Worldlines, Worldsheets, and Reparametrization Gauge

The relativistic particle is a one-dimensional generally covariant system. The field $h(\tau)$, the einbein, is not an ordinary dynamical force. It is the one-dimensional metric density on the path, and its job is to make the action insensitive to how we label points along the same curve. The price of this gauge symmetry is a constraint: varying $h$ imposes the mass-shell condition.

This page turns the same idea into a two-dimensional theory. A particle sweeps out a worldline. A string sweeps out a worldsheet. The square-root length of a worldline generalizes to the square-root area of a worldsheet, and the einbein generalizes to a worldsheet metric $g_{ab}(\xi)$. This is the passage from the Nambu–Goto action to the Polyakov action.

The conceptual punchline is simple and important:

$$\text{geometry of the parameter space is gauge, but its constraints are physical.}$$

For the worldline, gauge fixing leaves the proper-time modulus. For the worldsheet, gauge fixing by diffeomorphisms and Weyl transformations leads to conformal gauge, but the metric equation of motion survives as the pair of constraints

$$T_{++}=0, \qquad T_{--}=0.$$

These are the classical Virasoro constraints. They are the bridge between the geometric string picture and the two-dimensional CFT machinery developed earlier in the course.

Required background. Lesson 30 supplies the relativistic worldline action, einbein, and reparametrization symmetry used below.

Worldline gauge fixing revisited

Euclidean worldsheet and string-tension symbol

This page uses a Euclidean worldsheet unless explicitly stated otherwise. The worldsheet coordinates are $\xi^a$, with $a=1,2$. The embedding into a flat target space is $X^\mu(\xi)$, and $\partial_aX\cdot\partial_bX$ means contraction with the target-space metric. The string tension is denoted $\mathcal T$ to avoid confusing it with the stress tensor $T_{ab}$.

In Lorentzian worldsheet signature one inserts the usual minus signs, for example $\sqrt{\det h}\mapsto\sqrt{-\det h}$. None of the gauge-theoretic points depend on that choice.

For a relativistic particle, the Polyakov-like worldline action is

$$S[x,h] ={1\over2}\int_0^1d\tau\left({\dot x^2\over h}+m^2h\right).$$

The reparametrization symmetry is

$$x(\tau)\mapsto x_f(\tau)=x(f(\tau)), \qquad h(\tau)\mapsto h_f(\tau)=f'(\tau)h(f(\tau)),$$

where $f(0)=0$, $f(1)=1$, and $f'(\tau)>0$. The combination $h(\tau)d\tau$ is invariant in the sense that its integral is unchanged:

$$L=\int_0^1h(\tau)d\tau.$$

This number $L$ is the proper-time modulus of the worldline interval. A reparametrization can redistribute the local density $h(\tau)$, but it cannot change $L$.

A useful way to see the correct gauge fixing is to define a new parameter

$$s(\tau)={1\over L}\int_0^\tau h(u)du.$$

Then $s(0)=0$, $s(1)=1$, and in the $s$ coordinate the einbein is constant:

$$h_s(s)=L.$$

Thus the safe local gauge choice is

$$\dot h=0, \qquad h=L,$$

followed by an integration over $L$ in the path integral. By contrast, setting $h=1$ on the interval would also set $L=1$. That is not just gauge fixing; it discards the modulus. A weaker condition such as $\ddot h=0$ leaves a linear function $h(\tau)=a+b\tau$ and therefore does not fully fix the local reparametrization freedom.

Endpoint-preserving reparametrizations can make the einbein constant, but they cannot change $L=\int_0^1h(\tau)d\tau$. The condition $\dot h=0$ is a good local gauge; $h=1$ overfixes the interval by fixing the modulus.

After fixing $h=L$, the worldline action becomes

$$S[x,L] ={1\over2}\int_0^1d\tau\left({\dot x^2\over L}+m^2L\right).$$

Equivalently, with $t=L\tau$,

$$S[x,L] ={1\over2}\int_0^Ldt\left[\left({dx\over dt}\right)^2+m^2\right].$$

This is the form underlying the proper-time representation of the scalar propagator. The worldline lesson is now in place: first introduce a metric to make gauge symmetry manifest, then fix the metric carefully, leaving global moduli and constraints intact.

Spin and line geometry

The bosonic worldline action describes scalar propagation. To describe spin, one must add extra structure: worldline fermions, spin factors, or in special low-dimensional settings geometric phases related to the torsion of a path. Such terms can change the spin/statistics carried by the path integral. This page keeps only the bosonic embedding variables $X^\mu$; the more elaborate spinning-particle and spinning-string actions follow the same gauge-symmetry logic.

From lines to surfaces

A string is an extended one-dimensional object. As it evolves, it sweeps out a two-dimensional surface. Choose coordinates $\xi^1,\xi^2$ on this surface and describe its embedding into target space by

$$X^\mu=X^\mu(\xi^1,\xi^2).$$

The induced metric on the surface is

$$h_{ab}(\xi)=\partial_aX^\mu\partial_bX_\mu =\partial_aX\cdot\partial_bX.$$

The geometric area element is

$$dA=\sqrt{\det h}\,d^2\xi,$$

so the direct area action is

$$\boxed{ S_{\rm NG}[X] =\mathcal T\int d^2\xi\,\sqrt{\det h}. }$$

This is the Nambu–Goto action. It is the worldsheet analog of the square-root length action

$$S_{\rm length}=m\int ds.$$

The map $X^\mu(\xi)$ sends a coordinate domain to an embedded surface. The tangent vectors $\partial_aX$ define the induced metric $h_{ab}=\partial_aX\cdot\partial_bX$, and the Nambu–Goto action is the area weighted by the string tension $\mathcal T$.

The Nambu–Goto action is manifestly geometric. It does not depend on the particular coordinates $\xi^a$ used on the surface. Under a reparametrization

$$\xi^a\mapsto \xi'^a=f^a(\xi),$$

the embedding transforms as a scalar field on the worldsheet,

$$X^\mu(\xi)\mapsto X_f^\mu(\xi)=X^\mu(f(\xi)).$$

The induced metric transforms by pullback:

$$h_{ab}(\xi) \mapsto \partial_a f^c(\xi)\partial_b f^d(\xi)h_{cd}(f(\xi)),$$

and the area form $\sqrt{\det h}\,d^2\xi$ is invariant. This is the two-dimensional analog of the worldline statement that $\sqrt{\dot x^2}d\tau$ is invariant.

The drawback is also the same as before: the square root is awkward for quantization, and the action is nonlinear in a way that hides the relation to free two-dimensional fields. The cure is to introduce an auxiliary worldsheet metric.

The Polyakov action

The Polyakov action treats $g_{ab}(\xi)$ as an independent metric on the worldsheet:

$$\boxed{ S_P[X,g] ={\mathcal T\over2}\int d^2\xi\,\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX. }$$

The analogy with the worldline action is direct:

$$\text{einbein }h(\tau) \quad\longrightarrow\quad \text{worldsheet metric }g_{ab}(\xi).$$

Both are auxiliary gauge fields for reparametrization symmetry. Both make the action quadratic in derivatives of $X$. Both impose constraints when varied.

The Polyakov action has two local symmetries.

First, it is invariant under worldsheet diffeomorphisms. The embedding $X^\mu$ transforms as a scalar,

$$X^\mu(\xi)\mapsto X^\mu(f(\xi)),$$

while the metric transforms by pullback,

$$g_{ab}(\xi) \mapsto \partial_a f^c(\xi)\partial_b f^d(\xi)g_{cd}(f(\xi)).$$

Second, because the worldsheet is two-dimensional, it is invariant under Weyl rescalings

$$g_{ab}(\xi) \mapsto e^{2\omega(\xi)}g_{ab}(\xi).$$

Indeed, in two dimensions,

$$\sqrt g\mapsto e^{2\omega}\sqrt g, \qquad g^{ab}\mapsto e^{-2\omega}g^{ab},$$

so the product is invariant:

$$\sqrt g\,g^{ab} \mapsto \sqrt g\,g^{ab}.$$

This simple cancellation is one of the reasons strings are special among extended objects. For a $p$-brane, the worldvolume dimension is $p+1$. The same Polyakov-type action has Weyl invariance only when $p+1=2$, namely for strings.

The Polyakov action introduces an independent metric $g_{ab}$. Varying $X$ gives a harmonic-map equation. Varying $g_{ab}$ gives the stress-tensor constraint. In two dimensions that constraint sets $g_{ab}$ proportional to the induced metric, recovering the Nambu–Goto action classically.

Classical equivalence to Nambu–Goto

Varying $X^\mu$ in the Polyakov action gives

$$\delta_X S_P =-\mathcal T\int d^2\xi\, \delta X_\mu\,\partial_a\left(\sqrt g\,g^{ab}\partial_bX^\mu\right),$$

up to boundary terms. Thus

$$\boxed{ \partial_a\left(\sqrt g\,g^{ab}\partial_bX^\mu\right)=0. }$$

This says that $X^\mu$ is a harmonic map from the worldsheet metric $g_{ab}$ into target space. In flat conformal gauge it will become the free Laplace or wave equation.

Now vary the inverse metric $g^{ab}$. The useful identity is

$$\delta\sqrt g=-{1\over2}\sqrt g\,g_{ab}\delta g^{ab}.$$

Therefore

$$\delta_gS_P ={\mathcal T\over2}\int d^2\xi\,\sqrt g\, \left(\partial_aX\cdot\partial_bX -{1\over2}g_{ab}g^{cd}\partial_cX\cdot\partial_dX\right)\delta g^{ab}.$$

The metric equation is the vanishing of the worldsheet stress tensor:

$$\boxed{ \Theta_{ab} \equiv \partial_aX\cdot\partial_bX -{1\over2}g_{ab}g^{cd}\partial_cX\cdot\partial_dX =0. }$$

The trace vanishes identically in two dimensions:

$$g^{ab}\Theta_{ab} =g^{ab}\partial_aX\cdot\partial_bX -{2\over2}g^{cd}\partial_cX\cdot\partial_dX =0.$$

This tracelessness is the local Noether identity associated with Weyl invariance.

To recover the Nambu–Goto action, solve the metric equation for a nondegenerate induced metric. It implies that $g_{ab}$ is locally proportional to $h_{ab}$:

$$g_{ab}=e^{2\omega}h_{ab}.$$

The arbitrary Weyl factor $e^{2\omega}$ is gauge. Substituting into the Polyakov action gives

$$S_P[X,g=e^{2\omega}h] ={\mathcal T\over2}\int d^2\xi\,\sqrt h\,h^{ab}h_{ab}.$$

Since $h^{ab}h_{ab}=2$ in two dimensions,

$$S_P[X,g=e^{2\omega}h] =\mathcal T\int d^2\xi\,\sqrt h =S_{\rm NG}[X].$$

Thus Nambu–Goto and Polyakov are classically equivalent. Quantum mechanically, the Polyakov form is far more useful because it exposes the two-dimensional field theory and the gauge symmetries.

Reparametrization and Weyl gauge

A two-dimensional metric has three independent local components. Diffeomorphisms provide two local gauge functions, and Weyl symmetry provides one more. Locally, this is enough to put the metric in conformal form.

In Euclidean signature, we choose the conformal-factor convention

$$g_{ab}=e^{\phi}\delta_{ab}.$$

In light-cone or Lorentzian coordinates one often writes

$$ds^2=e^{\phi}d\xi^+d\xi^-.$$

Equivalently, up to conventional factors,

$$g_{++}=g_{--}=0, \qquad g_{+-}=e^{\phi}.$$

A two-dimensional metric has three local components. Diffeomorphisms remove two and Weyl rescaling removes one, so locally the metric can be written in conformal gauge. Residual conformal transformations remain and become the Virasoro symmetry of the gauge-fixed theory.

In conformal gauge, the Weyl factor cancels out of the classical matter action:

$$\sqrt g\,g^{ab}=\delta^{ab}$$

in Euclidean notation. Hence

$$\boxed{ S_P ={\mathcal T\over2}\int d^2\xi\,\partial_aX\cdot\partial_aX. }$$

The equation of motion becomes

$$\partial^2X^\mu=0.$$

In Lorentzian light-cone coordinates this is

$$\partial_+\partial_-X^\mu=0,$$

with the general local solution

$$X^\mu(\xi^+,\xi^-)=X_L^\mu(\xi^+)+X_R^\mu(\xi^-).$$

The gauge-fixed embedding looks like a collection of free massless scalar fields on the worldsheet. That statement is true but incomplete: the stress-tensor constraints must still be imposed.

Conformal gauge and Virasoro constraints

Gauge fixing simplifies the action, but it does not erase the metric equation of motion. The equations

$$\Theta_{ab}=0$$

survive as constraints on the free fields $X^\mu$.

In conformal light-cone coordinates, these constraints become

$$\boxed{ T_{++}=\partial_+X\cdot\partial_+X=0, \qquad T_{--}=\partial_-X\cdot\partial_-X=0. }$$

These are called the Virasoro constraints. They say that the left-moving and right-moving worldsheet stress tensors vanish. In Euclidean complex notation, the classical stress tensors are proportional to

$$T_{\mathrm{cl}}(z)\propto-\partial X\cdot\partial X, \qquad \bar T_{\mathrm{cl}}(\bar z)\propto-\bar\partial X\cdot\bar\partial X,$$

where the omitted coefficient depends on the normalization of complex derivatives. The classical constraints are

$$T_{\mathrm{cl}}(z)=0, \qquad \bar T_{\mathrm{cl}}(\bar z)=0.$$

After quantization these composite fields must be normal ordered, and the matter stress tensor must be combined with the ghost contribution in the BRST constraints.

Conformal gauge turns the Polyakov action into a free two-dimensional scalar theory for the target coordinates $X^\mu$. The metric equation remains as the two stress-tensor constraints $T_{++}=0$ and $T_{--}=0$.

On the equations of motion, the constraints are chiral. For example,

$$\partial_-T_{++} =2\partial_+X\cdot\partial_-\partial_+X =0,$$

and similarly

$$\partial_+T_{--}=0.$$

This is exactly the holomorphic stress-tensor structure encountered earlier in two-dimensional CFT. The difference is interpretational: in ordinary CFT the stress tensor generates conformal transformations as a global or local symmetry of correlation functions; in the Polyakov string it also enforces a gauge constraint. Physical states must satisfy the quantum version of these constraints.

A useful Euclidean way to read the conditions is to define

$$\partial_+=\partial_1-i\partial_2, \qquad \partial_-\equiv\partial_1+i\partial_2,$$

up to harmless factors of $2$. Then

$$(\partial_+X)^2=0$$

is equivalent to the pair of real conditions

$$(\partial_1X)^2=(\partial_2X)^2, \qquad \partial_1X\cdot\partial_2X=0.$$

Thus, when the induced metric is nondegenerate and positive definite, the two coordinate tangent vectors have equal length and are orthogonal. In other words, the constraints make the induced metric locally proportional to the flat metric. Together with the harmonic equation for $X^\mu$, they ensure that the gauge-fixed map describes a minimal surface rather than an arbitrary free-field configuration.

The path-integral viewpoint

The worldline representation of a scalar propagator has the schematic form

$$G(x,y) =\int_0^\infty dL\int_{X(0)=y}^{X(L)=x}\mathcal DX\,e^{-S_1[X,L]}.$$

For strings, the corresponding object is a sum over embeddings and worldsheet geometries:

$$Z =\int {\mathcal DX\,\mathcal Dg\over \operatorname{Vol}(\operatorname{Diff}\times\operatorname{Weyl})} \,e^{-S_P[X,g]}.$$

This formula should be read with care. The division by the gauge volume is schematic; a proper quantum treatment requires gauge fixing, Faddeev–Popov ghosts, an integral over moduli, and cancellation or explicit treatment of the Weyl anomaly. But the structure is already visible:

$$\text{particle propagator}: \sum_{\text{paths}}, \qquad \text{string amplitude}: \sum_{\text{surfaces}}.$$

The proper-time integral over worldlines generalizes to a worldsheet path integral over embeddings $X$ and metrics $g_{ab}$, modulo diffeomorphism and Weyl gauge redundancy. Gauge fixing produces constraints and, on higher-topology worldsheets, moduli.

The next page begins to unpack this formula. Once $g_{ab}$ is treated as a dynamical integration variable, the theory resembles two-dimensional gravity coupled to matter. The conformal factor that disappeared classically can reappear through the quantum measure as a conformal anomaly, leading toward Liouville theory and nonlocal effective actions.

Summary

The worldline action

$$S[x,h]={1\over2}\int d\tau\left({\dot x^2\over h}+m^2h\right)$$

teaches the first lesson: the metric on parameter space is gauge, but gauge fixing must leave global moduli such as

$$L=\int h\,d\tau.$$

The worldsheet generalization begins with the induced metric

$$h_{ab}=\partial_aX\cdot\partial_bX$$

and the Nambu–Goto area action

$$S_{\rm NG}=\mathcal T\int d^2\xi\sqrt{\det h}.$$

Introducing an independent worldsheet metric gives the Polyakov action

$$S_P={\mathcal T\over2}\int d^2\xi\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX.$$

Varying $g_{ab}$ gives the stress-tensor constraint

$$\Theta_{ab}=0,$$

which classically makes $g_{ab}$ proportional to the induced metric and recovers the Nambu–Goto action. In conformal gauge the action becomes a free scalar theory,

$$S_P={\mathcal T\over2}\int d^2\xi\,(\partial X)^2,$$

but the metric equation survives as

$$T_{++}=0, \qquad T_{--}=0.$$

These are the classical Virasoro constraints, and they are the point where worldsheet geometry meets two-dimensional conformal field theory.

Common pitfalls

Do not set $h=1$ on a finite worldline interval unless the proper-time modulus has already been handled. The invariant quantity $L=\int h d\tau$ must still be integrated over.

Do not confuse the induced metric $h_{ab}=\partial_aX\cdot\partial_bX$ with the independent Polyakov metric $g_{ab}$. They become proportional only after using the $g_{ab}$ equation of motion.

Do not gauge-fix the Polyakov action and then forget the metric equation. Conformal gauge makes $X^\mu$ look free, but the constraints $T_{++}=T_{--}=0$ are still part of the theory.

Do not treat Weyl symmetry as available for every extended object. The cancellation $\sqrt g\,g^{ab}\mapsto\sqrt g\,g^{ab}$ works only in two worldsheet dimensions for this action.

Do not promote the classical equations $T_{++}=T_{--}=0$ directly to operator identities for the matter CFT. Quantum string constraints are imposed on physical states through the total matter-plus-ghost stress tensor, with anomaly cancellation required for consistency.

Exercises

Exercise 1: Constant-einbein gauge and the modulus

Show that any positive einbein $h(\tau)$ on $0\le \tau\le1$ can be brought to the constant value $L=\int_0^1h(\tau)d\tau$ by an endpoint-preserving reparametrization.

Solution

Define

$$s(\tau)={1\over L}\int_0^\tau h(u)du, \qquad L=\int_0^1h(u)du.$$

Since $h>0$, the function $s(\tau)$ is monotone, with $s(0)=0$ and $s(1)=1$. The invariant line element is

$$h(\tau)d\tau=h_s(s)ds.$$

But

$$ds={h(\tau)\over L}d\tau.$$

Therefore

$$h_s(s)=L.$$

Thus the local shape of $h(\tau)$ can be gauged away, but the constant $L$ remains.

Exercise 2: Pullback of the induced metric

Under a reparametrization $\xi^a\mapsto f^a(\xi)$, show that the induced metric

$$h_{ab}=\partial_aX\cdot\partial_bX$$

transforms by pullback.

Solution

The transformed embedding is

$$X_f^\mu(\xi)=X^\mu(f(\xi)).$$

By the chain rule,

$$\partial_aX_f^\mu(\xi) =\partial_a f^c(\xi)\partial_cX^\mu(f(\xi)).$$

Therefore

$$h^{(f)}_{ab}(\xi) =\partial_aX_f\cdot\partial_bX_f =\partial_a f^c\partial_b f^d\, \partial_cX(f(\xi))\cdot\partial_dX(f(\xi)).$$

Thus

$$h^{(f)}_{ab}(\xi) =\partial_a f^c(\xi)\partial_b f^d(\xi)h_{cd}(f(\xi)),$$

which is the pullback transformation law for a metric.

Exercise 3: Why Weyl invariance selects worldsheets

Show that the Polyakov action

$$S_P={\mathcal T\over2}\int d^n\xi\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX$$

is Weyl invariant only when the worldvolume dimension is $n=2$.

Solution

Under

$$g_{ab}\mapsto e^{2\omega}g_{ab},$$

the determinant transforms as

$$\sqrt g\mapsto e^{n\omega}\sqrt g,$$

while the inverse metric transforms as

$$g^{ab}\mapsto e^{-2\omega}g^{ab}.$$

Therefore

$$\sqrt g\,g^{ab} \mapsto e^{(n-2)\omega}\sqrt g\,g^{ab}.$$

The action is invariant for arbitrary local $\omega(\xi)$ only if

$$n-2=0,$$

so $n=2$. A string has a two-dimensional worldsheet, so the Polyakov action has Weyl symmetry. A generic $p$-brane has $n=p+1$, and this simple Weyl symmetry is absent unless $p=1$.

Exercise 4: Stress tensor from metric variation

Derive the stress-tensor constraint from variation of the Polyakov action with respect to $g^{ab}$.

Solution

Start from

$$S_P={\mathcal T\over2}\int d^2\xi\sqrt g\,g^{ab}\partial_aX\cdot\partial_bX.$$

Use

$$\delta\sqrt g=-{1\over2}\sqrt g\,g_{ab}\delta g^{ab}.$$

Then

$$\delta S_P ={\mathcal T\over2}\int d^2\xi\sqrt g\, \left[ \partial_aX\cdot\partial_bX -{1\over2}g_{ab}g^{cd}\partial_cX\cdot\partial_dX \right]\delta g^{ab}.$$

Since $\delta g^{ab}$ is arbitrary, the metric equation of motion is

$$\partial_aX\cdot\partial_bX -{1\over2}g_{ab}g^{cd}\partial_cX\cdot\partial_dX=0.$$

This is the vanishing of the worldsheet stress tensor, up to the conventional overall sign and factor used in defining $T_{ab}$.

Exercise 5: Chiral conservation of the constraints

In conformal gauge, use the equation of motion $\partial_+\partial_-X=0$ to show that $T_{++}=\partial_+X\cdot\partial_+X$ is left-moving.

Solution

Compute

$$\partial_-T_{++} =\partial_-\left(\partial_+X\cdot\partial_+X\right).$$

Using the product rule,

$$\partial_-T_{++} =2\partial_+X\cdot\partial_-\partial_+X.$$

The conformal-gauge equation of motion is

$$\partial_+\partial_-X=0.$$

Hence

$$\partial_-T_{++}=0.$$

So $T_{++}$ depends only on $\xi^+$ locally. Similarly,

$$\partial_+T_{--}=0.$$

These are the classical chiral conservation equations for the two components of the worldsheet stress tensor.

Exercise 6: Geometry of the Virasoro constraint

Use the Euclidean definitions

$$\partial_+=\partial_1-i\partial_2, \qquad \partial_- =\partial_1+i\partial_2$$

to show that $(\partial_+X)^2=0$ is equivalent to

$$(\partial_1X)^2=(\partial_2X)^2, \qquad \partial_1X\cdot\partial_2X=0.$$

Solution

Expand

$$(\partial_+X)^2 =(\partial_1X-i\partial_2X)\cdot(\partial_1X-i\partial_2X).$$

This gives

$$(\partial_+X)^2 =(\partial_1X)^2-(\partial_2X)^2 -2i\,\partial_1X\cdot\partial_2X.$$

For this complex quantity to vanish, both its real and imaginary parts must vanish:

$$(\partial_1X)^2-(\partial_2X)^2=0,$$

and

$$\partial_1X\cdot\partial_2X=0.$$

Thus the coordinate tangent vectors have equal length and are orthogonal, which is exactly the statement that the induced metric is conformally flat in these coordinates.

Further reading

• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory (Springer, 1997), Chapters 4–6, for the stress tensor and Virasoro symmetry.
• M. B. Green, J. H. Schwarz, and L. Brink, “Superfield theory of type II superstrings,” Nuclear Physics B 219 (1983) 437–478, for historical context on locally supersymmetric worldsheet actions.
• M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Volume 1 (Cambridge University Press, 1987), Chapters 1–2, for Nambu–Goto and Polyakov actions, conformal gauge, and Virasoro constraints.
• J. Polchinski, String Theory, Volume 1 (Cambridge University Press, 1998), Chapters 1–2, for the Polyakov path integral, ghosts, moduli, and the Weyl anomaly.
• A. M. Polyakov, Gauge Fields and Strings (Harwood Academic Publishers, 1987), Chapters 9–10, for random paths, random surfaces, worldsheet metrics, and string amplitudes.