Two-dimensional quantum gravity

Chapter 1

Two-Dimensional QuantumGravity

Ivan Kostov

Institut de Physique Theorique CEA Saclay,

F-91191 Gif-sur-Yvette Cedex, France

Abstract

This chapter contains a short review the correspondence between large N matrix

models and critical phenomena on lattices with fluctuating geometry.

1.1 Introduction

One of the applications of the large N matrix models is that they can be used as

machines for generating planar graphs. The problem of counting planar graphs

was related to the saddle point solution of matrix integrals by Brezin, Parisi,

Itzykson and Zuber [BIPZ78]. The concept of the 1/N expansion for matrix

fields and its topological meaning was introduced by ’t Hooft in [tH74].

The ensembles of planar graphs exhibit universal critical behavior. By tun-

ing the coupling constants one can achieve a regime which is dominated by

planar graphs of diverging size. Seen from a distance, such planar graphs re-

semble surfaces with fluctuating metric. Therefore the sum over planar graphs

can be considered as a discretization of the path integral of 2d quantum gravity

[Pol81]. The interpretation of the ensembles of planar graphs as models of 2D

lattice gravity was suggested in [Dav85b, Kaz85, ADF85] and further studied

in [Dav85a, KKM85, BKKM86].

The 2d lattice gravity with matter fields was initially constructed as a

gaussian discretization of the path integral for the non-critical Polyakov string

[Pol81] with D-dimensional target space. However such a discretization leads

2 CHAPTER 1. AKEMANN ET AL.

to an exactly solvable model only in the case of D = 0 and D = −2 [KKM85,

BKKM86, KM87]. Another way to introduce a non-trivial target space is to

consider solvable spin models on planar graphs. The Ising model on a dy-

namical lattice was solved, after being reformulated as a two-matrix model, by

Kazakov and Boulatov [Kaz86, BK87]. This solution was used as a guideline

by Knizhnik, Polyakov and Zamolodchikov, who obtained the exact spectrum

of anomalous dimensions in the continuum theory of 2d gravity [KPZ88].

Soon after the the Ising model, all basic statistical models were solved on a

dynamical lattice: the O(n) model [Kos89a], the Q-state Potts model [Kaz88],

the SOS and ADE height models [Kos89b][Kos92b], as well as the 6-vertex

model [Kos00]. Particular cases of the Potts and O(n) models are the tree/bond

percolation [Kaz88] and self-avoiding polymer networks [DK88] [DK90]. In

general, the theories of discrete gravity admit an infinite number of multi-

critical regimes, which are first observed in the one-matrix model [Kaz89]. The

multi-critical point in the ADS models and the O(n) model were identified in

[Kos92b, KS92].

Practically all solvable 2d statistical models whose critical behaviour is de-

scribed by a c 6 1 CFT can be transfered from the plane to a dynamical lattice

and solved exactly. There can be several different discretizations of the same

continuous theory. For example, all rational theories of 2D gravity can be ob-

tained either by the ADE matric models, or as the critical points of the 2-matrix

model [DKK93].

For statistical systems coupled to gravity it is possible to solve analytically

problems whose exact solution is inaccessible on flat lattice. On the other hand,

the solution of a spin model on a dynamical lattice allows to reconstruct the

phase diagram and the critical exponents of the same spin model on the flat

lattice. Therefore the solvable models of 2d gravity represent a valuable tool

to explore the critical phenomena in two dimensions. This is especially true in

presence of boundaries, where the solution of the matrix model can give insight

about the interplay between the bulk and the boundary flows.

We review the solvable models of 2d lattice quantum gravity and the way

they are related to matrix models. We put the accent on the critical behavior

and the solution in the scaling limit near the critical points. We start with a

short description of the continuum world sheet theory, the Liouville gravity, and

derive the KPS scaling relation. Then we present in detail the simplest model

of 2d gravity and the corresponding matrix model, followed by less or more

sketchy presentation of the vertex/height integrable models on planar graphs

and their mapping to matrix models.

1.2. LIOUVILLE GRAVITY AND KPZ SCALING RELATION 3

1.2 Liouville gravity and KPZ scaling relation

The continuum approach to critical phenomena in presence of gravity was de-

velopped by Knizhnik, Polyakov and Zamolodchikov [KPZ88]. The confor-

mal weights from the Kac spectrum are converted by ‘gravitational dressing’

into new ones through the remarkable KPZ formula [KPZ88]. Subsequently,

a more intuitive alternative derivation of the KPZ formula was proposed in

[Dav88, DK89]. The conformal theory with quantum gravity has larger sym-

metry and hence simpler phenomenology: we are allowed to ask only questions

invariant under diffeomorphisms. The anomalous dimensions in such a theory

are perfectly meaningful but not the operator product expansion; only fields

integrated over all points make sense. Therefore, instead of the correlation

functions of the local operators we have to consider the corresponding suscepti-

bilites. The way they depend on the temperature and the cosmological constant

allows to determine the scaling dimensions.

1.2.1 Path integral over metrics

Let Zmatter[gab] be the partition function of the matter field on a two-dimensional

worldsheet M with Riemann metric gab. For the moment assume that the world

sheet is homeomorphic to a sphere. The partition function on the sphere, which

we denote by F(µ), is symbolically written as the functional integral

F(µ) =

∫

[Dgab] e−µA Z(sphere)

matter [gab], (1.2.1)

where the parameter µ, called cosmological constant, is coupled to the area of

the worldsheet,

A =

∫

Md2σ

√

det gab. (1.2.2)

For a worldsheet M with a boundary ∂M one can introduce a second pa-

rameter, the boundary cosmological constant µB coupled to the boundary length

ℓ =

∫

∂M

√

gab dσadσb . (1.2.3)

The partition function on the disk, which we denote by U(µ, µB), is defined as

a functional integral over all Riemann metrics on the disk,

U(µ, µB) =

∫

[Dgab] e−µA−µBℓ Z(disk)

matter[gab]. (1.2.4)

Here we assumed that the matter field satisfy certain boundary condition. We

will also consider more complicated situations where the boundary is split into


several segments with different boundary cosmological constants and different

boundary conditions for the matter field.

By the invariance of the functional measure under diffeomorphisms of the

world sheet one can impose the conformal gauge

gab = e2φ gab, (1.2.5)

where gab is some background metric, so that the fluctuations of the metric

are described by the local scale factor φ. At the critical points, where the

matter QFT is conformal invariant, the scale factor φ couples to the matter

field through a universal classical action induced by the conformal anomaly

cmatter. In addition, the conformal gauge necessarily introduces the Faddeev-

Popov reparametrization ghosts with central charge cghosts = −26 [Pol81] . The

effective action of the field φ is called Liouville action because the correspond-

ing classical equation of motion is the equation for the metrics with constant

curvature, the Liouville equation.

1.2.2 Liouville gravity

Liouville gravity is a term for the two-dimensional quantum gravity whose ac-

tion is induced by a conformal invariant matter field. The effective action of the

Liouville gravity consists of three pieces, associated with the matter, Liouville

and ghost fields. For our discussion we will only need the explicit form of the

Liouville piece Aµ,µB[φ]. This is the action of a gaussian field with linear cou-

pling to the curvature and exponential terms coupled to the bulk and boundary

cosmological constants:

Aµ,µB[φ] =

1

4π

∫

M

(

g (∇φ)2 + QφR)

+1

2π

∫

∂M

dσQφK

+µA + µBℓ. (1.2.6)

The area and the length are defined by (1.2.2) and (1.2.3), R the background

curvature in the bulk and by K the geodesic background curvature at the

boundary. The coupling to the curvature is through the imaginary background

charge iQ. We will assume that the coupling constant g of the Liouville field is

larger than one: g ≥ 1.

The Liouville coupling g and the background charge Q are determined by

the invariance with respect to the choice of the background metric gab. The

background charge of the Liouville field must be tuned so that the total con-

formal anomaly vanishes:

ctot ≡ cφ + cmatter + cghosts

= (1 + 6Q2/g) + cmatter − 26 = 0. (1.2.7)

1.2. LIOUVILLE GRAVITY AND KPZ SCALING RELATION 5

The ballance of the central charge gives

cmatter = 1 − 6(g − 1)2/g. (1.2.8)

The observables in Liouville gravity are integrated local densities

O ∼∫

MΦh e2αφ , (1.2.9)

where Φh represents a scalar (h = h) matter field1 and Vα is a Liouville ‘dressing’

field, which completes the conformal weights of the matter field to (1, 1). The

balance of the conformal dimensions, h + hLiouvα = 1, gives a quadratic relation

between the Liouville dressing charge and the conformal weight of the matter

field:

h +α(Q − α)

g= 1 . (1.2.10)

In particular, the Liouville interaction e2φ dresses the identity operator. The

balance of scaling dimensions (1.2.10) determines, for α = 1 and h = 0, the

value of the background charge Q:

Q = g + 1. (1.2.11)

Similarly, the boundary matter fields ΦBh with boundary dimensions hB are

dressed by boundary Liouville exponential fields,

OB ∼∫

∂MΦB

h eαBφ , (1.2.12)

where the boundary Liouville exponent αB is related to h as in (1.2.10).

The quadratic relation (1.2.10) has two solutions, α and α, such that α ≤Q/2 and α ≥ Q/2. The Liouville dressing fields corresponding to the two

solutions are related by Liouville reflection α → α = Q − α. Generically the

scaling limit of the operators on the lattice is described by smaller of the two

solutions of the quadratic constraint (1.2.10). The corresponding field Vα has

a good quasiclassical limit. Such fields are said to satisfy the Seiberg bound

α 6 Q/2, see the reviews [DFGZJ95, GM93].2

The simplest correlation functions in Liouville gravity (up to 3-point func-

tions of the bulk fields and 2-point functions of the boundary fields) factorize

into matter and Liouville pieces. This factorization does not hold away from

the critical points.

1The fields cannot have spin because any local rotation can be ‘unwound’ by a coordinatetransformation.

2Originally it was believed that the second solution is unphysical, until it was realized thatit describes specially tuned measure over surfaces [Kle95].


1.2.3 Correlation functions and KPZ scaling relation

The scaling of the correlation function with the bulk cosmological constant

µ follows from the covariance of the Liouville action (1.2.6) with respect to

translations of the Liouville field:

Aµ,µB[φ − 1

2 ln µ] = −12Qχ ln µ + A1, µB/

√µ[φ ] (1.2.13)

where χ = 2− 2#(handles) − #(boundaries) is the Euler characteristics of the

worldsheet. Applying this to the correlation function of the bulk operators Oj

and the boundary operators OBk , we obtain the scaling relation

⟨

∏

j

Oj

∏

k

OBk

⟩

µ,µB

= µ12χQ−

P

j αj− 12

P

k αBk f(µB/

√µ), (1.2.14)

where f is some scaling function. We see that for any choice of the local fields

the unnormalized correlation function dependence on the global curvature of the

world sheet through an universal exponent Q = 1+g = 2−γstr, where γstr = 1−g

is called string susceptibility exponent. The scaling of the bulk and boundary

local fields are determined by their gravitational anomalous dimensions ∆i and

∆Bj , related to the exponents αi and αB

j by

∆i = 1 − αi, ∆Bi = 1 − αB

i . (1.2.15)

Once the exponent γstr and the gravitational dimensions ∆i are known,

the central charge cmatter and conformal weights hi of the matter fields can be

determined from (1.2.7) and (1.2.10):

h =∆(∆ − γstr)

1 − γstr, cmatter = 1 − 6

γ2str

1 − γstr(γstr = 1 − g), (1.2.16)

and a similar relation for the boundary fields. This relation is known as KPZ

correspondence between flat and gravitational dimensions of the local fields

[KPZ88, Dav88, DK89].

The ‘physical’ solution of the quadratic equation (1.2.10) can be parametrized

by a pair of real numbers r and s, not necessarily integers,3

hrs(g) =(rg − s)2 − (g − 1)2

4g, αrs(g) =

g + 1 − |rg − s|2

. (1.2.17)

The corresponding gravitational anomalous dimension ∆rs = 1 − αrs are

∆rs(g) =|rg − s| − (g − 1)

2. (1.2.18)

3When r and s are integers, hrs are the conformal weights of the the degenerate fields ofthe matter CFT.

1.3. DISCRETIZATION OF THE PATH INTEGRAL OVER METRICS 7

We remind that our conventions are such that g > 1. The solution (1.2.17)

apply also for the boundary exponents, after replacing h → hB ,∆ → ∆B , α →β. It generalizes in obvious way to the case when each boundary segment is

characterized by a different boundary cosmological constant.

The string susceptibility exponent γstr is a measurable quantity in the dis-

crete models of 2D gravity and is determined by the the scaling behavior of the

susceptibility on the sphere u ≡ ∂2µF ∼ µ−γstr at the critical points. At different

critical points of the same model one can have different exponents γstr. Once

γstr is known, the matter central charge and the conformal weights of the local

fields can be determined by the KPZ relation (1.2.16), with the gravitational

demensions ∆i and ∆Bi defined through the scaling relation

⟨

∏

j

Oj

∏

k

OBk

⟩

µ,µB

= µ12χ(2−γstr)−

P

j(1−∆i)− 12

P

k(1−∆Bk

) f(µB/√

µ). (1.2.19)

1.3 Discretization of the path integral over metrics

The statement that the integral over Riemann metrics can be discretized by the

sum over a sufficiently large ensemble of planar graphs has no rigorous proof,

but it has passed a multitude of quantitative tests. In order to built models of

discrete 2D gravity it is sufficient to consider the ensemble of trivalent planar

graphs which are dual to triangulations, as the one shown in Fig. 1.1. To avoid

ambiguity the propagators are represented by double lines. The faces of such a

‘fat’ graph are bounded by the polygons formed by the single lines.

1/β

i ij

j

k

k

ij

ij

β

Figure 1.1: Left: A trivalent planar graph with the topology of a disk and its dultriangulation. Right: Feynman rules for the matrix model for pure gravity.

The local curvature of the triangulation is concentrated at the vertices r

of the triangulation. The curvature Rr (or the boundary curvature Kr if the

vertex is on the boundary) is expressed through the coordination number cr,

equal to the number of triangles meeting at the point r,

Rr = π6 − cr

3, Kr = π

3 − cr

3. (1.3.1)


The total curvature is equal to the Euler number χ(Γ):

1

2π

∑

r∈bulk

Rr +1

π

∑

r∈boundary

Kr = χ(Γ) . (1.3.2)

For each fat graph Γ we define the area and the boundary length as

|Γ| def= # (faces of Γ) , |∂Γ| def

= # (external lines of Γ) . (1.3.3)

Let us denote by {Sphere} and {Disk} the ensembles of fat graphs dual

to triangulations respectively of the sphere and of the disk. Assume that the

matter field can be defined on any triangulation Γ and denote its partition

function by Zmatter[Γ]. Then the path integrals for the partition function of the

sphere (1.2.1) and that of the disk (1.2.4) are discretized as follows:

F(µ) =∑

Γ∈{Sphere}

1

k(Γ)µ−|Γ| Z

(sphere)

matter [Γ]. (1.3.4)

U(µ, µB) =∑

Γ∈{Disk}

1

|∂Γ| µ−|Γ| µ−|∂Γ|B Z

(disk)

matter[Γ]. (1.3.5)

Here µ and µB are the the lattice bulk and boundary cosmological constants

and k(Γ) is the volume of the symmetry group of the planar graph Γ.

1.4 Pure lattice gravity and the one-matrix model

1.4.1 The matrix model as a generator of planar graphs

The solution of pure discrete gravity thus boils down to the problem of counting

planar graphs, reformulated in terms of a matrix integral in [BIPZ78]. The

matrix integral

Z(N,β) =

∫

dM eβ

“

−12TrM2+ 1

3TrM3

”

, (1.4.6)

where dM is the flat measure in the linear space of N ×N hermitian matrices,

can be considered as the partition function of a zero-dimensional QFT with

Feynman rules given in Fig. 1.1 (right). The integral is divergent but makes

sense as perturbative expansion around the gaussian measure. The logarithm of

the partition function (1.4.6), the ‘vacuum energy’ of the matrix field, is equal

to the sum of all connected diagrams.

The weight of a vacuum graph Γ is written, using the Euler relation, as

Weight(Γ) =1

k(Γ)β#vertices−#linesN#faces =

1

k(Γ)βχ(Γ) (N/β)|Γ|. (1.4.7)

1.4. PURE LATTICE GRAVITY AND THE ONE-MATRIX MODEL 9

Therefore we can identify the ratio µ = β/N with the lattice bulk cosmological

constant. In the large N limit with the ratio β/N fixed,

N → ∞, β/N = µ , (1.4.8)

only Feynman graphs dual to triangulations of the sphere (χ = 2) contribute

and their weight will be β2µ−|Γ|. As a result the sphere partition function

(1.3.4) of discrete 2D pure gravity is equal to the vacuum energy of the matrix

model in the the large N , or planar, limit

F(µ) = limN→∞

1

β2logZ(N,β) , µ = β/N . (1.4.9)

It is convenient to consider the disk partition function (1.3.5) for any complex

value of the boundary cosmological constant, which we denote by the letter z.

The disk partition function for µB = z is evaluated, up to a trivial logarithmic

term, by

U(µ, z) = − 1

β〈Tr log(z − M) 〉 . (1.4.10)

The derivative W = −∂zU , which is the partition function on the disk with a

marked point on the boundary, or the boundary one-point function on the disk,

is the expectation value of the resolvent of the random matrix,

W (z) =1

β〈 trW(z) 〉 , W(z)

def=

1

z − M. (1.4.11)

1.4.2 Solution at Large N

The resolvent (1.4.11) is the only solution of a non-linear Ward identity, known

also as loop equation, which follows from the translational invariance of the

matrix integration measure. For any matrix observable F we have the identity

1

β〈∂MF〉 = 〈tr

[

V ′(M)F]

〉 , ∂M Fdef= ∂Fij/∂Xij . (1.4.12)

Written for F = W(z), the equation states

W 2(z) =1

β

⟨

tr[

V ′(M)W(z)]⟩

. (1.4.13)

Now, using the factorization property of the traces, 〈 trA trB 〉 = 〈 trA 〉〈 trB 〉,which is true up to 1/N2 corrections, we obtain that the resolvent satisfies a

quadratic equation

W 2(z) − V ′(z)W (z) = P (z), (1.4.14)


where P is a polynomial. Therefore the function

J(z)def= W (z) − 1

2V ′(z), (1.4.15)

should be such that that J2(z) is a polynomial. The polynomial is determined

by the asymptotics at infinity,

J(z) = −12V ′(z) + (µz)−1 + O(z−2). (1.4.16)

The solution is

J(z) = (z − 1 + 12(a + b))

√

(z − a)(z − b), (1.4.17)

where the positions of the branch points are determined as functions of µ by

(b − a)2 = 8S(1 − S), a + b = 2S, 12S(1 − S)(1 − 2S) =

1

µ. (1.4.18)

It is also easy to determine the derivative ∂µJ(z), which has the same analytic

properties as J(z), but simpler asymptotics at infinity,

∂µJ(z) = ∂µW (z) = −µ−2z−1 + ∂µW1 z−2 + O(z−3), (1.4.19)

where W1 = 〈 trM 〉. The solution is obviously

µ2∂µJ(z) = − 1√

(z − a)(z − b). (1.4.20)

One can check that the compatibility of (1.4.20) and (1.4.17) gives (1.4.18).

Comparing (1.4.20) and (1.4.19) we get

∂µ 〈 trM 〉 = S. (1.4.21)

1.4.3 The scaling limit

We are interested in the universal critical properties of the ensemble of triangu-

lations in the limit when the size of the typical triangulation diverges. This limit

is achieved by tuning the bulk and boundary cosmological near their critical

values,

µ = µc + ǫ2 µ, z = µcB + ǫ x , (1.4.22)

where ǫ is a small cutoff parameter with dimension of length and x is propor-

tional to the renormalized boundary cosmological constant µB . The renormal-

ized bulk and boundary cosmological constants µ and x are coupled respectively

to the renormalized area A and length ℓ of the triangulated surface, defined as

A = ǫ2 |Γ|, ℓ = ǫ |∂Γ|. (1.4.23)

1.4. PURE LATTICE GRAVITY AND THE ONE-MATRIX MODEL 11

In the continuum limit ǫ → 0 the universal information is contained in the

singular parts of the observables, which scale as fractional powers of ǫ. This

is why the continuum limit is called also scaling limit. In particular, for the

partition functions (1.2.1) and (1.2.4) we have

F(µ) = regular part + ǫ2(2−γstr) F (µ),

U(µ, z) = regular part + ǫ2−γstr U(µ, x),

W (µ, z) = regular part + ǫ1−γstr w(µ, x). (1.4.24)

The powers of ǫ follow from the general scaling formula scaling (1.2.19) with

χ = 2 for the sphere and with χ = 1 for the disk. The partition functions

for fixed length and/or area are the inverse Laplace images with respect to x

and/or µ. For example

w(µ, x) = −∂xU(µ, x) =

∫ ∞

0dℓ e−ℓx w(µ, ℓ). (1.4.25)

Now let us take the continuum limit of the solution of the matrix model.

First we have to determine the critical values of the lattice couplings. From

the explicit form of the disk partition function, eqs. (1.4.17)-(1.4.18), it is clear

that the latter depends on µ only through the parameter S = (a + b)/2. The

singularities in 2D gravity are typically of third order and one can show that

S is proportional to the second derivative of the free energy of the sphere.

This means that the derivative ∂µS diverges at µ = µc. Solving the equation

dS/dµ = 0 gives

µc = 12√

3 . (1.4.26)

Furthermore the critical boundary cosmological constant coincides with the

right endpoint b of the eigenvalue distribution at µ = µc,

µcB = b(µc) = (3 +

√3)/6. (1.4.27)

The function J(z) is equal to W (z) with the regular part subtracted. In the

scaling limit (1.4.22)

J(µ, z) ∼ ǫ3/2 w(µ, x) (1.4.28)

w(x) = (x + M)3/2 − 32M (x + M)1/2, M =

√

2µ . (1.4.29)

We dropped the overall numerical factors which does not have universal mean-

ing. With this normalization of w and µ, the derivative ∂µW is given by

∂µw(x) = −3

4

1√x + M

. (1.4.30)


The scaling exponent 3/2 in (1.4.28) corresponds, according to (1.2.16) and

(1.2.19), to γstr = 1 − 3/2 = −1/2 and cmatter = 0. Finally, it follows from

(1.4.21) that M ∼ ∂2µF and the free energy on the sphere scales as F ∼ ǫ5 µ5/2.

This is not the only critical point for the one-matrix model. With a poly-

nomial potential of highest degree p + 2 one can tune p couplings to obtain a

scaling behavior w(z) ∼ xp+ 12 . On the other hand, the derivative ∂µw ∼ x−1/2

does not depend on p, which means that M ∼ ∂2µF and x scale as µ1/(p+1).

These multicritical points were discovered by Kazakov [Kaz89]. The tricriti-

cal point was first correctly identified by Staudacher [Sta90] as the hard dimer

problem on planar graphs, which is in the universality class of the Yang-Lee

CFT with cmatter = −22/5. The Liouville gravity with such matter CFT is

studied in [Zam07]. In general, the p-critical point corresponds to a negative

matter central charge given by (1.2.16) with g = p + 12 . In these non-unitary

models of 2d gravity the constant µ is coupled not to the identity operator, but

to the field with the most negative dimension.

1.5 The Ising model

This simplest statistical model is characterized by a fluctuating spin variable

σr = ±1 associated with the vertices of the planar graph. The nearest neighbor

spin-spin interaction is introduced by the energy

E[{σr}] =∑

<rr′>

K σrσr′ +∑

r∈Γ

Hσr (1.5.1)

and the partition function on the graph Γ is given by

ZIsing[Γ] =∑

{σr}e−E[{σr}]. (1.5.2)

The ferromagnetic vacuum is attained at K → +∞.

The Ising model coupled to 2g gravity is discretized as the on planar graphs

can be constructed by painting the assigning to all vertices a color which can

take two values. The corresponding matrix model involves two random matri-

ces, Mσ , σ = ±, and its partition function is [Kaz86]

ZIsing(N,β;K,H) =

∫

dM+dM− e−βTrV (M+,M−), (1.5.3)

V (M+,M−) =∑

σ=±

(

−12M

2σ + 1

3eσHM

3σ

)

+ e−2KM+M− . (1.5.4)

The model can be solved exactly in the large N limit and the solution

exhibits the following critical behavior. For generic K the sum over planar

graphs diverges at some critical value of µ = β/M . Again we can define the

1.5. THE ISING MODEL 13

scaling limit as in (1.4.22). Along the critical curve µ = µc(K,H) the partition

function is dominated by infinite planar graphs and we have the ‘pure gravity’

critical behavior F ∼ µ5/2. For H = 0 and there is a critical value of the Ising

coupling K = Kc for which the critical behavior of the free energy changes to

F ∼ µ7/3. At this point the correlations of the Ising spins on the infinite planar

graph become long range and are described by a matter CFT with cmatter = 1/2

(γstr = −1/3).

Boulatov and Kazakov [BK87] found that in the vicinity of the critical point

µ = µc,K = Kc,H = 0, parametrized by

µ − µc ∼ ǫ2µ, K − K0 ∼ −ǫ2/3t, H ∼ ǫ5/3h, (1.5.5)

the susceptibility u = ∂2µF satisfies the following simple algebraic equation:

µ = u3 + 32t u2 + 3

2

h2

(u + t)2. (1.5.6)

The coefficients in the equation correspond to some normalization of the scaling

couplings.

At the critical point t = h = 0 the susceptibility is u = µ1/3. The world

sheet theory at the critical point is that of Liouville gravity with the Ising

CFT (Majorana fermions) as a matter field. The primary fields of this CFT

are the energy density ε with dimension hε = 1/2 and the spin field σ with

dimension hσ = 1/16. For the corresponding gravitational dimensions we have

1 − ∆ε = 1/3 and 1 − ∆σ = 5/6, which explains the scaling (1.5.5). The

equation of state (1.5.6) describes a perturbation of the Liouville gravity with

Ising matter CFT by the action

δA(t, h) = t ε e2φ/3 + hσ e5φ/3. (1.5.7)

(The normalizations of the couplings are of course different.)

Apart from the Ising critical point there is a second non-trivial critical point

at purely imaginary value of the magnetic field: the gravitational Yang-Lee edge

singularity [Sta90, CGM90, ZZ06]. In order to localize the YL singularity let us

analyse the the equation of state (1.5.6). First we observe that assuming that

t > 0 and rescaling

u =u

t, h =

h

t5/2, µ =

µ

t3(1.5.8)

we can bring (1.5.6) to the scaling form

µ = u3 + 32 u2 + 3

2

h2

(u + 1)2. (1.5.9)


There is a line of critical points where the susceptibility u develops a square

root singularity, which is that of pure gravity (γstr = −1/2). The equation of

the critical line is obtained by differentiating the r.h.s. of (1.5.9) in u:

u(u + 1)4 = h2. (1.5.10)

Expanding (1.5.9) near the solution u = uc(h),

µ = µc + 32(1 + 5uc)(u − uc)

2 + . . . , (1.5.11)

we see that a tri-critical point occurs when uc = −1/5, which corresponds to

purely imaginary magnetic field hLY = ±i 42/55/2 and µLY = −7/50. This is

the Yang-Lee critical point for the gravitational Ising model. Near this point

the equation of state has the form

δµ = δu3 ± δh δu (1.5.12)

where δµ ∼ µ − µLY

, δu ∼ u − uLY

and δh ∼ i(h − hLY

).

The free energy per unit volume on the critical line (1.5.11), where the size

of the typical planar graph diverges, is equal to the critical value of the cos-

mological constant µc(h). This is a meromorphic function of h whose Riemann

surface consists of 5 sheets [Zam05]. The high temperature expansion goes in

the even powers of h = h/t5/2 and is convergent for |h| < |hLY |. On the ‘high-

temperature’ sheet it has two simple branch points on the imaginary axis at

h = ±hLY

. The two YL cuts connect the high-temperature‘sheet to a pair of

other sheets. These two sheets are in turn connected to another pair of sheets

by two Langer cuts which go from h = 0 to h = ±∞ and describe the low

temperature (t < 0) instability of the Ising model [FZ01, ZZ06].

1.6 The O(n) model (−2 ≤ n ≤ 2)

The local fluctuating variable in the O(n) loop model [Nie82] is an n-component

vector ~S(r) with unit norm, associated with the vertices r ∈ Γ. The partition

function of the O(n) model on the graph Γ can be represented a sum over

self-avoiding, mutually-avoiding loops as the one shown in Fig. 1.3 (left), each

counted with a factor of n:

ZO(n)

(T ; Γ) =∑

loops on Γ

T−length n#loops. (1.6.1)

The temperature coupling T controls the length of the loops. The advantage of

the loop gas representation (1.6.1) is that it makes sense also for non-integer n.

The model has a continuous transition at −2 ≤ n ≤ 2. A standard parametriza-

tion of n in this interval is

n = 2cos(πθ), 0 < θ < 1. (1.6.2)

1.6. THE O(N) MODEL (−2 ≤ N ≤ 2) 15

The O(n) matrix model [Kos89a] generates planar graphs covered by loops

in the same way as the one-matrix model generates empty planar graphs. The

fluctuating variables are the hermitian matrix M and an O(n) vector ~Y, whose

components Y1, . . . ,Yn are hermitian matrices. All matrix variables are of size

N × N . The partition function is given by the integral

ZO(n)

N (T ) ∼∫

dM dnY e−β tr( 1

2M

2+ T2

~Y2− 13M

3−M~Y2). (1.6.3)

This integral can be considered as the partition function of a zero-dimensional

QFT with Feynman rules given in Fig. 1.3 (right).

aa

i ij

j

k

k

i ij

j

k

k

ij

ij

i i

j

T1/β

j

β β1/β

Figure 1.2: Left: A loop configuration on a disk with ordinary boundary condition.Right: Feynman rules for the O(n) matrix model.

The basic observable in the matrix model is the resolvent (1.4.11), evaluated

in the ensemble (1.6.3), which is the derivative of the disk partition function

with ordinary boundary condition for the O(n) spins. In terms of loop gas,

the ordinary boundary condition means that the loops in the bulk avoid the

boundary as they avoid the other loops and themselves.

The matrix integral measure becomes singular at M = T/2. We perform a

linear change of the variables M = T(

12 + X

)

which sends this singular point

to X = 0. After a suitable rescaling of ~Y and β, the matrix model partition

function takes the canonical form

ZO(n)

N (T ) ∼∫

dX dnYeβtr[−V (X)+X~Y2] , (1.6.4)

where V (x) is a cubic potential V (x) = −13T (x+ 1

2 )3 + 12(x+ 1

2 )2. Accordingly,

we redefine the spectral parameter by

z = T(

x + 12

)

. (1.6.5)

Now the one-point function with ordinary boundary condition is

W (x) =1

β〈trW〉 , W

def=

1

x − X. (1.6.6)

After integration with respect to the Y-matrices, the partition function takes

the form of a Coulomb gas

ZO(n)

N (T ) ∼∫ N

∏

j=1

dxi e−V (xi)

∏

i<j(xi − xj)2

∏

i,j(xi + xj)n/2. (1.6.7)


1.6.1 Loop equation

As in the case of the one-matrix model, the resolvent can be determined by

Ward identities which follow from the translational invariance of the measure

in (1.6.3). Introduce the singular part J(x) of the resolvent by

W (x) = Wreg(x) + J(x), Wreg(x) =2V ′(x) − nV ′(−x)

4 − n2. (1.6.8)

The function J(x) satisfies the quadratic functional identity [Kos92b]

J2(x) + J2(−x) + n J(x)J(−x) = A + Bx2 + Cx4 . (1.6.9)

The coefficients A,B,C as functions of T , µ and W1 = 〈trX〉 can be evaluated

by substituting the large-x asymptotics

J(x) = −Wreg(x) +µ−2

x+

〈trX〉βx2

+ O(x−3) (1.6.10)

in (1.6.9). The solution of the loop equation (1.6.9) with the asymptotics

(1.6.10) is given by a meromorphic function with a single cut [a, b] on the first

sheet, with a < b < 0. This equation can be solved by an elliptic parametriza-

tion and the solution is expressed in terms of Jacobi theta functions [EK95]

[EK96] (see also [Kos06]).

The functional equation (1.6.9) becomes singular when A = B = 0. The

condition A = 0 determines the critical value µc of the cosmological constant.

The condition B = 0 determines the critical value Tc of the temperature cou-

pling, for which the length of the loops diverges:

Tc = 1 +√

2−n6+n . (1.6.11)

1.6.2 The scaling limit

We introduce a small cutoff ǫ and and define the renormalized coupling con-

stants as

µ − µc ∼ ǫ2µ , T − Tc ∼ ǫ2θt . (1.6.12)

We also rescale the boundary cosmological constant and the resolvent as

x → ǫx, J(x) → ǫ1+θw(x). (1.6.13)

The scaling resolvent w(x) has a cut [−∞,−M ], where M is a function of µ

and t. We can resolve the branch point at x = −M by the hyperbolic map

x = M cosh τ . (1.6.14)

1.6. THE O(N) MODEL (−2 ≤ N ≤ 2) 17

Then (1.6.9) becomes a quadratic functional equation for the entire function

w(τ) ≡ w[x(τ)]:

w2(τ + iπ) + w2(τ) + n w(τ + iπ)w(τ) = A1 + B1t M2 cosh2 τ . (1.6.15)

Here used the fact that near the critical temperature B ∼ T − Tc and the term

Cx4 vanishes. The unique solution, for some normalization of t, is

w(τ) = M1+θ cosh(1 + θ)τ + tM1−θ cosh(1 − θ)τ . (1.6.16)

The function M = M(µ, t) can be evaluated using the fact that the derivative

∂µw(x) depends on µ and t only through M . This is so if M satisfies the

transcendental equation

µ = (1 + θ)M2 − tM2−2θ. (1.6.17)

To summarize, the derivatives of the disk partition function in the scaling

limit, U(x, µ, t), are given in the following parametric form:

−∂xU |µ = M1+θcosh((1 + θτ) + t M1−θcosh(1 − θ)τ ,

−∂µU |x ∼ Mθ cosh θτ,

x = M cosh τ, (1.6.18)

with the function M(µ, t) determined from the transcendental equation (1.6.17).

The function M(t, µ) plays an important role in the solution. Its physical

meaning can be revealed by taking the limit x → ∞ of the bulk one-point

function −∂µU(x). Since x is coupled to the length of the boundary, in the

limit of large x the boundary shrinks and the result is the partition function

of the O(n) field on a sphere with two punctures, the susceptibility u(µ, t).

Expanding at x → ∞ we find

−∂µU ∼ xθ − M2θ x−θ + lower powers of x (1.6.19)

(the numerical coefficients are omitted). We conclude that the string suscepti-

bility is given, up to a normalization, by

u = M2θ. (1.6.20)

The normalization of u can be absorbed in the definition of the string coupling

constant gs ∼ 1/β. Thus the transcendental equation (1.6.17) for M gives the

equation of state of the loop gas on the sphere,

(1 + θ)u1θ − t u

1−θθ = µ . (1.6.21)


The equation of state (1.6.21) has three singular points at which the three-

point function of the identity operator ∂µu diverges. The three points corre-

spond to the critical phases of the loop gas on the sphere.

1) At the critical point t = 0 the susceptibility scales as u ∼ µθ. This is

the dilute phase of the loop gas, in which the loops are critical, but occupy an

insignificant part of the lattice volume. By the KPZ relation (1.2.16), the O(n)

field is described by a CFT with central charge (1.2.8) with g = 1 + θ.

2) The dense phase is reached when t/xθ → −∞. In the dense phase the

loops remain critical but occupy almost all the lattice and the susceptibility

has different scaling, u ∼ µ1−θ

θ . The O(n) field in the dense phase is described

by a CFT with a lower central charge (1.2.8) with g = 1 − θ. Considered on

the interval −∞ < t < 0, the equation of state (1.6.21) describes the massless

thermal flow relating the dilute and the dense phases [Kos06].

3) At the third critical point ∂µM becomes singular but M itself remains

finite. Around this critical point µ − µc ∼ (M − Mc)2 + · · · , hence the scaling

of the susceptibility is that of pure gravity, u ∼ (µ − µc)1/2.

1.6.3 Other boundary conditions

Besides the ordinary boundary condition, there is a continuum of integrable

boundary conditions, which are generated by anisotropic boundary term [DJS09]

The loop gas expansion on the disk with such boundary condition can be for-

mulated in terms of loops of two different colors with fugacities n(1) and n(2) ,

such that n(1) + n(2) = n. The bulk is color blind, while the boundary interacts

differently with the loops of different color. A loop of color (α) acquires an

additional weight factor λ(α) each time it touches the boundary. The operator

in the matrix model creating a boundary segment with anisotropic boundary

condition is defined as follows. Decompose the vector ~Y into a sum of an

n(1)-component vector ~Y(1) and an n(2)-component vector ~Y(2) . Then the disk

one-point function with anisotropic boundary conditions is evaluated by

H(x, λ(1) , λ(2)) =1

β〈 trH 〉 , H

def=

1

x − X− λ(1)~Y2

(1) − λ(2)~Y2

(2)

. (1.6.22)

Nontrivial boundary critical behavior is achieved by tuning the two matter

boundary couplings λ(1) and λ(2) . The boundary phase diagram is qualitatively

the same as the one on the flat lattice [BHK09]. The boundary two-point func-

tions D0(x, y) = 1β

⟨

tr[

W(x)H(y)]

⟩

can be found as solutions of a functional

equations. The analysis of these equations away from the criticality allows to

draw the qualitative picture of bulk and boundary flows in the O(n) model

[BHK09].

1.7. THE SIX-VERTEX MODEL 19

1.7 The six-vertex model

The matrix model for the Baxter’s six-vertex model on planar graphs is a de-

formation of the O(2) model. Its partition function is given by [Kos00]

ZO(n)

N (T ) ∼∫

dX dYdY† eβtr[−V (X)+eiνX~YY

†+e−iνX~Y†

Y] . (1.7.1)

1.8 The q-state Potts model (0 < q < 4)

The local fluctuating variable in the q-state Potts model takes q discrete values

and the Hamiltonian has Sq symmetry. The partition function of the Potts

model is a sum over all clusters with fugacity q and weight 1/T for each link

occupied by the cluster. The corresponding matrix model is [Kaz88, Kos89c]

ZN (T ) ∼∫

dM dqY e−β tr( 1

2M

2+Pq

a=1(T2Y

2a− 1

3(M+Ya)3). (1.8.1)

The solution of the q-state Potts matrix model in the large N limit boils down

to a problem of matrix mean field. After a shift Ya → Ya − M the matrix

integral can be written as

Zq PottsN (T ) ∼

∫

dM e−β T+q

2TtrM2−βqW [M]), (1.8.2)

eβW [M] =

∫

dY eβtr(− 12T

Y2+ 1

3Y

3+ 1TMY). (1.8.3)

The effective action W defined by the last integral depends only on the eigenval-

ues of the matrix M. It was evaluated in the large N limit by the methods used

previously for the mean field problem in the large N gauge theory [BG80]. In

the scaling limit one obtains, after a redefinition of the spectral parameter, the

same functional equation as that for the O(n) model with n = q − 2 [Kos89c].

The analysis of the solution is however more delicate than in the case of the

O(n) matrix model. The complete solution of the q-state Potts matrix model

was found in [Dau94, ZJ01, EB99].

1.9 SOS and ADE matrix models

The rational theories of 2d QG are discretized by ensembles of planar graphs

embedded in ADE Dynkin diagrams [Kos90, Kos89b, Kos92b]. The correspond-

ing matrix models are constructed in [Kos92a].

More generally, for a discrete target space representing a graph G one as-

sociates a hermitian matrix Xa with each node a ∈ G and a complex matrix

20 REFERENCES

a aa

b

aa

a b a

aij

ij

T1/β

j

β1/β

i i

j

i ij

j

k

k

β

i ij

j

k

k

Figure 1.3: Feynman rules for the matrix model with target space G.

Y<ab> = Y†<ba> with each link <ab>. The matrix model that generates planar

graphs embedded in the graph G is defined by the partition function

ZG

N (β, T ) =

∫

∏

a∈GdXa e−βtrV (Xa)

∏

<ab>

etr(XaY<ab>Y<ba>+Y<ab>XbY<ba>). (1.9.1)

For trivalent graphs the potential V is the same as that of the O(n) model.

The matrix model has an interesting continuum limit if the graph G is an ADE

or ADE Dynkin diagram. The SOS model, which is a discretization of the

gaussian field with a charge at infinity [Nie82], corresponds to an infinite target

space Z [Kos92b]. Matrix models with semi-infinite target spaces A∞ and D∞were studied in [KP07].

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