Infinite Dimensional Grassmannian Structure of Two-Dimensional Quantum Gravity

Commun. Math. Phys. 143, 371-403 (1992) Communications in

Mathematical Physics

9 Springer-Verlag 1992

Infinite Dimensional Grassmannian Structure of Two-Dimensional Quantum Gravity

Masafumi Fukuma 1,., Hikaru Kawai 1, **, and Ryuichi Nakayama 2' ***

1 Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

2 National Laboratory for High Energy Physics (KEK), Tsukuba-shi, Ibaraki 305, Japan Received March 22, 1991

Abstract. We study the infinite dimensional Grassmannian structure of 2D quantum gravity coupled to minimal conformal matters, and show that there exists a large symmetry, the W1 + ~ symmetry. Using this symmetry structure, we prove that the square root of the partition function, which is a z function of the p-reduced KP hierarchy, satisfies the vacuum condition of the W1 + ~o algebra. We further show that this condition is reduced to the vacuum condition of the Wp algebra when the redundant variables for the p-reduction are eliminated. This mechanism also gives a prescription for extracting the Wp algebra from the W1 + oo algebra.

1. Introduction

Recently great progress has been made in the understanding of the non- perturbative aspects of 2D quantum gravity by formulating it in terms of the matrix models [1]. In particular, it has been recognized that the 2D quantum gravity has a close connection with the KP hierarchy [2-4]. Furthermore, the analysis through the Schwinger-Dyson equation reveals the universal structures of 2D quantum gravity and from this one can obtain an analogue of the operator product expansion even in the theory of quantum gravity [5, 6]. In particular, it is shown [5, 6] that the Schwinger-Dyson equation for the 2D gravity coupled to the (p, q) conformal matters ((p, q) quantum gravity)1 is expressed as the vacuum condition of the W e algebra on the function z(x)= ~ , where

Z(x)

is the partition function of the 2D gravity with the action S = Z x,(9, and the x's

n * 0 (rood p)

* E-mail address: tkyvax$hepnet::fukuma

** E-mail address: tkyvax$hepnet::kawai

*** E-mail address: nakayama@jpnkekvm.bitnet

1 We use the convention different from that in [4] where the roles of p and q are interchanged

372 M. Fukuma, H. Kawai, and R. Nakayama control the renormalization group flows corresponding to the change of the parameter q.

Although this formulationby the Schwinger-Dyson equation gives a systematic description, it is not completely satisfactory for the purpose of revealing the universal structures of the 2D quantum gravity. In fact, in contrast to the case of the q flows which we can easily control by varying the location of the background sources x,, we must change the form of the constraint itself in order to control the p flows. For example, while two theories parametrized by (p, q)= (2, 3) and (p, q)

= (3, 2) should be identical, the relation between two sets of the scaling operators {(9,},.o(~o~2 ) and {(9,},.mmo~3) is not clear. Thus we need a m o r e general framework for the 2D quantum gravity where the two parameters p and q can be changed freely.

In this paper, as a first step towards this direction, we investigate the symmetry structure of the 2D quantum gravity and show the existence of a large symmetry, the W1 + ~ symmetry I-7]. Using this symmetry structure, one can easily prove that the z function automatically satisfies the vacuum condition of the W 1 + o~ algebra if it obeys the string equation. Furthermore, we show that this condition is reduced to the vacuum condition of the Wp algebra when some redundant variables are eliminated. In this sense, the Wp symmetry in the

(p, q)

quantum gravity can be regarded as a by-product of the W1 + ~o symmetry. Although we have not yet found a final framework in which the p flows can be controlled, we expect that the exploration along this line leads to the final answer.

This paper is organized as follows. In Sect. 2, we summarize the Sato theory that describes the K P equation in terms of the infinite dimensional Grassmann manifold, and show how the string equations are treated in this formalism. In Sect. 3, after explaining the rule for translating the operators acting on the z function into the one-body operators acting on the one-body wave function, we introduce the wa + ~ algebra as a set of the one-body differential operators. Then we show that the string equation takes a simple form when it is expressed in terms of the universal Grassmann manifold. In Sect. 4, we show that this wl + o~ algebra becomes its central extension, the WI + 0o algebra, when it is represented as a set of differential operators acting on the z function. Furthermore, we prove that a z function of the p-reduced K P hierarchy satisfies the vacuum condition of the W~ + o~

algebra when it obeys the string equation. This form of the vacuum condition includes the variables, xp, x2p, ..., which become redundant when we take the conditions of the p-reduction into account. In Sect. 5, we then show that after the elimination of these redundant variables the vacuum condition of the W~ + algebra is reduced to the vacuum condition of the Wp algebra with central charge c = p - 1. Section 6 is for some concluding remarks. In Appendix A, for the sake of completeness, we explain the relation between the Lax operator and the z function.

Appendices B and C are devoted to the details of the calculations in Sect. 5.

2. The ~ Functions of the KP Hierarchy and the String Equations

In this section, we summarize the Sato theory that describes the KP equation in terms of the infinite dimensional Grassmann manifold [8-10], and show how the string equations are treated in this formalism.

Let H be the set of formal Laurent series:

Two-Dimensional Gravity 373 and V be a subspace of H which is spanned by linearly independent functions ~(~

= y. ~0 z - , - 1/2 (i = 0,1, 2,...) in H; V = [r176 r ~r .,.]. We call V

r E Z + 1/2

comparable when V is linearly isomorphic to the subspace H _ = ~f(z)= ~ a,z -~- 1/2~, and the universal Grassmann manifold ( U G M ) i s

( ^{r < 0} J

defined as the set of such comparable subspaces:

U G M = { V C H I V ~ _ H _ } . (2.2)

See, for example, [10] for more mathematically complete definitions.

In order to express the ~ functions in a compact form, we introduce fermionic operators ~p~, ~Ps (r, s ~ Z + 1/2) satisfying the following anticommutation relations {~P~, ~Ps} = firs, {Ip~, lp~} = 0, {~p~, lp~} = 0, (2.3) and define the zero-particle state 1 - c o ) as the state that satisfies

~pr[-- oo) = 0 ( V r 6 Z + 1/2). (2.4) We then make a correspondence between a vector r in H and a fermionic operator toil] t in the following way:

r Z ~,z-r-1/z~-WE~]t= Z ~,tPt, 9 (2.5)

r ~ Z + I / 2 r ~ Z + 1/2

Furthermore, we associate a comparable subspace V = [~(~ ~(l)(z), ...] of H with a decomposable multi-fermion state [g) via 2

V=[~(~162 ...] ~ l g ) =~p [~(~ Ip[~(1)]t...I - oo). (2.6) Obviously this correspondence between the U G M and the set of all decomposable states is one-to-one up to an overall factor. We denote the subspace V associated with the decomposable state [g) by V o. We further define the vacuum as the state of Dirac sea filled up to r = - 1 / 2 and denote it by 10):

10) - ~pt ~/2tpt 3/2.-.1- oo). (2.7) This state satisfies ~p~J0)=0 ( r > 0) and v;~[0)= 0 (r < 0).

The -c function is now defined as follows. First, we introduce the normal ordering for fermionic operators by

[lp~*p, (s>O) (2.8)

: W~s : - ( _ ~p~, (s < 0)' and define the current operators J . as

J , - ~2 : ~P~-.lP,: (n e Z), (2.9)

r

or equivalently

J(z) = Z J . z - " - ~ = : w t ( z ) ~ p ( z ) : , (2.10)

n ~ Z

Z ~prZ-r-t/2 and ~pt(z)= Z ~ptrz-'-1/2. Then, the 9 func-

r e Z + 1/2 r e Z + 1/2

where ~p(z)=

2 A multi-fermion state [g) is called decomposable if it can be expressed in the form of the right- hand side of Eq. (2.6). This state is nothing but the one that can be written as a Slater determinant

374 M. Fukuma, H. Kawai, and R. Nakayama

tion associated with a decomposable state [g} is defined as a function of infinitely many variables

x = (xD x2, x3, ...):

v(x)= (01exp ( , ~ i

x,J,)lg).

(2.11) Hereafter we will consider only neutral states, that is, J o l g ) = 0 .

Note that this function z(x) can be reinterpreted as the bosonic coherent representation of the state Ig). In fact, if we introduce a free boson 4,(z) via

a ~ (z) = J(z) = : ~*(z) ~p (z): (2.12) or conversely

~p*(z) = : e*(~) :, (2.13)

~p(z) =: e-O(~) :, (2.14)

then (01exp (,~1

x.J,)is

nothing but the coherent state of the free boson. Thus, the

/

following relations hold:

f ~. X n J n

x,J, /0m(0l e"el (m>0)

-=' J - (2.15)

(0le

, , - ~ ~ x.s.

(re<O)"

[Imlxlml<Ole"='

The p-reduction of the K P hierarchy is defined by imposing the following additional constraint on the state Ig):

J,plg> = 0 (n= 1,2,3, ...). (2.16) Due to (2.15), Eq. (2.16) is equivalent to

O,pZ(X)

= 0 (n = 1,2, 3,...). (2.17) Now let us consider the following differential operators

~'.- 89 2 klxkXt+ 2 kxkS,+89 Z t?kOt,

(2.18)

k + l = - n k - l = - n k + l = n

which satisfy the Virasoro algebra with central charge c = 1:

[~vn, ~--Cam]

=(n-m)~n+m+l-~(nZ-n)~,+m,o .

(2.19) We then impose the following condition on a z function of the p-reduced K P hierarchy 3

~ _ f i ( x ) = 0 . (2.20)

This z function can be identified with the square root of the partition function

Z(x)

of the (p, q) quantum gravity with source terms Z

X,(gn:

n * 0 (rood p)

Z(x)

= z(x) 2 . (2.21)

3 Note that xp, xze, xap .... are included in Eq. (2.18), which are redundant parameters under the p-reduction. However, in Eq. (2.20) they appear in the form 2px2pOp + 3px3pa2e + .... so that we can neglect them in Eq. (2.20). This kind of remark should be kept in mind in the following discussions

Two-Dimensional Gravity 375 One way to see this is to expand Eq. (2.20) around a background source

0 _

x ~ = t, xp+q~ = const, Xothors-- 0. (2.22) Using the equations in (A.2.17), one can show that the corresponding pseudo- differential operator L for this background satisfies [5]

(Lq) - 1 . . . (Lq) - (v - 2) = 0,

(2.23) (Lq)-(p - 1) = const 9 Xl,

which leads to the string equation of the 2D gravity coupled to (19, q) minimal conformal matters [4]:

[L p, (L q ) + ] = const,

(L p) _ = O. (2.24)

Furthermore, as we will show in Sect. 4 in a more general framework, ifa z function of the p-reduced K P hierarchy satisfies the condition (2.20), then it automatically satisfies the Virasoro constraint

{ ~ , , + ~ ( p 2 - 1 ) 6 , . o) z(x) = 0 (n= - 1 , 0 , 1,2,...). (2.25) On the other hand, for the ease of the one-matrix models one can prove that the set of Schwinger-Dyson equations for

z(x)

= ~ is equivalent to the Virasoro constraint (2.25) with p = 2 [5]. Since the Schwinger-Dyson equations contain necessary and sufficient information of the system, it is

expected

that the function

z(x)

satisfying the Virasoro constraint for p = 2 automatically becomes a z function of the 2-reduced K P hierarchy. However, in the cases of p > 3 the Virasoro constraint alone does not determine the system completely unless we use the fact that the

z(x)

is a z function. In these cases we thus need additional constraints on the

z(x),

and it was shown in [5, 6] that these constraints should be the vacuum condition of the Wp algebra:

W~)T(x) = 0 (k=2, ...,p; n > - k + 1), (2.26) where the w(~k)'s are the generators of the Wp algebra constructed via quantum Miura transformation in terms of the Zv-twisted scalar fields (see Appendix C). In fact, these conditions generically determine the system up to several non- perturbafive parameters, and in the particular case of the topological gravity [11], (p, q)= (p, 1), they enable us to calculate all the correlation functions analytically [5].

In the following sections, we will prove that the K P hierarchy has the W1 + oo algebra as its fundamental symmetry under every reduction, and that the function satisfies the formal vacuum condition of the W1 + o~ algebra if it obeys the constraints (2.17) and (2.20). Furthermore, we will show that they are reduced to the vacuum condition of the

Wp

algebra (2.26) after we eliminate the redundant variables for the p-reduction.

Before going through the proof, we make a crucial remark that the string equations (2.17) and (2.20) for the (/9, q) quantum gravity are expressed in terms of local fermion bilinear operators. In fact, the reduction condition (2.16) is stated in terms of a local fermion bilinear operator, that is, the current operator

J(z)=:~pt(z)~p(z):.

Furthermore, due to Eq.(2.15) and the bosonization rule

376 M. Fukuma, H. Kawai, and R. Nakayama (2.12)--{2.14), the Virasoro operator acting on the z function in Eq. (2.20) also turns out to be a local bilinear operator acting on [g):

where

Z xmJm

~.z(x) = <01 e" L . l g ) , (2.27)

T(z) = E L , , z - " - 2

n

= 89 :

J(z) 2 :

: 89

(: ~C(z)~(z):-: ~p*(z)~p(z):).

Remark. Rigorously speaking, the multi-fermion state ]g) given in Eq. (2.6) is, in general, ill-defined. In fact, the z function for the topological gravity ((2.1) quantum gravity) has the form [-5]

z(xa, x a , 0 , 0 , . . . ) = c o n s t x f l / 2 4 e x p [ 18x3j'x~ ] (2.28) so that we can not set x = 0. However, since we can expand it around a generic background source x = x ~ the state [g) becomes well-defined if we define the function by

(x. + x~

v(x) = (0le "~' ]g).

Furthermore, if we introduce J'. by

(2.29)

-- ~, x O J m E x ~

J'. - e " J.e"

~J. (n > O) (

J, + nx ~ (n < 0)'

(2.30)

then we have

( E (x,~+xR)Jn

(x- + x~ ~ %n(0l e"~' (m>0) (2.31)

(01e"--"

Jm= / ~ (x,+~o)j, ( m < 0 ) ' [ Imlxlml(0le "~'

which allows us to still regard (OI exp ( ~. (x, + x~ as the bosonic coherent

\n__> t J

state. Although we will formally set x = 0 hereafter, the generalization to the cases 0

x~ 0 is easily carried out by replacing the operator 0 which acts on the state [g) by

- E x O j m IgxOjm

O ' = e " Oe ~ (2.32)

For example, as for the Virasoro generators in Eq. (2.27), it holds that

(xm + x ~

~e,z(x) = (O.[e m L',lg) (2.33)

Two-Dimensional Gravity for

- E x~ E xOjm L'. = e " L . e"

377

(2.34)

3. String Equations in Terms of UGM

As we have seen in the previous section, fermion bilinear operators such as the J.'s and the L.'s play essential roles in the two-dimensional gravity. We call operators

W(z) local fermion bilinear of spin k if they have the following form:

k - 1

W(z)= 5~ cj:OiWt(z)O k - l - j W ( z ) : , (3.1)

j = 0

where cj are complex numbers. In particular, J(z) and T(z) are local fermion bilinear operators of spin 1 and 2, respectively:

J(z) =

E

J . z - " - 1 = : ~ * ( z ) t 0 ( z ) : ,

neZ (3.2)

T(z) = Y~ L . z - " - 2 = 89 O*/(z)*p(z) :-- : *pt(z)&p(z):).

r i t z

We also call W, local fermion bilinear of spin k if it is the n th mode of such an operator:

W(z)= y. W,z -"-k . (3.3)

n

In this section, we investigate how such operators act on a decomposable state Ig) and rewrite the string equations in terms of the UGM.

First we consider fermion bilinear operators of the general form:

O = E : ~p~Or~p~ :. (3.4)

r,8

The crucial point is that bilinear operators can be regarded as one-body operators in terms of the first quantization. In fact, one can construct the one-body operator o that corresponds to a fermion bilinear operator O by taking a commutator with the fermion operators %:

[%, O] = E O,~s, (3.5)

s

which means that the one-body operator o maps a one-body wave function f ( z )

= ~ a r z - r - n 2 ~ H to

r

of(z)~---~r (~s OrsSs)Z-r-1/2 ( 3 . 6 )

Conversely, if a one-body operator 0 is given, we can construct a fermion bilinear operator O uniquely up to an ambiguity of additive constant caused by the operator ordering. Furthermore, if fermion bilinear operators O1 and 02 correspond to one-body operators 01 and 02, respectively, then the commutator of O1 and 02 corresponds to the commutator of 01 and 02:

0 1 + ' + O 1 , 0 2 4 " + 0 2 =::> [ - 0 1 , 0 2 1 1 - - + [ - 0 1 , 0 2 ] . (3.7)

378 M. Fukuma, H. Kawai, and R. Nakayama Thus we have

Lemma 3.1. I f a set of one-body operator forms a Lie algebra w, then the set of corresponding fermion bilinear operators forms a central extension of w.

Furthermore, for any value of e the action of exp(eO) on a decomposable multifermion state Ig) is represented over the U G M as

e~~ ~->e~~ = [e~~176 e'~ ...] , (3.8)

where Vg = [r176 ~(1)(z) .... ] is the subspace of H corresponding to I g). F r o m this fact, it is obvious that the following lemma holds:

Lemma 3.2.

Olg) --constlg) ~ oVaC V 0. (3.9)

This lemma will be frequently used below.

If we restrict ourselves to the local fermion bilinear operators, the correspond- ing one-body operators become local differential operators with respect to z. In fact, for an operator having the form

w. = ~ ~ i w" +~-lw(w), dw

W(w) = : O~v*(w)O k- 1 -hp(w) :, (3.10)

the commutator with ~p(z) gives dw ^{n + k -}

[~o(z),

W.] = - ! ~ w l~o(z) : ~hp*(w)~ ~- 1-hp(w):

d j _ ( d ~ k - l - ~

= ( - 1 ) ~ (~-zz) z " + k l \ ~ Z / W(Z). (3.11) This equation indicates that the corresponding one-body operator is given by 4

d ^j _ l ( d ~ k - l - j w " = ( - 1 ) J ( d z z ) z"+k \dzzJ

l=0 \dzzJ ' (3.12)

where [ m ] , - m ! / ( m - n ) ! . In particular, the one-body operators corresponding to the current J , and the energy-momentum tensor L, are as follows:

d, ~ w(. 1) = z",

Ln.._>W(n2)=__[zn+ld n + l J

dzz + - - 2 - z" . (3.13) If we take into account all modes of all the local fermion bilinear operators, they generate a Lie algebra called the W~ + ~ algebra. F r o m Eq. (3.12) it is easily seen 4 If we define the z function as (2.29), then this expression for w. must be replaced by w'.

OXp( ~

Two-Dimensional Gravity 379 that the corresponding one-body operators form a Lie algebra of differential operators that is spanned by

z" ( n e Z , l ~ Z + ={0, 1,2, ..}), (3.14) which we call the wl + co algebra. Obviously the W1 + ~o algebra is a central extension of the wl + | algebra. In fact, the central charge c of the Virasoro subalgebra of W1 + co is 1, whereas that of w~ + co is 0.

Using these one-body operators, we can express the string Eqs. (2.17) and (2.20) for the (p, q) quantum gravity in terms of the U G M . As is shown in the next section, these two equations are equivalent to the following weaker conditions on the z function:

0ff(x) = const z(x), 5e_ fi(x) = const z(x), or equivalently,

Jnlg) = const Ig), (3.15)

L_ pig)

= const ]g). (3.16)

Then Lemma 3.2 together with the relations in (3.13) tells that these equations are equivalent to

w(pl)Vo = zPVo

C V 0 , (3.17)

[z_p+ 1 d . - - p + l

]

w(_2) Va = + z - . / voc v.. (3.18)

i_

Therefore the subspace V 0 is invariant under any differential operator that is constructed by repeatedly taking products and linear combinations of w~ 1) and w~)p. In other words, if we define r § (p) as the associative subalgebra of differential operators that is generated by @1) and w~)p, we have

oVocV o for

Voer+(p).

(3.19)

Since the commutator of w~ 1) and w~)p is a c-number

[w(pl),lpw~)pl-

= 1, (3.20)

(1) n (2) l

and the differential operators

(wp) (w_.) (n,/=0,

1, 2,...) are all linearly inde- pendent to each other, any element of r (p) is uniquely expressed as

~ t,,,(1)~.~,,,~2)~l (3.21)

t, nl~r p ) ~VV_p] , n , l = O

where the c j s are complex numbers. Thus we have the following theorem:

Theorem 3.3.

Let z(x) be a z function of the KP hierarchy that satisfies the string equations of (p, q) quantum gravity

(3.15)

and

(3.16).

Then the corresponding element V o of the UGM satisfies

r+(p) V o = V o , (3.22)

~nltTVp ! t ' ~ - - p ] , C n l ~ C . n , I = O

380 M. Fukuma, H. Kawai, and R. Nakayama

As is shown in the next section, we can reinterpret Eq. (3.22) in terms of the z function by using Lemma 3.2 again. There we will see that the structure of the W1 + oo algebra arises in a natural way. Here, before going back to the T function, we analyze the Lie algebraic structures of the set r+(p) of one-body operators that appears in Eq. (3.22).

First we introduce the following notations for p____2:

r = w l + ~ 1 7 6 n~z /ez+2 Cnl Zn -- ~ , (3.23)

r + = w ~ + ~ = Z c . t : - ~ , (3.24) n + /eZ+

{

^{(t). (2) (3.25)}

r(P)=%+~176 ~z X c.t(wp ) w_v ,

n feZ+

{ oz

C [W (l,'n ( W (2, ~ ~

^-,:

r+(p)=w~+~(P) = Z § t~z+Z n~, p , J . (3.26) Here, r, r +, r(p), and r+(p) are, as a set, identical to %+oo, w~+,, Wl+oo(p), and w++ ~o(P), respectively. We introduce, however, different symbols for them in order to indicate whether we regard them as associative algebras or Lie algebras. In other words, when we call them r, we consider not only commutators but also products as differential o p e r a t o r s : In addition to the trivial relations

r + Cr, r+(p)Cr(p)Cr (as associative subalgebras),

(3.27)

W + + o o C W l + ~ ,

W~[+~o(p)Cwl+~o(p)Cwl+~o

(as Lie subalgebras),

we have the following isomorphisms, which indicate that the w~ + ~o algebra has an infinite-fold self-similar structure:

L e m m a 3.4.

r(p) ~- r, r +(p) ~ r + (as associative algebras),

(3.28) wl+o~(p)~_wl+oo, wl + o~(P) = wl + oo + ~ + (as Lie algebras).

Proof As is clear from the definition, r is generated by w~. ~) = z" (n 9 Z) and w~]

= - d/dz, and the structure of r is completely specified by the following relations among them:

w(1)_.,,m~. n - - \ ' V l 5 , (3.29) [,.m ,,(2) q _ 1 rv I , r V - l _ l - - ~ . (3.30)

a (2)

O n the other hand, r(p) is generated by w~lp ) = z "p (n e Z) nd W_p with the relations

( I ) _ (1) n

w . ~ - ( w ~ ) , (3.31)

[w~'),~w~)pJ = 1 . (3.32)

5 In fact, the so-calledlone-starproduct ofthe W1 + oo algebra [7] is nothingbut the usual productin r, when we translate the set offermion bilinear operat ors, W 1 + ~,int o the set of corresponding one-body operators, w 1 + oo

Two-Dimensional Gravity 381 Therefore

r(p)

is isomorphic to r as an associative algebra through the following mapping:

w i l ) ~ W(PI)I"~ N i l ) ' (3.33)

2 1

~(

-1--p

) = - w ~ w ~ .

Furthermore, since r+(p) is generated by @1) and wt_2)p, and r + is generated by w] 1) and

w~],

these two associative algebras are isomorphic under the above mapping. []

To express this isomorphism more explicitly, we note the following relation:

, )

z - P + 1 _ _ + Z - p = z ( P - 1 ) / 2 _ z - i v - 1)/2

p w _ . - - p dz

(3.34)

#]1) = w~l) = z p = z(p- 1)/2(2) z-{p- 1)/2, (3.35) where 2 = z p. Then we see that the

wl+oo(p)

algebra and the Wl+~ algebra are related via

Wl + ~(p) -- ztp - 1)/2(Wl + ~ Iwith z rophced by Z) z - {p - 1)/2 (3.36) In order to see how wl § ~(p) is imbedded in wl § o~ more explicitly, we introduce the following basis of the wl § ~ algebra:

( a ? - ' ,

k-1 1 ([k--1]l) 2

[n+k_l]lzn§ z

(3.37)

W~)=(--l)k-1 t~o l!

[ 2 k - 2 ] , \dzzJ ' which corresponds to the standard basis of the W1 § co algebra 6 [7]:

w(k)(z) = ~ w ( k ) z - n - k = 2k_l(2k_3)[! ~

(--1) j

. :~k-l-Jlp~f(z)~J~t)(z)'.

, j=o (3.38)

Then by using Eq. (3.36) the corresponding basis of Wl § ~(p) is expressed as

k - 1 ( d ~ k - l - I

1 - l Z - ( p - 1 ) / 2 .

1 ( [ k - 1 ] t ) 2

[n+k_1]tztp_~)/22,+k_ \ ~ J

(3.39) wtnk)=(--1)k-1 t=~0 l! [ 2 k - 2 ] t

The right-hand side of this equation can be expressed as a linear combination of the w~k)'s. For example,

~(1) = w(1) _n " t i p , (3.40)

2 1

~( )= - w r ( 3 . 4 1 )

n P "'np

w. y ~w~. + (p2_l)w~.~) , (3.42)

w-(4)= 1 ( ' ( a ) ~ 0 _{. b v}

~..

⁺ ( p 2 _ l ) w , 2 } . (3.43) 6 Here, (--1)!!---1

382 M. Fukuma, H. Kawai, and R. Nakayama As a final remark in this section, we point out the following properties of w~-+

and w++ ~o(P), which we use in the next section:

Lemma 3.5. Any element of w++| can be expressed as a commutator of two elements of w++ o~. The same statement holds for w~+ oo(P).

Proof For any element of w++ 00, we have

I

Z Ck, zZ k = Ck, l Z k+ 1 (3.44)

which proves the assertion for wi~+~. The same assertion holds for w++o~(p) because it is isomorphic to w~+ ~o. []

4 . W 1 + ~ Constraint

In the preceding sections we have investigated the structure of the string equation in terms of the U G M . In this section we will rewrite Eq. (3.22) as a set of differential equations for z(x).

As we have seen in the previous section, if a set of one-body operators is given, we can construct a corresponding set of fermion bilinear operators up to the operator ordering ambiguities. One way to fix the ambiguities is to introduce the usual normal ordering for the fermion operators (2.8). In fact, the W1 + ~o algebra can be regarded as the set of fermion bilinear operators constructed from w~ + ~o through this normal ordering. Similarly, we define W1 + | as the subset of W~ + ~o that is constructed from w~+ ~ by using the same normal ordering. As is clear from Eq. (3.12), WI++ o~ is spanned by w,~k)'s satisfying n > - k + 1. A crucial property of the W~++ | is the following:

Lemma 4.1. W1 + o~ forms a Lie algebra without a central term. In other words, Wa + oo closes under the commutator.

Proof Since w++ oo is a Lie algebra, it is obvious that W~ + ~ is a Lie algebra with possible central terms. In order to show that the central terms vanish, we consider the c-number term of the operator product expansion between fermion bilinear operators wIk)(z) and W(~ of spin k and l, respectively:

w(k)(z) W(l)(w) = const (z -- w) k +~ + (fermion bilinear operators), 1 (4.1)

which means that the e-number term of the commutator [W, ~k), W~ z)] is given by const[n+k--1]k+l_16,+,,,o=COnst(n+k--1)...(n--l+l)6,+m, o . (4.2) Then it is easy to see that one of the factors in Eq. (4.2) vanishes if n > - k + 1 and m > - l + 1. Thus we find that W~ + ~ closes under the commutator. []

We then define W1 + ~(p) and Wa++ ~o(P) as the sets of fermion bilinear operators that are constructed from wl + ~(P) and w~+ o~(P), respectively. Here again we have the operator ordering ambiguities for the 0 th modes of the fermion bilinear operators. We can, however, show the following:

Lemma 4.2. There exists such a proper definition of the 0 th modes that Wx++ ~(p) closes under the commutator. Furthermore, any element of W++ ~o(P) can be expressed as a commutator of two elements of W~+ ~(p).

Two-Dimensional Gravity 383

Proof

The isomorphisms (3.36) between w 1 + | and wl + o~(P) can be regarded as a conformal mapping from z to 2=zP:

(d2~ -1/2

,p'(2) =

\dzJ ,p(z).

(4.3)

Therefore, if we fix the operator ordering by introducing the new normal ordering as the subtraction of the singular part in the (p-sheeted) 2 plane, such as

~176 ~176 ='P't(Z')~P'(2)- 2 ' - Z' 1 (4.4) then the structure of the c-number terms of the operator product expansions for W1 + | in the 2 plane is exactly the same as that of W, + oo in the z plane. Thus the same argument as in Lemma 3.5 leads to the closedness of W1 + ~o(P) under the commutator. The latter assertion follows immediately from this fact and Lemma 3.5. []

As is well known, the difference between the 2 plane normal ordering and the usual normal ordering can be calculated as the Schwarzian terms associated with the transformation

z~--~Z=z v.

For example, the generators of the Wx+oo(p) corresponding to (3.40)-(3.43) take the following forms:

1~, (1)= W, ~ , (4.5)

ffV,(z)= ~ {W,(~) + l (p2-1)6,,o},

(4.6)

1~-'3)= l ~ W - ( a ) + I~(pE--1)Wn(vl) }

pZ ~ ,v

.

,

(4.7)

1 7 2 2 .

VVn(4)=-~(Wn~)W7(p2-1)Wn(2p)-b~(p-1)

6n, O} (4.8) Note the appearance of the additional c-number correction terms compared to (3.40)-(3.43). It is obvious from the construction that they satisfy the commutation relations of the I4"1 + co algebra with central charge c = p.

After the rather lengthy preparation given above, we can finally prove that a z function of the p-reduced K P hierarchy satisfies the vacuum condition of the W~ + oo(P) algebra when it obeys the string Eq. (2.20). Here, we can start with weaker assumptions

Qpv(x)

= const

z(x),

(4.9)

~e_ pz(x) = const

z(x),

(4.10)

or equivalently

Jplg> = const [g>,

L-pig>

= const Ig>-

In Theorem 3.3, we found that these equations are equivalent to r+(p) v. = w++ o (p) v. = v o.

(4.11) (4.12)

(4.13)

384 M. Fukuma, H. Kawai, and R. Nakayama By using Lemma 3.2 again, this means that [g) is a simultaneous eigenstate of w~+. o~(p):

O l g ) = c o n s t l g ) for voE w~++o~(p). (4.14) Furthermore, Lemma 4.2 asserts that all of these constants vanish. In fact, any element 0 of WI++ o~(P) can be expressed as

0 = EO1, 0 2 ] , 01, 02 ~ Wl% oo(P) 9 (4.15) Therefore, we have

Olg)= O1021g)-

0 2 0 1 1 g ) = 0, because Ig) is a simultaneous eigenstate of W1 + oo(P).

We thus have proved the following theorem.

Theorem 4.3.

Let z(x) be a T function of the KP hierarchy that satisfies the conditions

(4.9)

and

(4.10).

Then the decomposable fermion state

]g)

corresponding to the z function satisfies

O I g ) = 0

for vo~wl+oo(p),

(4.16)

that is,

g/,(k)lg)=0 ( k = 1 , 2 .... ; n > - k + l ) . (4.17) This theorem can be restated in terms of bosons as follows. By using the bosonization rule (2.12)-(2.14), the normal ordering of two fermion operators

: 8klp *(W) 0tip (z) : (4.18)

is expanded in powers of w - z as

_

k t ( 1 :e4,(w)_O(z) : 1 )

-OwOz w - z w L z

\ /

- - - Z Z ( - 1 ) ' 1

j=l ,=o

\ / j . ( j - l - k - r ) ! (W--Z)J-I-k-'oZ-'Pti)(Z)'

(4.19) where

PtJ)(z)

is defined by

P(J)(z)

- : e-r :, (4.20)

and :: stands for the normal ordering for bosonic operators. Considering the coefficient of

( w - z ) ~

in Eq. (4.19), we have

:ok~p*(z)t?~tO(z):= •

(--1) J - l - k

ok+Z-J+Ip(j)(z).

(4.21)

j=~+l -- - k

Thus, by substituting this expression into Eq. (3.38), we obtain the bosonic realization of the W1 +~o algebra:

k-1 (--1) / ([k--l]/) 2

Oip(k_l)(Z),

W(k)(Z)= ~=0Z

(k-1).l!

[ 2 k - 2 ] t

the first few of which are

W(l)(z)= J(z),

w ( : ) ( z ) = 89 : J(z) 2 :, W(3)(z) = 89 : J(z) 3 :,

w(4)(z) = 88 [: J ( z p : + ~ :J(z)O~J(z): - ~: (0J(z)) ~ :].

(4.22)

(4.23) (4.24) (4.25) (4.26)

Two-Dimensional Gravity 385 Then by using the equations such as (4.5)-(4.8), the generators of W1 + ~o(P) are expressed as follows:

1,~1) = j~p, (4.27)

pVg(e)=~a+~b=np:JaJb: q- 2~ (p2-- 1)On, O, (4.28) 1 ~, :JaJbJ c . + ~__~ (p2 __ 1)J,p, (4.29)

1 1

: - ~ "JaJbdcJd : - Z ( a + l ) ( b + l ) :JaJb"

P31'~(4) 4 a+b+c+d=np 4 a+b=np

+ ~ d ( n p + 2 ) ( n p + 3 ) F, "J,Jb: + (P 2 - 1 ) 2 "J,Jb:

a+b=np a+b=np

7 2

+ 960 (p - 1)z 6"'~ (4.30)

Hence, Eq. (4.17) is rewritten as a set of differential equations for the z function:

"~/',(k)z(x)=O (k= 1,2, 3, ...; n > - k + 1), (4.31) where the differential operator ~/~(k) is obtained by replacing all the J,'s in the ff-~k)

by

{~.

(n > 0)

J ~ = [nlxl, I (n<0)" (4.32) Note that the first equation ~/~,mz(x)= ~ z(x)=0 shows that the z(x) is a z function of the p-reduced KP hierarchy. As was discussed in [5] it is expected that a function z(x) satisfying Eq. (4.31) automatically becomes a z function of the KP hierarchy, since Eq. (4.31) determines the function z(x) completely at least for the case of topological fields [53.

5. W~ Constraint

We have shown that a z function of the KP hierarchy under the conditions (4.9) and (4.10) is a z function of the p-reduced KP hierarchy which satisfies the vacuum condition of the W~ + oo(P)- However, in the expressions (4.30)-(4.32) there appear redundant variables for the p-reduced KP hierarchy, that is, J,p (n e Z). In this section we show that after the elimination of these redundant variables the W1 + algebra with central charge c--p is reduced to the Wp algebra with c - - p - 1 .

As we have seen in the previous section, the generators of ~ + | have simple forms when they are expressed on the p-sheeted 2 plane which is the image space of the conformal transformation z w.2=zP. More explicitly, we first define the operators w,(k)(2) on the p-sheeted 2 plane by Eq. (4.22) with z replaced by 2:

k-~ (__l)t ([k__lJl)2c3~p(k_Z)(2) '

W(k)(2)= ~=0Y' ( k - l ) ' l ! [2k-2]~

(5.1)

P~J)(2) = ~ e-*(~)~{e *(~) ~ = ~ (~ + J(2)) ~ ~ 1. (5.2)

386 M. Fukuma, H. Kawai, and R. Nakayama Here, aa~b(2)=J(2)=

7~2-n/P-lJn,

1 ^{and o ~ o} ~ stands for the "minimal" normal ordering on the p-sheeted 2 plane, by which we mean the following procedure:

8J(2) o ~ = J(2), (5.3)

~J(2)2~ = limz'-,z

{J(2')J(2)-

(2 Z1 2) 2'} (5.4)

f

~J(2)3~ = lim

~J(2")J(2')J(2)

2"-'* A

(

2"~2

1 J(2) 1 J(Z) 1 j(2,,) t (5.5)

( , v , _ 2,) 2) ( 2 , - 2 ) _ ,

Since we are considering the p-sheeted 2 plane, each value of 2 corresponds to p different points, which we denote by 2~ .... ,2p. Then the generators of Wt + oo(P) are expressed as

~'r(k)(2) = E ~ ( k ) 2 - n - 1 = w ( k ) ( 2 1 ) + w ( k ) ( 2 2 ) n k . . " -t- w(k)(,~,p). (5.6)

n

Note that this expression gives a single-valued function of 2 because the right-hand side is invariant under the transformation: 2~-,exp[2rci]2, which generates a permutation of the 2fs.

In order to investigate further the structure of the

W~+oo(p)

constraint, we introduce the elementary symmetric polynomials of J(201 ..., J(2p) as

S(k)(A) = Y. J(2h)... J(2~k ) (1 < k__< p). (5.7)

l =<ii < i 2 < ... <ik<~p

Here, the ordering of the operators on the right-hand side need not be specified because we have for two different points 2 and 2' on the p-sheeted 2 plane

[J(2'),J(2")]=0 if 121=12'1. (5.8) Furthermore it is apparent that the S(k)'s are single-valued functions of 2 for the same reason as in Eq. (5.6). Next we introduce another type of product for two local operators on the single-sheeted 2 plane as

d2' 01(2')02(2 )

(0~(2), 02(2))= ! 2rci 2 ' - 2 (5.9)

Although this product is neither commutative nor associative, it plays a crucial role in the following argument 9

For any set 6 ~ of local operators on the single-sheeted 2 plane, we can construct an algebra R [6e] of operators by repeatedly taking A-derivatives, linear combi- nations and the product (5.9). Then it is expected that the following statement holds, although we do not have a complete proof:

Lemma 5.1 (conjecture).

R[{S(k)(2);k=l,2,...,p}]=R[{ff~(k)(2);k=l,2,...)].

(5.10)

In other words, tim) e R

[{s(k)(2); k = 1, 2,..., p}]

and S a) e R

[{ ff'(k)(2); k = 1,2,...}].

Two-Dimensional Gravity 387 This equation becomes obvious if we ignore all terms including 2-derivatives of J(2). In fact, we have

ff~k)(2 ) = 1 ~ oj(2i)ko ~ + terms including 0J(2), 02J(2), ..., (5.11)

/ = 1

and the product (5.9) is equal to the normal ordering up to terms with derivatives of J(2), e.g.,

(s(k)(2), S(1)(2)) = ~ s(k)(2)S(0(2) ~ + terms including ~J(2), 82J(2) ... (5.12) Therefore, if we keep terms only with the form of polynomials of J(2), the statement of Lemma 5.1 is nothing but the fundamental theorem for symmetric polynomials.

The remarkable fact is that the use of the product (5.9) gives Eq. (5.10) exactly. As is shown in Appendix B, we have checked this for operators of spin k = 1, 2 ... 6. The explicit relations between the ff'(k)'S and S(k)'s are given by, for example,

W O ) = S O ) '

2171,7( 2 ) = (S 0), S (1)) - 2S(2),

3 if-ca) = (S(1), (S(*), S t 1))) _ 3 (S (1), S (2)) + 3S ~3) ,

(5.13) 41~ (4) + 3 ~21~(2)

= { 89

(s"), s")))) + 89 s")),

(1) (1) (2) (1) (i) (2) (2) (I) (i)

- { 2 ( S ,(S ,S ))+((S ,S ),S ) + ( S ,(S ,S ))}

+ 4(S (1) , S (3)) + 2 ( S (2) , S (2)) - - 4S(4).

Assuming the correctness of Lemma 5.1, we have Theorem 5.2. The vacuum condition of the W 1 + 0o algebra

W~tk)/g) = 0 (k=1,2,3 .... ; n = - k + l ) (5.14) is equivalent to

S~)[g> = 0 ( k = l , 2 , 3 , . . . , p ; n > - k + l ) , (5.15) where stk)(2)= ~ 2-"-kS~ k).

n

Proof. We introduce such an abbreviation as u(k)lg> = O, ifa state Ig> is annihilated by the modes of spin k operator u(k)(2) with n___ - k + 1, that is,

U~,k)lg>----0 (for n _ _ > - k + l ) , (5.16) where U (k) = ~ 2-"-ku~,k). Obviously, we have

Ii

(i) U~*)lg> = 0 ~

(O'U~))Ig>

= 0 , (5.17)

(ii) utk)lg>=0, V(k)lg>=0 ~ (aU~k)+bV~k))lg>=O.

Furthermore, since the ^{n th} mode of the product of operators U (k) and V (~ of spin k a n d / , respectively, is given by

k) Z k) (5.18)

(U ~k), V~O). = y~ - . - q q _

q>=k+n r>= - k + 1

388 M. Fukuma, H. Kawai, and R. Nakayama and the left-hand side is an operator of spin k + l, we have

(iii) u(k)lg)=0, V ( ~ ~

(U (k),

V ~ (59

Lemma 5.1 together with (i), (ii), and (iii) proves the statement of this theorem 9 []

Finally we show that the condition (5.15) becomes the vacuum condition of the W v algebra after the elimination of the variables Jnp (n ~ Z) which are redundant for the p-reduced K P hierarchy. In order to eliminate these variables, we first expand S(k)(2) as a sum of operators with the usual normal ordering for the oscillators J,.

For example, we rewrite s(k)(2) a s p

i = 1

S~2)(2) =

S~3)(,~) =

s J(,~,)J(zj)

l~i<j<--p

Z :J(2i)J(2j): + 2 (J(2i)J('~j)), l <i<j<_p l <i<j<=p

l<_i<j<k<_p

1 < - i < j < k < = p 1 < = i < j < k < = p

+ (J(2~) J(2k)) J(2j) + (J(2/) J(2k) ) J(2~)},

(5.20)

(5.21)

(5.22) recall that J(2)= 1_

where :: is the usual normal ordering for the J,'s P

Zn 2-"/v-lJ,),

and ( ) is the usual vacuum expectation v a l u e ,

(JnJm) = n6, + m, o O(n).

We then define w(k)(2) by formally setting

J,p (n c

Z) to 0 in this normal ordered form of S(k)(2). Note that W")(2) vanishes identically, because S(1)(2) = Z 2 - " -

1j,p

contains only the variables to be dropped 9 As is shown in

n

Appendix C, these W(k)(2)'s are identified with the generators of the Wp algebra with Virasoro central charge c = p - 1 that is to be constructed from Z f t w i s t e d free bosons. Furthermore we have

Lemma 5.3.

The condition

S~,k)]g)=0 (k= 1, 2 .... , p; n > - k + 1)

is equivalent to {Jnplg) = 0 (n >= O)

(5.23) W,~)lg) = 0

( k = 2 , . . . , p ; n > - k + l ) .

Proof.

First we note that S~ ) can be expanded with respect to Jnp (n E Z) as

k

st, k)=w,(k)+ • ~

C(~-t~)l+...+,,)[na,...,nz] :J, lp...J,w:.

(5.24)

/ = 1 n l . . . n t ~ Z

Here the operators

t'(k-l)

"-',-(,i+... +,,)Enl .... ,nl] do not contain

J,p (ncZ),

and are obtained from S~ ) by repeatedly taking commutators with the J,p's (n ~ Z) and then setting

J,p (n ~ Z)

to 0:

c ( k - I) n - (nl + . . . + n,) [n l . . . . , n~] = const [J_,,p, [ .... [J-,~v,

s~,k)] ""

"]] [s.p : o ~, ~ z) (59 This implies that

,--(k- 0 . 9 : c o n s t W n _ ( , , + . . . + , , ) , (5.26)

~ n - (n~ . . . . + nl) I n 1, .., n t ] ( k - t)

Two-Dimensional Gravity 389 because we have

[Jnp, S# ) ] -- L-,F'~(1), --,,stk)lj = (P + 1 -- "'~"~'m+n~"~(k - 1), (5.27) which follows from

p + l - k ~k 1)

S(1)(2')S~k)(2)- ~ , ~ S - (2) + (regular terms). (5.28) The assertion follows immediately. []

Thus combining Theorem 5.2 and Lemma 5.3, we obtain the following theorem:

Theorem 5.4. Let z(x) be a 9 function of the K P hierarchy that satisfies the conditions (4.9) and (4.10). Then the corresponding state Ig) satisfies the conditions of the p-reduction

Jnplg) = 0 (n > 1) (5.29)

and the vacuum condition of the Wp algebra

W(,k)lg) = 0 ( k = 2 , 3 , . . . , p ; n > - k + l ) . (5.30) By reinterpreting the oscillators J,'s as the differential operators J , ' s acting on r(x) [see Eq. (4.32)], the set of Eqs. (5.29) and (5.30) are rewritten in the form of differential equations for v(x). Thus we have seen that the statement conjectured in [5, 6] follows naturally from the infinite dimensional Grassmannian structure of 2D quantum gravity. 7

6. Conclusion

In this paper, we showed that all the quantities of 2D quantum gravity can be expressed in terms of the infinite dimensional Grassmann manifold. Then, we found that every matrix model has the W1 + ~ algebra as its fundamental symmetry, and that the z function obeying the string equation satisfies the vacuum condition of the W1 + o~ algebra. Furthermore, if we restrict ourselves to the 2D gravity coupled to (p, q) conformal matters, then the vacuum condition of the W 1 + oo

algebra is reduced to the vacuum condition of the Wp algebra with central charge c-- p - 1. In this sense, it seems that the real universal symmetry of 2D gravity is the W1 + ~o algebra and that the Wp algebra which appears in the (p, q) quantum gravity is merely a by-product of this symmetry. Thus we expect that a deeper investigation of this structure will lead to a formalism in which we can control the renormalization group flows that correspond to the change of the parameter p.

Furthermore, it might be useful for writing down the exact solutions of the non- perturbative 2D gravity coupled to the c--1 conformal matters. Among other interesting problems are the generalization to the D- and E-type W1 + | algebras and their application to 2D quantum gravity. Moreover, it has been reported recently that the relations among the orthogonal polynomials of matrix models are reduced to the Toda lattice hierarchy [13] (see also [14]). Their observations should also be understood in terms of infinite dimensional Grassmann manifolds.

v See [12] where some analyses are made based on Hirota's bilinear equation

390 M. Fukuma, H. Kawai, and R. Nakayama Appendix A. Explicit Construction of the T Functions

In this appendix, we explain a method of constructing a z function directly from a solution of the K P hierarchy.

A.1. The K P Hierarchy and the Sato Equation

In this subsection, we define the K P hierarchy and derive the Sato equation which plays an important role when we interpret the K P hierarchy as a dynamical system over an infinite dimensional Grassmann manifold I-8-10].

First we introduce functions ui(t, x) (i=2, 3,...) of infinitely m a n y variables (t, x) = (t, xl, x:, xa,...) and define the Lax operator

L =. 0 + U2(t , X)~3-1 -b U3(t , X)0 - 2 + . . . (A.1.1) as a pseudo-differential operator with respect to O -= 0/0t. Here, 0-1 is defined so that 0 0 - 1 = 0-10 = 1. Explicitly, we have for any function f

o k . f = ~ (kl) Otf.Ok_t,(kl) k ( k - 1 ) . . . ( k - l + l )

l=o ~ - - l! (A.1.2)

We further introduce the potentials B, and B c as

B, - (L") +, B, = - ( L " ) _ , c - (n.l.3) where ( )+ (respectively ( ) _ ) is non-negative (negative) power part of a pseudo- differential operator with respect to 0. Then the K P hierarchy is defined as the set of the following differential equations:

- - L= [B c, L] (= [B,, L]). 0 (A.1.4) 0xn

Note that they satisfy the integrability conditions O2L/Ox,Ox,, = OZL/OxmOXn. The following theorem is fundamental.

Theorem 1. There exists a pseudo-differential operator W of the form

W = 1 + w l ( t , x ) ~ - 1 + Wz(t, x ) ~ - 2 + ...

such that

(A.1.5) (A.1.6) (A.1.7)

L= W a W -1 ,

- - W = B C W = B , W _ W O .. 0 OXn

Equation (A.I.7) is called the Sato equation.

Proof First we introduce a group f# and its Lie algebra Lief# as the set of elements of the following forms:

(r + va(t)O- l + v2(t)O- 2 +...} ,

Lie ff = { b l (t) 0 - 1 + b 2(t) 0- 2 + . . . }. (A. 1.8) Then the B ~ can be regarded as the components of a Lie fr 1-form f2 c on the space of the ~,'s:

0 c= • BCdx,. (A.I.9)

n>l

Two-Dimensional Gravity

Then the K P Eq. (A.1.4) is rewritten as d L = [ 0 c, L ] ,

391

d=- ~ d x , (A.I.10)

n ~ 1 ( ~ X n "

0 C

Furthermore, by using an identity ~y~. B k

=([BI,

BkC])_, one can easily show that Eq. (A.I.10) leads to the zero-curvature condition

d•C = ~ c ^ Oc. (A.1.11)

Thus the connection o c is a pure gauge and can be written in the form

~ c = _ V - ldV, (A.1.12)

where V is a if-valued function of the x,'s. It is also easy to see that d ( V L V - 1) = 0, which indicates that the V L V - 1 has the form

V L V - 1 = 0 + r2(t ) a - 1 + ra(t ) 0 - 2 + .... (A.1.13) Therefore, V L V - 1 can be expressed as

V L V - x = U O U - ~ (A.I.14)

using an element U of if which depends only on t:

U = 1 +pl(t)0 -1 +p2(t)O - 2 + . . . . (A.1.15) Hence, if we denote V- 1 U by W, that is,

W = _ V ( t , x ) - i U ( t ) - l + w l ( t , x ) O - l + W E ( t , x ) O - 2 + . . . , (A.1.16) then the following relations hold:

L = W O W - i , (A.1.17)

g2 c = - V - l d V = - W d W - 1 = d W . W - i. (A.I.18) The latter equation d W = O C w is nothing but the Sato equation

0

Ox, W = B C W = B " W - W O " " []

Remark. F r o m the identity Ba = (L)+ = O/Ot, we obtain 0 L = [ B ~ , L ] = ~

0x-~ ~ L . (A.1.19)

Thus every function f appearing in the K P equation depends on the variables (t, x) in the following manner:

f = f ( t + x l, x 2, x 3 .. . . ), (A.1.20)

so that we can (and will) set t = 0 without loss of generality. Then ~ is interpreted as a differential operator O/axl.

392 M. Fukuma, H. Kawai, and R. Nakayama

A.2. z Functions and an Infinite Dimensional Grassmann Manifold

In this subsection, by using the Sato equation, we show that the K P hierarchy is nothing but a d, ynamical system over an infinite dimensional Grassmann manifold [-8-10]. Furthermore we explain that all the unknown functions u~(x)'s can be described in terms of a single function, Hirota's z function [15]. In what follows, we assume that the functions wjx) can be taylor-expanded around a point

X 0 ~ (x~, xz,...). In particular, we restrict ourselves to the case x ~ 0. Generaliz- 0 0

ation to the cases x ~ 0 is straightforward.

Let H be the linear space consisting of pseudo-differential operators, which is isomorphic to C z.

o

and H_ be a linear subspace of H consisting of all the differential operators. Then all the linear subspaces of H which are linearly isomorphic to H_ make an infinite dimensional Grassmann manifold, and we denote it by U G M (Universal Grassmann Manifold). s

Now we make a mapping from the set of the solutions of the K P equation { W) into a set of orbits in the U G M in the following way. First we construct the vectors q~")(x) (n=0, 1,2 .... ) in H as those whose components are the coefficients of the pseudo-differential operators O"W:

a " w = Z ~ - 1 / ~ ( ~ ) 0~,

k ~ Z

.(")(x) = (q~"~(x))r z + 1/2 =

(A.2.2)

q ~ m ( x ) (A.2.3)

Then we define the subspace spanned by these vectors ~/~~ ~/~l)(x) .... and denote it by V(x):

V(x) = [q(~ q(1)(x),...]. (A.2.4)

Note that we can represent the subspace V(x) as a Z x Z + matrix

~(x) = [n~~ (A.2.5)

Obviously, this matrix has an ambiguity of the right-multiplication of GL(Z+), which corresponds to changes of the basis of V(x).

The time (x) evolution of the V(x) in U G M is, in general, determined from the Sato Eq. (A.1.7). This evolution can be represented in terms of the corresponding matrix in the following form:

- - Z x n A n

tl(X) = e "~_1 t 1(0) C(x), C(x) E GL(Z +), (A.2.6)

8 For more mathematically complete definitions, see [10]

Two-Dimensional Gravity 393 where A ⁼

(Akl)

⁼ ((~k, l + 1) E GL(Z). In fact, since the first term on the right-hand side ofEq. (A.1.7) has the form B , W = Y, b,4OJW, it does not change the subspace V(x)

1_>o

itself and gives rise to the factor C(x) in Eq. (A.2.6). Hence x-evolution of the V(x) comes only from the second term - WO" in Eq. (A.1.7), which is integrated into exp [ - ~ xnA"] in the matrix representation. Thus we have seen that the KP hierarchy is nothing but a dynamical system over the U G M whose time evolution is given in a simple form exp [ - ~ x.A"].

We next show that a solution of the KP equation, W, which has infinitely many unknown functions wj(x), can be expressed by a single function, z function. First let us introduce a matrix q(x), which also represents the subspace V(x), as

- - F, x n A n

q(x)-e

"-->'

t/(0), (A.2.7)

and further we decompose it in the following way:

q(x) = Lq-(x)J'

q + (X) ~-" (ll~n)(X))r > O,n >- O, q _ (X) ~ (ll~n)(x))t < O,n >- O" (A.2.9) Then the z function corresponding to the subspace V(x) is defined as

z(x) = det q_ (x). (A.2.10)

This ~ function completely reproduces the solution of the KP equation due to the following theorem:

Theorem 2. Let "c(x) be the z function which corresponds to a solution of the Sato equation, W = ~ wj(x)O -j. Then it holds that

j>=o

Y. w ,(x) k - i = z ( x - e(k- 1))/z(x) , (A.2.11)

j>=o

where x - e(k- 1)= (xl - 1/k, x 2 - 1/(2k2), x 3 - 1/(3k3), ...).

Proof First, if we rewrite q(x) as

q(x)= [q +(x)q-(x)- l] q_(x)- e(x)q_(x),

^(A.2.12)

then the matrix ~(x) also represents the subspace V(x). Noting that the vector having the form

1

il

3/2 1/2 - 1 / 2 - 3 / 2 - 5 / 2

(A.2.13)

394 M9 Fukuma, H. Kawai, and R. Nakayama does not exist in

V(x)

except for

. ( ~ = W~

wl

1

0 0

x; 3/2 x' 1/2

-1/2

- 3 / 2

- 5/2

(A.2.14)

we find that the first column vector of the matrix O(x) is nothing but t/(~ Next, if we calculate

q(x - e(k-

1)), then we can show from Eqs. (A.2.6), (A.2.7), and (A.2.9) that

~(x-e(k-~))=expI~ l(A)"l~(x)

= Z

-s e(x)q-(x)

j>_o

FO+(x, kl]

= LO-(x,

k)]

q_(x).

Since a straightforward calculation shows that

detO_(x,k)= 2 wj(x) k-j,

j > o

we conclude that

T(X - - g ( k - 1 ) ) = det q_ (x - e(k- a))

= det (0- (x, k). q_ (x))

= ( 2

wj(x)k-~).~(x). []

\j>=0 /

(A.2.15)

(A.2.16)

The set of Eqs. (A. 1.5)-(A. 1.7) and (A.2.11) yields useful formulas which express the second derivatives of In z in terms of the pseudo-differential operator L:

02

OxlOx,

In z = (L")_ 1,

aXEOX~ "

In z = 2(L") _ 2 + (L") _ 1,

~2 ~x ~

⁰²

OxaOx---n lnz=3(Ln)-3+3 1

(L")-2 + 1

(L")-l+3~x~ lnz'(L")-x'

9 ..etc., (A.2.17)

where the symbol ( )-k stands for the coefficient of 0 -k.

A.3. Free Fermion Representation of z Functions

In this subsection, we describe the method of expressing z functions in terms of free fermions.

Two-Dimensional Gravity 395 First, by using the correspondence between U G M and the set of all decomposable states [see Eqs. (2.5) and (2.6)], we construct a multi-fermion state ]g') from the matrix t/(0)= [t/(~ which represents the subspace V(0):

Ig') = Y~ n,o (0)~,~ (0) . . . (0) (1) .w,ow,1 t t I - ~ ) - (A.3.1)

r o , r l , . . .

Then, the multi-fermion state corresponding to the matrix ?/(x)= exp [ - ~

x,A"]

k A

~(0) in Eq. (A.2.7) is

n ~ 1 x n J n !

Ig'(x)) - e- = Ig ), (A.3.2)

where J, = 2" lp~_,~p,: (n e Z). Thus

r

z (x) = det g/_ (x) = (0[g'(x))

-- ]~ x n J n

=(0]e "~ Ig'). (A.3.3)

Note that this state [g') can be expressed in terms of the Clifford group Cliff:

Ig') = g'10), g' e Cliff, (A.3.4)

where the Clifford group is the set of the elements of the following form:

Cliff= {g=exp [ ~ " ~P,*brs~P~ :]}. (A.3.5) Thus the

z(x)

is also written as

-- Z X n J n

z(x)=(0le "-->~ g'10), g'eCliff. (A.3.6) The above expression (A.3.3) has a different sign of exponent from that in Eq. (2.11). We can change the former by carrying out the

CP

transformation

~Pr ~ P t - r, (A.3.7)

~P~ ~-W-~. (A.3.S)

In fact, under this mapping, J, changes its sign;

J, ~ - J . ,

while the vacuum 10) remains unchanged. Moreover, recalling that z functions can be calculated by using algebraic relations of fermions alone, we find that ~(x) is unchanged under the transformation. Thus, denoting the transformed element in Cliff by g, the z(x) can be reexpressed as follows:

-- ]~ x n J n ~, X n J n

"c(x)=(Ole ,~1 g, lO)__(Ole.el glO)

~- X n J n

=(Ole "~1 Ig) ( I g ) - g l 0 ) ) .

Appendix B. The Structure of R [{S(k); k + 1, 2,...,p}]

In this appendix, by explicit calculations we check Lemma 5.1 for operators of spin up to 6.

396 M. Fukuma, H. Kawai, and R. Nakayama First in order to simplify the calculations, we generalize the normal ordering ~ defined in Sect. 5 as follows. If we consider a small disk D on the single-sheeted 2 plane (D C C - {0}), it corresponds to p disks D ~ .... , Dp on the p-sheeted 2 plane. We denote by 2~ the point in D~ that corresponds to 2 e D. Furthermore we denote J(2~) by Ji(2). We then define the normal ordering of Ji(2) as that of p independent current operators on the complex 2 plane. For example, we have

J,(~)

= : J,(,l) o,

J~(2')Jj(2) - (2' -- 2) ~ 6ij +o o Ji(2 , ) J,(2) o, o 6~j

Ji()~")JJ(2')Jk(2) = ( 2 " - 2 ' ) 1 Jk(~')

5~k J j(2')+ 6jk . . .

+ (2o__2)2 (2t--2) 2 Ji(2") -[- oJi(2 )Jj(2)Jk(2)o. ^(B.1) Clearly the normal ordering considered in Sect. 5 is a special case of this one.

Next we make some formulas which relate this normal ordering to the product defined by Eq. (5.9). Suppressing 2 and denoting the summation for i by an overline as

p

Jt(O(Jq))"(Oz(J')) " = E J,(2)t(O(J,(2)q))"(az(J,(2)')) ", (B.2) i=1

we have the following formulas:

Formula.

(i) (S (', (S (', (S ( ' , . . . (S")~ A)...)))= ~ for any operator A(2),

( i i ) to ~ o ~ o . , o,o,, o j-- ~j2jk~ + k o O 2 j j k - t o , o 7 ~ o ~ _ - - -

k ( k - ~)

(iii) w,,t~176 oo~176 o j a j k o ~ + 23 k o O 2 ( j 2 ) j k _ ~ -f 8 o 0 4 j j k - Z o ,

(iv)

rot2o t o t 2 o oj2oo)) = oo(j2)3o \ v o o~\oo o~ q_6~O2jjj2~ .b2oo(~32j)2: -t- ~176 (v) ( : j 2 : , :02jj ) = o(02j)2oO + 89 o04jg: + o(~2jj j2o.

P r o o f (i) First we consider (S(1)(2), S(1)(2)). Recalling that S")(2) = ~ Ji(2), we have

s"~(2')s"~(2) = 2 J,(2') E j~2)

9 j

9 ,. 2' 2 + ~176

~ P + o~ ~ .

( 2 ' - 2 ) 2 Taking the coefficient of ( 2 ' - 2 ) ~ we obtain

(S(1)(2), S (1)(2)) = :S ( ~)(2)s(~)(2):.

(B.3)

(B.4)

Two-Dimensional Gravity 397 Similarly for any operator A(2) we have

= o o . ( B . 5 )

Using this equation repeatedly, we have the formula (i).

(ii) For example, we show the calculation for the simplest case, k = 2. Following the definition of the normal ordering, we have

i j

=Ei, j ((,~2--~) 4 ~ (~,_~)2

4~ij ~176176

^.

^(B.6)

Expanding this expression with respect to 2' around 2 and taking the coefficient of ( 2 ' - 4 ) ~ we have

o

~ J,(2) o,o Js(Z) o = 2 : E 0 2 J , ( 2 ) J , ( 2 ) o ~ + :

J,(Z)2s

^(B.7)

i " j

(iii)-(v) can be shown by similar calculations. []

Using these formulas, we can rather easily check the statement of Lemma 5.1 for operators of lower spins. In the following we abbreviate R [{S(k); k = 1 .... , p)] to R(p).

spin I

~:(1) = oo yoo = y = S(1) C R (v) . (B.8)

spin 2

or2o [Eqs. (5.1) and (5.6)1 21~ t2)=o,, o

= o~ (S(1)) 2 -- 2S(2) ~

=(S (1), S ( 1 ) ) - 2 S (2) [Formula (i)] e R (v) . (B.9)

spin 3

317V(3)= ~J3o~ [Eqs. (5.1) and (5.6)1

= ~ (S(1)) 3 _ 3S(1)St2) + 3S(3)~

= (S ~l), (S (1), S(1))) - 3 (S (l), S ~2)) + 3S (3) [Formula (i)] e R (v) . (B.10)

spin 4

It is easier to consider the following combination:

41~:(4)q-a~2W (2)-- ~ [Eqs. (5.1) and (5.6)1. (B.11) For the first term on the right-hand side, we have

~j4~ = ~(S(1))4 4(S(1))2S(2)q_4S(1)S(3)_4S(4).q_2(S(2))2~, . (B.12) By using Formula (i), it is obvious that each term on the right-hand side except for the last one belongs to R (v). Thus we have

o~ - 2o~ (modR(V)). (B.13)