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Loop Equations and Virasoro Constraints

arXiv:hep-th/9112058v1 19 Dec 1991Loop Equations and Virasoro Constraintsin Matrix ModelsYu. Makeenko∗Institute for Theoretical and Experimental PhysicsSU-117259 Moscow, USSRSeptember, 1991AbstractIn the first part of the talk, I review the applications of loop equations to thematrix models and to 2-dimensional quantum gravity which is defined as their con-tinuum limit.

The results concerning multi-loop correlators for low genera and theVirasoro invariance are discussed.The second part is devoted to the Kontsevich matrix model which is equivalentto 2-dimensional topological gravity. I review the Schwinger–Dyson equations forthe Kontsevich model as well as their explicit solution in genus zero.

The relationbetween the Kontsevich model and the continuum limit of the hermitean one-matrixmodel is discussed.Talk at XXVth International Symposium on the Theory of Elementary Particles, G¨osen,Germany, September 23–26

1IntroductionThe relevance of matrix models to the problem of genus expansion of Feynman graphsgoes back to the original work by ’t Hooft [Hoo74]. An explicit solution for the simplestcase of the hermitean one-matrix model had been first obtained by Br´ezin, Itzykson, Parisiand Zuber [BIPZ78] in genus zero and then extended in [Bes79, IZ80, BIPZ80] to nextfew genera.The modern interest in matrix models is associated with the context of statistical theo-ries on random lattices and discretized random surfaces [Kaz85, Dav85, ADF85, KKM85]as well as with the conformal field theory approach to 2D quantum gravity [Pol87, KPZ88,Dav88, DK89].

A connection between continuum limits of the matrix model and minimalconformal models had been conjectured by Kazakov [Kaz89] on the basis of genus-zeroresults.The whole genus expansion of 2D quantum gravity has been constructed in [BK90,DS90, GM90b] taking the ‘double scaling limit’ of the (hermitean) one-matrix model.Moreover, the specific heat turns out to obey a (non-perturbative) equation of theKorteweg–de Vries type so that a relation between the continuum limit of the matrixmodels and integrable theories emerges [GM90a, BDSS90].While these results were obtained using orthogonal polynomial technique, one moremethod — that of loop (or Schwinger–Dyson) equations — is custom in studies of ma-trix models. Loop equations had been proposed originally for Yang-Mills theory bothon a lattice [Foe79, Egu79, Wei79] and in the continuum [MM79, Pol80] (for a review,see [Mig83]) and then were applied [PR80, Fri81, Wad81] to matrix models.

A modernapproach to loop equation which is based on its interpretation as a Laplace equation onthe loop space can be found in [Mak88, Mak89, HM89]. The recent applications of loopequations to 2D quantum gravity have been initiated by Kazakov [Kaz89].

The role ofloops in 2D quantum gravity is played by boundaries of a 2-dimensional surface.Ref. [Kaz89] deals with genus zero.The whole set of loop equations for 2D quan-tum gravity was first obtained by David [Dav90] taking the ‘double scaling limit’of the corresponding equations for the hermitean matrix model.As was shown in[Mig83, Dav90, AM90], these equations can be unambiguously solved order by orderof genus expansion.

However, this solution is non-perturbatively unstable [Dav90] as itshould be for 2D euclidean quantum gravity.One of the most interesting results which are obtained with the aid of loop equationsis the fact that the partition function of 2D quantum gravity in an external backgroundis the τ-function of KdV hierarchy which is subject to additional Virasoro constraints[FKN91, DVV91]. This proves a conjecture of Douglas [Dou90].

The existence of Virasoroalgebra was extended to the case of the matrix model at finite N in [AJM90, GMM*91,IM91, MM90] while the relation to the continuum Virasoro algebra of [FKN91, DVV91]had been studied by M4 [MMMM91].The second application of loop equations concerns the relation between 2D quantumand topological [LPW88, MS89] gravities. As Witten [Wit90] had conjectured, these twogravities are equivalent.

This conjecture has been verified in genus zero and genus one

for 2D quantum gravity coincide with the recursion relations between correlators in 2Dtopological gravity which were obtained by Verlindes [VV91].From the mathematical point of view, a solution of 2D topological gravity is equivalent[Wit90] to calculations of intersection indices on the moduli space, Mg,s, of curves of genusg with s punctures. Interesting results for this problem have been obtained recently byKontsevich [Kon91] who has represented the partition function of 2D topological gravityas that of a (hermitean) matrix model in an external field.It is worth noting thatthe Kontsevich matrix model is associated with the continuum theory.Therefore, itshould be directly related to the ‘double scaling limit’ of the standard one-matrix model[BK90, DS90, GM90b].The Kontsevich model can be studied by the method of loop equations.

As has beenshown recently by Semenoffand the author [MS91a], the Schwinger-Dyson equations forthe hermitean one-matrix model in an external field, which is equivalent to the Kontsevichmodel, can be represented as a set of Virasoro constraints imposed on the partition func-tion. The large-N solution of these equations, which is known from the work of Kazakovand Kostov [KK89], solves the Kontsevich model in genus zero [MS91a] showing explicitlythe equivalence of 2D topological and quantum gravities to this order.The fact that the partition function of the Kontsevich model obeys the same setof Virasoro constraints [FKN91, DVV91] as the continuum limit of the hermitean one-matrix model has been proven recently by Witten [Wit91] using diagrammatic expansionand by (A.M.)3 [MMM91] using the Schwinger–Dyson equations.

This demonstrates anequivalence of 2D topological and quantum gravities to any order of genus expansion.In the first part of the talk, I review some works [AM90, AJM90, Mak90, MMMM91] onapplications of loop equations both to N×N matrix models at finite N and to 2D quantumgravity which is defined as their continuum limit. The results for multi-loop correlatorsin low genera and the Virasoro invariance both at finite N and in the continuum arediscussed.

The second part is devoted to the Kontsevich matrix model which is equivalentto 2-dimensional topological gravity. The Schwinger–Dyson equations for the Kontsevichmodel as well as their explicit solution in genus zero [MS91a] is reviewed.

Some originalresults concerning the Kontsevich matrix model are reported.

2Matrix Models and 2D Quantum Gravity2.1Loop equation for hermitean matrix modelThe hermitean matrix model is defined by the partition functionZHN =ZDM exp −tr V (M)(2.1)where M is the N × N hermitean matrix. V stands for a generic potentialV (p) =∞Xk=0tkpk.

(2.2)The coupling tk plays here the role of a source for the operator tr Mk while V (p) is asource for the Laplace image of the Wilson loop tr [1/(p −M)] :tr V (M) =Z +i∞+0−i∞+0dp2πiV (p) tr1p −M . (2.3)The correlators ⟨tr Mk1 .

. .

tr Mkm ⟩c , where the average is defined with the samemeasure as in (2.1), can be obtained differentiating log ZHN w.r.t. tk1, .

. .

, tkm while loopcorrelators can be obtained by applying1δδV (p) = −∞Xk=0p−k−1 ∂∂tk(2.4)so that the m-loop correlator readsW H(p1, . .

. , pm) ≡⟨tr1p1 −M .

. .

tr1pm −M ⟩c =δδV (p1) . .

.δδV (pm) log ZHN . (2.5)To calculate the actual values for the given model, say for the matrix model with cubicpotential, one should, after differentiations, put tk’s equal their actual values, say tk = 0for k > 3 in the case of cubic potential.The loop equation can be derived using the invariance of the integral under an (in-finitesimal) shift of M and readsZC1dω2πiV ′(ω)(p −ω)W H(ω) = (W H(p))2 +δδV (p)W H(p).

(2.6)The contour C1 encircles singularities of W H(ω) so that the integration is a projectorpicking up negative powers of p. Eq.

(2.6) is supplemented with the asymptotic conditionpW H(p) →N as p →∞(2.7)which is a consequence of the definition (2.5).Notice that one obtains the single (functional) equation for W H(p) . This is due to thefact that tr V (M) contains a complete set of operators.

Such an approach is advocated in[DVV91, Mak90]. The set of equations for multi-loop correlators (2.5), which is consideredin [Dav90, FKN91], can be obtained from Eq.

(2.6) by m−1 –fold application of δ/δV (pi).The system of the standard Schwinger-Dyson equations for the connected correlators⟨tr Mk1 . .

. tr Mkm ⟩c can be then obtained by expanding in powers of p−11 , .

. .

, p−1m .

2.2Solution in 1/NEq. (2.6) can be solved order by order of the expansion in 1/N2 (the genus expansion).The second term on the r.h.s.

represents the connected correlator of two Wilson loopsand is, in our normalization, of order 1 as N →∞while two other terms are of orderN2 since W H(p) and V (p) are of order N . Therefore one can omit it as N →∞(whichcorresponds to genus zero = the planar limit).The simplest (one-cut) solution of Eq.

(2.6) as N →∞reads [Mig83]2W H(0)(p) = V ′(p) −M(p)q(ω −x)(ω −y)(2.8)whereM(p) =ZC1dω2πiV ′(p) −V ′(ω)(p −ω)q(ω −x)(ω −y)(2.9)is a polynomial of degree K −2 if V (p) is that of degree K. The ends of the cut, x and y,are determined from the asymptotics (2.7):0 =ZC1dω2πiV ′(ω)q(ω −x)(ω −y); 2N =ZC1dω2πiωV ′(ω)q(ω −x)(ω −y)≡W(x, y). (2.10)For the even potential (V (−p) = V (p)), the first of these equations yields y = −x = √zwhich simplifies formulas.

This case is called the reduced hermitean matrix model.The multi-loop correlators in the planar (genus zero) limit can be obtained by varyingaccording to the r.h.s. of Eq.(2.5).

The 2-loop correlator reads [AJM90]W H(0)(p, q) =14(p −q)22pq −(p + q)(x + y) + 2xyq(p −x)(p −y)q(q −x)(q −y)(2.11)while an expression for the 3-loop correlator is given in [AJM90]. Note that the 2-loopcorrelator (2.11) depends on the potential, V , only via x and y but not explicitly.

This isnot the case for all other multi-loop correlators.To calculate 1/N2 correction to (2.8) one needs W H(0)(p, p) which enters the r.h.s. ofEq.(2.6).

Eq. (2.11) yieldsW H(0)(p, p) =(x −y)216(p −x)2(p −y)2.

(2.12)and one can now obtain W H(1)(p) by an iteration of Eq.(2.6). The resultW H(1)(p) =1q(p −x)(p −y)ZC1dω2πi1(ω −p)M(ω)(x −y)216(ω −x)2(ω −y)2(2.13)is unambiguous [Mig83, Dav90] provided that one requires analyticity of W H(1)(p) atzeros of M(p).

This procedure of iterative solution can be pursued order by order of

2.3Continuum loop equationThe continuum limit of the reduced hermitean matrix model is reached as N →∞whileK −1 conditions W(n)(zc) = 0 (W(z) ≡W(−√z, √z)) with n = 1, . .

. , K −1 are imposedon the couplings, tk, in addition to (2.10) at Kth multi-critical point.

2D quantum gravitycorresponds to K = 2. The ‘double scaling limit’ can be obtained if one expands aroundthe critical point:p →√zc +aπ2√zc; z →√zc −a√Λ2√zc,(2.14)so that π and Λ play the role of continuum momentum and cosmological constant, re-spectively.

The dimensionful cutoffa should depend on N such that the string couplingconstant G = N−2a−2K−1 would remain finite as N →∞[BK90, DS90, GM90b].To obtain the continuum limit of loop correlators (2.5), it is convenient to introducethe even partsW even(p1, . .

. , pm) ≡δδV even(p1) .

. .δδV even(pm) log ZreducedN(2.15)where ZreducedNand V even(p) means, respectively, (2.1) and (2.2) with t2k+1 = 0 :V even(p) =∞Xk=0t2kp2k ,δδV even(p) = −∞Xk=0p−2k−1 ∂∂t2k.

(2.16)W even(p1, . .

. , pm) differs from W H(p1, .

. .

, pm) by correlators of products of traces of oddpowers of M. Near the critical point, one getsW H(p1, . .

. , pm) →2m−1W even(p1, .

. .

, pm). (2.17)This formula can be proven analyzing loop equations or by a direct inspection of multi-loop correlators [AJM90].The continuum loop correlators can be obtained by the multiplicative renormalization[Dav90, AM90, FKN91]W H2N(p1, .

. .

, pm) →2ma−mG12m−1Wcont(π1, . .

. , πm) for m ≥3(2.18)while additional subtractions of genus zero terms are needed for m = 1 and m = 2 :W H2N(p) −12V ′(p) →1a√G(2Wcont(π) −J′(π)) ,(2.19)W H2N(p1, p2) →4a−2Wcont(π1, π2) +1a2(√π1 + √π2)2√π1π2.

(2.20)For latter convenience, W H2N(p1, . .

. , pm) on the l.h.s.’s of these formulas is the multi-loopcorrelator for the 2N × 2N reduced hermitean matrix model.J(π) on the r.h.s.

of Eq. (2.19) plays the role of a source for the continuum Wilsonloop:

J(π) =∞Xn=0Tnπn+ 12 ,δδJ(π) = −∞Xn=0π−n−3/2 ∂∂Tn(2.22)with Tk being sources for operators with definite scale dimension. Therefore, Eqs.

(2.18),(2.20) can be derived from Eq. (2.19) by varying w.r.t.

J(π).The continuum loop equation can be obtained from (2.6) substituting (2.19), (2.20):ZC1dΩ2πiJ′(Ω)(π −Ω)Wcont(Ω) = (Wcont(π))2 + GδWcont(π)δJ(π)+G16π2 + T 2016π. (2.23)This equation describes what is called the ‘general massive model’.

It corresponds toarbitrary J(π) and interpolates between different multi-critical points. For Kth multi-critical point, one puts, after varying w.r.t.

J(π), Tn = 0 except for n = 0 and n = K.2.4Genus expansionEq. (2.23) can be solved order by order in G (genus expansion) analogously to that ofSect.2.2 .If J(π) is a polynomial (Tn = 0 for n > K), K −1 lower coefficients ofthe asymptotic expansion of Wcont(π) are not fixed while solving in 1/π and should bedetermined by requiring the one-cut analytic structure in π.

The continuum analog of(2.8) reads2W (0)cont(π) =ZC1dΩ2πiJ′(Ω)(π −Ω)√π + u√Ω+ u(2.24)where u versus {T} is determined from the asymptotic behavior.This asymptotic relation can be obtained comparing 1/π terms in Eq.(2.23). Denotingthe derivative w.r.t.

x = −T0/4 by D , it is convenient to represent this relation as2x =ZC1dΩ2πiJ′(Ω)DWcont(Ω). (2.25)For the ansatz (2.24), one getsDW (0)cont(π) =1√π + u −1√π.(2.26)Eq.

(2.26) can be extended to any genera using the representation [GM90a, BDSS90]DWcont(π) = 2⟨x|π + u(x) −14GD2−1|x⟩−1√π = 2∞Xn=1Rn[u] ≡2R(π)(2.27)where the diagonal resolvent of Sturm-Liouville operator is expressed via the Gelfand-Dikiˇıdifferential polynomials [GD75]Rn[u] = 2−n−1G8 D2 −u −D −D−1uDn· 1. (2.28)Substituting the r.h.s.of Eq.

(2.27) into Eq. (2.25), one obtains the string equation[GM90a, BDSS90]∞1

The fact that the ansatz (2.27) does satisfy Eq. (2.23) is shown in [DVV91].

To thisaid, one applies the operator∆= −G16D4 + (u + π)D2 + 12(Du)D,(2.30)which annihilates Wcont(π) given by Eq. (2.27), to Eq.(2.23).

The result vanishes providedu satisfies Eq. (2.29) and−2δδJ(π)u = D2Wcont(π)(2.31)whose expansion in 1/π reproduces the KdV hierarchy ∂u/∂Tn = DRn+1[u] .Comparing (2.21) and (2.31), one concludes thatZcont = exp−2GZ x0 dy(x −y)u(y) + Φ(T1, T2, .

. .

)(2.32)where the perturbative solution of Eq. (2.29), which satisfies u(0) = 0, is chosen andan integration ‘constant’ Φ(T1, T2, .

. .) depends on all T’s except for T0.

At the givenmulti-critical point, this Φ is unessential so that the r.h.s. of Eq.

(2.32) coincides withthe continuum partition function which was obtained in [BK90, DS90, GM90b] using themethod of orthogonal polynomials.The general procedure of solving Eq. (2.23) order by order in G can be now formulatedas follows.

One should first solve Eq. (2.29) to find u versus x and {Tn} (this is perturba-tively unambiguous).

Then Eq. (2.27) determines DWcont(π) while Wcont(π) itself can becan be obtained by integratingWcont(π) = 2Z xdxR(π)(2.33)where the integration ‘constant’ can be expressed via Φ entering Eq.(2.32).

This constantbecomes unessential for Kth multi-critical point when Tn = 0 except for n = 0 and n = Kso that u depends only on the cosmological constant Λ:ΛK/2 =x4K+1(K! )2(2K + 1)(2K)!TK.

(2.34)2.5Multi-loop correlators in 2D quantum gravityA formula which is similar to Eq. (2.33) exists for the multi-loop correlators:Wcont(π1, .

. .

, πm) = 2Z xdxδδJ(π1) · · ·δδJ(πm−1)R(πm),(2.35)where the integration ‘constant’ depends again on T1, T2, . .

. .Since R(π) depends on T’s only implicitly via u, the following chain rule can be usedfor calculationsδ2Z +∞d DR( )δ(2 36)

with δ/δu(x) being the standard variational derivative. The expansion of Eq.

(2.36) in 1/πreproduces the standard (commuting) KdV flows. An advantage of Eq.

(2.36) is that itallows to obtain results without solving the string equation (2.29). Therefore, to calculatethe multi-loop correlator for a given multi-critical point, one can substitutes the solutionof Eq.

(2.29) only for this multi-critical point and should not solve it for arbitrary T’s.An alternative way of calculating correlators in 2D quantum gravity is to takethe continuum limit (2.14) of formulas of Sect.2.2 with the aid of the renormalization(2.18)–(2.20). For the case of pure gravity (the K = 2 critical point), the explicit form ofWcont(π) is known for genus zero [Dav90] and genus one [AM90]:W (0+1)cont (π) = −54T2(π −12√Λ)qπ +√Λ−−G90T2(π + 52√Λ)Λ(π +√Λ)5/2(2.37)where [.

. .

]−means subtraction of the term which diverges as π →∞as well as that oforder O(π−12).The multi-loop correlators are known [AM90, AJM90] for genus zero:W (0)cont(π1, π2) =qπ1 +√Λ −qπ2 +√Λ24(π1 −π2)2qπ1 +√Λqπ2 +√Λ−14(√π1 + √π2)2√π1π2,(2.38)W (0)cont(π1, . .

. , πm) ∝∂m−3∂Λm−3 1√ΛmYi=11(πi +√Λ)3/2!for m ≥3.

(2.39)As is mentioned above, the additional subtraction is needed only for m ≤2.Analogous formulas can be obtained for higher multi-critical points.For K = 3,Eq. (2.38) remain unchanged while the analog of Eq.

(2.37) reads [AM90]W (0+1)cont (π) = −74T3(π2 −12π√Λ + 38Λ)qπ +√Λ−−4G315T3(π + 74√Λ)Λ3/2(π +√Λ)5/2. (2.40)The above expressions for multi-loop correlators agree with those obtained recently[GL91, MMS91, MSS91, MS91b] for the Liouville theory.2.6Complex matrix modelThe complex matrix model is defined by the partition functionZCN =ZDMDM† exp −tr V even(MM†)(2.41)where the integral goes over N × N complex matrices and V even is given by (2.16).

As isseen from this formula, the complex matrix model resembles the reduced hermitean one.However, it differs by combinatorics as well as by the fact that averages of traces of oddpowers of M do not appear. The model (2.41) has been studied in [Mor91, AMP91] using

The variables t2k in (2.41) play the role of sources for the operators tr (MM†)k whileV even is a source for the Wilson loop tr [p/(p2 −MM†)] (comp. (2.3)):tr V even(MM†) =Z +i∞+0−i∞+0dp2πiV even(p) trp(p2 −MM†) .

(2.42)The analog of (2.5) readsW C(p1, . .

. , pm) =δδV even(p1) .

. .δδV even(pm) log ZCN(2.43)which leads to the following loop equation for the complex matrix model [Mak90]ZC1dω4πiV ′even(ω)(p −ω) W C(ω) = (W C(p))2 +δδV even(p)W C(p).

(2.44)This equation should be supplemented with the asymptotic condition same as (2.7).Comparison of Eq. (2.44) and loop equation for the reduced hermitean model yieldsfor genus zero:W CN (p1, .

. .

, pm) = 12W even2N (p1, . .

. , pm),(2.45)where the correlators W even for the reduced hermitean model are defined is Sect.2.3 .Notice that the correlator for N × N complex matrix model enters the l.h.s.

while thatfor 2N × 2N reduced hermitean one enters the r.h.s.. This guarantees the asymptoticcondition (2.7).

The coefficient 1/2 in Eq. (2.45) leads to the following relation betweenthe partition functions for genus zero:ZCN ∝qZreduced2N.

(2.46)Due to the relation (2.45), 4W C(p) to the leading order in 1/N is given by the r.h.s.of Eq. (2.8) with V replaced by V even and y = −x = √z .

The multi-loop correlator toleading order in 1/N can be then calculated by varying according to (2.43). The analogof (2.11), (2.12) readsW C(0)(p, q) =14(p2 −q2)2(2p2q2 −zp2 −zq2√p2 −z√q2 −z −2pq), W C(0)(p, p) =z216p2(p2 −z)2 .

(2.47)Moreover, an explicit expression for arbitrary multi-loop correlators exists [AJM90] forthe complex matrix model even far from the critical point (comp. (2.39)):W C(0)(p1, .

. .

, pm) = 1W′(z)∂∂z!m−312zW′(z)mYi=1z2(p2i −z)3/2for m ≥3. (2.48)As is proven in [Mor91, AMP91, Mak90, AJM90], the complex and hermitean matrixmodels belong to the same universality class in the ‘double scaling limit’.

This implies,in particular, that the continuum limit of all multi-loop correlators coincide with thoseof Sects.2.4,2.5 . They do not coincide, generally speaking, for higher genera far fromthe critical points.

Eqs. (2.45), (2.46) remain valid, however, to arbitrary order of genusexpansion near the critical points.

Using (2.17), one concludes that W CN (p1, . .

. , pm) hasH

2.7Loop equations as Virasoro constraintsThe loop equation (2.6) can be represented as a set of Virasoro constraints imposed onthe partition function. Eq.

(2.6) can be rewritten, using the definitions (2.2) and (2.4), as1ZHN∞Xn=−11pn+2LHn ZHN = 0(2.49)where the operators2LHn =∞Xk=0ktk∂∂tk+n+X0≤k≤n∂2∂tk∂tn−k(2.50)satisfy [AJM90, GMM*91] Virasoro algebra[LHn , LHm] = (n −m)LHn+m. (2.51)Therefore, Eq.

(2.6) is represented as the Virasoro constraintsLHn ZHN = 0 for n ≥−1. (2.52)These constraints manifest the invariance of the integral on the r.h.s.

of (2.1) under theshift of integration variable δM = ǫ · Mn+1 with n ≥−1 [AJM90, MM90].It is impossible, however, to make in (2.50), (2.52) the reduction to even times. Foreven n, this reduction can be done for the first term on the r.h.s.

of (2.50) but not forthe second one. Therefore, there exist no Virasoro constraints imposed on ZreducedNatfinite N.A set of Virasoro operators built up from the even times, t2k, arises for the complexmatrix model.

The loop equation (2.44) can be represented as Virasoro constraintsLCn ZCN = 0 for n ≥0;(2.53)LCn =Xk=0kt2k∂∂t2(k+n)+X0≤k≤n∂2∂t2k∂t2(n−k). (2.54)The Virasoro invariance is now related [AJM90, MM90] to the change δM = ǫ(MM†)nMwith n ≥0.Analogously, the continuum loop equation (2.23) can be represented as Virasoro con-straints which are imposed on Zcont defined by (2.21).Using (2.22), one proves thatEq.

(2.23) is equivalent to the continuum Virasoro constraints [FKN91, DVV91]LcontnZcont = 0 for n ≥−1;(2.55)Lcontn=∞Xk=0(k + 1/2)Tk∂∂Tk+n+ GX0≤k≤n−1∂2∂Tk∂Tn−k−1+ δ0,n16 + δ−1,nT 2016G . (2.56)The relation between the continuum Virasoro constraints (2.55), (2.56) and those atfinite N can be studied [MMMM91] without referring to loop equations.

IntroducingTn =X √Gan+1/2kt2kΓ(k + 1/2)(kn)!Γ(n + 3/2)−4N√Gaδn,0,(2.57)

or, vice versa,kt2k −2Nδk,0 =Xn≥k(−)k−n a−n−1/2TnΓ(n + 3/2)√G(n −k)!Γ(k + 1/2),(2.58)and rescaling the partition functionZCN →˜ZCN = e−12PAmn ˜Tm ˜TnZCN,(2.59)Amn = (−)n+mΓ(n + 3/2)Γ(m + 3/2)2π(n + m + 1)(n + m + 2)n!m!G−1a−m−n−1,(2.60)one gets from (2.52), (2.53)˜LCn ˜ZCN = 0 for n = −1;˜LCn ˜ZCN = (−)n116an ˜ZCN for n ≥0,(2.61)˜LCn =∞Xk=0(k + 1/2)Tk∂∂˜Tk+n+ GX0≤k≤n−1∂2∂˜Tk∂˜Tn−k−1+ δn,016 + δn,−1T 2016G . (2.62)The variables { ˜T} are related to {T} byTn = ˜Tn + ann + 1/2˜Tn−1 −4N√Gaδn,0(2.63)so that the difference disappears as a →0.Eqs.

(2.57), (2.58) are the standard transition [Kaz89] to operators with definite scaledimensions in the continuum. The rescaling (2.59) makes ˜ZCN finite as a →0 so that˜ZCN →Zcont.

While the operators ˜LCn tend to Lcontndefined by (2.56) as a →0, thea →0 limit is not permutable with differentiating ˜ZCN w.r.t. Tn .

This is why ˜LCn ˜ZCN arenonvanishing (even singular for n ≥1) as a →0. These terms do not appear [MMMM91],however, when ˜LCn ’s act one−12PAmn ˜Tm ˜TnqZreduced2N→Zcont(2.64)(comp.

(2.46), (2.59)). Thus, the l.h.s.

of Eq. (2.64) defines the proper continuum partitionfunction which is annihilated by the continuum Virasoro operators (2.56).

3Kontsevich Model and 2D Topological Gravity3.12D topological gravity as Kontsevich modelThe starting point in demonstrating an equivalence between 2D topological gravity andthe Kontsevich model is the Witten’s geometric formulation [Wit90] of 2D topologi-cal gravity. In this formulation, one calculates the correlation functions of s operatorsσn1(x1), .

. .

, σns(xs) with definite (non-negative integer) scale dimensions ni, living on a2-dimensional Riemann surface Σ of genus g. Those are expressed [Wit90] via the inter-section indices⟨σn1(x1) · · ·σns(xs)⟩g =Z Yic1(L(i))niN (ni)(3.1)where c1(L(i)) is the first Chern class of the line bundle (which is the cotangent spaceto a curve at xi) over the moduli space, Mg,s, of curves of genus g with s puncturesand the integral goes over¯Mg,s. The normalization factor Qi N (ni), which is relatedto the normalization of the operators σ, is to be fixed below.

The r.h.s. of Eq.

(3.1) isnon-vanishing only ifXini = 3g −3 + s,(3.2)i.e.the (complex) dimension of Mg,s.Notice the crucial property of correlators intopological theories — those depend only on the dimensions n1, . .

. , ns and genus g butnot on the metric on Σ and, therefore, not on positions of the punctures x1, .

. .

, xs.It is convenient to introduce the set of couplings tn which play the role of sources forthe operators σn. The genus g contribution to the free energy then readsFg[t] =*exp XntnZσn!+g(3.3)while the correlator on the l.h.s.

of Eq. (3.1) can be obtained by differentiating Fg[t] w.r.t.tn1, .

. .

, tns since the correlators do not depend on x1, . .

. , xs.

The total free energy canbe obtained from the genus expansionF[t; λ] =Xgλ2g−2Fg[t](3.4)with λ2 being the string coupling constant3. Note, that due to the relation (3.2), theλ-dependence of F can be absorbed by the rescaling of tn:F[.

. .

, tn, . .

. ; λ] = F[.

. .

, λ23(n−1)tn, . .

. ; 1].

(3.5)The Kontsevich approach [Kon91] to evaluate F[t; λ], given by Eq. (3.4), is based ona combinatorial calculation of the intersection indices on Mg,s.

Let us represent Eq. (3.3)asFg[t] =Xs≥0Xn1,...,ns1s!F (n1,...,ns)g,s(3.6)3We use in this part of the talk the Witten’s normalization [Wit90] of 2D topological gravity (N(n) =

whereF (n1,...,ns)g,s= ⟨σn1(x1) · · · σns(xs)⟩g tn1 · · · tns. (3.7)The last quantity can be interpreted as a contribution from a band graph (or a fat graphin Penner’s terminology [Pen88]) of genus g with s loops and three bands linked at eachvertex.

These graphs were introduced in quantum field theory by ’t Hooft [Hoo74]. Theoriginal Riemann surface with s punctures can be obtained from this band graph byshrinking the boundaries of bands (forming loops) into the punctures.As is proven by Kontsevich [Kon91],Xn1,...,ns⟨σn1(x1) · · ·σns(xs)⟩gYi(2ni −1)!

!N (ni)tr Λ−2ni−1 =Xgraphsg,s2−#(vert.)#(aut. )Ylinksi,j2Λi + Λj(3.8)where Λi are eigenvalues of a N × N hermitean matrix Λ and the sum goes over theconnected band graphs with #(vert.) vertices.

The product goes over the links of thegraph. Each link carries two indices i, j which are continuous along the loops while eachof s traces on the l.h.s.

corresponds to the summation over the index along one of s loops.The combinatorial factor #(aut.) in the denominator is due to a symmetry of the graph.Substituting Eq.

(3.8) into Eqs. (3.7),(3.6), identifyingtn = λ(2n −1)!

!N (n)tr (Λ−2n−1)(3.9)and making use of Eq. (3.5), one represents the r.h.s.of Eq.

(3.4) in the form of thelogarithm of the partition functionZKonts[Λ; λ] ≡R DXe tr (√λ6 X3−12ΛX2)R DXe−12 tr ΛX2(3.10)where the integral goes over the hermitean N × N matrix X. The original normalizationof [Kon91] corresponds to λ = −1.There is, however, a subtlety in the identification (3.9).The point is that, for anN × N matrix Λ−1, tr (Λ−k) are independent only for 1 ≤k ≤N while, say tr (Λ−N−1)is reducible.

All tr (Λ−2n−1) become independent, as it should be for the sources in 2Dtopological gravity, as N →∞. Thereforelog ZKonts[Λ; λ] →F[t; λ].

(3.11)only as N →∞. The equality (3.11) is valid in a sense of an asymptotic expansion atlarge Λ with each term being finite providing Λ is positively defined.Let us explain the Kontsevich results from the viewpoint of the standard analysisof the matrix model (3.10).

ZKonts[Λ; λ] admits the perturbative expansion in λ thatstarts from the term O(λ). This term corresponds to the contribution of three punctureoperators in genus zero [Wit90]:t3

where λ2 in the denominator emerges because of Eq.(3.9). The contribution of a genericgraph with #(vert.) vertices, #(link) links and s loops is proportional toλ#(vert.

)2Ns = (λN)sλ2g−2(3.13)in an agreement with Eq. (3.8).Notice that while N →∞, all terms of the perturbative expansion in λ contributeto F[t; λ] in contrast to the standard large-N expansion by ’t Hooft [Hoo74] when anexpansion in1N2 emerges so that N = ∞corresponds to planar graphs only.

The ’t Hooftcase can be reproduced if λ ∼N−1. Then F ∼N2 whileWKonts[Λ; 1N ] ≡1N2 log ZKonts[Λ; 1N ] →1N2F[t; 1N ](3.14)is finite.

Therefore, WKonts[Λ; 0] = F0[t] can be obtained in the ’t Hooft planar limit. Thisfact has been utilized in [MS91a] to solve the Kontsevich model in genus zero.3.2The Schwinger-Dyson equationsThe Kontsevich model can be studied using the custom methods of solving matrix models.Since Λ in Eq.

(3.10) is a matrix, the standard orthogonal polynomial technique can notbe applied. For this reason, the method of Schwinger-Dyson equations has been appliedto this problem [MS91a, MMM91].To derive the Schwinger–Dyson equations, it is convenient to make a linear shift of theintegration variable in the numerator on the r.h.s.

of Eq.(3.10). Modulo an unessentialconstant, one getsZKonts[Λ; λ] =YiqΛiYi>j(Λi + Λj) e−Λ33λ Z" Λ2(2λ)23#(3.15)whereZ[M] =ZDX e tr (−X33 +MX)(3.16)and the Gaussian integral in the denominator has been calculated.Z[M] which is defined by Eq.

(3.16) is the standard partition function of the hermiteanone-matrix model in an N ×N matrix external field M. This external field problem, whichis analogous to the corresponding problem [BN81, BG80, BRT81] for the unitary matrixmodel, has been studied recently in [GN91, MS91a].While the representation (3.10)is convenient for constructing the perturbation theory expansion, the partition functionZ[M] is convenient for deriving Schwinger-Dyson equations.The partition function (3.16) depends on N invariants, mi, — the eigenvalues of M.Let us perform the integral over angular variables in the standard way [IZ80, Meh81]to express Z[M] as the integral over xi — the eigenvalues of X. Modulo an irrelevantmultiplicative constant, the result readsN∆[ ]N3

where ∆[m] = Qi

. , N and n ≥−1 in full analogy to the matrix model withoutexternal field [AJM90, IM91, MM90].

Noticing that xi in the integrand can be replacedby∂∂mi when applied to ∆[m]Z[m], the set of Schwinger–Dyson equations can be writtenin the form [MS91a]Ln∆[m]Z[m] = 0forn ≥−1(3.18)withLn =Xi− ∂∂mi!n+3+ ∂∂mi!n+1mi + 12nXk=0Xj̸=i ∂∂mi!k ∂∂mj!n−k. (3.19)It is easy to verify by a direct calculation that these operators obey Virasoro algebra[Ln, Lm] = (n −m)Ln+m.

(3.20)The Virasoro generators (3.19) annihilate the totally antisymmetric function∆[m]Z[m].One can easily construct the generators Ln which annihilate Z[m] itself.Let us introduce for this purpose the ‘long’ derivatives∇i ≡∆−1[m] ∂∂mi∆[m] =∂∂mi+Xj̸=i1mi −mj(3.21)which commute one with each other. The Virasoro constraints (3.18), (3.19) now takethe form [MS91a]LnZ[m] = 0forn ≥−1(3.22)andLn =Xi−(∇i)n+3 + (∇i)n+1mi + 12nXk=0Xj̸=i(∇i)k(∇j)n−k.

(3.23)Due to the commutativity of ∇’s, the order in the last term in unessential. As followsfrom the definition (3.21), the generators (3.23) obey the Virasoro algebra commutationrelations, same as (3.20).The Virasoro constraints (3.22), (3.23) turn out to be equivalent to the followingequation(∂i)2 +Xj̸=i1mi −mj(∂i −∂j) −miNZ[m] = 0(3.24)which is called in [MS91a] the ‘master equation’.As is shown in [GN91, MMM91],Eq.

(3.24) results from shifting X in Eq. (3.16) by an arbitrary (hermitean) matrix whileEqs.

(3.18), (3.19) (or (3.22), (3.23)) result from the shift X →X + ǫnXn+1.3.3The genus-zero solution

Br´ezin and Gross [BG80] with the aid of the Riemann–Hilbert method. The correspondingsolution had been first found by Kazakov and Kostov [KK89] and is discussed in [GN91,MS91a].Substituting this solution into Eq.

(3.15) which expresses the partition function of theKontsevich model via that for the hermitean matrix in an external field, one gets in genuszeroF0=1NXi(13(Λ2i −2u)32 + uqΛ2i −2u + u36 −Λ3i3+12NXjlog (Λi + Λj) −logqΛ2i −2u +qΛ2j −2u(3.25)where u[Λ] is determined byu = 1NXi1q(Λ2i −2u). (3.26)The solution (3.25) is similar to the strong coupling solution of [BG80] while Eq.

(3.26)has emerged to guarantee correct analytic properties. It is important that the r.h.s.

ofEq. (3.25) is stationary w.r.t.

u due to Eq.(3.26). For a constant field Λi = (6g)−23, thissolution recovers the results of Br´ezin et al.

[BIPZ78] for the case of a cubic interaction.Eq. (3.26) can be rewritten in the form of the string equation of a ‘general massivemodel’ [BDSS90] in genus zero.

To this aim, let us expand the r.h.s. of Eq.

(3.26) in uand substitutetn = 1N(2n −1)!!n!XiΛ−2n−1i. (3.27)This equation is nothing but Eq.

(3.9) with N (n) = n! which fixes the normalization[Wit90] of 2D topological gravity and λ =1N as is prescribed by Eq.(3.14).

We rewriteEq. (3.26) finally asu =∞Xn=0tnun.

(3.28)The precise form of the genus-zero string equation can be obtained by the well-knownshift [DW90]: t1 →t1 + 1.3.4Relation to 2D topological and quantum gravitiesIt is instructive to compare the solution (3.25) of the Kontsevich model with knownresults for the partition functions of 2D topological and quantum gravities in genus zero.To obtain the perturbative expansion of F0[t], one solves Eq. (3.28) by iterations in u:u = t0 + t0t1 + t0t21 + t20t2 + .

. .

(3.29)with tn ∼t2n+10, and substitutes the result into the r.h.s. of Eq.

(3.25) which is expanded

While the complicated structure of the perturbative expansion of F0[t] represents thevariety of planar band graphs (taken with appropriate combinatorial coefficients), greatsimplifications occur for derivatives of F0[t]. Let us define D byD = 2Xi∂∂(Λ2i ).

(3.30)Then, by a direct differentiation of Eq. (3.25), one getsDF0 = t202 + u22 −∞Xk=0tkuk+1(k + 1).

(3.31)This expression is again stationary w.r.t. u due to Eq.

(3.28) so that one more applicationof D yieldsD2F0 = u −t0(1 + t1). (3.32)To compare Eq.

(3.25) with the known solution of 2D topological gravity in genus zero,let us notice that D defined by (3.30) can be rewritten using Eq. (3.27) asD = −∞Xn=1ntn∂∂tn−1.

(3.33)This is exactly the operator entering the puncture equation [DW90] which reads in genuszero:∂F0∂t0= t202 −DF0. (3.34)Therefore, one gets from Eq.

(3.31)∂F0∂t0=∞Xk=0tkuk+1(k + 1) −u22(3.35)which is a true formula that gives in particular∂2F0∂t20= u. (3.36)Since t0 is the cosmological constant, one sees from this formula u to be the string sus-ceptibility.Using Eq.

(3.36), one can immediately calculate the critical index γstring.For Kthmulti-critical point, when one puts all tn = 0 except for n = 0 and n = K, Eq. (3.28)yields u ∝x1K , γstring = −1K in full analogy to [Kaz89].

Notice, however, that the solution(3.25) is associated with the continuum interpolating model while in the standard caseone ‘interpolates’ by a matrix model whose couplings should be turned to critical valuesin order to reach the continuum limit.Finally, let us mention that the genus-zero solution (3.25) can be rewritten exactly inthe form of that for the hermitean one-matrix model in the continuum limit. Let us firstnote that Eq.

(3.35) can be viewed as an integrated version of Eq. (3.36):

where u(t0) is a solution of Eq. (3.28) which is considered as a function of t0 at fixedvalues of tn for n ≥1.

The integration constant is fixed by the fact that u(0) = 0 which isnothing but the condition that chooses the perturbative solution of the string equation.One more integration of Eq. (3.37) yieldsF0 =Z t00dx(t0 −x)u(x) + Φ(t1, t2, .

. .

). (3.38)which coincides with the representation (2.32) of the free energy for 2D quantum gravity.As has been proven recently [Wit91, MMM91], the Kontsevich model obeys the sameset of Virasoro constraints (2.55), (2.56) as 2D quantum gravity.

This demonstrates anequivalence of 2D topological and quantum gravities to arbitrary genus.4Concluding remarksLoop equations turned out to be a useful tool in studies of matrix models as well as of theircontinuum limit associated with 2D quantum gravity with matter. The point is that loopequations are literally the Virasoro constraint imposed on the partition function.

In thecontinuum limit, this Virasoro symmetry represents the underlying conformal invariance.The appearance of new symmetries of loop equations (as well as the very idea of the‘double scaling limit’ [BK90, DS90, GM90b]) is very interesting from the viewpoint ofmulti-dimensional loop equations (see [Mig83]). A step along this line has been done in[FKN91, DVV91, Goe91] where the W-algebras were associated with the continuum limitof multi-matrix models.

It would be interesting to find an analog of this symmetry formulti-matrix models at finite N.The fact that the Kontsevich matrix model is a solution of the continuum Virasoroconstraints (and, therefore, of the continuum loop equations) throws light on the originof Virasoro constraints. It is non-trivial that this matrix model appears as a solution ofthe continuum loop equation.

This seems to be analogous to the known property [Mig83]of multi-dimensional loop equations which possess solutions differing from the originalYang–Mills path integral.

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