The Low-Energy Effective Action

The massless closed string contains a metric fluctuation, a two-form field, and a scalar dilaton:

$$\epsilon_{\mu\nu} \quad\longrightarrow\quad G_{\mu\nu},\qquad B_{\mu\nu},\qquad \Phi.$$

This page explains how their low-energy spacetime action is inferred from string scattering and from general symmetry principles. The answer is not merely “Einstein gravity plus matter.” It is Einstein gravity written first in the natural string frame, with a universal dilaton prefactor.

What an effective action must reproduce

Consider tree-level scattering of massless closed-string states at momenta much smaller than the string scale:

$$\alpha' k_i\cdot k_j\ll 1.$$

The exact string amplitudes have two kinds of low-energy contributions:

1. Massless exchange poles, such as $1/s$, $1/t$, and $1/u$. These must be reproduced by ordinary Feynman diagrams built from cubic spacetime interactions.
2. Analytic contact terms, expandable in powers of $\alpha' k_i\cdot k_j$. These become local higher-derivative terms in the spacetime action.

The leading action must also respect the gauge symmetries found from vertex operators:

$$\begin{aligned} G_{\mu\nu} &\quad\text{has diffeomorphism invariance},\\ B_{\mu\nu} &\quad\text{has } B\mapsto B+d\Lambda,\\ \Phi &\quad\text{is a scalar}. \end{aligned}$$

These requirements strongly constrain the two-derivative answer.

Low-energy closed-string amplitudes split into massless exchange poles and local contact terms. The former determine the two-derivative interactions; the latter organize higher-derivative $\alpha'$ corrections.

The string-frame action

The universal two-derivative closed-string action is

$$\boxed{ S_{\rm S} = \frac{1}{2\kappa_D^2} \int d^D x\sqrt{-G}\,e^{-2\Phi} \left[ R +4\nabla_\mu\Phi\nabla^\mu\Phi -\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho} -\frac{2(D-26)}{3\alpha'} +O(\alpha') \right]. }$$

Here

$$H=dB, \qquad H_{\mu\nu\rho}=3\partial_{[\mu}B_{\nu\rho]}.$$

The subscript S means string frame. This is the frame in which the target-space metric $G_{\mu\nu}$ appears directly in the worldsheet sigma-model kinetic term

$$\frac{1}{4\pi\alpha'}\int d^2\sigma\sqrt\gamma\,\gamma^{ab}G_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu.$$

In the critical bosonic string, $D=26$, the central-charge-deficit term vanishes. In noncritical strings it becomes a dilaton potential or cosmological term in string frame.

The universal closed-string NS-NS fields are packaged into the string-frame action. The factor $e^{-2\Phi}$ is characteristic of closed-string tree level.

The $O(\alpha')$ terms are higher-derivative corrections. In the bosonic string, the first corrections include curvature-squared and related $H$-dependent terms. In Type II superstrings, supersymmetry removes many lower-order corrections, and the first purely gravitational correction appears much later, schematically as $\alpha'^3R^4$.

Why the dilaton multiplies the whole tree action

The dilaton couples to the intrinsic curvature of the worldsheet:

$$S_\Phi = \frac{1}{4\pi}\int_\Sigma d^2\sigma\sqrt\gamma\,R^{(2)}(\gamma)\,\Phi(X).$$

If the dilaton is constant, $\Phi(X)=\Phi_0$, the Gauss-Bonnet theorem gives

$$S_\Phi=\Phi_0\chi(\Sigma), \qquad \chi(\Sigma)=\frac{1}{4\pi}\int_\Sigma\sqrt\gamma R^{(2)}=2-2g$$

for a closed oriented genus-$g$ worldsheet. Thus the path integral includes

$$e^{-S_\Phi}=e^{-\Phi_0(2-2g)}=g_s^{2g-2}, \qquad \boxed{g_s=e^{\Phi_0}}.$$

This is why the expectation value of the dilaton controls the closed-string perturbation expansion.

For closed oriented worldsheets, the constant dilaton coupling gives the topology weight $g_s^{2g-2}$. A fluctuating dilaton therefore controls both a spacetime scalar and the local string coupling.

The same factor explains the string-frame prefactor $e^{-2\Phi}$ at sphere level. Roughly, the local version of the genus-zero weight $g_s^{-2}=e^{-2\Phi_0}$ is promoted to $e^{-2\Phi(x)}$ when the dilaton varies slowly in spacetime.

Equations of motion at leading order

Varying the string-frame action gives the leading spacetime equations. It is useful to write them in a form that will match the sigma-model beta functions on the next page:

$$\boxed{ R_{\mu\nu}+2\nabla_\mu\nabla_\nu\Phi -\frac14 H_{\mu\lambda\rho}H_\nu{}^{\lambda\rho} +O(\alpha')=0, }$$ $$\boxed{ \nabla^\lambda\left(e^{-2\Phi}H_{\lambda\mu\nu}\right)+O(\alpha')=0, }$$

and

$$\boxed{ 4\nabla^2\Phi-4(\nabla\Phi)^2+R -\frac{1}{12}H^2 -\frac{2(D-26)}{3\alpha'} +O(\alpha')=0. }$$

For $B=0$ and constant $\Phi$ in the critical dimension, these reduce to

$$R_{\mu\nu}+O(\alpha')=0.$$

Thus ordinary vacuum Einstein gravity is the first term in the closed-string low-energy expansion.

Einstein frame

The string-frame metric is natural from the worldsheet point of view. The Einstein-frame metric is natural from the spacetime gravitational point of view. Define

$$\boxed{ G^{\rm E}_{\mu\nu} = e^{-4(\Phi-\Phi_0)/(D-2)}G^{\rm S}_{\mu\nu}. }$$

Equivalently,

$$G^{\rm S}_{\mu\nu} = e^{4(\Phi-\Phi_0)/(D-2)}G^{\rm E}_{\mu\nu}.$$

Under this Weyl rescaling, the prefactor multiplying $R$ is removed. The leading action becomes

$$S_{\rm E} = \frac{1}{2\kappa_D^2}\int d^D x\sqrt{-G_{\rm E}} \left[ R_{\rm E} -\frac{4}{D-2}(\nabla\Phi)^2 -\frac{1}{12}e^{-8(\Phi-\Phi_0)/(D-2)}H^2 +\cdots \right].$$

A canonically normalized dilaton is obtained by defining

$$\varphi=\sqrt{\frac{8}{D-2}}\,(\Phi-\Phi_0),$$

so that the kinetic term inside the brackets becomes $-\frac12(\nabla\varphi)^2$.

String frame is natural for the worldsheet coupling. Einstein frame is natural for comparing with general relativity because the Einstein-Hilbert term has canonical normalization.

The low-energy and weak-coupling expansions

The effective action is organized by two expansions.

First, the worldsheet genus expansion is a string-loop expansion governed by $g_s=e^{\Phi_0}$. For closed oriented strings, genus $g$ contributes with weight

$$g_s^{2g-2}.$$

Second, the derivative expansion is governed by $\alpha'$. It is valid when the curvature radius and field-variation length scales are much larger than the string length

$$\ell_s=\sqrt{\alpha'}.$$

Schematically,

$$S_{\rm eff} = \frac{1}{2\kappa_D^2}\int\sqrt{-G}\,e^{-2\Phi} \left[ \underbrace{R+4(\nabla\Phi)^2-\frac{1}{12}H^2}_{\text{two derivatives}} + \underbrace{\alpha'\mathcal R^2+\alpha'^2\mathcal R^3+\cdots}_{\text{higher derivatives}} \right] + \text{string loops}.$$

Here $\mathcal R$ denotes curvature or field-strength tensors in a schematic way.

The two-derivative action is the first term in the $\alpha'$ expansion. Higher-derivative corrections probe string-scale structure, while higher genus gives quantum string-loop corrections.

This hierarchy is the precise sense in which string theory contains general relativity as a low-energy limit. At distances much larger than $\sqrt{\alpha'}$ and at weak coupling, the leading dynamics is classical gravity coupled to the two-form and the dilaton.

Summary

The leading closed-string spacetime action in string frame is

$$S_{\rm S} = \frac{1}{2\kappa_D^2} \int d^D x\sqrt{-G}\,e^{-2\Phi} \left[ R+4(\nabla\Phi)^2-\frac{1}{12}H^2+\cdots\right].$$

The metric $G_{\mu\nu}$ is the same metric that appears in the worldsheet sigma model. The dilaton expectation value sets

$$g_s=e^{\Phi_0},$$

and the Einstein-frame metric removes the dilaton prefactor from the gravitational term. The next page explains a deeper origin of these equations: they are the vanishing of worldsheet beta functions.

Exercises

Exercise 1: Genus weight from the constant dilaton

For a closed genus-$g$ worldsheet, show that a constant dilaton background gives the perturbative weight $g_s^{2g-2}$.

Solution

For $\Phi=\Phi_0$,

$$S_\Phi=\frac{1}{4\pi}\int\sqrt\gamma R^{(2)}\Phi_0=\Phi_0\chi.$$

For a closed oriented genus-$g$ surface, $\chi=2-2g$. Therefore

$$e^{-S_\Phi}=e^{-\Phi_0(2-2g)}.$$

With $g_s=e^{\Phi_0}$, this becomes

$$e^{-S_\Phi}=g_s^{2g-2}.$$

Exercise 2: The $B$-field equation of motion

Starting from

$$S_B=-\frac{1}{24\kappa_D^2}\int d^D x\sqrt{-G}\,e^{-2\Phi}H_{\mu\nu\rho}H^{\mu\nu\rho},$$

show that varying $B$ gives $\nabla^\lambda(e^{-2\Phi}H_{\lambda\mu\nu})=0$.

Solution

Since $H=dB$, the variation is

$$\delta H_{\lambda\mu\nu}=3\nabla_{[\lambda}\delta B_{\mu\nu]}.$$

The variation of the action is

$$\delta S_B =-\frac{1}{12\kappa_D^2}\int\sqrt{-G}\,e^{-2\Phi}H^{\lambda\mu\nu}\delta H_{\lambda\mu\nu}.$$

Using antisymmetry, this becomes

$$\delta S_B =-\frac{1}{4\kappa_D^2}\int\sqrt{-G}\,e^{-2\Phi}H^{\lambda\mu\nu}\nabla_\lambda\delta B_{\mu\nu}.$$

Integrating by parts gives

$$\delta S_B =\frac{1}{4\kappa_D^2}\int\sqrt{-G}\,\nabla_\lambda\left(e^{-2\Phi}H^{\lambda\mu\nu}\right)\delta B_{\mu\nu}.$$

Since $\delta B_{\mu\nu}$ is arbitrary, the equation of motion is

$$\nabla_\lambda\left(e^{-2\Phi}H^{\lambda\mu\nu}\right)=0.$$

Exercise 3: The Einstein-frame Weyl factor

Let $G^{\rm S}_{\mu\nu}=e^{2\omega}G^{\rm E}_{\mu\nu}$. Find $\omega$ such that

$$\sqrt{-G_{\rm S}}e^{-2\Phi}R(G_{\rm S})$$

has no dilaton prefactor multiplying $R(G_{\rm E})$.

Solution

Under $G^{\rm S}_{\mu\nu}=e^{2\omega}G^{\rm E}_{\mu\nu}$,

$$\sqrt{-G_{\rm S}}=e^{D\omega}\sqrt{-G_{\rm E}}, \qquad R(G_{\rm S})=e^{-2\omega}\left[R(G_{\rm E})+\cdots\right].$$

The prefactor of $R(G_{\rm E})$ is

$$e^{-2\Phi}e^{(D-2)\omega}.$$

To make this independent of the fluctuating dilaton, choose

$$(D-2)\omega=2(\Phi-\Phi_0), \qquad \omega=\frac{2(\Phi-\Phi_0)}{D-2}.$$

Therefore

$$G^{\rm S}_{\mu\nu}=e^{4(\Phi-\Phi_0)/(D-2)}G^{\rm E}_{\mu\nu},$$

or equivalently

$$G^{\rm E}_{\mu\nu}=e^{-4(\Phi-\Phi_0)/(D-2)}G^{\rm S}_{\mu\nu}.$$

Exercise 4: Flat space as a solution

Set $B=0$, $\Phi=\Phi_0$ constant, and $D=26$. Show that the leading equations of motion reduce to $R_{\mu\nu}=0$.

Solution

With $B=0$, $H=0$. With constant $\Phi$, all derivatives of $\Phi$ vanish. In $D=26$, the central-charge-deficit term is zero. The metric equation becomes

$$R_{\mu\nu}+O(\alpha')=0.$$

At leading order in $\alpha'$, this is simply $R_{\mu\nu}=0$. Minkowski space is the simplest solution.