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STRING THEORY IN TWO DIMENSIONS

arXiv:hep-th/9108019v2 25 May 2003PUPT-1271July 1991STRING THEORY IN TWO DIMENSIONS⋆Igor R. Klebanov†Joseph Henry LaboratoriesPrinceton UniversityPrinceton, New Jersey 08544ABSTRACTI review some of the recent progress in two-dimensional stringtheory, which is formulated as a sum over surfaces embedded in onedimension.1. INTRODUCTION.These notes are an expanded version of the lectures I gave at the 1991 ICTPSpring School on String Theory and Quantum Gravity.

Here I have attempted toreview, from my own personal viewpoint, some of the exciting developments in two-dimensional string theory that have taken place over the last year and a half. Becauseof the multitude of new results, and since the field is still developing rapidly, a com-prehensive review must await a later date.

These notes are mainly devoted to thematrix model approach1 that has truly revolutionized the two-dimensional Euclideanquantum gravity. Recently this approach has led to the exact solution of quantumgravity coupled to conformal matter systems with c ≤1.2−6These lectures are about the c = 1 model5,6 that is both the richest physicallyand among the most easily soluble.

This model is defined by the sum over geometries⋆Lectures delivered at the ICTP Spring School on String Theory and Quantum Gravity, Trieste,April 1991.† Supported in part by DOE grant DE-AC02-76WRO3072 and by an NSF Presidential YoungInvestigator Award.

embedded in 1 dimension.The resulting string theory is, however, 2-dimensionalbecause the dynamical conformal factor of the world sheet geometry acts as an extrahidden string coordinate.7In fact, this is the maximal bosonic string theory thatis well-defined perturbatively. If the dimensionality is increased any further, then atachyon appears in the string spectrum and renders the theory unstable.

The matrixmodel maps the theory of surfaces embedded in 1 dimension into a non-relativisticquantum mechanics of free fermions,8from which virtually any quantity can becalculated to all orders in the genus expansion. The second-quantized field theoryof non-relativistic fermions can be regarded here as the exact string field theory.

Atransformation of this theory to a rather simple interacting boson representation9,10will also be given.I will argue that the 2-dimensional string theory is the kindof toy model which possesses a remarkably simple structure, and at the same timeincorporates some of the physics of string theories embedded in higher dimensions.The simplicity of the theory is apparent in the matrix model approach, but much ofit remains obscure and mysterious from the point of view of the continuum Polyakovpath integral. Until we develop a better insight into the “miracles” of the matrixmodels, we may be sure that our understanding of string physics is very incomplete.These notes are mainly a review of published papers, but I also included a fewnew observations and results.

In section 2 I review the formulation of quantum grav-ity coupled to a scalar field as a sum over discretized random surfaces embedded in 1dimension. I will show that the matrix quantum mechanics generates the necessarystatistical sum.

In section 3 I exhibit the reduction to free fermions and define thedouble-scaling limit.5 The sum over continuous surfaces is calculated to all orders inthe genus expansion. In section 4 I show how to calculate the exact string amplitudesusing the free fermion formalism.

Section 5 is devoted to the continuum path integralformalism11where the Liouville field acts as an extra dimension of string theory.Some of the exact matrix model results will be reproduced, but this approach stillfalls far short of the power of the matrix model. In section 6 I discuss the specialstates which exist in the spectrum in addition to the massless “tachyon”.Thesestates, occurring only at integer momenta, are left-overs of the transverse excitations2

of string theory. Remarkably, they generate a chiral W1+∞algebra.

In section 7 Iuse the matrix model to formulate the exact string field theory both in the fermionicand in bosonic terms. I present some manifestly finite bosonic calculations which arein perfect agreement with the fermionic ones.

In section 8 I discuss the new physicaleffects that arise when the random surfaces are embedded in a circle of radius R: theR →1/R duality and its breaking due to Kosterlitz-Thouless vortices.6 This modelcan be regarded either as a compactified Euclidean string theory or as a Minkowskisignature string theory at finite temperature. In section 9 I show that in the matrixquantum the effects of vortices are implemented by the states in the non-trivial repre-sentations of SU(N).6,12 Including only the SU(N) singlet sector gives the vortex-freecontinuum limit where the sum over surfaces respects R →1/R duality.6 In section10 I use the thermal field theory of non-relativistic fermions to find exact amplitudesof the compactified string theory.13 Finally, in section 11 I use the matrix chain modelto solve a string theory with a discretized embedding dimension.6 Remarkably, whenthe lattice spacing is not too large, this theory is exactly equivalent to string theorywith a continuous embedding.2.

DISCRETIZED RANDOM SURFACESAND MATRIX QUANTUM MECHANICS.In this section I will introduce the discretized approach to summation over randomsurfaces embedded in one dimension and its matrix model implementation. If weparametrize continuous surfaces by coordinates (σ1, σ2), then the Euclidean geometryis described by the metric gµν(σ), and the embedding – by the scalar field X(σ).Thus, the theory of random surfaces in one embedding dimension is equivalent to 2-dquantum gravity coupled to a scalar field.

In the Euclidean path integral approach tosuch a theory, we have to sum over all the compact connected geometries and theirembeddings,Z =XtopologiesZ[Dgµν][DX]e−S . (2.1)The integration measure is defined modulo reparametrizations, i.e.

different func-3

tions describing the same geometry-embedding in different coordinates should not becounted separately. We assume the simplest generally covariant massless action,S = 14πZd2σ√g( 1α′gµν∂µX∂νX + ΦR + 4λ) .

(2.2)In 2 dimensions the Einstein term is well-known to give the Euler characteristic, thetopological invariant which depends only on the genus of the surface h (the numberof handles),14πZd2σ√gR = 2(1 −h) . (2.3)Thus, the weighting factor for a surface of genus h is (exp Φ)2h−2.

Since the sum overgenus h surfaces can be thought of as a diagram of string theory with h loops, weidentify the string coupling constant as g0 = eΦ.The main problem faced in the calculation of the Euclidean path integral of eq. (2.1) is the generally covariant definition of the measure for the sum over metrics[Dgµν] and of the cut-off.

One may attempt to do this directly in the continuum.Considerable success along this route has been achieved when metrics are described inthe light-cone gauge,14 or in the more traditional conformal gauge.15 We will carryout some comparisons with the continuum approach in the course of the lectures.The main subject of this section is a different approach to summing over geome-tries, which has so far proven to be far more powerful than the continuum methods.In this approach one sums over discrete approximations to smooth surfaces, and thendefines the continuum sum by taking the lattice spacing to zero.1We may, for in-stance, choose to approximate surfaces by collections of equilateral triangles of side a.A small section of such a triangulated surface is shown in fig. 1.

The dotted lines showthe lattice of coordination number 3, which is dual to the triangular lattice. Each faceof the triangular lattice is thought of as flat, and the curvature is entirely concentratedat the vertices.

Indeed, each vertex I of the triangular lattice has a conical singularitywith deficit angle π3(6−qI), where qI is the number of triangles that meet at I. Thus,4

IJjiFig. 1.

A small section of triangulated surface. Solid lines denote the trian-gular lattice Λ, and dotted lines – the dual lattice ˜Λ.the vertex I has a δ-function of curvature with positive, zero, or negative strengthif qI < 6, = 6, > 6 respectively.

Such a distribution of curvature may seem like a“poor man’s version” of geometry. However, in the continuum limit the size of eachface becomes infinitesimal, and we may define smoothed out curvature by averagingover many triangles.

In this way, the continuum field for the metric should appearsimilarly to how the quantum field description emerges in the continuum limit of themore familiar statistical systems, such as the Ising model, the XY model, etc. Lateron, we will show some strong indications that the discretized approach to summingover geometries is indeed equivalent to the continuum field-theoretic approach.The essential assumption of the discretized approach is that sum over genus hgeometries,R[Dgµν]h, may be defined as the sum over all distinct lattices Λ, with thelattice spacing subsequently taken to zero.R[DX] is then defined as the integral overall possible embeddings of the lattice Λ in the real line.

For simplicity, the latticesΛ may be taken to be triangular, but admixture of higher polygons should not affectthe continuum limit. In order to complete the definition of the discretized approach,5

we need to specify the weight attached to each configuration. This can be defined bysimply discretizing the action of eq.

(2.2) and counting each distinct configurationwith weight e−S. We will find it convenient to specify the embedding coordinatesX at the centers of the triangles, i.e.

at the vertices i of the dual lattice. Then thediscretized version of14πα′Rd2σ√ggµν∂µX∂νX is ∼P(Xi −Xj)2, where thesum runs over all the links < ij > of the dual lattice.

Similarly,Zd2σ√g →√34 a2V(2.4)where V is the number of triangles, or, equivalently, the number of vertices of thedual lattice ˜Λ. Thus, the discretized version of the path integral (2.1) isZ(g0, κ) =Xhg2(h−1)0XΛκVVYi=1ZdXiY⟨ij⟩G(Xi −Xj) ,(2.5)where Λ are all triangular lattices of genus h, κ ∼exp(−√3λa2/4π), and (for somechoice of α′) G(X) = exp(−12X2).

If the continuum limit of eq. (2.5) indeed describesquantum gravity coupled to a scalar field, then there should exist a whole universalityclass of link factors G, of which the gaussian is only one representative, that result inthe same continuum theory.A direct evaluation of the lattice sum (2.5) seems to be a daunting task, whichis still outside the numerical power of modern computers.

Fortunately, there existsa remarkable trick which allows us to exactly evaluate sums over surfaces of anytopology using only analytic tools. As was first noted by Kazakov and Migdal,16 astatistical sum of the form (2.5) is generated in the Feynman graph expansion ofthe quantum mechanics of a N × N hermitian matrix.⋆Consider the Euclidean path⋆Similar tools work for other simple matter systems coupled to two-dimensional gravity.

Forexample, as discussed in other lectures in this volume, in the case of pure gravity the sum overdiscretized surfaces is generated simply by an integral over an N × N hermitian matrix.6

integralZ =ZDN 2Φ(x) exp−βT/2Z−T/2dx Tr 12∂Φ∂x2+ U(Φ)!. (2.6)where x is the Euclidean time and U =12α′Φ2 −13!Φ3.

The parameter β enters as theinverse Planck constant. By a rescaling of Φ eq.

(2.6) can be brought to the formZ ∼ZDN 2Φ(x) exp−NT/2Z−T/2dx Tr 12∂Φ∂x2+ 12α′Φ2 −κ3!Φ3!,(2.7)where κ =pN/β is the cubic coupling constant. The connection with the statisticalsum (2.5) follows when we develop the graph expansion of Z in powers of κ. TheFeynman graphs all have coordination number 3, and are in one-to-one correspon-dence with the dual lattices ˜Λ of the discretized random surfaces (fig.

1). The latticesΛ dual to the Feynman graphs can be thought of as the basic triangulations.

Oneeasily obtains the sum over all connected graphs ln Z,limT→∞ln Z =XhN2−2h XΛκVVYi=1∞Z−∞dxiY⟨ij⟩e−|xi−xj|/α′ . (2.8)This is precisely of the same form as eq.

(2.5) which arises in two-dimensional quan-tum gravity! The Euclidean time x assumes the role of the embedding coordinateX in eq.

(2.5). We note that the role of the link factor G is played by the one-dimensional Euclidean propagator.

Only for this exponential G can we establish theexact equivalence with the matrix model. This does not pose a problem, however, aswe will find plenty of evidence that the continuum limit of the model (2.5) with theexponential G indeed describes quantum gravity coupled to a scalar, eq.

(2.1). It isevident from (2.8) that the parameter α′ sets the scale of the embedding coordinate.In fact, we have normalized α′ so that in the continuum limit it will precisely coincidewith the definition in eq.

(2.2). Whenever α′ is not explicitly mentioned, its valuehas been set to 1.7

Further comparing eqs. (2.8) and (2.5), we note that the size of the matrixN enters as 1/g0.

Let us show why. Each vertex contributes a factor ∼N, eachedge (propagator) ∼1/N, and each face contributes N because there are as manyindex loops as there are faces.

Thus, each graph is weighted by NV −E+F which,by Euler’s theorem, equals N2−2h. The expansion of the free energy of the matrixquantum mechanics in powers of 1/N2 automatically classifies surfaces according totopology.

It would seem that, in order to define a theory with a finite string coupling,it is necessary to consider finite N which, as we will see, is associated with severedifficulties. Fortunately, this naive expectation is false: in the continuum limit the“bare” string coupling 1/N becomes infinitely multiplicatively renormalized.

Thus, ifN is taken to ∞simultaneously with the world sheet continuum limit, then we mayobtain a theory with a finite string coupling. This remarkable phenomenon, known asthe “double-scaling limit”2,5, will allow us to calculate sums over continuum surfacesof any topology.In order to reach the continuum limit, it is necessary to increase the cubic couplingκ =pN/β ∼N0 until it reaches the critical value κc where the average number oftriangles in a surface begins to diverge.

In this limit we may think of each triangle asbeing of infinitesimal area√3a2/4, so that the whole surface has a finite area and iscontinuous. In the continuum limit κ →κc we define the cosmological constant∆= π(κ2c −κ2) .

(2.9)Recalling that κ = κc exp(−√3λa2/4π), we establish the connection between ∆andthe physical cosmological constant λ: λ ∼∆/a2.Above we have sketched the connection between matrix quantum mechanics andtriangulated random surfaces.It is easy to generalize this to the case where, inaddition to triangles, the surfaces consist of other n-gons: we simply have to add tothe matrix potential U(Φ) monomials Φn, n > 3. For consistency, the continuum limitshould not be sensitive to the precise manner in which the surfaces are discretized.In the next section we will give the exact solution of the matrix model and show thatthe continuum limit is indeed universal.8

3. MATRIX QUANTUM MECHANICS AND FREE FERMIONS.In the previous section we established the equivalence of the sum over connecteddiscretized surfaces to the logarithm of the path integral of the matrix quantummechanics.

On the other hand, in the hamiltonian language,Z =< f|e−βHT|i > . (3.1)Thus, as long as the initial and final states have some overlap with the ground state,the ground state energy E0 will dominate the T →∞limit of eq.

(3.1),limT→∞ln ZT= −βE0 . (3.2)The divergence of the sum over surfaces proportional to the length of the embeddingdimension arises due to the translation invariance.In order to calculate the sumover surfaces embedded in the infinite real line R1, all we need to find is the groundstate energy of the matrix quantum mechanics.

To this end we carry out canonicalquantization of the SU(N) symmetric hermitian matrix quantum mechanics.8,17 TheMinkowski time lagrangianL = Tr{12 ˙Φ2 −U(Φ)}(3.3)is symmetric under time-independent SU(N) rotations Φ(t) →V †Φ(t)V . It is con-venient to decompose Φ into N eigenvalues and N2 −N angular degrees of freedom,Φ(t) = Ω†(t)Λ(t)Ω(t)(3.4)where Ω∈SU(N) and Λ = diag(λ1, λ2, .

. .

, λN). The only term in the lagrangianwhich gives rise to dependence on ΩisTr ˙Φ2 = Tr ˙λ2 + Tr[λ, ˙ΩΩ†]2 .

(3.5)To identify the canonical angular degrees of freedom, the anti-hermitian traceless9

matrix ˙ΩΩ† can be decomposed in terms of the generators of SU(N),˙ΩΩ† =i√2Xi

Substituting eq. (3.6) into eqs.

(3.5) and (3.3), we findL =Xi12 ˙λ2i + U(λi)+ 12Xi

Because of the non-trivial Jacobian, thekinetic term for the eigenvalues becomes−12β2NXi=11∆2(λ)ddλi∆2(λ) ddλi. (3.8)It is left as an exercise for the reader to show that this can be rewritten as−12β2∆(λ)Xid2dλi2∆(λ) .

(3.9)It follows that the total hamiltonian isH = −12β2∆(λ)Xid2dλi2∆(λ) +XiU(λi) +Xi

These constraintsarise because transformations Ω→AΩ, where A is a diagonal SU(N) matrix, do notchange Φ.10

The constraints require that only those irreducible representations of SU(N)that have a state with all weights equal to zero are allowed in the matrix quan-tum mechanics.12For instance, the fundamental representations are excluded, andthe simplest non-trivial representation is the adjoint. It is not hard to classify all theirreducible representations allowed in the matrix quantum mechanics, i.e., the onesthat have states with zero weights.

In the standard notation, the Young tableauxof such representations consist of i + j columns, with the numbers of boxes in thecolumns (from left to right)(N −m1, N −m2, . .

. , N −mj, ni, .

. .

, n2, n1)(3.11)where n1 ≤n2 ≤. .

. ≤ni ≤N/2 and m1 ≤.

. .

≤mj ≤N/2.Since the angular kinetic terms in (3.10) are positive definite, it is clear that wehave to look for the ground state among the wave functions that are annihilatedby Πij and ˜Πij, i.e. in the trivial (singlet) representation of SU(N).

The singletwave functions are independent of the angles Ω, and are symmetric functions of theeigenvalues, χsym(λ).Because of the special form of the hamiltonian (3.10), theeigenvalue problem Hχsym = Eχsym assumes the remarkable form8 NXi=1hi!Ψ(λ) = EΨ(λ) ,hi = −12β2d2dλi2 + U(λi) ,(3.12)where Ψ(λ) = ∆(λ)χsym(λ) is by construction antisymmetric. Since the operatoracting on Ψ is simply the sum of single-particle non-relativistic hamiltonians, theproblem has been reduced to the physics of N free non-relativistic fermions movingin the potential U(λ).8 This equivalence to free fermions is at the heart of the exactsolubility of string theory in two dimensions.The N-fermion ground state is obtained by filling the lowest N energy levels of11

h,E0 =NXi=1ǫi . (3.13)The potential arising from triangulated random surfaces is U(λ) = 12λ2−13!λ3, which isunbounded from below and does not seem support any stable states.

Recall, however,that in order to find the genus expansion of the sum over surfaces, we are onlyinterested in the expansion about the classical limit β →∞(¯h →0) in powers of1/β2. Classically, there is a continuum of bounded orbits to the left of the potentialbarrier.

Semiclassically, these orbits are associated with energy levels (fig. 1) whosespacing is O(1/β), and whose decay time is exponentially long as β →∞.

Thisinstability cannot be seen in the context of the semiclassical expansion in powers of1/β2. Our goal, therefore, is to find the expansion of the ground state energy (3.13),where ǫi are the semiclassical energy levels.To leading order, ǫn are determined by the Bohr-Sommerfeld quantization condi-tion,Ipn(λ)dλ = 2πβ n(3.14)where pn(λ) =p2(ǫn −U(λ)), and the integral is over a closed classical orbit.

Inparticular, the position of the Fermi level µF = ǫN is determined byλ+Zλ−p2(µF −U(λ)) = πNβ ,(3.15)where λ−and λ+ are the classical turning points. Eq.

(3.15) shows that µF is anincreasing function of N/β. Clearly, both µF and E0 become singular at the pointwhere N/β →κ2c such that µF equals the height of the barrier µc.

Recall that thecubic coupling of the matrix quantum mechanics is κ =pN/β, and that we expectto find a singularity in the sum over graphs when κ becomes large enough that theaverage number of vertices in a Feynman graph (or equivalently, of triangles in a12

Fµ.....0βµ.Fig. 2a) N fermions in theasymmetric potential arisingdirectly from the triangulatedsurfaces.Fig.

2b) The double-scalinglimit magnifies the quadraticlocal maximum.random surface) begins to diverge. In the vicinity of this singularity the continuumlimit of quantum gravity can be defined.

Now we have identified the physics of thissingularity in the equivalent free fermion system: it is associated with spilling offermions over the barrier.We are interested in the singularity of E0 as a function of the cosmological con-stant ∆= π(κ2c −Nβ ). The parts of E(∆) analytic in ∆are not universal and can bedropped.

Indeed, if we calculate the sum over surfaces as a function of the fixed areaby inversely Laplace transforming E(∆), the analytic terms give contributions onlyfor zero area.Let us introduce the density of eigenvaluesρ(ǫ) = 1βXnδ(ǫ −ǫn),(3.16)in terms of which13

κ2 = Nβ =µFZ0ρ(ǫ)dǫ ;limT→∞−ln ZT= βE = β2µFZ0ρ(ǫ)ǫdǫ . (3.17)Differentiating the first equation, we find∂∆∂µ = πρ(µF ) ,(3.18)which can be integrated and inverted to obtain µ = µc −µF as a function of ∆.Differentiating the second equation, we find ∂E∂∆= 1πβ(µ −µc).

After addition to Eof an irrelevant analytic term, E →E + 1πβµc∆, we find∂E∂∆= 1πβµ(3.19)These equations show that the continuum limit of the sum over surfaces is determinedby the singularity of the single particle density of states near the top of the barrier.Using the WKB approximation, it was found in ref. 16 thatρ(µF ) = 1πλ+Zλ−dλp2(µF −U(λ))∼−1π ln µ + O(1/β2)(3.20)It follows that ∆= −µ ln µ + O(1/β2), and−βE = 12π(βµ)2 ln µ + .

. .

≈12πβ2∆2ln ∆+ . .

. (3.21)The critical behavior of ρ comes from the part of the integral near the quadraticmaximum of the potential.

In the continuum limit µ →0, the classical trajectory atµF spends logarithmically diverging amount of time near the maximum. Therefore,the WKB wave functions are peaked there.

In fact, we will show that, at any order14

...........0βµ. βµFig.3a) N fermions in adouble well potential.Fig.3b) The system rel-evant to the double-scalinglimit.in the WKB expansion, the leading singularity of ρ is entirely determined by thequadratic nature of the maximum.

To see this, let us rescale the coordinates √β(λ −λc) = y, and e = µc −ǫ. The single particle eigenvalue problem becomes512d2dy2 + 12y2 + O1/pβy3ψ(y) = βeψ(y)(3.22)For a finite βe, all the terms beyond the quadratic are irrelevant in the limit β →∞because they are suppressed by powers of β.

This demonstrates the universality ofthe continuum limit: any potential possessing a quadratic maximum will yield thesame sum over continuous surfaces. In effect, our rescaling has swept all the non-universal details of the potential out to infinity.This implies, for instance, that,instead of working with the potential of fig.

2, we can use the double well potentialU(λ) = −12λ2 + λ4 (fig. 3).

After rescaling √βλ = y, the Schr¨odinger equation againreduces to the motion in the inverted harmonic oscillator. The only difference is thatnow fermions fill both sides of the maximum.

To all orders of the WKB expansionthis simply multiplies the density of states, and therefore the free energy, by a factorof 2. The double well potential does not suffer from the non-perturbative instabilities:the ground state energy can in principle be calculated even for finite N and β. Wewill often adopt the symmetric case because it is more convenient for calculations,remembering that we have to divide by 2 to find the sum over triangulated surfaces.15

From eq. (3.21) we know the sum over all surfaces of spherical topology embeddedin R1.

In order to find the sums over surfaces of any topology, we need to develop asystematic WKB expansion of the ground state energy. According to eqs.

(3.18) and(3.19), all we need to know is the complete expansion of the single particle density ofstates ρ(µ). If we consider µ such that βµ is O(1), then this problem has a remarkablysimple solution,5because eq.

(3.22) shows that in this energy range the system ispurely quadratic. Let us write the density of states asρ(µ) = Tr δ(h0 + βµ) = 1πIm Tr1h0 + βµ −iǫ,h0 = −12d2dy2 −12y2 .

(3.23)We will evaluate the trace in position space in order to take advantage of the factthat the resolvent of the hamiltonian of a simple harmonic oscillator is well known,⟨yf|1−12d2dy2 + 12ω2y2 + βµ −iǫ|yi⟩=∞Z0dTe−βµTyfZyiDy(t)e−R T0 dt 12( ˙y2+ω2y2)=∞Z0dTe−βµTrω2πsinhωT e−ω(y2f+y2i ) cosh ωT−2yfyi/2 sinh ωT . (3.24)In our case, the frequency ω is actually imaginary (we have an inverted harmonicoscillator), so we must define the resolvent by analytic continuation.

We rotate ω →−i while simultaneously rotating T →iT, and the relevant resolvent is⟨yf|1h0 + βµ −iǫ|yi⟩= i∞Z0dTe−iβµTr−i2πsinhT ei(y2f+y2i ) cosh T−2yfyi/2 sinh T.(3.25)Now it is easy to derive the derivative of the density of states5∂ρ∂(βµ) = 1π∂∂(βµ)Im∞Z−∞dy⟨y|1h0 + βµ −iǫ|y⟩= 1πIm∞Z0dTe−iβµTT/2sinh(T/2) . (3.26)Note that only for the purposes of deriving the large βµ expansion can we reduce16

the problem to pure inverted harmonic oscillator. The integral representation (3.26)is not designed to correctly give the non-perturbative terms O(e−βµ).

In fact, suchterms depend on the details of the potential away from the quadratic maximum, andare not universal. While each term in the large βµ expansion is related to the sumover all geometries of a certain genus, we do not yet fully understand the geometricalor physical meaning of the non-perturbative terms.After expanding eq.

(3.26) and integrating once, we find the complete series ofcorrections to the leading order estimate in eq. (3.20),ρ(µ) = 1π(−ln µ +∞Xm=1(22m−1 −1)|B2m|m(2βµ)−2m).

(3.27)In solving eq. (3.18) for µ in terms of ∆, it is useful to introduce parameter µ0defined by∆= −µ0 ln µ0 = µ0| ln µ0| .

(3.28)Then,µ = µ0(1 −1ln µ0∞Xm=1(22m−1 −1)|B2m|m(2m −1) (2βµ0)−2m + O1ln2 µ0). (3.29)Integrating (3.19) we finally arrive at the complete genus expansion of the sum oversurfaces−βE = 18π((2βµ0)2 ln µ0 −13 ln µ0 +∞Xm=1(22m+1 −1)|B2m+2|m(m + 1)(2m + 1) (2βµ0)−2m).

(3.30)Clearly, the sensible way to approach the continuum limit is through “double scaling”β →∞and µ0 →0 in such a way that βµ0 = 1/gst is kept fixed.5,6 Thus, quiteremarkably, we have found closed form expressions for sums over continuous surfacesof any topology embedded in one dimension. The sum over genus h surfaces, givenby the coefficient of g2h−2st, grows as (2h)!.

This behavior is characteristic of closed17

string theories,18while in field theory we typically find that the sum over h-loopdiagrams exhibits a slower growth ∼h!. The badly divergent perturbation series(3.30) indicates that the theory is not well-defined non-perturbatively.Eq.

(3.30) shows explicitly how the bare string coupling 1/β gets multiplicativelyrenormalized. Compared to the c < 1 matter coupled to gravity,2−4 the new featurehere is that the renormalization is not simply by a power of the cosmological constant∆.

It appears that ∆itself is multiplicatively renormalized: ∆→µ0 ≈∆/| ln ∆|.Another peculiarity of the c = 1 matter is that the sums over spherical and toroidalsurfaces diverge in the continuum limit as | ln µ0|. In section 5 we will argue that allthese features have a natural explanation in the context of Liouville theory.

Beforewe do that, however, we further exhibit the power of the matrix model by calculatingsome correlation functions to all orders in the topological expansion.4. CORRELATION FUNCTIONS.The simplest kind of matrix model operator is Tr Φn(x) where n is any finiteinteger.

This operator inserts a vertex of order n in all possible places on the dualgraph ˜Λ. In terms of the basic lattice Λ, this amounts to cutting an n-gon hole inthe surface, pinning it at the embedding coordinate x, and integrating the positionof the hole all over the surface.

In the continuum limit, for any finite n the size ofthe hole becomes infinitesimal, and it is commonly referred to as “a puncture”. Thecontinuum expression for the operator above should be roughlyZd2σ√gδX(σ) −x) .

(4.1)By Fourier transforming, we derive the continuum translation of a related operatorZdxeiqx Tr Φn(x) →Zd2σ√geiqX ,(4.2)which resembles the basic “tachyon” operator of string theory. Below we outline theprocedure for calculating the correlation functions of such operators in the matrix18

model. In section 5 we show that the results can indeed be reproduced in the stringtheoretic formalism.Since the matrix model has been reduced to a system of free fermions, it is con-venient to perform calculations in the formalism of non-relativistic second quantizedfield theory.

The second quantized fermion field is defined asˆψ(λ, x) =∞Z−∞dνeiνxaǫ(ν)ψǫ(ν, λ) ,(4.3)where ψǫ(ν, λ) are the single fermion wave functions of energy ν, and the associatedannihilation operators aǫ(ν) satisfy {a†ǫ (ν), aǫ(ν′)} = δǫ,ǫ′δ(ν −ν′). Setting λ =√2yin eq.

(3.22), we find that ψǫ(ν, λ) are solutions of the Whittaker equation( d2dλ2 + λ24 )ψ = νψ. (4.4)and can therefore be expressed in terms of Whittaker functions W(ν, λ).19Theirspectrum is continuous and doubly degenerate.Here ǫ denotes the parity of thewavefunction ψǫ(ν, λ) and repeated ǫ are summed over.The Euclidean second-quantized field theory with chemical potential βµ is definedby the actionS =∞Z−∞dλ∞Z−∞dx ˆψ†(−ddx + d2dλ2 + λ24 + βµ) ˆψ(4.5)where x is the Euclidean time.

The ground state of the system (fig. 3b) satisfiesaǫ(ν)|βµ⟩= 0,ν < βµ ,a†ǫ(ν)|βµ⟩= 0,ν > βµ .

(4.6)Consider correlation functions of operators of the typeO(q) =Zdxeiqx Tr fΦ(x),(4.7)19

where f is any function. In the second quantized formalism, they translate intoO(q) =ZdxeiqxZdλf(λ) ˆψ† ˆψ(x, λ) .

(4.8)Thus, to find the connected correlation functions of any set of such operators, weneed to calculateG(q1, λ1; . .

. ; qn, λn) =nYi=1Zdxieiqixi⟨βµ| ˆψ† ˆψ(x1, λ1) .

. .

ˆψ† ˆψ(xn, λn)|βµ⟩c . (4.9)Application of Wick’s theorem reduces this to the sum over one-loop diagrams withfermion bilinear insertions in all possible orders around the loop.

A convenient formulafor the Euclidean Green function is19SE(x1, λ1; x2, λ2) =∞Z−∞dνe−(ν−βµ)∆x{θ(∆x)θ(ν −βµ) −θ(−∆x)θ(βµ −ν)}×ψǫ(ν, λ1)ψǫ(ν, λ2) =∞Z−∞dp2πe−ip∆x∞Z−∞dνip + i(ν −βµ)ψǫ(ν, λ1)ψǫ(ν, λ2)= i∞Z−∞dp2πe−ip∆xsgn(p)∞Z0dse−sp+iβµs⟨λ1|e2isH|λ2⟩(4.10)where ∆x = x1 −x2, and we have used19∞Z−∞dνe−iνsψǫ(ν, λ1)ψǫ(ν, λ2) = ⟨λ1|e2isH|λ2⟩=1√−4πi sh s exp−i4λ21 + λ22th s−2λ1λ2sh s(4.11)with H = 1/2p2 −1/8λ2. With the Green function (4.10), after integration over the20

loop momentum, Moore derived the representation∂∂µG(q1, λ1; . .

. ; qn, λn) = in+1δ(Xqi)Xσ∈Σn∞Z−∞dξeiµξǫ1∞Z0ds1 .

. .ǫn−1∞Z0dsn−1e−s1Qσ1−...−sn−1Qσn−1⟨λσ(1)|e2is1H|λσ(2)⟩.

. .⟨λσ(n)|e2i(ξ−Pn−11si)H|λσ(1)⟩(4.12)where Qσk = qσ(1) + .

. .

+ qσ(k), and ǫk = sgn[Qσk].In principle, any correlationfunction of operators of type (4.7) can be found by integrating G. A convenient wayof calculating correlations of puncture operators is to first introduce the operator thatcreates a finite boundary of length l, with momentum q injected into the boundary19O(l, q) =Zdxeiqx Tr e−lΦ(x) . (4.13)The puncture operator should be the leading term in the small l expansion of O(l, q).The correlations of puncture operators P(q) can be extracted according to*YiO(li, qi)+∼Yil|qi|i*YjP(qj)+.

(4.14)The details of the calculations are highly technical and can be found in ref. 19.

Herewe will simply state the results for the two, three, and four point functions.⟨P(q)P(k)⟩= −δ(q + k)[Γ(1 −|q|)]2µ|q|"1|q| −124(βµ)2(|q| −1)(q2 −|q| −1) + . .

.#. (4.15)⟨P(q1)P(q2)P(q3)⟩= δ(Xqi)3Yi=1Γ(1 −|qi|)µ|qi|2 1βµ"1−124(βµ)2(|q3| −1)(|q3| −2)(q21 + q22 −|q3| −1)+4Yr=1(|q3| −r)3(q41 + q42) + 10q21q22 −10(q21 + q22)|q3| −5(q1 −q2)2 + 12|q3| + 7×15760(βµ)2 + .

. .#,(4.16)21

where q1, q2 > 0.⟨P(q1)P(q2)P(q3)P(q4)⟩= −δ(Xqi)4Yi=1Γ(1 −|qi|)µ|qi|2×12(βµ)2(|q1 + q2| + |q1 + q3| + |q1 + q4| −2) + O(1/(βµ)2). (4.17)The new feature of the c = 1 correlators, compared to theories with c < 1, is that theyhave genuine divergences occurring at quantized values of the external momenta.20 Inthis sense, the c = 1 model is the most similar to critical string theory where theamplitudes are well-known to have divergences associated with the production of on-shell physical particles.

The remarkable feature of the c = 1 amplitudes is that allthe divergences factorize into external leg factors.Because of the divergences, eqs. (4.15)-(4.17) are strictly valid only if none ofqi are integer.If any of the momenta have integer values, then additional termsneed to be taken into account, which leads to cancellation of the infinity.20Thephysical meaning of this is simple: in the matrix model there is an explicit ultravi-olet and infrared cut-off, and therefore there are no genuine zeroes or poles of thecorrelation functions.

Instead, they are regularized according to 1/0 →| ln µ0|, and0 →1/| ln µ0|. Note that in eqs.

(4.15)-(4.17) µ can be replaced by µ0. Thus, thecorrelation functions scale as powers of µ0, and not as powers of ∆found in all thec < 1 theories coupled to gravity.2−4 This provides further evidence that, for c = 1,a renormalization of the cosmological constant takes place.

In the next section wesketch a Liouville theory explanation of some of the new effects peculiar to c = 1.22

5. LIOUVILLE GRAVITY COUPLED TO c = 1 MATTER.In this section we outline the continuum approach to evaluating the sum oversurfaces (2.1).

In order to integrate over all metrics, in 2 dimensions we may pick theconformal gauge gµν = ˆgµν(τ)e−φ where ˆgµν depends on a finite number of modularparameters collectively denoted as τ. For spherical topology, there are no moduli,and we may choose ˆgµν = δµν.

Eq. (2.2) shows that classically the Liouville field φ isnon-dynamical.

It is well-known, however,11 that the kinetic term for φ is inducedthrough quantum effects arising from dependence on φ of the path integral measure.Additional modification of the dynamics of φ has been proposed by David and byDistler and Kawai,15 who argued that we can simplify the integration measure at theexpense of further renormalization of the action for φ such that the ghost, X and φfields combine into a conformally invariant theory with net conformal anomaly equalto zero. Following their approach, we assume that the original path integral (2.1) istransformed in the conformal gauge intoZ =Xhg2h−20Z[dτ][DbDc][Dφ][DX]e−Sb,c−S ,S = 14πZd2σpˆg∂µX∂µX + ∂µφ∂µφ −4 ˆRφ + 4∆e−2φ,(5.1)and Sb,c is the standard free action for the b and c ghosts.

The exponential interactionterm originates from the cosmological term in the action (2.2).⋆With the action (5.1), the simplest set of conformally invariant operators areT(q) = 1πβZd2σpˆgeiqX+ǫ(q)φ(5.2)These operators inject embedding momentum q into the world sheet. They should bethought of as the Liouville theory implementation of the operators (4.2).

In a theory⋆Polchinski has argued21that, for c = 1, the interaction term is not simply an exponential.We will instead work with the exponential interaction, and will show that it is also possibleto explain correlation functions within this framework. Our argument does not substantiallydiffer from Polchinski’s, but we prefer working with the exponential because it was shown byCurtwright and Thorn22that with this form of interaction the conformal invariance can bemaintained exactly in the continuum limit.23

with ∆= 0, T(q) is conformally invariant for ǫ(q) = −2 ± |q|. For ∆> 0, however,it was shown in ref.

23, 24 that only the operator with ǫ(q) = −2 + |q| exists. Thisconclusion agrees with the scaling of correlation functions in the matrix model.

Letus now use the Liouville formalism to calculate the spherical correlation functions.As discovered in ref.25, in performing the path integral it is convenient todecompose φ = φ0 + ˜φ, and integrate first over the zero mode φ0. Then, since theinteraction is purely exponential, we obtain* NYi=1T(qi)+h= g2h−20δ(Xqi)12∆πsΓ(−s)Z[dτ][D ˜X][D ˜φ][DbDc]e−Sb,c−S0Zd2σpˆge−2˜φs Yi1πβZd2σpˆgeiqiX+(−2+|qi|)φ(5.3)whereS0 = 14πZd2σpˆg∂µ ˜X∂µ ˜X + ∂µ ˜φ∂µ ˜φ(5.4)and s = 2(1 −h) −N + 12PNi=1 |qi|.

The factor Γ(−s) indicates that correlationfunctions with non-negative integer s have a divergence from the volume of φ0. Thisexplains the divergences in the 0-point functions on the sphere and torus found ineq.

(3.30). Now, the difficulty is to deal with the insertion of T(0)s. Goulian andLee26 proposed to calculate eq.

(5.3) first for integer s, where the rules are perfectlyclear, and then to define it for non-integer s by analytic continuation. This givesanswers in agreement with the matrix model.26,27Now we show how the multiplicative renormalization ∆→µ0 takes place.

Al-though it seems that eq. (5.3) scales as ∆s, the free field correlator multiplying itis, formally, of order 0s.For integer s this can be shown through direct calcula-tion.

Consider, for instance, the tree level correlator of M tachyons with qM < 0,q1, . .

. , qM−1 > 0.

If we impose the condition s = 0, then qM = 2 −M. The integral24

for the coefficient of 12Γ(0) can be performed explicitly27 and givesβ2−M(M −3)!M−1Yi=1Γ(1 −|qi|)Γ(|qi|),(5.5)where we have set g20 = π3β2.If we now send q1, q2, . .

., qm →0, while keepingPM−11qi = M −2, we obtain a tachyon correlator with s = m. From eq. (5.5) weobtain*MYi=m+1T(qi)+= 12β2−M+mΓ(−m)∆m0m1(M −3)!M−1Yi=m+1Γ(1 −|qi|)Γ(|qi|)(5.6)The physical meaning of the zero of order m is that the “tachyon” decouples at zeromomentum.

This is a sign of the fact that the “tachyon” is actually a massless particlein two-dimensional string theory. We will show this more explicitly later on.Now we would like to argue that in a cut-offtheory 0s should be replaced by(1/| ln µ0|)s. If the cut-offis imposed, then the Liouville zero mode is enclosed in a boxof size ∼| ln µ0|.

Thus, the lowest accessible value of the “Liouville energy” 2 + ǫ(q)is ∼1/| lnµ0|, and there is no complete decoupling of T(0). Hence, the amplitudesshould scale as (∆/| ln µ0|)s = µs0, which is the sought for renormalization of thecosmological constant.

We have identified the physical origin of this renormalization:the decoupling of the massless particle at zero momentum.The continuation of eq. (5.6) to non-integer s gives the N-point tachyon ampli-tudes with qN < 0, q1, .

. .

, qN−1 > 0,27* NYi=1T(qi)+= 12β2−NNYi=1Γ(1 −|qi|)Γ(|qi|) ddµ0N−3µs+N−30. (5.7)This agrees with the matrix model results (4.15)-(4.17) if the matrix model andLiouville theory operators are related by momentum dependent normalization factorT(q) =1Γ(|q|)P(q)(5.8)With this correspondence, we can use the matrix model results to obtain tachyoncorrelation functions to all orders in the genus expansion.

This is remarkable because25

direct Liouville theory calculations of correlation functions have not yet been per-formed even for genus 1. For h > 3 we do not know the region of modular integrationand could not perform a direct Liouville calculation even in principle.

From eq. (4.15)we find, for instance,⟨T(q)T(k)⟩= −δ(q+k)Γ(1 −|q|)Γ(|q|)2µ0|q|"1|q| −124(βµ)2(|q|−1)(q2−|q|−1)+.

. .#.

(5.9)A natural continuation of this formula to q = 0 is to replace limq→0 |q| by 1/| ln µ0|.Then we obtain,⟨T(0)T(0)⟩∼∂2∂∆2 E . (5.10)This is a simple consistency check: differentiation of the path integral with respectto ∆inserts a zero-momentum tachyon.To conclude this section, we will describe the only h > 0 calculation that hasbeen performed to date.

This is the sum over toroidal surfaces with no insertions.28The reason this calculation is relatively straightforward is that s = 0, and the com-plicated insertion of T(0)s is absent. Sending s →0 in eq.

(5.3), and rememberingthe renormalization ∆→µ0, we obtain the free field path integralZ1 = −12 ln µ0ZFd2τZ[D ˜X][D ˜φ][DbDc]e−S0(5.11)where τ = τ1 + iτ2 is the modular parameter of the torus, and F denotes the funda-mental region of the modular group: τ2 > 0, |τ| > 1, −12 ≤τ1 < 12. This path integralis standard, and we findZ1(R/√α′) = −12 ln µ0ZFd2τ|η(q)|42τ2(2π√τ2|η(q)|2)−1Z(R/√α′, τ, ¯τ) ,(5.12)where η is the Dedekind function and q = e2πiτ.

In the above integrand, the firstterm is from the ghost determinant, the second is from the Liouville determinant, and26

Z(R/√α′, τ, ¯τ) is the partition function of the scalar field compactified on a circle,Z(R/√α′, τ, ¯τ) =R√α′1√τ 2|η(q)|2∞Xn,m=−∞exp−πR2|n −mτ|2α′τ2. (5.13)The double sum is over the soliton winding numbers about the two non-trivial cyclesof the torus.

From eqs. (5.12) and (5.13), we obtainZ1(R/√α′) = −12 ln µ0R4π√α′F(R/√α′) ,F(R/√α′) =ZFd2ττ22Xn,mexp−πR2|n −mτ|2α′τ2.

(5.14)The η-functions cancel out, and we end up only with the contributions of the zeromodes. This is due to the absence of particles corresponding to the transverse stringexcitations, and reflects the two-dimensional nature of this string theory.⋆Remark-ably, the integral in eq.

(5.14) can be easily performed using the trick of ref. 30.The idea is to trade the m-sum over the winding sectors of the string for a sum overmany inequivalent fundamental regions which together cover the strip −12 ≤τ1 < 12in the upper half-plane.

In 26 dimensions this gives a representation of the string freeenergy in terms of the free energies of all the modes of the string31. Similarly, eq.

(5.14) becomesF(R/√α′) =ZFd2ττ22+ 2∞Z0dτ2τ22∞Xk=1exp−πR2k2α′τ2. (5.15)The second term is simply the temperature-dependent one-loop free energy for asingle massless boson in 2 dimensions, expressed in the proper time representation.The first term is the one-loop cosmological constant of the massless boson, which hasbeen automatically supplied with a ‘stringy’ ultraviolet cut-off: here the τ integral⋆The isolated transverse states at discrete momenta, to be discussed in section 6, do not affectthe torus partition function.2927

is over the fundamental region, not over the strip. Performing the integrals in eq.

(5.15), we find F(R/√α′) = π3(1 + α′/R2) andZ1(R/√α′) = −124( R√α′ +√α′R ) ln µ0 . (5.16)As we will show in section 9, this answer agrees with the matrix model result.

Thiscalculation gives us hope that the Liouville path integral exactly describes sums overgeometries beyond the tree level.6. THE SPECIAL OPERATORS.The most striking feature of the tachyon correlation functions is the occurrenceof the external leg factors Γ(1−|qi|)Γ(|qi|) , which contain factorized poles whenever any ofthe momenta approaches an integer value.

Although this exact factorization is notyet fully understood, it was noted in ref. 20 that the poles are related to the presenceof other physical states in the theory besides the tachyons (5.2).

These states areremnants of the transverse excitations of the string and occur only at integer q.20,32,33The origin of these states can be traced to the pure c = 1 conformal field theorywhere there are special primary fields of the form34VJ,m(∂X, ∂2X, . .

. )e2miX(z)(6.1)with conformal weight J2.These states form SU(2) multiplets with spin J andmagnetic number m. The connection with SU(2) becomes apparent once we notethat the states (6.1) are the full set of primary fields in the compact c = 1 theorywith the self-dual radius R = 1, where there is a well-known level 1 SU(2) currentalgebra.

For each spin J, VJ,±J = 1, which gives a tachyon operator. Other mem-bers of the multiplet can be constructed by applying raising and lowering operators28

12πiHdz exp(±2iX(z)). After coupling to gravity, we can “dress” the special c = 1operators to obtain new (1, 0) operators of the formVJ,m(z) = VJ,me2miXe2(J−1)φ(z) .

(6.2)Recently, it was established that these operators are related to a W1+∞algebra.35 †Undoubtedly, this algebra plays an important role in determining the properties ofthe theory.In the matrix model, the physical (1, 1) operatorZd2zVJ,m(z)¯VJ,m(¯z)(6.3)was, up to normalization, identified with10,36∞Z−∞dxe2miX(λ −λc)2J(6.4)where λc is the coordinate of the quadratic maximum. Using this identification, thecorrelation functions of all operators with m = 0 were calculated in ref.

10. Recently,it was shown37,38 that the operators (6.4) generate a W1+∞algebra.

The connectionof this algebra with the algebra of special operators in Liouville theory has beenestablished by E. Witten.35The simplest new operator is the zero-momentum dilatonV1,0 ¯V1,0 = ∂X ¯∂X . (6.5)Thus, of the full dilaton field of the critical string theory only its zero-momentumpart remains here.

Similarly, the higher J excitations correspond to remnants of thehigher mass particles. The connection of the special operators to the divergences in† A related observation was made by I. Klebanov and A. Polyakov.29

the correlation functions is provided by the fusion rules. There are terms ∼1/|z−w|2in the fusion rules of the physical operators (6.3).

Integration over one operator nearanother can thus produce logarithmic divergences.Let us elaborate on this mechanism using, as an example, the tachyon 2-pointfunction.20The poles in it occur when |q| is integer, i.e. when tachyon operatorsbecome members of the SU(2) multiplets of the special operators.The operatorproduct expansion of two integer momentum tachyons with n > 0 iseinX+(−2+|n|)φ(z, ¯z)e−inX+(−2+|n|)φ(w, ¯w) ∼.

. .

+1|z −w|2V|n|−1,0 ¯V|n|−1,0 + . .

. .

(6.6)Integrating over z near w, we obtain a logarithmic divergence for any integer |q| > 0due to the appearance of a physical operator in the fusion rule. This explains theinfinite sequence of poles in the two-point function.7.

STRING FIELD THEORY FROM THE MATRIX MODEL.The success of the Liouville approach shows that the theory of surfaces embed-ded in one dimension can also be viewed as critical string theory embedded in twodimensions.7,21 The field φ, which starts out as the conformal factor of the world sheetmetric, assumes the role of another embedding dimension similar to X. Indeed, thepath integral (5.1) can be thought of as the sigma model for the bosonic string prop-agating in two-dimensional Euclidean target space with the metric Gµν = δµν, thedilaton background D(φ) = −4φ, and the tachyon background T(φ) = ∆e−2φ.

Thetwo-dimensionality of the string theory has many important physical consequences.First of all, the mass of the particle corresponding to motion of the string in itsground state, “the tachyon” ism2T = 2 −D6. (7.1)Thus, D = c+1 = 2 is the “critical dimension of non-critical string theory” where “thetachyon” is exactly massless.

Indeed, in the calculation of the Liouville path integral30

on the torus, we found a massless two-dimensional boson propagating around theloop. From the decoupling at zero momentum, observed in the study of the tachyoncorrelation functions, we deduce that the theory is invariant under constant shifts ofthe tachyon field.The second expected feature of two-dimensionality is the absence of transverseoscillation modes of the string.

Indeed, if we could pass to the light-cone gauge, thenall transverse oscillations would be eliminated and the entire spectrum of the stringtheory would consist of one massless field. In reality, we cannot pick the light-conegauge because of the lack of translation invariance in the φ-coordinate.

As a result,the spectrum of the string contains the transverse excitations constructed in section 6.These states are not full-fledged fields, however, because they only occur at discretemomenta |q| = n. Indeed, the string one-loop calculation of section 5 showed thatthere are no massive transverse fields in the theory. Thus, at the minimal level, weshould be able to formulate the string field theory with a single massless field, perhapscoupled to an infinite number of quantum mechanical degrees of freedom.In this section we show that a string field theory with the features anticipatedabove can be derived directly from the matrix quantum mechanics.This furtherstrengthens the connection of c = 1 quantum gravity with string theory in D = 2.The matrix quantum mechanics directly reduces to the exact string field theory, whichis the second-quantized field theory of non-relativistic fermions discussed in section 4.In this section we will simply recast this formalism so that some of its physical featuresbecome more transparent.

First, we will show that the non-relativistic field theorycan be expressed in the double-scaling limit as a theory of free quasi-relativistic chiralfermions.10,39 These chiral fermions have the kinetic term that is relativistic to orderg0st, but receives translationally non-invariant corrections of order gst. Further, we willcarry out the conventional bosonization40 of this fermionic hamiltonian, and obtainan interacting theory of massless bosons in D = 2 first derived by Das and Jevickiwith somewhat different methods.9 In conclusion, we will perform a few manifestlyfinite calculations in the bosonic formalism to show that it is fully equivalent to theoriginal representation in terms of non-relativistic fermions.31

7.1. Chiral Fermions.In order to exhibit our general method, we first consider fermions moving in anarbitrary potential U(λ).

When we later take the double-scaling limit, as expected itwill depend only on the existence of a quadratic local maximum of U(λ). The secondquantized hamiltonian for a system of free fermions with Planck constant 1/β isˆh =Zdλ 12β2∂Ψ†∂λ∂Ψ∂λ + U(λ)Ψ†Ψ −µF(Ψ†Ψ −N),(7.2)where µF is the Lagrange multiplier necessary to fix the total number of fermions toequal N. As usual, it will be adjusted so as to equal the Fermi level of the N fermionsystem, µF = ǫN.

The fermion field has the expansionΨ(λ, t) =Xiαiψi(λ)e−iβǫit ,(7.3)where ψi(λ) are the single particle wave functions and αi are the respective annihila-tion operators. The time t is now taken to be Minkowskian.All the known exact results of the c = 1 matrix model have been derived fromthis nonrelativistic field theory.

An important feature of this theory is that it is two-dimensional: the field Ψ(λ, t) depends on t, and also on the eigenvalue coordinateλ. This is the simplest way to see how the hidden dimension, originating from theLiouville mode of quantum gravity, emerges in the matrix model.Note that, as β →∞, the single-particle spectrum is approximately linear nearthe Fermi surface,ǫn ≈µF + 1βnω + O 1β2.

(7.4)This suggests that the behavior of excitations near the Fermi surface is quasi-relativistic.The spectrum of relativistic fermions in a box is, of course, exactly linear. We willtherefore find non-relativistic corrections arising from the violations of the linearity ofǫn due to higher orders of the semiclassical expansion.

As the Fermi level approaches32

the maximum, µ = µc −µF →0, the spacing of the spectrum vanishes, ω ≈π/| ln µ|.Thus, we expect that the size of the box in which the quasi-relativistic fermions areenclosed diverges ∼| ln µ|.Let us introduce new fermionic variables ΨL and ΨR,Ψ(λ, t) = e−iµF tp2v(λ)e−iβ R λ dλ′v(λ′)+iπ/4ΨL(λ, t) + eiβ R λ dλ′v(λ′)−iπ/4ΨR(λ, t), (7.5)where v(λ) is the velocity of the classical trajectory of a particle at the Fermi level inthe potential U(λ),v(λ) = dλdτ =p2(µF −U(λ)) . (7.6)We substitute (7.5) into (7.2) to derive the hamiltonian in terms of ΨL and ΨR.We drop all terms which contain rapidly oscillating exponentials of the form exp±2iβR λ v(x)dx, since these give exponentially small terms as β ∼N →∞and donot contribute to any order of semiclassical perturbation theory.

For the same reasonwe can restrict the coordinate λ to lie between the turning points of the classicalmotion. For µ > 0, we will consider the case where the fermions are localized onone side of the barrier.

If we considered the symmetric case of fig. 3, we would havefound two identical worlds decoupled from each other to all orders of the semiclassicalexpansion.

We will also discuss µ < 0 where the Fermi level is above the barrier, andthe fermions can be found on both sides.After some algebra we findH = 2βˆh =T/2Z0dτiΨ†R∂τΨR −iΨ†L∂τΨL +12βv2∂τΨ†L∂τΨL + ∂τΨ†R∂τΨR+ 14βΨ†LΨL + Ψ†RΨR v′′v3 −5(v′)22v4,(7.7)where v′ ≡dv/dτ. Here we see that the natural spatial coordinate, in terms of whichthe fermion has a standard Dirac action to leading order in β, is τ – the classical time33

of motion at the Fermi level – rather than λ. An important feature of the hamiltonianis that it does not mix ΨL and ΨR,H = HL + HR .

(7.8)The only mixing between the different chiralities is through the boundary conditions.In order to determine the boundary conditions, consider the leading semiclassicalexpression for Ψ(λ, 0),Ψ(λ, 0) =Xn>0ψnan +Xm≤0ψmb†m ,ψn =2√TvncosβλZdλ′vn(λ′) −π4vn =p2(ǫn −U(λ)) ,(7.9)where we have relabeled αi = ai , i > 0 , αi = b†i,i ≤0, and i is the number of thefermion energy level starting from the Fermi level µF . Expanding ǫn about µF , wefindΨ(λ, 0) =1√2veiβ R λ v(λ′)dλ′−iπ/4ΨR + e−iβ R λ v(λ′)dλ′+iπ/4ΨL,ΨR =r2T Xn>0e2πinτ/T an +Xm≤0e2πimτ/T b†m,ΨL =r2T Xn>0e−2πinτ/T an +Xm≤0e−2πimτ/Tb†m.

(7.10)Thus, ΨR and ΨL are expressed in terms of a single set of fermionic oscillators, anand bm. Semiclassically, they satisfy the boundary conditionsΨR(τ = 0) = ΨL(τ = 0),ΨR(τ = T2 ) = ΨL(τ = T2 ) .

(7.11)These ensure that the fermion number current not flow out of the finite interval, i.e.,that ¯Ψ(τ)γ1Ψ(τ) = Ψ†RΨR−Ψ†LΨL vanish at the boundary. They also guarantee that34

ΨR and ΨL are not independent fields and that we are including the correct number ofdegrees of freedom. The issue of dynamics at the boundary is quite subtle, however,since the corrections to the relativistic hamiltonian in eq.

(7.7) blow up preciselyat the boundary. The problem is that the semiclassical approximation breaks downat the points where v(τ) = 0 even as β →0.

One possibility of dealing with thisbreakdown is to carefully regularize the physics at the boundary points. This haslead to some success.41,42 We will suggest another method, which is problem-freefrom the beginning: to approach double-scaling limit with µ < 0.

As we will show,an equivalent procedure is to work with µ > 0 but to construct the theory in termsof Ψ(p, t), where p is the momentum conjugate to λ.We have succeeded in mapping the collection of N nonrelativistic fermions, whichdescribe the eigenvalues of Φ, onto an action which, to leading order, is just the two-dimensional Dirac action with rather standard bag-like boundary conditions. How-ever, the 1β corrections in eq.

(7.7) cannot be disregarded in the double scaling limitbecause near the quadratic maximum v(τ) = √2µsinh(τ). In the double scaling limitthe surviving correction to the relativistic hamiltonian is of order gst = 1/(βµ),Hgst =14βµ∞Z0dτsinh2(τ)|∂τΨi|2 + 12|Ψi|21 −52coth2(τ),(7.12)where i runs over L and R.This correction does not change the non-interactingnature of the fermions, but it does render the fermion propagator non-standard.In the double-scaling limit the boundary at T/2 recedes to infinity and is irrele-vant.

In fact, the correct expression for the hamiltonian would follow had we replacedU(λ) by −12λ2. However, the hamiltonian (7.12) still suffers from the divergence atτ = 0.

In view of this, let us modify the problem to approach the double-scalinglimit with µ < 0. Since string perturbation theory is in powers of (βµ)−2, analyticcontinuation to positive µ should not be problematic.

Now v2(τ) = 2|µ| ch 2τ, and35

we findH = =∞Z−∞dτiΨ†R∂τΨR −iΨ†L∂τΨL +14β|µ| ch 2τ∂τΨ†L∂τΨL + ∂τΨ†R∂τΨR+18β|µ| ch 2τ (1 −52 th 2τ)Ψ†LΨL + Ψ†RΨR. (7.13)By choosing µ < 0 we have eliminated two problems at once.

First, now the hamil-tonian has no divergence at τ = 0. Second, there is now no relevant boundary atall, ΨL and ΨR do not mix, and there is no need to impose boundary conditions.The fact that the two chiralities do not mix to all orders of the string perturbationtheory is remarkable.

Its impact on the scattering processes will be discussed afterwe introduce the bosonized formalism.7.2.The Bosonic FormalismIn this section we will explicitly confirm the expectation that the space-timepicture of the c = 1 string theory involves the dynamics of one massless scalar fieldin two dimensions. To this end we will bosonize the fields ΨL and ΨR following thestandard bosonization rules for Dirac fermions.

40 A two dimensional free masslessDirac fermion is equivalent to a single free massless scalar boson. In our case, althoughthe fermions are free, they are not truly relativistic beyond the semiclassical limit.This will give rise to interaction terms in the equivalent bosonic field theory.To bosonize the system we replace the fermion fields by40ΨL =1√2π : expi√πτZ(P −X′)dτ′: ,ΨR =1√2π : expi√πτZ(P + X′)dτ′: ,(7.14)where X is a massless two-dimensional periodic scalar field, and P is its canonicallyconjugate momentum.

The normal ordering in eq. (7.14) is in terms of the con-ventionally defined creation and annihilation operators, which we will utilize for the36

explicit calculations of section 6.3. To convert eq.

(7.7), we make use of the followingeasily derived expressions: Ψ†L∂τΨL −Ψ†R∂τΨR : = i2 : P 2 + (X′)2 :: Ψ†LΨL + Ψ†RΨR : = −X′√π: ∂τΨ†L∂τΨL + ∂τΨ†R∂τΨR : = −√π : PX′P + 13(X′)3 + 16πX′′′ :. (7.15)Substituting these into eq.

(7.7), we find: H :=12T/2Z0dτ :P 2 + (X′)2 −√πβv2PX′P + 13(X′)3 + 16πX′′′−12β√πX′v′′v3 −5(v′)22v4 : . (7.16)If we integrate by parts and discard the boundary terms, this reduces to [10]: H := 12T/2Z0dτ :P 2 +(X′)2 −√πβv2(PX′P + 13(X′)3)−12β√πX′ v′′3v3 −(v′)22v4 : .

(7.17)The boundary conditions obeyed by the field X are determined by those of Ψ, eq. (7.11), which ensure that fermion number not flow out of the τ box.

As we have shown,the current density ¯Ψ(τ)γ1Ψ(τ) = Ψ†RΨR −Ψ†LΨL vanishes at the boundary. Sincethis density is proportional to ∂tX(t, τ), we deduce that X(t, 0) and X(t, T2 ) mustbe constant, i.e., X satisfies Dirichlet boundary conditions.

The constraint on thetotal fermion number requires that X(t, 0) −X(t, T2 ) = 0. We are free to chooseX(t, 0) = X(t, T2 ) = 0.

These boundary conditions eliminate all the winding andmomentum modes of X. As a result, there is no need to worry about the periodicnature of X: it acts like an ordinary scalar field in a box with Dirichlet boundaryconditions.37

The hamiltonian (7.17) and the boundary condition agree with the collectivefield approach.9,41 Physically, the massless field X describes small fluctuations of theFermi surface. In the double-scaling limit the hamiltonian reduces to (for µ > 0): H := 12∞Z0dτ :P 2 + (X′)2 −√π2βµ sh 2τPX′P + 13(X′)3−1 −32 cth 2τ12βµ√π sh 2τ X′: .

(7.18)This hamiltonian, as its fermionic counterpart, suffers from a divergence at τ = 0.This divergence was regularized in ref. 41 with zeta-function techniques.

Alterna-tively, we may take the double-scaling limit with µ < 0. Then: H := 12∞Z0dτ :P 2 +(X′)2 −√π2β|µ| ch 2τPX′P + 13(X′)3−1 −32 th 2τ12β|µ|√π ch 2τ X′: ,(7.19)and the sickness at τ = 0 has disappeared.

The bosonized theory of non-relativisticfermions has a remarkable structure. In the double-scaling limit there is a single cubicinteraction term of order gst, and a tadpole of order gst.

Therefore, we can developan expansion of correlation functions in powers of gst using conventional perturbationtheory. As we will show, this expansion reproduces the genus expansion of the stringamplitudes.7.3.

Scattering Amplitudes.Scattering amplitudes of the X-quanta can be related to the Euclidean correlationfunctions in the matrix model.43 We will exhibit this relation for µ > 0 38 (the µ < 0case works similarly). The finite boundary operator O(l, q) from eq.

(4.13) can betranslated into12Zdxeiqx∞Z0dτe−lλ(τ) : Ψ†LΨL + Ψ†RΨR(τ, t) : ,(7.20)38

where λ(τ) is the classical trajectory at the Fermi level. Upon bosonization, we findO(l, q) ∼ZdxeiqxZdτe−lλ(τ)∂τX ∼i∞Z−∞dkF(k, l)k ˜X(q, k)(7.21)whereF(k, l) =∞Z0dτe−lλ(τ) cos(kτ) ,X(x, τ) =Zdxe−iqxZdk sin(kτ)X(q, k) .

(7.22)Evaluating F(k, l) with λ(τ) = √2µ ch τ, we find38F(k, l) = Kik(lp2µ) =π2 sin(ikπ)(I−ik(lp2µ) −Iik(lp2µ)) . (7.23)For small z, we will replaceIν(z) →(z/2)ν1Γ(ν + 1) .

(7.24)In calculating the Euclidean correlation functions, each operator will be connectedto the rest of the Feynman graph by the propagator 1/(q2 + k2). We will deformthe k-integral of eq.

(7.21) for each external leg, and pick up the residue of thepropagator pole. There is a subtlety here: each K-function has to be split into asum of two I-functions as in eq.

(7.23), and the allowed sense of deformation for thetwo is opposite. For I−ik we pick up the pole at k = i|q|, while for Iik – the pole atk = −i|q|.

As a result, we obtain the amputated on-shell Euclidean amplitude timesa factor for each external leg, which for small l is(lpµ/2)|q|πsin(π|q|)Γ(1 + |q|) = −(lpµ/2)|q|Γ(−|q|) . (7.25)Therefore, the correlation function of puncture operators is* NYi=1P(qi)+∼NYj=1(−µ|qj|/2Γ(−|qj|))A(q1, .

. .

, qN)(7.26)where A is the Euclidean continuation of a scattering amplitude of N X-quanta. The39

same factor for each external leg appears in the direct fermionic calculations, (4.15)-(4.17). The lesson from this rather technical exercise is that the Das-Jevicki fieldtheory assembles the correlation functions in a rather remarkable way.

Each scatteringamplitude of the X-quanta has to be multiplied by the external leg factors which arisebecause of the unusual form (7.21) of the external operators.43 On the other hand,these factors containing the poles have an important string theoretic meaning andcertainly cannot be discarded. Thus, we are still missing the precise interpretation ofthe Das-Jevicki field X in the string theory with Euclidean signature.After continuation of string amplitudes to the Minkowski signature, X →it, theconnection with the Das-Jevicki field theory is more straightforward.

The Euclideanexternal leg factor for the operator T(q) is −µ|q|/2Γ(−|q|)/Γ(|q|). Upon continuation|q| →−iE, this factor turns into a pure phase 13−exp−iE2 ln µ Γ(iE)Γ(−iE)(7.27)which does not affect any scattering cross-sections.

Since the external leg phases areunobservable, they can be absorbed in the definition of the vertex operators. Thus,we find that the Minkowskian scattering amplitudes of tachyons are given by thecorresponding amplitudes of the X-quanta.In fact, we may consider two kinds of scattering experiments.

The usual scatteringinvolves colliding right-moving and left-moving wave packets. In this theory such anexperiment yields trivial results since H = HL + HR 43, and the wave packets passthrough each other with no influence or time delay.44 Thus, there is no conventional“bulk” scattering, whose rate is finite per unit spatial volume when plane waves arebeing scattered.We can, however, consider another kind of scattering.44 For µ > 0 we may preparea left-moving wave packet incident on the boundary and wait for it to reflect.

Thiscan be interpreted as the scattering of n left-moving particles into m right-movingparticles. (Recall that the number of massless particles in one spatial dimension is40

a subtle concept that needs to be carefully defined. )For µ < 0, a left- (right- )moving wave packet stays left- (right- ) moving to all orders of perturbation theory ingst.

However, now there are two asymptotic regions, τ →±∞, and the wave packetundergoes some deformation as it passes through the interaction region near τ = 0.This deformation can be interpreted as a change of state (and number) of particles.It is this “non-bulk” scattering,44,41,42 whose rate is not proportional to the spatialvolume, that gives, upon Euclidean continuation, the matrix model correlators. Todemonstrate this explicitly, we will now calculate tree-level amplitudes for 2 →1 and2 →2 particle scattering.We will work with the µ < 0 hamiltonian of eq.

(7.19) and show that the calcula-tions are manifestly finite. Since the chiralities do not mix, we will consider scatteringof right-moving particles described by the hamiltonianHR = 14∞Z−∞:"(P −X′)2 +√π6β|µ| ch 2τ (P −X′)3 +1 −32 th 2τ12β|µ|√π ch 2τ (P −X′)#: .

(7.28)Following ref. 41, we will perform our calculation in the hamiltonian formalism.

Thisis advantageous here because bosonization maps the fermionic hamiltonian into thenormal-ordered bosonic one. Since fermionic calculations are finite, we expect thatthe normal ordering will remove all the divergences in the bosonic approach.

In otherwords, the hamiltonian approach provides a definite set of rules on how to handle thebosonized theory so that it is perfectly equivalent to the original fermionic theory.We introduce the canonical oscillator basisX(t, τ) =∞Z−∞dkp4π|k|a(k)ei(kτ−|k|t) + a†(k)e−i(kτ−|k|t),P(t, τ) = −i∞Z−∞dkp4π|k||k|a(k)ei(kτ−|k|t) −a†(k)e−i(kτ−|k|t),(7.29)such that [a(k), a†(k′)] = δ(k −k′). The hamiltonian assumes the form HR = H2 +41

H3 + H1.H2 =∞Z0dkka†(k)a(k) ,H3 =i24πβ|µ|∞Z0dk1dk2dk3pk1k2k3f(k1 + k2 + k3)a(k1)a(k2)a(k3)−3f(k1 + k2 −k3) : a(k1)a(k2)a†(k3) :+ h.c. ,H1 = −i48πβ|µ|∞Z0dk√kg(k)a(k) −a†(k),(7.30)wheref(k) =∞Z−∞dτ1ch 2τ eikτ =πksh (πk/2) ,g(k) =∞Z−∞1 −32 th 2τch 2τeikτ = π(k3 + 2k)4 sh (πk/2) . (7.31)Now we calculate the S-matrix,S = 1 −2πiδ(Ei −Ef)T .

(7.32)Each right-moving massless particle has energy equal to momentum, E = k. Considerthe amplitude for two particles with momenta k1 and k2 to scatter into a single particleof momentum k3,S(k1, k2; k3) = −2πiδ(E1 + E2 −E3)pE1E2E3 < k3|Hint|k1k2 >(7.33)where |k1k2 >= a†(k1)a†(k2)|0 >. Substituting the cubic interaction term, we findS(E1, E2; E3) = −gstδ(E1 + E2 −E3)E1E2E3 .

(7.34)The fact that this amplitude describes “non-bulk” scattering is evident from theabsence of a separate delta-function for momentum conservation. In order to relate42

this to the matrix model three-puncture correlation function, we perform Euclideancontinuation Ej →i|qj|. Including the external leg factors according to eq.

(7.26),the result is in agreement with the tree-level contribution to eq. (4.16).Now we proceed to the more complicated case of non-forward scattering of par-ticles of momenta k1 and k2 into particles of momenta k3 and k4.

The amplitude isgiven by second-order perturbation theory,T(k1, k2; k3, k4) =pE1E2E3E4Xi< k3k4|Hint|i >< i|Hint|k1k2 >E1 + E2 −Ei,(7.35)where the sum runs over all the intermediate states i. The s-channel contribution iseasily seen to be41S(s)(k1, k2; k3, k4) = −ig2st8π4Yj=1Ej∞Z0dkkf2(k1 + k2 −k)k1 + k2 −k + iǫ −kf2(k1 + k2 + k)k1 + k2 + k −iǫ,(7.36)where the first term arises from the one-particle intermediate state, and the secondone – from the five-particle intermediate state.

Eq. (7.36) can be expressed as anintegral over k from −∞to ∞.

The t- and u-channel contributions assume a similarform. Summing over the three channels, we obtain41S(k1, k2; k3, k4) = −iπg2st8 δ(E1+E2−E3−E4)4Yj=1EjF(ps)+F(pt)+F(pu), (7.37)where ps = k1 + k2, pt = |k1 −k3|, pu = |k1 −k4|, andF(p) =∞Z−∞dk(p −k)2sh 2(π(p −k)/2)kp −k + iǫ sgn(k) .

(7.38)Using1x −iǫ = P 1x + πiδ(x)(7.39)43

xxFig. 4a) The one-loop Og2stcontributions to the 1 →1amplitude.Fig.

4b) The tadpole graphsof order g2st.we find F(p) = −i4pπ −83π. The total amplitude thus becomesS(E1,2; E3,4) = −g2st2 δ(E1+E2−E3−E4)4Yj=1EjE1+E2+|E1−E3|+|E1−E4|−2i.

(7.40)Upon the Euclidean continuation Ej →i|qj|, and inclusion of the external leg factors,this precisely agrees with the non-relativistic fermion calculation of the 4-puncturecorrelator (4.17).Now we give an example of one-loop calculation. Following ref.

41, we calculatethe O(g2st) correction to the two-point function (1 →1 amplitude). Like the tree-level4-point function, this is given by second-order perturbation theory.

The contributionto T(k, k) from the one-loop graphs is18g2stk2(24π)2∞Z0dk1dk2 k1k2 f2(k1 + k2 −k)k −k1 −k2 + iǫ −f2(k1 + k2 + k)k + k1 + k2 −iǫ(7.41)where the first term is from the 2-particle intermediate state, and the second – fromthe 4-particle intermediate state (fig. 4a).

After changing variables to s = k1 + k2and k2, and integrating over k2, this reduces to3g2stk2(24π)2∞Z−∞dss3f2(k −s)k −s + iǫ sgn(s) = −g2st48πk2ik3 + 2k2 + 815(7.42)44

where we evaluated the integral similarly to that of eq. (7.38).This is not thecomplete answer because there are also the diagrams of fig.

4b) arising from thetadpole term H1. They contribute equally, giving12g2stk2(24π)(48π)∞Z0dk1k1f(k1)g(k1)−k1 + iǫ = −g2stk2384∞Z0dk1k41 + 2k21sh 2(πk1/2) = −g2st48π × 7k215 .

(7.43)Adding this to the one-loop contribution (7.42), we findT(E, E) = −g2st48πE2 iE3 + 2E2 + 1. (7.44)Upon the Euclidean continuation and inclusion of the external leg factors, this agreeswith eq.

(4.15).The agreement with the string one-loop calculation gives us further confidencethat the bosonic string field theory is finite and exact. This implies that the perturba-tion series exhibits the (2h)!

behavior, which is unlike the h! found in the conventionalfield theory.

This appears to be connected with the lack of translational invariance ofthe hamiltonian and of the scattering amplitudes.41 In general, owing to the expo-nential fall-offof the form-factors f(k) and g(k), the hamiltonian (7.30) is completelyfree of ultraviolet divergences. Thus, our innovation of taking the double-scaling limitwith µ < 0 has rendered the entire perturbative expansion manifestly finite.

On theother hand, the end results of our calculations do not change under the continuationµ →−µ. This suggests that the divergences for µ > 0, connected with the fact thatthere f(k) and g(k) did not fall offas k →∞,41are simply a result of using aninconvenient formalism.

This is also indicated by the fact that all divergences arecorrectly removed by zeta-function regularization.41 In fact, positive and negative µare related by the transformation that interchanges the classical coordinate λ withits conjugate momentum p. Thus, the equation(−12∂2∂λ2 −12λ2)ψ = −βµψ(7.45)45

can be written in terms of p as(−12∂2∂p2 −12p2)ψ = βµψ . (7.46)The lesson is that, while for µ < 0 it is convenient to regard λ as the coordinate, forµ > 0 the natural coordinate is p. If we define the original non-relativistic fermiontheory in terms of Ψ(p, t), then the formalism of this section goes through nicelyfor µ > 0: there are no boundaries, and the chiralities do not mix.

Thus, all thecalculations above can also be regarded as µ > 0 calculations which are now manifestlyfinite. Using this approach, the tree-level 4-point amplitude was calculated in thelagrangian formalism in ref.

42. Another technique, which works efficiently for all thetree amplitudes, but is not easily generalized to the loop corrections, was introducedin ref.

44.Despite the impressive progress in the formulation of the bosonic string fieldtheory of the c = 1 quantum gravity, there are many puzzles remaining. In particular,how should we interpret the coordinate τ and the field X(τ)?

Initially, it was arguedthat τ is essentially the zero mode of the Liouville field φ. If so, then it is naturalto think of X as the tachyon field.

However, it was shown recently that τ is notlocally related to the scale factor, but is instead a conjugate variable in a complicatedintegral transform.38 This transform is necessary to turn a string field theory, which ishighly non-local in φ-space, into a simple local hamiltonian in τ-space. Miraculously,the matrix model has automatically provided the coordinates in which the theorylooks the simplest.

However, it does not appear possible to interpret X simply as thetachyon. Recall that, in addition to the tachyon, the theory has a discrete infinityof other non-field degrees of freedom.

It appears that the theory has soaked up thediscrete degrees of freedom together with the tachyon into a single massless field X.The form of the matrix model operators for the discrete observables, eq. (6.4), indeedshows that they are mixed in the X-field together with the tachyon.

Clearly, we needto attain a much better understanding of the precise string theoretic meaning of thebosonized field theory of the matrix model.46

8. COMPACT TARGET SPACE AND DUALITY.In this section we consider the discretized formulation of the sum over surfacesembedded in a circle of radius R.6 We will adopt the basic definition (2.5), with thenew conditionG(x) = G(x + 2πR)(8.1)that ensures the periodicity around the circle.

In this model we encounter a new setof physical issues related to the R-dependence of the sum. The c = 1 conformal fieldtheory is symmetric under the transformation45R →α′R ;gst →gst√α′R.

(8.2)The change of gst, equivalent to a constant shift of the dilaton background, is necessaryto preserve the invariance to all orders of the genus expansion. A symmetry of thematter system, such as (8.2), is expected to survive the coupling to gravity.Inthis section we will find, however, that in the discretized formulation the duality isgenerally broken, and can only be reinstated by a careful fine-tuning.

This breakingof duality is due not to the coupling to gravity, but to the introduction of the explicitlattice cut-off. Indeed, a generic lattice formulation of the compact c = 1 model on afixed geometry has no dual symmetry.

The phase transition that separates the smallR (high temperature) phase from the large R (low temperature) phase is well-knownin statistical mechanics as the Kosterlitz-Thouless transition.46 Physically, it is dueto condensation of vortices, configurations that are ignored in the “naive” continuumlimit, but are included in the continuum limit of a generic lattice theory. The vorticesare irrelevant for R > RKT , but they condense and change the behavior of the theoryfor R < RKT .In this section we will demonstrate the lack of duality of the discretized sum oversurfaces.

We will also show that, if the lattice sum is modified to exclude the vortices,then the partition function is explicitly dual. These general results will be of interest47

when, in the next section, we consider the specific discretized sum generated by thematrix model. We will be able to isolate the effects of vortices in the matrix model,and will calculate the dual partition function in the vortex-free continuum limit.In the conformal field theory the target space duality (8.2) is generated by thedual transformation on the world sheet47∂aX →ǫab∂bX .

(8.3)Its lattice analogue is the transformation from the lattice Λ to the dual lattice ˜Λ.Recall that, when we sum over lattices Λ, we define the target space variables xi at thecenters of the faces of Λ, i.e. at the vertices of ˜λ.

We will carry out a transformation,after which the new variables, pI, are defined at the vertices of Λ.First, to each directed link < ij > of ˜Λ associate the link < IJ > of Λ whichintersects it, directed so that the cross product ⃗ij × ⃗IJ points out of the surface (fig.1). Now we define Dij = DIJ = xi −xj.

Let us change variables in the integral (2.5)so that the integral is over DIJ instead of xi. There are E links but, apart from thezero mode, there are only V −1 independent x’s.⋆Therefore, the DIJ’s are not allindependent but must satisfy F −1 + 2h constraints.

The constraint associated witheach face is that P⟨ij⟩Dij = 0, where the sum runs over the directed boundary ofa face of ˜Λ. This is equivalent to the condition that the sum of the DIJ’s emergingfrom each vertex of Λ must vanish, PJ DIJ = 0.

Similarly, for each independentnon-contractable loop on ˜Λ, of which there are 2h, we find Ploop Dij = 0. In termsof λ this means that P⟨IJ⟩ǫaIJDIJ = 0, where the symbol ǫaIJ is non-zero only if thelink IJ intersects the specially chosen non-contractable loop a on ˜Λ.

We direct theloop a, and define ǫaIJ = 1 if ⃗IJ × ⃗a points into the surface, and −1 if it points out.The theorem of Euler V −E + F = 2 −2h,insures that the net number ofindependent variables remains unchanged. Introducing Lagrange multipliers pI for⋆Here V , E and F are, respectively, the numbers of vertices, edges and faces of ˜Λ, and h is thegenus.48

each face of ˜Λ and la for each non-contractable loop, the integral over the variablesxi, for each discretization of the surface, can be replaced withZ Y⟨IJ⟩dDIJG(DIJ)Z YIdpI exp"ipIXJDIJ# Z2hYa=1dla expilaX⟨IJ⟩ǫaIJDIJ=Z YIdpI2hYa=1dlaY⟨IJ⟩˜G(pI −pJ + laǫaIJ)(8.4)where ˜G is the Fourier transform of G. We see that, after the transformation (8.4),the Lagrange multipliers pI assume the role of new integration variables residing atthe vertices of Λ. On surfaces of genus > 0 there are 2h additional l-integrations.Geometrically speaking, one introduces a cut on the surface along each canonicallychosen non-contractable loop, so that the values of p undergo a discontinuity of laacross the a-th cut, and subsequently integrates over la.

On a sphere these additionalintegrations do not arise, and one simply includes a factor ˜G(pI −pJ) for each linkof the lattice Λ.The discussion above literally applies to the string on a real line. When G isdefined on a circle of radius R, then the dual variables la and pI assume discretevalues nR.

The integrals over pI and la in eq. (8.4) are replaced by discrete sums,and the partition function, when expressed in terms of the dual variables, bears littleresemblance to the original expression.

In fact, the new variables pI describe theembedding of the world sheet in the discretized real line with a lattice spacing 1/R.On a sphere, where la do not arise, the resulting lattice sum is precisely what isneeded to describe string theory with such an embedding. For higher genus, the fewextra variables la spoil the precise equivalence, but this does not alter the essential“bulk” physics.We have found that the transformation to the dual lattice, unlike the transforma-tion (8.3) in the “naive” continuum limit, does not prove R →α′/R duality.

Instead,it establishes that string theory embedded in a compact dimension of radius R isessentially equivalent to string theory embedded in a discretized dimension of lattice49

spacing 1/R. To conclude that R is equivalent to 1/R is just as counterintuitive asto argue that theories with very small lattice spacing in target space have the samephysics as theories with enormous lattice spacing.

In section 11 we will directly studythe theory with the discrete target space. We will show that, if the lattice spacingis less than a critical value, then the target space lattice is “smoothed out”, but ifit exceeds the critical value, then the theory drastically changes its properties.

Thetwo phases are separated by a Kosterlitz-Thouless phase transition. Equivalently, inthe circle embedding the massless c = 1 phase for R > RKT is separated by the K-Ttransition from the disordered (massive) c = 0 phase for R < RKT .

The mechanismof this phase transition involves deconfinement of vortices.46Consider a fixed geometry ˆgµν, and assume that in the continuum limit the actionreduces to eq. (2.2).

A vortex of winding number n located at the origin is describedby the configurationX(θ) = nRθ(8.5)where θ is the azimuthal angle. This configuration is singular at the origin: the valuesof X have a branch cut in the continuum limit.

On a lattice, however, there is nosingularity, and the vortex configurations are typically included in the statistical sum.If we introduce lattice spacing √µ, then the action of a vortex is Sn = n2R2| ln µ|/4α′,and it seems that the vortices are suppressed in the continuum limit. We will nowshow that this expectation is only true for large enough R.Let us consider thedynamics of elementary vortices with n = ±1.

Although each one is suppressed bythe action, it has a large entropy: there are ∼1/µ places on the surface where itcan be found. Thus, the contribution of each vortex or antivortex to the partitionfunction is of the order1µe−S1 = µ(R2/4α′)−1 .

(8.6)It follows that, for R > 2√α′, the vortices are irrelevant in the continuum limit. Onthe other hand, for R < 2√α′ they dominate the partition function, invalidating the“naive” continuum limit assumed in the conformal field theory.46In fact, in this50

phase the proliferation of vortices disorders X and makes its correlations short range.The critical properties of this phase are those of pure gravity. Is there any way tosalvage the “naive” continuum limit?

The answer is yes, but only at the expense offine tuning the model, so that the vortices do not appear even on the lattice. Belowwe find an explicit example of a model without vortices, and show that it does exhibitthe R →1/R duality.First, we have to give a clear definition of a vortex on a lattice.

Loosely speaking,a face I of lattice ˜Λ contains v units of vortex number if, as we follow the boundary ofI, the coordinate x wraps around the target space circle v times. This does not quitedefine the vortex number because x is only known at a few discrete points along theboundary and cannot be followed continuously.

In order to define vortex number ona lattice, it is convenient to adopt the Villain link factor 48G(x) =∞Xm=−∞e−12(x+2πmR)2(8.7)where the sum over m renders G periodic under x →x+2πR. Now, eq.

(2.5) becomesZ(g0, κ) =Xhg2(h−1)0XΛκAreaVYi=12πRZ0dxi√2πY⟨ij⟩∞Xmij=−∞e−12(xi−xj+2πmijR)2. (8.8)In this model the number of vortices inside a face I can be defined asvI =X∂Imij ,(8.9)where ∂I is the directed boundary of I.

After the dual transformation, one obtains adiscrete sum with˜G(pI −pJ + laǫaIJ) = e−12(pI−pJ+laǫaIJ)2pI = nIR , la = naR. (8.10)This sum is closely connected to the sum over all possible vortex numbers inside eachface of ˜Λ.4851

We shall now change the definition of the partition function (8.8) to exclude thevortex configurations. It is helpful to think of mij as link gauge fields.

The vortexnumbers vI defined in eq. (8.9) are then analogous to field strengths.

If there are novortices, the field strength is zero everywhere. However, we still need to sum over allpossible windings around non-contractable loops on the lattice ˜Λ, lA = Ploop a mij.Thus, the space of mij we need to sum over ismij = ǫAijlA + mi −mj,(8.11)where mi range over all integers and play the role of gauge transformations.

ǫAij isdefined above and is non-zero only for the links ⟨ij⟩intersecting the specially chosennon-trivial loop A on the lattice Λ. Thus, lA is the winding number for the non-trivialcycle a on ˜Λ which intersects the loop A. Summing over mij from eq.

(8.11) only,the partition function becomesZ(g0, κ) = 2πRXhg2(h−1)0XΛκVV −1Yi=1∞Z−∞dxi√2π∞XlA=−∞Y⟨ij⟩e−12(xi−xj+2πlAǫAijR)2,(8.12)where the factor of 2πR arises from integration over the zero mode of x.Aftertransforming to the dual lattice we obtainZ(g0, κ) = 1RXhg0√2πR2h−2 XΛκVF−1YI=1∞Z−∞dpI√2π∞Xla=−∞Y⟨IJ⟩e−12(pI−pJ+laǫaIJ/R)2. (8.13)Thus, after elimination of the vortices, the transformation to the dual lattice clearlyexhibits duality under√2πR →1√2πR,g0 →g0√2πR .

(8.14)Remarkably, the duality is manifest even before the continuum limit is taken. Inthe next section we consider the matrix model for random surfaces embedded in a52

circle.We will find a clear separation between the vortex contributions, and the“naive” vortex-free continuum sum. This will allow us to calculate explicitly the dualpartition function of the vortex-free model.9.

MATRIX QUANTUM MECHANICSAND THE CIRCLE EMBEDDING.The Euclidean matrix quantum mechanics is easily modified to simulate the sumover discretized random surfaces embedded in a circle of radius R: we simply definethe matrix variable Φ(x) on a circle of radius R, i.e. Φ(x + 2πR) = Φ(x).

The pathintegral becomes6Z =ZDN 2Φ(x) exp−β2πRZ0dx Tr 12∂Φ∂x2+ U(Φ)!. (9.1)The periodic one-dimensional propagatorG(xi −xj) =∞Xm=−∞e−|xi−xj+2πmR|(9.2)gives the weight for each link in the random surface interpretation.

In terms of thehamiltonian (3.10), eq. (9.1) is simply a path integral representation for the partitionfunctionZR = Tr e−2πRβH ,(9.3)so that 2πR plays the role of the inverse temperature.

The problem of calculatingthe finite temperature partition function seems to be drastically more complicatedthan the zero temperature problem, where only the ground state energy was relevant.Now we have to know all the energies and degeneracies of states in arbitrarily highrepresentations of SU(N). We also have to explain the sudden enormous jump inthe number of degrees of freedom as we slightly increase the temperature from zero.53

In view of the discussion in the last section, the reader should not be surprised if weclaim that the new degrees of freedom, incorporated in the non-trivial representationsof SU(N), are due to the K-T vortices. Since the vortices are dynamically suppressedfor R > RKT , the problem of calculating ZR is not as complicated as it may seem.Let us begin by calculating the contribution to ZR from the wave functions inthe trivial representation of SU(N).

Since the singlet spectrum is that of N non-interacting fermions moving in the potential U, its contribution to the partition func-tion can be calculated with the standard methods of statistical mechanics.Instead of working with a fixed number of fermions N, we will take the well-knownroute of introducing a chemical potential µF adjusted so thatκ2 = Nβ =∞Z0ρ(ǫ)11 + e2πRβ(ǫ−µF )dǫ . (9.4)In the thermodynamic limit, N →∞, the free energy satisfies∂F∂N = µF.

(9.5)Shifting the variables, µ = µc −µF , e = µc −ǫ, we can write the equations for thesingular part of F as∂∆∂µ = π˜ρ(µ) = π∞Z−∞deρ(e) ∂∂µ11 + e2πRβ(µ−e)∂F∂∆= 1πβµ ,(9.6)to emphasize resemblance with eqs. (3.18) and (3.19) in the R = ∞case.

Indeed,we will calculate F(∆) using the same method. The change introduced by a finite Ronly affects the first of the equations, determining µ(∆).

We can think of ˜ρ(µ) as the54

temperature-modified density of states. Differentiating it, we find1β∂˜ρ∂µ =∞Z−∞de 1β∂ρ∂eπRβ2 cosh2[πRβ(µ −e)].

(9.7)Now, substituting the integral representation (3.26) and performing the integral overe, we find1β∂˜ρ∂µ = 1πIm∞Z0dTe−iβµTT/2sinh(T/2)T/2Rsinh(T/2R) ,(9.8)up to terms O(e−βµ) and O(e−βµ2πR) which are invisible in the large-β asymptoticexpansion. Thus, this integral representation should only be regarded as the generatorof the correct expansion in powers of gst.

Integrating this equation and fixing theintegration constant to agree with the WKB expansion, we find∂∆∂µ = Re∞Zµdtt e−itt/2βµsinh t/2βµt/2βµRsinh(t/2βµR) . (9.9)This relation has a remarkable duality symmetry underR →1R,β →Rβ .

(9.10)This is precisely the kind of duality expected in the vortex-free continuum limit.This strongly suggests that, in discarding the contributions of the non-singlet statesto ZR, we have suppressed the vortices. Later on we will offer additional argumentsto support this claim.To demonstrate the duality in the genus expansion, consider the asymptotic ex-pansion of eq.

(9.9)∂∆∂µ ="−ln µ +∞Xm=12βµ√R−2mfm(R)#,fm(R) = (2m −1)!mXk=0|22k −2| |22(m−k) −2| |B2k| |B2(m−k)|(2k)! [2(m −k)]!Rm−2k .

(9.11)Note that the functions fh(R) are manifestly dual: f1 = 16(R + 1/R), f2 = 160(7R2 +55

10 + 7R−2), and so forth. Solving for µ(∆) and integrating eq.

(9.6) we find the sumover connected surfaces Z = −2πRβF:Z = 14((2βµ0√R)2 ln µ0 −2f1(R) ln µ0 +∞Xm=1fm+1(R)m(2m + 1)(2βµ0√R)−2m). (9.12)Comparing this with eq.

(3.30), valid in the case of infinite radius, we find that thecoupling constant 1/(βµ0) has been replaced bygeff(R) =1βµ0√R,(9.13)and that the contribution of each genus h contains the function fh(R). Since bothgeff(R) and fh(R) are invariant under the dual transformation (9.10), we have con-firmed that the genus expansion of the sum over surfaces is manifestly dual.

As in thecritical string theory, the effective coupling constant depends on the radius. Indeed,the transformation R →1/R keeping βµ0 fixed is not a symmetry, as it interchangesa weakly and a strongly coupled theory.From eq.

(9.12) we find that the sum over toroidal surfaces is −112(R+1/R) ln µ0.This agrees with the direct calculation in the continuum, eq. (5.16).The extrafactor of 2 in the matrix model is due to the doubling of free energy for models withsymmetric potentials.

To get the basic sum over triangulated surfaces, we have todivide eq. (9.12) by 2, which then gives perfect agreement with Liouville theory.This provides additional evidence that the singlet free energy in the matrix modelevaluates the sum over surfaces in the vortex-free continuum limit.

To see this moreexplicitly, we have to consider the non-singlet corrections to the free energy and showthat, like the vortices, they are irrelevant for R > RKT .This problem was discussed in ref. 12, where it was argued that the total partitionfunction can be factorized asTr e−2πRβH = Trsinglet e−2πRβH XnDne−2πRβδEn!

(9.14)where δEn is the energy gap between the ground state and the lowest state in the nth56

representation, and Dn is the degeneracy factor. Let us discuss the leading correctionto the free energy coming from the adjoint representation.

Although the degeneracyfactor Dadj = N2 −1 diverges as N →∞, an estimate of ref. 12 shows that theenergy gap also diverges in the continuum limit,βδEadj = c| ln µ| .

(9.15)As a result, the correctionDadje−2πRβδEadj ∼N2µ2πRc(9.16)is negligible for R >1πc in the continuum limit where Nµ is kept constant. Takingthe logarithm of eq.

(9.14), we find that the total free energy isF = Fsinglet + ON2µ2πRc+ . .

. (9.17)Thus, on the one hand, the higher representations are enhanced by enormous degen-eracy factors; on the other hand, they are suppressed by energy gaps which divergelogarithmically in the continuum limit.

This struggle of entropy with energy is pre-cisely of the same type as occurs in the physics of the Kosterlitz-Thouless vortices, asdiscussed above. In fact, we will now argue that the leading non-singlet correction ineq.

(9.17) is associated with a single vortex-antivortex pair. We perturb the actionof the continuum theory with the operator which creates elementary vortices andanti-vortices,Ov =Zd2σpˆg cos Rα′(XL −XR)(9.18)where XL,R are the chiral components of the scalar field X.

In string theory language,this is the sum of vertex operators which create states of winding number 1 and −1.The insertion of such an operator on a surface creates an endpoint of a cut in thevalues of X. The conformal weight of Ov is h = ¯h =R24α′; therefore, Ov becomesrelevant for R < 2√α′, causing a phase transition at RKT = 2√α′49.

The phasetransition is connected with the instability towards creation of cuts in the values ofX, i.e., with vortex condensation.57

This picture of the Kosterlitz-Thouless phase transition can be easily adaptedfor coupling to two-dimensional quantum gravity. The sum over genus zero surfacescoupled to the periodic scalar reduces to the path integralZ = N2Z[DX][Dφ] exp−SL −S0 −kZd2σpˆgeβφ cos Rα′(XL −XR)(9.19)where SL is the Liouville gravity action for c = 1, and β = −2+ R√α′ so as to make theconformal weight of the perturbing operator (1, 1).

The gravitational dimension ofthe perturbation is thenR2√α′ −1. For small values of the cut-offµ, and for R > 2√α′,ZN2 = −µ2R ln µ + Ok2µR/√α′+ Ok4µ(2R/√α′)−2+ .

. .

,(9.20)which demonstrates that the Kosterlitz-Thouless transition occurs at Rc = 2√α′, thesame value as in a fixed gravitational background. The term O(k2) in Z originatesfrom the 2-point function of dressed winding mode vertex operators, i.e.

from surfaceswith one cut.This leading correction has the same form as eq. (9.17) found inthe context of matrix quantum mechanics.The further corrections, coming fromadditional vortex-antivortex pairs, are even more suppressed for R > RKT in thecontinuum limit.For R < RKT , however, the expansion in k diverges badly, indicating an instabil-ity with respect to condensation of vortices.

In this phase we expect the field X tobecome massive, and the continuum limit of the theory should behave as pure gravity.In the matrix model this is easy to show for very small R. The argument is standard:at very high temperature a d + 1-dimensional theory reduces to d-dimensional theorybecause the compact Euclidean dimension of length 2πR = 1/T becomes so small thatvariation of fields along it can be ignored. Thus, for small R, the matrix quantummechanics (9.1) reduces to the integral over a matrix (zero-dimensional theory)50ZDN 2Φ exp[−2πRβ Tr U(Φ)] ,(9.21)well-known to describe pure gravity.58

We have accumulated a considerable amount of evidence that the non-singletwave functions of the matrix quantum mechanics implement the physical effects ofthe Kosterlitz-Thouless vortices.For R > RKT their corrections to the partitionfunction are negligible, and the singlet free energy (9.12) gives the sum over surfaces.For R < RKT the non-singlets dominate the partition function, and the matrix modelno longer describes c = 1 conformal field theory coupled to gravity. In this phase thesinglet free energy gives the sum over surfaces in the vortex-free “naive” continuumlimit.

As we showed in section 8, this continuum limit can be achieved at the expenseof a careful fine tuning of the lattice model.In the matrix model the equivalentoperation turns out to be simple: just discard the non-singlet states. We conclude,therefore, that for any R the singlet sector of the matrix quantum mechanics describesc = 1 conformal field theory coupled to gravity.

This remarkable fact means that, likethe R = ∞theory, the compact theory is exactly soluble in terms of free fermions. Inthe next section, we will use this to calculate compact correlation functions exactlyto all orders in the genus expansion.10.

CORRELATION FUNCTIONS FOR FINITE RADIUS.In this section we evaluate the correlation functions of the compact c = 1 modelcoupled to gravity using the formalism of free non-relativistic fermions at finitetemperature.13We have already used this approach to find the sum over surfaceswith no insertions, eq. (9.12).

The calculation of correlation functions has a verysimple relation to the zero-temperature (R = ∞) calculation described in section 4.We again consider general operators of type (4.7), which reduce to fermion bilinears.Thus, the goal is to calculate the generating function G(q1, λ1; . .

. ; qn, λn) of eq.

(4.9)in the compact case.The free fermions are now described by the thermal Euclidean second-quantizedactionS =∞Z−∞dλ2πRZ0dx ˆψ†(−ddx + d2dλ2 + λ24 + βµ) ˆψ ,(10.1)59

and the fermion field satisfies the antiperiodic boundary conditions ˆψ(λ, x + 2πR) =−ˆψ(λ, x). Compactifying the Euclidean time is a standard device for describing thetheory at a finite temperature T = 1/(2πR).

The thermal vacuum satisfies⟨βµ|a†ǫ (ν)aǫ′(ν)|βµ⟩= δǫǫ′f(ν) ≡δǫǫ′1e(βµ−ν)/T + 1(10.2)where f(ν) is the Fermi function and µ is the Fermi level. The thermal EuclideanGreen function may be obtained by replacing θ(ν −βµ) by f(ν) in the T = 0 Greenfunction of eq.

(4.10),SE(x1, λ1; x2, λ2) =∞Z−∞dνe−(ν−βµ)∆x{θ(∆x)f(ν) −θ(−∆x)(1 −f(ν))}×ψǫ(ν, λ1)ψǫ(ν, λ2) =i2πRXωne−iωn∆xsgn(ωn)∞Z0dse−sωn+iβµs⟨λ1|e2isH|λ2⟩. (10.3)The only change compared to the R = ∞formula (4.10) is that the allowed fre-quencies become discrete, ωn = (n + 12)/R, where n are integers.

This ensures theantiperiodicity of SE around the compact direction.As in the zero-temperature case, the calculation of eq. (4.9) reduces to a sumover one-loop diagrams.

Since the frequencies are now discrete, the single integrationover the loop momentum is replaced by a sum. This is the only change introduced bythe finite temperature.

As we show below, its effects are easy to take into account.First of all, each external momentum is quantized, qi = ni/R.As a result,the momentum conserving delta function δ(P qi) occurring in the non-compact caseis replaced by R times the Kroenecker function δPqi. Consider a diagram whichcorresponds to a given ordering of the external momentum insertions, which we denote60

as q1, q2, . .

. , qn.

Factoring out the delta-function, the contribution of this diagram isinRXωnsgn(ωn)∞Z0dα1e−α1ωn+iβµα1⟨λn|e2iα1H|λ1⟩sgn(ωn+q1)∞Z0dα2e−α2(ωn+q1)+iβµα2×⟨λ1|e2iα2H|λ2⟩. .

.sgn(ωn+Pn−11qi)∞Z0dαne−αn(ωn+Pn−11qi)+iβµαn⟨λn−1|e2iαnH|λn⟩. (10.4)It is necessary to break the sum over ωn into parts where the sgn functions in theexponents are constant.

Thus, we form pk = −Pk1 qk and put the p’s in the increasingorder. If p and p′ are two consecutive p’s, then the summation that needs to beperformed is, defining ξ = Pn1 αi,1Rp′−12RXωn=p+ 12Re−ωnξ = e−pξ −e−p′ξξξ/2Rsh (ξ/2R) .

(10.5)This factor multiplies the integrand of an n-fold αi-integral. Comparing this withthe R = ∞case, we find that there the loop momentum integration between p andp′ gives the factor (e−pξ −e−p′ξ)/ξ multiplying the same n-fold integrand.

It followsthat, in every single contribution to the final answer, the compactification of thetarget space simply inserts the extra factorξ/2Rsh (ξ/2R) into the integrand. Thus, fromeq.

(4.12) we arrive at1β∂∂µGR(q1, λ1; . .

. ; qn, λn) = in+1RδPqiXσ∈Σn∞Z−∞dξeiβµξξ/2Rsh (ξ/2R)ǫ1∞Z0ds1 .

. .ǫn−1∞Z0dsn−1e−s1Qσ1−...−sn−1Qσn−1⟨λσ(1)|e2is1H|λσ(2)⟩.

. .⟨λσ(n)|e2i(ξ−Pn−11si)H|λσ(1)⟩(10.6)This formula allows us to calculate correlation functions of any set of operators of type(4.7) in the compact c = 1 model.

The integral representation of any such correlator61

is obtained from that in the non-compact case by insertion of the factorξ/2Rsh (ξ/2R) intothe integrand, and by replacement δ(P qi) →RδPqi. A special case of this rule isevident in passing from the R = ∞equation (3.16) to the finite R equation (9.8).In general, comparing eqs.

(10.6) and (4.12), we find that, if⟨O1(q1) . .

. On(qn)⟩R=∞= δ(Xqi)F(q1, .

. ., qn; µ, β)then⟨O1(q1) .

. .

On(qn)⟩R = RδPqiFR(q1, . .

. , qn; µ, β) ,FR(q1, .

. .

, qn; µ, β) =12Rβ∂∂µsin(12Rβ∂∂µ)F(q1, . .

. , qn; µ, β) .

(10.7)This remarkable relation exists because the entire dependence on µ is through thefactor eiβµξ in the integrands of eqs. (4.12) and (10.6).

Therefore, a function f(ξ)can be introduced into the integrand simply by acting on the integral with operatorf(−iβ∂∂µ).Since µ has the interpretation of the renormalized cosmological constant, in eachsum over surfaces of fixed area the compactification simply introduces the non-perturbative factorA/2Rβsin(A/2Rβ).The remarkable feature of this modification, and,equivalently, of the operator in eq. (10.7), is that they are entirely independent ofwhich operators are inserted into the surface.

To demonstrate the power of eq. (10.7),we will use the non-compact correlators of eqs.

(4.15)-(4.17) to develop the corre-sponding genus expansions in the compact case with virtually no extra work. To thisend, we expand the operator in eq.

(10.7) as12Rβ∂∂µsin(12Rβ∂∂µ)= 1 +∞Xk=1(1 −2−2k+1)|B2k|(βR)2k(2k)! ∂∂µ2k.

(10.8)Substituting this expansion into eq. (10.7), we determine the finite R genus g corre-62

lators in terms of the R = ∞genus ≤g correlators,F g=0R(qi; µ) = F g=0(qi; µ) ,F g=1R(qi; µ) = F g=1(qi; µ) +124R2∂2∂µ2F g=0(qi; µ) ,F g=2R(qi; µ) = F g=2(qi; µ) +124R2∂2∂µ2F g=1(qi; µ) +75760R4∂4∂µ4F g=0(qi; µ) , etc. (10.9)In particular, apart from the discretization of momenta and the change in the overallδ-function, all spherical correlation functions are not affected by the compactification.This was, of course, expected in the Liouville approach.

The advantage of the matrixmodel formalism is that the entire genus expansion of any correlator can be calculated.Surprisingly, as we have demonstrated, the compact case is hardly any more difficultthan the non-compact case. From the point of view of the Liouville approach, thisconclusion is quite remarkable.

There the finite R calculations have the additionalcomplication of having to perform 2h sums over the winding numbers around the non-trivial cycles of a genus h surface. Thus, eqs.

(10.7) and (10.9), which are completelygeneral, pose a new challenge for the Liouville approach.Below we give genus expansions for some correlation functions of puncture oper-ators P(q). For the two-point function up to genus three we get⟨P(q)P(−q)⟩= −R[Γ(1 −|q|)]2µ|q|"1|q| −(√Rβµ)−224(|q| −1)×R(q2 −|q| −1) −1R+ (√Rβµ)−457603Yr=1(|q| −r)×R2(3q4 −10|q|3 −5q2 + 12|q| + 7) −10(q2 −|q| −1) + 7R2−(√Rβµ)−629030405Yr=1(|q| −r)R3(9q6 −63|q|5 + 42q4 + 217|q|3 −205|q| −93)−21R(3q4 −10|q|3 −5q2 + 12|q| + 7) + 147R (q2 −|q| −1) −93R3+ .

. .#(10.10)We see that the external leg factors are the same as in the non-compact case.

The63

relation between the tachyon operators T(q) and P(q) is also the same, eq. (5.8).

Forthe same reason as in the non-compact case, eq. (10.10) is, strictly speaking, validonly for non-integer q, where we may replace µ by µ0.

As in the sum over surfaces, wefind that the effective string coupling constant is geff ∼1√Rβµ0. The R-dependenceof the correlator at each genus is not dual: the duality is broken by insertions ofmomentum into the world sheet.

In the limit qi →0 the duality is restored, and thetachyon correlators reduce to derivatives of the sum over surfaces Z(µ0, R),⟨T(0)T(0)⟩∼∂2∂∆2 Z(µ0, R) . (10.11)The three-point function for the case q1, q2 > 0 is, up to genus two,⟨T(q1)T(q2)T(q3)⟩= RδPqi3Yi=1Γ(1 −|qi|)Γ(|qi|)µ|qi|2 1βµ"1−(√Rβµ)−224(|q3| −1)(|q3| −2)R(q21 + q22 −|q3| −1) −1R+4Yr=1(|q3| −r)×R23(q41 + q42) + 10q21q22 −10(q21 + q22)|q3| −5(q1 −q2)2 + 12|q3| + 7−10(q21 + q22 −|q3| −1) + 7R2(√Rβµ)−45760+ .

. .#(10.12)In the case of the four-point function we must consider two independent kinematicconfigurations which will give amplitudes that are not related by analytic continuation.19 Thesimplest case is when q1, q2, q3 > 0,64

⟨T(q1)T(q2)T(q3)T(q4)⟩= −RδPqi4Yi=1Γ(1 −|qi|)Γ(|qi|)µ|qi|2(βµ)−2"(|q4| −1)−(√Rβµ)−2243Yr=1(|q4| −r)R(q21 + q22 + q23 −|q4| −1) −1R+ (√Rβµ)−457605Yr=1(|q4| −r)R23(q41 + q42 + q43) + 10(q21q22 + q21q23 + q22q23)−10|q4|(q21 + q22 + q23) −5(q21 + q22 + q23) + 10(q1q2 + q1q3 + q2q3) + 12|q4| + 7−10(q21 + q22 + q23 −|q4| −1) + 7R2+ . .

.#(10.13)The other kinematic configuration is q1, q2 > 0, q3, q4 < 0 with min(|qi|) = q1,max(|qi|) = q2. This leads to a much more complicated formula, so we just give theanswer out to genus one⟨T(q1)T(q2)T(q3)T(q4)⟩= −RδPqi4Yi=1Γ(1 −|qi|)Γ(|qi|)µ|qi|2 q2 −1(βµ)2"1 −(√Rβµ)−224×RS3(S −2q2 −6) + q22(S2 + q23 + q24) + 13S2q2 −8Sq22 −5q2(q23 + q24)+ 6S(S −2q2) + 6(q23 + q24) + 10q22 + S −2q2 −6−1R(S −2)(S −3)+ .

. .#(10.14)where we have defined S = 12P |qi|.Using the matrix model, we have directly calculated the correlation functions ofthe tachyon operators for the momentum modes,T(q) = 1βZd2σpˆgeiq(X+ ¯X)e(−2+|q|)φ ,(10.15)where X and ¯X are the holomorphic and anti-holomorphic parts of the X-field.

Instring theory we also have the winding operators˜T(˜q) = 1βZd2σpˆgei˜q(X−¯X)e(−2+|˜q|)φ ,(10.16)65

where ˜q is quantized as nR. Although these operators are hard to introduce in thematrix model directly, their correlators are obtained from those of the momentumoperators through the dual transformationX + ¯X →X −¯X ,R →1R ,β →βR.

(10.17)Carrying out the dual transformation (10.17) on eqns. (10.10)-(10.14), we findD˜T(˜q) ˜T(−˜q)E=RΓ(1 −|˜q|)Γ(|˜q|)2µ|˜q|"1|˜q|−(√Rβµ)−224(|˜q| −1) 1R(˜q2 −|˜q| −1) −R+ .

. .#,(10.18)etc.

In general, if⟨T(q1) . .

. T(qn)⟩R=∞= δ(Xqi)F(q1, .

. ., qn; µ, β) ,thenD˜T(˜q1) .

. .

˜T(˜qn)ER = Rn−1δP˜qi12β∂∂µsin( 12β∂∂µ)F(˜q1, . .

. , ˜qn; µ, βR) .

(10.19)This formula was obtained by carrying out the dual transformation on the similarformula (10.7) for the momentum states. Since we do not have a direct matrix modelcalculation of the winding operator correlations, we do not yet know the expressionsfor correlators where both the momentum and the winding operators are present.Is there any real time interpretation of the exact correlation functions we havecalculated?

In the R = ∞case we found that, after the continuation to Minkowskisignature, the external leg factors turn into pure phases, and the correlation functionsbecome scattering amplitudes of the X-quanta of the Das-Jevicki field theory. We canperform the same continuation, |qj| →−iEj, for the finite R correlation functions,expecting to obtain the same field theory at a finite temperature T = 1/(2πR).

Theexternal leg factors again turn into phases and can be absorbed in the definition of66

the vertex operators, and we find the scattering amplitudes at temperature T. Forinstance,S(E1, E2; E3) = −gstrδ(E1 + E2 −E3) E1E2E3"1+g2str24 (1 + iE3)(2 + iE3)1 −iE3 + E21 + E22 + (2πT)2+ . .

.#(10.20)Is the concept of finite-temperature S-matrix well-defined? In the usual theories theanswer is negative, because interaction with the heat bath stops a particle before itreaches the interaction region.

However, in the hamiltonian (7.19) the interaction isexponentially localized in a finite region of space near τ = 0. Thus, the interactionwith the heat bath is negligible at infinity, and the concept of S-matrix still appearsto be meaningful.

It would be interesting to compare the fermionic results like eq. (10.20) with bosonic finite temperature calculations with the hamiltonian (7.19).In conclusion, we should also mention the fascinating speculation due to Wittenthat the c = 1 matrix model describes physics in the background of a two-dimensionalblack hole.51 ⋆The black hole has finite temperature, and its Euclidean descriptionis, perhaps, related to the compact c = 1 model with a definite radius R. It wouldbe remarkable if the formulae reported in this section could be interpreted in termsof massless particle scattering offthe black hole and in terms of Hawking radiation.⋆For a detailed discussion, see H. Verlinde’s lectures in this volume.67

11. DISCRETIZED TARGET SPACE.In this section we will study another interesting model: the theory of surfacesembedded in a discretized real line with lattice spacing ǫ.

As shown in section 8,the lattice duality transformation relates it to the model of surfaces embedded in acircle of radius 1/ǫ. Although for genus > 0 this transformation generates 2h extravariables which violate the precise equivalence of the two theories, the basic physicaleffect – the Kosterlitz-Thouless phase transition – is of the same nature.In fact, the model with discretized target space is very interesting in its own right.Here we can test how string theory responds to introduction of a small lattice spacingin space-time.

It has been argued52 that in this respect string theory is very differentfrom any field theory: if the lattice spacing is smaller than some critical value, thenthe theory on a lattice is precisely the same as the theory in the continuum. This is tobe contrasted with any known field theory, where the continuum behavior may onlybe recovered as the lattice spacing is sent to zero.

The exactly soluble c = 1 stringtheory is an ideal ground for testing this remarkable stringy effect. Below we give theexact solution of the model with the discretized target space and confirm that, forǫ < ǫc, the lattice is smoothed out so that the theory is identical to the embeddingin R1.The matrix model representation of the partition function is now in terms of anintegral over a chain of M matrices with nearest neighbor couplings53Z(ǫ) =MYi=1ZDN 2Φi exp"−βXi 12ǫ Tr(Φi+1 −Φi)2 + ǫ Tr W(Φi)#.

(11.1)It is easy to check that the perturbative expansion of this integral gives the statisticalsum of the formlimM→∞ln Z(ǫ)M=Xhg2(h−1)0XΛκVV −1Yi=1XniY⟨ij⟩e−ǫ|ni−nj| ,(11.2)which is precisely what is needed to describe the embedding in a discretized real linewith lattice spacing ǫ. As in the matrix quantum mechanics (2.6), the integral (11.1)68

can be expressed in terms of the eigenvalues of the matrices Φi. The only modificationhere is that, instead of quantum mechanics of N identical non-interacting fermions,we now find quantum mechanics with a discrete time step ǫ. Thus,limM→∞ln Z(ǫ)M=NXi=1ln µi ,(11.3)where µi are the N largest eigenvalues of the transfer matrixµifi(x) =∞Z−∞dyK(x, y)fi(y),K(x, y) = β2πǫ1/2exp−β2(x −y)2ǫ+ ǫW(x) + W(y).

(11.4)Although it is hard to solve this problem exactly, as usual things simplify in the con-tinuum limit where we expect to find universal behavior controlled by the quadraticmaximum of W(x). Indeed, if we takeW(x) = −12x2 + O(x3)(11.5)and rescale x√β = z, we findK(z, w) =1√2πǫ exp−(z −w)22ǫ+ 14ǫz2 + w2 + O 1√β(z3 + w3).

(11.6)Thus, the terms beyond the quadratic order are suppressed by powers of β and are,therefore, irrelevant. The quadratic problem can be solved exactly through finding theequivalent quantum mechanical problem with Planck constant 1/β′ and hamiltonianH(ǫ) such thatK(x, y) = ⟨x| e−ǫβ′H(ǫ) |y⟩.

(11.7)Then, µi = exp(−ǫβ′ei) where ei are the N lowest eigenvalues of H(ǫ).For the69

quadratic transfer matrix,K(x, y) = β2πǫ1/2exp−β2(x −y)2ǫ−12ǫ(x2 + y2),(11.8)the hamiltonian is also quadratic H = p22 −ω2x22 . Comparing⟨x| e−ǫβ′H |y⟩= mωβ′2π sin ωǫ1/2exp−mωβ′2(x2 + y2) cot ωǫ −2xy sin−1 ωǫ(11.9)with eq.

(11.8), we findcos ǫω = 1 −12ǫ2 ,β′ = βsin(ωǫ)ωǫ. (11.10)Thus, we have reduced the problem of random surfaces embedded in discretized targetspace to quantum mechanics of N fermions moving in an inverted harmonic oscillatorpotential, i.e.

to random surfaces embedded in R1! Eq.

(11.10) dictates how the cur-vature of the potential and β′ depend on ǫ. Recalling that α′ = 1/ω2 and gst =1β′µ0,we find that the target space scale and the coupling constant of the equivalent con-tinuum string theory depend on ǫ. Thus, changing the lattice spacing in target spacesimply changes the parameters of the equivalent string theory embedded in continu-ous space.

This effect was originally demonstrated in higher-dimensional string theoryusing completely different techniques.52 It appears to be a general property of stringtheory.Intuitively, we do not expect the equivalence with the continuum string theoryto be there for arbitrarily large ǫ. Indeed, eq.

(11.10) shows that, if ǫ exceeds ǫc = 2then the solution for ω becomes complex, which is a sign of instability. Since largelattice spacing corresponds, in the dual language, to small radius, it is clear that theinstability is with respect to condensation of vortices.

Thus, for ǫ > ǫc, we expectthe vortices to eliminate the massless field on the world sheet corresponding to the70

embedding dimension, so that the pure gravity results. This is easy to check for verylarge ǫ where the sites of the matrix chain become decoupled,limM→∞ln Z(ǫ)M→ZDN 2Φe−βǫ Tr W (Φ),(11.11)and we obtain the one-matrix model well-known to describe the pure gravity.We will now give a continuum explanation of the Kosterlitz-Thouless transitionfrom the small ǫ phase, where the lattice is irrelevant, to the large ǫ phase, whereit has the dominant effect.

We will replace the discretized real line by a continuousvariable X with a periodic potential V (X) = Pn>0 an cos(2πnǫ X), which implementsthe effects of the lattice. The conformal weight of the operatorRd2σ cos(2πnǫ X) ish = π2n2α′ǫ2.

Thus, for small ǫ all the operators perturbing the action have h > 1and are irrelevant. This is the essential reason why the string does not feel a latticein target space with spacing < O(√α′).

The phase transition to lattice-dominatedphase takes place when the perturbation with n = 1 becomes relevant, i.e., forǫc = π√α′(11.12)We can compare this result with the position of the K-T transition in the matrixchain model, where we found ǫc = 2 and√α′(ǫc) = 2/π. The agreement of thesevalues with the continuum relation (11.12) strengthens our explanation of the phasetransition in the matrix model.

It is satisfying that we have found the matrix modelconfirmation of a general effect, smoothing out of a target space lattice, which appliesto string theories in any dimension.71

12. CONCLUSION.I hope that I have convinced the reader that the 2-dimensional string theory isa highly non-trivial toy model for string theory in higher dimensions.Its matrixmodel formulation as a sum over surfaces embedded in 1 dimension is an example ofa perfectly regularized generally covariant definition of the Polyakov path integral.It also turns out to be remarkably powerful, giving us the exact solution of a non-trivial string theory.

Many of its physical features, such as the R →1/R duality(and its breaking due to the vortices), smoothing out of the target space discreteness,presence of poles in the correlation functions, etc., carry over to string theories inhigher dimensions. Also, even though this string theory is two-dimensional, it hassome interesting remnants of transverse excitations, manifested in the isolated statesat integer momenta which generate the W1+∞algebra.

Thanks to the matrix models,we have acquired a wealth of exact information on the perturbative properties ofthis string theory. Unfortunately, we still do not have a detailed understanding orinterpretation of most of it in the conventional continuum formalism.

Hopefully, thisgap will be closed in the not too distant future.Acknowledgements: I am grateful to D. Gross, D. Lowe, M. Bershadsky and M. New-man, in collaboration with whom much of this material was developed. I am indebtedto A. Polyakov for many helpful discussions and comments.

I also thank D. Lowefor reading the manuscript, and R. Wilkinson for his help in making figures. I amgrateful to ICTP and the organizers of the Spring School for their hospitality.72

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