The Polyakov Path Integral and Moduli

The Polyakov formulation turns perturbative string theory into a path integral over two kinds of fields:

$$X^\mu:\Sigma\to \mathbb R^{1,D-1}, \qquad h_{ab}\text{ on }\Sigma .$$

The maps $X^\mu$ describe the embedding of the worldsheet into spacetime. The metric $h_{ab}$ describes the intrinsic geometry of the worldsheet. Since two metrics related by a worldsheet diffeomorphism and a Weyl rescaling represent the same physics, the path integral must divide by

$$\mathrm{Diff}(\Sigma)\times\mathrm{Weyl}(\Sigma).$$

This division is not merely formal bookkeeping. It produces the $bc$ ghost system, the finite-dimensional integral over moduli space, and the familiar rule that on the sphere three punctures may be fixed by $SL(2,\mathbb C)$.

The unfixed Polyakov integral

For a Euclidean closed bosonic string in flat spacetime, the matter action is

$$S_X[X,h] = \frac{1}{4\pi\alpha'} \int_\Sigma d^2\sigma\,\sqrt h\,h^{ab} \partial_aX^\mu\partial_bX_\mu .$$

A genus-$g$ contribution to an $n$-point amplitude has the schematic form

$$\mathcal A_{g,n} = g_s^{2g-2+n} \int \frac{\mathcal D h\,\mathcal D X}{\mathrm{Vol}(\mathrm{Diff}\times\mathrm{Weyl})} \,e^{-S_X[X,h]} \prod_{i=1}^n \mathcal V_i .$$

The precise power of $g_s$ depends on the normalization of external states. The topological dependence is universal: a closed oriented worldsheet of genus $g$ carries Euler characteristic

$$\chi(\Sigma_g)=2-2g,$$

so the vacuum worldsheet contribution is weighted by $g_s^{-\chi}=g_s^{2g-2}$.

The quotient by gauge symmetries means that we should not integrate separately over every metric. Instead we must choose one representative on each orbit and include the corresponding Faddeev-Popov determinant.

The intrinsic metric is integrated modulo diffeomorphisms and Weyl transformations. Gauge fixing chooses a slice through this infinite-dimensional space, while true moduli remain as finite-dimensional integration variables.

Metric fluctuations: gauge directions and moduli

Let $\widehat h_{ab}(m)$ be a reference metric depending on moduli $m^I$. An infinitesimal metric fluctuation decomposes as

$$\delta h_{ab} = (P_1v)_{ab} + 2\delta\omega\,h_{ab} + \sum_I\delta m^I\,\mu_{Iab}.$$

Here $v^a$ is a vector field generating a diffeomorphism, $\delta\omega$ is a Weyl variation, and $\mu_{Iab}$ are representatives of tangent vectors to moduli space. The operator $P_1$ extracts the traceless part of the diffeomorphism variation:

$$(P_1v)_{ab} = \nabla_av_b+\nabla_bv_a-h_{ab}\nabla_cv^c.$$

The trace of $P_1v$ vanishes in two dimensions:

$$h^{ab}(P_1v)_{ab}=0.$$

So the three pieces in $\delta h_{ab}$ have distinct meanings:

$$\begin{array}{ccl} (P_1v)_{ab} &:& \text{diffeomorphism gauge direction},\\ 2\delta\omega\,h_{ab} &:& \text{Weyl gauge direction},\\ \delta m^I\mu_{Iab} &:& \text{true complex-structure deformation}. \end{array}$$

The decomposition of $\delta h_{ab}$ separates redundant gauge variations from true deformations of the Riemann surface. The latter become coordinates on moduli space.

For a closed oriented surface with $n$ punctures, the stable cases obey

$$\dim_\mathbb C\mathcal M_{g,n}=3g-3+n, \qquad 2g-2+n>0.$$

The sphere and torus have special conformal Killing symmetries. For the sphere, the group of globally defined conformal maps is

$$SL(2,\mathbb C): \qquad z\mapsto \frac{az+b}{cz+d}, \qquad ad-bc=1.$$

This group has complex dimension $3$. Therefore an $n$-punctured sphere has

$$\dim_\mathbb C\mathcal M_{0,n}=n-3$$

moduli. Equivalently, three punctures may be fixed, conventionally to $0$, $1$, and $\infty$.

The Möbius group removes three complex positions on the sphere. The remaining puncture positions are genuine moduli to be integrated over.

Conformal gauge and the Faddeev-Popov determinant

Conformal gauge chooses

$$h_{ab}=e^{2\omega}\widehat h_{ab}(m).$$

At the classical level, the matter action is independent of the conformal factor $\omega$. If Weyl invariance survives quantization, the $\omega$ integral cancels against the Weyl gauge volume and can be discarded. What remains is the integral over moduli $m^I$ and a Faddeev-Popov determinant.

The determinant is produced by the change of variables from general metric fluctuations to gauge parameters plus moduli:

$$\mathcal D h \longrightarrow \mathcal Dv\,\mathcal D\omega\,\prod_I d m^I\, J[h].$$

The Jacobian $J[h]$ is represented by anticommuting ghosts. The ghost $c^a$ is the fermionic version of the diffeomorphism parameter $v^a$, while the antighost $b_{ab}$ pairs with traceless metric variations. In complex coordinates the holomorphic part is

$$S_{bc} = \frac{1}{2\pi} \int d^2z\,b_{zz}\bar\partial c^z,$$

with an antiholomorphic copy

$$S_{\widetilde b\widetilde c} = \frac{1}{2\pi} \int d^2z\,\widetilde b_{\bar z\bar z}\partial \widetilde c^{\bar z}.$$

The field $b$ has conformal weight $2$ and $c$ has conformal weight $-1$. Their OPE is

$$b(z)c(w)\sim \frac{1}{z-w}.$$

This is the same $bc$ system introduced as a conformal field theory. The path integral explains its geometric origin.

Conformal Killing vectors and fixed punctures

A conformal Killing vector is a diffeomorphism that preserves conformal gauge. On the Riemann sphere, the holomorphic conformal Killing vectors are generated by

$$1, \qquad z, \qquad z^2.$$

The corresponding ghost zero modes are the three modes

$$c_{-1}, \qquad c_0, \qquad c_1.$$

A nonzero sphere ghost correlator must soak up these zero modes. In the holomorphic sector,

$$\langle c(z_1)c(z_2)c(z_3)\rangle = z_{12}z_{13}z_{23}, \qquad z_{ij}=z_i-z_j.$$

For the closed string, the antiholomorphic ghosts give the complex conjugate factor.

The standard tree-level closed-string prescription is therefore

$$\mathcal A_{0,n} = g_s^{n-2} \left\langle c\widetilde c V_1(z_1,\bar z_1) \,c\widetilde c V_2(z_2,\bar z_2) \,c\widetilde c V_3(z_3,\bar z_3) \prod_{i=4}^n \int d^2z_i\,V_i(z_i,\bar z_i) \right\rangle_{S^2}.$$

The three unintegrated vertices fix the $SL(2,\mathbb C)$ gauge freedom. The remaining vertex positions are integrated because they are genuine moduli of the punctured sphere.

For open strings on the disk or upper half-plane, the analogous statement uses the real group $SL(2,\mathbb R)$ and three boundary $c$ ghosts.

Moduli and $b$-ghost insertions

Higher-genus amplitudes require integration over the moduli space of Riemann surfaces. The tangent directions to moduli space are represented by Beltrami differentials

$$\mu_I{}^z{}_{\bar z}.$$

Each modulus is paired with a $b$-ghost insertion

$$(b,\mu_I) = \int_\Sigma d^2z\,b_{zz}\mu_I{}^z{}_{\bar z}.$$

The antiholomorphic sector contributes

$$(\widetilde b,\overline\mu_I) = \int_\Sigma d^2z\,\widetilde b_{\bar z\bar z}\overline\mu_I{}^{\bar z}{}_{z}.$$

The schematic closed-string genus-$g$ measure is

$$\mathcal A_{g,n} = \int_{\mathcal M_{g,n}} \prod_{I=1}^{3g-3+n} d^2m^I\, \left\langle \prod_{I=1}^{3g-3+n} (b,\mu_I)(\widetilde b,\overline\mu_I) \prod_i U_i \right\rangle,$$

where the $U_i$ are vertex operators in a convenient integrated or unintegrated representation. The formula is schematic because special cases with conformal Killing vectors require separate treatment, but the principle is robust: moduli are integrated, and each modulus brings a $b$-ghost insertion.

The insertion $(b,\mu_I)$ is the finite-dimensional remnant of the Faddeev-Popov determinant after conformal gauge fixing. It pairs an antighost with a Beltrami differential representing a modulus.

The torus is the simplest higher-genus example. A complex torus is

$$\Sigma_\tau=\mathbb C/(\mathbb Z+\tau\mathbb Z), \qquad \operatorname{Im}\tau>0,$$

with one complex modulus $\tau$, modulo modular transformations. The one-loop vacuum amplitude therefore contains an integral over a fundamental domain of $SL(2,\mathbb Z)$, together with the corresponding ghost and matter determinants.

The gauge-fixed recipe

The full story can be summarized as follows.

Gauge fixing converts the original metric integral into a CFT correlator with ghost insertions and an integral over moduli space.

In words:

1. Start with the Polyakov integral over $X^\mu$ and $h_{ab}$.
2. Divide by $\mathrm{Diff}\times\mathrm{Weyl}$.
3. Choose conformal gauge $h_{ab}=e^{2\omega}\widehat h_{ab}(m)$.
4. Exponentiate the Faddeev-Popov determinant using $bc$ ghosts.
5. Fix conformal Killing symmetries using $c$ ghosts.
6. Integrate over true moduli using $b$-ghost insertions.

This is the bridge from the classical Polyakov action to the practical rules for string perturbation theory. The next page explains the last missing consistency condition: Weyl invariance must survive quantization. Its failure is the Weyl anomaly, and its gauge-fixed remnant is BRST symmetry.

Exercises

Exercise 1. Tracelessness of $P_1v$

Show that

$$(P_1v)_{ab}=\nabla_av_b+\nabla_bv_a-h_{ab}\nabla_cv^c$$

is traceless in two dimensions.

Solution

Contract with $h^{ab}$:

$$h^{ab}(P_1v)_{ab} = 2\nabla_av^a-h^{ab}h_{ab}\nabla_cv^c.$$

In two dimensions $h^{ab}h_{ab}=2$, so

$$h^{ab}(P_1v)_{ab} = 2\nabla_av^a-2\nabla_av^a=0.$$

Exercise 2. Punctures on the sphere

Use the dimension of the Möbius group to explain why the moduli space of an $n$-punctured sphere has complex dimension $n-3$.

Solution

Each puncture has one complex coordinate, so before quotienting there are $n$ complex parameters. The globally defined conformal maps of the sphere are Möbius transformations,

$$z\mapsto \frac{az+b}{cz+d},$$

which form a complex three-dimensional group. For generic marked points this group can fix three punctures. The remaining number of complex parameters is therefore

$$n-3.$$

Exercise 3. Three $c$ ghosts on the sphere

Assuming that $c(z)$ is a primary of weight $-1$, use global conformal invariance and antisymmetry to determine

$$\langle c(z_1)c(z_2)c(z_3)\rangle.$$

Solution

The correlator must be antisymmetric under exchange of any two insertion points because $c$ is fermionic. It must also transform with weight $-1$ in each argument. The unique expression with these properties, up to normalization, is

$$\langle c(z_1)c(z_2)c(z_3)\rangle =C z_{12}z_{13}z_{23}.$$

With the standard ghost vacuum normalization, $C=1$.

Exercise 4. Why $b$ ghosts accompany moduli

Explain why a modulus $m^I$ is paired with an insertion of the form

$$(b,\mu_I)=\int d^2z\,b_{zz}\mu_I{}^z{}_{\bar z}.$$

Solution

A variation of the complex structure is represented by a Beltrami differential $\mu_I{}^z{}_{\bar z}$. In the Faddeev-Popov construction, the antighost $b_{zz}$ pairs with traceless metric variations. The finite-dimensional part of the determinant associated with changing moduli therefore inserts the natural pairing between $b$ and $\mu_I$:

$$(b,\mu_I)=\int d^2z\,b_{zz}\mu_I{}^z{}_{\bar z}.$$

This insertion also soaks up $b$ zero modes associated with moduli.