘) First we will discuss the Z, twisted realization of the basic representation L (iiO) over Ai1 ). Let g be the finite dimensional simple Lie algebra sl( 3, C) with generators
We may in fact gather the Heisenberg generators in formal expansions, such that we obtain the familiar vertex realization of A$’ ) with two bosons who have a Fourier expansion
Zh.Y2k-
Q= [er, Q], f;= (P,)’ and h,= [e,, f;]. Let g=Ai’) be the associated affine Kac-Moody algebra, g=I ,,,tL g@,I’“OCkO@d, where k is the central charge and d the derivation. The algebra g contains an intinite-dimensional Heisenberg subalgebra s generated by the central charge k and 180
&r“p2h
1 - ~ L-i-‘/2pzA+, 2k+ 1
>I )
>I ,
where we have chosen o+ (z) to be the untwisted
(9)
(10) bo-
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son (¢ + ( z e 2m) = ¢ + ( Z ) ) and ¢2 (z) to the Z2 twisted boson (¢_ (z e 2m ) = - - ¢ _ (2")). The bosons have the following OPE exhibiting the boundary condition
¢+ (z)O+ ( z ' ) ~ l o g ( l - Z,) ,
(11)
O_ ( . ) ¢ _ ( z ) ~ 1 o g ( . ~ + V~7) .
(12)
Expressions for the other current algebra generators can now in principle be found using the expressions for the vertex operators e , ( z ) = :e("~): = ~ (e,)kz - ~ - I ,
(13)
k
where the : : refer to the normal ordering of the modes of 0.+_, and the inner product ( ' ) is fixed by defining the direction of the twisted boson (for which we took a3). The gradation d is set equal to Lo, the scaling generator of the Sugawara tensor. In particular we find ~1
d" (Xm®e nxf~x°) = ( ½m+ 3n2) (x,,, ® e "'/5~°) .
(14)
Finally, the polynomial space admits a natural and unique hermitian form ( , ) given by
( P(x) e m'fSx°, Q(x) e "',/s~° ) = P ( D ~ ) ( ~ ( x ) ]x=o&..,
(15)
where ( 0 1 0 1 0 ) Dx= 0xl'20x2'30x3'"'
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r ( x ) e LAo ~ r(X (l)) ®r(X (2)) ~ L(2Ao) c L(Ao)®L(Ao) •
If we denote the orthogonal complement of L(2Ao) with respect to the hermitian form by L (2Ao) •, then for all elements PsL(2Ao) •, (P, r ® r ) =0. It is not difficult to see that L (2Ao) • contains arbitrary polynomials in the sum variables x=x (l~ + x (2). From the inner product (15) it then follows that the condition for r ( x ~ l ~ ) N r ( x ~2~) turns into a differential equation bilinear in r ( x ) . In this way one may construct the r functions as the subset of symmetric elements in L (2Ao), but we are only interested in the polynomials that define the differential equations, the so-called Hirota polynomials. These polynomials depend only on the difference variables y=xC1)--X (2) [3]. From the symmetry properties of the bilinear equation finally one obtains that only the polynomials in the symmetric tensor product of L(Ao) can give rise to non-trivial differential equations. The number of Hirota polynomials ZHir can thus be determined from the decomposition of L(Ao)®sL(Ao) with respect to the 7/2 twisted Heisenberg algebra, which can be determined by first calculating the branching A~ l ) ~ Al I ) × U ( 1 ) and then twisting the AI l) The result of this rather lengthy calculation is )~Hir~---q3/2 -t- q7/4 q_2q2 +
+2qll/4+5q3+ (16) "
3. The bilinear formulation In this section a few of the Hirota bilinear equations are constructed. The basic input is that the z functions are elements of the orbit O.o of the highest weight vector VAo, and
q9/4 + 3qS/2
....
(18)
We are now set to calculate the Hirota polynomials of lowest degree. Apparently, the first one occurs at degree 3/2. At this degree there are two independent polynomials ~2 in y that satisfy the symmetry property: YzVland cosh (x/3 Yo). Acting with (e2) _ ~(el) _ 1 on the vacuum of L(Ao)® L (Ao) gives the polynomial .~f3y2yl + c o s h ( ~ 3 y o ) which is an element of L(2Ao). Constructing the orthoplement this uniquely fixes the one polynomial in L(2Ao) • at this particular degree, (Hit)3/2 =
~ Note that the structure of (8) is almost identical to the basic representation L(Ao) over A} ~) in the untwisted (homogeneous) realization. The only difference is the zero mode spectrum, which in our notation would be 2n, neZ. Therefore the string functions start as weight n 2.
(17)
-
l ~ y 2 f l I +cosh(x/3
yo) .
(19)
Proceeding in this way for the next few degrees we found ~2 This includes exponentsin the variables Xo and Yo.
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(Hir)7/4 = ~x/3y2 exp( ~x/3 x0) sinh( ½x/3 yo) ~Yi exP(½x/3 Xo) cosh(½x/3 Yo) ,
(20)
(Hir)2,, =Y3Y,-~Y'~-½Y, sinh(,,/3 yo),
(21)
1 sinh(xf~yo) (Hir)2,2 =1jY22 -- gYt
(22)
(Hir)9/4 = - ~Xf3 y, y2 exP(½x/3Xo) cosh(½xf3 yo) - 2 ( y 3 - ~ 2 y 3) exp(½x/3 Xo ) sinh(½xfl3 yo). (23) Let x=xj, flow
t2 =2Xf3X2,
then (19), (20) define the t2
d~' = - ~ ' l ,, + ~u~,, J
(24)
d~ dr2
(25)
dl2
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neric and in fact quite unpleasant feature of the bilinear construction. As we already see from (18), the number of Hirota polynomials grows rapidly with the degree. Nevertheless, it is always possible to eliminate all equations but one equation using the equations of lower degree. Of course, this indicates a deeper structure, which, in the bilinear formulation, is still unravelled. The other thing that points in the direction of a deeper structure is the magic substitution (27), (28). It turns out that both issues have a natural solution when they are stated in the zero curvature formulation, which will be discussed in the next section.
4. The zero curvature formulation
du
dt2
!,,,,_ !2 ,~," F, --2"F
-
,
(26)
(¢~V)',
where u= - 2 log"r .... ~'m-- I
qJ=- --, l"m
(27) ~'m+ 1
¢=--
Tm
(28)
The first two equations (24), (25) are Schr6dinger equations, which are coupled non-linearly through the third equation. We do not know whether this particular Schr6dinger equation has any physical relevance. From ( 21 ), (23 ) the equations defining the 13flow are constructed ( t 3= X3) , d~' = ~u'"- 3 u ~ ' - 43u ' ~ + 3 ~,20, dt3
( 29 )
__de =¢,,,_ ~u¢_ ]u,¢_ 3i~u~_2 ,
(30)
d/3 du _ [¼u'- ~3 u -3 - ~-(~,'¢/- ~u¢') l t dr3
Vvac(/)=exp(
k~o~>~Pk& )
,
(31)
(32)
where now the tk are genuine complex variables. Vv,c(t) is a vertex operator since Vv,c(t)r(x)= r(x+t). Therefore, we obtain the function r(t) by calculation of the "vacuum expectation value" of r(x+t), making use of the inner product (15), i.e.
~(t) = ( 1, T-~Vv,c(t)h(x) . 1 ) ,
Notice that by putting c/and ¢7to zero we get the KdV equation from (31 ). In that case we essentially forget about the untwisted U( 1 ) and reduce to the 22 twisted realization of A} ~ The equation resulting from (23) is trivially satisfied using eqs. ( 2 4 ) - ( 2 6 ) and is therefore redundant. This redundancy of Hirota polynomials is age182
The way the hierarchy of equations is reformulated in terms of zero curvature relations is generally referred to as Wilson's dressing method [ 12 ]. The zero curvature relations arise when differentiating the r function directly. Since the group orbit is a non-linear space such an operation should be treated a bit carefuly. Therefore we first replace the r ( x ) (an element of the representation space L(Ao) ) by a genuine differentiable function z (t). This comes about as follows. Following Wilson [12], we introduce the vertex operator
(33)
where h(x) is the (non-unique) group element that produced the r function. The t dependence of the group element V~=T-IVvac(t)h(x) is now quite simple:
dV~ dr,
=p,V~.
(34)
Any arbitrary group element h in the Kac-Moody group Sin (C [2, 2-1 ] ) admits a factorization
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PHYSICS LETTERS B
Volume 274, number 2
h=h_ hoh+ ,
(35)
where h_ = 1 + 2 - IMI + 2 - 2 M 2 + .... h+ = 1 +2N~ +22N2 + ...,
(36)
where the N~, Ms are arbitrary elements of sln, and
ho =N_ wHN+ ,
(37)
where N_ ( N + ) is the exponentiation of a strictly lower (upper) matrix, w is a Weyl group element and H is the exponentiation of an element of the Cartan subalgebra. Using the Birkhoff decomposition
In (42) one can recognize a gauge equivalent form of the constrained S13 current algebra giving rise to the W~ 2) algebra as defined in ref. [ 11 ]. The U ( 1 ) current of the W~ 2) algebra is missing though. It should not be confused with the U ( 1 ) current v, as v can be gauged away. This fixes the gauge freedom in the definition of h in (33 ), transforming T ' to
T=T'+v2-v ' .
(43)
The fields in (42) can actually be related directly to the r function. For instance, by projecting (42) on the h3 subspace one finds V-----½(/].-lh3, )[-1 ( ~ / _ ) - l p l ~//_ ) = (2-1e3, g/_ ) = l o g ' z t
I7,= (Vl) _ ( Vt)o,+
while for T ' one gets
a differential equation for (Vt)o,+ can be derived, which reads
T ' = ( 2 - 'f3, 2 - ' (~d_)- 'pl q/_ )
(~--~-[(V,)=~P,(V,)_]+)(V,)o.+ =0
Di (V,)o.+ i = 1 , 2 .....
(38)
As the r function (33) is merely the vacuum expectation value of Vo.+, (38) gives rise to a hierarchy of differential equations for ft. Finally, compatibility of the equations leads to the zero curvature conditions [D,, Dj] = 0 .
(39)
To be able to appreciate (38) we consider the simplest equation that occurs for tl = x . Explicitly, the covariant derivative in (38) then takes the form
D~=[Ox-(Vt)zl pI(V,)_1+ = a x - (2f3 +e3 + [ (M,) l,f31 ) .
(40)
The matrix (Mj) ~ has arbitrary form, and for convenience is parametrized like
__--_(,~- lh3, qfl_ > = --log"r-- (1og'z) 2 .
(44)
(45)
For both calculations we only used the contravariance of the inner product and the explicit polynomial representation (3). Combined with (43), (45) and (44) yield T = - 2 log"r,
(46)
The calculation of G -+ makes explicit reference to the translation operator Twhen we use T - ~q/= V t+ ~such that G + = <2-~fl, 2 -~ ( q / _ ) - ' p , g / > = _ <,~- le2 ' eft/_ ) =-(T-1
,q/)=-
r~_~ r~
(47)
and likewise G - = rt+ ~/rt. Apparently, the substitutions ( 27 ), (28 ) are motivated by the underlying zero curvature structure.
5. Solution of the zero curvature conditions
-2a
G+
,
d
a+½r'
(41)
where we have dropped the indices l for notational convenience. This brings (40) to the form ( Dx=0x-
v G+ \T'+2
0 0 G-
_' ) O . v
(42)
In this section the starting point are the zero curvature conditions for (38). To be precise, we will focus on the curvatures involving the x derivative, i.e. the equation
dJx =[Ji, Jx]+J'i, dt,
i = 0 , 1.....
(48)
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where ,Ix is a n sl 3 valued matrix that is now constrained to the full W ~ 2 ) form [ 11 ], i.e. including the U( 1 ) current j,
J~=
G+ T+2
-2j G-
,
(49)
=\
be c,
-2r, de
.
(50)
re-at/
dj _½q+ G+_½q;_ G_ +~r,' dt,
(51)
+ I 3 G ± O+G+'+3jG±lhe,
(52)
dT dt - I -½ ~ 3 + 2 T ~ + T ' + 2 2 01te +I3G + O+~G+'-3jG+Iq? +I~G- O+½G-'+3jG-Iqi-.
(53)
These equations determine the Poisson structures [9]. Besides the W (32) algebra which is the second hamiltonian structure of this hierarchy, we now also
find the first Hamilton structure of the hierarchy {G+(x), G - ( y ) } ( , ) = - 6 ( x - y ) ,
(54)
{T(x), T ( y ) } ~ , ) = - 2 6 ' ( x - y ) .
(55)
All other brackets vanish. Clearly, the current j is a central element of the algebra, which fixes the current j to be constant under all time flows. For notational convenience and without loss of generality it is set to zero. Writing (48) like
and recursion relation j = l .... , i ,
(58)
dJ - ~ 2 R , = CJ:( ~ff t .@2)R,_2 . d6
(59)
Finally, using the bi-Hamilton structure, the hamiltonians can be found explicitly from the defining equation dF - { H i + , , F}(2)= {H,+3, F}(l) , dr,
(60)
where F is any functional of the currents and Hi has scaling dimension i. Using ( 5 1 ) - ( 5 3 ) , the initial condition (57) reads
(56)
(61)
This has two non-trivial solutions, to= 1 and ho= 1. Since the initial condition will fix the scaling behaviour of the currents rood 2 it will give rise to two essentially different and mutually commuting hierarchies. Straightforward application of the algebraic algorithm just described yields the first two (trivial) sets of equations
dG±_-7½G±, dto dG * = G +', dtl
d~T=0, dto
(62)
dT -T'. dtl
--
(63)
More interesting are however dG + - T - ~ 1G dt2
+
tt
+_½TG~,
(64)
dT -- = (G+G-) ' dt2
(65)
and dG + dt3
_ _
2~2J,+2+~,oJ,
(57)
h ; = q ~ =qff = 0 .
dG + 1 + _ + _ I - ( O + _ 3 j ) 2 + T + 2 I q T T-~G r, dte
184
~0Ro=0
reads
Due to the constraints in J~, J~ can only be of a vex3, restricted form. This fixes at, b~, c, and d, in terms of 2 and the "source" fields re, q,+, q~- and h,. This in turn fixes the flows for all currents in J~. Explicitly, they are
dJ -dte
and expanding J, = ,~~~-=o v ' ~e-ko,,k, the solution to (48) in terms of the initial condition
~oRj=~2Rj_2,
while Je is parametrized like
Z
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=
- - - ~ T3G - - 3+ Tt ' G
G +lit
+
+~G±2G
+
,
(66)
Volume 274, number 2 dT
dt~ =[¼T"-~T2+3(G+'G--G+G-')]"
PHYSICS LETTERS B
(67)
which as advertised are in complete accordance with eqs. ( 2 4 ) - ( 3 1 ) found in the bilinear formulation with u = T, q/= G + a n d ~ = G - .
6. Discussion and conclusion In the preceding sections, Hirota polynomials of A~ ~> in the 772twisted realization are constructed. The polynomials give rise to n o n - l i n e a r differential equations for r functions in a bilinear form. The r functions are related explicitly to the currents of the algebra W~ 2) , It explicitly verifies the conjectured relation between the intermediate realization associated with the m i n i m a l sl 2 subalgebra and the corresponding reduction of the s13 current algebra. A general proof of the relation for arbitrary reductions and realizations is still lacking. Though the second h a m i l t o n i a n structure of the hierarchy is identical to the one used in ref. [ 13 ], the resulting hierarchy is different. This is clearly due to the different first h a m i l t o n i a n structure. In fact, as explained in ref. [ 14 ] the resulting fractional KdV hierarchy is related to the conjugacy class of the Weyl group that one usually associates with the principal realization. The modified hierarchy in ref. [14] should be related to the one we have found by a Miura transformation. The zero curvature formulation is an entirely classical formulation in the sense that the currents satisfy Poisson algebras a n d are functions, c o m m u t i n g with respect to ordinary multiplication. It is still an open problem how to quantize these algebras in general, i.e. how to replace the Poisson algebras by c o m m u tator algebras. It is interesting to compare this with the bilinear formulation, where one is dealing with a q u a n t u m algebra, be it at fixed level. It is still unclear if this relation can be used to construct unitary representations of the q u a n t u m version of the algebra W~2)
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along the lines of ref. [ 15 ]. This is u n d e r current investigation.
Acknowledgement The author wishes to thank Sander Bais and Fons ten Kroode for fruitful discussions, Koos de Vos for guidance through the bilinear literature, and the "Stichting voor F u n d a m e n t e e l Onderzoek der Materie ( F O M ) " for financial support.
References [ ! ] M. Douglas, Phys. Lett. B 238 (1990) 176. [2] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Proc. Jpn. Acad. A 57 ( 1981 ) 342; J. Phys. Soc. Jpn. 50 ( 1981 ) 3806; Physica D 4 (1982) 343; Publ. RIMS 18 (1982) 1111; J. Phys. Soc. Jpn. 50 (1981) 3813; Publ. RIMS 18 (1982) 1077. [3]V.G. Kac, Infinite dimensional Lie algebras, 2nd Ed. (Cambridge Univ. Press, Cambridge, 1985). [4] M.J. Bergvelt and A.P.E. ten Kroode, J. Math. Phys. 29 (1988) 1308. [5] I. Frenkel and V.G. Kac, Invent. Math. 62 (1980) 23; G. Segal, Commun. Math. Phys. 80 (1981) 301. [6] V.G. Kac, Funct. Anal. Appl. 3 (1969) 252. [7] V.G. Kac and M. Wakimoto, Proc. Symp. Pure Math. 49 (1989) Part 1. [8] V. Drinfeld and V. Sokolov, J. Sov. Math. 30 (1984) 1975. [9] F.A. Bais, T. Tjin and P. van Driel, Covariantly coupled chiral algebras, Nucl. Phys. B 357 ( 1991 ) 632. [ 10] P. Bouwknegt, J. Math. Phys. 30 (1988) 571. [ 11 ] M. Bersbadsky, Conformal field theories via hamiltonian reduction, Princeton preprint IASSNS-HEP-90/44. [ 12 ] G. Wilson,Compt. Rend. Aead. Sci. Paris 1299 ( 1985) 587. [13] I. Bakas and D. Depireux, A fractional KdV hierarchy, Maryland preprint UMD-PP91-168. [ 14] M. de Groot, T. Hollowoodand J. Miramontes,Generalized Drinfeld-Sokolov hierarchies, Princeton preprint IASSNSHEP-91/19. [ 15 ] F.A. Baisand K. de Vos, The algebraicstructureofintegrable hierarchies in bilinear form, in: Proc. Trieste Conf. on Recent developments in conformal field theory (October 1989).
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