Nuclear Physics B335 (1990) 635—654 North-Holland
STRINGS IN LESS THAN ONE DIMENSION
Michael R. DOUGLAS and Stephen H. SHENKER
Department of Physics and Astronomy, Rutgers University, Piscalaway, NJ 08854, USA Received 24 October 1989
Starting from the random triangulation definition of two-dimensional euclidean quantum gravity, we define the continuum limit and compute the partition function for closed surfaces of any genus. We discuss the appropriate way to define continuum string perturbation theory in these systems and show that the coefficients (as well as the critical exponents) are universal. The universality classes are just the multicritical points described by Kazakov. We show how the exact non-perturbative string theory is described by a non-linear ordinary differential equation whose properties we study. The behavior of the simplest theory, c 0 pure gravity, is governed by the Painlevé transcendent of the first kind.
1. Introduction
Despite much work, the theory of strings in non-critical dimensions is still poorly understood. Besides the original motivation for these strings as alternate descrip-
tions of the three-dimensional Ising model and of the large-N Yang—Mills theory, we might also hope they will shed more light on strings in the critical dimensions, in two ways. One is that the Liouville degree of freedom in the non-critical string [1] is not strictly speaking a conformal field theory; if we can understand it, we may be able to describe new solutions of the critical string which also are not conformal field theories
[2—41.
A more important point is that these string theories are, at least for dimensions less than 1, much simpler than the critical string. They can be non-perturbatively defined and — as we shall see below — solved exactly, yet they share many of the formal properties of the critical string. A useful analogy is the relation in field theory between QCD and interacting scalar field theory in zero or one dimension.
Although the degrees of freedom and the symmetries are completely different in the two cases, the theories can be defined using the same functional integral techniques, and there are interesting properties of the theories (such as the relation between the perturbation series and the exact amplitudes) for which the analysis is very similar.
We can therefore hope that the exact solution of a lower-dimensional string will provide ideas which could be used to make an exact definition of critical string theory and give some information about its non-perturbative behavior.
636 M. R. Douglas, S. II. Shenker/ Strings in less than one dimension
2. Continuum world-sheets
The continuum theory of two-dimensional gravity has been notoriously difficult to understand, but recent progress has been made for the case of gravity coupled to conformal field theory with c s~1, by using a new light-cone gauge [5,6] and in conformal [1] gauge [7, 8]. Even in these dimensions, the theory has not been solved completely, but we cite some very useful results which agree between the two formalisms.
The main result is a scaling relation which can be derived for any amplitude on a surface of fixed area and gives the power-like area dependence,
Z~eAA+~_2)/2, (2.1)
where y = ~-(D— 1 — ~Iii~—2~) is the string susceptibility,
x
is the Euler characteristic, and ~.tis the renormalized cosmological constant. A useful heuristic derivation of this formula has been given in refs. [7, 8] where it is related to the shift in the R~2~4term of the Liouville action,SL= 28~ Jd2
(gaa~ab~
+ +~(e~ —1)). (2.2)An important feature of this result is that for D> 1, the exponents become complex, signalling the breakdown of this approach.
Another important result comes from considering the spectrum and Virasoro representation theory. We will state it for the covariant formalism, where it has not to our knowledge been proven* —an analogous statement in light-cone gauge is discussed in refs. [9,10]. We can define a stress-tensor for the Liouville+ matter system by varying the reference metric ~ it gives two commuting Virasoro algebras.
We define reparameterization-invariant states as those satisfying L~)= L~)= 0, n >0, and physical states as those which satisfy this and L0= = 1. Then given unitary matter, the reparameterization-invariant state space has no negative norm states, and the only physical states (of positive norm) are of the form
Liouville primary) 0 matter primary), (2.3) where the conformal dimensions add to (1, 1). This is a non-trivial statement because there are Liouville primaries with arbitrarily negative dimension and is analogous to the no-ghost theorem in critical strings. In particular, for a unitary model with D < 1, there are finitely many physical states. This is another indication of the simplicity of these theories.
*M.R.D. has checked it at low levels.
MR. Douglas, S.H. Shenker / Strings in less than one dimension 637
Finally, we mention that in the Liouville system there seems to be no sensible perturbation theory around ~u= 0. On the other hand, if p.> 0, then all but a few low-genus amplitudes have a Liouville potential with a single minimum, and there
seems to be no reason the theory should not be well defined (for D < 1).
3. Discretized world-sheets
A second approach to non-critical string theory has been to discretize the world-sheet [11—14].There are many ways to do this, but the one we will use here, proposed in refs. [11—13],is especially simple. It is a variation of Regge calculus in which we sum over triangulations of a R.iemann surface, assigning all links the same length. This is a reparameterization-invariant discretization of the sum over geome- tries — the geometrical degree of freedom is the number of triangles which meet at each vertex, which is a discrete form of the gaussian curvature. One great advantage of this model is that the dual graphs to triangulations are ~ field theory Feynman diagrams, so pure two-dimensional gravity amplitudes are related to the perturba- tion theory for a zero-dimensional field theory [16, 17]. To distinguish amplitudes for world-sheets of different topology, we use ‘tHooft’s topological description of large-N perturbation theory [18]. Let 4 be an NXN hermitian matrix, and consider the path integral
Z=eF=fd~exp[_Tre~BV(~)], (3.1)
~ ~ (3.2)
k>3
The contractions of the matrix indices define faces on each Feynman diagram, and with the couplings scaling as above, each graph comes with a factor
= NX. (3.3)
So the order N’~term in F is the number of connected diagrams of Euler character
x
(with symmetry factors divided out). We have generalized slightly by allowing many couplings gk; in terms of the original graphs, we are allowing not just triangles but arbitrary polygons, each with specified weight. /LB is the bare cosmo- logical constant — we are assigning every polygon the same area (this is not essential). For simplicity in the following we will restrict ourselves to even potentials V; the same techniques apply for odd potentials and all our conclusions hold for them as well.Note that to get a graph counting problem with positive weights, all of the couplings gk must be negative. The integral must then be defined by analytic continuation, which restricts our ability to use this functional integral representation
638 A’!.R. Douglas, S. H. Shenker/ Strings in less than one dEmension
to directly study the behavior of Z. On the other hand, if one is willing to sacrifice manifest positivity, there are continuum string theory limits in which the potential is bounded below.
We note in passing [19] that one way to couple matter to the theory is to make the matrix t~a function on a set S, and calculate
f fT
d~iexP{_[
i,JES~ c11Tr~+ i~S~
TrV(~)}.
(3.4)This will sum over graphs where each vertex is labelled with a member of S, and there is an interaction between adjacent faces.
Returning to the case of pure gravity, the saddle-point evaluation of the one-matrix integral was explained in ref. [20]; the results for an arbitrary potential are given explicitly in ref. [22]. The first step is to change variables 4 = U ‘aU, with U unitary and a diagonal satisfying a ~ a~for i<j. Neither the original integrand nor the jacobian depend on U, so the U integration gives a factor which depends only on N, and we drop this factor. The path integral is now
ff1
dajfl(a1_aj)2exp[_~et~V(ai)]. (3.5)If we exponentiate the jacobian and rescale a1—~‘/~ict,,every term in the exponent is of order N2, so the order N2 term in F is the value of the action at the saddle point.
The simplest way to find this term is to explicitly describe the saddle point by a spectral density
u(a) = ~~(a—a~). (3.6)
In the large-N limit this becomes a smooth function of a, which satisfies a simple variational equation. We can then evaluate the action at the saddle point, and the leading 0(N) term in the expectation value of (Tr41) as well.
Rather than describe this in detail we will instead outline below an alternative method due to Bessis, Itzykson and Zuber [21,22] which can be used to get subleading terms in 1/N as well. Let us first state some properties of the results. At each order in 1/N, F is analytic near zero coupling, unlike the finite N result. If we consider the dependence on a single coupling, say p.8, we find the singularity nearest
Is!. R. Douglas, S. H. Shenker / Strings in less than one dimension 639
the origin has the same location p~at every order in 1/N; letting
F(~.L~g~)=N21~+F1+N2F2+ , (3.7)
2
= + c~(~—
~
+c2(!1B —~
+z0(
~B —~)
+ ... , (3.8)Fl=ZIloS(~sB—~tC)+ , (3.9)
Fg= Zg(!~B— ~t~) X(l±l/2m) + ... , (3.10)
~ depends on V. as does the integer m. In general m is 2, but by tuning k parameters in V one can reach a point with m= 2+ k.
For m= 2, the exponents of the leading singular terms are exactly the Liouville critical exponents for D = 0 calculated in refs. [6—8]and the agreement between the two formalisms for low-genus surfaces has been noted for pure gravity in ref. [13].
The multicritical points were found by Kazakov [23]. He has made the very interesting conjecture that an m > 2 model is equivalent to gravity coupled to a minimal model with c= 1— 6/m(m + 1).
So far we have been talking about lattice gravity. Now we take a continuum limit —introduce an explicit constant a with dimensions of length to play the role of a cut-off, introduce appropriate powers of a into the results above, and take the dominant terms in the limit a—* 0. We identify
~R= (p.~—p.~)/a2, (3.11)
where p.~,the renormalized cosmological constant, is held fixed by tuning p.~to p.~
as a—~0. The dominant terms have the correct continuum scaling except for J~,the genus-U contribution. By transforming from fixed cosmological constant to fixed area, one can see that terms analytic in~.tR do not contribute to the amplitude for a surface of large area. We therefore interpret these terms as additional remnants of the cut-off, and consider the continuum limit to be the first non-analytic term insiR, which also satisfies the correct scaling relation.
We point out that the agreement between the continuum scaling prediction and the lattice result has a highly non-trivial consequence. The lattice result comes from the sum over all metrics, including those at the boundary of moduli space, the continuum scaling is an analysis that ignores the boundary. A divergence in the lattice integrated amplitudes at the boundary would show up as a more singular lattice spacing dependence and hence a deviation from continuum scaling. All orders agreement (demonstrated below) implies that the integrated amplitudes are finite at all orders.
This scaling also necessitates a change in our definition of string coupling constant from the naive one of 1/N. The constant icX that multiplies the genus-g
640 M. R. Douglas, S. H. Shenker/ Strings in less than one dimension
amplitude is given by
I
(3.12) This diverges in the naive continuum limit, a signal that the string coupling in non-critical string theory has a dimension and must be renormalized. Define the true string coupling A to be A= a~2~~’m)/N,held fixed by letting N— oc as a—~0. The dimensionless coefficient ic now becomes
IC = A/~l~/2m (3.13)
finite in the continuum limit. The existence of a dimensionful string coupling distinguishes non-critical from critical strings. It suggests the impossibility of defining a weakly coupled non-critical string theory at zero cosmological constant.
The rescaling also makes problematic naive interpretations of 1/N as a string coupling in large-N QCD.
Physical quantities only depend on K so a change in the string coupling can be compensated for by a change in a world-sheet coupling, a situation reminiscent of the Fischler—Susskind mechanism [24].
We expect, because the continuum Liouville plus conformal minimal model lagrangian has no other free parameters, that no matter which lattice model (choice of V(4) in a universality class) we use to take the continuum limit, the coefficients Zg should be the same up to an overall factor ,c2~2.We show this by explicit calculation in the lattice model, and determine the coefficients Zg.
Given the solution of the theory on an arbitrary finite-genus surface, one is naturally led to ask whether one can use this information to define an exact theory at finite values of the string coupling. Our techniques for solving the lattice model produce the perturbative amplitudes as coefficients in an asymptotic expansion of a solution of a universal (for each class of model) differential equation. We have as yet no derivation of this equation from an equation of motion or other more familiar physical concept, so we will simply refer to it as the “string equation”. We will solve this equation below for the D = U theory, finding that it has an especially simple structure if we use for the string coupling not ic2, which counts ioop orders, but instead u ic4”5, the cosmological constant measured in the length scale set by A. The strong coupling limit now becomes the u —‘U limit, and we will find that
physical quantities are analytic everywhere in the finite u plane.
One can equally well couple massive matter to gravity. The induced gravitational action becomes more complicated, and little work has been done on this case in the continuum formalism, but the lattice formulation is only slightly more compli- cated — for example the free energy of the Ising model coupled to gravity can be computed exactly as a function of both temperature and magnetic field [25]. We
M.R. Douglas, S. H. Shenker/ Strings in less than one dimension 641
discuss below how such calculations can be done in the scaling limit in the multicritical models.
4. The I/N expansion
We now give a condensed description of the method of Bessis et a!. A detailed exposition of this method is given in ref. [22]. The starting point is to define a series of orthogonal polynomials with respect to the measure dp.(a)= da exp[ —e~V(a)],
hn~nm=fd~(a)Pn(a)Pm(a), P~(a)=a~+~ (4.1),(4.2) These are useful because we can write the jacobian (3.5) above as
fT(a1_aj)2=det2la~_hI =det2~P1_i(a1)~, (4.3)
1<1
from which follows
N-i
z=
flh1 (4.4)i~-O
(times factors independent of the couplings gk and p.).
We can now derive recurrence relations for the P1. and ultimately for the h1.
Define the matrix R by
csP~(a)=~Rn,mPm(a). (4.5)
For V(a) even, we have
Rnm = ~n+1m + ~n~i,mhn/hn_i, (4.6)
so the matrix R determines Z= e’~.Normalizing byFat zero coupling,
F= N log(h0(g)/h0(U)) + ~ (N —k)log(R~~1(g)/R~~1(U)).(4.7)
We can find an equation which determines R by starting with the identity
nhn=fdaexp[_e~~BV(a)1aP,:(a)Pn(a), (4.8)
642 Al’.R. Douglas, S. 1-1. Shenker / Strings in less than one dimension
replacing aP~(a)with RnmPm(a), and integrating by parts. The result is
= e~Bn (4.9)
(no sum over n), a non-linear recurrence for the non-zero entries of R.
So far everything has been exact. We now make the expansion in 1/N. Write
= 1/N, x = k/N and
k R~~1(g)
— / \ . (4.10)
N Rkk...1~O)
Forexample, consider V(a)= ~a2+(g/N)a4. UsingRk,kl(U) = e~k, the recur- rence relation becomes
e~Bx= r((x,g) + 4gt(x, g)[i~(x— E,g) + r((x,g) +~(x +~, g)]. (4.11) A ~2k term inVwill give a sum of rj’ terms. ,~is symmetric about f = U, a general feature. So, expand
r~(x,g) r0(x, g) + 2r1(x, g) + ~4r2(x, g) + . . . (4.12) r0 satisfies a simple equation found by setting c= U in the recursion relation,
(2p)!
W(r0(x))=e~Bx, W(r)~r+ ~ g2~, ~ 1~1r~’.(4.13),(4.14)
p.2
One can see that p r0(1)is the square of the largest eigenvalue of the original matrix 4’. at the saddle point, by calculating the quantity (Tr 41) with this formal- ism. We will see as we continue the calculation that F0 will be analytic in p, and its
singularities will be purely a consequence of the non-analytic dependence of p on the parameters. Higher genus .E~’sare not analytic in p, but their only singularities are of the form W’(p)~.In all cases, the singular behavior in j.t~is governed by the point with W’(r)= U nearest r U. A point where the first m— 1 derivatives
a
~W/ ôrk vanish will be an mth-order critical point, near which we can writeW(r) =e~o(1—$(p~—r)m+...). (4.15) By shifting p.8, we can set p.~to zero. Let us assume this has been done in the following.
The leading term in F is now
1 r0(x)
=
f
dx (1 — x)log . (4.16)0 x
M.R. Douglas, S. H. Shenker / Strings in less than one dimension 643
We can drop the uninteresting leading analytic terms and simplify the calculation by computing derivatives of F. Changing variables to r= r,3(x) we find
dF 1
8r0(x
)
0 =
f
dre~BW’(r)(1_e~W(r))_ (4.17)dp.~ ~ r
pdr
=
—f
_e~(1_e~BW(r))W(r), (4.18)or
d2F0 pdr
2 = —
f
o _(e~BW(r)r — 2e2~BW(r)2), (4.19) plus a boundary term which vanishes since W(p)= e~. The integrand is polyno- mial in r (since Wcz r), so the second derivative is a polynomial in p, proving that 1~does not have terms more singular than 4+1/rn Finally,a
d2F0 1——---~-=—+O— (4.20)
dpdp.B ~
and
ap.8/ap
cr (~— p~)m_l, ~F = 1 4±1/rn + analytic+ ... . (4.21)
0 (2 + 1/m)(1 + l/m)$~~~mp~
We can absorb the non-universal constants into the string coupling; let us do so, defining
1
K2 ~/3l/rnpp.~_(2+l/m) (4.22)
To get higher orders we must expand the recursion relation in 1/N, and use the Euler—Maclaurin formula on the sum which gives F. The recurrence relation at finite N can be expressed as an infinite-order differential equation, by expanding each appearance of ..~(x+ n~)in x derivatives. To express these terms, we define a generalization of the function W(r) as follows. Let W~b (r) be a series whose r~
term is the sum of terms in eq. (4.9) with p appearances of r~(x),an appearance of an appearance of abr~/axb,and so on. For example, we have
1 akw
WOOk times... = . (4.23)
644 M. R. Douglas, S. H. Shenker/ Strings in less than one dimension
The recurrence is now
w(~)
+~
~(~)(~~~)
((b~~) ... = e~x. (4.24)We now replace p.B with a2j.t~and ~ with Aa2±~min preparation for the continuum limit a—~0. Just as in genus U, the non-analytic p.~dependence of the result will come from the limit x—s 1 of the integrand. Therefore, it will be convenient to blow up this region, with the change of variables
1 — a~z, (4.25)
which maps the region xE[0,1] to zE[a2, p.~],for small a. Now c
a/ax
becomes—al/mA
a/az,
so if we can show that i does not have too large a derivative, we will be able to drop all but finitely many derivative terms in the limit. Furthermore, near x= 1 we have(4.26) so r0— a2/m. Assume this is true beyond genus U and write
— ~ a2~~~~lR(z). (4.27)
If R <<a—2/rn we will be able to drop almost all products of R ‘s as well.
Now, in the continuum limit, we are only interested in W(R) for R small, so we can substitute in eq. (4.15) for W. Furthermore, we can use the results of appendix A to get the functions W1. It is shown there that the functions W1(r) are given by linear differential operators L1 acting on W(r). The behavior of W1(R) near R = 0 is then determined by the term of highest order of L1, which is shown to be
flk(1~)= ~kT
arp+k
W(r), (4.28)where the c are constants and
P’~T>flk. (4.29)
If ~— a2/m, then J4’~..,~(R) a(m_~2,’1~.
Consider the case of pure gravity, m = 2. Now z—~a2, R a and ~
a/ax
— a’~2.Evidently only terms with at most two derivatives can contribute. Using the
M. R. Douglas, S. H. Shenker / Strings in less than one dimension 645
appropriate results from appendix A, we find
1 82R 1
a2$R2+ ~a2$A2(p~+aR)~ + ~a3W~ff(pc)A2(~) + =a~z. (4.30)
Taking the a— U limit gives
R2 + ~A2p~R”= /3~’z, (4.31)
an ordinary differential equation for R(z). All dependence on the higher-order terms in W(r) has dropped out. The only remnants of the detailed form of the potential are $ and p~.They can be eliminated by rescaling A and z (z will be set to so this is equivalent to rescaling ic as in eq. (4.22)). We have thus demonstrated universality.
Universality means that we can derive eq. (4.31) from any given potential. A particularly simple derivation follows from applying the scaling argument above to eq. (4.11).
Given R(z), we find F by evaluating the sum (4.7) with the Euler—Maclaurin formula. This gives (eq. (7.6) of ref. [22]),
1 1 h0(g/N)
F=
J
dx(1—x)log~(x)— ~[2log h0(U) _lo~~(0)] (4.32)I
a
1— 12N2 ~—[(1 —x)log4(x)]10 (4.33)
1
a~
1+ 6!N4 .—[(i —x)log~(x)}~0+... , (4.34) where
~(x) = ~(pe+a2/mR(x)). (4.35)
Change variables to z. The first observation is that starting with N2, successive terms in the series are each down by N2(1 — x)2. Since
1—x=a2 (4.36)
1—
this is ~2a4, which in the scaling limit goes to zero. We can therefore drop all terms after the 1/N term. The 1/N term consists of 4(U) with no p. dependence, and terms in h0, a quantity defined in the N= I problem, which therefore is not
646 Al. R. Douglas, S. H. Shenker/ Strings in less than one dimension
singular in p. either. This leaves only the integral. We can simplify it by observing that
p +r0(x) a2/mR_,~
log4=log C +log 1+ (4.37)
x p~+r0(x)
p +r0(x) a2/mR_r0
=log C + +O(a4/m). (4.38)
x PC
So,
F=F0+_f0dz(z_p.~)(a2/mR(z)_r0(z)), (4.39)
Pc P~R
d2F a4
= ($_1/mp.1~rn+ (a2//mR(p.~) _rO(p.R))). (4.40)
p.R Pc
The combination (a2/’mR(z)—r0(z)) — a2~’mfor all z in the scaling limit. One might think that the r0 we subtracted in the first step is exactly compensated by the F0” we added back in the last step, and that R is the second p. derivative of the string free energy. However, eq. (4.26) has m possible solutions,
r0= + $h/’ma21”mz~~~m. (4.41) Clearly we must take the branch of the root which is given by the limit of R(z) as z— oc. This may or may not be the same sign as given in the calculation of F0. In the case of m= 2 it is opposite and R is F” with the sign of the sphere term reversed!
We return to m = 2 and eq. (4.31), rescaling to set p~and ,8 to one. We have
R2+ ~A2R”=z. (4.42)
One way to solve this equation is by writing R as a series in A,
R(z) = ~RgA2~z(l_s~)/2, (4.43)
generating the usual string perturbative expansion. To start, we need to know the sign of R0. The natural sign from the lattice model is in fact negative. This is because the original boundary condition on was re(x = 0) = U. Although this is infinitely far away in the continuum limit, and we have dropped terms which are important at x= 0, we must take the solution which matches smoothly onto the
M.R. Douglas, 5ff. Shenker/ Strings in less than one dimension 647
solution of the full equation there, and this goes as R(z) — —~ So,
R0= —1, Rg+i= ~(25g2 —1) Rg+ ~ ~RkRg+1_k. (4.44), (4.45)
For A2> U, all of the coefficients Rg, g> U are positive. Free energy derivatives are written in terms of R(z = sift). Since the expansion variable is A2z512 we see that this is really an expansion in K2, as expected. We can see the validity of the various
scaling assumptions to all orders.
From the series solution (4.45), the amplitudes as a function of genus grow asymptotically like
Rg__B_2~(2g)!, B= ~/~6/25 . (4.46)
In the lattice definition, this behavior is simply a consequence of the factorial growth of perturbation theory in the matrix model coupling, combined with the fact that higher orders in 1/N first come in at higher orders of the coupling [26,27].
For comparison we note that this growth is faster than the lower bound found for the critical string [28] and is of the same order as the number of top dimension simplices in the triangulation of moduli space given by Witten’s open string field theory [29].
Clearly this expansion gives an asymptotic series, useful only for small ic.We are not restricted to this series though, and can directly analyze the equation at finite coupling. In fact, eq. (4.31) is a rather well-studied equation; up to rescalings, it is the equation satisfied by the first Painlevé transcendent. We will return to the finite coupling analysis later in the paper.
The procedure we used above to get the string equation will work for any m, producing an equation with a number of terms which grows exponentially in m.We now argue that the equations are universal for each m;the higher-order terms in the expansion of W(r) about PC drop out, and the $ and ~ dependence can be absorbed into A. We have z— a2, R — a2,’m and e 3/3x — al/rn. We will therefore get terms
a
~1’~2’ R (4.47)
— ~ k+ ~n~/2 ~ m, (4.48)
k+ ~n~/2>m. (4.49) The terms in which the leading /3(r—p)rn piece does not survive go to zero in the
648 Al. R. Douglas, S. H. Shenker/ Strings in less than one dimension
continuum limit. p~can clearly be absorbed into A2, and the redefinition R —
will put $1/rn where it can be absorbed.
If we are interested in a finite genus g, only terms with up to 2g derivatives will contribute. As an example we compute the genus-i amplitude for general rn. Taking the limit we describe above, we get the subset of the string equation,
R”~+ ~rn(rn — 1)A2p~Rm2R~~+ ~grn(m— 1)(rn—2)A2p~Rm_3(R~)2= $~z+ 0(A4).
(4.SU) Inserting R = (z//3)l~/m +A2R1 and solving for R1 gives
R1= j~(i— 1/rn)p~z2, Z1= ~g(i— 1/rn). (4.51), (4.52)
5. Massive matter
The multicritical models described above are conformally invariant theories coupled to gravity. It is also possible to take the continuum limit of massive matter theories where the mass scale is in some fixed ratio to the scale of the renormalized cosmological constant. To do this one takes a lattice model with potential chosen to put a small relevant perturbation in W, for example
W(r)= 1 +ya2/m(r_ p~)m~+ (r— p~)m+... (5.1) near the critical point where y is the dimensionless ratio of scales.
We note that the asymptotic expansion in l/y determines a set of observables in the infrared conformal field theory plus gravity system that cannot be simply described by the finite number of physical states mentioned earlier. Naively, one would associate them with irrelevant operators in the matter sector, but in at least one situation these are all Virasoro descendants.
6. Analysis of the D= 0 equation
We now study the behavior of the string equation (4.42). Letting
u=(A2/3)2/5z, p= _(A2/3)’/5R, (6.1)
the equation becomes
p”(u)=p2(u)—u. (6.2)
This is the equation for the Painlevé transcendent of the first kind. In evaluating
M.R. Douglas, S. H. Shenker / Strings in less than one dimension 649
free energy derivatives we set z= p.~and so u becomes (3)2/51c_4/5. We will use these expressions for u interchangeably.
Known properties of the Painlevé transcendent [30—32]imply that the only singularities any solution has in the u plane are double poles, plus an essential singularity at cc that is the cluster point of an infinite number of poles. In fact p can be written as [30]
(6.3)
where ~ is an entire function. Although p is single valued on the u plane, because
= 3u5”~ there will be a cut beginning at the origin in the ic2 plane. The discontinuity in F across the cut is invisible in perturbation theory. Formal manipulations with the asymptotic behavior of the perturbation series (eq. (4.45)) determine the discontinuity to be — exp( —2 B/ic). We see that the ic~plane is five-fold branched, the discontinuity given by p(u) — w’p(w2u) where o~= exp(2iri/5).
We now turn to the important question of boundary conditions. We ask which values of p (U), p’(U) when evolved by the differential equation give functions that describe the string. We first demand that p(u) join the asymptotic series in ic2 as u— cc, K— U. The asymptotic series approximates the solution of eq. (6.2), p(u)—
+ ~ u— cc. Linearizing around this behavior we find two solutions to the homogeneous problem [32],
(u)~”8exp(+ ~ (u)”8exp(— ~/~u5/4). (6.4) Note that the argument of the exponent is just ±2 B/ic. One solution grows exponentially as u—cc, the other decays exponentially. We will only be able to match onto the desired large-u behavior if we kill the exponentially growing solution. This is one constraint on the boundary conditions. But there remains a one-parameter family of solutions that track onto the asymptotic series with deviation — exp( —2B/sc).
There appears to be nothing universal that sets this remaining boundary condi- tion. Of course any specific sequence of matrix models used to construct the scaling limit will give an unambiguous answer. The terms in the exact matrix model equation that set this boundary condition are lattice spacing dependent and so there is no reason to expect them to be universal. If this is really the case then there is a one-parameter family of acceptable theories. There is also a five-fold discrete family because of the branched sc2 plane. Only one of these will be real, however. That any branch would be real is somewhat surprising since the simplest potentials at finite N are unstable.
650 M.R. Douglas, S.H. Shenker / Strings in less than one dimension
7. Discussion
The ambiguity noted above is very interesting. A second free parameter has appeared in the non-perturbative string theory whose effect is invisible in weak coupling perturbation theory, as all solutions differ from the asymptotic series by O(exp( —2 B/ic)). This situation is reminiscent of the 8 parameter in Yang—Mills theory*.
It is possible that there is some subtle consistency condition, automatically embodied in the matrix models, that fixes this free parameter. Because the string equation is an equally valid non-perturbative definition of the theory, it seems a more fruitful approach to calculate interesting physical quantities like loop expecta- tion values [23] and integrated matter correlation functions in this formalism and then explore the effect and consistency of various values of this 8 parameter. Such a continuum formulation is possible because the theory is finite.
At the higher (order rn) multicritical points the string equation is a 2(rn— I) order differential equation. So there will be 2(rn— 1) modes in the linearization around the asymptotic solution. We have looked at rn = 3 where there are two exponentially increasing and two exponentially decreasing modes. This implies that there are two 8 parameters in this model. We conjecture that there are rn— 1 0 parameters at the order rn multicritical point. This growing number of potentially free parameters certainly needs to be understood.
There are other aspects of the non-perturbative solutions that are noteworthy. As mentioned earlier at large ic quantities are simple in the variable u. There is a convergent expansion in u around u= U. This variable is just p.R measured in units of the string coupling. Since small p.~means large world-sheet area we might guess that we are taking some kind of thermodynamic limit of a gas of handles. This qualitative feature might persist in the D ~ 1 non-critical strings.
In general we need a way of interpreting the string equation and its solutions. As a first step we could try to understand the various non-perturbative effects in terms of matrix model variables. The effect of other saddle points in the eigenvalue space seems worth exploring**.
If some kind of string field theory functional integral description exists we might expect to be able to describe the solution by a positive measure. This is a non-trivial constraint that should be tested. In this connection let us note that eq. (6.3) for p(u), which is essentially a two-point function, is given by (~
2/a
u2)log ~T.Even better, ~ is entire. The five-fold branched structure of the theory might very well be an important clue. In any event, we hope that unravelling the mysteries of strings in less than one dimension will shed some light on the properties of the more complicated string theories that may describe our world.* We thank N. Seiberg for this analogy.
** We thank V. Kazakov for discussions on this point.
Al. R. Douglas, S. TI. Shenker / Strings in less thanonedimension 651
We would like to acknowledge helpful discussions with T. Banks, F. David. D.
Friedan, P. Griffin, D. Gross, G. Harris, V. Kazakov, E. Martinec, P. Mende, A.
Migdal, S. Rey, N. Seiberg, M. Staudacher, S. Trivedi, and P. Windey as well as the hospitality of the Aspen Center for Physics and the MIT Artificial Intelligence
Laboratory where parts of this work were done.
An introduction to related work in the mathematical literature can be found in ref. [33].
This research was partially supported by grants to Rutgers University from the Department of Energy and the National Science Foundation.
Note added in proof
The discussion immediately following equation 4.41 is incorrect, and R= F”.
After submission of this manuscript we received preprints from E. Brézin and V.
Kazakov [34], and D. Gross and A. Migdal[351 that have substantial overlap with the present work.
Appendix A
The main equations in the orthogonal polynomial technique are the non-linear equations
V’(R)ki,kRk,k_l=k (A.1)
for the matrix R, which give a recurrence relation for the coefficients Rk.kt. We take the N— cc limit of each equation, by writing Rk+,,./,+fl_i = r(x + cn) and expanding in powers of ~
a/ax.
Each equation is linear in the couplings g21 of the potential V. so let us compute the limit for V=The leading ° term is proportional to the k— 1,k matrix element of the matrix M ~ +
~
raised to the 2j— 1 power, or in other words the number of random walks on a one-dimensional lattice which cover one step after timet = 2/— 1.This is simply a binomial coefficient. Adding the answers for different couplings gives the function W,
W(r) r+ ~ g2~,~ 1~, (A.2)
I>2 1).
Higher-order terms are given by expanding every occurrence of r using
r(x+cn)=>J~—!- n—i r(x). (A.3)
ax
Let us take the r2t i~ term of W~ ~(r) as defined in the text. It will be given by the same random walk as for W, with a sum over all ways of inserting
DPi, flP2, and so on, on downward steps of the random walk (downward because
only the lower diagonal entries of the matrix R contain non-constant coefficients:
652 MR. Douglas, S.H. Shenker/ Strings in less than one dimension
the upper diagonal entries are 1). We can write this using the lattice propagator,
1 (A.4)
1— w(z +z’)
whose wtzl term counts walks for time t which go Isteps. With the insertions and the condition that the walk travel one step, the quantity of interest is the generating function
dz k
a ~
fp(w)~fl(P(z~w)(z~) _)~. (A.5)
Then
Wp(r) = w-~-—wfp(w)j~2..,.
a
(A.6)The first claim is that Wp(r) is given by some linear differential operator Lp(r,
a/ar)
acting on W(r). From this it follows that the singularities of W~(r)are determined by those of W(r), and in particular that the order of a singularity is determined by the order of the operator L11,. The considerations in the text will then show that the only quantity which survives the continuum limit is the coefficient of the highest-order term of each L1. These we can compute by taking the continuum limit of the random walk. The final result is essentially that the coefficients of the continuum equation for R are generated by the Green function G(x, x;t) of a one-dimensional particle moving in an arbitrary potential.The first step in evaluating the function f~(w)would be to eliminate the z
a/az
factors. We do this using the relations
(z
a~)2P
= 2(1— 4w2)P3 — 3P2 + P, ((za2)~)2
= (1 — 4w2)P4— 2P3 +P2, (A.7), (A.8) to eliminate all but at most one za/az.
This one can be dealt with by integration by parts. The result is a sum of termsdz w k
fk,l,m(14W)
~(-;-)
pi, (A.9)where
I> k
+ rn/2. This is because our starting point hadI> k
+rn/2, and each step in the calculation preserves this condition. We then use1
a
pfl+l = —w—.-— +1 P’~ (A.1U)
~ aw
M. R. Douglas, S.H. Shenker/ Strings in less than one dimension 653
to reduce the integral to the original random walk. The final result for frn, k, I when 1> k + rn/2 differs from the original random walk by a polynomial in n, and therefore gives a contribution to W~which can be represented by a linear differen- tial operator acting on W.
We now give the continuum limit for the random walk problem, which is a practical method for calculating the W~.The propagator becomes 1/(p2 + E), the z
a~
insertion becomesi a/ap,
and to get coincident initial and final points we integrate over p. The power of E in the result corresponds to the order of L~—dimensional analysis givesorderL~1~2. ,pkk ~ (A.11)
Finally converting from the E to the t representation gives an overall constant. The result is that (inserting the factors l/p! from the original expansion),
a
E1112a
Lp=cp(r~-)
(_)
F(1/2) ~ 1
cP= F(k+~p~/2+l/2)J_~dPp2+l
k 1
a ~
1xfl
~ 2 peven~ p) ap p +1
=U, podd. (A.12)
In particular,
c2— ~, ~ll — 12 ‘ ~4 —i-— 55’.
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