R,(u) ~Rk(u), Eanz", t2, t4, .... Z(t, t3, ...) ~ k__Eo t'R,(.) (2) (xl a2/ax2+ u(x) Geometric interpretation of the partition function of 2D gravity

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Volume 257, n u m b e r 3,4 PHYSICS LETTERS B 28 March 1991

Geometric interpretation of the partition function of 2 D gravity

V. K a c

Massachusetts Institute o f Technology, Cambridge, MA 02139, USA

a n d

A. S c h w a r z

University o f California at Davis, Davis, CA 95616, USA

Received 30 N o v e m b e r 1990; revised manuscript received 31 December 1990

We construct explicitly the subspace in the infinite dimensional grassmannian corresponding to the r-function o f the 2D topological gravity. This allows us to give a simple proof of some conjectures on the equations defining this function.

It has been discovered recently that the reduction to N × N m a t r i x models, N-~oo, can be used to obtain the complete non-perturbative solution o f 2D gravity and of string theory coupled with matter fields having central charge c < l [ 1-3]. The answer for the partition function was expressed in terms o f r-functions. Later is was shown that the same partition function arises in 2D topological gravity [4,5].

Namely the generating function o f amplitudes in topological gravity r(tj, t3, ...) and the partition function Z ( t , t3, ...) o f the one-matrix model at the k = 1 crit- ical point are connected by the formula

Z ( I , , 1 3 . . . . ) = r 2 ( / , , t 3 . . . . ) • (1) The function r u b t3,---, t2n+j, ... ) satisfies the so called string equation and the following equations o f the K d V hierarchy:

Ou k c~

a t 2 k + ~ - ( - 2 ) ~ R k ( u ) ,

3 2

u = - 2 O~12 In r .

Here the R , ( u ) are defined by the formula

¢~ Supported in part by NSF grant DMS-8802489 and DOE grant DE-FG02-88ER25066.

(xl exp{ - t[ - a2/ax2+ u(x) ]} I x )

~

1

k__Eo t'R,(.) (2)

for t - , 0 (i.e. Rk ( u ) are Seeley coefficients for the operator - 02/0x 2 + u ( x ) ) . Recall that the r-function o f the K P hierarchy can be defined as a function o f an infinite n u m b e r o f variables tl, t2 .... satisfying bilinear Hirota equations. Such a function can be considered as a r-function o f the K d V hierarchy if it does not depend on the even variables t2, t4, .... Every r- function o f the KP hierarchy corresponds to a point o f the Sato infinite-dimensional grassmannian Gr [6]. (The points o f Gr are linear subspaces o f the space H consisting o f the formal Laurent series Eanz", a~ = 0 for n >> 0. The subspace V ~ H belongs to Gr if the natural projection zc+ of V into the space H+

spanned by z n, n/> 0, is a Fredholm operator. The big cell o f Gr consists o f those V for which 7r+ is an isomorphism; we denote it by Gr °. ) The r-functions o f the K d V hierarchy correspond to the points o f Gr ob- eying z2V ~ V; we denote this subset o f Gr by Gr(2) and let Gr~2) -- Gr(2~ c~ Gr °.

O u r aim is to describe the point o f Gr corresponding to the r-function arising in 2D gravity. The de- scription will be based on refs. [ 7,8 ]. It is proved in

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Volume 257, n u m b e r 3,4 P H Y S I C S L E T T E R S B 28 March 1991

these papers that the r-function in ( 1 ) obeys ( 3 J 2 n + , "+'Ln_ l ) r = 0 , / 7 > / 0 , ( 3 ) where the L . are Virasoro generators given by the formula

L~ = ¼ Z JpJz~_p + ~ fino . (4)

p= 2k+ I

Here Jp acts as multiplication by ( - - p ) t _ p for p~<0 and as O/Otp for p > 0. O u r formula ( 3 ) differs slightly from the formulas in refs. [7,8] because the r-function ( 1 ) is obtained from the r-function studied in these papers by the shift t3--, t3 + 1. Such a shift is nec- essary for our aim because we want to consider the r- function as a formal power series. (The r-function o f refs. [ 7,8 ] is singular at t~ = t3 . . . 0. ) Using ( 3 ) we will probe below that the r-function under consideration corresponds to the subspace V(2) from the big cell Gr~)2) satisfying

AV~2~ c V ~ , ~ , (5)

where 1 d

A = 3 z + 2 z d z - 1 4 z - 2 (6)

The condition (5) singles out a unique point in the big cell Gr~2). We use the notation V~2) having in mind the generalization to the case o f Wh-gravity. It is very plausible [7,8] that in this case the partition function is connected with a r-function as well. The point of the grassmannian corresponding to this r- function will be denoted by V(h).

Let us sketch the p r o o f that the element Vc2)e Gr~2) satisfying (5) is unique. Since the projection of V(2) on H+ is an isomorphism, there exists a (0~V~2) o f the form (0= 1 + 32,~> ~ Gz -~. Furthermore A ~(0 = z" + lower degree terms, hence these functions with n >/0 form a basis o f V. Therefore z 2(0 is a linear combination o f the A"(0; it is now easy to check that z2(0 is proportional to A2(0. This determines (0 uniquely.

The equation A 2(0 = const. × z2(0 can be reduced to the Airy equation by means o f the substitution 1 , = z - l / Z e x p ( - - ~ z 3 ) ( 0 , X = ]A2z 2 , ]A3~__ - - ~ , 3 This permits us to describe the subspace V(2) c H as follows. It is well known [9] that there exists a solution N//XKl/3(2x 3/2) of the Airy equation y " = x y

having for x ~ + o0 the asymptotic expansion y = x ~ x - , / 4 e x p ( - ~ x 3/2) amX -3m/2 , (7)

\ m = O /

where the a,n are found from the recurrent formula 4 8 m a , , = ( - 1 ) m ( 6 m - 1 ) ( 6 m - 5 ) a m _ ~ . (8) Then

p ( z ) = 1 +al ( ] z z ) - 3 + a 2 ( ] z z ) - 6 + .... (9) Thus, the space V(2) is the subspace o f H spanned by (0, A(0, 22(0, 7.2A(p ... Z2n(0, z2"A(0 , .... (10) It is easy to check that z Zn(0 = z 2 , + lower order terms, 22nA(0 = z2n+ I -b lower order terms. Therefore the natural projection zr+ of V(2) into H+ is an isomorphism. We see that V~2 ~ belongs to Gr~2). The invari- ance o f V(2) with respect to the operator A follows from the equation A 2(0= const. × z2(0.

O u r arguments give a rigorous p r o o f that there exists a r-function o f the K d V hierarchy satisfying (3).

Therefore the combination o f our results and the results of refs. [7,8 ] gives a new derivation o f the con- nection between the partition function o f the matrix model and the r-function.

It is important to note that the space V~2) belongs to the Sato grassmannian Gr [ 6 ] but does not belong to the Segal-Wilson modification o f the Sato grass- m a n n i a n [ 10 ]. (Recall that in the Segal-Wilson definition the space H is replaced by the space L2(S ~ ) o f all square integrable functions.)

It remains to prove the characterization of the r- function under consideration by (5) (see the appendix for an alternative p r o o f ). We will use in the p r o o f the fermionic representation o f Gr. Let us consider the representation of canonical a n t i c o m m u t a t i o n relations [q&, qJ,+]+--fin,,, [~u.,

I//m]+=[I//n+,~//m+]+

= 0 in the Fock space .Y with v a c u u m vector qb satisfying ~,n~=0, n < 0 , q/~ qb=0, m>~0. Let us define G L ( ~ ) as a group with the Lie algebra consisting o f operators a+Zc,~, :m,,~u+:. Here a is a complex number, cm,, = 0 for I m - n [ >> 0, dots denote the normal product with respect to the v a c u u m q~:

~,,~,+ = :~,,~,+ . + (~,,,~,~+) ,

where ( ~ , , q ; + ) = 1 i f m = n < 0 , (gzmq/+ ) = 0 in all other cases.

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Volume 257, number 3,4 PHYSICS LETTERS B 28 March 1991

In .~- one can introduce the operators d , , = Y.~u,~,k+ ~ . " + - satisfying the canonical c o m m u t a t i o n relations

[Am, A , , ] = m 8 ... (11)

This permits us to perform the bosonization proce- dure in the following way. Let us associate with every vector ga~ .~- a function

p ~ ( t ~ , t 2 , . . . ) = exp t, An t P , ~ . (12)

1

It is easy to check that by this correspondence the operators A,, transform into the operators J , (i.e.

If the vector g~ belongs to G L ( ~ ) . q ~ (to the GL(o~ )-orbit o f the v a c u u m vector) the function q~"

is a > f u n c t i o n o f the KP-hierarchy. Conversely every

>function can be represented in this form (recall that we consider the z-function as a formal power series).

There exists a natural m a p p o f the orbit G L ( ~ o ) . q~

onto Gr; the z-function z ~ coincides (up to multi- pliers) with the z-function corresponding to p ( ~ ) e G r . It is easy to check that in the case when the vector ~ satisfies the condition ( a + ~ c ...

X :~u,,,q/, +" ) 7a=0 the subspace p ( ~ ) c: H is invariant with respect to the operator (" acting in H. (Here denotes the linear operator in H having the matrix

~,,,,, = c,,,,, in the basis z", n = 0, _+ 1 .... ). This assertion is evident in the case ~ = @, p( 7 a) = H + . One can re- duce the general case to this particular case using the fact that the map p is compatible with the action o f the group G L ( o o ) .

Let us introduce the operators

L n = a ~ J 2 , , _ x J a . + f~ 8 ,,o

=½ Z Je,,_kJ~ +~J,,J,,+ ~68,,o . (13)

k > n

Note that the two expressions for Ln in (13) are formally equivalent; however only the second form leads to a well defined operator. In the fermionic representation the operators (13) correspond to the operators

L;, = ½ ~ A2,,_kAk + i 2 ~ A,, + ~68,,0

k > ' n I

- 2 E : ~ ' , ~ ' ~ " +

_ _ . ~ / . , ~ , a .

f l - - r ~ + d - - 7 = 2 n //-- ee < a" ;,

• + . l

+~ ~ : ~ u ~ - : .~'~&'a . + ~8~o • (14) Using Wick's theorem one can represent Ln in the normal form. We obtain in this way a sum o f three terms: L n = L ~,, 1~ + L~ 2~ + L}, 3) where L~ ~) is a quartic expression with respect to ~u, ~,+, L~, 2) is a quadratic expression and L~ 3~ is a scalar. It is important that all quartic terms cancel: L}/) = 0 . To prove the can- cellation we have to note that ½ .~'~'~" + ~'~,~ua+" cancels with ½ :q/~'J-~u~4u~-" if 7 - f l < c ~ - & both ½ :~,y~u~

× ~<~ ~u3" if a - 8 < 7 - fl and with ~ :~uy ~'~- ~u~ ~'J-" + :~u~ ~ua ~ ~u:, ~,~" if y - fl= c~ - 8. Wick's theorem gives the following expression for the quadratic term:

L ~,2) = ~_c,~a:qj~t~. where c~a= 0 if c5-c~ # 2n,

2 / / < 2 o : + 2 n

1 E ( ~ , ~ , ~ ) ~ ( ~ , ~ . + ) . (15)

- - - - ~ / o / - n

2 / / > 2{~ + 2 n

It follows from ( 15 ) that c~.~+2,-- ~ ( ~ - n) + ~. For n # 0 the scalar term L}, 3) vanishes; this follows from Wick's theorem. Namely ( ~ u ~ , f ) (~'~-~5) can be non-zero only in the case o~=/3, fl= 7, and therefore f l - e e + 8 - 7 = 2 n = O . We see that for n # 0

L ; , = E ~ . ~/,-e ~//~t + 2 n • + + ( ½ n + ~ ) E : ~ l l t , ~ - / o e + 2 n . . + "

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In a slightly different language this result is well known.

One can give another p r o o f of this formula.

Namely, it is easy to check that the operators (16 satisfy the relations

= ~kA~+2,~.

The same relations are fulfilled for the operators (14).

(This follows from the c o m m u t a t i o n relation ( 11 ). ) Therefore the operators (14) and (16) coincide up to an additive constant. Formula (16) permits us to assert that the operator :q/,~c~,q,,~" where

C o z / / = ~ S f l _ c , _ 3 1 _1_ ( 120~__ 1 ) 8 l l o ? + 2 , ( 1 7 )

annihilate the vector ~P corresponding to the z-func- tion satisfying (3) with n=O. The matrix o f the op-

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Volume 257, number 3,4 PHYSICS LETTERS B 28 March 1991 erator A in the basis z ~, n = 0 , _+, ..., can be obtained

from (17) by transposition, therefore the space V(2~ = p ( T ) satisfies (5).

Note that as a b y p r o d u c t o f our consideration we obtain a p r o o f o f conjecture 1 of ref. [ 8 ]. N a m e l y we see that the r-function of the K d V hierarchy satisfying (3) for n = 0 obeys also (3) for every n >/0. In- deed, eq. (3) for the r-function o f the K d V hierarchy is equivalent to the relation A (")V ~ V for the corresponding point V o f Gr for a certain linear operator A ~) in H. It is easy to check that A(")=z2"A (see appendix) and therefore the relation A (")V c V follows from AV ~ V and z2V = V. In particular, the point V (2) constructed above satisfies A ( ~ ~V~2) ~ V~2) and therefore the corresponding r-function obeys (3).

The results above can be generalized in the following way. Let us consider the operators

_ 1 2 h 2 - 1 -

L C h ) = 1 ~ Jh, , J , + ~ J h ~ / 2 + ~ 6 ~ o (18)

h h n < 2 l --

acting on functions ~0(tt, t2, ...). ( O f course one has to omit the term containing Jh,,/2 ifhn is odd. ) These operators generate a Virasoro algebra. For n = 2 they coincide with the operators (13); the methods used for n = 2 can be applied in the general case too. We obtain that in the fermionic representation the operators (18) can be written as

E~ ~) = a .~'~ ~,,+,~.

• + ")

+ ½ ( n h + l ) ~ . q ~ q ~ ÷ , ~ . . (19)

The simplest way to prove this fact (up to an additive constant) is to check that the operators ( 18 ) and

( 19 ) satisfy the c o m m u t a t i o n relations

[Jz, L~h)]=~J~+kh , [ A ~ , / S ~ ) ] = A¢+~. (20)

It was conjectured in refs. [7,8] that the partition function o f Wh gravity is connected with the r-function r(~)(t~, t2, ...) that does not depend on th, t2h, ...

and satisfies the condition

--~-J~.+,+L~2~

^{r = 0} forn>~0. (21) (More precisely, in the definition o f operators (18)

given in refs. [7,8 ] I runs over only integers that are not multiples o f h, however in (21 ) this restriction is irrelevant.) It follows from (21) that V(h)~Gr corresponding to the r-function Z~h) satisfies the conditions

A ~ h ) V ( h ) ~ V ( h ) , z h V ( h ) ~ V ( h ) , where

A~,,>= - - U - z + ~ z - A + ' - - ½ ( h - 1 ) z -~

dz

The space V(h) is spanned by the elements A~'h)~0eH, n = 0 , I, 2 . . . . , where the formal series

~o= 1 + X,>~ ~ b,z-' satisfies the equation

A ~h) ~0 = const. X zh~o. ( 22 )

The solution of this equation can be reduced to the solution o f the "generalized" Airy equation y(h) = xy.

Consider its solution with the asymptotic expansion

y = x - ( h - l ) / 2 h

Then the power series ~o(z) _ v ~ - - ~ n = O . ~ h ) , , - . . - ( h + ~ . l,~ n taL ~

where/1= ( h + 1 ) / h and the a~ h) may be found from the recurrent formula na~ ~) =~k=~h-~ c~h)~n~_¢h)~ J " , - k , i s a unique power series (in z - I ) solution of (22) with ao ~h) = 1. (For example, 48c12)(n) = ( 6 n - 1 ) ( 6 n - 5 ) . ) Indeed, ( 2 2 ) w i t h ~o(z)=ZT=obnz-" is equivalent to a recurrent formula o f the form nb, =

We are deeply indebted to M. Sato for the expla- nation that this definition o f grassmannian is appro- priate for our aims.

Appendix. We show here how the above construction fits in the representation theory of arbitrary simply laced affine algebras [ 11,12 ].

(A1) Let E be a subset o f the set of non-zero integers and let h be a positive integer such that E = - E and E = h + E . Let E + = { m ~ E I m > 0 } , Eo=

{m~ElO<m<h}, l= DEol.

We construct twisted Virasoro operators Ln in terms o f the oscillator algebra [Jm, J, ] = m6m,_~ ( m, n~ E):

1 1

L o = ~ ~ J .... J,~+ ~ Z J ( h - J ) ,

m~E+ )~E0

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L , = ~-~ y~ J, . . . . Jm if n # 0 .

m ~ E

We have

[J,., L . ] = ~- J , . + . e , m

[L.,, L . I = ( m - n ) L , . + . + ~ ( m 3 - m ) l d . . . Examples. (a) let E = 7\hTL T h e n the operators L~

are the ones appearing in the equations ofrefs. [6,7 ].

When h - - 2, these are the operators ( 13 ). On the other hand, they are the twisted Virasoro operators in the basic representation o f the affine algebra slh [ 1 l, 12 ].

(b) Let E={m~<~...<~ml} be the exponents o f a simply laced simple Lie algebra g, and let h=m~+ 1 be the Coxeter number. Let E = {Eo+jh, jeT7}. Then the L , are the twisted Virasoro operators in the basic representation ofO [ 11,12].

(A2) Recall that the basic representation of the affine ~ = (H ® g ) @ C K is its unique irreducible representation ~z in a vector space V such that k = 1 and there exists a non-zero vector qb such that

~r(H+ ® g ) q ~ = O .

Turning to g = slh, consider the following matrices in (for all other g the construction is similar, see ref.

[ 1 1 1 ) :

h - - I

& = ~ ei,,+t +zeh, l, sn=sT ,

i = l

h

p = ½ ~ ( h - 2 i + 1)eii.

i = l

Note that s, e ~ if n ~ h;Y and that [ Sm, S~ ] = m~)m,_ n ( m, n E E ). This is called the principal subalgebra o f 8.

The basic representation ~r o f ~ is constructed in the space o f formal power series in indeterminates t,, n e E + such that S,,=Jn and 1 = q~. T h e representation o f the rest o f the generators o f O is achieved by use o f twisted vertex operators (ref. [ 11 ], ch. 14).

Let G be the Lie group with Lie algebra g and let C J = G ( H ) . Here and further for any ring R, SL~ ( R ) = { (a,j) I a,ae R, det (a,j) = 1 }; the definition o f G ( R ) for arbitrary G is similar. T h e formal power series from the orbit n ( G ) . l are called (formal) r- functions. This condition can be rewritten as an infi-

nite hierarchy o f partial differential equations in the bilinear form o f Hirota (see ref. [ 12] ).

Introduce the following operators on 8:

D ~ ( a ( z ) ) = z " + , d a ( z ) dz

1

- - + -h [ z " p , a ( z ) ] ,

a ( z ) e f i .

Then we have D~(sm)= ~Sm+~h, m

and c o m p a r i n g with (A1) we have [ T t ( a ( z ) ) , L , ] = n ( D n ( a ( z ) ) ) , a ( z ) e g .

(A3) Recall yet another construction of 7r for

= sl n- Consider the natural representation offi (with k = 0 ) and o f G on the space o f h-vectors H h. Con- sider the associated infinite wedge representation .~

o f g (ref. [ 1 1 ] , ch. 14), (This is actually the same Fock space ,~ considered earlier. ) Then k = 1 and the v a c u u m vector q~e.Y corresponding to the subspace H+ generates the representation zt in V ~ .N. U n d e r the b o s o n - f e r m i o n correspondence .~(o) (respectively V) gets identified with the space o f formal power series in t j; j = 1, 2, 3 .... (respectively in tj, j~ E + ); here .N(°~ is the subspace o f zero charge vec-

tors in 2 .

For each subspace W e G r ( h ) = G - H + we thus get a

> f u n c t i o n rweG" 1 and the key remark is that for an operator B, B r w = 0 if and only i f B W ~ W, B W ~ W .

(Note that the big cell is SLh ( H _ ) . H +. ) Finally consider the operator (cf. (21) )

, d 1 . j

AlZl = ~ f ~ + ~ - p + , s , ~ + ~ ,

w h e r e p is a free parameter. It follows from (A2) that [A {~l, ~ ( a ( z ) ) 1 = - L ° _ , + ~ , ~ + ~ .

Considering the m a p H e ~ H defined by .... , £ ) - ~ z ~- 'A ( z h) + z ~ - ~ A ( z h ) + ...

+ L ( z ~) , the operator A/~I operator

1 - h + t d A(h, = ~ z dz

gets transformed to our basic

h - 1

2h z - h + ltz '

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a n d t h e o p e r a t o r s A/~,I t o t h e z"hA(h).

Note added. T h e d e s c r i p t i o n o f a b a s i s o f t h e ele- m e n t o f t h e g r a s s m a n n i a n c o r r e s p o n d i n g t o t h e ~- f u n c t i o n i n q u e s t i o n is e q u i v a l e n t t o c e r t a i n i n f o r - m a t i o n a b o u t t h e B a k e r f u n c t i o n . W e a r e g r a t e f u l t o t h e r e f e r e e w h o p o i n t e d o u t t h a t t h i s i n f o r m a t i o n i n t h e c a s e h = 2 c a n b e e x t r a c t e d f r o m a r e c e n t p r e p r i n t b y M o o r e [ 1 3 ] .

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