---

Strings with Discrete Target Space

arXiv:hep-th/9112059v2 21 Jul 2003hep-th/9112059Strings with Discrete Target SpaceIvan K. Kostov ∗Service de Physique Th´eorique † de Saclay, CEN-Saclay, F-91191 Gif-sur-Yvette , FranceWe investigate the field theory of strings having as a target space an arbitrary discrete one-dimensional manifold. The existence of the continuum limit is guaranteed if the target space isa Dynkin diagram of a simply laced Lie algebra or its affine extension.

In this case the theorycan be mapped onto the theory of strings embedded in the infinite discrete line ZZ which is thetarget space of the SOS model. On the regular lattice this mapping is known as Coulomb gaspicture.

Introducing a quantum string field Ψx(ℓ) depending on the position x and the length ℓof the closed string, we give a formal definition of the string field theory in terms of a functionalintegral. The classical string background is found as a solution of the saddle-point equation whichis equivalent to the loop equation we have previously considered [1].

The continuum limit existsin the vicinity of the singular points of this equation. We show that for given target space thereare many ways to achieve the continuum limit; they are related to the multicritical points of theensemble of surfaces without embedding.

Once the classical background is known, the amplitudesinvolving propagation of strings can be evaluated by perturbative expansion around the saddlepoint of the functional integral. For example, the partition function of the noninteracting closedstring (toroidal world sheet) is the contribution of the gaussian fluctuations of the string field.

Thevertices in the corresponding Feynman diagram technique are constructed as the loop amplitudesin a random matrix model with suitably chosen potential.Nuclear Physics B 376 (1992) 539-598SPhT/91-1429/91∗on leave of absence from the Institute for Nuclear Research and Nuclear Energy, BoulevardTrakia 72, BG-1784 Sofia,Bulgaria†Laboratoire de la Direction des Sciences de la Mati`ere du Commissariat `a l’Energie Atomique.

1. IntroductionThe recent progress in the theory of non-critical strings has been prepared by theconjecture that the functional integral of 2d gravity can be discretized as a sum overplanar graphs [2] .

Such a discretization allowed to apply powerful methods of calculationborrowed from the theory of random matrices [3].Some of the corresponding matrixmodels can be solved exactly after being reduced to a problem of non-interacting fermionsin a common potential. This fermionic representation proved to be very helpful in practicalcalculations but its connection with the original model seems quite formal.

It is not clear,for example, whether the nonperturbative phenomena in the fermion problem can be givenan interpretation in terms of strings.It is therefore desirable to have a formalism in which the string excitations appearexplicitely.An example of such a formalism applied to the string propagating in theone-dimensional space IR is the collective field approach of Das and Jevicki [4].In this paper we develop a similar formalism for string theories with discrete targetspaces. Bosonic string with discrete target spaces are interesting mainly because of theirinterpretation as theories of two-dimensional gravity coupled to matter.

The (discrete)degrees of freedom of the matter field are the points of the target space. Because of theenormous symmetry of the problem, it is much easier to investigate the critical behaviour ofstatistical systems on a fluctuating surface than on the plane.

The exact results obtainedfor a fluctuating surface can be translated according to [5], [6] to the case of a frozengeometry of the world sheet.The string models we are investigating are a generalization of the ADE strings con-structed in ref. [1].

They are closely related to the SOS and RSOS solvable statistical mod-els on a regular planar lattice [7]. The target spaces of these models are one-dimensionaldiscrete manifolds representing Dynkin diagrams of simply laced Lie algebras or their affineextensions.

All such target spaces can be mapped onto a master target space which is theinfinite discretized line ZZ.Since in a one-dimensional embedding space there is no room for transverse excitations,the states of the closed string is completely determined by the two global modes - the centerof masses x and its length L. The string field Ψ(L) is an operator creating on the worldsheet a boundary of length L and position x in the target space. The path integral forthe correlation functions of the string field involve embeddings of the world sheet suchthat each connected component of its boundary is mapped into a single point of the targetspace.The string models related to the SOS model are remarkable with the possibility toreduce the string path integral to a simpler one involving only the global modes x andL, even before going to the continuum limit.

As a result one obtains an effective two-dimensional QFT for the field Ψx(L).The reduction of the string path integral stems from the loop, or polygon represen-tation of the SOS-type statistical systems [7] which has been generalized to the case ofirregular lattices in [1]. A map of the world sheet of the string in the target space can berepresented by a collection of nonintersecting loops (domain walls) separating the domainsof constant coordinate x.

We assume that the discontinuity across a domain wall is ±1.Each loop configuration is characterized by its topology, the lengths of the loops, and the1

coordinates of the connected domains. It can be imagined as a “stroboscopic picture” of athe evolution of one or several closed strings, the time direction on the world sheet beingat all points orthogonal to the domain walls.

The evolution of the string is decomposedinto elementary processes represented by the domain walls and connected domains. Thedomain walls describe elementary steps in the X-space without changing the world-sheetgeometry.

On the other hand, the connected domains describe topology-changing processeswithout propagation in the embedding space. This nice factorization renders the modelexactly solvable.A loop configuration is represented by a graph with lines corresponding to domainwalls and vertices corresponding to the connected domains on the world sheet.Thereare vertices of all orders including tadpoles describing processes of creation or decay of aclosed-string state.

Each vertex is characterized also by the number H of handles of theconnected domain on the world sheet. Vertices with H > 0 describe processes involvingcreation and annihilation of H virtual closed strings at the same point x of the embeddingspace.The string path integral can be written as a sum over all loop configurations, andthe entropy of a loop configuration factorizes to a product of weights associated withthe domains.

The weight of each domain is proportional to the entropy of a nonembeddedrandom surface with the corresponding topology. The latter can be calculated, for example,by using the matrix model formalism.

In this way the the graphs representing the loopconfigurations can be interpreted as Feynman diagrams for an ordinary field theory in thespace of global modes (x, L) and the string path integral is reduced to a functional integralfor the field Ψx(L).The first (and in a sense the most difficult) problem to solve is to find the classicalstring background Ψclx (L) which is the solution of the classical equation of motion inthe effective field theory.Then the propagation of the noninteracting closed string isdetermined by the gaussian fluctuations of the string field around the classical solutionand the vertices are given by the higher order terms. The accomplishment of this programwill be the subject of this paper.We are going to present in details the results announced in three short publications[8], [9], and [10], as well as a few new results as the microscopic construction of the diluteADE models and a description of their multicritical points [11].The paper is organized as follows.In section 2 we define the path integral for a string theory with discrete target spaceand establish its equivalence with an ordinary field theory for the string field Ψx(L).The measure in the space of world sheet geometries will be discretized by planar graphs.Therefore we have first to define the statistical systems related to the SOS model (knownalso as interaction-round-a-face, or IRF models [7]) on an arbitrary irregular lattice S .

Insect. 2.1 we present a generalized version of the construction of these models worked out in[1].

The present version allows two nearest neighbour sites s and s′ of S to have coordinates(heights) x and x′ which either or coincide or are nearest neighbours in the X space. Inthis way we are able to achieve both the dense and dilute phases of the IRF models.

Theimportant special case X = ZZ is discussed in sect. 2.2. where we consider a nonunitaryversion of the SOS model with complex Boltzmann weights.

This model renormalizes at2

large distances onto a gaussian field with a linear term coupled to the local curvature of thediscretized world sheet. The mapping of the IRF models onto this SOS model known asCoulomb gas picture, can be constructed in the same way as in the case of a lattice withoutcurvature [7], [12], [13].

The Coulomb gas picture is based on the loop expansion which isexplained in sect. 2.3.

The partition function of the statistical system with target spaceX equals the sum over the embeddings of a world sheet with fixed geometry. In section2.4 we define the string path integral as a sum over the world sheet geometries and reviewits description by means of a two-component gaussian field [14][5][6].

The normalization ofthe electric and magnetic charges is fixed by the correspondence with the SOS model. Insection 2.5 we establish the equivalence between the string path integral and the Feynmandiagram series for the field Ψx(L) describing the dynamics of the global modes of thestring.

By means of a shift Ψx(L) →Ψx(L) + Ψclx (L) we define an improved diagramtechnique containing no tadpoles. The classical string background Ψclx (L) is determinedby the saddle-point equation.

In sect. 2.6 we show that this equation is equivalent to theloop equation considered in [1].In section 3 we study the continuum limit of the string field theory.

For this purposewe introduce a cutoffa with dimension of length and replace the dimensionless lengthL with a renormalized length ℓ= aL. The continuum limit can be achieved if the bareparameters of the string are tuned to a critical point.

The critical points can be foundas the singularities of the loop equation for the classical string background. The latterallows an interpretation as the partition function of a gas of nonintersecting loops on afluctuating surface with the topology of a disc.

In the vicinity of a critical point the volumeof the world sheet always diverges and the different critical regimes are distinguished bythe behaviour of the loops. The latter is gouverned by a parameter (energy) coupled tothe total length of the loops.

If the energy exceeds the entropy, then the loops remainsmall, and the possible critical regimes are the m-critical points of nonimbedded surfaces(sect. 3.1).

This is the phase of noncritical loops. In the opposite case, considered in sect.3.2, the energy of the loops is not sufficient to compensate their entropy, and the loopsform a dense critical phase filling the world sheet (sect.

3.2). Finally, the dilute phase ofthe loop gas (sect.

3.3) is achieved when the energy of the loops is tuned to its criticalvalue. In the dilute phase the loops are still critical but they cover only a small fractionof the points of the lattice.

In this phase the critical behaviour is sensitive to the choice ofthe integration measure in the space of world sheet geometries. By tuning the Boltzmannweights of the planar graphs we can achieve an infinite sequence of multicritical points ofthe dilute phase.

The continuum limit of the loop equation (sect. 3.4) is universal ; thedifferent critical regimes are classified by the possible asymptotics of the solution at smalllengths.Once the string background is known, it is not difficult to find explicit expressions forthe Feynman rules in the continuum limit.

The vertices in the improved diagram technique(i.e., dressed by tadpoles) can be calculated as the loop correlators in a special matrixmodel constructed in sect. 4.1.

The physical meaning of the coupling constant of thismatrix model is discussed in sect. 4.2.

It is coupled to a local operator which has negativedimension if the theory is not unitary. Its dimension is related to the fractal dimensionof the connected domains on the world sheet.

In sect.4.3 we give explicit expressions for3

the planar (H = 0) vertices in the continuum limit. Since the nonplanar loop amplitudesin the one-matrix model are not yet known, we have only fixed the general structure ofthe nonplanar vertices.

The expression for a vertex with H handles and n legs contains3H + n −3 unknown coefficients. The Feynman rules simplify in the momentum space(p, E) where the propagator of the string field diagonalizes.

The spectrum of on-shellsstates discussed in sect. 4.4 is given by the light cone in the (p, E) space.

The latterhas the geometry if a half-infinite cylinder since the discreteness of the X space leads toperiodicity in the p-direction. Finally, in sect.

4.5 we check that the partition functionof the noninteracting string (closed surfaces with the topology of a torus) is obtained byintegrating over the gaussian fluctuations around the saddle point.2.Statistical systems on planar graphs and kinematics of strings with discretetarget space2.1. The IRF height models on an arbitrary irregular latticeIt is well known [15] that the two-dimensional rational conformal-invariant QFT withcentral charge C < 1 are classified by the simply laced Lie algebras (i.e., these of the clas-sical series An, Dn, E6, E7, E8).

Each of these theories can be constructed microscopicallyas a lattice statistical model whose local degrees of freedom are labeled by the points ofthe Dynkin diagram of the corresponding Lie algebra [16]. The statistical models associ-ated with dynkin diagrams of A, D, E type, or shortly ADE models represent a naturalgeneralization the RSOS face models considered by Andrews, Baxter and Forrester [17].The local degrees of freedom are attached to the sites of the lattice and interact through“interactions-round-a-face” (IRF) around each plaquette.Similarly, the extended ˆA ˆD ˆE Dynkin diagrams describe conformal invariant QFT withC = 1 and discrete spectrum of conformal dimensions.

In fact, Pasquier’s construction [16]can be applied to any one-dimensional discrete manifold X , i.e.,a set of points x and links⟨xx′⟩with the structure of a one-dimensional simplicial complex. The requirement that Xis an (extended) Dynkin diagram of ADE type guarantees the existence of a scaling limit.Another important target spase is the infinite discretized line ZZ.

This is the target spaseof the model known as SOS (solid-on-solid) model.Before presenting the definition of the IRF model with target space X we are goingto describe the space of excitations of this model.The one-dimensional discrete manifold X is defined by the set of its points x and theadjacency matrix Cxx′, x, x′ ∈XCxx′ = [the number of links connecting x and x′](2.1)We assume that the target space X is represented by a nonoriented graph which impliessymmetric adjacency matrix.The Hilbert space of states of the X-field in a fixed “time slice” consists of all closedpaths in X with given length. Therefore in order to identify the ground state and theexcited states we have to solve the problem of random motion in X.

The propagation4

kernel for a random walk on X consisting of n steps is just the n-th power of the adjacencymatrix C. Introducing the eigenvectors V x(p)Xx′Cxx′V x′(p) = βpV x′(p)(2.2)we can write the kernel Kxx′ = (Cn)xx′ as a sum of projectors on the eigenstates(Kn)xx′ =Xp(βp)nV x(p)V x′(p)(2.3)If X is an ADE Dynkin diagram (Fig.1), then the eigenvalues of C have the formβp = 2 cos(πp), p = m/h(2.4)where h is the Coxeter number and the integer m is one of the Coxeter exponents of theDynkin diagram 1 . The eigenvectors V x(p) define the Fourier transform from coordinatesx to the discrete momenta p = m/h.The ground state corresponds to the maximaleigenvalue of the adjacency matrix Cβ ≡βp0 = 2 cos(πp0),p0 = 1/h(A, D, E)(2.5)Name of the4567algebraName of thealgebraE8123456789^E7^23456781Dinkin diagramDinkin diagramCoxeter numberh=n+1h=12h=2(n+1)h=18h=30n+1E81234567823An12...nn+1n−2... n−1nADnn^^12...nn−1E6^12345673exponents m=1, 2, ..., n , (0) m=1, 3,..., 2n−3, n−1, (0) m=1, 4 , 5 , 7 , 8 , 11 , (0) m=1 , 5 , 7 , 9 , 11 , 13 , 17 , (0) m=1 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , (0) XXhmDn12...n−2n−1nE6123456E7123Fig.1 : Dynkin diagrams and their Coxeter exponentsIf X is an extended ˆA ˆD ˆE Dynkin diagram, then the spectrum of Coxeter exponentsincludes m = 0 and the maximal eigenvalue of C isβ = 2,p0 = 0( ˆA, ˆD, ˆE)(2.6)1 The inverse statement reads: if all the eigenvalues of the adjacency matrix are less than 2,then the graph X is an either an ADE dynkin diagram or a quotent A2n/Z2 [18]5

In order to simplify the notations we shall always denote the ground state wavefunctionbySx = V x(p◦)(2.7)Since the entities of the adjacency matrix are nonnegative, by the Peron-Frobenius theoremSx ≥0 for all x ∈X.The propagation kernel (2.3) becomes in the limit of large proper time n a singleprojection operator to the ground state Sx(Kn)xx′ ∼Sx (β)n Sx′(2.8)It is convenient to define the wavefunctions of the excited states asχx(p) = V x(p)/Sx(2.9)They satisfy a closed algebraχx(p)χx(p′) =Xp′′Cpp′p′′χxp′′(2.10)and orthogonality conditions of the formXxS2xχx(p)χx(p′) = δpp′,Xpχx(p)χx′(p) = δx,x′(SxSx′)−1(2.11)Let us give two simplest examples.i) The Dynkin diagram of the algebra Ah−1 is the chain of h−1 points x = 1, 2, ..., h−1.This is the target space of the RSOS models [17]. The eigenvectors of C areV x(p) =r2h sin(πpx), p = 1/h, 2/h, ..., (h −1)/h;Sx =r2h sin(πx/h)(2.12)In this case the ground state corresponds to p0 = 1/h.ii) The extended Dynkin diagram corresponding to the affine Kac-Moody algebraˆA2h−1 is the ring ZZ2h made of 2h points.

It can be represented by the set of integersmodulo 2h. Then the connectivity matrix Cxx′ readsCxx′ = δ(2h)x,x′+1 + δ(2h)x,x′−1(2.13)where δ(2h) is the Kronecker symbol modulo 2h.

The eigenvectors of the matrix ˆC areV x(p) = (2h)−1/2 exp(iπpx), p = 0, ± 1h, ..., ±h −1h, 1. (2.14)The target space is translational invariant and the spectrum of momenta contains the pointp = 0.

The ground state is Sx = V x(0) = 1/√2h. The vertex operators (order parameters)χx(p) = eiπpx(2.15)6

satisfy a closed algebra with fusion rules representing the momentum conservation modulothe period 2 of the momentum spaceCpp′p′′ = δ(2)(p + p′ + p′′)(2.16)The microscopic definition of the IRF models on a general irregular lattice was pre-sented in [1] as a direct generalization of Pasquier’s construction [16], the only new pointbeing the explicit dependence of the Boltzmann weights on the scalar curvature.Theeffects of the curvature have been also studied in [13]. Below we present a more generalconstruction including both the dense and the dilute versions of the IRF models.Let S be a two-dimensional discrete manifold such that all its faces are squares (weshall call them also plaquettes).

It can be constructed from an arbitrary planar graph andits dual one by taking the points of both graphs and adding bonds connecting all pairs ofpoints which serve as extremities of mutually dual lines. A small section of such lattice isshown in Fig.2Each configuration of the x-field on S defines a map S →X.

Sometimes the localvalues of the x-field are called heights. The map is continuous in the sense that nearestneighbours in X are images of nearest neighbours in S. In the original papers a strongercondition has been imposed, namely, that the links ⟨ss′⟩of S are mapped into links ⟨xx′⟩of X.

Here we relax this condition by allowing two nearest neighbours in S to be mappedinto the same point x ∈X.Fig. 2: A section of the discretized world sheetThe statistical weight of each field configuration is a product of factors associated withthe plaquettes and sites of the lattice S. We assume that the weights of the plaquettesare symmetric under cyclic permutations.This property will be of crucial importance forthe generalization of the models to the case of an irregular lattices.

The weight of theplaquette (s2s4s1s3) is defined byxx 1x23x4xx 1x23x4xx 1x23x4xx 1x23x4xx 1x23x4xx 1x23x4x++++++,x 1 x23x4xΩ( , , , )x=1x23x4(2.17)7

with1x2xx4x3= δx1x2δx2x3δx3x4δx4x11x23x4xx= 1T Cx1x3Cx3x4δx4x2δx2x14sSx1Sx31x23x4xx= 1T 2 Cx1x3Cx3x4Cx4x1Cx1x2 δx2x44sSx1Sx3Sx2Sx4(2.18)where T is the temperature of the statistical system.The factors associated with the sites s ∈S reflect the response of the x-field to thelocal curvature ˆRs which is concentrated at the sites. On an irregular lattice the localGaussian curvature at the point s is defined as twice the deficit angleˆRs = π(4 −qs)(2.19)where qs is the coordination number at the site s (= the number of plaquettes sharing thissite).

We use the standard normalization such thatXs∈SˆRs = 4πχ(2.20)where χ = 2 −2H is the Euler characteristic of the surface S. The weight associated witha site s isΩ(x) = (Sx)ˆRs/4π(2.21)The factors (2.21)are trivial on a regular square lattice where all coordination numbers areequal to 4.Collecting all factors, we define the partition function of the model with target spaceX on the lattice S as the sum over all maps S →XF[S] =XS→XYsΩ(x(s))Y(s2s3s1s4 )Ω(x(s1), x(s2), x(s3), x(s4))(2.22)In the limit T →0 the r.h.s. of (2.22) is the partition function of the critical ADEmodels considered in [1] .

In the high temperature limitT →∞the x-field freezesto a constant and there are no long range correlations. At some critical temperature Tcthere should be a phase transition between the low-temperature (critical) and the high-temperature (noncritical) phase.

At the point Tc a new critical regime can be achieved.We call this phase, by reasons which will become clear later, dilute phase. The criticallow-temperature phase will be refered also as the dense phase of the IRF models.8

2.2.The nonrestricted SOS model with a false vacuumIt is known [19] , [20] , that the dynamics of the two-dimensional conformal theoriescan be described in terms of a Gaussian massless field with a linear term coupled to thelocal curvature. Let ξa, a = 1, 2, be the local coordinates on a surface S with metricˆgab(ξ); a = 1, 2 and local curvature ˆR(ξ).

Then the gaussian field x(ξ) is defined by theactionA[x; ˆg] = 14Zd2ξpdetˆg [πg ˆgab∂ax(ξ)∂bx(ξ) + iα0x(ξ) ˆR(ξ)](2.23)where g is the coupling constant and α0 is the background electric charge.One can map all conformal field theories with conformal anomaly C ≤1 onto thisgaussian field in the sense that the partition and correlation functions can be interpretedin terms of distributions of electric and magnetic charges in the gaussian theory. Thismapping is known as Coulomb gas picture [21], [22], [23], [24].

The electric charge e iscarried by the vertex operator Vα(ξ) = exp(iπαx(ξ)) and the charge µ at the point ξcreates a discontinuity 2µ along a line starting at the point ξ (in fact, the discontinuitycan be distributed among several lines forming a star with center at ξ).The (++)-component of the energy-momentum tensor isT(ξ) = −gπ2∂ξx∂ξx + iπα0∂2ξx(ξ)(2.24)where we have introduced the complex variable ξ± = ξ1 ± iξ2. From the o.p.e.

[19], [23]T(ξ)T(ξ′) = 121(ξ −ξ′)4 [1 −6α20g ] +2(ξ −ξ′)2 T(ξ2) + ...(2.25)T(ξ)Vα(ξ′) = α(2α0 −α)4g1(ξ −ξ′)2 Vα(ξ′) + ...(2.26)one reads that the conformal anomaly C of the field (2.23) and the conformal dimensionsof the vertex operators Vα arec = 1 −6α20g ;∆α = α(α −2α0)4g(2.27)The operators Vα and V2α0−α have the same conformal dimension and therefore can beconsidered as related by charge conjugation. Their two-point function is⟨Vα(ξ)V2α0−α(ξ′)⟩= |ξ −ξ′|α(2α0−α)/g(2.28)where g is the coupling constant in (2.23).

The charge neutrality is restored by the presenceof the background charge −2α0. The construction of higher correlation functions involvesthe so called screening operators that carry nonzero electric charge and have conformaldimension 1.

By eq. (2.27) there are two such operators which are charge-conjugated toeach otherα±(α± −2α0) = 4g;α+ + α−= 2α0(2.29)9

The allowed charges are labeled according to the number of screening charges needed toneutralize the 4-point function [23]αrs = 1 −r2α+ + 1 −s2α−(2.30)The conformal dimensions of the corresponding vertex operators form the Kac spectrum∆rs = (rα+/2 + sα−/2)2 −α204g(2.31)Finally, the vortex operator with magnetic charge (discontinuity) µ has conformal dimen-sion˜∆µ = g2µ2/4 −α204g(2.32)The gaussian field dominance in the two-dimensional critical phenomena has its mi-croscopic equivalent. It happens that all target spaces with dimension not greater thanone can be mapped onto the target space of the nonrestricted SOS model which is thediscretized real line Z.

The corresponding connectivity matrixCxx′ = δx,x′+1 + δx,x′−1(2.33)has a continuous spectrum of excitationsβp = 2 cos(πp),−1 < p ≤1(2.34)and its eigenvectors are plane wavesV x(p) = eiπpx(2.35)The momentum space (the dual of the SOS target space) is therefore a circle with perimeterequal to two.The large distance behaviour of this SOS model is argued [23][24] to be described bythe action (2.23) with an appropriate choice of the background charge α0 and the couplingconstant g. The background charge in the SOS model is introduced by taking an excitedstate with momentum p0 as a vacuum stateSx = V x(p0) = eiπp0x(2.36)Then the curvature dependent-factor in the definition of the partition function can bewritten as an exponentYsSx ˆR(s)/4π = expip04XsˆR(s)x(s)(2.37)10

which is the microscopic realization of the curvature-dependent term in (2.23). Knowingthat the global curvature does not renormalize, we conclude thatα0 = p0 + even integer(2.38)Since the charge in the SOS model is determined up to an even integer, we can assumethat α0 = p0.Now it remains to fix the coupling constant g in the gaussian theory.First we observe that the spectrum of allowed momenta in the target space of the SOSmodel is p = mp0, m =integer.

The operators (2.35) corresponding to momenta outsidethis spectrum have vanishing correlators because the electric neutrality is not fulfilled. Onthe other hand, due to the discreteness of the target space of the SOS model all charges ofthe form α ± 2n in the gaussian model are indistinguishible in the SOS model.

Therefore,the spectrum of allowed electric charges in the gaussian model isα = rp0 + 2n;r, n ∈Z(2.39)The way the coupling constant g depends on the background charge α0 is fixed by thecompatibility of (2.29) and (2.39)α0 = g −1;α+ = 2g,α−= −2(2.40)Then the spectrum of allowed charges (2.39) coincides with (2.30) and the dimensions ofthe Kac spectrum (2.31) read∆rs = (rg −s)2 −(g −1)24g(2.41)The corresponding central charge (2.27)C = 1 −6(g −1)2/g(2.42)is symmetric under g →1/g.By (2.40) the coupling constant g of the gaussian field is related to the “vac-uum”eigenvalue β of the connectivity matrixβ ≡2 cos(πp0) = −2 cos(πg)(2.43)The branch of the multivalued function g = 1π arccos(−β) is determined by the dynamics,that is, by the choice of the critical regime for the SOS model. The analysis of the O(n)model [22] which is in a sense dual to the SOS model with β = n, suggests that the interval0 < g < 1 describes the dense phase and the interval 1 < g < 2 describes the dilute phaseof the SOS model.

Sasha Zamolodchikov conjectured [25] that the other branches of (2.43)( 2 < g < ∞) correspond to multicritical regimes of the SOS model. This is shown tobe the case for the SOS model on a fluctuating lattice [11].

We will give a sketch of the11

analysis of ref. [11] in one of the next sections.

The m-critical point corresponds to g inthe interval (m −1, m).The order parameters of the SOS model are represented by vertex operators in thegaussian theoryχx(s)(p) = eiπ(p−p0)x(s)↔Vα(ξ) = eiπαx(ξ),α = p −p0(2.44)The operators carrying charges p = rp0, r = 0, ±1, ... have conformal dimensions in theKac spectrum∆electric(p)= p2 −p204g= (g −1)2(r2 −1)4g= ∆rr(2.45)The vortex operators in the SOS model can have only integer magnetic charge.Theconformal dimension of a magnetic operator with charge µ = m, m = 0, ±1, ... is∆magneticm= (mg/2)2 −(g −1)24g= ∆m0(2.46)The presence of negative dimensions means that the theory is not unitary. There are twocharge conjugated identity operators (p = ±p0).2.3.

The loop expansionThe partition functions of the IRF models on a surface with the topology of a sphereis equal to the partition function of a gas of nonintersecting loops on the surface [22], [16].The loops have the meaning of domain walls separating domains of constant x.The microscopic construction of the loop expansion goes as follows. Write the partitionfunction as a sum of monomials, choosing for each plaquette only one of the seven terms onthe r.h.s.

of (2.17). The ensemble of field configurations S →X is divided in this way intosubsets, each characterized by a given decoration of the plaquettes of S. The decorationsof all plaquettes form a set of nonintersecting polygons on the lattice S∗dual to S (Fig.3).

The polygons play the role of boundaries of the domains of constant x.12

Fig.3: A loop configuration formed by decorated plaquettes.The partition function is equal to the sum over all configurations of polygons on theworld sheet and all allowed values of x in the connected domains bounded by polygons.The Boltzmann weight of each configuration is equal to T −L where L is the total length ofall polygons, times a product of factors associated with the domains bounded by polygons.Let us consider for simplicity a lattice with the topology of a disc with and impose aDirichlet boundary condition x = constant. The weight of a domain of constant height xwith n boundaries is (Sx)2−n.

To prove this we can use the freedom to redistribute theBoltzmann weights between plaquettes and vertices. The new plaquette weights are givenby (2.18) with1x2xx4x3= δx1x3δx3x4δx4x2δx2x11Sx41x23x4xx= 1T Cx1x3Cx3x4δx4x2δx2x11Sx41x23x4xx= Cx1x3Cx3x4Cx4x2Cx2x11Sx4(2.47)and the new vertex weights areΩ(x) = Sx(2.48)Using the fact that the number of bonds is twice the number of faces, one can easily showthat the total power of Sx is equal to the Euler characteristic χ = 2 −n of the domain.

Itis convenient to distribute the weight S2−nxas follows: a factor Sx for the outer boundaryand a factor S−1xfor each of the inner boundaries.Next, for a given configuration of polygons we wish to sum over all possible heights ofthe domains bounded by these polygons. Let us start with the domains with the topologyof a disc, i.e., having no inner boundaries.The sum over the allowed heights of suchdomain is performed using the definition (2.2).

The result is βSx where x is the heightof the surrounding domain. The factor Sx cancels with the factor S−1xin the Boltzmannweight of the surrounding domain, associated with this boundary.

Proceeding this way,from the inside out, we find that the Boltzmann weight of a polygon configuration is13

obtained by assigning to each polygon a factor β = 2 cos(πp0). There will be also anoverall factor Sx where x is the height of the most outer domain .

Note that the polygonshave no orientation. One can introduce an orientation and split the weight of a polygon intotwo phase factors exp(±iπp0) for the two possible orientations.

Note that the Boltzmannweights of the loops do not depend on the local curvature.The correlation function of two order parameters, or vertex operators (2.9) is equal,up to a normalization, by the partition function of the loop gas where the loops enclosingonly one of the two points have different fugacity βp. The correlation function of twodesorder parameters, or vortex operators with magnetic charge m is given by the partitionfunction of the loop gas in presence of m nonintersecting lines having as extremities thetwo points.

We have seen that the partition function on a surface with the topology of adisc or sphere depends on the model only through the momentum p0 of the ground state inthe target space. The mapping onto the SOS model is trivial in this case.

For higher genussurfaces the partition function will depend on the spectrum of allowed momenta (torus),fusion rules (double torus), etc. The mapping onto the SOS model then becomes moreand more involved and requires the introduction of a system of distributed electric andmagnetic charges [26] [13].

For more complicated geometries it is convenient to formulatethe loop expansion using a special diagram technique which will be considered later in thissection.2.4. Summing over the world-sheet geometriesNow we come to the problem of the evaluation of the string path integral.Thepartition function of the string is defined formally as an integral in the space of all embeddedsurfaces .

In our case the integration with respect to the intrinsic geometry is equivalentto the average of the partition function in the ensemble of all possible lattices S. Thedual graphs S∗are generated by the perturbative expansion of a φ4 matrix field theory.In this approach the length of a loop on the graph is an integer (all links of the graph aresupposed to be of unit length). In what follows we prefer to consider the length L as acontinuous parameter, in order to simplify the notations.

However, the equivalence withthe discrete formulation is complete.For the moment both the length L and the area A are dimensionless quantities. Sincewe are interested in the scaling limit A →∞, L →∞, later we are going to introduce therenormalized quantitiesℓ= aL,A = a2νA(2.49)where a will be a cut-offparameter (elementary length).

It is natural to assume that thedimension of the world sheet is 2; then 1/ν gives the fractal dimension of the loops whichis determined by the dynamics. This is also the fractal dimension DB of the boundary ofthe world sheet with Dirichlet boundary condition on x.

It can be any number between 1and infinity.Let S be any closed connected discrete manifold whose cells are squares. Introducingthe bare “cosmological constant” K0 coupled to the area (= number of plaquettes) A(S)and the bare “string interaction constant” κ0 coupled to the genus (= number of handles)14

H(S) , we define the canonical partition function as a sum over all such SF(K0, κ0) =XSκ2−2H(S)0e−K0A(S)F(S)(2.50)where F(S) is the partition function for frozen geometry defined by (2.22). This partitionfunction is well defined for K0 larger than some critical K∗where the continuum limit mayexist.

In what follows the cosmological constant will be introduced implicitely through themeasure in the space of empty surfaces.The loop expansion acquires a new significance in the case of fluctuating geometry. Ithelps to reduce the problem to the problem of surfaces without embedding.

The conditionfor this is the factorization of the Boltzmann weight of a polygon configuration to a productof weights associated with the connected domains.This property is very restrictif.Itexcludes, for example, terms in the action quadratic in the scalar curvature.The integration measure over surfaces is controlled by two parameters: the cosmolog-ical constant K0 coupled to the area of the world sheet and the string interaction constantκ0 coupled to its topology. If F (H)(A) is the partition function of surfaces with fixed areaA and topology of a sphere with H handles, then the canonical partition function readsF(K0, κ0) =∞XH=0κH0Z ∞0dA e−K0AF [H](A)(2.51)Note that K0 and κ0 are dimensionless constants which will be renormalized in the con-tinuum limit A →∞.

There are different ways to achieve the continuum limit dependingon the choice of the measure in the space of surfaces.The mapping of the SOS model onto a gaussian field can be constructed also in thecase of fluctuating geometry. The sum over the geometries means functional integrationw.r.

to the metric ˆgab in (2.23). In conformal gaugeˆgab(ξ) = ˆgab0 (ξ)e2πνφ(ξ)(2.52)where ˆgab0 (ξ) is some fiducial metric, the functional integral leads to a theory of Liouvillegravity coupled to the matter fields [27].

As has been demonstrated by David, Distlerand Kawai [6] , it is consistent to treat the Liouville field φ as a gaussian field (withrenormalized parameters) and the Liouville interaction as a perturbation. The fields x(ξ)and φ(ξ) combine into a two-component gaussian field with conformal anomaly 26 definedby the actionA[x, φ] = 14Zd2ξpdetˆg0 [πg ˆgab0 (∂ax∂bx + ∂aφ∂bφ) + ˆR(ξ)(ip0x(ξ) −ε0φ(ξ))]+ ΛZd2ξpdet ˆg0 e2πνφ(ξ)(2.53)where Λ ∼K0 −K∗0 is the renormalized cosmological constant.15

The vertex operators dressed by the fluctuations of the metric areV(p,ε)(ξ) = eiπ(p−p0)x(ξ)−π(ε(p)−ε0)φ(ξ)(2.54)In particular, the puncture operator ( = identity operator + gravitational dressing) isrepresented byP(ξ) = e−π(ε(p0)−ε0)φ(ξ) = e2πνφ(ξ)(2.55)The conformal anomalies of the two components of the gaussian field are cx = 1 −6α20/g, cφ = 1 + 6ε20/g and the condition cx + cφ = 26 impliesε20 −p20 = 4g(2.56)We choose the positive solutionε0 = g + 1, p0 = g −1(2.57)so that the two screening charges can be represented asα± = p0 ± ε0 = g −1 ± (g + 1) ⇒α+ = 2g, α−= −2(2.58)The condition that the conformal dimension of the operator (2.54) is one∆x + ∆φ = p2 −p204g−ε(p)2 −ε204g= 1(2.59)combined with (2.56) leads to the relationε(p)2 −p2 = 0(2.60)which can be interpreted as a mass-shell condition of the same form as the mass-shellcondition for a light particle in a two-dimensional space-time.In principle, the vertex operator (2.44) dressed by the fluctuations of the metric is alinear combination of two operators (2.54) corresponding to the two solutions of (2.60)V(p)(ξ) →A+Vp,ε(ξ) + A−Vp,−ε(ξ)(2.61)The two terms will have the same dimension if the coefficient A−contains a positivepower a2|p| of the cutoffa. Therefore the second term is irrelevant when p ̸= 0.

Sincethe minimal momentum in the theories with C < 1 is positive (p0 = 1/h), all physicaloperators correspond to the positive branch of (2.60)ε(p) = |p|(2.62)Another motivation for the choice (2.62) is based on the quasiclassical treatment of theLiouville theory [28].16

In a theory with C = 1 (g = 1, p0 = 0) both terms become essential in the limit p →0and the Liouville interaction in (2.53) which comes from the gravitational dressing of theidentity operator can be not simply an exponential [29].By (2.57) and (2.62) the Liouville charge of the identity operator (2.55) in the areaterm of (2.53) equals ε0 −ε(p0) = 2ν whereν = 12(g + 1 −|g −1|) =g,if g < 1 ;1,if g > 1(2.63)The gravitational dimension of the vertex operator with electric charge αrs = prs +p0 = r −sg + (g −1) is [5],[6]δrs = 1 −ε0 −ε(prs)ε0 −ε(p0) = |r −gs| −|g −1||g + 1| −|g −1|(2.64)In particular,δelectric(p)= |p| −|p0|2ν,δmagneticm= mg/2 −|p0|2ν(2.65)Finally, the so-called string susceptibility exponent γstr is related to the conformal anomalyof the matter field byC = 1 −6γ2str1 −γstr(2.66)It can be determined as twice the dimension δ00 in the Kac spectrum of gravitationaldimensions (2.64)γstr = −2|g −1|g + 1 −|g −1| = −ε(p0)ν;ν(2 −γstr) = ε0(2.67)2.5. Diagram techniqueWe have seen that a map S →X can be described by a collection of self-avoidingnonintersecting loops dividing the world sheet into domains of constant x.

The energyof each such loop (domain wall) is proportional to its length L. In order to simplify thenotations, from now we shall consider the length as a continuous quantity. The Boltzmannweight of a domain wall isT −L = exp(−2P0L)(2.68)Further, the weight of each domain of constant height x depends on the geometry of thedomain through its Euler characteristic.

Indeed, the product of the Boltzmann weights(2.47) associated with the sites and squares of the domain is(Sx)sites−squares = (Sx)2−2H−n(2.69)where H is the number of handles and n is the number of boundaries of the domain.In this way the partition function of an ADE model on a surface S with arbitrarygeometry has been reformulated as the partition function of a gas of nonintersecting loops17

on this surface. The Boltzmann weights of the loops depend on the geometry of the surfaceonly through homotopic invariants.

In particular, they do not feel the local curvature. Theloops sensible to the topology of the surface are the noncontractible ones that wrap aroundthe handles of the surface S. Proceeding as in subsec.

2.3 one can check that all contractibleloops have fugacity β = 2 cos(πp0) while the fugacity of the loops wrapping around a handleis βp = 2 cos(πp) where p is the momentum of the matter-field excitation propagating alongthis handle. Thus the mapping of the ADE models onto the loop gas depends, in the caseof a surface with arbitrary topology, both on the spectrum of the order parameters and theway they interact.

This mapping can be systematically formulated using a special diagramtechnique which has been first considered in [16] and developed further in [1]. In the caseof fluctuating world-sheet geometry this diagram technique gives rise to the Feynman rulesfor the corresponding string field theory, to be discussed below.The sum over the embeddings of a world sheet with fixed geometry is a formidableproblem.

It can be avoided by reorganizing the measure over embedded surfaces. First wetake the sum over the world-sheet geometries with given configuration of domain walls.Each domain-wall configuration is determined by its topology and the lengths of the loops.It can be described by a Feynman diagram with vertices corresponding to the domains andlines corresponding to the domain walls (Fig.

4).6961234517978234587Fig. 4: A loop configuration on a toroidal surface and the corresponding Feynman diagramThe heights x and the lengths L are associated with the vertices and lines, corre-spondingly.

The partition function (2.51) of the string is equal to the sum of all connecteddiagrams without external lines.The weight of a diagram is a product of the weights associated with its vertices andpropagators. A vertex is defined by the coordinate x, the number of handles H, and thelengths L1, ..., Ln of the boundaries of the corresponding domain.

Its weight factorizes toa coordinate- and length-dependent componentsV(H)◦(x|L1, ..., Ln) = (κ0/Sx)n−2+2He−P0(L1+...Ln) W (H)◦(L1, ..., Ln)(2.70)where W (H)◦(L1, ..., Ln) is the partition function of nonembedded (empty of loops) surfaceswith H handles and n boundaries with lengths L1, ..., Ln.The “propagator” associated with a line L isG0(x, L|x′, L′) = Cxx′ δ(L −L′)(2.71)18

The integration over the lengths is performed with the measure dL/L. This measure isconsistent with the convention that each boundary has a marked point on it.It is convenient to introduce a quantum field theory in which all these graphs appearas Feynman diagrams.The latter can be formulated as a functional integral over theconfigurations of a loop field Ψx(L) defined in the direct product of the target space Xand the positive real axis.

The interactions of the loop field are given by the vertices (2.70).It is not difficult to see that the grand canonical partition function Z[J] = exp(F[J])(i.e., the one where also disconnected surfaces are allowed) can be written as the followingfunctional integralZ[J] =ZDΨeS[Ψ]S(Ψ) = −12Xx,x′Z ∞0dLL Ψx(L)C−1xx′Ψx′(L)+Xx∞Xn=1∞XH=01n!h Z ∞0dLJx(L)Ψx(L)+XxZ ∞0dL1L1...dLnLnV(H)◦(x|L1, ..., Ln)nYk=1Ψx(Lk)i(2.72)The multiloop correlation functions are formally obtained as derivatives with respect tothe source J(L)⟨Ψx1(L1)...Ψxn(Ln)⟩= nYk=1δδJxk(Lk)F[J]!J=0(2.73)The functional integration measure corresponds to the norm∥Ψ∥2 =XxZ ∞0dLLΨx(L)2(2.74)Formally, eq. (2.72) defines the string field theory corresponding to the target space X.However, in order to make this definition more explicit, a lot of work has to be done.

First,the diagram technique following from the domain wall representation contains tadpoles.This means that the string field has a nontrivial background which can be found as thesolution of the saddle-point equationΨclx (L) = δF[J]δJx(L)κ0=0(2.75)This is the partition function of surfaces with the topology of the disc and bounded by acontour of length ℓsituated at the point x.19

It is easier to solve the classical equation of motion (2.75) after going to the momentumspace. For this we expand the string field as a linear combination of eigenstates (2.2)Ψx(L) =XpV x(p)Ψ(p)(L)(2.76)Let us denote by V(H)◦(p1, ..., pn|L1, ..., Ln) the vertices (2.70) in the momentum space.They again have a factorized formV(H)◦(p1, ..., pn|L1, ..., Ln) = κn−2+2H0N (H)p1...pne−P0(L1+...+Ln) W (H)◦(L1, ..., Ln)(2.77)whereN (H)p1...pn =XxnYk=1V x(pk)(Sx)2−n−2H=Xx(Sx)2χx(p1)...χx(pn) Xpχx(p)χx(p)H(2.78)The p-dependent part of the vertices (2.78) has a natural interpretation in terms of stringstates.

The factor (Sx)2 is related to the measure in the X space, each leg is multipliedby an order parameter χ(p) and each handle can be considered as the result of contractingtwo legs 2 (in the last line of (2.78) we have used the orthogonality relations (2.11)). Forall target spacesN (0)pp′ = δ(p, p′),N (0)pp′p′′ = Cpp′p′′(2.79)where Cpp′p′′ are the fusion rules (2.10).By applying several times the fusion (2.10)the coefficient (2.78) can be represented as a sum of products of fusion coefficients.

Forexample,N (1)p=Xp′Cpp′p′, N (1)p1p2 =Xp,p′Cp1p2pCpp′p′, N (0)p1p2p3p4 =XpCp1p2pCpp3p4(2.80)In general, the expression for the coefficient N (H)p1...pn can be represented by a “Feynman dia-gram” with n−2+2H vertices (to each vertex is assigned a fusion coefficient) and H loops.2 Each handle therefore contributes a factor proportional to the volume of the target space. Inorder to define the topological expansion for the string embedded in ZZ, we have first to introducea cutoffin the dual momentum space.

The simplest way to do this is to replace the continuumspectrum of momenta by a discrete one with a small spacing δp. Such a discretization appearsautomatically in the matrix-model realization of the string embedded in IR [30].

The fact thatthe vertices depend explicitely on the cutoffdoes not contradict the general covariance. Indeed,the renormalized string interaction constant κ depends explicitely on the cutoffa in the space oflenghts ℓ: it vanishes as 1/ log a.

If we choose δp ∼1/ log a, then the partition function will becutoff-independent.20

Thus the p-dependent component of a multiple vertex can be decomposed into elementaryinteractions involving only three strings. Note that all “Feynman diagrams”representingthe possible decompositions of a multiple vertex have the same contribution (duality prop-erty).

3Now we can write the functional integral (2.72) in the formeF [J] =ZdΨeA[Ψ]A[Ψ] = −12XpZ ∞0Ψ(p)(L)(2 cos(πp))−1Ψ(p)(L)dLL+∞Xn=1∞XH=0Xp1,...,pn1n!Z ∞0V(H)◦(p1, ..., pn|L1, ..., Ln)nYk=1Ψ(pk)(Lk)dLkLk(2.81)and the classical equations of motion (2.75) are equivalent toΨcl(p)(L) = δp,p0W (0)(L)(2.82)W(L) =∞Xn=0βnn!Z ∞0nYk=1dLkLke−P0LkW(Lk)W (H=0)◦(L, L1, ..., Ln)(2.83)By the loop expansion of subsec. 2.3 the quantity W (H)(L) is the partition functionof the gas of nonintersecting loops on a disc with boundary of length L and a fluctuatingmetric.

The index (0) means planar topology of the world sheet (H = 0). Eq.

(2.83) isgraphically represented in Fig. 5.Fig.

5: A diagrammatic representation of the saddle-point equation.Suppose that we have solved the equation for the string backgroung (2.82). Shiftingthe field in the functional integral (2.81) by its classical valueΨ(p)(L) →Ψcl(p)(L) + Ψ(p)(L)(2.84)we arrive at a new diagram technique without tadpoles.

The new vertices are obtainedfrom the original ones by dressing with tadpoles (Fig. 6).3 We believe that the ℓ-dependent component of a multiple vertex can be decomposed as wellinto elementary interactions by means of Witten’s factorization arguments [31]21

Fig. 6: Graphical representation of the dressing of the vertices with two and three legs.V(H)(p1, ..., pn|L1, ..., Ln) = κn−2+2H0N (H)p1...pnW (H)(L1, ..., Ln)(2.85)W (H)(L1, ..., Ln) =nXk=01k!Z ∞0n+kYr=n+1W (0)(Lr)e−P0Lr dLrLre−P0(L1+...+Ln)W (H)◦(L1, ..., Ln, Ln+1, ..., Ln+k)(2.86)Note that for n = 1 (2.86) coincides with the saddle-point equation (2.83).

The quantityW (H)(L1, ..., Ln) is the partition function of the loop gas on a fluctuating surface with Hhandles and n boundaries with the condition that only contractible loops are allowed.The gaussian part of the new action (the inverse propagator) is equal to the sumof the inverse original propagator (2.71) and the two-point vertex V(0)(p, p′|L, L′) =δpp′W (0)(L, L′). (It is convenient to treat the vertices V(H)(p1, p2|L1, L2) with H > 0as part of the interaction.) The propagator G(p)(L, L′) is therefore determined by theequationG(p)(L′, L′′) = 2 cos(πp)hδ(l′, l′′) + 2 cos(πp)Z dLL W (0)(L′, L)G(p)(L, L′′)i(2.87)whose solution is graphically represented in Fig.

7.+ ...+=++Fig. 7: The full propagator expressed in terms of two-point vertices.The symmetric function W (0)(L, L′) can be considered as a kernel of a symmetricoperator in the space of the real square-integrable functions defined on the positive realline.

This operator describes the evolution of the string in the space of the world-sheetgeometries. Let us introduce its eigenstates which we assume orthonormalizedZ ∞0dLL W (0)(L, L′)⟨L|E⟩= Ω(E)⟨L′|E⟩(2.88)22

Then the propagator (2.87) is diagonalized in the same basis and its eigenstates are˜G(E, p) =2 cos(πp)1 −2 cos(πp)Ω(E)(2.89)The eigenstates ⟨L|E⟩describe the excitations in the L-space in the same way as theeigenstates of the connectivity matrix describe the excitations in the x-space. The wavefunctions of the closed string states are labeled by the “energy-momentum” (E, p) andhave a factorized form⟨E, p|L, x⟩= V x(p)⟨E|L⟩(2.90)We have seen that the string background depends on the structure of the X-spaceonly through the momentum p0 of the ground state (p0 = 1/h for the Dynkin diagramsand p0 = 0 for the extended Dynkin diagrams) and therefore the same factorization takesplace for the vertices (2.85) describing interactions due to joining and splitting of closedstrings.

This is the miracle which makes possible to solve exactly the string theory in ourformalism. Note that in our case the separation of the two global modes occurs beforetaking the continuum limit, which is not the case for the standard discretization of theD = 1 string path integral [32].

It is worth to try to understand better the origin of thissymmetry.Let us mention that there exists a simpler model such that the only allowed momentumis p0. This means that the vertices have trivial coordinate part.

A microscopic definitionof this model is given by the O(n) model on a fluctuating lattice with n = −2 cos(πg) [33].2.6.Loop equationIn order to find the explicit expressions for the dressed vertices (2.86) we have firstto solve the saddle-point equation (2.82).Imagine that the n-loop amplitudes for theemply random surface are constructed as the connected correlation functions of an N × NHermitean random matrix Φ with potential U◦(Φ)∞XH=0N 2−2HW (H)◦(L1, ..., Ln) =nYk=1δδJ(Lk)F◦[J](2.91)eF◦[J] =ZdΦeNtrU◦(Φ)+R ∞0dLJ(L)tr exp(LΦ)(2.92)Then the loop amplitudes W (H)◦(L1, ..., Ln) satisfy the following system of integral equa-tions [34], [35]U ′◦(∂/∂L)W (H)◦(L, L1, ..., Ln)+HXH′=0XI+J={1,...,n}Z L0dL′W (H′)◦(L′; Li|i∈I)W (H−H′)◦(L −L′; Lj|j∈J)W (H−1)(L −L′, L′, L1, ..., Ln) +nXk=1W (H)◦(L + Lk; Ls|s̸=k) = 0(2.93)23

Inserting (2.92) into (2.83) we arrive at the following closed equation for the classical stringfield [8], [9]U ′◦(∂/∂L)W (0)(L) =Z L0dL′W (0)(L′)W (0)(L −L′)+ βZ ∞0dL′W (0)(L′)W (0)(L + L′)e−2P0L′(2.94)which should be completed with the condition that W (0)(L) is analytic at the point L = 0W (0)(L) =∞Xk=0Lkk! Wk(2.95)In the limit L →∞the function W(L) is expected to behave asW (0)(L) ∼L−bePRL(2.96)where the entropy per unit length PR depends on the parameters of the theory.Eq.

(2.94) has a transparent geometrical meaning [36]. A small deformation of theworld sheet at the marked point is equivalent to a local deformation of the boundary.

Itsexact form depends on the measure in the space of the world-sheet geometries.In general,a local deformation of the boundary is equivalent to a differential operatorU ′◦(∂/∂L) = g2∂/∂L + g3∂2/∂L2 + ...(2.97)On the other hand, the deformation of the world sheet cancels part of the integrationmeasure in the space of surfaces and produces boundary (or contact) terms. The standardcontact term, which appears also in pure gravity, is due to the surfaces with degeneratedworld sheet such that another point of the boundary approaches the marked point.

Herewe encounter a new contact term due to one of the loops approaching the marked pointat the boundary.The loop equation (2.94) has been derived originally [1]in terms of the coefficients WkXk≥2gkWn+k−2 =n−2Xk=0WkWn−2−k + β∞Xp,q=0(p + q)!p!q!e−2Po(p+q+1)WpWn+q−1(2.98)We arrive at this form of the loop equation if the gas of loops is considered on a randomgraph, the length of the loops being the number of bonds they occupy. The formulationin terms of continuous lengths has been proposed by Kazakov [36].It is convenient to introduce the Laplace imageˆW(P) =Z ∞0dL W (0)(L)e−P L(2.99)24

which can be interpreted as the partition function of surfaces with the topology of a discand “boundary cosmological constant” P. In order to simplify the notations we do notassign an index (0) to ˆW(P). The condition of analyticity (2.95) impliesˆW(P) = W0P + W1P 2 + ...,P →∞(2.100)(In what follows we assume the standard normalization W0 = 1.) The asymptotics (2.96)means that the series (2.100) converges for |P| > PR; in the vicinity of the point PR itbehaves as (P −PR)b−1.We are interested in the solutions of the loop equation having a single cut [PL, PR]along the real P-axis.

The loop equation (2.94) in terms of P-variables readsˆW 2(P) =ICdP ′ ˆW(P ′)2πi(P −P ′)[−U ′◦(P ′) −β ˆW (2P0 −P ′)](2.101)where the contour C encloses the cut [PL, PR] of the Riemann surface of the function ˆW(P).The contour integral makes sense only if the two cuts [PL, PR] and [2P0 −PR, 2P0 −PL]of the Riemann surface of the integrand function do not overlap, that is, for P0 > PR.Thepositions PL and PR of the branchpoints are functions of the potential U◦(P) and thetemperature P0 of the loop gas. Eqn.

(2.101) implies the following conditions on the realand imaginary part of ˆW(P) [1]2Re ˆW(P) + β ˆW (2P0 −P) + U ′◦(P) = 0,P ∈[PL, PR]Im ˆW(P) = 0,P ̸∈[PL, PR](2.102)3.Critical behaviour3.1. Critical surfaces, noncritical loopsThe partition function (2.50) is defined in certain domain D in the space of the param-eters P0, K0,... .

The critical behaviour is achieved at the boundary ∂D of this physicaldomain, where the partition function and the observables develop singularities. There aretwo types of singularities in the space of parameters.

The first is related to the divergingarea of the fluctuating surface. It appears when the coupling constants g1, g2, ... in the po-tential U◦(P) which defines the integration measure in the space of nonembedded surfaces,are tuned in a special way.

The loops on the world sheet are small and do not affect thescaling properties. The critical behaviour of the model is the (multicritical) behaviour ofpure gravity.

The m-critical points of the nonembedded random surface [36] have been in-terpreted [37],[38], [39], [40] as pure gravity coupled to a conformal field theory (2, 2m−1).It can be mapped onto the Coulomb gas (2.53) with g = m−1/2. The m-critical potentialhas been chosen in [36] as an even polynomial of degree 2mU◦(P) =X(−)k−1(k −1)!m!(2k)!

(m −k)!P 2k(3.1)25

However, if the symmetry P ↔−P is abandoned, it suffices to take a polynomial of degreem + 1. The scaling limit is achieved by replacing L, P by the renormalized quantities ℓ, zL = ℓ/a,P = P∗+ az(3.2)At the m-critical point the scaling part ˆw(z) of the planar loop amplitude is defined byˆW(P) = ˆW(P∗) + am−1/2 ˆw(z)(3.3)In the matrix-model realization of the multicritical points the renormalised cosmologicalconstant λ scales as a−m which could mean that the fractal dimension of the boundaryof the world sheet is 2/m.

For m > 2 this is less than the classical dimension 1. Theresolution of this paradox comes from the observation [37][38]that for m > 2 λ is coupledto the operator with minimal (negative) gravitational dimension δm1 = 1 −m/2 andtherefore does not measure the area of the world sheet.

The true cosmological constant Λhas dimension a−2 for m ≥2 and a−1 for m = 1. Finally, the string interaction constantscales asκ0 = κ a1−γstr/2;γstr = 3/2 −m(3.4)3.2.

Dense phaseWhen P0 approaches its critical value P∗, long-range effects due to the diverging size ofthe loops change the critical behaviour of the model. For generic potential U◦the randomsurface grows only due to the critical loops which are densely packed on the world sheet.Its area is essentially equal to the total length of the loops.

This is the dense phase of theloop gas. If both the potential U◦and the temperature P0 are tuned, a more complicatedpicture arises.The regime of critical loops coupled to noncritical surfaces describes the dense phaseof the loop gas model.

In this regime the loops are densely packed and fill almost allsurface of the world sheet. The nonrenormalized area of the world sheet is therefore equalto the total (nonrenormalized) length of the loops: A = Ltot and P◦plays the role of acosmological constant.It is easy to see that the condition for complete compensation of the entropy and theenergy of the loops is PR = P0 .

First let us remind that W(L) behaves when L is largeaccording to (2.96). Let us choose one of the random loops on a surface with the topologyof a sphere.

It splits the sphere into two discs, each contributing a factor ePRL when Lgoes to infinity. Thus the entropy 2PRL of the loop totally compensate its energy 2P0Lwhen PR = P0.The vicinity of the critical point is defined by introducing a cutoffparameter a withdimension of length.

The variable z dual to the renormalized length ℓ(the renormalizedboundary cosmological constant) is introduced byP = P∗+ az,PR = P∗−aM(3.5)The parameter M is the contribution of the fluctuations of the surface to the renormalizedboundary cosmological constant. It defines the position of the cut in the z-plane.26

The singular part of the loop amplitude W(P) behaves at the critical point PR =P0(M = 0) as zg where g is the solution of 2 cos(πg) + β = 0 , 0 < g < 1 [41],[1].Therefore we define the scaling part ˆw(z) of the loop amplitude asˆW(P) = ˆW∗+ (constant)ag ˆw(z)(3.6)where ˆW∗is the value of the bare loop amplitude for P = P∗, P0 = P∗.Now we are going to find the scaling law for the cosmological constant Λ which isdefined asP0 = P∗−(constant)a2νΛ(3.7)If we assume that the dimension of the world sheet is 2, then the exponent ν has themeaning of inverse fractal dimension of the boundary of the world sheet. The inducedboundary cosmological constant M is a function of P0 which vanishes at P0 = P∗.

Thereforethe renormalized bulk and boundary cosmological constants are related byΛ = (constant)M 2ν(3.8)The scaling of Λ can be extracted from a nonlinear algebraic equation which followsfrom the loop equation (2.101). In order to get rid of the contour integration we use thefollowing trick.

First write again eq. (2.101) with the variables P, P ′ replaced by theirimages w.r.t.

the reflection P →2P0−P. The integrand remains the same but the contourC is replaced by a contour ¯C enclosing the cut [2P0 −PR, 2P0 −PL] of ˆW(2P0 −P).

Thenadd the two equations and apply the Cauchy theorem to the integral along the contourC + ¯C. The result is the following functional equation for ˆW(P)[−U ′◦(P) ˆW(P) −Xk≥0P k(gk+2W0 + gk+3W1 + ...) −ˆW 2(P)]+ [P →2P0 −P] = β ˆW(P) ˆW(2P0 −P)(3.9)At the point P = P0 this functional equation becomes algebraic(2 + β) ˆW 2(P0) + 2U ′◦(P0) ˆW(P0) −2Xk≥0P k0 (gk+2 + gk+3W1 + ...) = 0(3.10)Eq.

(3.10) becomes very simple in the limit P0 →−∞where the only allowed loopconfigurations are those with no space between loops. This extremely dense limit can beachieved by choosing gaussian potential (g2 = −1, g3 = ... = 0).

Then eq. (3.10) involvesonly the coefficient W0 = 1 of the expansion (2.100) and can be solved instantlyˆW(P0) = 2/(P0 +qP 20 −2(2 + β))(3.11)This quantity has a singularity at P0 =p2(2 + β) = P∗in accord with the exact resultobtained with more elaborated technique [41].27

By the definitions (3.6) and (3.7) we findW(P0) = W∗+ (const. )ag ˆw(0) = W∗−(const.

)√Λ(3.12)By its dimension ˆw(0) ∼M g and eq. (3.12) implies ν = g. ¿From now we will fix thenormalizations of Λ and M so thatˆw(0) = −cos(πg/2)M g,Λ = M 2g(3.13)(Note that the sign of ˆw(M) should be negative.) This scaling law persists for genericpotential U◦(P), i.e., in a domain D1 of codimension zero in the space of coupling constantsg2, g3, ... .Indeed, let us repeat the above argument for a generic potential U◦(P) =g2P 2 + g3P 3 + ....

Expanding eq. (3.10) around the point P0 = P∗we find a relationbetween ˆw(0) = −cos(πg/2)M g and Λ.

In the limit a →0 the most singular part of thecoefficients Wk is the v.e.v. of the puncture operator P = −∂/∂ΛWk ∼a2ν(1−γstr)⟨P⟩∼(a2νΛ)1−γstr(3.14)Inserting this in (3.10) we find a relation of the type(aM)2g =Xk≥1Ak(a2νΛ)k + B(a2νΛ)1−γstr(3.15)where A1, A2, ... and B are numerical coefficients.In the limit a →0 only the smallest power in a will survive.

For a generic potentialall the coefficients Ak are nonzero. The strongest singularity is that of the A1-term andthe matching of powers gives2g = min[2ν, 2ν(1 −γstr)](3.16)Sinse the string susceptibility exponent is allways non-positive, this implies ν = g. In orderto fix the string susceptibility exponent, we use the fact that it is related to the dimensionof the loop amplitudeν(2 −γstr) −1 = g(3.17)Thus in the dense phaseν = g,γstr = 1 −1/g;0 < g < 1(dense phase)(3.18)In the interval 1/2 < g < 1 the difference P∗−P0 vanishes faster than a and the pointP0 in the definition of the scaling variables can be replaced with P∗.

This is not the case,however, if g < 1/2. In this case the cut of the loop amplitude appears at z = −M −a2g−1Λand in the limit a →0 the singularity of the loop amplitude is gone to −∞.

We have avery poor understanding of this case at the moment. Let us only mention that the fugacityof the loops is negative and the fractal dimension of the boundary is larger than two.

Thepoints g = 1/m, m = 1, 2, 3, ... have integer γstr and should describe topological theories.A very interesting discussion on the physics at these points has been given by H. Saleur[42] (see also [11]).The limiting case g = 1/2 corresponds to the gaussian model. In this case there is nosingularity at all and the choice of the distance M is arbitrary.28

3.3. Dilute phaseNow let us consider the case when both kinds of singularities are present.If thepotential U◦(P) is critical (m = 2) , then at the point M = M∗both the length of theloops and the area of the space between them become infinite.

This phase is known as thedilute phase of the loop gas. The multicritical potentials will lead,further, to new typesof critical behaviour of the gas of loops [11] which we are going to discuss briefly.As we have seen above, the dense phase exists in a domain D1 of codimension zero inthe space of coupling constants g1, g2, ....

The generic point on the boundary D2 = ∂D1of this domain corresponds to the next critical regime. The simplest way to achieve it isto adjust the constant g3 in a cubic potential so that the coefficient A1 in (3.15) vanishes.Then the condition of matching of powers reads2g = min[4ν, 2ν(1 −γstr)](3.19)If we are in the interval 1 < 1 −γstr < 2, this condition means 2g = 2ν(1 −γstr) .

Thenfrom (3.17) we findν = 1,γstr = 1 −g;1 < g < 2 (dilute phase)(3.20)Proceeding in the same way , we can achieve the m-critical point by adjusting the cou-pling constants g3, ..., gm+1 so that the coefficients A1, ..., Am−1 vanish. The condition ofmatching of powers then gives [11]2g = min[2mν, 2ν(1 −γstr)](3.21)Combining (3.21) with (3.17) we findν = 1,γstr = 1 −g;m −1 < g < m(m−critical dilute phase)(3.22)Let us assume that the loop gas with given g describes at the critical point a conformalfield theory coupled to gravity.

Then, by the KPZ-DDK argument [5], [6] its conformalanomaly is given byC = 1 −6γ2str1 −γstr= 1 −6(g −1)2g(3.23)Thus the interval 0 < g < ∞covers twice the spectrum −∞< C < 1 of the central charge.Note that even if the central charge is symmetric w.r.t. g →1/g, the fractal dimension ofthe boundary is not.

In the dilute phase it takes the “classical” value 1 and in the densephase it is greater than one.The m-critical quantum gravity coupled to a gas of critical loops exhibits a specificcritical behaviour characterized by the fugacity β of the loops.One can say that the(2m+1, 2)-conformal theory of matter coupled to the gas of loops with fugacity β producean effective matter field with central charge C = 1 −6(g −1)2/g where g is the branch ofthe function g = −(1/π) arccos(β/2) determined by m −1 < g < m.29

The change of the critical regime occurs at the integer points g = m, m = 1, 2, 3, ....Let us consider the vicinity of the point g = m dividing the m-critival and m + 1-criticaldilute phases of the loop gas. In this case eq.

(3.17) readsM 2 = Λ[Am(a2Λ)m−g + B]1/g(3.24)The first term on the r.h.s. vanishes when g < m because the power of a is positive as wellas when g < m because then Am = 0.

However, when g = m both terms survive and ourprevious arguments may not be applicable. The limits g →m and a →0 do not commuteand one has to solve the loop equation for g = m and then take the continuum limit a →0.Exact solutions are known for the cases g = 1 [43] and g = 2 [44], to be discussed below.

Atg = 2 the the scaling is the same as in the whole dilute phase, Λ = M 2. At the point g = 1which is the endpoint of the dilute phase, the scaling law receives logarithmic corrections:Λ = (M log M)2.3.4.

The loop equation and the string background in the continuum limitWe are going to solve the loop equation (2.101) in the continuum limit. Let us definethe vicinity of the critical point as followsP = P0 + azPR = P0 −aMˆW(P) = (2U ′◦(P0 + az) −βU ′(P0 −az))/(4 −β2) + (const.

)ag ˆw(z)(3.25)Here we used as a reference point P0 instead of P∗; we shall see that if 1/2 < g < 1 thismakes no difference in the limit a →0.After the vicinity of the critical point is blown up by the change of variables (3.25) ,the cut [PL, PR] of the Riemann surface of W(P) is replaced by the cut [−∞, −M] alongthe negative z-axis. Eq.

(2.102) implies the following conditions on the real and imaginaryparts of ˆw(z)2Re ˆw(z) + β ˆw(−z) = 0,z ≤−MIm ˆw(z) = 0,z ≥−M(3.26)Eq (3.26) becomes more transparent when written in terms of the variable τ which wehave introduced in [8],[9]z = M cosh(τ)(3.27)Then the z-plane cut along the interval −∞< z < −M is mapped into the half-stripReτ ≥0, −π ≤Imτ ≤π(3.28)The two sides of the cut are mapped to the horizontal boundaries [τ = t ± iπ, t > 0] of theτ-strip. Therefore , if z is real and positive, then −z ± i0 = M cosh(τ ± iπ).

Since ˆw(z) isa real function ( ( ˆw(z))∗= ˆw(z∗)), the first of the eqs. (3.26) can be written as[cos(π∂/∂τ) −cos(πg)] ˆw(z(τ)) = 0(3.29)30

Eq. (3.29) has a single solution (up to a constant factor) which is real along the verticalboundary of the τ-strip and behaves at infinity as zgˆw(z) = −M g cosh(gτ)(3.30)If we return to the original variable z, the solution readsˆw(z) = −12[(z +pz2 −M 2)g + (z −pz2 −M 2)g](3.31)It can be expanded as an infinite series in fractional (in general) negative powers of z withradius of convergence 1/z = 1/Mˆw(z) =Xn≥0w±n M g(M/z)2n∓g(3.32)and dimensionless coefficientsw±n = ±Γ(2n ± g)Γ(n + 1 ± g)n!= ±(2n ± g)(2n ± g −1)...(n + 2 ± g)n!

(3.33)Even if each of the terms in (3.32) has a cut −∞< z < 0, the whole series defines afunction which is analytic for |z| < M.The v.e.v. of the loop operator is determined by the same equation (3.26) both in thedense and dilute phases.

It is given by the same analytic function (3.31) with g rangingfrom 0 to ∞. The m-critical behaviour of the loop amplitude is described by the branchm −1 < g < m of the parametrization (2.43).The function ˆw(z) has a square-rootsingularity at z = −M .

It is meromorphic in the z-plane with a cut from z = −M toz = −∞where it behaves as zg. For nonrational values of g the Riemann surface of ˆw(z)is infinitely foliated and has two cuts −∞< p < −M and M < p < ∞.

All sheets exceptthe first one have two cuts. We will always consider the function ˆw(z) on the first sheet.For g = p/q the Riemann surface of w(z) has a branch point of order q at infinity.

Thenthe first and the last sheet have only one cut.If we introduce a renormalized length ℓ= aL, then the inverse Laplace image of ˆw(z)w(ℓ) =Z i∞−i∞dzeℓz ˆw(z);ˆw(z) =Z ∞0dℓe−zℓw(ℓ)(3.34)is the Bessel function [45]w(ℓ) = M gℓKg(Mℓ). (3.35)It satisfies the following loop equationZ ∞0dℓ′w(ℓ′)w(ℓ−ℓ′) + βZ ∞0dℓ′w(ℓ′)w(ℓ+ ℓ′) = 0(3.36)31

which is the renormalized version of (2.94). The renormalized string background (3.34)behaves differently at large and small lengthsw(ℓ) = ℓ−1M gKg(Mℓ) ∼M g−1/2ℓ−3/2e−Mℓ,if ℓ≫M −1;ℓ−g−1,if ℓ≪M −1(3.37)The small-ℓasymptotics (3.37) can be taken as a subsidiary condition to eq.

(3.36) inorder to have a unique solution. The large-ℓasymptotics describes a loop of length muchlarger than the characteristic length ¯ℓ= 1/M .

The area of the world sheet is zero inthis limit and the power of ℓis the same as in the topological gravity (g = 1/2). Thesmall−ℓasymptotics describes the critical point where the area of the world sheet andthe characteristic length of the loops are infinite.

We see that the square-root singularitychanges generically to a nonrational one in the limit M →0.In order to to reproduce the regime of critical surfaces, noncritical loops we have totake the limit Λ →0 , keeping M finite, i.e., to leave the trajectory Λ = M 2ν. However,we can achieve the critical points of nonembedded surfaces by tending the fugacity β ofthe loops to zero.

This limit is achieved at the half-integer values of g. By the scaling ofthe loop amplitude ˆw(z) these points can be identified with the multicritical points of puregravity [36]. The explicit expression for the loop average ˆw(z) in the m-critical point isˆw(z) = −mXk=0(1 + (−)k)2m−1/2(2m −1)!k!

(2m −1 −k)! (z + M)m−1/2−k/2(z −M)k/2(3.38)Eq.

(3.38) reproduces the known results (see, for example, [34]) for g = 3/2 (pure gravity)and g = 1/2 (gaussian model)ˆw(z) =−2(2z −M)p(z + M)/2,g = 3/2−2p(z + M)/2,g = 1/2(3.39)At the integer points g = m, m = 1, 2, 3, 4, ... the planar loop amplitude (3.31) is apolynomial and has no cut at all. However,this is not always the physical solution becausethe limit a →0 is taken prior to the limit g →m.

The exact solution for the case g = 2[44] lead to the following expression in the continuum limit 4ˆw(z)g=2 = −X+,−z ±pz2 −M 22logz ±pz2 −M 2=h ddg ˆw(z)ig=2,Λ = M 2(3.40)4The equivalence between the gas of dilute loops with fugacity β = −2 (g = 2) and the bosonicstring embedded in -2 dimensions can be established as follows. If the loops are considered asoriented, then their fugacity is -1.

Further, it is easy to see that the condition of nonintersectioncan be abandoned, since the total contribution of the configurations containing intersecting loopsvanishes because of cancellations. Therefore the partition function can be written as the expo-nential of minus the entropy of a single oriented loop, with no restriction to its configurations.This is exactly the determinant of the Laplace operator in the lattice S, which can be writtenas a Gaussian integral with respect to a couple of grassmanian fields x, ¯x defines on S. Takingthe derivative w.r. to g we subtract the zero mode of the Laplacian.

By the Kirchofftheorem,32

The case g = 1 has been solved by M. Gaudin [43] for the gaussian potential U0 =−1/2Φ2. The imaginary part of ˆw(z) along the cut −∞< z < −M has been found asthe solution of an integral equation with Cauchy kernel.

From this it is not difficult toreconstruct the meromorphic function ˆw(z)ˆw(z)g=1 = −M[τ 2 cosh τ + 2τ log(aM) sinhτ]= −2pz2 −M 2 log(aM) logz +√z2 −M 2M−zhlogz +√z2 −M 2Mi2=h d2dg2 ˆw(z)ig=1 + (const) z,Λ = [M log(aM)]2(3.41)The origin of the logarithmic corrections in these two cases is quite different. In the caseg = 1 this is the appearance of a massless excitation (the “tachyon”).

The divergences dueto this zero mode produce the logarithmic factor in the relation between Λ and M. Thelogarithmic scaling violation is a well known phenomenon in the string theory with X = IR[36], [47]. In the case g = −2 there are no divergences which can produce logarithmiccorrections in the scaling of M. The logarithmic factors in the loop amplitude are relatedto the fact that the partition function vanishes and therefore the loop amplitude is actuallya derivative [42].

Within the matrix-model representation, the vanishing if the partitionfunction can be explained with the occurence of a supersymmetry at the point g = −2[48],[44]. This has nothing to do with the sum over geometries.

The partition functionfor fixed world-sheet geometry is equal to the determinant of the Laplace operator on thecorresponding lattice and vanishes due to its zero mode.As for the higher integer points, g = 3, 4, ..., it is possible that there are no logarithmicsingularities 5 These points will be discussed in [11].We would like to understand why the loop amplitude of the g = 1 string is equal to thesecond derivative in g of the generic solution. For the moment this is just an experimentalfact.4.

The multi-loop amplitudes in the scaling limit4.1. An effective matrix model for the dressed verticesIn order to be able to exploit the functional integral formulation (2.72) of the ADEthe finite part of the determinant is equal to the sum of connected spanning trees on the worldsheet.

Since each spanning tree defines a dense loop on the world sheet [46], the derivative of thepartition function can be also expressed in terms of the gas of dense loops with fugacity β →0.We see that the derivatives of the models g = 1/2 and g = 2 are identical and describe the stringin -2 dimensions.5 Saleur [42] considered the topological points g = 1/2, 1/3, ... which are in a dual to the pointsg = 2, 3, .... He argued that a self-consistent definition of the theory as a derivative in g existsonly for g = 1/2.33

and SOS strings we need explicit expressions for the vertices (2.85). Using the random-matrix representation (2.91),(2.92) for the bare vertices, the L-dependent part (2.86) ofthe dressed vertices can be calculated by means of a random matrix model with a specialpotential.

Namely, the generating function of the dressed verticesU[J] =∞XH=0∞Xn=oN 2−2H−nn!ZW (H)(L1, ..., Ln)nYk=1J(Lk)dLkLk(4.1)is equal to the free energy of the effective matrix modeleU[J] =ZdΦe(NtrU(Φ)+R ∞0trΦ(L)J(L)dL/L(4.2)with non-polinomial potentialU(Φ) = U◦(Φ) + βZ ∞0dLL e−2P0LW (0)(L)treLΦ(4.3)The potential U(P) is analytic in the the complex plane cut along the line 2P0 −PR

For example, loops going around a handle or aboundary are forbidden.The one-loop amplitude in the matrix model (4.2) coincides in the planar limit N →∞with the string background (2.82) (the partition function of the loop gas on a disc)⟨N −1treLΦ⟩= W (0)(L),N →∞(4.5)In order to prove this, it will be sufficient to show that (4.5) satisfies the loop equation(2.94). Indeed, the planar loop amplitude ˆW(P) in any matrix model is related to thepotential U(P) byReW(P) = −12U ′(P),PL < P < PRImW(P) =0,P < PL or P > PR(4.6)With the choice (4.4) of the potential, (4.6) is equivalent to (2.102) and therefore to (2.94).We are interested in the continuum limit of the dressed verticesw(H)(ℓ1, ...ℓn) = a−(g+1)(n−2+2H) W (H)(ℓ1/a, ..., ℓn/a),a →0(4.7)Below we give a method for their evaluation using the random matrix representation (4.2).The idea is to replace the potential (4.3) with a simpler one such that the correspondingmatrix model will generate directly the scaling limit (4.7) of the dressed vertices.34

Such a potential will represent a function u(z, M) with the following scaling propertyu(z, M) = ραu(ρz, ρM)(4.8)Comparing (4.6) and the functional equations (3.26) for the renormalized classicalloop field ˆw(z) , we see that a possible choice of the scaling potential (4.8) isu(z) = βZ ∞0dℓℓw(ℓ)ezℓ;ddz u(z) = β ˆw(−z)(4.9)Let us mention that the inverse problem of finding the potential given the loop amplitudehas many solutions. For example, the multicritical points of pure gravity can be obtainedfrom the polynomial potentials of Kazakov [36] as well as from the non-polynomial singularpotentials of Gross and Migdal [49].We have shown that the dressed vertices in the scaling limit (4.7) can be calculatedas the multi-loop correlators in the effective matrix model with potential (4.9)∞XH=0w(H)(ℓ1, ..., ℓn)N 2−n−2H =ZdΦeβN R ∞0(dℓ/ℓ)w(ℓ)tr exp(ℓΦ)nYk=1treℓkΦ(4.10)The one- and two-loop correlators in the planar limit arew(0)(ℓ) = 1ℓM gKg(Mℓ)(4.11)w(0)(ℓ, ℓ′) =√ll′l + l′ e−M(l+l′)(4.12)The general formula for the planar multi-loop correlators in the matrix model [50][45]readsw(0)(ℓ1, ..., ℓn) =√ℓ1...ℓn(ℓ1 + ...ℓn)−∂∂yn−2e−f(y;M)(ℓ1+...+ℓn)=pℓ1...ℓn∂n−2∂yn−2Z ∞f(y;M)dze−z(ℓ1+...+ℓn)(4.13)where the derivatives are taken at the point y = 0 and the function f(y; M) is determinedby the diffusion equation (Appendix A) corresponding to the potential (4.9)y = limN→∞βZ ∞0dℓw(ℓ)⟨y|ez(−f(t)+(d/dt)2)|y⟩=Z ∞0dℓℓhβM gKg(Mℓ)ihe−f(y;M)ℓ√ℓi(4.14)We assume that the infinite constant produced by the singularity at small distances (ℓ∼a)is subtracted so thatf(0; M) = M(4.15)35

The finite part of the integral can be extracted by means of analytic continuation w.r.to a regulating parameter α. First we multiply the integrand by ℓα+1/2, then calculate theintegral (4.14) in the domain Reα > g where it converges, using the formula [51]Z ∞0dℓℓα−1e−zℓKg(Mℓ)=r π2M (M + z)1/2−α Γ(α + g)Γ(α −g)Γ(α + 1/2)2F1(1/2 + g, 1/2 −g; 1/2 + α; M −z2M)=r π2M (M + z)1/2−α∞Xk=0Γ(1/2 + g + k)Γ(1/2 −g + k)Γ(1/2 + α + k)k!M −z2Mk(4.16)and finally perform analytic continuation to α = 1/2.The result isy =βM gr π2M (M + f)∞Xk=1Γ(1/2 + g + k)Γ(1/2 −g + k)k!

(k −1)!M −f2Mk=2(2π)3/2(g2 −14)M g+1/2∞Xk=11k! (k −1)!Γ(k −1/2 + g)Γ(k −1/2 −g)Γ(1/2 + g)Γ(1/2 −g)M −f2Mk(4.17)(the k = 0 term disappears because of the Γ(0) in the denominator.

)The parameter y is the coupling constant of the matrix model; in terms of the loopgas it is the coupling of a special local operator, to be discussed below.4.2. Some critical exponentsUntil this moment we have considered two of the critical exponents of the loop gasmodel - the string susceptibility γstr and the fractal dimension of the loops DB = 1/νwhich is the same as the fractal dimension of the boundary of the world sheet.

Now weare going to introduce a third exponent DC which has the meaning of fractal dimensionof the connected domains bounded by loops.By construction, the vacuum energy of the matrix model in the planar limit is thepartition function of the gas of nonintersecting loops on the sphere, with one labeled do-main.One can imagine it as the partition function of the gas of small spots (baby universes)floating in the labeled domain. A spot of perimeter ℓrepresents the partition function ofthe loop gas on a disc and have a Boltzmann weight u(ℓ) = βw(ℓ).

The size of thesebaby universes is of the order of the cut-offa because of their entropy being divergent atℓ→0. The measure in the space of world-sheet metrics is tuned to the m-critical phase ofnonembedded random surfaces.

The parameter y makes sense of “cosmological constant”for the labeled domain.36

Let AC and A be the area of the labeled connected domain and the total area of theworld sheet, correspondingly. (Of course, the labeled domain is equivalent to any otherconnected domain on the world sheet.) Then the fractal dimension of a single connecteddomain is determined by the way these two areas grow near the critical point M = 0AC ∼ADC/2(4.18)The area of a single connected domain is measured by the matrix-model puncture operatorPC = −∂∂y(4.19)The coupling constant y of the matrix model scales as M g+1/2; this is sufficient to determinethe fractal dimension DC.

It is convenient to consider the dense and the dilute phasesseparately. (i) Dense phaseIn this phasePC = −∂∂y ⇒AC,P = ∂∂Λ ⇒A(4.20)The cosmological constant Λ is related to y by y = M g+1/2 = Λ1/2+1/(4g) and the fractaldimension of the connected area isDC = 1 + 12g(4.21)For positive β (g > 1/2) all Boltzmann weights are also positive and the gas of loopsallows a statistical interpretation.

Then the area of a connected domain diverges moreslowly than the total areaAconn ∼A1/2+1/4g(4.22)In the dense phase A coincides with the total length of the loops on the world sheet,and Aconn - with the total length of the loops forming the boundary (hull) of a connecteddomain.The scaling (4.22) is confirmed by the following argument based on the Coulomb gaspicture.The susceptibility ∂2U/∂y2 of the matrix model can be interpreted as the two-pointfunction of a special local operator defined on the whole world sheet and not only on thelabeled connected domain. The two-point function of this operator is zero when the twopoints are in different domains and one if they are in the same domain.

This operatorcan be constructed as a vertex operator with electric charge e = g −1 ± 1/2 [52]. Thecorresponding gravitational scaling dimensionδ0,1/2 = 1/2 −|g −1|2g= 1/2 −1/(4g)(4.23)is positive when 1/2 < g < 1.

The dimension 1 −δ0,1/2 = DC/2 of the coupling constantof this operator indeed coincides with the dimension of y.37

(ii) Dilute phaseIn this phase the puncture operator PC of the matrix model scales as Λ−(g+1/2)/2. Inthe m-critical phase PC = Am/2Cand the two areas are related byAC ∼ADC/2;DC = (2g + 1)/m(4.24)In the Coulomb gas picture, the constant y is coupled to the same local operator withgravitational scaling dimensionδ0,1/2 = 1/2 −|g −1|2= 3/4 −g/2(4.25)This time 1 −δ0,1/2 = (m/4)DC.Knowing the area of a connected domain, we can evaluate the characteristic numberof connected domains on the world sheet.

By the Euler theorem, this is also the numberof loops N (including the most external loop representing the boundary). ThereforeN = A/AC ∼A1−DC=A1/2−1/(4g),dense phaseA(m−g−1/2)/m,m-critical dilute phase(4.26)In the interval 1/2 < g < 3/2 where the loop gas has statistical interpretation (all Boltz-mann weights are positive, including these related to the measure over random surfaces)the total number of loops tends to infinity when the critical point M = 0 is approached.The density of loops is proportional to 1/AC and goes down to zero at the critical point.The number of loops becomes one at the half-integer points g = m−1/2, m = 1, 2, 3, ...which are the solutions of the equation β = 0.

These are the multicritical points of puregravity [36] where the only loop is the boundary of the world sheet. At m = 1 and m = 2the dimension δ0,1/2 of the local operator coupled to y is zero and therefore PC = P. In thehigher multicritical points with β = 0 the dimension of the operator PC fits the dimensionδm−1,1 of the Kac table of the model (2, 2m −1)δ0,1/2 = 1/2 −|m −3/2|2= (m −1) −(m −1/2) −|m −3/2|2= δm−1,1(4.27)Therefore PC can be identified with the operator with the most negative dimension in them-critical regime of pure gravity [37][38][40].4.3.

Feynman rulesNow we are able to formulate the Feynman diagram technique for the ADE stringsin a more explicit way.The vertices (2.77) and (2.85) read, in the continuum limitv(H)(x|ℓ1, ..., ℓn) =κSx2H−2+nw(H)(ℓ1, ..., ℓn)(4.28)38

v(H)(p1, ..., pn|ℓ1, ...ℓn) = κn−2+2HN (H)p1...pnw(H)(ℓ1, ..., ℓn)(4.29)where w(H) are the loop amplitudes in the one-matrix model with potential u(z) = β ˆw(−z),ˆw(z) given by (3.31), and the renormalized string interaction constant κ is defined byκ = a−ν(2−γstr)κ0 = a−g−1κ0(4.30)The explicit form of the vertices without handles (H = 0) is given by (4.17) and (4.13).To our knowledge, the explicit formula for the general macroscopic loop amplitudes in theone-matrix model is not yet found. Below we give an argument allowing to imagine thegeneral form of the matrix-model loop amplitudes with H > 0.In the general case the loop amplitudes in the effective matrix model can be con-sidered as loop amplitudes for the gaussian model dressed by tadpoles representing thenongaussian part of the potential (see eq.

(2.86)). From the explicit form of the gaussianloop amplitudes (Appendix B) we conclude thatw(H)(ℓ1, ..., ℓn) =pℓ1...ℓn P (H)n(ℓ1 + ...ℓn)e−M(ℓ1+...+ℓn)(4.31)where P (H)n(ℓ) is a polynomial of degree n −3 + 3H with coefficients depending on g andMP (H)n(ℓ) =n−3+3HXk=0A(H)n;k (g, M)ℓk(4.32)(Note that n −3 + 3H is the complex dimension of the moduli space for surfaces with Hhandles and n punctures).Let us denote the n-loop amplitude in the continuum limit by⟨nYk=1Ψxk(ℓk)⟩(4.33)where Ψx(ℓ) is the loop operator creating a boundary of length ℓat the point x.

For thetree approximation (H = 0) we will use the symbol ⟨...⟩0. The v.e.v.

of the loop operatoris, in the tree approximation,⟨Ψx(ℓ)⟩= 1ℓM gKg(Mℓ)Sx(4.34)The two-loop correlator satisfies the equation⟨Ψx1(ℓ1)Ψx2(ℓ2)⟩=XxCx1xZ ∞ℓ=0dℓw(0)(ℓ1, ℓ)⟨Ψx(ℓ)Ψx2(ℓ2)⟩(4.35)where w(0)(ℓ, ℓ′) is the two-loop amplitude in the effective matrix model. The latter isdiagonalized by Bessel functions [51]w(ℓ, ℓ′) =√ℓℓ′ℓ+ ℓ′ e−M(ℓ+ℓ′) =Z ∞0dE⟨ℓ|E⟩12 coshπE ⟨E|ℓ′⟩(4.36)39

⟨ℓ|E⟩= 2πpπE sinh(πE)KiE(Mℓ);KiE(Mℓ) =Z ∞0dτe−Mℓcosh τ cos(Eτ)(4.37)where the states ⟨ℓ|E⟩form a complete orthonormal system [45]. On the other hand, theconnectivity matrix is diagonalized byCxx′ =XpV x(p)2 cos(πp)V x′(p)(4.38)where the sum includes the allowed momenta p = m/h.Thus the wave functions ofthe closed string states diagonalizing the two-loop correlator are labeled by the “energy-momentum” (E, p) and have a factorized form⟨E, p|ℓ, x⟩= V x(p)pE sinh(πE)KiE(Mℓ)(4.39)The equation for the two-loop correlator becomes algebraic after the diagonalizationand its solution is [10]⟨Ψx(ℓ)Ψx′(ℓ′)⟩0 =Z ∞0dEXp⟨ℓ, x|E, p⟩12 cosh(πE) −2 cos(πp)⟨E, p|ℓ′, x′⟩(4.40)For p = 1/2 (pure gravity) and p = 1/3 (Ising model) the two-loop correlator has been cal-culated from matrix models [45], [53].

One can check that (4.40) reproduces the expressionsfound for these particular cases.Remarkably, the two-loop correlator is universal; it depends on the model only throughits spectrum of allowed momenta. This makes possible to map the Hilbert spaces of theADE strings onto the Hilbert space of the SOS string whose spectrum of excitations coversthe whole interval −1 < p ≤1.

The two-loop amplitude is essentially the propagator inthe string diagram technique. The latter is given by the continuum limit of (2.89)G(p)(ℓ, ℓ′) =Z ∞0dE⟨ℓ|E⟩˜G(E, p)⟨E|ℓ′⟩(4.41)˜G(E, p) = 2 cosh(πE) cos(πp)cosh(πE) −cos(πp)(4.42)The (E, p) space has the geometry of a semi-infinite cylinder.

It is represented by theinfinite (E, p) plane factored by the relation of equivalence(E, p) ≡(E, p + 2)(4.43)The periodicity of the propagator (4.39) is a direct consequence of the discreteness of thecoordinate space X.It follows from the explicit form of the eigenstates (4.37) that the τ-parametrizationof the boundary cosmological constant z (eq. (3.27)) provides a local coordinate dual tothe energy E.40

The configuration space of the string field theory is a direct product of the X andτ spaces.The relation between the ℓand τ representations has been discussed in ref.[53]. It has been argued that the τ variable makes sense of the time variable in the Das-Jevicki approach, and that the transformation from ℓto τ is analogous to the B¨acklundtransformation in the Liouville theory.

This transformation reads explicitely [53]KiE(Mℓ) =Z ∞0dτe−Mℓcosh τ cos(Eτ)cos EτE sinh πE =Z ∞0dℓℓe−Mℓcosh τKiE(Mℓ)(4.44)and does not relate delta-function normalized bases of wavefunctions.Let us consider the case when the target space is the discretized real line ZZ ⊂IR. Theinverse propagator (4.42) acts in the (x, τ) space as a differential operator of infinite orderˆG−1 = [cosh(π∂/∂x)]−1 −[cos(∂/∂τ)]−1(4.45)This operator can be also interpreted as a finite-difference operator in the x and iτ di-rections.

Thus Minkowski rotation τ →t = iτ of the ZZ string is described by a latticeHamiltonian. The saddle-point equation (3.29) means that the classical string backgroundis annihilated by the Hamiltonian (4.45).It will be convenient to apply the Feynman rules directly in the (E, p) space.

TheFourier image of the ℓ-dependent part of the vertices (4.31) can be found using the inte-gration formulaZ ∞0dℓℓ√ℓe−Mℓℓk KiE(Mℓ) =√π2Mk+1/2Γ(k + 1/2 + iE)Γ(k + 1/2 −iE)Γ(k + 1)(4.46)which follows from (4.16). The result is˜v(H)(p1, ..., pn|E1, ..., En) = κn−2+2HNp1...pn ˜w(H)(E1, ..., En)(4.47)˜w(H)(E1, ..., En) =Z ∞0nYk=1dℓkℓk⟨Ek|ℓk⟩w(H)(ℓ1, ..., ℓn)=n−3+3HXm=0Xm1+...+mn=m; mk≥0m!

A(H)n;m(g, M)×nYk=1pEk sinh(πEk) Γ(1/2 + mk + iE)Γ(1/2 + mk −iE)mk! mk!

(4.48)The vertices in the (E, p) space are given by the product of the r.h.s.of (4.48)and (2.78). Each vertex represents a symmetric polynomial in E1, ..., En times a factorpπEk sinh(πEk)/ cosh(πEk) for each line.41

It is convenient to absorbe the non-polynomial factors in the propagators; then thenew propagator and vertices have the following form˜Gnew(E, p) =psinh(πE)√πE cosh(πE)2 cos(πp) cosh(πE)cosh(πE) −cos(πp)psinh(πE)/Ecosh(πE)= G♥(E, p) −G♥(E, 1/2)(4.49)G♥(E, p) = sinh(πE)πE2cosh(πE) −cos(πp) =∞Xn=−∞1E2 + (p + 2n)2(4.50)˜V (H)new (p1, ..., pn|E1, ..., En) = N (H)p1...pnn−3+3HXm=0Xm1+...+mn=m; mk≥0m! A(H)n;mnYk=1Ek(mk!

)2mk−1Ys=0[E2k + (s + 1/2)2](4.51)With the Feynman rules (4.49), (4.51) the evaluation of the loop amplitudes can be doneas in an ordinary quantum field theory 6. The Greens functions of this effective field theoryare the correlation functions of the Fourier-transformed loop field˜Ψ(E, p) =Xx(Sx)2Z ∞0dℓℓχx(p)Ψx(ℓ)(4.52)˜Ψ(E, p)new =√E cosh(πE)psinh(πE)˜Ψ(E, p)(4.53)Let us calculate, for example, the three-loop correlator of the loop field Ψ(p)(ℓ) =Px χx(p)Ψx(ℓ).

We start with the corresponding Greens function in the (E, p) space whichis given by a single Feynman diagram⟨3Yk=1Ψ(Ek, pk)⟩0 = Cp1p2p3M −(1+g)3Yk=1˜G(Ek, pk)(4.54)Then we integrate w.r. to the E variables associated with the three legs according to theformulaZ ∞0dEE sinh(πE)cosh(πE) −cos(πp)KiE(Mℓ) = MℓK1−|p|(Mℓ)(4.55)to find⟨3Yk=1Ψ(pk)(ℓk)⟩= Λ−(1−γstr/2)Cp1p2p33Yk=1ℓkK1−|pk|Mℓk(4.56)In the particular case p1 = p2 = p3 = p0 corresponding to three loops associated with theidentity operator, eq. (4.56) is in accordance with the the formula conjectured in [9].6The form of the propagator (4.49) suggests an interpretation of the string field as a collectionof two modes: a bosonic relativistic particle with compactified momentum space in the space-likedirection and a ghost excitation (twist) with spectrum of momenta p = ±1/242

4.4. SpectroscopyThe spectrum of on-shell states is given by the poles of ˜G(E, p).

They form the lightcone in the (E, p) cylinder (Fig. 8)E = ±iεn(p),εn(p) = |p + 2n|; n = 0, ±1, ±2, ...(4.57)The periodicity in the momentum space is a consequence of the discreteness of thecoordinate Xspace.

If the X-space is compact, then the momentum space is discreteas well and the on-shell states form an infinite two-dimensional lattice on the space-timecylinder.−1pE10Fig. 8: The light cone for a compact momentum space.The edges of the strip correspond to identical momenta p = ±1.Each on-shell state creates a “microscopic loop” on the world sheet and correspondsto a local scaling operator.

The sequence of poles corresponding to a given momentum pdefines the set of local operators O(p);n; n = 0, ±1, ±2, ... having nonvanishing correlationswith the vertex operatorV(p) ≡O(p);0 =Xxχx(p)Px(4.58)where Px is the operator creating a puncture in the world sheet with coordinate x. Theseoperators are the coefficients in the expansion of the off-shell loop field Ψ(p)(ℓ) as a seriesof on-shell wave functions [45]Ψ(p)(ℓ) =∞Xn=−∞(|p| + 2n)M ||p|+2n|I||p|+2n|(Mℓ)O(p);n(4.59)The expansion of the two-loop correlator as a sum of products of on-shell wave func-tions reads⟨Ψ(p)(ℓ)Ψ(p)(ℓ′)⟩=∞Xn=−∞|p| + 2nsin(πp) K|p|+2n(Mℓ) I|p+2n|(Mℓ′),ℓ′ < ℓ(4.60)43

Eqs. (4.59) and (4.60) imply⟨O(p);nΨ(p′)(ℓ)⟩= M ||p|+2n|K|p|+2n(Mℓ)(4.61)A direct derivation of (Laplace transform of ) (4.61) using the loop equations can be foundin [1] and [9].The gravitational dimension δn(p) of the local operator V(p);n can be found from (4.61)M ||p|+2n| = Λδn(p)−γstr/2 ⇒δn(p) = ||p| + 2n| −|p0|2ν(4.62)where p0 = g −1 is the background momentum (2.57) (see (3.18) , (3.22)).The dimensions of the operators O(p);n with p = kp0 , k - integer, are contained inthe KPZ-DDK spectrum (2.64)δ(Orp0;n) = δk+2n,k(4.63)4.5.

Partition function of the noninteracting stringThe Feynman rules in the continuum limit are generated by the functional integraleF [J] =ZdΨeA[Ψ]A[Ψ] =−12XpZ ∞0dℓℓΨ(p)(ℓ)Z ∞0dE⟨ℓ|E⟩1cos πp −1cosh πE⟨E|ℓ′⟩Ψ(p)(ℓ′)+∞Xn=2∞XH=0Xp1,...,pn1n!Z ∞0κn−2+2HN (H)p1,...,pnP (H)n(ℓ1 + ... + ℓn)nYk=1Ψ(pk)(ℓk) dℓk√ℓk(4.64)which formally defines the string field theory with target space X. The partition functionF (1) of the free string (topology of a torus) is closely related to the vacuum energy of thisfield theory.The loop expansion for the embedded surfaces with the topology of a torus involvesloop configurations with contractible and noncontractible loops.

It is easy to see that thelogarithm of the determinant of the kernel Gxx′(ℓ, ℓ′) is equal to twice the contribution ofthe surfaces in the loop expansion containing at least one noncontractible loop. Indeed, ifwe return to the original diagram technique (subsec.

2.5) and retain only the gaussian partof the action, the contribution of the gaussian fluctuations of the Ψ-field to the vacuumenergy will be given by the sum of all Feynman diagrams with the topology of a tree withone cycle. Each cycle corresponds to a closed path in X and consists of even number ofpoints (bare vertices).

A cycle with 2k points should be taken with a symmetry factor 2k.44

Let us denote by F[k;p] the contribution of the toroidal surfaces (H = 1) containingexactly k noncontractible loops with momentum p. Then the partition function for fixedmomentum p is equal toF(p) =∞Xk=0F[k;p]=F[0] + 12∞Xk=112kXpZ ∞adℓℓZ ∞0dEπ ⟨l|E⟩ cos(πp)cosh(πE)2k⟨E|ℓ⟩=F[0] −Z ∞adℓℓ|⟨ℓ|E⟩|2 log1 −cos2(πp)cosh2(πE)(4.65)The contribution of the surfaces without noncontractible loops is equal to the trace ofthe identity operator in the X-space (= the number of its points) times the 1/N 2-correctionto the free energy of the effective matrix model considered in the beginning of this section.A simple calculation (see, for example Appendix C of [49]) leads toF[0] = g −1/224log(aM)(4.66)The integral over the length ℓin (4.65) yields a factor log[1/(aM)] (this is the di-agonal value of the kernel of the regularized identity operator in the E-space) and theE-integration givesZ ∞0dE log1 −cos2(πp)cosh2(πE)= π2 (1/2 −|p|)2(4.67)Inserting this in (4.65) we find for contribution to the partition function of an excitationwith momentum pF(p) = log(aM)g −1/2 + 6(1/2 −|p|)224(4.68)Let us give some examples. (i) The SOS string compactified on ZZ2hSumming over the allowed momenta p = m/h; m = 0, ±1, ..., ±(h −1), h in theembedding space X = ZZ2h we findF2h(g) = log(aM)h2 + 224h+ 2h g −1/224= log(aM)gh + 1/h12(4.69)The ZZ2h string is well defined if the background momentum p0 belongs to the spectrumof allowed momenta.

Therefore the possible values of g areg = m ± kh;m = integer, 0 ≤k < h(4.70)45

The partition function of the string embedded in ZZ2h is symmetric under the dualitytransformationg →1/g, h →1/h, M →M g(4.71)relating the dense and dilute phases. Due to the different scaling in these two phases, thecosmological constant is invariant under the duality transformation.

The self-dual point isg = 1, h = 1. It corresponds to the level-one representation of an SU2 current algebra.

TheKosterlitz-Thouless transition may occur at g = 1, h = 2 and its dual point g = 1, h = 1/2.The case h = 2 describes the ZZ4 model which is a particular case of the Ashkin-Tellermodel. At this point the magnetic (vortex) operator with discontinuity m = 4 (= theperiod in the X space) becomes marginal: its conformal dimension (2.46) is 1.

The dualmodel is formally the ZZ1 model having as a target space a graph with one point and oneloop. The space of paths in this space is not trivial because each step is made by traversingthe loop in one of the two possible directions.

This ZZ1 string is equivalent to the n = 2limit of the O(n) string introduced in [33]. The vortex operators in the O(n) string are theelectric operators in the ZZ1 string.

The spectrum of momenta consists of all integers andthe marginal operator is the vertex operator with p = 2 (= the period in the momentumspace). (ii) The RSOS stringsThe embedding space X = Ah−1 has spectrum of momenta p = m/h; m = 1, 2, ..., h−1and the partition function reads 7FAh−1 =(h −1)(h −2)12hlog(aM)=h −224log(a2(1−1/h)Λ),dense phase (g = 1 −1/h)(4.72)FAh−1 =h(h −1)12hlog(aM)=h −124log(a2Λ),dilute phase (g = 1 + 1/h)(4.73)In both critical regimes the partition function of the Ah−1 string is equal to the differencebetween the partition functions of the ZZ2h and ZZ2 stringsFAh−1 = 12F2h(g = 1 ± 1/h) −F2(g = 1 ± 1/h)(4.74)The relation (4.74) takes place even before performing the sum over the world sheet ge-ometries [1].In a similar way one can calculate the partition functions of the D and E strings.In the A and D cases the expressions will coincide with these obtained from the matrixmodels in the formulation of M. Douglas [54] by Di Francesco and Kutasov [55].

All thesepartition functions can be expressed as linear combinations of partition functions of ZZnstrings [1]. The formulas are the same as these for the regular lattice [12].

The Coulombgas calculation of the genus-one partition function of the ADE strings was presented in[56].7 Here we restrict ourselves to the unitary theories with 0 < g < 2.46

5. Concluding remarksWe have shown that the propagation and interactions of strings embedded in a discreteone-dimensional target space are described by string field theory with propagator (4.42)and vertices (4.48).We were not able to calculate explicitely the vertices with highertopology but they are defined unambiguously by the matrix integral (4.2).Our diagram technique based on the loop-gas representation of the IRF models, pro-vides a natural decomposition of the moduly space into elementary cells.

The verticesrepresent the topology-changing amplitudes involving no propagation in the target space.The fact that the vertices are themselves loop amplitudes guarantees the “stringy” largeorder behaviour [57]. The vertices included in a Feynman diagram cut a number of holes inthe moduli space such that the rest of it is a direct product of one-dimensional spaces (onefor each propagator).

The degenerated surfaces appear when the proper time for someof the propagators becomes infinite (the proper time is measured by the number of thenoncontractible loops along the corresponding cylindric surfaces). This decomposition ofthe moduli space is well defined only if all proper times are nonzero (i.e., if there is at leastone noncontractible loop in each of the channels.

The situation when one of the propertimes vanish is taken into account by replacing the two involved vertices with a new vertexof higher topology. Thus the interaction of the closed string is described by a collectioninfinitely many vertices, one for each topology.

This is possibly a general feature of theperturbative expansion for any string theory [58].It would be interesting to compare our diagram technique for the loop amplitudes tothe one for the string embedded in IR. The two-loop correlator for the string embedded inIR [45][59], [53].˜G(E, p) =Esinh(πE)1E2 + p2(5.1)is compatible with (4.40) only in the limit of small pandE.

The propagator of theIR-string contains an infinite sequence of poles at integer imaginary values of E which havebeen interpreted as the special stationary states 8 discovered by A. Polyakov [60]. Thesestates are created by infinitesimal motion in the X space; in the Coulomb gas picture theyare constructed by taking the derivatives of the x-field.

Therefore, such states should notexist in a string theory with discrete target space. The special states in the IR-string arerelated to the closed classical trajectories in the imaginary time direction.The infinite spectrum of states in our case comes from the fact that the discreteness ofthe X-space leads to periodicity in the momentum space.

As a consequence, the light conemaps an infinite set of energies to the same momentum. These energies make the towerof gravitational descendents of the vertex operator with given momentum.

The classicalmotion in the imaginary time direction is gouverned by a finite-difference Hamiltonian.The fact that the difference between the embedding spaces ZZ and IR survives in thecontinuum limit can be explained. If we construct the D=1 string as an infinite chainof coupled matrices [61] [62], then the IR-string will describe the low temperature phase8Since these “states” can be elliminated by redefinition of the vertices, their physical meaningis not very clear to us.47

of the matrix chain and the ZZ-string will describe the Kosterlitz-Thouless point wherethe distance between two nearest neighbours on the chain remains finite in the continuumlimit. The change in the critical behaviour of the matrix chain near the Kosterlitz-Thoulesspoint is due to the liberation of the angular excitations [63] [64]AcknowledgementsI thank M. Douglas, P. Di Francesco, B. Duplantier, G. Moore, A. Polyakov, S.Shenker, M. Staudacher and A. Zamolodchikov for many stimulating discussions.

I amgrateful to J.-B. Zuber for a critical reading of the manuscript.Appendix A.

The method of orthogonal polynomialsThe integral over Hermitean N × N matriceseN2F =ZdΦeNtrU(Φ)(A.1)can be written as an integral w.r.t. the eigenvalues φn, n = 1, ..., N, of the matrix ΦeN2F =ZNYk=1dφkeU(φk) Yi

The one-fermion wave functions are of the form⟨φ|n⟩= Pn(φ)eU(φ)/2(A.3)where Pn(φ) is a polynomial of degree n. The polynomials are fixed by the condition oforthonormality⟨m|n⟩=ZdφPm(φ)Pn(φ)eU(φ) = δmn(A.4)The ground state of the system of fermions is equal to the antisymmetrized product of thestates |n⟩, n = 1, ..., NΞ(φ1, ..., φN) = 1N!Xσ∈SN(−1)[σ]⟨1|φσ1⟩...⟨N|φσN⟩(A.5)The operator Φ is represented in the space of one-fermion states is represented by thetridiagonal Jacobi matrix which we denote by the same letterΦ|n⟩= Rn+1|n + 1⟩+ Rn|n −1⟩+ Sn|n⟩(A.6)The nonzero matrix elements Sn and Rn = An−1/An of this matrix are fixed by therecurrence relations⟨n|ΦU ′(Φ)|n⟩= (2n + 1)/N(A.7)48

⟨n|U ′(Φ)|n⟩= 0(A.8)which can be interpreted in terms of diffusion of a particle with n-dependent hoppingparameter. In the limit N →∞we can replace Rn, Sn byR = limn→∞Rn, S = limn→∞Sn(A.9)and the resolvent of the Jacobi matrix is given by the propagator of a one-dimensionalrelativistic particleRnn′(P) = ⟨n|1P −Φ|n′⟩= P −S −p(P −S)2 −4R22R!|n−n′|1p(P −S)2 −4R2(A.10)Another useful quantity is the inverse Laplace transform of (A.10)⟨n|eLΦ|n′⟩= eSLI|n−n′|(2RL)(A.11)where In(z) is a modified Bessel function.

All quantities characterizing the radial part ofthe random matrix can be expressed in terms of (A.10) or (A.11). For example, the one-and two-loop correlators are given byW(L) = 1NNXn=1⟨n|eLΦ|n⟩= eSL 1NNXn=1I0(2RnL)(A.12)W(L, L′) =Xn>N,n′

Using the explicit expression (A.10) forthe resolvent we findZ PRPLdPPU ′(P)p(P −S)2 −4R2 = π(1 −y)(A.16)Z PRPLdPU ′(P)p(P −S)2 −4R2 = 0(A.17)49

The positions of the branchpoints of W are therefore given byPL = S −2R,PR = S + 2R(A.18)It follows by (A.12) thatW(L) = −Z 10dy eS(y)LI0(2R(y)L)(A.19)W(P) = −Z 10dy1p(P −PL(y))(P −PR(y))(A.20)The two-loop correlator is obtained immediately from (A.13)W(L, L′) = eS(L+L′)∞Xn=1nIn(2RL) In(2RL′)(A.21)Its Laplace image depends only on the positions of the endpoints of the cut and is givenbyW(P, P ′) =q(P −PR)(P ′−PL)(P ′−PR)(P −PL) +q(P ′−PR)(P −PL)(P −PR)(P ′−PL) −24(P −P ′)2. (A.22)In the continuum limitL = ℓ/a,P = P ∗R + az,PR(y) = P ∗R −af(y),f(0) = M(A.23)eqs.

(A.16) and (A.17) implyZ ∞f(y)dzpa(z + M)u′(−z) = y(A.24)where u(z) = U(P ∗R + az). If the potential u(z) scales as aα, the dependence on the cutoffa can be elliminated by rescaling u(z) →aαu(z), y →aα−1/2y.

Eq. (A.24) is then theLaplace image of (4.14) for the potential (4.9) .Finally, the two-loop correlator in the scaling limit readsw(ℓ, ℓ′) =√ℓe−M(ℓ+ℓ′)ℓ+ ℓ′√ℓ′(A.25)w(z, z′) =√z+M√z′+M +√z′+M√z+M −24 (z −z′)2= −∂∂z∂∂z′ log(√z + M +√z′ + M)(A.26)In the case of gaussian potential U(Φ) = −Φ2/2 the one-loop amplitude can be cal-culated explicitely in all orders in κ in the continuum limit [65] [66]w(ℓ) =Z −M−∞dy⟨y|el(y−κ2(∂/∂y)2)|y⟩= ℓ−3/2e−Mℓ+κℓ3.

(A.27)50

References[1]I. Kostov, Nucl. Phys.

B 326 (1989)583[2]V. Kazakov, Phys. Lett.

150B (1985) 282; F. David, Nucl. Phys.

B 257 (1985) 45; V.Kazakov, I. Kostov and A.A. Migdal, Phys. Lett.

157B (1985),295; J. Ambjorn, B.Durhuus, and J. Fr¨ohlich, Nucl. Phys.

B 257 (1985) 433[3]E. Br´ezin, C. Itzykson, G. Parisi and J.-B. Zuber, Comm.

Math. Phys.

59 (1978) 35;D. Bessis, C. Itzykson and J.-B. Zuber, Adv.

Appl. Math.

1 (1980) 109; C. Itzyksonand J.-B. Zuber, J.

Math. Phys.

21 (1980) 411[4]S. Das and A. Jewicki, Mod. Phys.

Lett. A5 (1990) 1639; K. Demeterfi, A. Jevicki andJ.

Rodrigues, Brown Univ. preprint, BROWN-HET-795, February 1991[5]V. Knizhnik, A. Polyakov and A. Zamolodchikov, Mod.

Phys. Lett.

A3 (1988) 819.[6]F. David, Mod.

Phys. Lett.

A3 (1988) 1651; J. Distler and H. Kawai, Nucl. Phys.

B321 (1989) 509[7]R. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982[8]I.Kostov, “Strings embedded in Dynkin diagrams”, Lecture given at the Cargese meet-ing, Saclay preprint SPhT/90-133[9]I. Kostov, Phys. Lett.

B 266 (1991) 317.[10]I. Kostov, Phys.

Lett. B 266 (1991) 42.[11]I.

Kostov and M. Staudacher, to be published[12]P. di Francesco, H. Saleur, and J. B. Zuber, Journ.

Stat. Phys., 49 (1987) 57[13]O.Foda and B. Nienhuis, Nucl Phys.B 324 (1989) 643[14]A. Polyakov, Phys.

Lett. 103 B (1981) 207, 211.[15]A.

Cappelli, C. Itzykson and J. B. Zuber, Comm.

Math. Phys.

113 (1987) 113[16]V. Pasquier, Nucl. Phys.

B285 (1987) 162; J. Phys. 35 (1987) 5707[17]G. Andrews, R. Baxter , and P. Forrester, J. Stat.

Phys. 35 (1984) 35[18]F.M.

Goodman, P. de la Harpe and V.F.R. Jones, Coxeter - Dynkin diagrams andtowers of algebras, vol.

14, Mathematical Research Institute Publications, SpringerVerlag 1989[19]A. Belavin, A. Polyakov, and A. Zamolodchikov, Nucl. Phys.

B 241 (1984) 333[20]V.Dotsenko and V. Fateev, Nucl. Phys.

B 240 (1984) 312[21]L.Kadanoff, J. Phys. A 11 (1978)1399; H. Knops, Ann.

Phys. 128 (1981) 448; M. denNijs, Phys.

Rev. B27 (1983)1674[22]B. Nienhuis, in Phase Transitions and Critical Phenomena, Vol.

11, C.C. Domb andJ.L.

Lebowitz, eds.) Ch.

1 (Academic Press, 1987)[23]V.Dotsenko and V. Fateev, Nucl. Phys.

B 240 (1984)312; 251 (1985)691[24]J.-L. Gervais and A. Neveu, Nucl. Phys.

B 199 (1982) 50[25]A.B. Zamolodchikov, unpublished[26]P. di Francesco, H. Saleur, and J.-B.

Zuber, J. Stat. Phys.

49 (1987) 5751

[27]A.M. Polyakov, Phys.Lett. 103 B (1981) 207[28]N. Seiberg, Rutgers preprint RU-90-29[29]J. Polchinski, Nucl.

Phys. B346 (1990) 253[30]I. Kostov, Phys.

Lett. 189 B (1987) 118[31]E. Witten, Nucl.

Phys. B 340 (1990)281; R. Dijkgraaf and E. Witten, Nucl.

Phys.B342 (1990) 486[32]V. Kazakov and A.A. Migdal, Nucl. Phys.

B 311 (1988) 171[33]I. Kostov, Mod. Phys.

Lett. A4 (1989)217[34]F. David, Mod.

Phys. Lett.

A, No.13 (1990) 1019[35]A.A. Migdal, Phys. Rep. 102 (1983) 199[36]V. Kazakov, Mod.

Phys. Lett.

A 4 (1989) 2125[37]M. Staudacher,Nucl. Phys.

B 336 (1990) 349[38]E. Br´ezin, M. Douglas, V. Kazakov, S. Shenker, Rutgers preprint RU-89-47 (1989)[39]D. Gross and A. Migdal, Nucl. Phys.

B340 (1990) 333[40]C. Crnkovic, P. Ginsparg, and G. Moore, Phys. Lett.

237 B (1990) 196[41]M. Gaudin and I. Kostov, Phys. Lett.

B220(1989)200[42]H. Saleur, Chicago preprint EFI 90-92, December 1990; Nucl. Phys.

B 360 (1991) 219[43]M. Gaudin, unpublished[44]I. Kostov and M. Mehta, Phys. Lett.

B189 (1987)118[45]G.Moore, N.Seiberg and M. Staudacher, Nucl. Phys.

B 362 (1991)665[46]B. Duplantier and F. David, J. Stat. Phys.

51 (1988) 327[47]E. Br´ezin, V. Kazakov and Al. Zamolodchikov, Nucl.

Phys. B 338 (1990) 673; D. Grossand N. Miljkovic, Phys.

Lett. 238 B (1990) 217; P. Ginsparg and J. Zinn-Justin, Phys.Lett.

240 B (1990) 333; G. Parisi, Phys. Lett.

238 B (1990) 209[48]F. David, Phys. Lett.

B 159 (1985) 303[49]D. Gross and A. Migdal, Phys. Rev.

Lett. 64 (1990) 127.[50]J.

Ambjorn, J. Jurkiewicz and Yu. Makeenko, Phys.

Lett. B 251 (1990)517[51]I.Gradshtein and I. Ryzhik, Table of Integrals, Series, and Products, Academic press,1965[52]B.Duplantier and H. Saleur, Phys.

Rev. Lett.

63 (1989) 2536; B. Duplantier, Phys.Rev. Lett.

64 (1990)493[53]G. Moore and N. Seiberg, “From loops to fields in 2D quantum gravity” , preprintRU-91-29 and YCTP-P19-91[54]M. Douglas, Phys. Lett.

238B (1990) 176.[55]P. di Francesco and D. Kutasov, Nucl.

Phys. B 342 (1990) 589[56]M. Bershadski, I. Klebanov, Nucl.

Phys. B 360 (1991) 559[57]S. Shenker, Rutgers preprint RU-90-47 (1990)[58]H. Sonoda and B. Zweibach, Nucl.

Phys. B 331 (1990) 592; B. Zweibach, MIT preprintCPT-1926, December 199052

[59]G. Moore, Rutgers preprint RU-91-12[60]A. Polyakov, Mod. Phys.

Lett. A6 (1991) 635[61]D. Gross and I. Klebanov, Nucl.

Phys. B 344 (1990) 475[62]G. Parisi, Phys.

Lett. B 238 (1990) 213[63]D.Gross and I. Klebanov, Nucl.

Phys. B 354 (1991)459[64]V. Kazakov and D. Boulatov, ENS preprint, LDT-ENS-91/24 and KUNS-1094 HE(TH) 91/14, August 1990[65]F. David, unpublished[66]E. Br´ezin and V. Kazakov, Phys.

Lett. B 236 (1990) 14453

---

출처: arXiv:9112.059 • 원문 보기