Classical integrability and higher symmetries of collective string field theory

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Physics Letters B 266 ( 1991 ) 35-41 North-Holland

PHYSICS LETTERS B

Classical integrability and higher symmetries of collective string field theory J e a n A v a n a n d A n t a l Jevicki Department of Physics, Brown University, Providence, R! 02912, USA

Received 29 April 1991

We demonstrate the complete integrability of classical collective field theory. An exact equivalence to the classical N-body problem of Calogero type is described. A Lax pair is constructed for the general continuum collective equations. A form of background independence is pointed out. An infinite set of commuting currents is constructed in the general case. A large symmetry algebra is discovered and shown to take the form ofa W~ algebra.

1. Introduction The collective field theory [1,2] of one-dimensional matrix models was suggested to represent a field theory o f one-dimensional strings [ 1-5 ]. Its properties have been investigated recently in some detail. For example in refs. [ 6-8 ] the amplitudes and higher loop [ 7 ] effects were considered with results in perfect agreement with perturbative calculations using other methods [4,9,10 ]. The collective field theory is written as a field theory o f tachyons, but is seen to include the effect of an infinite sequence of additional discrete states. These states were exhibited as poles in the integral representation [ 7] of the fourpoint amplitude and also in the study of the two point function [ 11 ]. They are remnants o f the graviton and higher string states [12] in 1 + 1 dimension. Studying the classical dynamics of the collective field theory will certainly shed some further light [ 13 ] on their dynamics. There are certain indications that the collective string field theory is completely integrable. In the fermionic language conserved currents were suggested in ref. [ 5 ] and exact results for the amplitudes can be written in integral form [ 9 ]. An exact classical solution was recently given in ref. [ 8 ] and an analogy with Work supported in part by the Department of Energy under contract DE-AC02-76ER03130-TaskA. Elsevier Science Publishers B.V.

an integrable N-body system was pointed out long ago [141. In what follows we proceed to establish explicitly the integrability of the cubic collective field theory. First addressing the question of the classical dynamics of matrix eigenvalues we give an exact correspondence [ 14 ] with an integrable N-body system of Calogero type [ 15 ]. String theory is driven by a harmonic critical potential and for this particular case the Calogero system is known to have a Lax pair and an infinite number of conserved charges [ 16 ]. This gives an exact solution to the eigenvalue dynamics. We then proceed to investigate directly the continuum classical equations of the collective field theory. We first exhibit a reparametrization which is shown to transform away the harmonic background potential and provide an equivalence to the theory without an explicit potential. This in string theory context represents a demonstration of background independence. In the general case with an arbitrary background potential, we can write down an infinite sequence of conserved currents. Their Poisson brackets reveal a further structure; we construct an infinite parameter algebra which is shown to take a form of the W ~ algebra. The conserved quantities are seen to represent a particular Cartan subalgebra of this W ~ algebra. Our study is done at the classical level and quantum implications will be left for the future. 35

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2. Eigenvalue dynamics, integrability Consider the lagrangian of the collective field theory

f z'(

22 August 1991

and the fact that the integral in (2.5) is a Kronecker delta 6ij. We see therefore that the classical collective field theory, in a discretized form [eq. (2.2)], is equivalent to an N-body problem,. 2

2k

LN =

¢

,=,

~ Xi +

, (x,

xj)

+v(xi)

'

(2.7)

(2.1) Since the collective field represents the density ofeigenvalues of the matrix model 0(x, t ) = ~

O(x-x~(t))=Tr3(x-M(t)),

(2.2)

one can address the problem of the classical dynamics of these eigenvalues. This question was considered some time ago [14] with the following result: One has to solve the classical matrix model equations

~:l+v(m) = 0

(2.3)

with J,b = seen the

the constraint that the angular momentum be i[M, 3~/] = 1 -Oat,. This particular constraint was [14] to translate into the cubic interaction of collective field. For a harmonic potential v(M)=½co2MZ one has the matrix solution M ( t ) =A cos cot + B sin cot. It provides implicitly a classical solution of the collective theory. More explicitly, we can show that dynamics of eigenvalues is given by an N-body Calogero problem. To demonstrate this equivalence one writes the cubic interaction as

which describes the dynamics of the eigenvalues x,(t). For the special case of a harmonic potential v ( x ) = ½co2x2 this is the Calogero model [15]; it is completely integrable with known exact classical solutions [16]. To summarize one takes for the Lax matrix L=k,~o + i 1 - ~ °

(2.8)

xi-xj

and denotes

X= Diag (x,). A relation to the matrix model is given by and M(t)=U(t)L(t) × U - 1(t), where M is precisely of such a form that the above mentioned constraint on the angular momentum is fulfilled. The equations of motion are then solved by

M(t)=U(t)X(t)U-l(t)

M(t) = X ( 0 ) cos cot+ 1 L ( 0 ) sin cot

(2.9)

co

with arbitrary initial values for x~(0) and .~, (0), i = 1, 2 ..... N. This in turn provides a general solution to the collective theory through identification ¢~(x,

t)=Yr6[x-M(t)l. An infinite sequence of conservation laws follows from taking traces: The kinetic term is simply [ dx 3

I ~ = T r [L +icoX)

(0-1~)2

~,(x, t) - [ I ( x - x , ( t ) )

:;9

i N

1

- ~ Z k , 2,

(2.5)

i

= Y, C,e'"~°'HN_,(x).

(2.11)

rl~O

where we have used

~(x, t) = - Y~ ~c,c~'(x-x,) i

36

(2.10)

One can also write down still another form for the solution in terms of Hermite polynomials:

¢

-- ~ ~ .;c, f dx ~(x-x,)~(x-xj)O(x)

(L-icoX) f t .

(2.6)

This follows from the fact that gt(x, t) obeys the linear equation [ 15 ]

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iql t "4-ql,xx -- X~,x + Nq/= 0.

(2.12)

22 August 1991

~0(q, z) =O(x, t) ch t

(3.6)

Here the C,'s are arbitrary (initial value) constants providing a solution of the collective initial value problem.

coupled with a coordinate reparametrization

3. String theory, background independence

The integration measure becomes

String theory is obtained from the matrix model in the double scaling limit where N ~ and ~t--,0 with /z being the Fermi level. The potential responsible for a critical behavior and the continuum limit is an inverted oscillator v(x) = - ix2. The continuum string field theory is then driven by the interaction

~int = fdx [Ig2~3(X,I)--½X2~(X,t)].

(3.1)

X

q=cht'

r=tht.

(3.7)

d q d z = f d x d t ( c h t ) -3

(3.8)

and by a chain rule O ~r tP(q, r )

- ~r{O,[O(x,t) c h t ] + x c h t G ~ ( x , t ) } .

(3.9)

The linear tadpole term produces a classical vacuum background n~o(X)=(lz+x2) ~/2. This induces a background metric

Integrating

G~,~= ( 1/0 2, ~2~2 ( x ) ) ,

The kinetic term of the initial lagrangian (3.5) then becomes

( 3.2 )

since in the quadratic approximation the lagrangian becomes ~2)-

1 t)2 2 ~o(X)

½~¢%(x)(0xt/)2 "

(3.3)

The metric signifies a presence of gravitational degrees of freedom. One might wonder whether a background independence formulation is possible. In the Calogero model there exist a remarkable connection between the solutions of the model with an oscillator potential and the free Calogero model. It is given by (cf. ref. [ 16] )

X,( t ) = 2i( l tg ~ot) cos o~t .

(O-l(9)~2-17~2~3(q,c')].tp

Consider a field transformation

(3.10)

(G-,6) ~ ½f d q d r ! ~ dx dt ch-2t (x sh t~+ ch t ~ - ~ ) 2 -2 j 0 2

\ch t/ x2~ + 2xO~- ' ~ th t]

-- 21 ~ OX d/((0"~-~-~)2 +x2f~(x,t) ) .

(3.11)

(3.4)

We now show that in the collective string field theory there is a corresponding transformation. It will relate the harmonic theory to the formulation without a potential term, namely a theory with only a cubic interaction:

So= fdr;dq[~

O~l(~=-[x~(x,t) s h t + c h t O z ~ ] c h t .

(3.5,

In the last step, we have made use of partial integrations. What we have shown is that through a coordinate transformation the kinetic term generates a nontrivial oscillator potential term. This implies that the fundamental theory is that with no background potential [ eq. ( 3.5 ) ]. This property of the collective field theory obviously has far-reaching implications. First it strengthens its status as a closed string field theory. In any string theory one has gravitational and higher degree of freedom; a theory of gravity should have a 37

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background independent formulation. On the practical side the transformation given allows one to generate solutions of the full theory from those of the theory without an external potential.

4. Lax pair, conserved currents and a Woo algebra The fact that the Calogero model possesses a Lax pair and an infinite sequence of commuting conserved quantities leads us to investigate the same issues directly in the continuum. In addition, these currents belong to a larger infinite symmetry algebra. These will be described in what follows. Consider the continuum hamiltonian

H= f dx{½HxOHx+~r2~3+[v(x)-it]O}

(4.1)

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the equations without the harmonic potential O+c~± = Oxt~2 . These are easily solved by

a+ (x, t) =f+ ( x - a + t) ,

(4.7)

where f± are arbitrary functions. The classical collective field for the full harmonic problem is then given by 2~z ch t c~+

,tht

We can write down a Lax pair for the collective equations. The equations for the components a ± ( x , t) =Hx+nO(x, t) were seen to decouple and can be written as

and the classical equations

O,a(x, t) = 0x(½a2+ v).

& x , t ) = {/-/, 0} = 0x(Or/,),

This is seen to take the Lax form:

//(x, t) = {H, H } _- ! H22 ,x -t- 1/[ 2 ~ 2 -}- l) ,

(4.2)

arising from the Poisson brackets {H(x), ~(x)} = a ( x - y ) . Differentiating the second equation, and forming linear combinations a± (x, t ) = H x + re0 (x, t)

(4.3)

f ( 11-~x (a3+ -

o~3_) + ~1 (o~+ - a _ ) v(x) ) .

For the case of an inverted oscillator potential [ v ( x ) = - x 2] an exact solution was obtained recently by Polchinski [9 ]. It is given in a parametric form:

ch[t-b(a)] ,

p= - a ( a ) sh [t - b ( a) ] .

(4.6)

This is clearly related to the eigenvalue solution described above [eq. (2.9) ]. We record here one other general solution. It is based on the observations of section 2. Consider first 38

+Oqx+V+O~ 2

(4.11)

(4.4)

(4.5)

x=-a(a)

d

£ = Ux +e~(x, t) , 37/= ~ dx -2o~

This also follows from the fact that the hamiltonian can be written as H=

(4.10)

Simply take

d2

the classical equations separate,

O,ol+ =ct+xo~ ± +Vx.

d / ~ = [£, f l ] . dt

(4.9)

and the statement follows. We note that the L-operator is that of the modified KdV equation [28]. It is interesting to note that the above Lax pair exists in the case of an arbitrary potential v(x). In the discrete Calogero case integrability is present only with a quadratic potential v ( x ) = +_x2. Clearly the continuum formulation achieves a considerable simplification. We now consider the continuum equation further. From the fact that the continuum theory is written as a Lax equation one expects that there is an infinite number of conserved currents in the collective field theory. Consider either one of c~+ equations. There will be two independent sets of currents for the _+ components respectively. It is easy to find the currents when the potential v = 0 so that the last term is absent in the equation. Then

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J~)=a",

(4.12)

J}")-

n an+l n+l

22 August 1991

{f dx f da (oe2+2v)", ~ dy~ doe (a2+2v) ''}

for n = 1,2, .... Clearly

O,a"=n&a"-l=na~a"=

= f d x d y ( a 2 + 2 v ) " ( x ) ( a 2 + 2 v ) " ( y ) ~ ' (x-y)

?/

n-~ 0x(a"+' )

(4.13)

and the currents obey the conservation equation 0Pr~") - 0 ~ J l ") = 0 .

(4.14)

Next we extend these conservation equations to the general case with an arbitrary potential v(x). A recursive construction leads to the following ansatz for the Jo component:

o:(x,t) J6k)=

J

da'

(4.15)

[O/'2+2V(X)] k .

This in more explicit form reads k

j~k)_~ E O['2(k-n)+lVTx) .=o

2"k!

(k-n)!n[

[ 2 ( k - n ) + 1] (4.16)

Taking the time derivative gives

0~I~k~=&( a2 + 2v)k= ( aa,x + v,,) ( a2 + 2v) * 1

- 2 ( k + 1~ O~(a2+2v)k+'

(4.17)

and we find a divergenceless current with j}k)

1

2 ( k + ~ ) (O~2+2V)k+l

(4.18)

Moreover, the conserved charges fdxJo(x, t) are easily seen to Poisson-commute up to boundary terms in the general case. From the Poisson structure for a +_, deduced from the Poisson structure for H and (we scale a + --,x/~ oz+ ),

{a+_(x),a±(y)}=a'(x-y),

{a_+, at_ } = 0 , (4.19)

one computes the Poisson bracket of two conserved charges:

= f dx2n(otOZx+Vx)(Ol'2+2V)n+m-l(X)

= ~ dXax(n(a2+2v)n+"'~/ = 0 .

(4.20)

Here we have constructed a set of conserved commuting charges of the form

Q(n'= f dx ; doe (ot2+2v)".

(4.21)

The conservation and commutation of these charges is true up to boundary terms. At the quantum level again these currents can in principle result in conservation laws. This will happen only if the surface terms in the spatial integral die out. For string theory the interaction grows at the boundary and clearly such surface terms will have a most important effect [ 17 ]. It must be noted that the divergenceless currents and conserved quantities were constructed without using the Lax operator L. In fact, since L in (4.11 ) does not contain v while (4.21 ) does, we expect that the conserved currents arise as trace identities for a different Lax representation; however, it is quite sufficient to have the commuting conserved quantities (which is all that is needed to establish integrability, at least when one has a finite-dimensional system [ 18 ] ). A Lax representation is useful in practice when one wishes to use inverse scattering to solve a given integrable model; we shall not consider this point here. We now describe an infinite-dimensional algebra present in the collective field theory. We have seen that in general the conserved currents consist of terms of the form v(x)~aP(x, l). For polynomial potentials these become terms of the type x~a p. Consider now the algebra of these objects. Define the generators

H~ = fJ dxx m-I -0Lm-n m--n

(4.22)

with the fact that for m=n we define ot°/O-loga. An explicit computation using (4.19) gives us the algebra

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n2 {H~,~,, Hm2}

,l+n2 = [ (m2-- 1 )n,-- (m l - 1 )n2]Hm,+,,,2-2.

(4.23)

This is recognized as a version of a classical W~ algebra [25]. We have a W~ algebra for both sectors c~+ and a _ , and since they commute with each other we have W~®Wo~. A particular subalgebra for m = 2 consisting of generators of type L n = f x . . . . /~2-n)=H~ is the classical Virasoro algebra {L,,Ln2 } = (nl -nz)Ln~+,2 •

(4.24)

The conserved charges including the hamiltonian are also obviously members of the Wo~ algebra; they are given by HI' in the simplest case when v=0. This is a Cartan subalgebra of Wo~. The general case with an arbitrary potential v(x) can be set in this framework. We have seen that the commuting conserved currents are given in general by eq. (4.21). For a polynomial potential is a linear combination of generators of our Woo algebra. In particular for v(x) = x 2 we have

dx J~ ~

2~k' dx, _o ( k - n ) ! n I [ 2 ( k - n ) + k

2"k! rl4,,-zk - - , =y~o ( k - n ) ! n ! "'2"+j "

x2na2(u_n)+l l] (4.25)

In general and up to boundary terms, the conserved charges, including the hamiltonian (4.1), build therefore a commuting subalgebra of the (Wo~) 2 algebra which may be maximal. One should therefore try to obtain explicitly all such Cartan-like subalgebras of Woo in order to define different integrable hierarchies of hamiltonians for the collective fields (in the weak sense of existence of commuting conserved quantities). The study of representations of the W~ algebra then may provide explicit solutions of these field theories. A few concluding remarks on the results of this chapter must now be made. The main part in the construction of the W~o algebra is to check that the Poisson brackets of the generators close. Hence the partial integrations triggered by the non-ultralocal Poisson brackets (4.19) must take a closed form. As shown above, this is the case for the monomials 40

22 August 1991

~a~_x'" and a'~x% however, mixed monomials an+ a'"_x p in general do not lead to closed partial integrations. A combination of such monomials nevertheless can cancel the non-integrable terms and lead to closed Poisson brackets. From an SU ( 1.1 ) algebra present in the Calogero model [ 18], it follows that the continuum limit generators defined as {Ho = H(4.5 ), H1 = fx[/,x¢}, H2 fx20} build an SU( 1, 1 )algebra in (W~) 2. However, this is also true and can be explicitly checked when Ho is replaced by H~ = H o + f(3c~ 2 a + - 30~2+ct_ ). Although H~ does not belong to W 2 it has closed brackets with Hi and 112. We also expect therefore that other symmetry algebras (may be also infinite) can be constructed and help to solve this field theory. =

5. Conclusions We have demonstrated the complete integrability of classical collective string field theory. At the discrete eigenvalue level we have seen that the dynamics of eigenvalues is given by an integrable N-bode system of Calogero type. This leads to exact classical solutions for the eigenvalues of the matrix model and provides a solution for the continuum collective field. An application of the discrete solution could be to search for the eigenvalue instanton suggested in ref.

[201. In the continuum we have shown that the classical collective field equation has an associated Lax pair. We have exhibited a transformation to a field theory free of an external potential. This in gravitational context implies background independence. We have constructed an infinite sequence of conserved currents for the continuum theory for arbitrary background potential. The conserved charges extend to an infinite parameter algebra of W~ type. It is interesting to note that such algebras arose in some previous studies of large-N Yang-Mills theories [21 ]. In the present context the presence of a W~ algebra in the collective representation establishes a closer connection with the string equation approaches at D < 1. It has been noted that the infinite chain model corresponds to a KP equation with a W~ algebra [22-26 ]. One might wonder what the relevance of all this structure is in the scaling limit. Clearly the conserved charges will have contributions from the boundary

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since in the string t h e o r y the i n t e r a c t i o n grows at the b o u n d a r y . I n d e e d , o n e has particle p r o d u c t i o n and the S - m a t r i x is not o f simple factorized f o r m [ 7 ]. T h e c o n s e r v e d currents must, h o w e v e r , h a v e d y n a m i c a l effects i m p o s i n g n o n t r i v i a l c o n s t r a i n t s on the a m p l i tudes. It is interesting to ask f u r t h e r m o r e w h a t is the c o n n e c t i o n b e t w e e n the c o l l e c t i v e a c t i o n a n d the universal action o f ref. [ 2 7 ] . T h i s was g i v e n in t e r m s o f K P operators and has been shown recently by Y o n e y a to h a v e a Woo algebra [ 2 3 ] . Finally, to c o m e back to the s t r o n g e r n o t i o n o f L i o u v i l l e integrability [ 18 ] we s h o u l d c o n s t r u c t angle v a r i a b l e c o n j u g a t e to the c o m m u t i n g c o n s e r v e d action v a r i a b l e s ( 3.19 ). Such a c o n t r i b u t i o n c o u l d be m a d e easier by using the W ~ algebra c o n s t r u c t e d above; n o t e that ref. [ 19] c o n t a i n s an explicit construction o f a set o f " c o n j u g a t e " v a r i a b l e s to the conserved variables. We h o p e to r e t u r n to this and o t h e r p r o b l e m s in the future.

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[7] K. Demeterfi, A. Jevicki and J. Rodrigues, Brown preprint HET-795. [ 8 ] J. Polchinski, Texas preprint UTTG-06-91. [9] G. Moore, Rutgers/Yale preprint RU-91-12 ( 1991 ). [ 10 ] P. DiFrancesco and D. Kutasov, Princeton preprint PUPT1237. [ 11 ] D.J. Gross, I. Klebanov and J. Newman, Nucl. Phys. B 350 (1991) 621. [ 12] A. Polyakov, Landau/Princeton preprint ( 1991 ). [ 13 ] E. Witten, IAS preprint ( 1991 ); G. Mandal, A. Sengupta and S. Wadia, IASSNS-HEP-91/ 10; M. Ro~ek et al. to appear. [ 14] A. Jevicki and H. Levine, Phys. Rev. Lett. 44 (1980) 1443. [ 15] F. Calogero, J. Math. Phys. 12 ( 1971 ) 419. [ 16] A.M. Perelomov, Sov. J. Part. Nucl. 10 (4) (1979) 336; M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 71 (1981) 313. [ 17 ] A. Neveu, Les Houches ( 1982); L.D. Faddeev and L.M. Takhtajan, Hamiltonian methods in the thoery of solitons (Springer, Berlin, 1987); J.M. Maillet, Phys. Lett. B 162 (1985) 137. [ 18] J. Liouville, J. Mat. (Liouville) 20 (1855) 137; V. Arnold, Mathematical methods of classical mechanics (Springer, Berlin, 1979 ). [ 19] G. Barucchi and T. Regge, J. Math. Phys. 12 (1977) 1149. [20] S. Shenker, Rutgers preprint RU-90-47 (1990). [21 ] E. Floratos and J. lliopoulos, Phys. Lett. B 217 (1989) 285; J. Hoppe, Ph.D. Thesis MIT (1982). [22] M. Fukuma, H. Kawai and N. Nakyama, Tokyo preprint UT-562. [23] T. Yoneya, UT-Komaba preprint ( 1991 ). [24] M.A. Awada and S.J. Sin, Florida preprint HEP-91-3. [25 ] 1. Bakas and E.B. Kiritsis, LBL preprint (August 1990). [26] H. Itoyama and Y. Matsuo, Stony Brook preprint ITP-Sb91-10. [ 27 ] A. Jevicki and T. Yoneya, Mod. Phys. Lett. A 5 (1990) 1615. [28] V.G. Drinfeld and V.V. Sokolov, J. Sov. Math. 30 (1985) 1975.

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