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EXACTLY SOLVABLE STRING THEORY

arXiv:hep-th/9111064v1 30 Nov 1991PUPT-1290November 1991ON LOOP EQUATIONS IN KdVEXACTLY SOLVABLE STRING THEORYSimon DalleyJoseph Henry Laboratories, Princeton University,Princeton, New Jersey 08544, U.S.A.AbstractThe non-perturbative behaviour of macroscopic loop amplitudes in the exactlysolvable string theories based on the KdV hierarchies is considered. Loop equa-tions are presented for the real non-perturbative solutions living on the spectralhalf-line, allowed by the most general string equation [ ˜P, Q] = Q, where ˜Pgenerates scale transformations.

In general the end of the half-line (the ‘wall’)is a non-perturbative parameter whose rˆole is that of boundary cosmologicalconstant. The properties are compared with the perturbative behaviour andsolutions of [P, Q] = 1.

Detailed arguments are given for the (2, 2m −1) mod-els while generalisation to the other (p, q) minimal models and c = 1 is brieflyaddressed.1

1IntroductionSoon after the seminal works on non-perturbative 2D gravity appeared [1], anumber of authors discussed the behaviour of macroscopic string amplitudes inthose theories [2, 3, 4]. In genus perturbation theory these are the surfaces withboundaries whose lengths scale in the continuum limit.

It is possible however todiscuss such correlators independently of this weak coupling expansion. Sincethe latter is only asymptotic and typically not Borel resummable one must inputsome non-perturbative information in order to even begin discussing such ques-tions.

The mathematical possibilities are of course in principle endless. It is notso easy however to come up with a systematic formalism.

In what follows theextra information that is required will be taken to be the principle that the KdVstructure organising genus perturbation theory is present non-perturbatively1.This point of view was taken to its logical conclusion in refs. [5, 6] where itwas found that new non-perturbative solutions were allowed that had not beenpreviously considered.

It is expedient to briefly review the main points of thatwork. Consider the (2, 2m−1) minimal models, for which one has the KdV flows∂tku = −∂t0Rk+1[u] (notation is hopefully standard), which define the isospec-tral symmetries u →u+ǫ∂tku leaving invariant the spectrum of the hamiltonianoperator −∂2t0 + u.

Once the string susceptibility u has been determined theseflows yield correlators of the local operators with couplings tk. The minimalphysical requirement that one can impose on u is its scaling equation under achange of the length scale in the theory.

For the set of parameters {u, {tk}} theKdV flows determine this equation to be [5];LR = 0(1)whereR=Xk≥0(k + 1/2)tkRk=t0 −Xk≥1(k + 1/2)tk(D−1L)k−1u(2)D = ν ∂∂t0,L = −14D3 + 12uD + 12Du(3)and ν is the renormalised value of 1/N. The string equation (1) may be writtenas the canonical commutation relation on the half-line, [ ˜P, Q] = Q, for co-1This choice clearly has aesthetic and computational qualities but is nevertheless not, atpresent, a scientific argument.2

ordinate (Lax) operator Q = D2 −u and ˜P = Pk≥0 tkQk+1/2+the generator ofscale transformations of these co-ordinates, in a double-scaled Dyson gas withpolynomial potential.To reproduce the ν-perturbative results as specified by the matrix modelsone must choose a solution which is asymptotically a solution of R = 0 att0 →−∞. One possibility is to choose an exact solution of R = 0, for whichthere are two types of real solution u.

Those accumulating poles as t0 →∞[1](type 1), and solutions of k odd critical points and non-singular flows aroundthose points [3] (type 2), for which u has a real asymptotic expansion at t0 →∞.There is one other type of real solution to (1) with correct asymptotic behaviouras t0 →−∞(type 3). This has no poles2 and has u →0 as t0 →∞(R →t0).It is realised by a Dyson gas restricted to lie on the spectral half-line i.e.

thereis an infinite potential ‘wall’ at the critical edge of the charge distribution. Inthe following section this representation will be used to derive Dyson-Schwingerloop equations.

Figure 1 shows the numerical result for the type 3 susceptibilityfor pure gravity found in ref. [5].2Type 3 Loop equationsThe scaling symmetry that led to (1) is one of the non-isospectral symmetriesof the KdV hierarchy (see e.g.[9]).

Together with the isospectral symmetries(KdV flows) it implies a family of constraint equations by applying the recursionoperator LD−1;(LD−1)nLR = 0,n ≥0(4)These take the form of Virasoro constraints Lnτ = 0, u = −2D2 log τ [10].There is an additional non-isospectral symmetry which, if applied, constrains thesolution u further. Invariance under this transformation is the L−1 constraintDR = 0, which is the hermitian matrix model string equation.

This is theGalilean transformation: A translation of the Dyson gas potential, given ininfinitesimal form by;u−→˜u = u −ǫtk−→˜tk = tk + ǫ(k + 3/2)tk+1(5)is an invariance of R but is not respected by (1).2The pole-free nature of the solutions was verified for the k = 1, 2 critical points in ref. [5]and has also been checked for k = 3 and the Ising model [7]3

In the case of type 3 there is a further parameter in the theory which hasso far been neglected [6]. For the solution on the half-line one should allowthe infinite potential wall to be at some arbitrary scaled position, σ say.

Thismodifies the canonical position and momentum to Q + σ and ˜P + σP, whereP = Pk≥1 tkQk−1/2+is the translation operator. In this way the string equationis modified to [6];[ ˜P + σP, Q + σ] = Q + σ(6)equivalentlyLR −σDR = 0(7)Since σ has the same dimension as u and (7) is the scaling equation one identifiesthe evolution with respect to σ: ν∂σu = −DR.

This is the differentiated form ofthe L−1 condition in the present case: L−1τ = ∂στ. The Virasoro constraintsare the expression of diffeomorphism invariance of the spectral line and thepresence of a ‘wall’ has induced a boundary term on the right hand side.

Similarboundary terms appear in the other constraints when σ ̸= 0, since varying theboundary as σ →σ + ǫσn+1 implies thatLnτ = σn+1 ∂τ∂σ(8)The Galilean and higher KdV symmetries are now respected when the correcttransformation properties of σ are taken into account. In particular to (5) onemust add σ −→σ −ǫ.

(7) becomes an equation for ˜u in terms of ˜t, independentof σ. The constraints (8) may be rewritten in the old form for n ≥0 by usingthe invariance of the Virasoro algebra underLn −→˜Ln = e−σL−1LneσL−1(9)Given L−1τ = ∂στ the equations (8) are easily seen to be equivalent to ˜Lnτ = 0.The transformation (9) describes a finite Galilean transformation by σ, and thepossibility of making redefinitions of this sort can be viewed as another reasonfor introducing σ in the most general framework.The parameter σ transforms in the same way as tm−1/tm under (5), wheretk = 0, k > m and tm is invariant in this case.

Following the reasoning of ref. [11]it suggests that its rˆole is that of boundary cosmological constant.

Indeed if onetakes the usual expression for the renormalised macroscopic loop wavefunction[2];=Z ∞−t0 dz(10)4

a finite Galilean transformation by σ exhibits the e−σl dependence of the loop.The net effect of the transformation is to define away the non-exponential de-pendence of < w(l) > upon σ. In this case one can now say that σ couples tothe boundary operator [11] in the sense that ∂∂σ˜t=Xili (11)where ∂∂σ˜t= ∂∂σt−Xk≥1(k + 1/2)tk∂∂tk−1(12)∂nu∂σn˜t=δn1(13)It is worth noting that there is a certain amount of ambiguity in the definitionof the loop wavefunction, the choice (10) being perhaps the simplest.Tworeasonable requirements might be that it agree with the results of Liouvilletheory and have a local (KdV) operator expansion at small l. The former ismeaningful perturbatively in ν and the latter perturbatively in l. This does notfix contributions non-perturbative in both of these parameters however.

In thisletter the definition (10) will always be assumed.The arguments up to now have been somewhat heuristic so it is instructiveto give a more careful treatment of some of the points. First a proof of thefact that (7) is the string equation of a Dyson gas on [σ, ∞) will be given.To agree with the conventions for non-universal constants adopted implicitlyearlier, the following calculation is performed with an even polynomial actionNV/Λ on the interval [−2, 2], the ends of the charge distribution coincidingwith infinite walls at ±2 (in un-scaled variables).

The scaling regions around±2 thus furnish identical copies of the system on the half-line. Introducing aninfinitesimal cutoffδ, let the scaled positions of the walls be ±2 ∓σδ2.

Furtherrenormalised parameters are defined by;Λ =1 + t0δ2m1N = νδ2m+1ΛnN =1 −zδ2mRn = 1 −u(z)δ2, n ∼N(14)for the neighbourhood of the mth critical point. The equations of motion in theorthogonal polynomial formalism [12] are;(2n + 1)ΛN−=2ΛN (2 −σδ2)P 2n(15)5

nΛN −pRn =2ΛNpRnPnPn−1(16)The notation Pn is shorthand for Pn(2)exp(−NV (2)/2Λ) where Pn(λ) are poly-nomials orthogonal on [−2, 2]. Terms involving Pn in (15)(16) are boundaryterms picked up when one integrates by parts standard identities to derive theequations of motion.

The left hand side of (15) is;(2n + 1)ΛN−pRn+1 −pRn (17)=2ΛN (pRn+1PnPn+1 +pRnPnPn−1)(18)using (16). From the work on polynomial potentials [1];nΛN −pRn = δ2m(R(z) + O(δ2))(19)From (15)(17) and (19) one has;δ2mY 2=2ΛN (2 −σδ2)P 2n(20)Y=(R(z) + R(z + νδ) + O(δ2))1/2(21)and eliminating Pn, Pn−1 from (18) gives the string equation at the first non-trivial order in δ;ν22 RR′′ −ν24 (R′)2 −(u −σ)R2 = 0(22)Differentiating once yields (7).

The Dyson gas also supplies the appropriateboundary conditions u →σ as t0 →∞since u marks the edge of the chargedistribution in the leading WKB approximation. In fact a BIPZ analysis [14,5] shows that in this approximation the charge density aquires a square-rootdivergence at the wall (t0 > 0).

More generally at t0 < 0 this divergence hasexponentially small residue, as can be most easily seen from the conventionalform of the Dyson-Schwinger equations. These can be derived in the usual way[13] since the type 3 solutions have a path integral representation, and shouldcorrespond to the Virasoro constraints described earlier when Taylor expandedalong the lines of refs.[10].

Starting from the partition function on [−Σ, Σ] say,in unscaled variables;Z =Z Σ−ΣNYi=1dλi ∆2(λ)e−N Pi V (λi)(23)6

and defining the loop generating function and its Laplace transform (markedloops are used for convenience);W(L)=1NXieLλi(24)˜W(T )=1NXi1T −λi(25)one may perform an infinitesimal change of variables λi →λi+ǫ/(T −λi) in (23).The only new contribution to the standard analysis [13] is from the variation ofthe boundaries, givingǫ∂Z∂Σ1T −Σ −1T + Σ(26)The first loop equation is thenV′(T ) < ˜W(T )> +Π(T ) =< ˜W(T ) ˜W(T )> +2ΣN 2(T 2 −Σ2)∂Z∂Σ(27)where Π(T ) is a linear combination of < W(0) >,< W′(0) >, . .

.. In terms ofW(L) one has the equation of motion (introducing connected correlators);V′ ∂∂Lc=Z L0dL′ ( 1N 2 c+cc) −2sinhLΣN 2∂log Z∂Σ(28)showing a new source-like term for the loop generating function.Following ref.

[4] it is a simple matter to take the continuum limit by intro-ducing, in the neighbourhood of the mth critical point, renormalised parameters;Σ = Σc −σδ2,T = −Σc −τδ2< ˜W(T )>c=12V′(T ) + δ2m−1 < ˜w(τ)>c(29)c=δ4m+2−M(2m+3) cEquation (27) becomes< ˜w(τ)>2c +ν2 < ˜w(τ) ˜w(τ)>c= π(τ) +ν2σ + τ∂log Z∂σ(30)For completeness the higher loop equations are2 < ˜w(τ)>c< ˜w(τ)MYi=1˜w(τi)>c +ν2 < ˜w(τ) ˜w(τ)MYi=1˜w(τi)>c7

+XI,J< ˜w(τ)YiǫI˜w(τi)>c< ˜w(τ)YjǫJ˜w(τj)>c+MXi=1< ˜w(τ1) . .

. ∂∂τi˜w(τi) −˜w(τ)τi −τ.

. .

˜w(τM)>c=c +ν2σ + τ∂∂σ c(31)where O0 is the (bulk) puncture operator and the index sets I, J are such thatI ∪J = {1, . .

. , M}: I, J ̸= ∅= I ∩J.

π(τ) is a polynomial in τ, of degree2m −1 at the mth critical point, which determines the structure of the chargedensity in the scaling region, given byi2πDisc < ˜w(τ) >ν=0 in the saddle-pointapproximation. The last term in (30) should be1σ+τRRdt0 from the discussionfollowing (7), which is exponentially small in ν at t0 →−∞but is O(1) fort0 →∞.The physical meaning of the divergence as τ + σ →0+, is thatlarge loops become unsuppressed, since ˜w(τ) is the Laplace transform w.r.t.

τof the renormalised macroscopic loop wavefunction (10). Because of the e−σldependence of (10) the Laplace transform only exists for τ > −σ.3 As indicatedin the discussion following (22), a concomitant divergence appears in the chargedensity as τ + σ →0−.3DiscussionThe previous calculations have shown that type 3 solutions yield very simplee−σl behaviour for macroscopic loops.

This exponential behaviour is much likethe ν-perturbative result. At each order of a WKB expansion the charge densityhas support on the half-line (−∞, −u] of the real τ axis, specified in the leadingapproximation by the (single) cut inpπ(τ).

The rest of the real τ axis describesthe loop function ˜w(τ) which on inverse Laplace transformation is seen to havee−ul times power law behaviour. At genus zero w(l) contains universal termswith inverse powers of l, as emphasised in refs.

[15], since π(τ) is polynomial:< w(l) >= 1νXk≥0k!tkl−k−1/2 + O(l1/2)(32)The Laplace transform does not exist because of these terms, but one can pro-ceed by differentiating w.r.t. τ a sufficient number of times.

It is in this sense3In fact there are also negative powers of l in w(l) preventing naive Laplace transformation,but these come from low genus and can be systematically isolated.8

that w(l) and ˜w(τ) are transforms of one another. Alternatively one can workat fixed ‘area’ instead of fixed bulk cosmological constant.

The offending termsare analytic in the latter and so do not contribute to an inverse transform tofixed ‘area’. They correspond to finite loops spanned by infinitesimal surfaces.The ν-non-perturbative exponential behaviour in l of loops for type 1 andtype 2 solutions is more complicated.

For type 1 the discreteness of the spectrumof −D2 + u implies that < w(l) > behaves like an infinite sum of exponentialswith different arguments [2]. Only in the l →∞limit does one recover a simplee−e0l behaviour, where e0 is the lowest eigenvalue.

It is sometimes suggestedthat a solution satisfying a loop equation such as (30), derived from a pathintegral representation, is ‘physical’ and that, by implication, one that does notis ‘unphysical’. There is no known path integral formula for type 1 solutions.This is not (presently) known to contradict any physical principle however.

Thehermitian matrix model is the path integral representation of type 2 solutions,and shows that the loop expectation is always diverging as l →∞for suchsolutions [3]. This markedly different behaviour is due to the fact that the chargedensity has support on the whole spectral line, in particular it has an exponentialtail.

For a tail ρ(e) ∼exp(−|e|p) as e →−∞a dimensional argument showsthat ∼expl1+1/(p−1) as l →∞. This means that one cannot define theloop function ˜w(τ) in this case.To conclude one may note some possible generalisations of the results ofthis letter to other models with c ≤1.

For the (p, q) minimal models describedby the generalised KdV hierarchy [16], the string equation (scaling equation)[ ˜P, Q] = Q will provide new solutions analoging those of type 3. Generally oneonly knows how to treat macroscopic loops embedded at a single point in the lineof q−1 points.

By the same argument [11], shifting the non-derivative part of Q,one can identify a parameter coupling to a boundary operator. Perturbatively itappears that only one parameter can be generated in this way (e.g.

for the Isingmodel (4,3) it is the boundary magnetic field), which led the authors of ref. [11]to conclude that certain operators could not be expressed in the KdV formalism.However in the case of type 3, non-isospectral evolution equations also play a rolein defining couplings and one can imagine restricting the q −1 types of chargeto different half-lines i.e.

the wall becomes ‘time’-dependent. The necessaryargument in terms of the W-constraints to confirm or deny the validity of thisnaive picture is a little involved.

A detailed account of the [ ˜P, Q] = Q versionof other minimal models will be given by the authors of ref.[7]. At c = 1 the9

picture is similar. By placing a wall in the scaling region given by the inversequadratic potential [17] one has a stable quantum mechanical system providinga non-perturbative definition of the theory.

More particularly the macroscopicloop amplitude at time t is non-perturbatively well-defined;=Z ∞σdλ ψ†(λ, t)e−λlψ(λ, t)(33)ψ(λ, t)=Z EFdEeiEtΨ(E, λ)(34)where the wavefunctions Ψ vanish at the wall λ = σ and the matrix modelcorresponds to the continuation to euclidean time. It exhibits a dependenceexp −lσ, at the expense of introducing a linear term in the potential.

Theextra parameter σ more generally has a continuous argument since again it maybe time-dependent. Understanding of the possible flow structure at c = 1 isstill hazy.

Is stabilisation by a simple wall, possibly fluctuating in position, theonly quantum mechanical problem non-perturbatively compatible with a flowstructure organising perturbation theory, akin to c < 1?Acknowledgements: The influence of Tim Morris on the early stages ofthis work cannot be overstated.I am also grateful to Clifford Johnson forhelpful information, and to Tim Morris and Andrea Pasquinnuci for commentson the manuscript.This work was supported by an SERC studentship andpost-doctoral fellowship RFO/B/91/9033.Note Added: After this letter was typed a preprint appeared [18] wherequantum mechanics on the half-line is discussed with regard to the requirementof unitarity of tachyon scattering at c = 1.10

Figure CaptionFigure 1: This shows the (unique) numerical solution of type 3 for pure gravity.The Hamiltonian −D2 + u has continuous spectrum down to the value of theleft asymptote of the potential u (zero in the figure shown). Those eigenvaluesare more-or-less directly related to the positions of the Dyson gas charges on thespectral line, which should lie on the positive halfline.

Note that although thereis a small well in u, the previous identification indicates that it is too shallow tosupport bound states below the continuum. A rough numerical estimate usingthe proportions of the figure confirms this.

As is explained in section 2, moregenerally the left asymptote can be u = σ if the Dyson gas is restricted to[σ, ∞). It is possible that non-perturbative subleties arise as σ turns negativee.g.

through an instability in u similar to that found in ref. [8].11

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