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Non-Perturbative Two-Dimensional

arXiv:hep-th/9108016v1 23 Aug 1991SHEP-90/91-35AugustNon-Perturbative Two-DimensionalQuantum Gravity, Again1Simon Dalley2 , Clifford Johnson and Tim MorrisDepartment of Physics, University of Southampton,Southampton SO9 5NH, U.K.1Talk given by SD at workshop on Random Surfaces and 2D Quantum Gravity, Barcelona 10-14 June1991, proceedings to appear in Nucl. Phys.

B Proc. Suppl.2Address after 15th September 1991: Joseph Henry Laboratories, Princeton University, Princeton NJ08544, U.S.A.1

This is an updated review of recent work done by the authors on a proposal for non-perturbatively stable 2D quantum gravity coupled to c < 1 matter, based on the flows ofthe (generalised) KdV hierarchy[1, 2].Since the discovery of the continuum limit of 2D quantum gravity coupled to minimalmatter [3, 4] to all orders in the genus expansion, the question of a non-perturbativedefinition of these theories has remained. In refs.

[1, 2] we have presented a ‘physical’ non-perturbative definition by elevating the fundamental (generalised) KdV flow symmetry,known to exist perturbatively, to a non-perturbative principle. Concentrating on KdV forsimplicity, these flows are [5];∂u∂tk= R′k+1[u] = ξk+1u(1)whereXk(k + 1/2)tkRk −z = R = 0.

(2)Here the string susceptibility u = Γ′′, prime is d/dz where z = µ/ν, while µ and ν arethe renormalised ‘cosmological constant’ and 1/N. The parameters tk/ν couple to thematrix–model integrated local operators Ok on the surfaces and Rk are the Gelfand-Dikiidifferentials [6].

The flows determine correlators of the local operators:−∂2∂z2 < Ok1 . .

. Okn >=∂∂tk1.

. .∂∂tknu(3)The string equation (2) is the usual condition imposed by the hermitian matrix model(HMM) [3], leading to non-perturbative instability in particular for pure gravity andprobably all unitary c < 1 matter.Let us relax this condition and ask for the mostgeneral string equation compatible with the flows (1).

To derive it note that dimensionallyif [u] = 1 then [z] = −1/2 and [tk] = −(k + 1/2) as follows from (1). Then u as a functionof its dimensionful arguments must have the scaling symmetry3;∞Xk=0(k + 1/2) tk∂u∂tk+ 12zu′ + u = 0(4)Using (1) and the recurrence relation [6];R′k+1 = 14R′′′k −uR′k −12u′Rk(5)this becomes:14R′′′ −uR′ −12u′R = 0(6)Multiplying by R and integrating gives the string equation [1]:uR2 −12RR′′ + 14 (R′)2 = 0(7)3We assume initially that no new dimensionful parameter arises at the non-perturbative level.2

The constant of integration (which is an order ν2 term) has been set to zero by handin order that the HMM equation R = 0 satisfies (7) asymptotically. This is necessaryto reproduce the perturbative (genus) expansion at z →+∞.

Note that the asymptoticexpansion of the solution to (7) is uniquely determined once we have fixed the sphericalapproximation.As an illustrative case we can consider pure gravity tm = tmδm2. In the sphericalapproximation, where we can neglect derivatives, (7 ) becomes:u(u2 −z)2 = 0(8)As z →+∞we take u = √z and the corrections will yield standard genus expansion,by construction.

If we require u to have a real asymptotic expansion at z →−∞thenwe must take the root u = 0. The susceptibility then begins at the ‘torus’ level and infact has non-vanishing contributions only every 2nd genus.

More generally it is every mthgenus4 [1]:u = −1z2∞Xn=0Cn1z2m+1n; C0 = 1/4 ; z →−∞(9)It is highly likely that all other non-perturbative solutions to (7) reproducing the standardgenus expansion at +∞are complex or have poles on the real axis and are thus physicallyunacceptable[7]. The z →±∞behaviour is now enough to fix completely the boundaryconditions for (7) [2] and the (unique) numerical solution we found for pure gravity isshown in figure 1.

On the question of uniqueness we note also that at tm = tmδm1, (7)is reducible to Painlev´e II in χ by the transformation u = 2χ2 + z. With our physicalboundary conditions it has already been shown analytically that there is a unique, real,pole-free solution [10].Thus, starting from the KdV flows and requiring physically acceptable behaviour inchoosing boundary conditions, we are led to the solution for pure gravity shown in figure1.

Considering u as the potential for a hamiltonian operator H, whose eigenvalues governthe non-perturbative positions of charges in a Dyson gas, we see that the spectrum iscontinuous and bounded below. In fact our analysis is equivalent to taking the continuumlimit of an appropriate (critical) Dyson gas on the positive real line, as we show in amoment.

We can compare our stabilisation with results provided by the SUSY D=1 /stochastic quantisation of the HMM investigated in refs.[8]. The non-perturbative differ-ence between the two at positive z is obvious from the manifestly different behaviour ofthe solutions at z →−∞.

For the model investigated in [8], since u is real and ∝√−z atz →−∞[11, 12], the susceptibility cannot satisfy (7) and so violates the KdV flows non-perturbatively. Also there still appear to be non-perturbative ambiguities in the approachof [8] in its most general form, while there are none for our approach.Our analysis so far has dealt directly with the continuum limit and it is natural to askwhether there is an appropriate formulation in terms of a Dyson gas.

The universality classof the critical behaviour in such a gas can be conveniently characterised by the structure4The same asymptotic behaviour has also recently been found in the unitary matrix model with‘external fields’ [9]3

of the eigenvalue density in the scaled neighbourhood of the end of an arc of eigenvalues,in the spherical approximation[13]. Figure 2(a) shows this region as appropriate for puregravity (m = 2) at z > 0.

Eigenvalues are concentrated on the cut (wavy line) ended bya square root branch point, and formally extending the expression for the density offthiscut, its first integral has the interpretation of effective potential for one eigenvalue, Veff[12]. For the mth critical point there are m −1 extra zeros of the density in the scaledneighbourhood of the branch point, shown as dots in the figure.

The behaviour as afunction of z is easily calculated by using the WKB approximation for the spectrum of theco-ordinate operator H, with the appropriate ‘potential’ u from (8). This also facilitatesa comparison of the extremal values of Veffwith the leading exponential corrections to theasymptotic solutions to (7) [2].

As z →0 the branch point and the extra zero in figure2(a) collide and at first it is difficult to see how a real, sensible answer is maintainedfor z < 0. A solution is shown in figure 2(b) whereby the density now diverges, ratherthan vanishes, like a square root at the branch point, thus generating an extra zero.

Thisscenario is thoroughly natural if there is a ‘wall’ in the problem, which the branch pointhits at z = 0. In this case the two zeros depart into the complex plane perpendicular tothe real axis as z becomes more negative.

The structure shown in figure 2(a) to the left ofthis wall is then in the sense of analytic continuation (Veff= ∞strictly). Simply speakingthe wall will prevent eigenvalues leaking out of the arc and this is in fact the origin ofnon-perturbative stability in our formulation, as we now show.We can derive the full continuum limit `a la Douglas [4] which also indicates the inclu-sion of the generalised KdV hierarchy.

Insertions of the eigenvalue operators −λ, −d/dλhave, for the mth critical point of the one-matrix model, the continuum limits:−H = Q = d2 −u;P =2m−1Xi=0αidi;d = d/dz(10)The eigenvalue space of the HMM is IR and the canonical momentum operator is P(translations). The c.c.r.

[P, Q] = 1 determines the αi :i < 2m −1 uniquely (α2m−1is an undetermined non-universal constant), the order d0 of this relation being R′ = 0.Our stabilisation scheme is equivalent to the restriction to IR+ i.e. the imposition of a‘wall’ in the scaling region.

To show this we note that the canonical momentum on IR+is ˜P, the generator of scale transformations, satisfying the new c.c.r. [ ˜P, Q] = Q.

If Qis conjugate to macroscopic loop length then these local scale transformations are relatedto physical scale transformations of the string/one-dimensional universe. We can find ˜Pby introducing the fractional-power pseudo-differential operator and its differential part:Qm+1/2 = d2m+1 + (m + 1/2){u, d2m−1} + · · · + {Rm+1[u], d−1} · · ·(11)[Qm+1/2+, Q] = R′m+1(12)We expect ˜P = P2m+1i=0αidi but Qm+1/2+does not quite do the job.

To obtain a term d2on the r.h.s. of the c.c.r.

we must use most generally:˜P = α2m+1Qm+1/2+−z2d(13)4

giving the (differential of the) string equation;[ ˜P, Q] = α2m+1R′m+1 + zu′2 + d2 = d2 −u. (14)This is (6) at the mth critical point, modulo α2m+1 which may be absorbed into the stringcoupling.

By explicitly taking the continuum limit of the Dyson gas on IR+, realised forexample by the complex matrix model [1], one finds that the constant arising from anintegration of (14) is zero as in (7). Moreover the physical boundary conditions we choseearlier are precisely the ones appropriate to this gas.

In the spherical approximation u isthe scaling part of the end of the eigenvalue density. For z > 0 the ‘wall’ has no affect onthe WKB expansion, in particular the leading term is √z as in the HMM.

At z < 0 theend of the density remains fixed at the wall so u = 0 in the spherical approximation.It is natural to suppose that these considerations have analogues for the generalisedKdV hierarchy, related to the full set of (p, q) minimal models [4, 14]. One would generaliseto:Q = dq +q−2Xi=0uidiP = Qp/q+(15)˜P = Q1+p/q+−zqd(16)Some details of this will be explored in [15].Let us finally consider the Dyson-Schwinger equations5 of the one-matrix models, atleast that part of them which may be represented as formal Virasoro constraints [16].Our differentiated string equation (7) follows from the constraint L0τ = 0 (where u =−2d2 log τ and z →z + t0) on using the KdV flow (1).

This is not suprising in that it isprecisely the expression of scale invariance. The higher Virasoro constraints:Lnτ = 0:n ≥1(17)whereLn =∞Xk=0k + 12tk∂∂tk+n+ 14nXk=1∂2∂tk−1∂tn−k(18)follow by applying the recursion operator which generates higher symmetries of the KdVhierarchy [17]6.

The HMM string equation derives from L−1τ = 0 and one might askwhat happens to this constraint in our formulation. Since we are working on IR+ thereis no translation invariance and the L−1 constraint is naively absent.

More precisely therelevant DS equation picks up a boundary term from the ‘wall’:L−1τ = ∂τ∂σ(19)where σ is the scaled position of the ‘wall’, which up to now we have taken as the origin.In fact there is no reason why we should choose σ = 0 and more correctly we might5Some of what follows was elaborated while these notes were being prepared.6H.Kawai informed me that the DS constraints {Lnτ = 0 : n ≥0} should in turn imply that τ is aτ-function of the KdV hierarchy subject to L0τ = 0 (see his contribution to these proceedings).5

consider it as an extra (non-perturbative) parameter u = u(z, tk, σ). The correct canonicalmomentum gets shifted now to ˜Pσ = ˜P + σP and thus[ ˜Pσ, Q + σ] = Q + σ(20)implying14R′′′ −uR′ −12u′R + σR′ = 0(21)which may now be integrated to give the string equation.

But σ, which has the samedimension as u, now contributes to the scaling equation (4) a term σ∂u/∂σ. Comparingwith (21) we identify∂u∂σ = R′(22)which is equivalent to (19).

The other Virasoro constraints also pick up boundary termsat general σLnτ = σn+1 ∂τ∂σ(23)as follows simply from varying the boundary of the eigenvalue integration as σ →σ+ǫσn+1.The parameter σ can be set to zero by an analytic redefinition of the tk together withu−σ →u. In this sense it is redundant and in fact if we assume, following the HMM, thatthe non-perturbative loop expectation is Ψ(l) =< e−lH >, σ corresponds to a boundarycosmological constant [18].

The loop wavefunction in the scaling limit isΨ(l) ∝Z ∞σρ(ψ)e−lψdψ(24)whereρ(ψ) ∝Z ∞µ< x | δ(ψ −H) | x > dx(25)Hence for σ > 0 the wavefunction is guaranteed to have sensible exponentially decreasinglarge l behaviour.Acknowledgements: S.D. would like to thank the organisers for the opportunity topresent this work and workshop participants for their interest.

Financial support fromthe S.E.R.C. is acknowledged by S.D.

and C.J.6

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