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Minimal models from W-constrained hierarchies via the Kontsevich-Miwa transform
Physics Letters B 288 (1992) 38-46 North-HoUand
PHYSICS LETTERS B
Minimal models from W-constrained hierarchies via the Kontsevich-Miwa transform B. G a t o - R i v e r a CERN, CH-1211 Geneva 23, Switzerland and lnstituto de Fisica Fundamental, Serrano 123, E-28006 Madrid, Spain and A.M. Semikhatov Theory Division, P.N. Lebedev Physics Institute, Leninsky prospect 53, 117 924 Moscow, Russian Federation Received 18 May 1992
A direct relation between the conformal formalism for 2D quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used to establish an identification between W constraints on the KP tau function and decoupling equations corresponding to Virasoro null vectors. The Kontsevich-Miwa transform maps the Wt°-eonstrained KP hierarchy to the (p', p) minimal model, with the tau function being given by the correlator of a product of (dressed) (l, 1) [or ( 1, l) ] operators, provided the Miwa parameter ni and the free parameter (an abstract bc spin) present in the constraints are expressed through the ratio p ' / p and the level/.
1. Introduction M a t r i x models, w h i c h a p p e a r e d as a " d i s c r e t i z e d " a p p r o a c h to 2 D q u a n t u m gravity [ 1-3 ], h a v e resulted at the c o n t i n u u m level in integrable h i e r a r c h i e s s u b j e c t e d to V i r a s o r o [4,5 ] a n d possibly higher W [5,6 ] constraints. H o w e v e r , their r e l a t i o n w i t h the L i o u v i l l e theory, at least w i t h that d e s c r i b e d in the f o r m a l i s m o f refs. [ 7,8 ], seems, a l t h o u g h h a r d l y d i s p u t a b l e in principle, s o m e w h a t obscure. In a recent p a p e r [ 9 ] the K o n t s e v i c h M i w a t r a n s f o r m was used to establish a r e l a t i o n b e t w e e n the V i r a s o r o c o n s t r a i n t s i m p o s e d on the tau f u n c t i o n s o f the K P hierarchy, a n d the d e c o u p l i n g e q u a t i o n c o r r e s p o n d i n g to the null v e c t o r at level two in m i n i m a l c o n f o r m a l field t h e o r i e s e x t e n d e d by a scalar c u r r e n t ~. M o r e o v e r , the field c o n t e n t o f the D a v i d - D i s t l e r - K a w a i f o r m a l i s m for 2 D q u a n t u m gravity was r e c o v e r e d in this way, w i t h the extra scalar playing a role s i m i l a r to that o f the L i o u v i l l e field in the f o r m a l i s m o f r e f s . [ 7,8 ]. T h e essence o f the m e t h o d based on the K o n t s e v i c h - M i w a t r a n s f o r m was t h e r e f o r e to relate the V i r a s o r o - c o n s t r a i n e d K P h i e r a r c h y a n d 2 D q u a n t u m gravity, bypassing the m a t r i x m o d e l technology. N o w the q u e s t i o n arises as to w h e t h e r the h i g h e r W constraints on the K P h i e r a r c h y are a m e n a b l e to the a b o v e scheme, a n d w h e t h e r a bridge can be established b e t w e e n m a t t e r i n t e r a c t i n g w i t h the c o n t i n u u m D D K q u a n t u m gravity a n d the W - c o n s t r a i n e d K P hierarchies. In this letter we give a p o s i t i v e a n s w e r to this q u e s t i o n , b a s e d on the results o b t a i n e d for the W ~3) c o n s t r a i n t s in a d d i t i o n to the p r e v i o u s results for the W ~2) ( V i r a s o r o ) case. ~ The effect of the Kontsevich-Miwa transformation was to replace the time parameters of the hierarchy with the spectral parameter, or, in different words, to create a world-sheet out of the time parameters. 38
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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By analogy with the level-two Virasoro case, one could expect that under the Kontsevich-Miwa parametrization of the KP times, the higher (l>~ 3) W constraints I/V~l) z = 0 , n >~ - l + 1, would lead to higher-level (l~> 3) null vector decoupling equations ~2. This argument, however, is fraught with an unpleasant "paradox": the algebra of the W (t) constraints contains the lower W (r), l' < l as well, which apparently should then give rise, by the same mechanism, to decoupling equations at the lower levels, thereby leading to an over-determined system. Fortunately, this does not happen; it turns out that a single level-I decoupling equation corresponds to the complete set of W (t) constraints (including the Virasoro ones) W ~ c ) z = O , n >t - l ' + 1, 2 <~l'<~ l, while these do not lead back to lower-level decoupling equations. More precisely, the Kontsevich-Miwa transform maps the W (1)constrained KP hierarchy to the (p', p) minimal model, with the tau function being given by the correlator of a product o f (dressed) (l, 1 ) [or ( 1, l) ] operators, provided (i) the Miwa parameter n, satisfies r
nZ=(l--1)2~p, and (ii) the free parameter (an abstract bc spin) J present in the constraints is determined from 1- 1 2n~ 2 J - 1= - n~ - l - 1 ' Although we have obtained the complete results only for the W (3) case (in addition to W (2) =Virasoro), it is hard to believe that so simple a result, the W-decoupling correspondence, following after rather tedious calculations, could be a mere coincidence and would not allow generalizations to W (>~4) algebras. We would like to stress that, rather than taking the decoupling equation in its naive form, one should first "dress" the theory by tensoring it with an extra U ( 1 ) current. The extra degrees of freedom which thus appear are then killed by projecting onto a special form of the null state. Therefore, although the null vector in the matter sector is the same, the specific form of the equation for correlation functions changes. We begin in the next section by introducing the KP hierarchy and the notion of the Kontsevich-Miwa transform. Then, in section 3, we introduce the dressed decoupling equation. For purely technical reasons, we describe in this paper the coincidence between W constraints and decoupling equations proceeding in the direction .from the decoupling equations to the W constraints: we show in section 4 how to "un-Kontsevich" conformal field-theoretic data into the W (3) constraints. It will be clear from the derivation that (and how) one can reverse the argument and thus establish equivalence of the two sets of data, the W(3)-constrained KP hierarchy and (appropriately dressed) minimal models.
2. The KP hierarchy and the Kontsevich-Miwa transform We start with the KP hierarchy [ 12 ]. It can be described either in an evolutionary form, as an infinite set of equations on (coefficients o f ) a pseudodifferential operator, or as (Hirota) bilinear relations on the tau function. The tau-functional description may be considered as having the more direct relevance to "physics", being related to the partition function, while the evolutionary form has all the usual advantages due to the introduction of a spectral parameter and the associated wavefunction. The wavefunction and the adjoint wavefunction depend on the spectral parameter z via exp [ + ~ (t, z) ] r (t ~ [ z - l ] ) / r (t), where ~(t, z) = ~ r>~l trZ r and t + [z -~] = (tl -+z -1, t2-+ ½z -2, t3-+ ½z -3 .... ) .
(2.1)
Here t = ( x = tb t2, t3, ...) are the time parameters of the hierarchy. This form of the z dependence is "generalized" by the following presentation for the tr, known as the Miwa parametrization [ 13,14 ] : ,2 A relation between Virasoro null vectors and W algebras has been noted in a different approach in ref. [ 10] and, more recently, in ref. [11]. 39
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tr= 1 ~,, njzfr, r j
PHYSICS LETTERS B
r>~ 1 ,
20 August 1992
(2.2)
where {zj) is a set of points on the complex plane and the Miwa parameters n: are integer classically; we will need, however, to continue offthe integer values. As we will see, the Miwa parameters acquire the role of U( 1 ) charges with respect to the scalar field. By the Kontsevich-Miwa transform we will understand the dependence, via eq. (2.2), Oftr on the z:. However, it is important that the presentation of the form (2.2) be considered for arbitrary nj (while, on the other hand, Kontsevich has originally used a parametrization of this type for all the nj equal [ 15 ], see also refs. [ 16-19 ]. As was noted in section 1, an important result of the development of matrix models was the discovery of an "integrable" counterpart of the 2D gravity + matter theories, in the form of Virasoro-constrained hierarchies. Moreover, the fact that appropriately constrained hierarchies provide a description of 2D gravity coupled to matter, may be promoted to a first-principle, in which case it is natural to consider on an equal footing various constraints more general than the Virasoro ones, including those whose derivation from a specific matrix model is not known. This applies, first of all, to the extension of Virasoro constraints to W ones, and also to the introduction of a parameter into the Virasoro generators acting on the tau function [20]. This parameter J (which would have parametrized the central charge as 2 (6J 2 - 6 J + 1 ), had the L,,~ _ 2 generators been present as well), can be thought of as the "spin" (dimension) of an abstract bc system underlying the Virasoro generators, ~n~z Lnz-n-2 ~ ( 1 - J ) Obc-Jb Oc. We repeat that although J-dependent Virasoro constraints may not have been derived from a matrix model, for us all of them are equally good starting points. The corresponding constraints,
Lnz=O,
n>_.-1,
(2.3)
p-1 02 Lp>o = ½,=Z Otp_,Ots + Lo=s~lsts~s,
,>~ZISts ~
0
0 + ( J - ½) (p-t- 1) Otp '
L_I=s>ZI (s+l)ts+,--,Ots
(2.4)
were shown in ref. [ 9 ] to take, after the Kontsevich-Miwa transform, the form of a decoupling equation corresponding to the level-two null vector in the (p', p) minimal model determined by p'/p=2n~, provided the Miwa parameter ni was determined from
1/ni-2ni=2J- 1-Q
(Virasoro) .
(2.5)
Our task in this paper is to extend this correspondence to include W generators. Let us start therefore with the W (3) constraints imposed on the tau function in a way similar to (2.4); we will write out the constraints explicitly later, see (4.7). Now, by virtue of the Kontsevich-Miwa transform (2.2), the tau function becomes a function r{zj} of the zj. We assume for it the ansatz ~3 r{zj}= lim < ~'(z,)...~t'(z~)) ,
(2.6)
with < > and kUbeing, respectively, the chiral correlation function and a (primary) field operator in a conformal field theory on the z plane. The operator and the theory itself should be specified according to the fact that the tau function satisfies the W (3) constraints. The ansatz will be justified by the fact that, as we are going to show, imposing these constraints amounts to the condition that ~ be a "31" operator in a (p', p) minimal model [21,22] for p'/p determined by the precise form of the constraints. That is, we will find the relation p'/p = ½n2, where n, is the Miwa parameter which in its own turn is related to a "spin" J, read off from the Virasoro part of the constraints, by ~3 Clearly, there have to be infinitely many points z~ in the Kontsevich-Miwa transform in order for the times tr to be independent.
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(W (3)).
(2.7)
As was noted above, we find it more convenient to present in this paper the derivation of the correspondence W constraints ~ decoupling equations in the inverse direction: we start from 7-, being the " 3 1 " operator and then show how to "un-Kontsevich" it into the W ° ) constraints. We therefore proceed in the next section with introducing the necessary conformal field-theoretic ingredients.
3. " D r e s s i n g " the decoupling equation
Now we introduce the second ingredient: null-vector decoupling equations in minimal models o f conformal field theory. To the KP hierarchy we will return at the very end. In addition to the e n e r g y - m o m e n t u m tensor T ( z ) = Z ~ z L n z -~-2, consider a U ( 1 ) current j ( z ) = Zn~zjnz-n-~. The commutation relations read, in the standard manner,
[jm,jn]=m~m+n,O,
[Lm,L~]=(m-n)Lm+n+~(d+l)(m3-m)~m+~,o,
[Zm,jn]=--njm+n.
(3.1)
The central charge is parametrized as d + 1, with 1 being the U ( 1 ) contribution. Now, let ~Ube a primary field with conformal dimension A and U ( 1 ) charge q. We are interested in null vectors at level three [21 ]. It is straightforward to see that a null vector will result from the action on I ~ > o f the following operator: 1
3
fiL_s-2L_2L_~+~-~L_~+ 3q
6+1 j-~L2-~-
d+l+
2
2d-1
.
fi+?q j2_lL_~+q-~--~J_2
L
_~+2qj_~L_2
~2+~+2~q2--q2 ~ + l + q 2 .3 fi(~--l) . 6+1 J-~J-2-q - ~ J-~-q~-J-3,
(3.2)
where
c~=A- ½q2 ,
3.3)
provided
c~= 7 - d -T-x/ ( 1 - d ) ( 25-d)
3.4)
6 [It is understood that for the (p', p) minimal model, d= 1 - 6 (p' _p)2/p,p. ] The operator (3.2) allows too much arbitrariness, as the values of A (or J, which is the matter dimension of T) and q2 cannot be fixed separately. However, the Kontsevich-Miwa transform suggests a more special form of the operator (3.2), which can be arrived at as follows. One first converts the expression of the null vector into the decoupling equation for correlation functions of the form
(~(z,) j#~ 1-1 C~(zJ)t ,
(3.5)
where, obviously, at least one operator insertion must be that of ~. (From now on, the corresponding insertion point z~ will therefore be singled out from the rest of the zj. The other insertions, of dimensions ~:4 Aj and U ( 1 ) charges qj, may or may not coincide with ~u.) Then, analyzing the decoupling equation corresponding to (3.2), one finds that the terms coming from j 3_, ~4 Dimensions refer to those as evaluated in the combined theory, by operator product expansions with the energy-momentum tensor T(z). On the other hand, separating away the current contribution by writing L~ = l~ + (Sugawara)n, would leave us with the matter sector in which the dimensions are found from the energy-momentum tensor composed of the/'s.
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cannot be recovered from the Kontsevich-Miwa transform of W constraints on the KP hierarchy ~5. To kill these terms, we have to choose 6=-l-q
2.
(3.6)
Further, with this condition satisfied, there are terms in the decoupling equation of the form
j,k,j~s i,k v~ i ( Z i - - Z j ) ( Z k - - Z i )
[ - (q2+2)qk--2qAk] T(z,) I-I 6)(zJ) ,
2
(3.7)
which are not acceptable either. Thus, similarly to the Virasoro case [ 9 ], we restrict ourselves to a subsector of those operators whose dimensions and U ( 1 ) charges satisfy
4 = -½(q2+2)qj/q.
(3.8)
This implies fixing a dressing prescription: the extra scalar 0(z) ~ f-'j enters vertex operators with an exponent depending on that of the matter part. As a result of conditions (3.6) and (3.8), the decoupling equation takes a very simple form: [
1 03
l+q2
qj 0 ) 1 ( 02 q~zi + j2~ , #~- , 2 -O~s - - -az, 3qJ
(0
q20z~+j~i(zj--zl) 2 ~zj
+2 s¢,EksiE( z j - z i )qk (z,-z,)
(qO
~z;
02 ) qO-~-,2
_ q , ~ ] ( ~ V ( z , ) I-I (~)(z,))=O.
oz,/_l\
j,,
(3.9)
In the chosen normalization [ + sign on the RHS of the first equation in (3.1) ], the q's must be imaginary ( q 2 < 0 ) , so let us define
q j = x / - ~ nj, j ¢ i ,
q = x / - ~ ni.
(3.10)
We claim that these are the n's from the Kontsevich-Miwa transform! In the next section we show how to "unKontsevich" eq. (3.9) into the full set o f W (3) constraints. Note that in a similar analysis of the level-two/Virasoro case [9], one obtains at this stage a generalized master equation (cf. ref. [23 ] ): [
10 2
1
1
2n~ Oz~ + ~i j~ai E z~_z, 3-i
(n
0
j-~z -n,
~Zj)] (
)
~P(z,) jv~t" f] C)(zj) = 0 ,
(3.11)
in which ~ is the "21" operator. The null vectors in the matter sector, underlying each one of these equations, are of course the same as for the respective standard decoupling equations [ 21 ]. However, the specific form that the decoupling equation takes in the presence of the current, is crucial for the derivation of W generators.
4. From the decoupling equation to the W (3) constraints
Having arrived in the last section at the decoupling operator ~5 The terms
qJq~q' l~V(z,)+~, ]7 6)(zj)), Z ~ Z (zj-z,)(zk-z,)(z~-~,)
j#ikCtl#t
correspondingto j3_1,are not in the imageof the Kontsevich-Miwatransformof the W constraints. 42
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-ni~g-----
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1-n 2 ( 0
+
-2j~iEks,E (zj-zi)(zk-z,)
nj O)
n,
20 August 1992
1 ( 02 j~- j - , azj Ozi
nj 02~ n~ 8z~ J
J~z, "
(4.1)
Let us now interpret the zj and nj as the ingredients of the Kontsevich-Miwa transform (2.2), and see whether the operator f can indeed be rewritten in terms of the time parameters tr. This is by no means automatic, and in particular would not be true had we not imposed the restrictions (3.6) and (3.8) ! A trivial part will be to express O/Oz's in terms of 0/0t's. The non-trivial part is to get rid of the zj themselves, as eq. (2.2) does not allow this to be done straightforwardly. As an example let us take the second term from (4.1). Using (2.2) to express derivatives with respect to z s, we find
r/j 0 ) _ l - - n 2 - - z T ' - ' ~i ~ = s+iX~j__ i #Tjr~l ~'~ Zj --Z i-Zjr-I OtrO
1-n 2 (0
J+iE (Zj__Zi)2 OZj =-(
1-n2)">_._,~
0
nj
+(1--n~) ~
0
q~l,q+,~,~ (P+2lzEp-3qtqo~p+ +(1-n~ln, p~>,2½ ( P + l (p+2)zTP-3otp Y. z S - Z ( p + l )
J-~t"7j. - - ZI p>~ 1
0-~-
8tp
(4.2)
"
Similarly, in the last term in (4.1) we can also divide by
- 2 k4-i E Zk nk Z ninj r>~ ~1 - - Z i jv~i =_2n, Z
nj
following way:
z T r - i - z f "-I O zj-zi 8tr
~
jv~i Z j - - Z i
zj- z~in the
Z
st~z;_p_2 0 + 2 n ~
p>~--l s>~l,p+s>~l
Otp+s
"
nj
~ (p+l)zF p - 2 ~ Otp
jvLi Z j - - Z i p>~l
(4.3) "
Computing in a similar manner the terms with the second and the third derivatives and collecting everything together, we get 3~/'=
E
z~q .....
q,,,s>~~
3
03
Otq at, ot~
"~
3--2n,
~
~ z[- . . . . 3 ( r + l ) _ _ Otr St,
r.,>~ l
+ (~i1 + 2n2 - - - 2) ~, ~ (r+l)(r+2)zT~_ 3 Ot~r 0 + -l ---n~ Z tit p>~--I +2
p-- 1
Z z- p-3 E p~>0
+2 Z
E
qtq
s = - - I q>~l,q>~l--s
nj
82
8tp_sOtq+s
+2 Z
nj
E
q>~l,q>~l--p
0 ZTP--3(p+2)qtq Otp+q
E Lp(Z-P-2-zf p-z)
j~iZj--Zip>~--I
~ Lpzf p-2,
(4.4)
jvai Zj - - Z l p>~ _ 1
with p--l
Lp>_= , ½~
02
0
,=,E 8tp ,Ot-------~+ ,>l,Zst, O~p+s
2-n 2
0 ( p + 1) --Sip,
(4.5)
and Lo and L_ ~ are the same as in (2.4). Notice that the Virasoro constraints ( 4 . 5 ) are different from the ones in (2.4), according to the value of J in terms of the Miwa parameter (assuming it is the same). The last term has been added to and subtracted from the RHS of (4.4). With the combination ( z J -2 -zfP-2)/(zj-z~) we proceed as above, and in this way we finally arrive at 43
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E
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nj
z,-~-3wg3>r+2E
p>~--2
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Y z)-p-2Lp ~,
(4.6)
j # i Z j - - Z i p>~--I
with
W(_3~= 2
~ r~>l
rt~L~_2, + 1--n2i ~ rtr O---~--
W(-3~=2 E rtrl-r-I--2niL-I r>~l
ni
r>~2
Otr--I
W;3)=2 ~ rt, Lr-4niLo+2 l-n2i 2 rtr r>~l
ni
+2 2
r>~l
r>~2 rtr
1 -n~
la/~3),, p~ l = 2 r>~E~rt, L , + p - 2 ( p + 2 ) n i l _ p + ( p + 2 ) ~ g
+(p+l)(p+2)
1
n2
1
a +
3 -2ni
-
Otl Otr_l a
~-'
a2
,>~Elrtr -~,+p + 2 5--E-, ,> l ,>~E~- s rt, atp_~ at,+s
).,
,~, ( r + l ) -at,- Otp_, +
03
2 . q,r,s>~l.q+,+s=pOtqatrOts
(4.7)
The final step consists in d e m a n d i n g that the W ~3) and the/. generators annihilate the tau function separately ,6: L:=0,
n>_--l,
I/V~3)z=0,
n~>-2.
(4.8)
Thus we have recovered the full set o f W ~3) constraints (i.e., t4/},3) and L,) with the coefficients in front o f the different terms d e p e n d i n g on n~ a n d therefore, via eq. (2.7), on the corresponding m i n i m a l model. Recall also that, according to (3.6), (3.8), (3.4) and (3.10), the Miwa p a r a m e t e r n, is equal to the cosmological constant: ni=~7og+_ ~ o "
(
-- ~ ~]
3
- 2
'
(4.9)
where a 2 = 1 a n d the u p p e r / l o w e r sign corresponds to that in (3.4). Stated differently, for the ( p ' , p ) m i n i m a l model we have n,.2= 2 p ' / p . Conversely, starting from the full set o f W ° ) constraints, one constructs the c o m b i n a t i o n (4.6) (however, unnatural it m a y seem from the point o f view o f the t-variables) and then, fixing the coefficients as explained and inverting the previous steps, one arrives at the decoupling equation. This proves the equivalence between the W o ) constraints and the level-three decoupling equation in the ( p ' , p ) m i n i m a l model. A r e m a r k is in order concerning why the Virasoro part of the derived constraints does not lead to the leveltwo decoupling equations as was the case in ref. [ 9 ]. The point is that the possibility to transform the Virasoro constraints into a decoupling equation d e p e n d s crucially on the relation between the " s p i n " J present in the Virasoro generators, and the value o f the Miwa p a r a m e t e r n, (recall that the ith point, being the position o f the insertion o f ~ = ~3~, plays a special role). However, the spin J as read off from the generators (4.5), eq. (2.7) ~7, is not the same as required by (2.5) and therefore the generators (4.5) do not allow a transformation into the z-variables.
#6 According to the form of the dependence on z~and zj in (4.6); in particular, the combination Zj~,nj/(zj-z,) cannotbe rewritten in terms of the time parameters, while all the rest of eq. (4.6) is expressed through the times, so that the two terms (those involving W o) and/. respectively) are "linear independent", hence the vanishing of each one. #7 Note that it follows from (2.7) that J - ½= ( + ) ½ ~ =- ½( + )Qm, where Qm is the matter background charge, which therefore coincides with Q---2J- 1.
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5. D r e s s i n g prescription and generalizations
The dressing prescription (3.8) corresponding to W (3) constraints, can be rewritten as Aj = _+ ½aQnj and in this form it coincides with the dressing prescription derived from the Virasoro constraints on the KP hierarchy [ 9 ]. Therefore the sector singled out from the m a t t e r ® U ( 1 ) theory by the dressing prescription is the same for the two cases. Using another "universal" relation, c~j= Aj + ~F/,]l 2, which gives the matter dimensions by subtracting away the Sugawara part, we arrive at
nj2 ++_aQnj- Z6j=O .
(5.1)
The value of nj found from here differs from the D D K dressing by the cosmological constant oz+ = ½( - QL +- Q). A general pattern which generalizes the two lowest cases, W (2) and W (3), can be formulated as follows: To the W (~) constraints on the KP hierarchy we associate the decoupling equation corresponding to the irreducible module built on the (dressed) (l, 1 ) [or ( 1, l) ] state in the (p', p) minimal model. The value o f p ' / p is optional, and depends on the "spin" J entering in the Virasoropart of the W (t) constraints:
p'/p=l+¼Q2+¼Q
Qx/~,
Q=2J-I.
(5.2)
According to the standard formulae, the primary field ~uf~has dimension (in the matter sector)
3=~( 4+QZ+Q~)(12-1)+
½(1-l) .
The dressing of ~vt~ is specified by its U ( 1 ) charge q = ~ parameter, determined from l+ 1 n 2
~= l-~
(5.3) 1 ni with nt, which at the same time is the Miwa
1 -l
2 + ~-
(5.4)
[ eq. (3.6) is recovered for l = 3 ]. This gives for the "bulk" dimension of the dressed ~P¢~operator j=
O - ~in , 2 = n~ + ½ ( 1 - l ) l-1
(5.5)
The dimensions and U ( 1 ) charges of all the other operators satisfy ni = l
i
'!) ~n
n~.
(5.6)
With n~ determined from the above formulae as ns=(1-/).-~(Q+
Qx/~8),
(5.7)
we find the relation between the Miwa parameter ni and the spin J present in the Virasoro part of the W (~) constraints: l-1
ni
2hi = 2 J - 1
(W ~t~).
(5.8)
l- 1
For the ( 1, l) operator the formulae would receive the obvious sign modifications. To conclude let us note that beyond level four it would be difficult to give a direct derivation of the correspondence between W (z) constraints and the (l, 1 ) decoupling equations, so one has to rely on more general arguments (cf., however, ref. [24] ). Also, it would be interesting to understand the role in this picture of the W-algebra null vectors, as well as of the decoupling equations other than those considered in ref. [24]. It is also 45
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unclear how integer values o f p ' and p could be singled out in the KP formalism. These problems are under investigation and will be the subject of a future publication.
Acknowledgement We would like to thank A. Berkovich, V. Fock, E. Kiritsis and W. Lerche for useful discussions.
References [ 1 ] E. Brrzin and V.A. Kazakov, Phys. Lett. B 236 (1990) 144. [2] M.R. Douglas and S.H. Shenker, Nucl. Phys. B 335 (1990) 635. [3] D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [ 4 ] M. Fukuma, H. Kawai and R. Nakayama, Intern. J. Mod. Phys. A 6 ( 1991 ) 1385; Explicit solution for p - q duality in two-dimensional quantum gravity, Tokyo University preprint UT-582 (May 1991 ). [ 5 ] R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B 348 (1991 ) 435. [ 6 ] E. Gava and K.S. Narain, Phys. Lett. B 263 ( 1991 ) 213. [7] F. David, Mod. Phys. Lett. A 3 (1988) 1651. [8] J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. [ 9 ] A.M. Semikhatov, Kontsevich-Miwa transform of the Virasoro constraints as null-vector decoupling equations, preprint, hepth@xxx/ 9111051; Corrected and extended version: Solving Virasoro constraints on integrable hierarchies via the Kontsevich-Miwa transform, preprint [hepth@xxx/9204063 ], submitted to Nucl. Phys. B. [ 10 ] M. Bauer, Ph. Di Francesco, C. Itzykson and J.-B. Zuber, Nucl. Phys. B 362 ( 1991 ) 515. [ 11 ] VI.S. Dotsenko, M. Picco and P. Windey, private communication. [ 12] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, in: Proc. RIMS Syrup. on Non-linear integrable systems, eds. M. Jimbo and T. Miwa (World Scientific, Singapore, 1983 ) p. 39. [ 13 ] T. Miwa, Proc. Jpn. Academy 58 ( 1982 ) 9. [14] S. Saito, Phys. Rev. D 36 (1987) 1819; Phys. Rev. Lett. 59 (1987) 1798. [ 15] M. Kontsevich, Funkts. Anal. Prilozh. 25 ( 1991 ) N2, 50. [ 16 ] E. Witten, On the Konsevich model and other models of two-dimensional gravity, Princeton preprint IASSNS-HEP-91-24 ( 1991 ). [ 17] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov and A. Zabrodin, Towards unified theory of 2-d gravity, Lebedev Institute preprint FIAN-TD-10-91 (October 1991 ). [ 18 ] D.J. Gross and M.J. Newman, Unitary and hermitian matrices in an external field II: the Kontsevich model and continuum Virasoro constraints, Princeton University preprint PUPT-1282 ( 1991 ). [ 19 ] C. Itzykson and J.-B. Zuber, Combinatorics of the modular group II: the Kontsevich integrals, Saclay preprint SPhT/92-001. [20] A.M. Semikhatov, Intern. J. Mod. Phys. A 4 (1989) 467. [21 ] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [ 22 ] V1.S. Dotsenko and V.A. Fateev, Nucl. Phys. B 240 (1984) 312. [ 23 ] Yu. Makeenko and G. Semenoff, Properties of hermitian matrix model in an external field, British Columbia University preprint 91-0329 (July 1991 ). [ 24 ] L. Benoit and Y. Saint-Aubin, Phys. Lett. B 215 ( 1988 ) 517.
46
PHYSICS LETTERS B
Minimal models from W-constrained hierarchies via the Kontsevich-Miwa transform B. G a t o - R i v e r a CERN, CH-1211 Geneva 23, Switzerland and lnstituto de Fisica Fundamental, Serrano 123, E-28006 Madrid, Spain and A.M. Semikhatov Theory Division, P.N. Lebedev Physics Institute, Leninsky prospect 53, 117 924 Moscow, Russian Federation Received 18 May 1992
A direct relation between the conformal formalism for 2D quantum gravity and the W-constrained KP hierarchy is found, without the need to invoke intermediate matrix model technology. The Kontsevich-Miwa transform of the KP hierarchy is used to establish an identification between W constraints on the KP tau function and decoupling equations corresponding to Virasoro null vectors. The Kontsevich-Miwa transform maps the Wt°-eonstrained KP hierarchy to the (p', p) minimal model, with the tau function being given by the correlator of a product of (dressed) (l, 1) [or ( 1, l) ] operators, provided the Miwa parameter ni and the free parameter (an abstract bc spin) present in the constraints are expressed through the ratio p ' / p and the level/.
1. Introduction M a t r i x models, w h i c h a p p e a r e d as a " d i s c r e t i z e d " a p p r o a c h to 2 D q u a n t u m gravity [ 1-3 ], h a v e resulted at the c o n t i n u u m level in integrable h i e r a r c h i e s s u b j e c t e d to V i r a s o r o [4,5 ] a n d possibly higher W [5,6 ] constraints. H o w e v e r , their r e l a t i o n w i t h the L i o u v i l l e theory, at least w i t h that d e s c r i b e d in the f o r m a l i s m o f refs. [ 7,8 ], seems, a l t h o u g h h a r d l y d i s p u t a b l e in principle, s o m e w h a t obscure. In a recent p a p e r [ 9 ] the K o n t s e v i c h M i w a t r a n s f o r m was used to establish a r e l a t i o n b e t w e e n the V i r a s o r o c o n s t r a i n t s i m p o s e d on the tau f u n c t i o n s o f the K P hierarchy, a n d the d e c o u p l i n g e q u a t i o n c o r r e s p o n d i n g to the null v e c t o r at level two in m i n i m a l c o n f o r m a l field t h e o r i e s e x t e n d e d by a scalar c u r r e n t ~. M o r e o v e r , the field c o n t e n t o f the D a v i d - D i s t l e r - K a w a i f o r m a l i s m for 2 D q u a n t u m gravity was r e c o v e r e d in this way, w i t h the extra scalar playing a role s i m i l a r to that o f the L i o u v i l l e field in the f o r m a l i s m o f r e f s . [ 7,8 ]. T h e essence o f the m e t h o d based on the K o n t s e v i c h - M i w a t r a n s f o r m was t h e r e f o r e to relate the V i r a s o r o - c o n s t r a i n e d K P h i e r a r c h y a n d 2 D q u a n t u m gravity, bypassing the m a t r i x m o d e l technology. N o w the q u e s t i o n arises as to w h e t h e r the h i g h e r W constraints on the K P h i e r a r c h y are a m e n a b l e to the a b o v e scheme, a n d w h e t h e r a bridge can be established b e t w e e n m a t t e r i n t e r a c t i n g w i t h the c o n t i n u u m D D K q u a n t u m gravity a n d the W - c o n s t r a i n e d K P hierarchies. In this letter we give a p o s i t i v e a n s w e r to this q u e s t i o n , b a s e d on the results o b t a i n e d for the W ~3) c o n s t r a i n t s in a d d i t i o n to the p r e v i o u s results for the W ~2) ( V i r a s o r o ) case. ~ The effect of the Kontsevich-Miwa transformation was to replace the time parameters of the hierarchy with the spectral parameter, or, in different words, to create a world-sheet out of the time parameters. 38
0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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By analogy with the level-two Virasoro case, one could expect that under the Kontsevich-Miwa parametrization of the KP times, the higher (l>~ 3) W constraints I/V~l) z = 0 , n >~ - l + 1, would lead to higher-level (l~> 3) null vector decoupling equations ~2. This argument, however, is fraught with an unpleasant "paradox": the algebra of the W (t) constraints contains the lower W (r), l' < l as well, which apparently should then give rise, by the same mechanism, to decoupling equations at the lower levels, thereby leading to an over-determined system. Fortunately, this does not happen; it turns out that a single level-I decoupling equation corresponds to the complete set of W (t) constraints (including the Virasoro ones) W ~ c ) z = O , n >t - l ' + 1, 2 <~l'<~ l, while these do not lead back to lower-level decoupling equations. More precisely, the Kontsevich-Miwa transform maps the W (1)constrained KP hierarchy to the (p', p) minimal model, with the tau function being given by the correlator of a product o f (dressed) (l, 1 ) [or ( 1, l) ] operators, provided (i) the Miwa parameter n, satisfies r
nZ=(l--1)2~p, and (ii) the free parameter (an abstract bc spin) J present in the constraints is determined from 1- 1 2n~ 2 J - 1= - n~ - l - 1 ' Although we have obtained the complete results only for the W (3) case (in addition to W (2) =Virasoro), it is hard to believe that so simple a result, the W-decoupling correspondence, following after rather tedious calculations, could be a mere coincidence and would not allow generalizations to W (>~4) algebras. We would like to stress that, rather than taking the decoupling equation in its naive form, one should first "dress" the theory by tensoring it with an extra U ( 1 ) current. The extra degrees of freedom which thus appear are then killed by projecting onto a special form of the null state. Therefore, although the null vector in the matter sector is the same, the specific form of the equation for correlation functions changes. We begin in the next section by introducing the KP hierarchy and the notion of the Kontsevich-Miwa transform. Then, in section 3, we introduce the dressed decoupling equation. For purely technical reasons, we describe in this paper the coincidence between W constraints and decoupling equations proceeding in the direction .from the decoupling equations to the W constraints: we show in section 4 how to "un-Kontsevich" conformal field-theoretic data into the W (3) constraints. It will be clear from the derivation that (and how) one can reverse the argument and thus establish equivalence of the two sets of data, the W(3)-constrained KP hierarchy and (appropriately dressed) minimal models.
2. The KP hierarchy and the Kontsevich-Miwa transform We start with the KP hierarchy [ 12 ]. It can be described either in an evolutionary form, as an infinite set of equations on (coefficients o f ) a pseudodifferential operator, or as (Hirota) bilinear relations on the tau function. The tau-functional description may be considered as having the more direct relevance to "physics", being related to the partition function, while the evolutionary form has all the usual advantages due to the introduction of a spectral parameter and the associated wavefunction. The wavefunction and the adjoint wavefunction depend on the spectral parameter z via exp [ + ~ (t, z) ] r (t ~ [ z - l ] ) / r (t), where ~(t, z) = ~ r>~l trZ r and t + [z -~] = (tl -+z -1, t2-+ ½z -2, t3-+ ½z -3 .... ) .
(2.1)
Here t = ( x = tb t2, t3, ...) are the time parameters of the hierarchy. This form of the z dependence is "generalized" by the following presentation for the tr, known as the Miwa parametrization [ 13,14 ] : ,2 A relation between Virasoro null vectors and W algebras has been noted in a different approach in ref. [ 10] and, more recently, in ref. [11]. 39
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tr= 1 ~,, njzfr, r j
PHYSICS LETTERS B
r>~ 1 ,
20 August 1992
(2.2)
where {zj) is a set of points on the complex plane and the Miwa parameters n: are integer classically; we will need, however, to continue offthe integer values. As we will see, the Miwa parameters acquire the role of U( 1 ) charges with respect to the scalar field. By the Kontsevich-Miwa transform we will understand the dependence, via eq. (2.2), Oftr on the z:. However, it is important that the presentation of the form (2.2) be considered for arbitrary nj (while, on the other hand, Kontsevich has originally used a parametrization of this type for all the nj equal [ 15 ], see also refs. [ 16-19 ]. As was noted in section 1, an important result of the development of matrix models was the discovery of an "integrable" counterpart of the 2D gravity + matter theories, in the form of Virasoro-constrained hierarchies. Moreover, the fact that appropriately constrained hierarchies provide a description of 2D gravity coupled to matter, may be promoted to a first-principle, in which case it is natural to consider on an equal footing various constraints more general than the Virasoro ones, including those whose derivation from a specific matrix model is not known. This applies, first of all, to the extension of Virasoro constraints to W ones, and also to the introduction of a parameter into the Virasoro generators acting on the tau function [20]. This parameter J (which would have parametrized the central charge as 2 (6J 2 - 6 J + 1 ), had the L,,~ _ 2 generators been present as well), can be thought of as the "spin" (dimension) of an abstract bc system underlying the Virasoro generators, ~n~z Lnz-n-2 ~ ( 1 - J ) Obc-Jb Oc. We repeat that although J-dependent Virasoro constraints may not have been derived from a matrix model, for us all of them are equally good starting points. The corresponding constraints,
Lnz=O,
n>_.-1,
(2.3)
p-1 02 Lp>o = ½,=Z Otp_,Ots + Lo=s~lsts~s,
,>~ZISts ~
0
0 + ( J - ½) (p-t- 1) Otp '
L_I=s>ZI (s+l)ts+,--,Ots
(2.4)
were shown in ref. [ 9 ] to take, after the Kontsevich-Miwa transform, the form of a decoupling equation corresponding to the level-two null vector in the (p', p) minimal model determined by p'/p=2n~, provided the Miwa parameter ni was determined from
1/ni-2ni=2J- 1-Q
(Virasoro) .
(2.5)
Our task in this paper is to extend this correspondence to include W generators. Let us start therefore with the W (3) constraints imposed on the tau function in a way similar to (2.4); we will write out the constraints explicitly later, see (4.7). Now, by virtue of the Kontsevich-Miwa transform (2.2), the tau function becomes a function r{zj} of the zj. We assume for it the ansatz ~3 r{zj}= lim < ~'(z,)...~t'(z~)) ,
(2.6)
with < > and kUbeing, respectively, the chiral correlation function and a (primary) field operator in a conformal field theory on the z plane. The operator and the theory itself should be specified according to the fact that the tau function satisfies the W (3) constraints. The ansatz will be justified by the fact that, as we are going to show, imposing these constraints amounts to the condition that ~ be a "31" operator in a (p', p) minimal model [21,22] for p'/p determined by the precise form of the constraints. That is, we will find the relation p'/p = ½n2, where n, is the Miwa parameter which in its own turn is related to a "spin" J, read off from the Virasoro part of the constraints, by ~3 Clearly, there have to be infinitely many points z~ in the Kontsevich-Miwa transform in order for the times tr to be independent.
40
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(W (3)).
(2.7)
As was noted above, we find it more convenient to present in this paper the derivation of the correspondence W constraints ~ decoupling equations in the inverse direction: we start from 7-, being the " 3 1 " operator and then show how to "un-Kontsevich" it into the W ° ) constraints. We therefore proceed in the next section with introducing the necessary conformal field-theoretic ingredients.
3. " D r e s s i n g " the decoupling equation
Now we introduce the second ingredient: null-vector decoupling equations in minimal models o f conformal field theory. To the KP hierarchy we will return at the very end. In addition to the e n e r g y - m o m e n t u m tensor T ( z ) = Z ~ z L n z -~-2, consider a U ( 1 ) current j ( z ) = Zn~zjnz-n-~. The commutation relations read, in the standard manner,
[jm,jn]=m~m+n,O,
[Lm,L~]=(m-n)Lm+n+~(d+l)(m3-m)~m+~,o,
[Zm,jn]=--njm+n.
(3.1)
The central charge is parametrized as d + 1, with 1 being the U ( 1 ) contribution. Now, let ~Ube a primary field with conformal dimension A and U ( 1 ) charge q. We are interested in null vectors at level three [21 ]. It is straightforward to see that a null vector will result from the action on I ~ > o f the following operator: 1
3
fiL_s-2L_2L_~+~-~L_~+ 3q
6+1 j-~L2-~-
d+l+
2
2d-1
.
fi+?q j2_lL_~+q-~--~J_2
L
_~+2qj_~L_2
~2+~+2~q2--q2 ~ + l + q 2 .3 fi(~--l) . 6+1 J-~J-2-q - ~ J-~-q~-J-3,
(3.2)
where
c~=A- ½q2 ,
3.3)
provided
c~= 7 - d -T-x/ ( 1 - d ) ( 25-d)
3.4)
6 [It is understood that for the (p', p) minimal model, d= 1 - 6 (p' _p)2/p,p. ] The operator (3.2) allows too much arbitrariness, as the values of A (or J, which is the matter dimension of T) and q2 cannot be fixed separately. However, the Kontsevich-Miwa transform suggests a more special form of the operator (3.2), which can be arrived at as follows. One first converts the expression of the null vector into the decoupling equation for correlation functions of the form
(~(z,) j#~ 1-1 C~(zJ)t ,
(3.5)
where, obviously, at least one operator insertion must be that of ~. (From now on, the corresponding insertion point z~ will therefore be singled out from the rest of the zj. The other insertions, of dimensions ~:4 Aj and U ( 1 ) charges qj, may or may not coincide with ~u.) Then, analyzing the decoupling equation corresponding to (3.2), one finds that the terms coming from j 3_, ~4 Dimensions refer to those as evaluated in the combined theory, by operator product expansions with the energy-momentum tensor T(z). On the other hand, separating away the current contribution by writing L~ = l~ + (Sugawara)n, would leave us with the matter sector in which the dimensions are found from the energy-momentum tensor composed of the/'s.
41
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cannot be recovered from the Kontsevich-Miwa transform of W constraints on the KP hierarchy ~5. To kill these terms, we have to choose 6=-l-q
2.
(3.6)
Further, with this condition satisfied, there are terms in the decoupling equation of the form
j,k,j~s i,k v~ i ( Z i - - Z j ) ( Z k - - Z i )
[ - (q2+2)qk--2qAk] T(z,) I-I 6)(zJ) ,
2
(3.7)
which are not acceptable either. Thus, similarly to the Virasoro case [ 9 ], we restrict ourselves to a subsector of those operators whose dimensions and U ( 1 ) charges satisfy
4 = -½(q2+2)qj/q.
(3.8)
This implies fixing a dressing prescription: the extra scalar 0(z) ~ f-'j enters vertex operators with an exponent depending on that of the matter part. As a result of conditions (3.6) and (3.8), the decoupling equation takes a very simple form: [
1 03
l+q2
qj 0 ) 1 ( 02 q~zi + j2~ , #~- , 2 -O~s - - -az, 3qJ
(0
q20z~+j~i(zj--zl) 2 ~zj
+2 s¢,EksiE( z j - z i )qk (z,-z,)
(qO
~z;
02 ) qO-~-,2
_ q , ~ ] ( ~ V ( z , ) I-I (~)(z,))=O.
oz,/_l\
j,,
(3.9)
In the chosen normalization [ + sign on the RHS of the first equation in (3.1) ], the q's must be imaginary ( q 2 < 0 ) , so let us define
q j = x / - ~ nj, j ¢ i ,
q = x / - ~ ni.
(3.10)
We claim that these are the n's from the Kontsevich-Miwa transform! In the next section we show how to "unKontsevich" eq. (3.9) into the full set o f W (3) constraints. Note that in a similar analysis of the level-two/Virasoro case [9], one obtains at this stage a generalized master equation (cf. ref. [23 ] ): [
10 2
1
1
2n~ Oz~ + ~i j~ai E z~_z, 3-i
(n
0
j-~z -n,
~Zj)] (
)
~P(z,) jv~t" f] C)(zj) = 0 ,
(3.11)
in which ~ is the "21" operator. The null vectors in the matter sector, underlying each one of these equations, are of course the same as for the respective standard decoupling equations [ 21 ]. However, the specific form that the decoupling equation takes in the presence of the current, is crucial for the derivation of W generators.
4. From the decoupling equation to the W (3) constraints
Having arrived in the last section at the decoupling operator ~5 The terms
qJq~q' l~V(z,)+~, ]7 6)(zj)), Z ~ Z (zj-z,)(zk-z,)(z~-~,)
j#ikCtl#t
correspondingto j3_1,are not in the imageof the Kontsevich-Miwatransformof the W constraints. 42
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-ni~g-----
PHYSICS LETTERS B
1-n 2 ( 0
+
-2j~iEks,E (zj-zi)(zk-z,)
nj O)
n,
20 August 1992
1 ( 02 j~- j - , azj Ozi
nj 02~ n~ 8z~ J
J~z, "
(4.1)
Let us now interpret the zj and nj as the ingredients of the Kontsevich-Miwa transform (2.2), and see whether the operator f can indeed be rewritten in terms of the time parameters tr. This is by no means automatic, and in particular would not be true had we not imposed the restrictions (3.6) and (3.8) ! A trivial part will be to express O/Oz's in terms of 0/0t's. The non-trivial part is to get rid of the zj themselves, as eq. (2.2) does not allow this to be done straightforwardly. As an example let us take the second term from (4.1). Using (2.2) to express derivatives with respect to z s, we find
r/j 0 ) _ l - - n 2 - - z T ' - ' ~i ~ = s+iX~j__ i #Tjr~l ~'~ Zj --Z i-Zjr-I OtrO
1-n 2 (0
J+iE (Zj__Zi)2 OZj =-(
1-n2)">_._,~
0
nj
+(1--n~) ~
0
q~l,q+,~,~ (P+2lzEp-3qtqo~p+ +(1-n~ln, p~>,2½ ( P + l (p+2)zTP-3otp Y. z S - Z ( p + l )
J-~t"7j. - - ZI p>~ 1
0-~-
8tp
(4.2)
"
Similarly, in the last term in (4.1) we can also divide by
- 2 k4-i E Zk nk Z ninj r>~ ~1 - - Z i jv~i =_2n, Z
nj
following way:
z T r - i - z f "-I O zj-zi 8tr
~
jv~i Z j - - Z i
zj- z~in the
Z
st~z;_p_2 0 + 2 n ~
p>~--l s>~l,p+s>~l
Otp+s
"
nj
~ (p+l)zF p - 2 ~ Otp
jvLi Z j - - Z i p>~l
(4.3) "
Computing in a similar manner the terms with the second and the third derivatives and collecting everything together, we get 3~/'=
E
z~q .....
q,,,s>~~
3
03
Otq at, ot~
"~
3--2n,
~
~ z[- . . . . 3 ( r + l ) _ _ Otr St,
r.,>~ l
+ (~i1 + 2n2 - - - 2) ~, ~ (r+l)(r+2)zT~_ 3 Ot~r 0 + -l ---n~ Z tit p>~--I +2
p-- 1
Z z- p-3 E p~>0
+2 Z
E
qtq
s = - - I q>~l,q>~l--s
nj
82
8tp_sOtq+s
+2 Z
nj
E
q>~l,q>~l--p
0 ZTP--3(p+2)qtq Otp+q
E Lp(Z-P-2-zf p-z)
j~iZj--Zip>~--I
~ Lpzf p-2,
(4.4)
jvai Zj - - Z l p>~ _ 1
with p--l
Lp>_= , ½~
02
0
,=,E 8tp ,Ot-------~+ ,>l,Zst, O~p+s
2-n 2
0 ( p + 1) --Sip,
(4.5)
and Lo and L_ ~ are the same as in (2.4). Notice that the Virasoro constraints ( 4 . 5 ) are different from the ones in (2.4), according to the value of J in terms of the Miwa parameter (assuming it is the same). The last term has been added to and subtracted from the RHS of (4.4). With the combination ( z J -2 -zfP-2)/(zj-z~) we proceed as above, and in this way we finally arrive at 43
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O=~T=
E
PHYSICS LETTERS B
nj
z,-~-3wg3>r+2E
p>~--2
20 August 1992
Y z)-p-2Lp ~,
(4.6)
j # i Z j - - Z i p>~--I
with
W(_3~= 2
~ r~>l
rt~L~_2, + 1--n2i ~ rtr O---~--
W(-3~=2 E rtrl-r-I--2niL-I r>~l
ni
r>~2
Otr--I
W;3)=2 ~ rt, Lr-4niLo+2 l-n2i 2 rtr r>~l
ni
+2 2
r>~l
r>~2 rtr
1 -n~
la/~3),, p~ l = 2 r>~E~rt, L , + p - 2 ( p + 2 ) n i l _ p + ( p + 2 ) ~ g
+(p+l)(p+2)
1
n2
1
a +
3 -2ni
-
Otl Otr_l a
~-'
a2
,>~Elrtr -~,+p + 2 5--E-, ,> l ,>~E~- s rt, atp_~ at,+s
).,
,~, ( r + l ) -at,- Otp_, +
03
2 . q,r,s>~l.q+,+s=pOtqatrOts
(4.7)
The final step consists in d e m a n d i n g that the W ~3) and the/. generators annihilate the tau function separately ,6: L:=0,
n>_--l,
I/V~3)z=0,
n~>-2.
(4.8)
Thus we have recovered the full set o f W ~3) constraints (i.e., t4/},3) and L,) with the coefficients in front o f the different terms d e p e n d i n g on n~ a n d therefore, via eq. (2.7), on the corresponding m i n i m a l model. Recall also that, according to (3.6), (3.8), (3.4) and (3.10), the Miwa p a r a m e t e r n, is equal to the cosmological constant: ni=~7og+_ ~ o "
(
-- ~ ~]
3
- 2
'
(4.9)
where a 2 = 1 a n d the u p p e r / l o w e r sign corresponds to that in (3.4). Stated differently, for the ( p ' , p ) m i n i m a l model we have n,.2= 2 p ' / p . Conversely, starting from the full set o f W ° ) constraints, one constructs the c o m b i n a t i o n (4.6) (however, unnatural it m a y seem from the point o f view o f the t-variables) and then, fixing the coefficients as explained and inverting the previous steps, one arrives at the decoupling equation. This proves the equivalence between the W o ) constraints and the level-three decoupling equation in the ( p ' , p ) m i n i m a l model. A r e m a r k is in order concerning why the Virasoro part of the derived constraints does not lead to the leveltwo decoupling equations as was the case in ref. [ 9 ]. The point is that the possibility to transform the Virasoro constraints into a decoupling equation d e p e n d s crucially on the relation between the " s p i n " J present in the Virasoro generators, and the value o f the Miwa p a r a m e t e r n, (recall that the ith point, being the position o f the insertion o f ~ = ~3~, plays a special role). However, the spin J as read off from the generators (4.5), eq. (2.7) ~7, is not the same as required by (2.5) and therefore the generators (4.5) do not allow a transformation into the z-variables.
#6 According to the form of the dependence on z~and zj in (4.6); in particular, the combination Zj~,nj/(zj-z,) cannotbe rewritten in terms of the time parameters, while all the rest of eq. (4.6) is expressed through the times, so that the two terms (those involving W o) and/. respectively) are "linear independent", hence the vanishing of each one. #7 Note that it follows from (2.7) that J - ½= ( + ) ½ ~ =- ½( + )Qm, where Qm is the matter background charge, which therefore coincides with Q---2J- 1.
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5. D r e s s i n g prescription and generalizations
The dressing prescription (3.8) corresponding to W (3) constraints, can be rewritten as Aj = _+ ½aQnj and in this form it coincides with the dressing prescription derived from the Virasoro constraints on the KP hierarchy [ 9 ]. Therefore the sector singled out from the m a t t e r ® U ( 1 ) theory by the dressing prescription is the same for the two cases. Using another "universal" relation, c~j= Aj + ~F/,]l 2, which gives the matter dimensions by subtracting away the Sugawara part, we arrive at
nj2 ++_aQnj- Z6j=O .
(5.1)
The value of nj found from here differs from the D D K dressing by the cosmological constant oz+ = ½( - QL +- Q). A general pattern which generalizes the two lowest cases, W (2) and W (3), can be formulated as follows: To the W (~) constraints on the KP hierarchy we associate the decoupling equation corresponding to the irreducible module built on the (dressed) (l, 1 ) [or ( 1, l) ] state in the (p', p) minimal model. The value o f p ' / p is optional, and depends on the "spin" J entering in the Virasoropart of the W (t) constraints:
p'/p=l+¼Q2+¼Q
Qx/~,
Q=2J-I.
(5.2)
According to the standard formulae, the primary field ~uf~has dimension (in the matter sector)
3=~( 4+QZ+Q~)(12-1)+
½(1-l) .
The dressing of ~vt~ is specified by its U ( 1 ) charge q = ~ parameter, determined from l+ 1 n 2
~= l-~
(5.3) 1 ni with nt, which at the same time is the Miwa
1 -l
2 + ~-
(5.4)
[ eq. (3.6) is recovered for l = 3 ]. This gives for the "bulk" dimension of the dressed ~P¢~operator j=
O - ~in , 2 = n~ + ½ ( 1 - l ) l-1
(5.5)
The dimensions and U ( 1 ) charges of all the other operators satisfy ni = l
i
'!) ~n
n~.
(5.6)
With n~ determined from the above formulae as ns=(1-/).-~(Q+
Qx/~8),
(5.7)
we find the relation between the Miwa parameter ni and the spin J present in the Virasoro part of the W (~) constraints: l-1
ni
2hi = 2 J - 1
(W ~t~).
(5.8)
l- 1
For the ( 1, l) operator the formulae would receive the obvious sign modifications. To conclude let us note that beyond level four it would be difficult to give a direct derivation of the correspondence between W (z) constraints and the (l, 1 ) decoupling equations, so one has to rely on more general arguments (cf., however, ref. [24] ). Also, it would be interesting to understand the role in this picture of the W-algebra null vectors, as well as of the decoupling equations other than those considered in ref. [24]. It is also 45
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unclear how integer values o f p ' and p could be singled out in the KP formalism. These problems are under investigation and will be the subject of a future publication.
Acknowledgement We would like to thank A. Berkovich, V. Fock, E. Kiritsis and W. Lerche for useful discussions.
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