Classical integrability and higher symmetries of collective string field theory

7 Pages·1991·0.422 MB·English

by  Jean Avan, Antal Jevicki

Physics Letters B 662 ( 1991 ) 35-41
North-Holland PHYSICS LETTERS B
Classical integrability and higher symmetries
of collective string field theory
Jean Avan and Antal Jevicki
Department of Physics, Brown University, Providence, R! 02912, USA
Received 92 April 1991
eW demonstrate the complete integrability of lacissalc collective field theory. An exact equivalence to the classical N-body
problem of Calogero type si described. A Lax pair si constructed for the general continuum collective equations. A form of back-
ground independence si pointed out. An infinite tes of commuting currents si constructed in the general case. A large symmetry
arbegla si discovered and shown to take the form ofa W~ .arbegla
1. Introduction an integrable N-body system was pointed out long ago
141.
The collective field theory 1,2 of one-dimen- In what follows we proceed to establish explicitly
sional matrix models was suggested to represent a field the integrability of the cubic collective field theory.
theory of one-dimensional strings 1-5 . Its proper- First addressing the question of the classical dynam-
ties have been investigated recently in some detail. ics of matrix eigenvalues we give an exact corre-
For example in refs. 6-8 the amplitudes and higher spondence 14 with an integrable N-body system of
loop 7 effects were considered with results in per- Calogero type 15 . String theory is driven by a har-
fect agreement with perturbative calculations using monic critical potential and for this particular case
other methods 4,9,10 . The collective field theory the Calogero system is known to have a Lax pair and
is written as a field theory of tachyons, but is seen to an infinite number of conserved charges 16 . This
include the effect of an infinite sequence of addi- gives an exact solution to the eigenvalue dynamics.
tional discrete states. These states were exhibited as We then proceed to investigate directly the contin-
poles in the integral representation 7 of the four- uum classical equations of the collective field theory.
point amplitude and also in the study of the two point We first exhibit a reparametrization which is shown
function 11 . They are remnants of the graviton and to transform away the harmonic background poten-
higher string states 12 in 1 + 1 dimension. Study- tial and provide an equivalence to the theory without
ing the classical dynamics of the collective field the- an explicit potential. This in string theory context
ory will certainly shed some further light 13 on their represents a demonstration of background
dynamics. independence.
There are certain indications that the collective In the general case with an arbitrary background
string field theory is completely integrable. In the fer- potential, we can write down an infinite sequence of
mionic language conserved currents were suggested conserved currents. Their Poisson brackets reveal a
in ref. 5 and exact results for the amplitudes can be further structure; we construct an infinite parameter
written in integral form 9 . An exact classical solu- algebra which is shown to take a form of the W~ al-
tion was recently given in ref. 8 and an analogy with gebra. The conserved quantities are seen to represent
a particular Cartan subalgebra of this W~ algebra. Our
Work supported in part by the Department of Energy under study is done at the classical level and quantum im-
contract ksaT-03130RE67-20CA-ED .A plications will be left for the future.
reiveslE Science Publishers .V.B 35
emuloV ,662 number 2,1 SCISYHP SRETTEL B 22 tsuguA 1991
2. Eigenvalue dynamics, integrability and the fact that the integral in (2.5) is a Kronecker
delta .ji6 We see therefore that the classical collective
Consider the lagrangian of the collective field field theory, in a discretized form eq. (2.2), is
theory equivalent to an N-body problem,.
f z'(
2
LN = ~ iX + +v(xi) (2.7)
2k ¢
,=, , (x, xj) '
(2.1)
which describes the dynamics of the eigenvalues x,(t).
Since the collective field represents the density ofei-
For the special case of a harmonic potential
genvalues of the matrix model
v(x)= 2x2oc½ this is the Calogero model 15; it is
0(x, t)= ~ O(x-x~(t))=Tr3(x-M(t)), (2.2) completely integrable with known exact classical so-
lutions 16. To summarize one takes for the Lax
one can address the problem of the classical dynam- matrix
ics of these eigenvalues. This question was consid-
ered some time ago 14 with the following result: L=k,~o +i 1-~° (2.8)
xi-xj
One has to solve the classical matrix model equations
and denotes
~:l+v(m) =0 (2.3)
X= Diag (x,).
with the constraint that the angular momentum be
J,b = iM, /~3 = 1 -Oat,. This particular constraint was A relation to the matrix model is given by
seen 14 to translate into the cubic interaction of M(t)=U(t)X(t)U-l(t) and M(t)=U(t)L(t)
the collective field. For a harmonic potential × U- 1 (t), where M is precisely of such a form that
v(M)=½co2MZ one has the matrix solution the above mentioned constraint on the angular mo-
M(t) =A cos toc + B sin .toc It provides implicitly a mentum is fulfilled. The equations of motion are then
classical solution of the collective theory. solved by
More explicitly, we can show that dynamics of ei-
genvalues is given by an N-body Calogero problem. M(t) =X(0) cos cot+ 1 L(0) sin toc (2.9)
To demonstrate this equivalence one writes the cu- co
bic interaction as
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)=Yr6x-M(t)l.
An infinite sequence of conservation laws follows
from taking traces:
The kinetic term is simply I~=Tr L +icoX) (L-icoX) ft. (2.10)
(0-1~)2
dx One can also write down still another form for the
3 ¢ solution in terms of Hermite polynomials:
-- ~ ~ ,c;. f )dxx( O)jx-x(~),x-x(~ 9;: ~,(x, t) - I (x-x,(t))
i
N
1 -~ Zk, 2, (2.5) = Y, C,e'"~°'HN_,(x). (2.11)
i rl~O
where we have used
This follows from the fact that gt(x, t) obeys the lin-
~(x, t) = - ~Y '~c,c~ (x-x,) (2.6) ear equation 51
i
36
Volume 266, number 2,1 PHYSICS LETTERS B 22 August 1991
iql t -4" ,lq xx -- X~,x + Nq/= 0. (2.12) ,q(0~ z) =O(x, t) ch t (3.6)
Here the C,'s are arbitrary (initial value) constants coupled with a coordinate reparametrization
providing a solution of the collective initial value
X
problem.
q=cht' r=tht. (3.7)
3. String theory, background independence The integration measure becomes
String theory is obtained from the matrix model in dqdz= f dxdt (cht) -3 (3.8)
the double scaling limit where N~ and ~t--,0 with
z/ being the Fermi level. The potential responsible for and by a chain rule
a critical behavior and the continuum limit is an in-
O
verted oscillator v(x) = - ix2. The continuum string ~r tP(q, r )
field theory is then driven by the interaction
~int = fdx Ig2~3(X,I)--½X2~(X,t). (3.1) - ~r{O,O(x,t) cht+xchtG~(x,t)}. (3.9)
The linear tadpole term produces a classical vacuum Integrating
background n~o(X)=(lz+x2) .2/~ This induces a
O~l(~=-x~(x,t) sht+chtOz~cht. (3.10)
background metric
~,~G = ( 1/0 2, ~2~2 (x)) , ( 3.2 ) The kinetic term of the initial lagrangian (3.5) then
becomes
since in the quadratic approximation the lagrangian
(G-,6) ~
becomes
½ f dqdr
1 )t 2
~2)- 2 ~o(X) ½~¢%(x) (0xt/)2 " (3.3)
! ~ dx dt ch-2t (x sh t~+ ch t ~-~)2
-2 j 0
The metric signifies a presence of gravitational de-
2
grees of freedom. One might wonder whether a back-
\ch t/ x2~
ground independence formulation is possible.
In the Calogero model there exist a remarkable
connection between the solutions of the model with
+ -~Ox2 ' ~ th t
an oscillator potential and the free Calogero model.
It is given by (cf. ref. 16 )
-- 1 ~ XO d/((0"~-~ -~)2 )
-2 +x2f~(x,t) . (3.11)
X,( t ) = 2i( l tg ~ot) cos t~o . (3.4)
In the last step, we have made use of partial integra-
We now show that in the collective string field theory tions. What we have shown is that through a coordi-
there is a corresponding transformation. It will relate nate transformation the kinetic term generates a non-
the harmonic theory to the formulation without a po- trivial oscillator potential term. This implies that the
tential term, namely a theory with only a cubic fundamental theory is that with no background po-
interaction: tential eq. ( 3.5 ) .
This property of the collective field theory ob-
So= fdr ;dpqt~ .)c',q(3~2~71-2~)9(l-O( (3.5, viously has far-reaching implications. First it
strengthens its status as a closed string field theory.
Consider a field transformation In any string theory one has gravitational and higher
degree of freedom; a theory of gravity should have a
37
Volume 266, number 1,2 PHYSICS LETTERS B 22 August 1991
background independent formulation. the equations without the harmonic potential
On the practical side the transformation given al- ±~c+O = ~txO 2 . These are easily solved by
lows one to generate solutions of the full theory from
a+ (x, t) =f+ (x-a+ t) , (4.7)
those of the theory without an external potential.
where f± are arbitrary functions. The classical collec-
tive field for the full harmonic problem is then given
4. Lax pair, conserved currents and a ooW algebra
by
The fact that the Calogero model possesses a Lax
z~2 ch t +~c ,tht
pair and an infinite sequence of commuting con-
served quantities leads us to investigate the same is-
sues directly in the continuum. In addition, these
currents belong to a larger infinite symmetry algebra.
These will be described in what follows. We can write down a Lax pair for the collective equa-
Consider the continuum hamiltonian tions. The equations for the components a±(x,
t) =Hx+nO(x, t) were seen to decouple and can be
H= f dx{½HxOHx+~r2~3+v(x)-itO} (4.1) written as
O,a(x, t) = 0x(½a2+ v). (4.9)
and the classical equations
This is seen to take the Lax form:
&x, t)= {/-/, 0} = 0x(Or/,),
//(x, t) = {H, H}_ -!H2 2 x, -t- 1/ 2 ~ 2 -}- l) , (4.2) d/~= £, fl . (4.10)
dt
arising from the Poisson brackets {H(x), ~(x)}
= a(x-y). Differentiating the second equation, and Simply take
forming linear combinations
d
£= Ux +e~(x, t) ,
a± (x, t)=Hx + 0er (x, t) (4.3)
the classical equations separate, 2d
37/= ~ dx -2o~ +Oqx+V+O~ 2 (4.11)
+lo,O =ct+xo~ ± +Vx. (4.4)
and the statement follows. We note that the L-opera-
This also follows from the fact that the hamiltonian
tor is that of the modified KdV equation 28. It is
can be written as
interesting to note that the above Lax pair exists in
f(1 1 ) the case of an arbitrary potential v(x). In the discrete
H= 1-~x (a3+ - _3~o ) + ~ (o~+ - a_ ) v(x) .
Calogero case integrability is present only with a qua-
(4.5) dratic potential v(x)= x_+ .2 Clearly the continuum
formulation achieves a considerable simplification.
For the case of an inverted oscillator potential
We now consider the continuum equation further.
v(x)=-x 2 an exact solution was obtained re-
From the fact that the continuum theory is written as
cently by Polchinski 9 . It is given in a parametric
a Lax equation one expects that there is an infinite
form:
number of conserved currents in the collective field
x=-a(a) cht-b(a) , theory. Consider either one of +~c equations. There
will be two independent sets of currents for the +_
p= -a(a) sh t-b( a) . (4.6)
components respectively.
This is clearly related to the eigenvalue solution de- It is easy to find the currents when the potential
scribed above eq. (2.9) . v=0 so that the last term is absent in the equation.
We record here one other general solution. It is Then
based on the observations of section 2. Consider first
38
emuloV ,662 number 2,1 PHYSICS LETTERS B 22 August 1991
J~)=a", J}")- n an+l (4.12)
{f dx f da (oe2+2v)", ~ dy~ eod (a2+2v) ''}
n+l
for n = 1,2, .... Clearly
(x-y)
= f dxdy (a2+2v)"(x)(a2+2v)"(y)~ '
?/
O,a"=n&a"-l=na~a"=
n-~ 0x(a"+' ) (4.13)
= f dx2n(otOZx+Vx)(Ol'2+2V)n+m-l(X)
and the currents obey the conservation equation
~rP0 ") -0~Jl ") =0. (4.14) = ~ dXax(n(a2+2v)n+"'~
/=0. (4.20)
Next we extend these conservation equations to the
v(x).
general case with an arbitrary potential A re-
Here we have constructed a set of conserved com-
cursive construction leads to the following ansatz for muting charges of the form
the Jo component:
Q(n'= f dx ; eod (ot2+2v)".
)t,x(:o (4.21)
J6k)= J da' O/'2+2V(X) k . (4.15) The conservation and commutation of these charges
is true up to boundary terms. At the quantum level
This in more explicit form reads
again these currents can in principle result in conser-
k 2"k! vation laws. This will happen only if the surface terms
~_)k~j E )xTVl+)n-k(2'O
.=o (k-n)!n 2(k-n) + 1 in the spatial integral die out. For string theory the
(4.16) interaction grows at the boundary and clearly such
surface terms will have a most important effect 17 .
Taking the time derivative gives
It must be noted that the divergenceless currents
~I~0 ~k =&( a2 + 2v)k= ( aa,x + v,,) ( a2 + 2v)
* and conserved quantities were constructed without
using the Lax operator L. In fact, since L in (4.11 )
1
- 2(k+ ~1 O~(a2+2v)k+' (4.17) does not contain v while (4.21 ) does, we expect that
the conserved currents arise as trace identities for a
and we find a divergenceless current with different Lax representation; however, it is quite suf-
ficient to have the commuting conserved quantities
1
j}k) 2(k+~) (O~2+2V)k+l (4.18) (which is all that is needed to establish integrability,
at least when one has a finite-dimensional system
fdxJo(x, t)
Moreover, the conserved charges are 81 ). A Lax representation is useful in practice when
easily seen to Poisson-commute up to boundary terms one wishes to use inverse scattering to solve a given
in the general case. From the Poisson structure for integrable model; we shall not consider this point
a ,_+ deduced from the Poisson structure for H and here.
(we scale a+ --,x/~ zo + ), We now describe an infinite-dimensional algebra
present in the collective field theory. We have seen
{a+_(x),a±(y)}=a'(x-y),
{a_+, at_ }=0 ,
that in general the conserved currents consist of terms
(4.19) of the form v(x)~aP(x, l). For polynomial potentials
x~a .p
these become terms of the type Consider now
one computes the Poisson bracket of two conserved
the algebra of these objects. Define the generators
charges:
H~ f dxx m-I 0 n-mL
= - - (4.22)
J m--n
m=n ot°/O-loga.
with the fact that for we define
An explicit computation using (4.19) gives us the
algebra
39
emuloV ,662 number 2,1 SCISYHP SRETTEL B 22 tsuguA 1991
,~,~H{ , Hm2} n2 ~a~_x'" and a'~x% however, mixed monomials
+na a'"_x p in general do not lead to closed partial in-
= (m2-- 1 )n,-- (m l- 1 .2-2,,,+,mH2n) 2n+l, (4.23)
tegrations. A combination of such monomials never-
This is recognized as a version of a classical W~ al- theless can cancel the non-integrable terms and lead
gebra 25. We have a W~ algebra for both sectors to closed Poisson brackets. From an SU ( 1.1 ) algebra
+~c and a_, and since they commute with each other present in the Calogero model 18, it follows that
we have W~®Wo~. A particular subalgebra for m = 2 the continuum limit generators defined as
consisting of generators of type oH{ = H(4.5 ), 1H = ,}¢x,/xf H2 }02xf build an SU( ,1
=
Ln=fx .... /~2-n)=H~ is the classical Virasoro 1 )algebra in (W~) .2 However, this is also true and
algebra can be explicitly checked when Ho is replaced by
H~ =Ho+ f(3c~ 2 a+ - +2~03 ct_ ). Although H~ does
{L,,Ln2 } = (nl -nz)Ln~+,2 • (4.24)
not belong to W 2 it has closed brackets with Hi and
The conserved charges including the hamiltonian .211 We also expect therefore that other symmetry al-
are also obviously members of the ~oW algebra; they gebras (may be also infinite) can be constructed and
are given by HI' in the simplest case when v=0. This help to solve this field theory.
is a Cartan subalgebra of .~oW
The general case with an arbitrary potential v(x)
can be set in this framework. We have seen that the 5. Conclusions
commuting conserved currents are given in general
by eq. (4.21). For a polynomial potential is a linear We have demonstrated the complete integrability
combination of generators of our ooW algebra. In par- of classical collective string field theory. At the dis-
ticular for v(x) =x 2 we have crete eigenvalue level we have seen that the dynamics
of eigenvalues is given by an integrable N-bode sys-
dx J~ tem of Calogero type. This leads to exact classical so-
lutions for the eigenvalues of the matrix model and
~ 2~k' x2na2(u_n)+l provides a solution for the continuum collective field.
dx, _o (k-n)!nI2(k-n)+ l
An application of the discrete solution could be to
search for the eigenvalue instanton suggested in ref.
k 2"k!
rl4,,-zk (4.25) 201.
--,=o ~y (k-n)!n! "'2"+j "
In the continuum we have shown that the classical
In general and up to boundary terms, the con- collective field equation has an associated Lax pair.
served charges, including the hamiltonian (4.1), build We have exhibited a transformation to a field theory
therefore a commuting subalgebra of the (Wo~) 2 al- free of an external potential. This in gravitational
gebra which may be maximal. One should therefore context implies background independence. We have
try to obtain explicitly all such Cartan-like subalge- constructed an infinite sequence of conserved cur-
bras of ooW in order to define different integrable rents for the continuum theory for arbitrary back-
hierarchies of hamiltonians for the collective fields ground potential. The conserved charges extend to an
(in the weak sense of existence of commuting con- infinite parameter algebra of W~ type. It is interest-
served quantities). The study of representations of ing to note that such algebras arose in some previous
the W~ algebra then may provide explicit solutions studies of large-N Yang-Mills theories 21 . In the
of these field theories. present context the presence of a W~ algebra in the
A few concluding remarks on the results of this collective representation establishes a closer connec-
chapter must now be made. The main part in the con- tion with the string equation approaches at D < .1 It
struction of the o~W algebra is to check that the Pois- has been noted that the infinite chain model corre-
son brackets of the generators close. Hence the par- sponds to a KP equation with a W~ algebra 22-26 .
tial integrations triggered by the non-ultralocal One might wonder what the relevance of all this
Poisson brackets (4.19) must take a closed form. As structure is in the scaling limit. Clearly the conserved
shown above, this is the case for the monomials charges will have contributions from the boundary
40
Volume 266, number 2,1 PHYSICS LETTERS B 22 August 1991
since in the string theory the interaction grows at the 7 K. Demeterfi, A. Jevicki and J. Rodrigues, Brown preprint
boundary. Indeed, one has particle production and HET-795.
8 J. Polchinski, Texas preprint UTTG-06-91.
the S-matrix is not of simple factorized form 7 . The
9 G. Moore, Rutgers/Yale preprint RU-91-12 ( 1991 ).
conserved currents must, however, have dynamical
01 P. DiFrancesco and D. Kutasov, Princeton preprint PUPT-
effects imposing nontrivial constraints on the ampli- .7321
tudes. It is interesting to ask furthermore what is the 11 D.J. Gross, I. Klebanov and J. Newman, Nucl. Phys. B 350
(1991) 621.
connection between the collective action and the uni-
21 A. Polyakov, Landau/Princeton preprint ( 1991 ).
versal action of ref. 27. This was given in terms of
31 E. Witten, IAS preprint ( 1991 );
KP operators and has been shown recently by Yoneya G. Mandal, A. Sengupta and .S Wadia, IASSNS-HEP-91/
to have a Woo algebra 23. ;01
Finally, to come back to the stronger notion of M. Ro~ek et al. to appear.
14 A. Jevicki and H. Levine, Phys. Rev. Lett. 44 (1980) 1443.
Liouville integrability 18 we should construct an-
15 .F Calogero, J. Math. Phys. 21 ( 1791 ) 419.
gle variable conjugate to the commuting conserved 16 A.M. Perelomov, Sov. J. Part. Nucl. 01 (4) (1979) 336;
action variables ( 3.19 ). Such a contribution could be M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 17
made easier by using the W~ algebra constructed (1981) 313.
71 A. Neveu, Les Houches ( 1982);
above; note that ref. 19 contains an explicit con-
L.D. Faddeev and L.M. Takhtajan, Hamiltonian methods
struction of a set of"conjugate" variables to the con-
in the thoery of solitons (Springer, Berlin, 1987);
served variables. We hope to return to this and other J.M. Maillet, Phys. Lett. B 261 (1985) 137.
problems in the future. 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. 21 (1977) 1149.
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41