Random Lattices, Matrix Models, and Random Surfaces

The previous page emphasized a conceptual problem that becomes unavoidable in gravity: if geometry is dynamical, a fixed coordinate lattice is not an innocent regulator. A square lattice presupposes a background coordinate system. A diffeomorphism-invariant theory wants something subtler: a sum over geometries, or at least a regulator whose combinatorics does not hardwire one geometry as the preferred one.

Random lattices are the cleanest discrete version of that idea. Instead of putting fields on one fixed lattice, we sum over many lattices, weighted by area, topology, and matter degrees of freedom. In two dimensions this becomes more than a formal dream. Matrix integrals generate precisely the right class of diagrams: ribbon graphs, whose fattened edges define oriented surfaces. The large-$N$ expansion then becomes a topological expansion by genus.

This page develops three connected ideas:

1. a random lattice is a discrete sum over geometries;
2. matrix models generate random two-dimensional surfaces through ribbon Feynman diagrams;
3. the continuum limit is a critical point whose nonanalyticity is controlled by arbitrarily large maps.

The payoff is a new interpretation of Feynman diagrams. In ordinary QFT a diagram is drawn in a pre-existing spacetime. In a matrix model, a diagram can itself be a discretized spacetime.

From a fixed lattice to a sum over lattices

A fixed lattice is usually introduced as a short-distance cutoff. If the lattice spacing is $a$, momenta above order $1/a$ are absent, and the functional integral becomes finite-dimensional. For an ordinary statistical system this is exactly what we want: microscopic details are part of the definition, and universality tells us when large-distance physics stops caring about them.

For a gravitational theory the logic changes. A lattice spacing defined with respect to a coordinate grid is not itself diffeomorphism-invariant. The physical distance between neighboring sites depends on the metric. A regulator that is supposed to approximate fluctuating geometry should not keep the coordinate grid fixed while only the fields fluctuate. It should allow the lattice itself to fluctuate.

A schematic random-lattice partition function has the form

$$Z(\mu)=\sum_T {1\over |\operatorname{Aut}T|}\,e^{-\mu A(T)}\,Z_{\rm matter}(T),$$

where $T$ runs over triangulations, quadrangulations, or another chosen class of combinatorial surfaces. The automorphism factor prevents overcounting of lattices with discrete symmetries. The factor $e^{-\mu A(T)}$ is the discrete analogue of the cosmological term

$$e^{-\mu\int d^2\xi\sqrt g}.$$

The matter partition function $Z_{\rm matter}(T)$ is computed on the graph $T$. For example, a scalar field on vertices gives a discrete Gaussian action built from edge differences.

A random-lattice regulator replaces a fixed coordinate grid by a sum over combinatorial geometries. The continuum limit is approached by tuning the lattice fugacity so that large maps control the singular part.

The key point is that $T$ is not embedded in a background plane unless we explicitly add embedding fields. Its connectivity is the geometry. A triangle knows which triangles neighbor it. A path length is the number of links along a path, multiplied by the lattice spacing. Curvature is encoded by the deficit angle around vertices. This is the discrete analogue of intrinsic geometry.

For a triangulated closed surface with equilateral triangles, curvature is concentrated at vertices. If $q_v$ triangles meet at a vertex $v$, the deficit angle is

$$\delta_v = 2\pi-{\pi\over3}q_v.$$

The discrete Gauss–Bonnet theorem reads

$$\sum_v \delta_v = 2\pi\chi,$$

where

$$\chi=V-E+F=2-2h$$

is the Euler characteristic of a genus-$h$ orientable closed surface. Already at this elementary level, the topology of a surface is encoded combinatorially.

Ordinary graphs versus ribbon graphs

A scalar zero-dimensional integral, understood as a formal power series in $g$,

$$Z(g)=\int d\phi\,\exp\left(-{1\over2}\phi^2+{g\over4!}\phi^4\right)$$

generates ordinary quartic Feynman graphs. Expanding in $g$ gives

$$Z(g)=\sum_{V=0}^\infty {1\over V!}\left({g\over4!}\right)^V \int d\phi\,e^{-\phi^2/2}\phi^{4V}.$$

Wick’s theorem pairs the $4V$ factors of $\phi$, and each pairing is a graph with $V$ four-valent vertices. This is a useful graph-counting device, but the graphs are not yet two-dimensional surfaces. An ordinary edge is one line; it has no memory of which side is left or right.

For the surface-counting problem, take a Hermitian $N\times N$ matrix and the formal integral

$$Z_N(g)=\int dM\,\exp\left[-N\operatorname{Tr}\left({1\over2}M^2-{g\over4}M^4\right)\right].$$

The sign of the quartic term makes the expansion in $g$ count quartic graphs with positive weights. The potential is unbounded for real $g>0$, so this equation defines a formal power series; equivalently, one may obtain the series by analytic continuation from a stable potential. Normalize the connected vacuum generating function by

$$\mathcal F_N(g)=\log {Z_N(g)\over Z_N(0)}.$$

The logarithm selects connected vacuum graphs, while division by $Z_N(0)$ removes the empty Gaussian vacuum. With this normalization the Gaussian contraction is

$$\langle M_{ij}M_{kl}\rangle_0={1\over N}\delta_{il}\delta_{jk}.$$

A matrix field changes this. The field $M_{ij}$ carries two indices. Its propagator has two Kronecker deltas,

$$\langle M_{ij}M_{kl}\rangle_0={1\over N}\delta_{il}\delta_{jk},$$

so one draws it as a double line. The vertex

$$\operatorname{Tr}M^4=M_{i_1i_2}M_{i_2i_3}M_{i_3i_4}M_{i_4i_1}$$

has a cyclic ordering of its four half-edges. This cyclic ordering is the missing data. It tells us how to thicken the graph into a ribbon graph, and a ribbon graph determines an oriented surface.

A Hermitian matrix propagator carries two index lines. Vacuum diagrams therefore thicken into orientable surfaces. The dual of a quartic ribbon graph is a random quadrangulation.

For the quartic one-matrix model, every interaction vertex is four-valent. The dual surface is therefore a quadrangulation: every vertex of the ribbon graph becomes a square face of the dual lattice. A cubic matrix model instead generates triangulations. The precise polygon type is not universal; different microscopic polygons can belong to the same continuum universality class.

The important object is not the thin graph but the fat graph. Once the graph is thickened, the number of faces $F$ is the number of closed index loops. Those faces are not interaction vertices. They are the regions swept out by index lines, and they carry factors of $N$.

The large-N genus expansion

The power of $N$ attached to a connected vacuum ribbon graph is purely topological. Suppose the graph has $V$ quartic vertices, $E$ propagators, and $F$ index faces. From the action normalization,

• each vertex contributes $N g$;
• each propagator contributes $1/N$;
• each closed index loop contributes $N$.

Thus the total weight is

$$(Ng)^V\left({1\over N}\right)^E N^F = g^V N^{V-E+F}.$$

But

$$V-E+F=\chi=2-2h$$

for a connected orientable closed surface of genus $h$. Therefore

$$\boxed{ \text{connected graph of genus }h \quad\Longrightarrow\quad \text{weight }N^{2-2h}. }$$

This is the same topological organization as closed-string perturbation theory, where the string coupling $g_s$ weights a genus-$h$ worldsheet by $g_s^{2h-2}$. In the matrix model, the identification is

$$g_s\sim {1\over N}.$$

A connected ribbon graph carries the factor $N^\chi=N^{2-2h}$. Planar graphs dominate at large $N$; handles are suppressed by powers of $1/N^2$.

The connected generating function therefore has the formal expansion

$$\mathcal F_N(g)=\sum_{h=0}^\infty N^{2-2h}\mathcal F_h(g),$$

where $\mathcal F_h(g)$ sums connected ribbon graphs of genus $h$. The leading term $\mathcal F_0$ is the planar or spherical contribution. The next term $\mathcal F_1$ is the torus contribution, and so on.

This is the reason matrix models are not just a cute graph-counting trick. They give a controllable topological expansion. The combinatorics of matrix indices knows about surfaces.

Why planar diagrams are random surfaces

A planar ribbon graph can be drawn on the sphere without crossing its ribbons. Its dual graph is a discretized sphere made of polygons. If there are $A$ interaction vertices in the ribbon graph, the dual surface has $A$ polygonal faces. The coefficient of $g^A$ in the planar free energy counts such surfaces with area $A$, weighted by symmetry factors.

Equivalently,

$$\mathcal F_0(g)=\sum_{A\ge1}\mathcal N_0(A)g^A,$$

where $\mathcal N_0(A)$ is the weighted number of spherical quadrangulations with $A$ faces. Tuning $g$ is therefore tuning a discrete cosmological constant. If $g$ is small, large surfaces are suppressed. As $g$ approaches its critical value $g_c$, the large-area tail controls the nonanalytic behavior.

The continuum limit is not obtained by taking a single triangulation and making it smoother. It is a scaling limit of the large-$A$ tail near the first singularity $g=g_c$. Care is needed when turning this statement into an expectation value: the unmarked sphere series need not have a divergent mean area because its coefficients carry a strong negative power of $A$.

A convenient positive ensemble has two marked faces. Its partition series is

$$\mathcal Z_0^{\bullet\bullet}(g) = \left(g{\partial\over\partial g}\right)^2\mathcal F_0(g) =\sum_{A\ge1}A^2\mathcal N_0(A)g^A.$$

The mean area in this marked ensemble is

$$\langle A\rangle_{\bullet\bullet} =g{\partial\over\partial g}\log \mathcal Z_0^{\bullet\bullet}(g).$$

For pure gravity it diverges as $g\to g_c^-$, so the lattice spacing can be sent to zero while the physical area is kept fixed:

$$A_{\rm phys}=a^2 A.$$

This is the same Wilsonian logic as before, but now applied to geometry. The continuum surface is a scaling limit of large random maps.

The number of maps grows exponentially with area, with a universal power-law correction. Tuning the graph fugacity to its critical value makes large maps control the singular part; in a two-marked ensemble the mean area diverges.

There is a small convention trap here. The unmarked spherical connected generating function has the standard singular form

$$\mathcal F_0(g)_{\rm sing}\sim (g_c-g)^{2-\gamma_{\rm str}},$$

so its coefficients behave as

$$\mathcal N_0(A) \sim g_c^{-A}A^{\gamma_{\rm str}-3}.$$

If instead we count surfaces with two marked points or faces, applying $(g\partial_g)^2$ multiplies each coefficient by $A^2$, giving

$$\mathcal N_0^{\bullet\bullet}(A) \sim g_c^{-A}A^{\gamma_{\rm str}-1}.$$

Both forms appear in the literature. The exponent $\gamma_{\rm str}$ is called the string susceptibility exponent. For pure two-dimensional gravity, the exact value is

$$\gamma_{\rm str}=-{1\over2},$$

so unmarked planar maps grow like

$$\mathcal N_0(A)\sim g_c^{-A}A^{-7/2}$$

up to a nonuniversal constant. The exponential factor depends on microscopic choices such as triangulations versus quadrangulations. The power is universal within the continuum theory.

Diagonalizing the matrix integral

The one-matrix model can also be studied without drawing diagrams. Since the action is invariant under

$$M\mapsto U M U^\dagger,$$

we diagonalize

$$M=U\operatorname{diag}(\lambda_1,\ldots,\lambda_N)U^\dagger.$$

The measure becomes

$$dM=C\,dU\prod_{i=1}^N d\lambda_i\,\Delta(\lambda)^2,$$

where

$$\Delta(\lambda)=\prod_{i<j}(\lambda_i-\lambda_j)$$

is the Vandermonde determinant. Therefore

$$Z_N=\int\prod_i d\lambda_i\, \exp\left[-N\sum_i V(\lambda_i)+2\sum_{i<j}\log|\lambda_i-\lambda_j|\right],$$

where

$$V(\lambda)={1\over2}\lambda^2-{g\over4}\lambda^4$$

for the formal quartic model.

The eigenvalues behave like a one-dimensional gas. For a stable potential, $V(\lambda)$ confines them, while the logarithm repels coincident eigenvalues. In the formal quartic counting model, the same equations are interpreted perturbatively or by analytic continuation. The saddle-point equation is

$$N V'(\lambda_i)=2\sum_{j\ne i}{1\over \lambda_i-\lambda_j}.$$

At large $N$ we introduce a density

$$\rho(\lambda)={1\over N}\sum_i\delta(\lambda-\lambda_i), \qquad \int d\lambda\,\rho(\lambda)=1.$$

Then the saddle equation becomes the singular integral equation

$$\boxed{ V'(\lambda)=2\,\mathrm{P}\!\int d\lambda'\,{\rho(\lambda')\over \lambda-\lambda'}. }$$

After diagonalization, the Vandermonde determinant turns matrix integration into a Coulomb gas of eigenvalues. The large-$N$ saddle is described by a continuous density supported on one or more cuts.

The resolvent

$$\omega(z)=\int d\lambda\,{\rho(\lambda)\over z-\lambda}$$

is the most efficient way to solve the saddle. Across a cut, its discontinuity gives the density:

$$\rho(\lambda)=-{1\over2\pi i} \left[\omega(\lambda+i0)-\omega(\lambda-i0)\right].$$

For the Gaussian potential $V(\lambda)=\lambda^2/2$, the solution is Wigner’s semicircle,

$$\rho(\lambda)={1\over2\pi}\sqrt{4-\lambda^2}, \qquad -2\le\lambda\le2.$$

For the quartic model, the endpoint behavior changes as $g$ approaches a critical value. At criticality the eigenvalue density develops a higher-order zero at the endpoint of its support. That endpoint singularity is the eigenvalue-language version of large-area random surfaces.

Double scaling and the continuum surface

The planar limit keeps $N=\infty$ first, so only the sphere survives. The continuum surface limit tunes $g\to g_c$, where large maps control the singular part and marked-area moments diverge. The most interesting limit combines the two. Near criticality, a genus-$h$ contribution behaves schematically as

$$\mathcal F_h(g)_{\rm sing}\sim (g_c-g)^{(2-2h)(2-\gamma_{\rm str})/2}$$

for the simplest one-matrix universality classes. The exact exponent depends on the matter coupled to gravity, but the important point is structural: higher-genus terms become more singular as $g\to g_c$.

The double-scaling limit sends

$$N\to\infty, \qquad g\to g_c,$$

while keeping the appropriate combination

$$\kappa^{-1} \sim N\,(g_c-g)^{(2-\gamma_{\rm str})/2}$$

fixed. This retains contributions from all genera:

$$\mathcal F\sim \sum_{h=0}^\infty \kappa^{2h-2}\mathcal F_h^{\rm cont}.$$

This is why matrix models historically gave a nonperturbative handle on two-dimensional string theory. They produce a sum over random worldsheets, and the double-scaling limit makes the sum continuum while preserving topology-changing effects.

For the present course, the conceptual message is more important than the exact exponent in the double-scaling variable. The matrix size $N$ is not merely a technical parameter. It is the inverse string coupling. The matrix coupling $g$ is a lattice fugacity, or equivalently a cosmological constant. The critical point is the continuum limit.

Matter on a random lattice

A random surface can carry matter. On a triangulation $T$, put a target-space coordinate $X_v^\mu$ at every vertex $v$. A natural discrete Dirichlet action is

$$S_T[X] = {K\over2}\sum_{\langle vv'\rangle}w_{vv'}(X_v-X_{v'})^2,$$

where the sum is over edges and the dimensionless weights $w_{vv'}$ encode the local geometry. For an equilateral triangulation one may take all weights equal and absorb their common value into $K$; on a general piecewise-flat mesh, cotangent weights give the finite-element discretization. There is no extra $a^{-2}$ in two dimensions: the factor $a^2$ from the cell area cancels the $a^{-2}$ in the squared finite difference. Then

$$Z(\mu)=\sum_T {e^{-\mu A(T)}\over |\operatorname{Aut}T|} \int\prod_{v\in T}d^D X_v\, \exp[-S_T[X]].$$

This is the discrete version of the formal Polyakov path integral for embedding fields at fixed topology,

$$Z= \int {\mathcal Dg\,\mathcal DX\over \operatorname{Diff}\times\operatorname{Weyl}} \exp\left[-{1\over4\pi\alpha'}\int d^2\xi\sqrt g\,g^{ab}\partial_aX^\mu\partial_bX_\mu -\mu\int d^2\xi\sqrt g\right].$$

The first term measures how the worldsheet is embedded in target space. The second term counts its intrinsic area. The quotient is schematic: gauge fixing the diffeomorphism and Weyl redundancies introduces Faddeev–Popov ghosts, moduli integrals, and, when the total Weyl anomaly does not cancel, a dynamical conformal factor.

Assigning target-space coordinates $X_i$ to the vertices of a triangulation gives a weighted discrete Dirichlet action for an embedded random surface. The sum over triangulations is the discrete analogue of the sum over worldsheet metrics.

This expression is the meeting point of several earlier themes. The worldline representation of a particle sums over one-dimensional paths. The string path integral sums over two-dimensional surfaces. Random lattices give a nonperturbative discretization of that surface sum.

There is also a crucial anomaly constraint. In conformal gauge,

$$g_{ab}=e^{2\phi}\hat g_{ab},$$

the Weyl factor $\phi$ decouples only when the total central charge vanishes. The reparametrization ghosts contribute $c_{bc}=-26$, so a theory containing only $D$ free embedding scalars requires

$$D=26$$

for anomaly cancellation. Away from this critical dimension, or with additional worldsheet matter, the conformal anomaly generates Liouville dynamics for $\phi$. Thus the random-surface measure is not just a combinatorial detail; it knows about the same conformal anomaly discussed earlier.

Integrated vertex operators and observables

If the surface is embedded in a target space, we can define target-space observables by integrating over the worldsheet. A basic example is the target-space density of the surface,

$$V(x)=\int d^2\xi\sqrt g\,\delta^{(D)}(x-X(\xi)).$$

Its Fourier transform is the integrated vertex operator

$$V(p)=\int d^2\xi\sqrt g\,e^{ip\cdot X(\xi)}.$$

These operators are natural because they do not ask for the value of $X$ at a preferred worldsheet coordinate. They integrate over the surface. In conformal gauge, however, reparametrization invariance alone is not enough: the local matter operator must have conformal weights $(1,1)$, possibly after Liouville dressing, so that its integral is Weyl-invariant. For the free-boson normalization used above,

$$\left\langle e^{ip\cdot X(\xi)}e^{-ip\cdot X(0)}\right\rangle \sim |\xi|^{-\alpha' p^2},$$

and the exponential has weights

$$h=\bar h={\alpha' p^2\over4}.$$

Here $p^2$ is the Euclidean free-boson norm; analytic continuation to Lorentzian target signature gives the corresponding signature-dependent mass-shell relation. Thus the bare integrated exponential is marginal only when $h=\bar h=1$; more general physical vertices include oscillator factors, ghosts in the unintegrated form, or Liouville dressing. This is the worldsheet origin of string mass-shell constraints.

Integrated vertex operators are natural observables on a fluctuating surface because they do not depend on a chosen worldsheet coordinate. Weyl invariance imposes the on-shell condition for physical string amplitudes.

A more geometric target-space observable is the local tangent tensor density

$$V_{\mu\nu}(x)= \int d^2\xi\sqrt g\,g^{ab}\partial_aX_\mu\partial_bX_\nu\, \delta^{(D)}(x-X(\xi)).$$

Such operators probe how the random surface is distributed in target space. They are reparametrization-invariant because all worldsheet coordinates are integrated over with the invariant measure. Their precise renormalization is subtle, because composite operators on a fluctuating surface mix under short-distance regularization.

Factorial growth versus planar criticality

It is tempting to say that matrix integrals “sum all diagrams,” but this phrase hides two different growth problems.

For an ordinary zero-dimensional integral, the number of Wick contractions grows factorially. The perturbation series is typically asymptotic. This is not special to matrices; it is the usual fate of perturbation theory.

Matrix models add a topological organization. At fixed topology, especially in the planar sector, the number of maps with area $A$ has the controlled form

$$\mathcal N_0(A) \sim g_c^{-A}A^{\gamma_{\rm str}-3}.$$

The exponential factor sets the radius of convergence. The power-law factor encodes continuum critical behavior. Thus the planar series has a genuine critical point, and the continuum limit is extracted from the singular part near that point.

This distinction matters. The statement “all diagrams grow factorially” is true for unrestricted perturbation theory. The statement “planar maps have a critical continuum limit” is also true, but it refers to the fixed-topology, large-$N$ sector. The large-$N$ expansion separates these issues.

Summary

Random lattices discretize the idea of summing over geometries. Instead of placing fields on a fixed grid, one sums over graphs or triangulations, weighted by their area and matter partition functions. In two dimensions, matrix integrals generate these sums automatically.

The reason is index structure. A matrix propagator has two lines, and a matrix vertex has a cyclic ordering. Feynman diagrams therefore thicken into ribbon graphs, which define oriented surfaces. The power of $N$ attached to a connected diagram is $N^\chi=N^{2-2h}$, so the large-$N$ expansion is a genus expansion.

The continuum random surface is reached at a critical coupling where arbitrarily large maps control the nonanalytic part of the generating function; in the two-marked ensemble, the mean area diverges. The singularity of the planar connected generating function encodes the string susceptibility exponent. Diagonalizing the matrix model gives an equivalent eigenvalue-gas description, where criticality appears as a special endpoint behavior of the eigenvalue density.

Adding vertex variables $X_v^\mu$ produces embedded random surfaces, the discrete form of the Polyakov string path integral. Integrated vertex operators are the natural observables because they respect worldsheet reparametrization invariance. Weyl invariance then supplies the on-shell conditions of string theory.

Common pitfalls

Connectivity, not appearance. A random lattice is not a randomly distorted drawing of a fixed lattice. Its connectivity is itself summed over.

Ribbon data matter. A matrix Feynman diagram is not just an ordinary graph with labels. The double-line structure gives a cyclic ordering and hence a surface.

Large $N$ is not the continuum limit. Large $N$ selects topology, usually the sphere. The continuum limit comes from tuning the area fugacity to criticality.

Marking changes area powers. The exponent in $\mathcal N(A)$ depends on whether surfaces are marked. The unmarked sphere generating function has coefficients $A^{\gamma_{\rm str}-3}$, while two marked points shift the power to $A^{\gamma_{\rm str}-1}$. Consequently, a claim that “the mean area diverges” must specify the ensemble.

Integration does not ensure Weyl invariance. An integrated worldsheet operator is automatically reparametrization-invariant, but not automatically physical. Its local integrand must have weights $(1,1)$, possibly after dressing, so that the insertion is compatible with Weyl invariance.

Exercises

Genus weight of a ribbon graph

Consider the Hermitian matrix model with Gaussian propagator

$$\langle M_{ij}M_{kl}\rangle_0={1\over N}\delta_{il}\delta_{jk}$$

and quartic interaction $Ng\operatorname{Tr}M^4/4$. Show that a connected ribbon vacuum graph with $V$ vertices, $E$ propagators, and $F$ index faces carries the factor

$$g^V N^{V-E+F}.$$

Then interpret this power in terms of the genus $h$ of the corresponding surface.

Solution

Each quartic vertex comes from the interaction term and contributes a factor proportional to $Ng$. Therefore $V$ vertices contribute

$$(Ng)^V.$$

Each propagator contributes a factor $1/N$, so $E$ propagators contribute

$$N^{-E}.$$

Every closed index loop is freely summed from $1$ to $N$, so every face contributes a factor $N$. Hence $F$ faces contribute

$$N^F.$$

Multiplying the factors gives

$$(Ng)^V N^{-E}N^F=g^V N^{V-E+F}.$$

For a connected orientable ribbon graph, the thickened graph is a closed orientable surface with Euler characteristic

$$\chi=V-E+F=2-2h.$$

Thus the graph carries the topological weight

$$g^V N^{2-2h}.$$

Planar diagrams have $h=0$ and scale as $N^2$. A one-handle correction has $h=1$ and scales as $N^0$.

Eigenvalue saddle equation

Diagonalize a Hermitian matrix $M=U\Lambda U^\dagger$, with $\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$. Assuming the measure contains the Vandermonde factor $\Delta(\lambda)^2$, derive the large-$N$ saddle equation

$$V'(\lambda)=2\,\mathrm{P}\!\int d\lambda'\,{\rho(\lambda')\over \lambda-\lambda'}.$$

Solution

The eigenvalue representation of the matrix integral is

$$Z_N\propto\int\prod_i d\lambda_i\, \exp\left[-N\sum_i V(\lambda_i)+2\sum_{i<j}\log|\lambda_i-\lambda_j|\right].$$

The exponent is

$$S_{\rm eff}=-N\sum_i V(\lambda_i)+2\sum_{i<j}\log|\lambda_i-\lambda_j|.$$

The saddle equation for $\lambda_i$ is

$$0={\partial S_{\rm eff}\over\partial\lambda_i} =-N V'(\lambda_i)+2\sum_{j\ne i}{1\over\lambda_i-\lambda_j}.$$

Thus

$$V'(\lambda_i)={2\over N}\sum_{j\ne i}{1\over\lambda_i-\lambda_j}.$$

At large $N$, define

$$\rho(\lambda)={1\over N}\sum_i\delta(\lambda-\lambda_i).$$

The sum becomes a principal-value integral because the $j=i$ term is omitted:

$${1\over N}\sum_{j\ne i}{1\over\lambda_i-\lambda_j} \longrightarrow \mathrm{P}\!\int d\lambda'\,{\rho(\lambda')\over\lambda-\lambda'}.$$

This gives

$$V'(\lambda)=2\,\mathrm{P}\!\int d\lambda'\,{\rho(\lambda')\over\lambda-\lambda'}.$$

Critical coefficients and singular behavior

Suppose the unmarked planar map coefficients behave as

$$\mathcal N_0(A)\sim C\,g_c^{-A}A^{\gamma_{\rm str}-3}.$$

Show that the singular part of

$$\mathcal F_0(g)=\sum_A\mathcal N_0(A)g^A$$

has the form

$$\mathcal F_0(g)_{\rm sing}\sim (g_c-g)^{2-\gamma_{\rm str}},$$

up to analytic terms and a nonuniversal constant.

Solution

Near $g=g_c$, define

$$x={g\over g_c}=e^{-t}, \qquad t\approx {g_c-g\over g_c}.$$

The large-$A$ tail controls the nonanalytic part. Its model sum is a polylogarithm:

$$\sum_{A\ge1}A^{\gamma_{\rm str}-3}e^{-tA} =\operatorname{Li}_{3-\gamma_{\rm str}}(e^{-t}).$$

After subtracting the finitely many terms analytic in $t$, the small-$t$ expansion contains

$$\Gamma(\gamma_{\rm str}-2)t^{2-\gamma_{\rm str}}.$$

The same result follows by replacing only the large-$A$ tail by a Laplace integral and treating its lower endpoint by analytic subtraction. Therefore

$$\mathcal F_0(g)_{\rm sing}\sim t^{2-\gamma_{\rm str}}.$$

Since $t$ is proportional to $g_c-g$ near criticality,

$$\mathcal F_0(g)_{\rm sing}\sim (g_c-g)^{2-\gamma_{\rm str}}.$$

For noninteger $2-\gamma_{\rm str}$ this is the nonanalytic critical behavior. At exceptional integer exponents the continuation instead produces logarithms. Analytic terms depend on small $A$ and are not fixed by the coefficient asymptotics.

Reparametrization invariance of an integrated vertex

Show that the integrated vertex operator

$$V(p)=\int d^2\xi\sqrt g\,e^{ip\cdot X(\xi)}$$

is invariant under worldsheet reparametrizations, assuming $X^\mu(\xi)$ is a scalar field on the worldsheet.

Solution

Under a reparametrization $\xi\mapsto\xi'=f(\xi)$, the scalar field satisfies

$$X'(\xi')=X(\xi).$$

The area element transforms as a density:

$$d^2\xi\sqrt g=d^2\xi'\sqrt{g'}.$$

Therefore

$$\int d^2\xi'\sqrt{g'}\,e^{ip\cdot X'(\xi')} = \int d^2\xi\sqrt g\,e^{ip\cdot X(\xi)}.$$

So $V(p)$ is reparametrization-invariant. This does not yet guarantee Weyl invariance. For a physical string vertex operator, the integrand must also have the correct conformal weight.

Continuum limit of the lattice Dirichlet action

For a triangulation $T$ with vertex variables $X_v\in\mathbb R^D$, consider

$$S_T[X]={K\over2}\sum_{\langle vv'\rangle}w_{vv'}(X_v-X_{v'})^2.$$

Explain why this is the natural discrete analogue of

$${1\over4\pi\alpha'}\int d^2\xi\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX.$$

Solution

On a fine triangulation, neighboring vertices are separated by a proper distance of order the lattice spacing. If $X$ varies slowly across the lattice, then along an edge

$$X_v-X_{v'}\approx a\,e^a\partial_aX,$$

where $e^a$ is a unit tangent direction on the triangulation. Each squared difference is therefore of order $a^2(\partial X)^2$. A two-dimensional region of fixed area contains order $a^{-2}$ edges, so these powers cancel. With finite-element weights, the sum converges to the Dirichlet energy:

$$\sum_{\langle vv'\rangle}w_{vv'}(X_v-X_{v'})^2 \longrightarrow \text{const}\times\int d^2\xi\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX.$$

The precise constant depends on the lattice and on the weights and is absorbed into $K$, which is matched to $1/(2\pi\alpha')$ in the continuum normalization. An additional factor $a^{-2}$ would double-count the derivative scaling in two worldsheet dimensions.

References

• Di Francesco, P., Ginsparg, P., and Zinn-Justin, J. “2D Gravity and Random Matrices.” Physics Reports 254 (1995): 1–133. https://doi.org/10.1016/0370-1573(94)00084-G.

• Eynard, B. Counting Surfaces. Progress in Mathematical Physics 70. Basel: Birkhäuser, 2016.

• Ginsparg, P., and Moore, G. “Lectures on 2D Gravity and 2D String Theory.” In Recent Directions in Particle Theory: From Superstrings and Black Holes to the Standard Model, TASI 1992, 277–469. Singapore: World Scientific, 1993. arXiv:hep-th/9304011.

• Polyakov, A. M. Gauge Fields and Strings. Contemporary Concepts in Physics 3. Chur: Harwood Academic Publishers, 1987.

• Zinn-Justin, J. Quantum Field Theory and Critical Phenomena. 4th ed. Oxford: Clarendon Press, 2002.