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A nonperturbative solution of D = 1 string theory
Volume 238. number 2,3,4
PHYSICS LETTERS B
5 April 1990
A N O N P E R T U R B A T I V E S O L U T I O N O F D = 1 S T R I N G T H E O R Y ¢~ David J. GROSS and Nikola MILJKOVI(~ Joseph ttenry Laboratories, Princeton University, Princeton, NJ 08544, USA
Received 15 January 1990
We derive a nonperturbative solution of D= 1 string theory, based on a double scaling limit of the one dimensional random matrix model. We derive an exact expression for the partition fimction in terms of the string couplingconstant. The weak coupling expansion suffers from infrared divergences, which we attribute to massless tadpoles. The continuum limit seems to be well defined, however, in a strong coupling expansion. This could correspond to a different stable nonperturbative vacuum.
1. Introduction Recently great progress has been made in the nonperturbative analysis of toy models of string theory, defined in terms of matrix models as discrete sums of r a n d o m surfaces with matter attached. However, all of this has so far only been accomplished for theories whose matter content has central charge less than one. From the point of view of string theory this corresponds to dimensions of space-time less than one. We are interested in extending these results to higher dimensions for m a n y reasons. These include the possible application of string theory in four d i m e n s i o n s to QCD, as well as the insight we hope to gain as to the n o n p e r t u r b a t i v e structure of critical string theory. c = 1 seems to represent a kind of barrier, beyond which many interesting complications arise. First, the straightforward c o n t i n u u m analysis ofconformal field theories coupled to two dimensional gravity breaks down at c = 1. The treatment that works for c < 1 produces, in this region, unphysical singularities [ 1-3 ]. This might be an indication, as m a n y people have speculated, of a possible phase transition to a new, strong coupling phase in which the string degenerates to a branched polymer or is in a crumpled phase. Alternatively these problems might be related to the fact that at c = 1 one expects to find a massless particle that could give rise to infrared singularities. This is the well known tachyon of the d-dimensional bo¢~ Research supported in part by NSF grant PHY80-19754.
sonic string theory, whose mass precisely vanishes for d = 1, according to the famous evaluation [4] of the ground state of a d-dimensional string, m 2 = ~ ( 1 - d ) .1 c = 1 also represents a borderline case for the applicability of the matrix model approach. Standard large N techniques break down for dimensions greater than one. This is not too surprising since the models with c < 1 are truly models with a finite n u m b e r of degrees of freedom but this n u m b e r diverges as c-~ 1. One might expect that the matrix representation of string theory for c > 1 would be as difficult to solve as large N QCD. Fortunately the case of c = 1 can still be treated by large N techniques and, as we shall see, solved exactly. The case of c = 1 matter, coupled to two dimensional q u a n t u m gravity, was solved on the sphere by Kazakov and Migdal [ 5 ], who made use of the beautiful solution of Br6zin, Itzykson, Parisi and Zuber [6] of the large N one dimensional matrix model. They discovered a scaling form for the free energy albeit with mysterious logarithms. Kostov studied this model further and calculated the two-point function on the sphere, showing the existence of a massless mode in the c o n t i n u u m limit [ 7 ]. We shall show in this paper that the appropriate
"~ The factor d - 1, which is simply the number of degrees of freedom of the string, differs from the number d-2, of transverse degrees of freedom, because the longitudinal mode does not decouple in noncritical string theory.
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double scaling limit of this model, where both N ~ o o and the bare cosmological constant is carefully adjusted, yields an expansion of the free energy which can be evaluated to all orders in the string coupling constant. The mysterious logarithms now appear in full force. We argue that they are simply manifestations of the infrared divergences that appear in higher orders due to the massless mode. Each successive handle that is added to the surface can attach through a massless propagator, which yields a logarithmically divergent factor. This means that the genus expansion, which is an expansion in powers of the string coupling, is very badly behaved. Not only is the series divergent (its coefficients growing like (2n)!), but the nth term contains a factor ln(ov) n. This means that no matter how small we make g s2t r i n g , w e a r e driven in the continuum limit, to large coupling and cannot trust the asymptotic perturbative expansion. In effect the number of handles becomes infinite and the picture in terms of well behaved surfaces breaks down. Fortunately the full sum has a natural nonperturbative definition which we can use to study the theory for large coupling. Here we find a convergent expansion, which leads in the continuum limit to a simple and finite result. We interpret this strong coupling result as equivalent to a shift of the vacuum o f the theory, which was unstable due to massless tadpoles.
2. The matrix representation The representation of the sum over random surfaces, or string theory, in terms of matrix models is well known by now [ 8 ]. Here we are interested in the coupling of a one dimensional continuous matter degree of freedom to the surface. This can be described by a one dimensional model of N by N hermitian matrices, q~(t), Zs(fl)
=
f
DNZ~(t)
5 April 1990
the partition function. Each given order in 1/N 2 corresponds to a sum over the surfaces of definite genus, generated by the Feynman graphs of that genus. It is easy to show that each graph is weighted by a factor (N/fl) Area. In addition the graph contains an integral over a product of propagators, exp(-mlti-t,I ), connecting adjacent ~ ( t ) ' s on the graph. It is believed that in the continuum limit, i.e. take N~ fl~g--+gcritical SO that the perturbation series diverges and is dominated by infinite area terms, this coincides with the continuum definition o f a gaussian variable coupled to two dimensional gravity ~2. From the point of view of string theory this corresponds to a noncritical string in one dimension, i.e. the theory in which the two dimensional world sheet is mapped onto one dimensional time t. As was nicely shown in the classic paper [6] the problem of finding the ground state energy In Z(fl) can be greatly simplified. Since the ground state is U (N) invariant one can essentially integrate out the angular variables leaving just the eigenvalues of 4 , which become fermionic. The problem is then equivalent to finding the ground state o f N fermions living in the potential U(x), i.e. of the hamiltonian ~__
1 d2 2fl 2 d x 2
(2.2)
"t- U ' ( X ) .
If we denote the energy levels of such a hamiltonian by ek, than the ground state energy of the model is Egs/fl=e~+e2+e3+...+ex, where we fill the energy levels up to the Fermi level, #F= eu. We have normalized so that f l ~ N plays the role of the inverse of Planck's constant. It is useful to introduce the density of states, 1
p(e)= Z a(e.-e), in terms of which the equations that determine the ground state energy are
g= ~ =
p(e) de , Eg~=B 2 p(e)ede. 0
0
(2.3)
× e x p ( - f l f dt Tr[½~2(t)+ U(Clg(t) ) ]) . (2.1)
The sign that we are in the continuum limit is that the energy, Egs, is a singular function of the cosmo-
The connection with D = 1 string theory is established by considering the perturbative expansion of
,2 The difference between the gaussian propagator and the exponential one should not matter in the continuum limit.
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logical constant. This we achieve by adjusting the potential so that its m a x i m u m , which we define to be ~ a ~ =/t~i,~,~, approaches the Fermi level from above. At this point the levels will begin to spill over the well and A=g~,~a~-g will be a singular function o f ~t-=/tc~iti~,~- ~tv ~3. A is the r e n o r m a l i z e d cosmological constant, in the sense that the surfaces are weighted by g A r e a = ( 1 --Z] )Area ~ e--AArea. In principle we should solve the first equation for /~ in terms o f A and then use the second to solve for Egs. It is often more convenient to work with the derivatives o f these 0g - -p(~), 0~t 0Egs 0// --
0g fl2~FP('I'IF)=flZ/tv 0-~"
( 2.4 )
Kazakov and Migdal solved this p r o b l e m in the leading 1 / N a p p r o x i m a t i o n , i.e. for the sphere, where one can use the W K B a p p r o x i m a t i o n . In this case the eigenvalues can be taken to be continuous and the density o f states is given by (_+x~ are the turning points of the classical trajectory,/t~iti~ = U( _+Xc) ) + Xc
p(e)=~
1 "j
dx
- x/~e-U(x)]"
(2.5)
The singularity o f this integral at e = ¢tV, which occurs when ~t~0, arises from the region near the turning points where U(x) can be a p p r o x i m a t e d by a quadratic expression, U ( x ) ~ / J c - - 2 ( Y c - - X ) 2 ~4. It then follows that
1 P(l~v) = - ~ I n / x + regular t e r m s .
(2.6)
Therefore 0 g / 0 / l ~ l n / ~ , or A = 1 - g ~ - ~ t l n / ~ . Since OEgs/Og~-fl2/J~fl2d/lnd we finally derive (as N-~ oo ) that Eg~
~/~2
N2zj2
lnA
lnA
lnd 2:R./~2ZI 2 "
(2.9)
We shall see that this is the case. The above calculation is totally universal, i,e. ind e p e n d e n t of the form o f the potential U(q~), as long as it has a quadratic m a x i m u m somewhere. It is easy to generalize this calculation to potentials that do behave differently a r o u n d their m a x i m u m [9]. I f the first k - 1 derivatives of U(x) vanish at Xc, then Og/O~,U -1/2+1/k, which leads to 7 s t r - - - - ( k - 2 ) /
(k+2). 3. The nonperturbative solution
(2.7)
This is in agreement with the expectation from the c o n t i n u u m theory that Egs ~ N 2 A 2+~'s'~ ,
and the K P Z calculation [ 1 ] that yielded ~str-~-0. The extra logarithm that appears is presumably due to the existence o f massless modes o f excitation in the theory. In this theory we have one s p a c e - t i m e coordinate and can form a continuous set of local operators labeled by a m o m e n t u m p. Indeed, Kostov evaluated the two-point function in the leading 1IN approxim a t i o n and found that the theory contains a sequence o f evenly spaced excitations, with masses m n = n¢o, which vanish in the c o n t i n u u m limit where (o ~ 7t/ln A. According to the principles o f string theory these massless particles cannot cause any real problem in the spherical limit since conformal invariance implies that they cannot be emitted as tadpoles. One way of arguing this is to note that when summing over all spheres one is automatically summing over genus zero tadpoles attached to the sphere, so that all infrared divergences associated with v a c u u m instability are automatically dealt with. But, as in critical string theory, we expect real infrared divergences to a p p e a r in the next order, where the massless tadpoles can couple to a torus. This would lead us to expect that the genus k contribution would behave as g~:~ ( l n A ) k, where the string coupling constant, gs~, is defined from the leading term as g2 _
-- Xc
5 April 1990
(2.8)
~3 We will always normalize so that gcritieal=I. ~4 The choice of U" (x~) = - 4 does not affect the critical behavior.
We now turn to the n o n p e r t u r b a t i v e solution o f the model. Again, we let N ~ but also take A-~0 fast enough so that the higher orders o f the genus expansion are not suppressed. The expectation [2,3] is that the free energy scales, in kth order, as ( l / N ) 2k X ( 1/A) 2 y,)k, so that if we keep gst fixed as N-~oo, the expansion will be an expansion in powers Ofgst. We 219
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shall see this is correct, up to logarithms of N which arise due to infrared divergences. First, let us note that the calculation of the singularity in the density of states is only sensitive, as it was on the sphere, to the behavior of the potential near its m a x i m u m . This can be seen by examining the expression for the density of states
p(ltv) =
;
1 Im T r / 7 - / ~ v - i ~ '
near the m a x i m u m of the potential. Expand the denominator about y = x ~ - x ~ O , and note that h -/zF ~ - ( 1/2,82 )02 + / z - 2 y 2 + O ( y 3 ) . In our scaling limit we take ,8~oo and/~ ~ 1/,8. Thus i f y 2 ~ 1/t8, then the denominator will scale as 1/,8, and the cubic and higher powers o f y will be suppressed. The coefficients R~ are local functions of U and its derivatives by powers of 1/,8. Even though the scaling will be spoiled by logarithms this does not affect this power suppression of all but the quadratic part of the potential! The hamiltonian that survives this scaling limit is simply an inverted harmonic oscillator. The calculation ofp(/zv) is then straightforward. We use the following trick. Consider the density of states of the normal harmonic oscillator of frequency o~,
5 April 1990
The density of states can be calculated by expressing the trace over states in the coordinate basis, 1 Ix'>. p ( e ) = 2~fllmnf dx' ( x ' l _82+2f12[U(x)_e]
(3.3) This integral is imaginary only for Ixl (x,, and is singular near the turning points. The diagonal matrix element of the resolvent can be expanded in inverse half integer powers of e - U(x' ) following Gelfand and Dikii [ 10 ],
), 7t"
l=o
Rl{2fl 2 [ U(x) - U(x' ) 11 ,= ~'
×
{2[e_U(x,)]}l+l/2
The coefficients Rt are local functions of U and its derivatives and are given in ref. [10]. If one evaluates this expansion for x' - x c it follows immediately that the only terms that survive in the scaling limit come from the quadratic expansion of U(x) in the vicinity of its maximum. The first few terms for p(/tF) are easily calculable and yield
1 1
~( 1 Im ~
p(E)
P(~v)=
1
~ (n+ ½) o ) h - E - i e '
and continue o) to imaginary frequency. This yields, in our case, the expression
1 Re~
P(/@)= ~
l
2 n + 1 +i,8/~ "
(3.1)
This expression is divergent; however, the divergent part is/t independent and will not affect the critical behavior. We can fix it by demanding agreement with (2.6) in the limit o f , 8 " ~ . This yields p(/tv) = ~-n Re[C(1, ½(1 + i , 8 / O ) - o o ]
1
- 2 n [ln(½fl) - R e ~/(½ (1 + i,8/t) )]
,
7
1
- - I n # + 6 f12p2 + 60 fl4~4
311
+ 126
)
(3.5)
fl6]~6 -{- ....
Let us now use the exact expression for p(;Zv) to derive the complete asymptotic expansion 1
p(l~v) = 2~n Re[C(I, ½(1 +ifl~) ) - 2 _--
1 ln/l+~im 2n
~
]
1
,,=0 1 - i ( 2 n +
1 )/,8#
- 2nl [-lng (3.2)
where C(z, q) is the zeta function, C ( z , q ) = Z~=o 1 / ( n + q ) - - a n d ~'(z)=I" (z) / l ' ( z ) . Since this derivation is rather formal we present a check of its validity by a more rigorous calculation of the asymptotic expansion o f p ( e ) in powers of 1/,8. 220
(3.4)
•
+2 ~=,
?k
\fl,uJ (22k ' - l ) C ( 1 - 2 k )
], (3.6)
where the C-functions arise from the (divergent) series En ( 2 n + 1 )*. These C-functions are given in terms of Bernoulli numbers B2m, so that
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+ ~_, (22~_,_1)IB2.,,I .,=, rn
1 )"
(flff)2m
(3.7)
It is easy to verify now that the first few terms are in complete agreement with (3.5). The Bernoulli numbers grow rapidly as IB2ml ~ (2rn)!, leading to a highly divergent expansion for P(/*F)- It is precisely of the same form that was found in the solution of the string theories in D < 1 [ 11-14 ]. N o w let us return to the evaluation of the free energy in terms of the cosmological constant 3. Unfortunately the best we can do is derive a parametric representation of 3 and Eg~ as functions of/2. It follows from (2.4) that Cq-~=P(/2v) ,
=fl2/*+regu lar terms ,
(3.8)
where p(/2V), is given by (3.2) as a function of/*. These equations can be solved perturbatively to express Eg~ as a function of 3. First, integrate (3.2) to obtain
5 April 1990
which behaves in the continuum limit (as A-~0) as G (p = 0) ~ - In A [ 7 ]. This means that the weak coupling expansion is very sick. Not only does it diverge badly but it cannot be used as an asymptotic expansion, since no matter how small we take gst the effective coupling is gst In A, which blows up in the continuum limit. This can be traced back to the fact that we evaluated P(/*F) in an asymptotic expansion in powers of 1/fl/2. However, we found that in the scaling limit
ln3 ~gs,~f-ln3
1
fl/2
(3.12)
f13
and thus we can never be made small! This situation is similar to the case in ordinary field theory where we are driven away, in the continuum limit, from an unstable fixed point at gs,=0. We therefore must sum the series exactly, to all orders in gs,, before taking the continuum limit. Fortunately we possess an expression for the exact form of P(/2v) which we can use to explore the theory for large fl/2. Previously we expanded 03 1 /20 =P(/2F) = ~ Re [In ½fl-- ~,( ½( 1 +ifl/2) ) ] 4-in a power series in 1/fl/2. Now let us expand
-
~ m
=
(22m-~--1) 1
IB2mI m(2m-l)
l
)
(flff)2m
"
(3.9)
Then invert this expression to write ~*=-2re3~ ln/2+f(fl/2)/flln/2 as a function o f d . This produces a series of the form 2)zd I I + /*=
-
~
~ ,.=,-
c
(-lnA'~n(-lnd)r~ 1 ....
k
where we neglect double logarithms. Now we integrate (3.8) to derive
l(
Eg,,= -~- 1+ gs~
~=1 m=l
c,,,mgs2"(--lnd) ''
(3.10)
)
P(/*F) = -
2~ In/~
1
+-R ~ 'eo 2 n~_ +l~z
1
1
l+ifl/2/(2n+l)
(3.13)
in a power series in fl/*. This is straightforward and yields 0A 1 0~ =P(/2F) = -- ~ In # 1
+ -~k~=l (--1)k(1--2--(ak+l))((2k+l)(fl,U) 2k" .
(3.11)
This weak coupling expansion exhibits the logarithmic infinities that we expected, i.e. the powers of ( - I n J ) ~ that occur in nth order. We associate these with infrared divergences that occur, say, when n genus one tadpoles couple to the sphere. Each tadpole connects through a zero m o m e n t u m propagator,
(3.14) This series is much better behaved. First, the (functions that appear are evaluated at positive values and do not increase at all (((z)_,~oo-, 1 ). In fact, this series is absolutely convergent for I fl/2[ < 1. Second, we expect that the expansion parameter, fl/2, will now be small in the continuum limit. We now integrate 221
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this expression for A and invert it to w r i t e / t as an expansion in A,
~=-i~
1-WS+ ....
This yields ( ~ f l 2 L2 j n
2hA (
d .... \ ~ n ~ j ?1~ | t;'l~tl+
gsl
1-~
n=l
re=n+1
I
Onm 2nlnmA ' gst
m
/
.
Op
1
i
t / fla
(4.1)
0~ ( / I v ) = 2 ~ I m
(3.16)
In fact if we were to directly Borel transform the asymptotic expansion of (Op/Ol~) (/~F) we would obtain the same integral, but with the contour o f integration rotated to lie along the imaginary axis, i.e.
4. Discussion There are a few points regarding our solution that are worthy o f comment. ( 1 ) W h a t is the nature of the strong coupling solution? N o r m a l l y when one encounters infrared divergences due to massless tadpoles one must shift the v a c u u m to obtain a finite theory with a stable ground state. Presumably that is what is occurring in D = 1 string theory. We have found a consistent nonperturbative c o n t i n u u m solution with no infrared divergences. It, however, does not have a weak coupling expansion. This could mean that we have, by summing over surfaces with an infinite n u m b e r o f handles, shifted to a stable vacuum. In the new v a c u u m the free energy no longer has a natural expansion in terms o f surfaces o f finite genus. Perhaps one cannot even describe it in terms o f two-dimensional surfaces at all, just as in o r d i n a r y field theory o f pointlike particles a nontrivial, n o n p e r t u r b a t i v e v a c u u m cannot be described in terms of the original F e y n m a n graphs. A complete u n d e r s t a n d i n g o f this phase requires evaluation of the correlation functions o f physical operators. This should be possible to do exactly and work in this direction is in progress. (2) S u m m i n g divergent series. It is interesting to ask how is it that we have m a n a g e d to sum the origi-
dte
-i'sinh(t/flll ) .
(3.15)
This series is now extremely well behaved. In fact it is reminiscent of an asymptotically free theory, since now when we go to the c o n t i n u u m limit, A--,0, only the first term survives.
222
nal, highly divergent and non-Borel s u m m a b l e perturbation expansion. The weak coupling expansion for, say, (8p/Oll)(ltv), is o f the typical non-Borel s u m m a b l e series, ~g2"(2n)!, that one gets in string theory [ 15 ]. The exact result that we derived can be written as an integral which looks like a Borel transform,
3)-m )
(ln
Then we integrate (3.8) to obtain
Egs= ~ -
5 April 1990
o
Op (a~)=
Ol~
1 i
2~
t/fl/2
(4.2)
dte-'sin(t/fll~)"
o
This integral does not exist since the Borel transform has an infinite n u m b e r of poles on the real axis at t= nnflCt, although it has an identical asymptotic expansion. To specify the integral we must a d d a prescription for integrating about each pole, a potential source o f an infinite n u m b e r of arbitrary parameters. In the recently solved string theories for D < 1 one found differential equations that could be used to recover perturbation theory and provide a nonperturbative definition o f the theory [ 11-14 ]. F o r example the one matrix model, at the kth multicritical point, is described by a 2 k - 2 order differential equation, whose solution apparently has k - 1 free parameters. Perhaps the poles in the Borel transform are the analog, here, o f these potential ambiguities. Nonetheless the n o n p e r t u r b a t i v e solution we constructed somehow resolves all these ambiguities at once and gives a unique result. This would suggest that the parameters for D < 1 are not really free either. ( 3 ) Multicritical models. We can repeat the nonperturbative a p p r o a c h for the case where U(x) has a kth order m a x i m u m at Xc. The result is now that the energy is given by a series, which for k > 2, behaves as
Egs=Z22 gst
1+ }~ n
(gstA
)
.
(4.3)
In this case we find that in the c o n t i n u u m limit A-,O,
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the e f f e c t i v e c o u p l i n g v a n i s h e s and the t h e o r y bec o m e s trivial. It is n o t at all clear w h a t is the physical i d e n t i f i c a t i o n o f these m o d e l s .
Acknowledgement We w o u l d like to t h a n k A. M i g d a l a n d A. P o l y a k o v for discussions.
Note added As this p a p e r was b e i n g t y p e d we r e c e i v e d a p a p e r f r o m Brezin, K a z a k o v a n d Z a m o l o d c h i k o v [ 16 ], in which the o n e d i m e n s i o n a l string t h e o r y was s o l v e d along the s a m e lines. T h e s e a u t h o r s do not, h o w e v e r , discuss a possible strong c o u p l i n g phase.
References [1 ] V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819.
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[2] [3] [4] [5] [6]
J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. F. David, Mod. Phys. Lett. A 3 (1988) 1651. L. Brink and H. Nielsen, Phys. Lett. B 45 (1973) 332. V, Kazakov and A. Migdal, Nucl. Phys. B 311 (1989) 171. E. Brezin, C. Itzykson, G. Parisi and J. Zuber, Commun. Math. Phys. 59 (1978) 35. [ 7 ] I. Kostov, Phys. Lett. B 215 ( 1988 ) 499. [8] V. Kazakov, Phys. Len. B 150 (1985) 282; F. David, Nucl. Phys. B 257 (1985) 45; V. Kazakov, I. Kostov and A. Migdal, Phys. Len. B 157 (1985) 295. [9] D. Gross and N. MiljkoviC unpublished; S. Das, A Dhar, A. Sengupta and S. Wadia, Tata Institute preprint TIFR/TH/89-71. [ 10] I. Gelfand and L. Dikii, Usp. Mat. Nauk 30 (1975) 5. [11 ] D,J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [12] M. Douglas and S. Shenker, Rutgers preprint RU-89-34 (October 1989 ). [ 13 ] E. Br6zin and V. Kazakov, Ecole Normale preprint (October 1989). [ 14 ] D.J. Gross and A.A. Migdal, Princeton preprint PUPT- 1159 (December 1989); E. Br6zin, M. Douglas, V. Kazakov and S. Shenker, Rutgers preprint RU-89-47 (December 1989); C. Crnkovid, P. Ginsparg and O. Moore, Yale preprint YCTP-P20-89 (December 1989). [ 15 ] D. Gross and V. Periwal, Phys. Rev. Len. 60 ( 1988 ) 2105. [16] E. Brdzin, V. Kazakov and A.B. Zamolodchikov, Ecole Normale preprint LPS-ENS-89-182.
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A N O N P E R T U R B A T I V E S O L U T I O N O F D = 1 S T R I N G T H E O R Y ¢~ David J. GROSS and Nikola MILJKOVI(~ Joseph ttenry Laboratories, Princeton University, Princeton, NJ 08544, USA
Received 15 January 1990
We derive a nonperturbative solution of D= 1 string theory, based on a double scaling limit of the one dimensional random matrix model. We derive an exact expression for the partition fimction in terms of the string couplingconstant. The weak coupling expansion suffers from infrared divergences, which we attribute to massless tadpoles. The continuum limit seems to be well defined, however, in a strong coupling expansion. This could correspond to a different stable nonperturbative vacuum.
1. Introduction Recently great progress has been made in the nonperturbative analysis of toy models of string theory, defined in terms of matrix models as discrete sums of r a n d o m surfaces with matter attached. However, all of this has so far only been accomplished for theories whose matter content has central charge less than one. From the point of view of string theory this corresponds to dimensions of space-time less than one. We are interested in extending these results to higher dimensions for m a n y reasons. These include the possible application of string theory in four d i m e n s i o n s to QCD, as well as the insight we hope to gain as to the n o n p e r t u r b a t i v e structure of critical string theory. c = 1 seems to represent a kind of barrier, beyond which many interesting complications arise. First, the straightforward c o n t i n u u m analysis ofconformal field theories coupled to two dimensional gravity breaks down at c = 1. The treatment that works for c < 1 produces, in this region, unphysical singularities [ 1-3 ]. This might be an indication, as m a n y people have speculated, of a possible phase transition to a new, strong coupling phase in which the string degenerates to a branched polymer or is in a crumpled phase. Alternatively these problems might be related to the fact that at c = 1 one expects to find a massless particle that could give rise to infrared singularities. This is the well known tachyon of the d-dimensional bo¢~ Research supported in part by NSF grant PHY80-19754.
sonic string theory, whose mass precisely vanishes for d = 1, according to the famous evaluation [4] of the ground state of a d-dimensional string, m 2 = ~ ( 1 - d ) .1 c = 1 also represents a borderline case for the applicability of the matrix model approach. Standard large N techniques break down for dimensions greater than one. This is not too surprising since the models with c < 1 are truly models with a finite n u m b e r of degrees of freedom but this n u m b e r diverges as c-~ 1. One might expect that the matrix representation of string theory for c > 1 would be as difficult to solve as large N QCD. Fortunately the case of c = 1 can still be treated by large N techniques and, as we shall see, solved exactly. The case of c = 1 matter, coupled to two dimensional q u a n t u m gravity, was solved on the sphere by Kazakov and Migdal [ 5 ], who made use of the beautiful solution of Br6zin, Itzykson, Parisi and Zuber [6] of the large N one dimensional matrix model. They discovered a scaling form for the free energy albeit with mysterious logarithms. Kostov studied this model further and calculated the two-point function on the sphere, showing the existence of a massless mode in the c o n t i n u u m limit [ 7 ]. We shall show in this paper that the appropriate
"~ The factor d - 1, which is simply the number of degrees of freedom of the string, differs from the number d-2, of transverse degrees of freedom, because the longitudinal mode does not decouple in noncritical string theory.
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double scaling limit of this model, where both N ~ o o and the bare cosmological constant is carefully adjusted, yields an expansion of the free energy which can be evaluated to all orders in the string coupling constant. The mysterious logarithms now appear in full force. We argue that they are simply manifestations of the infrared divergences that appear in higher orders due to the massless mode. Each successive handle that is added to the surface can attach through a massless propagator, which yields a logarithmically divergent factor. This means that the genus expansion, which is an expansion in powers of the string coupling, is very badly behaved. Not only is the series divergent (its coefficients growing like (2n)!), but the nth term contains a factor ln(ov) n. This means that no matter how small we make g s2t r i n g , w e a r e driven in the continuum limit, to large coupling and cannot trust the asymptotic perturbative expansion. In effect the number of handles becomes infinite and the picture in terms of well behaved surfaces breaks down. Fortunately the full sum has a natural nonperturbative definition which we can use to study the theory for large coupling. Here we find a convergent expansion, which leads in the continuum limit to a simple and finite result. We interpret this strong coupling result as equivalent to a shift of the vacuum o f the theory, which was unstable due to massless tadpoles.
2. The matrix representation The representation of the sum over random surfaces, or string theory, in terms of matrix models is well known by now [ 8 ]. Here we are interested in the coupling of a one dimensional continuous matter degree of freedom to the surface. This can be described by a one dimensional model of N by N hermitian matrices, q~(t), Zs(fl)
=
f
DNZ~(t)
5 April 1990
the partition function. Each given order in 1/N 2 corresponds to a sum over the surfaces of definite genus, generated by the Feynman graphs of that genus. It is easy to show that each graph is weighted by a factor (N/fl) Area. In addition the graph contains an integral over a product of propagators, exp(-mlti-t,I ), connecting adjacent ~ ( t ) ' s on the graph. It is believed that in the continuum limit, i.e. take N~ fl~g--+gcritical SO that the perturbation series diverges and is dominated by infinite area terms, this coincides with the continuum definition o f a gaussian variable coupled to two dimensional gravity ~2. From the point of view of string theory this corresponds to a noncritical string in one dimension, i.e. the theory in which the two dimensional world sheet is mapped onto one dimensional time t. As was nicely shown in the classic paper [6] the problem of finding the ground state energy In Z(fl) can be greatly simplified. Since the ground state is U (N) invariant one can essentially integrate out the angular variables leaving just the eigenvalues of 4 , which become fermionic. The problem is then equivalent to finding the ground state o f N fermions living in the potential U(x), i.e. of the hamiltonian ~__
1 d2 2fl 2 d x 2
(2.2)
"t- U ' ( X ) .
If we denote the energy levels of such a hamiltonian by ek, than the ground state energy of the model is Egs/fl=e~+e2+e3+...+ex, where we fill the energy levels up to the Fermi level, #F= eu. We have normalized so that f l ~ N plays the role of the inverse of Planck's constant. It is useful to introduce the density of states, 1
p(e)= Z a(e.-e), in terms of which the equations that determine the ground state energy are
g= ~ =
p(e) de , Eg~=B 2 p(e)ede. 0
0
(2.3)
× e x p ( - f l f dt Tr[½~2(t)+ U(Clg(t) ) ]) . (2.1)
The sign that we are in the continuum limit is that the energy, Egs, is a singular function of the cosmo-
The connection with D = 1 string theory is established by considering the perturbative expansion of
,2 The difference between the gaussian propagator and the exponential one should not matter in the continuum limit.
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logical constant. This we achieve by adjusting the potential so that its m a x i m u m , which we define to be ~ a ~ =/t~i,~,~, approaches the Fermi level from above. At this point the levels will begin to spill over the well and A=g~,~a~-g will be a singular function o f ~t-=/tc~iti~,~- ~tv ~3. A is the r e n o r m a l i z e d cosmological constant, in the sense that the surfaces are weighted by g A r e a = ( 1 --Z] )Area ~ e--AArea. In principle we should solve the first equation for /~ in terms o f A and then use the second to solve for Egs. It is often more convenient to work with the derivatives o f these 0g - -p(~), 0~t 0Egs 0// --
0g fl2~FP('I'IF)=flZ/tv 0-~"
( 2.4 )
Kazakov and Migdal solved this p r o b l e m in the leading 1 / N a p p r o x i m a t i o n , i.e. for the sphere, where one can use the W K B a p p r o x i m a t i o n . In this case the eigenvalues can be taken to be continuous and the density o f states is given by (_+x~ are the turning points of the classical trajectory,/t~iti~ = U( _+Xc) ) + Xc
p(e)=~
1 "j
dx
- x/~e-U(x)]"
(2.5)
The singularity o f this integral at e = ¢tV, which occurs when ~t~0, arises from the region near the turning points where U(x) can be a p p r o x i m a t e d by a quadratic expression, U ( x ) ~ / J c - - 2 ( Y c - - X ) 2 ~4. It then follows that
1 P(l~v) = - ~ I n / x + regular t e r m s .
(2.6)
Therefore 0 g / 0 / l ~ l n / ~ , or A = 1 - g ~ - ~ t l n / ~ . Since OEgs/Og~-fl2/J~fl2d/lnd we finally derive (as N-~ oo ) that Eg~
~/~2
N2zj2
lnA
lnA
lnd 2:R./~2ZI 2 "
(2.9)
We shall see that this is the case. The above calculation is totally universal, i,e. ind e p e n d e n t of the form o f the potential U(q~), as long as it has a quadratic m a x i m u m somewhere. It is easy to generalize this calculation to potentials that do behave differently a r o u n d their m a x i m u m [9]. I f the first k - 1 derivatives of U(x) vanish at Xc, then Og/O~,U -1/2+1/k, which leads to 7 s t r - - - - ( k - 2 ) /
(k+2). 3. The nonperturbative solution
(2.7)
This is in agreement with the expectation from the c o n t i n u u m theory that Egs ~ N 2 A 2+~'s'~ ,
and the K P Z calculation [ 1 ] that yielded ~str-~-0. The extra logarithm that appears is presumably due to the existence o f massless modes o f excitation in the theory. In this theory we have one s p a c e - t i m e coordinate and can form a continuous set of local operators labeled by a m o m e n t u m p. Indeed, Kostov evaluated the two-point function in the leading 1IN approxim a t i o n and found that the theory contains a sequence o f evenly spaced excitations, with masses m n = n¢o, which vanish in the c o n t i n u u m limit where (o ~ 7t/ln A. According to the principles o f string theory these massless particles cannot cause any real problem in the spherical limit since conformal invariance implies that they cannot be emitted as tadpoles. One way of arguing this is to note that when summing over all spheres one is automatically summing over genus zero tadpoles attached to the sphere, so that all infrared divergences associated with v a c u u m instability are automatically dealt with. But, as in critical string theory, we expect real infrared divergences to a p p e a r in the next order, where the massless tadpoles can couple to a torus. This would lead us to expect that the genus k contribution would behave as g~:~ ( l n A ) k, where the string coupling constant, gs~, is defined from the leading term as g2 _
-- Xc
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(2.8)
~3 We will always normalize so that gcritieal=I. ~4 The choice of U" (x~) = - 4 does not affect the critical behavior.
We now turn to the n o n p e r t u r b a t i v e solution o f the model. Again, we let N ~ but also take A-~0 fast enough so that the higher orders o f the genus expansion are not suppressed. The expectation [2,3] is that the free energy scales, in kth order, as ( l / N ) 2k X ( 1/A) 2 y,)k, so that if we keep gst fixed as N-~oo, the expansion will be an expansion in powers Ofgst. We 219
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shall see this is correct, up to logarithms of N which arise due to infrared divergences. First, let us note that the calculation of the singularity in the density of states is only sensitive, as it was on the sphere, to the behavior of the potential near its m a x i m u m . This can be seen by examining the expression for the density of states
p(ltv) =
;
1 Im T r / 7 - / ~ v - i ~ '
near the m a x i m u m of the potential. Expand the denominator about y = x ~ - x ~ O , and note that h -/zF ~ - ( 1/2,82 )02 + / z - 2 y 2 + O ( y 3 ) . In our scaling limit we take ,8~oo and/~ ~ 1/,8. Thus i f y 2 ~ 1/t8, then the denominator will scale as 1/,8, and the cubic and higher powers o f y will be suppressed. The coefficients R~ are local functions of U and its derivatives by powers of 1/,8. Even though the scaling will be spoiled by logarithms this does not affect this power suppression of all but the quadratic part of the potential! The hamiltonian that survives this scaling limit is simply an inverted harmonic oscillator. The calculation ofp(/zv) is then straightforward. We use the following trick. Consider the density of states of the normal harmonic oscillator of frequency o~,
5 April 1990
The density of states can be calculated by expressing the trace over states in the coordinate basis, 1 Ix'>. p ( e ) = 2~fllmnf dx' ( x ' l _82+2f12[U(x)_e]
(3.3) This integral is imaginary only for Ixl (x,, and is singular near the turning points. The diagonal matrix element of the resolvent can be expanded in inverse half integer powers of e - U(x' ) following Gelfand and Dikii [ 10 ],
), 7t"
l=o
Rl{2fl 2 [ U(x) - U(x' ) 11 ,= ~'
×
{2[e_U(x,)]}l+l/2
The coefficients Rt are local functions of U and its derivatives and are given in ref. [10]. If one evaluates this expansion for x' - x c it follows immediately that the only terms that survive in the scaling limit come from the quadratic expansion of U(x) in the vicinity of its maximum. The first few terms for p(/tF) are easily calculable and yield
1 1
~( 1 Im ~
p(E)
P(~v)=
1
~ (n+ ½) o ) h - E - i e '
and continue o) to imaginary frequency. This yields, in our case, the expression
1 Re~
P(/@)= ~
l
2 n + 1 +i,8/~ "
(3.1)
This expression is divergent; however, the divergent part is/t independent and will not affect the critical behavior. We can fix it by demanding agreement with (2.6) in the limit o f , 8 " ~ . This yields p(/tv) = ~-n Re[C(1, ½(1 + i , 8 / O ) - o o ]
1
- 2 n [ln(½fl) - R e ~/(½ (1 + i,8/t) )]
,
7
1
- - I n # + 6 f12p2 + 60 fl4~4
311
+ 126
)
(3.5)
fl6]~6 -{- ....
Let us now use the exact expression for p(;Zv) to derive the complete asymptotic expansion 1
p(l~v) = 2~n Re[C(I, ½(1 +ifl~) ) - 2 _--
1 ln/l+~im 2n
~
]
1
,,=0 1 - i ( 2 n +
1 )/,8#
- 2nl [-lng (3.2)
where C(z, q) is the zeta function, C ( z , q ) = Z~=o 1 / ( n + q ) - - a n d ~'(z)=I" (z) / l ' ( z ) . Since this derivation is rather formal we present a check of its validity by a more rigorous calculation of the asymptotic expansion o f p ( e ) in powers of 1/,8. 220
(3.4)
•
+2 ~=,
?k
\fl,uJ (22k ' - l ) C ( 1 - 2 k )
], (3.6)
where the C-functions arise from the (divergent) series En ( 2 n + 1 )*. These C-functions are given in terms of Bernoulli numbers B2m, so that
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+ ~_, (22~_,_1)IB2.,,I .,=, rn
1 )"
(flff)2m
(3.7)
It is easy to verify now that the first few terms are in complete agreement with (3.5). The Bernoulli numbers grow rapidly as IB2ml ~ (2rn)!, leading to a highly divergent expansion for P(/*F)- It is precisely of the same form that was found in the solution of the string theories in D < 1 [ 11-14 ]. N o w let us return to the evaluation of the free energy in terms of the cosmological constant 3. Unfortunately the best we can do is derive a parametric representation of 3 and Eg~ as functions of/2. It follows from (2.4) that Cq-~=P(/2v) ,
=fl2/*+regu lar terms ,
(3.8)
where p(/2V), is given by (3.2) as a function of/*. These equations can be solved perturbatively to express Eg~ as a function of 3. First, integrate (3.2) to obtain
5 April 1990
which behaves in the continuum limit (as A-~0) as G (p = 0) ~ - In A [ 7 ]. This means that the weak coupling expansion is very sick. Not only does it diverge badly but it cannot be used as an asymptotic expansion, since no matter how small we take gst the effective coupling is gst In A, which blows up in the continuum limit. This can be traced back to the fact that we evaluated P(/*F) in an asymptotic expansion in powers of 1/fl/2. However, we found that in the scaling limit
ln3 ~gs,~f-ln3
1
fl/2
(3.12)
f13
and thus we can never be made small! This situation is similar to the case in ordinary field theory where we are driven away, in the continuum limit, from an unstable fixed point at gs,=0. We therefore must sum the series exactly, to all orders in gs,, before taking the continuum limit. Fortunately we possess an expression for the exact form of P(/2v) which we can use to explore the theory for large fl/2. Previously we expanded 03 1 /20 =P(/2F) = ~ Re [In ½fl-- ~,( ½( 1 +ifl/2) ) ] 4-in a power series in 1/fl/2. Now let us expand
-
~ m
=
(22m-~--1) 1
IB2mI m(2m-l)
l
)
(flff)2m
"
(3.9)
Then invert this expression to write ~*=-2re3~ ln/2+f(fl/2)/flln/2 as a function o f d . This produces a series of the form 2)zd I I + /*=
-
~
~ ,.=,-
c
(-lnA'~n(-lnd)r~ 1 ....
k
where we neglect double logarithms. Now we integrate (3.8) to derive
l(
Eg,,= -~- 1+ gs~
~=1 m=l
c,,,mgs2"(--lnd) ''
(3.10)
)
P(/*F) = -
2~ In/~
1
+-R ~ 'eo 2 n~_ +l~z
1
1
l+ifl/2/(2n+l)
(3.13)
in a power series in fl/*. This is straightforward and yields 0A 1 0~ =P(/2F) = -- ~ In # 1
+ -~k~=l (--1)k(1--2--(ak+l))((2k+l)(fl,U) 2k" .
(3.11)
This weak coupling expansion exhibits the logarithmic infinities that we expected, i.e. the powers of ( - I n J ) ~ that occur in nth order. We associate these with infrared divergences that occur, say, when n genus one tadpoles couple to the sphere. Each tadpole connects through a zero m o m e n t u m propagator,
(3.14) This series is much better behaved. First, the (functions that appear are evaluated at positive values and do not increase at all (((z)_,~oo-, 1 ). In fact, this series is absolutely convergent for I fl/2[ < 1. Second, we expect that the expansion parameter, fl/2, will now be small in the continuum limit. We now integrate 221
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this expression for A and invert it to w r i t e / t as an expansion in A,
~=-i~
1-WS+ ....
This yields ( ~ f l 2 L2 j n
2hA (
d .... \ ~ n ~ j ?1~ | t;'l~tl+
gsl
1-~
n=l
re=n+1
I
Onm 2nlnmA ' gst
m
/
.
Op
1
i
t / fla
(4.1)
0~ ( / I v ) = 2 ~ I m
(3.16)
In fact if we were to directly Borel transform the asymptotic expansion of (Op/Ol~) (/~F) we would obtain the same integral, but with the contour o f integration rotated to lie along the imaginary axis, i.e.
4. Discussion There are a few points regarding our solution that are worthy o f comment. ( 1 ) W h a t is the nature of the strong coupling solution? N o r m a l l y when one encounters infrared divergences due to massless tadpoles one must shift the v a c u u m to obtain a finite theory with a stable ground state. Presumably that is what is occurring in D = 1 string theory. We have found a consistent nonperturbative c o n t i n u u m solution with no infrared divergences. It, however, does not have a weak coupling expansion. This could mean that we have, by summing over surfaces with an infinite n u m b e r o f handles, shifted to a stable vacuum. In the new v a c u u m the free energy no longer has a natural expansion in terms o f surfaces o f finite genus. Perhaps one cannot even describe it in terms o f two-dimensional surfaces at all, just as in o r d i n a r y field theory o f pointlike particles a nontrivial, n o n p e r t u r b a t i v e v a c u u m cannot be described in terms of the original F e y n m a n graphs. A complete u n d e r s t a n d i n g o f this phase requires evaluation of the correlation functions o f physical operators. This should be possible to do exactly and work in this direction is in progress. (2) S u m m i n g divergent series. It is interesting to ask how is it that we have m a n a g e d to sum the origi-
dte
-i'sinh(t/flll ) .
(3.15)
This series is now extremely well behaved. In fact it is reminiscent of an asymptotically free theory, since now when we go to the c o n t i n u u m limit, A--,0, only the first term survives.
222
nal, highly divergent and non-Borel s u m m a b l e perturbation expansion. The weak coupling expansion for, say, (8p/Oll)(ltv), is o f the typical non-Borel s u m m a b l e series, ~g2"(2n)!, that one gets in string theory [ 15 ]. The exact result that we derived can be written as an integral which looks like a Borel transform,
3)-m )
(ln
Then we integrate (3.8) to obtain
Egs= ~ -
5 April 1990
o
Op (a~)=
Ol~
1 i
2~
t/fl/2
(4.2)
dte-'sin(t/fll~)"
o
This integral does not exist since the Borel transform has an infinite n u m b e r of poles on the real axis at t= nnflCt, although it has an identical asymptotic expansion. To specify the integral we must a d d a prescription for integrating about each pole, a potential source o f an infinite n u m b e r of arbitrary parameters. In the recently solved string theories for D < 1 one found differential equations that could be used to recover perturbation theory and provide a nonperturbative definition o f the theory [ 11-14 ]. F o r example the one matrix model, at the kth multicritical point, is described by a 2 k - 2 order differential equation, whose solution apparently has k - 1 free parameters. Perhaps the poles in the Borel transform are the analog, here, o f these potential ambiguities. Nonetheless the n o n p e r t u r b a t i v e solution we constructed somehow resolves all these ambiguities at once and gives a unique result. This would suggest that the parameters for D < 1 are not really free either. ( 3 ) Multicritical models. We can repeat the nonperturbative a p p r o a c h for the case where U(x) has a kth order m a x i m u m at Xc. The result is now that the energy is given by a series, which for k > 2, behaves as
Egs=Z22 gst
1+ }~ n
(gstA
)
.
(4.3)
In this case we find that in the c o n t i n u u m limit A-,O,
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the e f f e c t i v e c o u p l i n g v a n i s h e s and the t h e o r y bec o m e s trivial. It is n o t at all clear w h a t is the physical i d e n t i f i c a t i o n o f these m o d e l s .
Acknowledgement We w o u l d like to t h a n k A. M i g d a l a n d A. P o l y a k o v for discussions.
Note added As this p a p e r was b e i n g t y p e d we r e c e i v e d a p a p e r f r o m Brezin, K a z a k o v a n d Z a m o l o d c h i k o v [ 16 ], in which the o n e d i m e n s i o n a l string t h e o r y was s o l v e d along the s a m e lines. T h e s e a u t h o r s do not, h o w e v e r , discuss a possible strong c o u p l i n g phase.
References [1 ] V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod. Phys. Lett. A 3 (1988) 819.
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[2] [3] [4] [5] [6]
J. Distler and H. Kawai, Nucl. Phys. B 321 (1989) 509. F. David, Mod. Phys. Lett. A 3 (1988) 1651. L. Brink and H. Nielsen, Phys. Lett. B 45 (1973) 332. V, Kazakov and A. Migdal, Nucl. Phys. B 311 (1989) 171. E. Brezin, C. Itzykson, G. Parisi and J. Zuber, Commun. Math. Phys. 59 (1978) 35. [ 7 ] I. Kostov, Phys. Lett. B 215 ( 1988 ) 499. [8] V. Kazakov, Phys. Len. B 150 (1985) 282; F. David, Nucl. Phys. B 257 (1985) 45; V. Kazakov, I. Kostov and A. Migdal, Phys. Len. B 157 (1985) 295. [9] D. Gross and N. MiljkoviC unpublished; S. Das, A Dhar, A. Sengupta and S. Wadia, Tata Institute preprint TIFR/TH/89-71. [ 10] I. Gelfand and L. Dikii, Usp. Mat. Nauk 30 (1975) 5. [11 ] D,J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127. [12] M. Douglas and S. Shenker, Rutgers preprint RU-89-34 (October 1989 ). [ 13 ] E. Br6zin and V. Kazakov, Ecole Normale preprint (October 1989). [ 14 ] D.J. Gross and A.A. Migdal, Princeton preprint PUPT- 1159 (December 1989); E. Br6zin, M. Douglas, V. Kazakov and S. Shenker, Rutgers preprint RU-89-47 (December 1989); C. Crnkovid, P. Ginsparg and O. Moore, Yale preprint YCTP-P20-89 (December 1989). [ 15 ] D. Gross and V. Periwal, Phys. Rev. Len. 60 ( 1988 ) 2105. [16] E. Brdzin, V. Kazakov and A.B. Zamolodchikov, Ecole Normale preprint LPS-ENS-89-182.
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