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Loop Equations and the Topological Phase

arXiv:hep-th/9108014v1 22 Aug 1991YCTP-P25-91RU-91-36Loop Equations and the Topological Phaseof Multi-Cut Matrix ModelsˇC. Crnkovi´c1M.

Douglas2G. Moore32 Dept.

of Physics and AstronomyRutgers UniversityPiscataway, NJ 08855-08491,3 Department of PhysicsYale UniversityNew Haven, CT 06511-8167We study the double scaling limit of mKdV type, realized in the two-cut Hermitianmatrix model. Building on the work of Periwal and Shevitz and of Nappi, we find anexact solution including all odd scaling operators, in terms of a hierarchy of flows of 2 × 2matrices.

We derive from it loop equations which can be expressed as Virasoro constraintson the partition function. We discover a “pure topological” phase of the theory in which allcorrelation functions are determined by recursion relations.

We also examine macroscopicloop amplitudes, which suggest a relation to 2D gravity coupled to dense polymers.August 15, 19911 (crnkovic@yalphy.hepnet, or @yalehep.bitnet), Address after Sept. 1991: CERN, TheoryDivision, CH-1211, Geneva 23, Switzerland. On leave of absence from Institute “Ruder Boˇskovi´c,” Zagreb, Yugoslavia.2 (mrd@ruhets.rutgers.edu)3 (moore@yalphy.hepnet, or @yalehep.bitnet)1

1. IntroductionRecent studies of two-dimensional Euclidean gravity have yielded a host of surprises.These surprises have come from lattice definitions [1–6], continuum path-integral defini-tions[7–9], and topological field theory definitions[10–12] of what is presumably thesame theory.Among the surprises from the lattice/matrix-model approach has been the discoveryof theories which look very similar to the solved models of two-dimensional gravity coupledto matter, but for which no continuum or topological interpretation has been found.

Forexample, a particularly natural family of matrix models can be derived from the integralover a unitary matrix. These have a double scaling limit whose exact solution is strikinglysimilar to that of the lattice gravity models, with the KdV hierarchy being replaced by themKdV hierarchy [13,14].

One is naturally led to ask whether these theories have equallysimple world-sheet interpretations.The simplest way to attack this problem would be to find an interpretation of theperturbation expansion for the theory as a lattice gravity theory, possibly with matter andappropriate weights in the sum over graphs, and directly take the world-sheet continuumlimit. This can be difficult since the same continuum theory can be obtained from manydifferent matrix models.

The important point is not the type of matrix but rather thestructure of the saddle-point eigenvalue distribution near its points of non-analyticity.The mKdV scaling limits come from tuning a coupling to a critical point where two end-points of the eigenvalue distribution collide.This is generic for a unitary matrix, butis easy to arrange for a Hermitian matrix as well, for example by taking a double-wellpotential [15,16]. Thus there are many possible graphical formalisms which one mighttry to interpret.Perturbation theory for the unitary matrix integral can be done bychanging variables U = exp iH; the resulting graphs have vertices of arbitrary order aswell as fermionic lines coming from a fermionic representation of the Jacobian [17].

Wewill discuss this expansion further in the conclusions. Alternatively one could sum overthe expansions about the various minima of the double-well problem.It is not obvious how to interpret any of these lattice theories, because we do notknow how to take the continuum limit directly on the world-sheet.

The simplest lattice2

gravity [1–6] is exceptional since it has the correct degrees of freedom and an everywherepositive measure. Nevertheless, already with non-positive measures there are subtleties inthe continuum limit which can reproduce gravity coupled to matter [18,19].

The measuresin the present problem are too complicated for heuristic arguments. All this underscoresthe importance of developing a more precise understanding of this continuum limit.Since we cannot deduce the continuum theory directly from the lattice we are forced tocompare the structure of the exact solution of the lattice theory with the structure of knowncontinuum theories.

With this motivation, we describe below the exact solution of thelattice theory in some detail. In section two we derive from the lattice the complete stringand flow equations for two-cut models, including all even and odd perturbations in thepotential.

In section three we show how these equations can be rewritten as Virasoro-typeconstraints. This reformulation suggests the existence of a topological phase of the theory,discussed in section four.

In section five we make some brief remarks about macroscopicloops in these theories. In the conclusions we attempt to draw some inferences about whatthe continuum formulation of these models might be.2.

Multicritical Matrix Model Potentials and Physical ObservablesIn this section we derive the string equation and flow formalism, generalizing [13],[14], [16] and [20]. The analysis is rather technical but the results are summarized in thestring equation (2.23), recursions (2.8), observables (2.19) and flows (2.11).To derive the full continuum loop equations for unitary matrix models, we will workwith the two-cut Hermitian matrix model and study the most general scaling potential.The general even multicritical potential resulting in a two-cut eigenvalue distribution wasobtained in [16]:V ′m(λ) = k(m)λ2m+11 −1λ21/2+,m = 1, 2, .

. .,(2.1)where the subscript + means we keep the polynomial part in an expansion about infinityand the constant k(m) = 22m+1(m + 1)!

(m −1)!/(2m −1)!.The universality class of a potential is determined by the behaviour of the saddle-pointeigenvalue density ρ(λ) around the critical point. The eigenvalue density corresponding3

to Vm vanishes at the critical point λ = 0 as λ2m. To obtain the most general scalingbehaviour, we look for the potentials resulting in ρ(λ) ∼λ2m+1 for λ →0.

Such ρ(λ)are not positive definite and we will introduce them through perturbations around someVm(λ), in the spirit of [21]. The required behaviour is˜ρn(λ) = 0for|λ| > 1,˜ρn(λ) ∝λn−1forλ →0,∝(1 −λ)1/2forλ →1−,∝(1 + λ)1/2forλ →−1+,Z 1−1dλ˜ρn(λ) = 0.

(2.2)This is satisfied by˜ρn(λ) = ddλ[λn(1 −λ2)3/2],n = 1, 2, . .

.and the corresponding potential is˜Vn = v(n)N c(n)λn(λ2 −1)3/2|+,with v(n) and c(n) constants to be fixed.The scaling ansatz appropriate to the potential Vm isRn = rc + (−1)na1/mf(x) + · · · ,Sn = sc + (−1)na1/mig(x) + · · · ,nN = 1 −a2(x −µ),Na2+1/m = 1. (2.3)sc is of the order of odd perturbations, which are supressed by a power of N. The or-thonormal polynomials Pn(λ) tend to two scaling functions, depending on parity:(−1)nP2n(a1/mλ) = a1/2mp+(x, λ),(−1)nP2n+1(a1/mλ) = a1/2mp−(x −a1/m, λ).

(2.4)From (2.3) and (2.4) it follows that the multiplication by the eigenvalue λ ≡a1/m˜λ isrepresented on Ψ =p+p−by a 2 × 2 matrix (after a rescaling of f and g and a rotationof Ψ):˜λJ3Ψ = (D + q)Ψ,(2.5)4

where D ≡d/dx, q = f(x)J1 + g(x)iJ2, and Ji are matrices satisfying [Ji, Jj] = iεijkJk.Finally, redefining Ψ →J3Ψ we obtain (dropping tilde from ˜λ)λΨ = 4(D + q)J3Ψ ≡AΨ.We will need the resolventR ≡1(D + q)J3 −λ4.In their work on the resolvents of matrix differential operators [22] Gelfand and Dikii showthat R satisfiesJ3R′ = [J3R, q −λJ3],(2.6)and that it has an asymptotic expansionR(x, λ) =∞Xk=0Rkλ−k.From (2.6) it is clear that, up to constants, the most general expansion of R isJ3R =∞Xk=0grad ˆHk−1λ−k,(2.7)where grad ˆHk ≡−J1Gk −iJ2Fk +J3Hk. Plugging (2.7) into (2.6) results in the recursionrelations determining Gk, Fk, Hk.

Setting G−1 = F−1 = 0, H−1 = 1, we obtainFk+1 = G′k + gHkGk+1 = F ′k + fHkH′k = gGk −fFk,(2.8)for all k ≥0. The first few areF0 = g,F1 = f ′,F2 = g′′ + 12g(g2 −f 2),G0 = f,G1 = g′,G2 = f ′′ + 12f(g2 −f 2),H0 = 0,H1 = 12(g2 −f 2),H2 = gf ′ −fg′,(2.9)F3 = f ′′′ + 32f ′(g2 −f 2),G3 = g′′′ + 32g′(g2 −f 2),H3 = gg′′ −12(g′)2 −ff ′′ + 12(f ′)2 + 38(g2 −f 2)2.5

This determines the expansion of the resolventR =∞Xk=0(−iJ2Gk−1 −J1Fk−1 + Hk−1/4)λ−k.There is an SO(1, 1) symmetry in our system coming from possible choices in the originaldefinition of q. Observables will be invariant under this symmetry.A related derivation of (2.8) would start from two Hamiltonian structures defined in[23] for the system with the Lax operatorL = D + q −λJ3. (2.10)For the flow in the coupling tk, with dimension k, their compatibility givesddtkq = [grad ˆHk, J3] = [grad ˆHk−1, D + q],(2.11)which is easily seen to result in (2.8).We are now ready to start determining the observables.

We will calculate the 2-pointfunctions of the puncture operator P with the scaling operators σn corresponding to theperturbations by ˜Vn. We will follow the presentation of [21], beginning with the one-pointfunction∂F∂tn= v(n)N c(n)ICdy2πiyn(y2 −1)3/2|+tr1y −AΠN,(2.12)where(ΠN)ij =δijif 0 ≤i, j ≤N −1,0otherwise,and C surrounds all of the eigenvalues of A.

By deforming C to a circle at infinity one cansee that the subscript + in (2.12) can be dropped. To obtain a two-point function, westudy a variation δ∂F/∂tn and use [21]:δtr1y −AΠN= 12tr1y −A[S(δV (A)), ΠN],S(B)ij = ε(i −j)Bij,ε(j) = sign of j,j ̸= 0,= 0,j = 0.6

We choose the simplest even operator as the puncture operator. The corresponding vari-ation of the potential is δV (A) = N cP A2, and we obtain (to leading order in N):∂2F∂x∂tn= v(n)2N cP +c(n)ICdy2πiyn(y2 −1)3/2tr1y −A[S(A2), ΠN],= v(n)rcN cP +c(n)ICdy2πiyn(y2 −1)3/21y −AN,N−2+1y −AN−1,N+1.

(2.13)The scaled quantities arey = a1/mλ,A = a1/m4(D + q)J3,|2n⟩= (−1)na1/2m|x, +⟩,|2n + 1⟩= (−1)na1/2m|x −a1/m, −⟩,(2.14)and (2.13) becomes (again to leading order)∂2F∂x∂tn= v(n)rc4 a(n+1)/mN cP +c(n)ICdλ2πiλn⟨x, +|R|x, +⟩+ ⟨x, −|R|x, −⟩x=µ,= v(n)rc2 a(n+1)/mN cP +c(n)Hn(µ). (2.15)The simplest non-trivial case in this formalism is m = 1 potential (2.1) with simplestodd perturbation [20]V = V1 + Naetr(φ).

(2.16)(Note that, in the context of the unitary matrix model as 2D lattice Yang-Mills theory thiscoupling corresponds to a discretized version of log detU mod2π, and hence corresponds toa lattice version of the theta angle [24].) Using (2.3) with sc = −ae/16 in the recursionrelations gives string equations10 = xf + eg′ + 2(f ′′ + 12f(g2 −f 2)),0 = xg + ef ′ + 2(g′′ + 12g(g2 −f 2)).

(2.17)1 To obtain these string equations we rescale e, f and g and shift x. In order to compare with(2.9) the rescalings of f and g have to be the same as the ones used to obtain (2.5).7

The specific heat is⟨PP⟩= −x4 + e248 + 14(g2 −f 2). (2.18)The first two terms are non-universal and the universal piece is given by H1/2.

Therefore,we can identify x ≡t1, σ1 ≡P. This identification shows that, in order to obtain a finiteresult in (2.15), we need c(n) = n/(2m + 1).

2 Finally, for later convenience, we choosev(n) = 1/rc for every n. In conclusion, in the scaling limit of the two-cut matrix modelwe have obtained the following observables∂2F∂x∂tn≡⟨Pσn⟩= 12Hn. (2.19)The odd flows F2k and G2k have terms with no derivatives.

However, an insertion ofan odd operator is still suppressed by 1/N.To derive the string equations we will usen = R1/2nZV ′PnPn−1 = R1/2nICdy2πiV ′(y)1y −An,n−1,(2.20)V ′ = Nk(m)y2m(y2 −1)1/2|+ +2mXk=1tk1rcNk2m+1 yk−1(y2 −1)3/2|+ + · · · ,(2.21)In (2.21) + · · · denotes terms of higher order in a upon introduction of scaling variablesthrough n = N(1 −a2x) and (2.14).We want to replace the y-integral along C, a contour surrounding all of the eigenvaluesof A, by an integral over λ, i.e. over infinitesimally small y’s.

An obstacle to passing to a λ-integral comes from the A-eigenvalues outside of the critical region. Their contributions willnot have scaling dependence on coupling constants, but can be large (O(N)) nonetheless.In order to be able to neglect consistently eigenvalues of A at large y, instead of scaling(2.20), we will scale its derivative with respect to tl, l ̸= 1.

Namely, the tl dependence ofV appears only for y very small (O(a1/m)), while for y ∼O(1) it is suppressed by a powerof (large) N. The result is0 = δδtlIdλ2πiXkktkλk−1(∓)⟨x, ±|∞Xr=0(−iJ2Gr−1 −J1Fr−1 + Hr−1/4)λ−r|x, ∓⟩, (2.22)2 The factor of Na in front of tr(φ) in (2.16) is precisely N 2/3 appearing in ˜V2. We use thefact that the scaling limit of a generic odd (even) potential like tr(φ) (tr(φ2)) is given by ˜V2 ( ˜V1).8

where the upper (lower) sign corresponds to n even (odd). After doing the integral andtaking the 1, 2 (2, 1) element of the resulting matrix, we have0 =Xk≥1ktkGk−1 ≡Sg,0 =Xk≥1ktkFk−1 ≡Sf.

(2.23)The integration constant, which is independent of all tl, l ̸= 1, can be seen to be zero bycomparison with (2.17). (It was shown in [25] that by modifying the action to includeboundaries, or “quark” degrees of freedom one can introduce a nonzero constant on theleft hand side of (2.23).

)The string equation generalizes (2.17). Moreover, it may be equivalently written as aflatness condition, generalizing that in [16]:L, ddλ −M= 0(2.24)whereM =Xk≥1ktkMk−1,Mk = grad ˆHk−1 + λ grad ˆHk−2 + · · · + λk−1grad ˆH0 + λkJ3,(2.25)and M0 = J3.

M is chosen to make the left hand side of (2.24) independent of λ, asdescribed in [23].The flatness condition (2.24) can be interpreted as the isomonodromic deformationcondition for the monodromy defined by a solution of ddλ −MΨ = 0. (2.26)Furthermore, the partition function is again the isomonodromic τ-function.

The proof ofthese statements follows the steps already outlined in [26,16]. 3 Moreover, the solutions3 The form of recursion relations useful in the proof is (2.8) with the third equation replacedbyHk+1 = 12kXr=0(FrFk−r −GrGk−r −HrHk−r).9

to the string equations chosen by the matrix model will in general lead to nontrivial Stokesdata. As described in [26] this implies that the τ function does not correspond to a pointin the Segal-Wilson Grassmannian, although it does correspond to a point in the SatoGrassmannian.3.

Loop Equations.The solutions of one-matrix models satisfy Schwinger-Dyson equations which can bederived by variation of the functional integral [27].These have a double scaling limit[28,29] which in the KdV-type systems can be rewritten as Virasoro constraints on thepartition function [29][30].Since the same Schwinger-Dyson equations apply (before taking the continuum limit)to the two-cut Hermitian matrix model, and since the unitary matrix model has analogousSchwinger-Dyson equations, one would expect continuum loop equations to exist in thiscase as well.Previously we had derived a subset of these loop equations (with two non-zero cou-plings) by directly scaling the matrix model Schwinger-Dyson equations [31]. Here wederive the complete loop equations from the string equation/mKdV formalism.

The argu-ment is similar to that used in [30] for the KdV theories. Using the flow equations, thestring equation can be reformulated as a differential constraint on the partition functionas a function of the couplings.

In principle we can then derive a series of constraints asfollows: the same recursion relation which takes the n’th flow to the n+1’st can be appliedto the n’th constraint to find a new constraint.Since the constraint operators all annihilate the partition function, they form a leftideal. Therefore commutators of constraints are also constraints.

Guided by our expecta-tion that the constraints form a Virasoro algebra, we will find a minimal set of constraintswhich generate this algebra. The completeness of the resulting algebra will then follow ifwe can argue that there is a unique solution of the resulting constraints.

This will be truein a certain sense to be discussed below.The scaling operator corresponding to the coupling tn will be called σn. σ1 will alsobe called P, though one should probably think of the P as standing for “primary” and not10

“puncture” for reasons explained below. The string equation is now Sf = Sg = 0, and ourconstraints will be derived from combinations of these.We start with0 = gSg −fSf=Xkktk⟨σk−1PP⟩= D2 Xk≥2ktk⟨σk−1⟩,(3.1)which we interpret as the second derivative of an L−1 constraint.

The explicit dependenceon t1 has cancelled out.The recursion relations imply that gGk+1−fFk+1 = gF ′k−fG′k, so we can get anotherequation from0 = gS′f −fS′g=Xkktk⟨σkPP⟩+ (g2 −f 2)=Xkktk⟨σkPP⟩+ 2⟨PP⟩= D2 Xkktk⟨σk⟩(3.2)the second derivative of an L0 constraint.We can continue in this vein using the recursions to construct tkd/dtk+n in termsof differential polynomials of the string equations.We will skip L1 (which is similar)and proceed to the L2 constraint, since it will generate the complete algebra. Using therecursions three times, we haveI2 =Xkktk⟨σk+2PP⟩= 12Xkktk(gGk+2 −fFk+2)= 12Xkktk(gF ′k+1 −fG′k+1)= 12Xkktk(gG′′k −fF ′′k + (g2 −f 2)H′k + (gg′ −ff ′)Hk).= 12Xkktk(gF ′′′k−1 −fG′′′k−1 + Xk−1 + (g2 −f 2)H′k + (gg′ −ff ′)Hk).

(3.3)11

with Xk−1 = g(fHk−1)′′−f(gHk−1)′′. The X piece is identically zero by the L−1 constraint(3.1) and its derivatives:Dn Xk≥2ktk⟨σk−1⟩=XkktkDn−1Hk−1.

(3.4)Using S′′′f = P ktkF ′′′k−1 + 3g′′ = 0 (resp. for g), this isI2 = −32(gg′′ −ff ′′) + 2Xkktk(2⟨PP⟩⟨σkPP⟩+ ⟨PPP⟩⟨σkP⟩)(3.5)and using derivatives of the L0 constraintI2 = −8⟨PP⟩2 −2⟨PPP⟩⟨P⟩−32(gg′′ −ff ′′).

(3.6)The other terms we expect in an L2 constraint are⟨PPPP⟩= 12gg′′ −ff ′′ + (g′)2 −(f ′)2,(3.7)⟨σ3P⟩= 12gg′′ −ff ′′ −12(g′)2 + 12(f ′)2 + 38(g2 −f 2)2. (3.8)These combine nicely to give0 =Xkktk⟨σk+2PP⟩+ 2⟨σ3P⟩+ ⟨PPPP⟩+ 2⟨PP⟩2 + 2⟨PPP⟩⟨P⟩.

(3.9)Defining Z = eF , this is the second derivative of the expected constraint L2Z = 0 withL2 =Xk≥1ktk∂∂tk+2+ ∂2∂t21. (3.10)Note that to get to (3.5) we assumed that the undifferentiated L−1 and L0 constraintswere satisfied, i.e.

that they contained no integration constant. Under this assumption,we can now write the result as Virasoro constraints for a two-dimensional free masslessboson.

Define∂φ(z) ≡Xk≥1kzk−1tk +Xk≥1z−1−k ∂∂tk,(3.11)an untwisted boson; then the constraints are the modes Ln for n ≥−1 of the usualuntwisted stress-tensor (∂φ)2 with zero ground-state energy.12

The integration constants in the constraints must be compatible with the Virasoroalgebra, otherwise the commutator of two constraints would generate new constraintswhich would force Z to be trivial. An obvious possibility would be to include the zeromode of the boson as well,∂φ(z) ≡Xk≥1kzk−1tk + z−1q +Xk≥1z−1−k ∂∂tk.

(3.12)q would be a new parameter of the theory, not associated with any operators or flows. Inthe “usual” phase of the theory with all odd couplings set to zero, it is clear that q = 0,since all the other terms in the L−1 constraint are zero, but we will find a use for non-zerovalues shortly.It should also be possible to derive these loop equations directly by double scalingthe finite N Schwinger-Dyson equations.

In particular, the theory of the boson (3.12)should emerge from the formulation of the lattice theory using second quantized fermions.Unfortunately, the currently available derivations of this “field theory on the spectralcurve,” are merely heuristic.4. A Topological PhaseIn the series of multicritical models including pure gravity, described by the stringequationx = t1u + t2(u′′ + 3u2) + .

. .

,(4.1)the simplest point is not pure gravity but rather the “pure topological” theory with t1 = 1and tn = 0 for n > 1. This theory has a conserved charge (“ghost number”) with a distinctbackground charge for each genus surface, so a specific correlation function can be non-zerofor at most one genus.

In particular ⟨σ0σ0σ0⟩= u′ = 1 is non-zero only on the sphere.With purely even couplings, there is no analogous model in the mKdV hierarchy, butwith odd couplings there is a precisely analogous model with t2 = 1 and tn = 0 for n > 2.The string equation is thenxg + f ′ = 0xf + g′ = 0. (4.2)13

with solutiong + f = C1e−x2/2g −f = C2ex2/2. (4.3)C1 and C2 are constants of integration.

Although these solutions are trancendental, phys-ical quantities are not: for example,⟨σ1σ1⟩= 12(g2 −f 2) = 12C1C2,(4.4)so this model appears to have “γstring = 0”. (The Baker functions, which are Airy functionsin topological gravity, are parabolic cylinder functions in this theory.

)The SO(1, 1) symmetry insures that all correlation functions in this model are poly-nomial in x. Assign the combinations f + g and f −g SO(1, 1) charge 1 and −1; thencorrelation functions must be neutral.

All charge-neutral polynomials in these functionsare constant, while appearances of d/dx will produce positive powers of x.The structure of the topological phase is seen most clearly in the loop equations. Justas for topological gravity, expanding the loop equations around these couplings gives aseries of recursion relations, one for each operator in the theory, expressing an operatorinsertion in terms of correlators of lower total degree.

For example, the L−1 constraintgives us⟨σ1Yiσni⟩= −12Xini⟨σni−1Yj̸=iσnj⟩(4.5)(where σ0 ≡0) for all cases except one:⟨σ1σ1⟩= −q2. (4.6)To get non-trivial answers in the topological phase, we must take the parameter q = −C1C2non-zero.Just as for the KdV models, the existence of a phase in which all operator insertionssatisfy recursion relations, means that the expansion of the partition function to all ordersis uniquely determined by the constraints.They can therefore have only one analyticsolution.

Unlike the KdV models, since the parameter q is not a coupling of the model,we cannot flow from this topological phase to the phases with q = 0. This is probably14

connected with the difficulty of defining this pure topological phase in the matrix model– our odd perturbations of the potential (2.2) really only make sense as perturbations ofeven potentials. On the other hand the structure of the pure topological phase seems veryclear from the solution, so we will not let this stop us from studying it.The ghost number conservation law follows most directly from the loop equations.If we write them in terms of our boson ∂φ(z), they are homogeneous in z, so the ghostnumber assignment of each operator and coupling must be just the associated power of zin a mode expansion, up to a possible constant shift.

We can use this constant shift tomake the ghost number of one coupling of our choice zero. If we choose t2, setting t2 = 1will not break ghost number conservation.For topological gravity this is the end of the story.

The special role of t2 in this theoryis that it is the only choice of non-zero coupling for which all operator insertions have anassociated recursion relation. In the present case, both t2 and q must be set non-zero, andthey have different ghost number, q having that of “t0”.

A simple example of a correlationfunction which shows that q must be assigned ghost number is⟨σ3σ4⟩= −48λ5qx −144λq3x + 168λ3q2x3 −24λ5qx5(4.7)where λ counts derivatives (powers of 1/N).We can maintain a ghost number conservation law if we take q non-zero but treat itas the coupling of a new operator in the theory. If the operator σn is given ghost number2 −n, then q will couple to an operator Q with ghost number 2.

Now the basic correlationfunction on the sphere is⟨σ1σ1Q⟩= −12. (4.8)Q is an unusual operator in that it does not have descendants.

The ghost numbercounting would suggest that it could be renamed σ0 and that the other operators are itsdescendants, but this is misleading. It clearly appears in a very different way in the stringequation; in the loop equations, it is different in that its conjugate d/dq does not appear.The necessity to include this operator is the most serious difference with the topologicalgravity coupled to matter of [10].15

5. Macroscopic Loops for Unitary-Matrix ModelsIt has recently been shown that the study of macroscopic loop amplitudes [32] isparticularly well-suited for the comparison of continuum-Liouville and matrix model for-mulations of 2D quantum gravity [33] [34].

In particular, the Wheeler-DeWitt equationplays a central role in making such comparisons. In this appendix we derive analogousmacroscopic loop amplitudes for the unitary-matrix model, and show that they too obeya Wheeler-DeWitt-like equation.

This equation gives a hint about the proper worldsheetinterpretation of these theories.The macroscopic loop operator is defined to be Ψ†e−ℓLΨ in the fermionic formulation,where Ψ is a two-component fermi field (obtained from scaling the even and odd indexedorthogonal polynomials) and L is the Lax operator representing multiplication by λ. Theone-point function is therefore⟨w(ℓ)⟩=Z ∞t1dx⟨x|tr(e−ℓD−f −gf −g−D)|x⟩(5.1)We obtain the genus zero approximation by evaluating the matrix element with the WKBapproximation to get⟨w(ℓ)⟩=Z ∞t1dxZ ∞−∞dp tr(e−ℓip−f −gf −g−ip)=Z ∞t1dxZ ∞−∞dpe−ℓ√g2−f 2−p2 + eℓ√g2−f 2−p2= 2Z ∞t1dxZ ∞−∞dp cos(ℓpp2 −H1)= −πZ ∞t1dxh1/21J1(ℓh1/21)(5.2)where h1 ≡−H1 is positive (for large x) and J1 is a Bessel function.Using the flowequations, specialized to the case of genus zero, we therefore obtain the wavefunctions ofthe even flow operators:⟨σ2k−1w(ℓ)⟩= −πℓZ ∞t1dxhk−11∂h1∂x J0(ℓh1/21)= −πℓZ ∞h1dyyk−1J0(ℓy1/2)(5.3)16

The operators for the odd flows have zero wavefunction at tree level.The integral in(5.3) only converges in a distributional sense. We may imagine regularizing the originalmacroscopic loop operator so as to obtain an additional factor of e−ǫ√y in (5.3).

It is theneasy to show that (5.3) and subsequent formulae below possess a finite unambiguous limitas ǫ →0+. In particular, using the identities 2νz Jν(z) = Jν+1(z) + Jν−1(z) andlimǫ→0+Z ∞1dye−ǫ√y(y −1)k−1J0(a√y) = Γ(k)2k(−a)−kJk(a)(5.4)we may prove (as in [33]) the existence of a linear, upper-triangular, analytic change ofoperators σν →ˆσν which have the wavefunctions:ψν(ℓ) ≡⟨ˆσνw(ℓ)⟩= hν/21Jν(ℓh1/21)(5.5)for ν = 2k −1, with k = 1, 2, ....The wavefunctions of the ˆσ operators satisfy the Bessel equation, which, in the one-cut phase has been interpreted as the Wheeler-DeWitt equation for gravity coupled toconformal matter in the minisuperspace approximation.

Assuming a similar interpretationholds in this case we see that the Bessel equation should be interpreted as"ℓ∂∂ℓ2 −µγ4 ℓ2 −8γ2Q28 + ∆−1#ψν = 0(5.6)where γ, Q, µ are parameters in the Liouville theory, following the conventions of [35], and∆is the dimension of the matter field being dressed.It follows that the cosmologicalconstant must be considered to be negative. Accordingly, for a convergent path integralthe worldsheet should have a Minkowskian signature [36].

Moreover, it follows from (5.6)that the central charge and a subset of dimensions of the matter theory must satisfyν2 = 8γ2 (1 −c24+ ∆)γ2 = 16(13 −c −p(1 −c)(25 −c))(5.7)where ν is any odd integer.We can now use (5.5)(5.7) to learn about the continuum theory. Let us assume that thestructure of wavefunctions (5.5) is exactly analogous to that described in [33].

In particular,17

we assume that the multicritical phases of the matrix model defined by k = 1, 2, 3, . .

. arecharacterized by the identification of the wavefunction of the cosmological constant withψν0 for ν0 = 2k −1.

Plugging into (5.7) we find a series of matter theories with centralchargesc = 1 −6ν20ν0 + 1(5.8)Again using (5.7) we discover that the ψν are wavefunctions of operators whose flatspacedimensions are∆ν = ν2 −ν204(ν0 + 1)(5.9)The above critical exponents are consistent with the identification of the matter theoryas a special case of the O(n) model [37]. In particular, the O(n) model for n = −2cosπg isthought [38] to have a hierarchy of multicritical points, for a fixed n, with central chargesc = 1 −6(g −1)2g(5.10)Moreover, the spectrum of the model includes operators with weights given by [39]∆ν = ν2 −(g −1)24g(5.11)for ν ∈ZZ.The operator algebra respects the ZZ2 grading defined by the parity of ν.It is possible to identify the ZZ2 eigenspaces as Ramond and Neveu-Schwarz sectors ofa conformal field theory [40].

In view of these results [38][39][40] we may identify g =ν0 + 1, an even integer, so that n = −2. Only the “Ramond operators” with ν odd havenonvanishing wavefunctions at genus zero, but it is natural to identify all the σν as thegravitationally dressed versions of the O(n = −2) operators with dimensions (5.11).6.

No ConclusionsIn the previous sections we have described in detail the solution of the unitary-matrixmodel, including string equations, flows, Virasoro constraints and macroscopic loop ampli-tudes. Returning to the questions mentioned in the introduction, we can now make somecomparisons with candidate continuum formulations.

We will discuss three candidate the-ories below. Unfortunately, in all three cases the available evidence remains inconclusive.18

The first candidate is 2D supergravity. The multicritical points would correspondto supergravity coupled to the (2, 4k) superminimal matter theories, as already discussedin [31].

Evidence for this is (1) the structure of even and odd scaling operators and theirdimensions match the (2, 4k) series of super-minimal matter; (2) correlators of even (Neveu-Schwarz) operators agree with those of bosonic gravity at tree level [41]; (3) there is nospace-time supersymmetry or Grassmann couplings, as one might expect for supergravitycoupled to low c minimal models with no possibility of GSO projection.Clearly the above is not very compelling evidence. One would like to have, for ex-ample, tree level Ramond correlators or an understanding of the super-WdW equation tomake a more definite statement.

Higher genus correlators could also settle the questionbut problems associated with sums over spin structures make these difficult to calculate.In particular, the calculation of the one-loop partition function from the continuum the-ory could be revealing. The contribution for the even spin structures has been carefullydetermined in [42].

There is a further point which is rather difficult to reconcile with thisinterpretation. The two-cut model in one dimension is the same to all orders in pertur-bation theory as the one-cut model [43] [44].

This would be rather surprising for ˆc = 1matter coupled to supergravity. Nevertheless, the supergravity interpretation has not yetbeen definitely ruled out.The second candidate is dense polymers or the O(n = −2) fermionic model, (thesetwo being intimately related non-unitary matter theories with c = −2 [39]) coupled toLorentzian metric gravity.

The principle evidence for this was described in section fiveand follows from assuming that the macroscopic loop amplitudes satisfy the Wheeler-DeWitt equation of 2D gravity. A further hint that this is the correct interpretation comesfrom examining in detail the perturbation series of the simplest unitary-matrix modelusing the action in [17].

The Feynman perturbation series can be written as a sum oversurfaces together with a sum over self-avoiding loop configurations (the fermion loops) onthose surfaces. Unfortunately, the Boltzman weights for the loops are similar to, but notprecisely equal to those of the O(n) model and it is not clear if the difference is important atcriticality.

Moreover, in this interpretation it is also rather mysterious that the partitionfunctions at every genus should be positive [13].Finally, the ZZ2-odd loops should be19

calculated and compared with the NS sector of the polymer problem.The third candidate is a candidate for the topological phase discussed in section four.The similarity of the structure of the topological phase to the KdV-type systems stronglysuggests that the theory has a topological interpretation. Ghost number counting rules outthe obvious possibility that it is Osp(2|1) topological gravity.

Another idea follows fromrecalling the identification of t2 with the “Yang-Mills θ angle” described below (2.16). Inthe continuum this operator becomes simplyRtrF which could serve as an action for atopological Yang-Mills matter theory, which could then be coupled to topological gravity[45] [46][10].However, there are some troubles with any candidate topological mattertheory.

If we have a theory of topological gravity coupled to topological matter, then, asshown by Dijkgraaf and Witten there must be a “puncture operator” P which satisfies a“puncture equation” expressing an insertion of P in terms of correlation functions of lowerdegree. The natural candidate for the puncture equation would be the L−1 constraint.However, this interpretation would make all operators σn descendants of a single primary,σ1, which seems incompatible with the uniqueness of the known TFT with a single primary.In addition, ghost number conservation required us to introduce a new operator Q withno descendants, which does not fit at all into the topological gravity framework.Although it is possible that a different interpretation of the model exists, which ismore compatible with the original topological gravity framework, we did not find such adescription and suspect that it does not exist.

If this is true, clearly it would be veryinteresting to find the correct world-sheet topological description.AcknowledgementsIt is a pleasure to thank T. Banks, V. Kazakov, D. Kutasov, E. Martinec, Y. Mat-suo, H. Saleur, N. Seiberg, S. Shenker, M. Staudacher, and A. Zamolodchikov for severaluseful discussions relevant to these matters. M.R.D.

especially thanks the Laboratoirede Physique Th´eorique de l’Ecole Normale Sup´erieure, where this work was begun, forits warm hospitality. G.M.

also thanks the Rutgers Department of Physics for hospitalitywhere some of this work was done. ˇC.C.

and G.M. are supported by DOE grant DE-AC02-76ER03075, and G.M.

is supported by a Presidential Young Investigator Award. M.R.D.20

is supported by DOE grant DE-FG05-90ER40559, an NSF Presidential Young InvestigatorAward, and a Sloan Fellowship.21

References[1]J. Ambjørn, B. Durhuus and J. Fr¨ohlich, “Diseases of triangulated random surfacemodels, and possible cures,” Nucl.Phys. B257 [FS14] (1985) 433.[2]F.

David, “Planar diagrams, two-dimensional lattice gravity and surface models,”Nucl.Phys. B257 [FS14] (1985) 45.[3]V.

A. Kazakov, I. K. Kostov and A. A. Migdal, “Critical properties of randomlytriangulated planar random surfaces,” Phys.

Lett. 157B (1985) 295.[4]E.

Br´ezin and V. Kazakov, “Exactly solvable field theories of closed strings,” Phys.Lett. B236 (1990) 144.[5]M.

Douglas and S. Shenker, “Strings in less than one dimension,” Nucl. Phys.

B335(1990) 635.[6]D. Gross and A. Migdal,“Nonperturbative two dimensional quantum gravity,” Phys.Rev.

Lett. 64 (1990)[7]A.M. Polyakov, “Quantum geometry of bosonic strings,” Phys.

Lett. 103B (1981)207.[8]A.M.

Polyakov, “Quantum gravity in two dimensions,” Mod. Phys.

Lett. A2 (1987)893;V. Knizhnik, A. Polyakov and A. Zamolodchikov, “Fractal structure of 2d-quantumgravity,” Mod.

Phys. Lett.

A3 (1988) 819.[9]F. David, “Conformal field theories coupled to 2-D gravity in the conformal gauge,”Mod.

Phys. Lett.

A3 (1988) 1651; J. Distler and H. Kawai, “Conformal field theoryand 2-D quantum gravity or Who’s afraid of Joseph Liouville?,” Nucl. Phys.

B321(1989) 509.[10]E. Witten, “On the structure of the topological phase of two dimensional gravity,”Nucl.

Phys. B340 (1990) 281.[11]J.

Distler, “2D quantum gravity, topological field theory and multicritical matrixmodels,” Nucl. Phys.

B342 (1990) 523.[12]R. Dijkgraaf and E. Witten, “Mean field theory, topological field theory, and multi-matrix models,” Nucl.

Phys. B342(1990)486.[13]V.

Periwal and D. Shevitz, “Unitary-Matrix Models as Exactly Solvable String The-ories,” Phys. Rev.

Lett. 64 (1990) 1326.[14]V.

Periwal and D. Shevitz, “Exactly solvable unitary matrix models: multicriticalpotentials and correlations,” Nucl. Phys.

B344 (1990) 731.[15]M. Douglas, N. Seiberg, and S. Shenker, “Flow and instability in quantum gravity,”Phys.

Lett. B244(1990)381[16]ˇC.

Crnkovi´c and G. Moore, “Multicritical multi-cut matrix models,” Phys. Lett.B257(1991)32222

[17]H. Neuberger, “Scaling regime at the large N phase transition of two dimensionalpure gauge theories,” Nucl. Phys.

B340 (1990) 703.[18]V. A. Kazakov, Mod.

Phys. Lett.

A4 (1989) 2125.[19]M. Staudacher, “The Yang-Lee Edge Singularity on a Dynamical Planar RandomSurface,” Nucl.

Phys. B336(1990)349.[20]C.

Nappi, “Painlev´e II and Odd Polynomials,” IASSNS-HEP-90/61[21]H. Neuberger, “Regularized String and Flow equations,” Rutgers preprint RU-90-50.[22]I.M. Gelfand and L.A. Dikii, “The Resolvent and Hamiltonian Systems,” Funct.

Anal.and Appl. [23]Drinfeld and Sokolov, “Equations of Korteweg-de Vries type and simple Lie algebras,”Sov.

Jour. Math.

(1985)1975[24]N. Seiberg, “Topology in Strong Coupling,” Phys. Rev.

Lett. 53(1984)637.[25]J.A.

Minahan, “Matrix Models with Boundary Terms and the Generalized Painlev´eII Equation,” UVA preprint; “Schwinger-Dyson Equations for Unitary Matrix Modelswith Boundaries,” UVA-HET-91-04[26]G. Moore, “Geometry of the string equations,” Commun. Math.

Phys. 133(1990)261;“Matrix Models of 2D Gravity and Isomonodromic Deformation,” in CommonTrends in Mathematics and Quantum Field Theories Prog.

Theor. Phys.

Suppl. 102(1990)255.[27]S.R.

Wadia, “Dyson-Schwinger equations approach to the large-N limit - model sys-tems and string representation of Yang-Mills theory,” Phys. Rev.

D 24 (1981) 970; A.A. Migdal, Phys. Rep. 102 (1983) 199.[28]F.

David, Mod. Phys.

Lett. A5 (1990) 1019.[29]M.

Fukuma, H. Kawai and R. Nakayama, “Continuum Schwinger-Dyson equationsand universal structures in two-dimensional quantum gravity,” Tokyo preprint UT-562, May 1990.[30]R. Dijkgraaf, E. Verlinde and H. Verlinde, “Loop equations and Virasoro constraintsin non-perturbative 2-D quantum gravity,” Nucl.

Phys. B348 (1991) 435.[31]ˇC.

Crnkovi´c, M. Douglas and G. Moore, “Physical Solutions for Unitary Matrix Mod-els,” Nucl. Phys.

B360(1991)507[32]T. Banks, M. Douglas, N. Seiberg, and S. Shenker, “Microscopic and macroscopicloops in non-perturbative two dimensional gravity,” Phys. Lett.

B241(1990)279[33]G. Moore, N. Seiberg, and M. Staudacher, “From Loops to States in 2D QuantumGravity,” Rutgers/Yale preprint RU-91-11/YCTP-P11-91[34]G. Moore and N. Seiberg, “From Loops to Fields in 2D Quantum Gravity,” Rut-gers/Yale preprint RU-91-29/YCTP-P19-91[35]N. Seiberg, “Notes on Quantum Liouville Theory and Quantum Gravity,” in CommonTrends in Mathematics and Quantum Field Theories Prog. Theor.

Phys. Suppl.

102(1990) 319.23

[36]The negative cosmological constant theory in Minkowski space has been studied inunpublished work of N. Seiberg. See also E. Martinec and S. Shatashvili, “Black HolePhysics and Liouville Theory,” Chicago preprint EFI-91-22.

[37]We are grateful to H. Saleur for explaining to us relevant results in the theory of densepolymers and for many discussions related to the unitary-matrix model.[38]A. Zamolodchikov, unpublished.[39]B.

Duplantier and H. Saleur, Nucl. Phys.

B290(1987)291[40]The fermionic description of dense polymers and O(n = −2) models, in particular thedetailed meaning of R and NS sectors will be described in a forthcomming paper ofH. Saleur.[41]P.

Di Francesco, J. Distler, and D. Kutasov, “Superdiscrete series coupled to 2Dsupergravity,” Princeton preprint PUPT-1189, June 1990.[42]M. Bershadsky and I. Klebanov, “Partition functions and physical states in two-dimensional quantum gravity and supergravity,” HUTP-91/A002,PUPT-1236[43]V. A. Kazakov, “Bosonic strings and string field theories in one dimensional targerspace,” to appear in the proceedings of the Carg´ese workshop on random surfaces andquantum gravity.[44]G.

Moore, “Double-Scaled Field Theory at c=1,” YCTP-P8-91, RU-91-12[45]J.M.F. Labastida, M. Pernici, and E. Witten, “Topological gravity in two dimensions,”Nucl.

Phys. B310 (1988)611.[46]D.

Montano and J. Sonnenschein, “The topology of moduli space and quantum fieldtheory,” Nucl. Phys.

B324 (1989)348.24

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