Worldsheet Symmetries, Boundary Conditions, and Conformal Gauge

The Polyakov action introduced an independent worldsheet metric $h_{\alpha\beta}$. At first sight this looks like we have enlarged the theory: instead of the embedding fields $X^\mu(\tau,\sigma)$ alone, we now also integrate or vary over a two-dimensional metric. The point of the Polyakov formulation is that the extra fields are pure gauge, up to subtle global information that will become important in string perturbation theory.

In flat target space the Lorentzian Polyakov action is

$$S_P = -\frac{1}{4\pi\alpha'} \int_\Sigma d^2\sigma\,\sqrt{-h}\,h^{\alpha\beta} \partial_\alpha X^\mu\partial_\beta X_\mu.$$

Equivalently, with $T=1/(2\pi\alpha')$,

$$S_P = -\frac{T}{2} \int_\Sigma d^2\sigma\,\sqrt{-h}\,h^{\alpha\beta} \partial_\alpha X\cdot\partial_\beta X.$$

This page explains why one can locally choose the simple conformal gauge

$$h_{\alpha\beta}=\eta_{\alpha\beta} =\begin{pmatrix}-1&0\\0&1\end{pmatrix},$$

and what equations remain after doing so. The most important lesson is this: even after setting $h_{\alpha\beta}$ to the flat metric, one must still impose the metric equation of motion

$$T_{\alpha\beta}=0.$$

These are the classical Virasoro constraints.

Conventions

Worldsheet coordinates are

$$\sigma^\alpha=(\tau,\sigma), \qquad \partial_\tau X^\mu=\dot X^\mu, \qquad \partial_\sigma X^\mu=X^{\prime\mu}.$$

The target-space metric is mostly plus,

$$\eta_{\mu\nu}=\operatorname{diag}(-,+,\ldots,+),$$

and target-space contractions are denoted by $A\cdot B=\eta_{\mu\nu}A^\mu B^\nu$.

The three classical symmetries

The Polyakov action has three basic symmetries. Two are gauge symmetries of the worldsheet description; the third is an ordinary global spacetime symmetry.

Worldsheet diffeomorphisms

A change of worldsheet coordinates

$$\sigma^\alpha\mapsto \widetilde\sigma^\alpha(\sigma)$$

is a gauge redundancy. The embedding coordinates are worldsheet scalars,

$$\widetilde X^\mu(\widetilde\sigma)=X^\mu(\sigma),$$

while the worldsheet metric transforms as a tensor,

$$\widetilde h_{\alpha\beta}(\widetilde\sigma) = \frac{\partial \sigma^\rho}{\partial \widetilde\sigma^\alpha} \frac{\partial \sigma^\lambda}{\partial \widetilde\sigma^\beta} h_{\rho\lambda}(\sigma).$$

Because $d^2\sigma\sqrt{-h}$ is invariant and $h^{\alpha\beta}\partial_\alpha X\cdot\partial_\beta X$ is a scalar, $S_P$ is invariant. This is the string analogue of reparametrization invariance of the relativistic particle.

In infinitesimal active notation, with a vector field $\xi^\alpha(\sigma)$,

$$\delta_\xi X^\mu=-\xi^\alpha\partial_\alpha X^\mu,$$

and

$$\delta_\xi h_{\alpha\beta} =-\nabla_\alpha\xi_\beta-\nabla_\beta\xi_\alpha.$$

The sign convention is not important; what matters is that two arbitrary functions $\xi^\tau$ and $\xi^\sigma$ are available for gauge fixing.

Weyl transformations

The special feature of the string worldsheet is that it is two-dimensional. In two dimensions, the Polyakov action is invariant under local rescalings of the metric,

$$h_{\alpha\beta}(\sigma) \mapsto e^{2\omega(\sigma)}h_{\alpha\beta}(\sigma).$$

Indeed,

$$h^{\alpha\beta}\mapsto e^{-2\omega}h^{\alpha\beta}, \qquad \sqrt{-h}\mapsto e^{2\omega}\sqrt{-h},$$

so the two factors cancel. This is Weyl invariance.

Weyl invariance has a crucial consequence: the stress tensor obtained by varying the metric is classically traceless,

$$T^\alpha{}_{\alpha}=0.$$

Quantum mechanically this statement is subtle. Weyl invariance may be anomalous, and the requirement that the anomaly vanish will eventually lead to the critical dimension of the bosonic string, $D=26$.

Global spacetime Poincare symmetry

In flat target space the action is also invariant under

$$X^\mu\mapsto \Lambda^\mu{}_\nu X^\nu+a^\mu, \qquad \Lambda^T\eta\Lambda=\eta.$$

This is not a worldsheet gauge symmetry. It is the usual global spacetime Poincare symmetry. Its Noether currents give the conserved spacetime momentum and angular momentum of a string; those currents will be useful in the next lecture note when we discuss rotating string solutions and Regge trajectories.

The Polyakov action has two local worldsheet gauge symmetries, diffeomorphism and Weyl invariance, plus global target-space Poincare invariance.

Why conformal gauge works locally

A symmetric two-dimensional metric has three independent local components:

$$h_{\alpha\beta}=h_{\beta\alpha} \quad\Longrightarrow\quad (h_{\tau\tau},h_{\tau\sigma},h_{\sigma\sigma}).$$

Worldsheet diffeomorphisms provide two arbitrary local functions, and Weyl invariance provides one more. Thus, locally, the gauge freedom is large enough to remove all three components of $h_{\alpha\beta}$.

More geometrically, every two-dimensional metric is locally conformally flat. Therefore we can write

$$h_{\alpha\beta}(\tau,\sigma) =e^{2\phi(\tau,\sigma)}\eta_{\alpha\beta}$$

in Lorentzian signature, or

$$h_{\alpha\beta}(\sigma^1,\sigma^2) =e^{2\phi(\sigma^1,\sigma^2)}\delta_{\alpha\beta}$$

in Euclidean signature. Weyl invariance then removes the conformal factor $e^{2\phi}$.

So, locally, we may choose

$$h_{\alpha\beta}=\eta_{\alpha\beta}.$$

This is conformal gauge.

A two-dimensional metric has three local components. Two diffeomorphism functions and one Weyl function locally reduce it to the flat metric. Global moduli are not shown in this local counting.

Local gauge fixing is not the whole story

Conformal gauge does not say that every worldsheet is globally equivalent to the flat plane. On a cylinder, torus, or higher-genus surface, there can be global moduli that are not removed by diffeomorphisms and Weyl transformations. These moduli are precisely the integration variables of string perturbation theory. For the classical equations on a strip or cylinder, however, conformal gauge is the right starting point.

Residual conformal transformations

Conformal gauge does not completely fix the gauge. Some coordinate transformations change the flat metric only by a Weyl factor, and the Weyl factor can be undone by a Weyl transformation.

Introduce Lorentzian light-cone coordinates on the worldsheet,

$$\sigma^+=\tau+\sigma, \qquad \sigma^-=\tau-\sigma.$$

Then

$$ds^2=-d\tau^2+d\sigma^2 =-d\sigma^+d\sigma^-.$$

The transformations

$$\sigma^+\mapsto f(\sigma^+), \qquad \sigma^-\mapsto g(\sigma^-)$$

give

$$ds^2\mapsto -f'(\sigma^+)g'(\sigma^-)\,d\sigma^+d\sigma^-.$$

This is just the same flat metric multiplied by a local scale factor, so Weyl invariance restores the gauge condition. These residual symmetries are the classical origin of left- and right-moving conformal transformations.

In Euclidean signature one often uses

$$z=\sigma^1+i\sigma^2, \qquad \bar z=\sigma^1-i\sigma^2,$$

and residual conformal transformations are holomorphic and antiholomorphic maps,

$$z\mapsto f(z), \qquad \bar z\mapsto \bar f(\bar z).$$

This is the bridge from the Polyakov action to two-dimensional conformal field theory.

The conformal-gauge action

In conformal gauge the action becomes a free two-dimensional field theory,

$$S_{\rm conf} = -\frac{1}{4\pi\alpha'} \int d\tau d\sigma\,\eta^{\alpha\beta} \partial_\alpha X\cdot\partial_\beta X.$$

Since $\eta^{\alpha\beta}=\operatorname{diag}(-1,1)$,

$$S_{\rm conf} = \frac{1}{4\pi\alpha'} \int d\tau d\sigma\, \left(\dot X^2-X^{\prime 2}\right).$$

The Euler-Lagrange equation is simply the two-dimensional wave equation,

$$(\partial_\tau^2-\partial_\sigma^2)X^\mu=0.$$

Equivalently,

$$\partial_+\partial_-X^\mu=0, \qquad \partial_\pm\equiv \frac{1}{2}(\partial_\tau\pm\partial_\sigma).$$

Hence every classical solution locally splits into left- and right-moving pieces,

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

Boundary conditions and periodicity determine how these pieces are related.

After Wick rotation $\tau=-i\tau_E$, the Euclidean action is positive for the non-time target coordinates,

$$S_E= \frac{1}{4\pi\alpha'} \int d^2\sigma\, \partial_a X^\mu\partial_a X_\mu,$$

with the usual caveat that the target-space time coordinate carries the Lorentzian minus sign before analytic continuation. The Euclidean action is the one used in conformal field theory path integrals.

Boundary terms and open strings

The conformal-gauge action is free in the bulk, but the boundary variation is highly meaningful. Varying

$$S_{\rm conf} = \frac{1}{4\pi\alpha'} \int d\tau d\sigma\, (\dot X^2-X^{\prime 2})$$

gives

$$\delta S_{\rm conf} = -\frac{1}{2\pi\alpha'} \int d\tau d\sigma\, \delta X\cdot(\partial_\tau^2-\partial_\sigma^2)X - \frac{1}{2\pi\alpha'} \int d\tau\,[\delta X\cdot X']_{\sigma=0}^{\sigma=\pi},$$

up to possible endpoint terms in $\tau$, which vanish for fixed initial and final configurations.

The bulk term gives the wave equation. The boundary term gives the boundary condition. For a closed string there is no boundary; periodicity makes the endpoint contribution vanish. For an open string, at each endpoint and in each target-space direction, one can impose either of the following conditions.

Neumann boundary condition

If the endpoint is free to move in the $\mu$ direction, then $\delta X^\mu$ is arbitrary at the boundary. The boundary term vanishes only if

$$X^{\prime\mu}=0 \qquad \text{at }\sigma=0,\pi.$$

This is a Neumann boundary condition. It says that no momentum flows off the end of the string in that target-space direction.

Dirichlet boundary condition

Alternatively, we may hold the endpoint fixed in the $\mu$ direction. Then

$$\delta X^\mu=0 \qquad \text{at }\sigma=0,\pi.$$

Equivalently, the endpoint obeys

$$X^\mu(\tau,0)=y_0^\mu, \qquad X^\mu(\tau,\pi)=y_\pi^\mu,$$

or a suitable generalization if the two endpoints lie on different hypersurfaces. This is a Dirichlet boundary condition.

Neumann boundary conditions allow the endpoint to move freely along a direction and impose $X'=0$. Dirichlet boundary conditions fix the endpoint position and impose $\delta X=0$.

D-branes from boundary conditions

A particularly important mixed boundary condition is the following. Let

$$a=0,1,\ldots,p, \qquad i=p+1,\ldots,D-1.$$

Impose Neumann boundary conditions along the $p+1$ directions $X^a$,

$$\partial_\sigma X^a=0 \qquad (a=0,1,\ldots,p),$$

and Dirichlet boundary conditions along the remaining transverse directions,

$$X^i=y^i \qquad (i=p+1,\ldots,D-1).$$

The open-string endpoint is then confined to a $(p+1)$-dimensional spacetime hypersurface, or equivalently a $p$-dimensional spatial object. This object is a D$p$-brane. The letter “D” stands for Dirichlet.

A D$p$-brane is the hypersurface swept out by directions with Neumann boundary conditions. The open-string endpoint is pinned in the transverse Dirichlet directions.

At this point D-branes are simply allowed boundary conditions of the open-string variational principle. Later they become dynamical objects carrying gauge fields, scalar fields, tension, and Ramond-Ramond charge.

Counting directions

A D$p$-brane has $p$ spatial directions plus time. Thus it has $p+1$ Neumann directions and $D-p-1$ Dirichlet directions.

Stress tensor constraints in conformal gauge

Gauge fixing $h_{\alpha\beta}=\eta_{\alpha\beta}$ does not erase the equation obtained by varying $h_{\alpha\beta}$. The worldsheet stress tensor is

$$T_{\alpha\beta} =\partial_\alpha X\cdot\partial_\beta X -\frac{1}{2}h_{\alpha\beta} h^{\rho\lambda} \partial_\rho X\cdot\partial_\lambda X.$$

The metric equation of motion is

$$T_{\alpha\beta}=0.$$

In conformal gauge this becomes

$$T_{\tau\tau}=T_{\sigma\sigma} =\frac{1}{2}\left(\dot X^2+X^{\prime 2}\right), \qquad T_{\tau\sigma}=\dot X\cdot X'.$$

Therefore the constraints are

$$\dot X^2+X^{\prime 2}=0, \qquad \dot X\cdot X'=0.$$

Equivalently,

$$(\dot X+X')^2=0, \qquad (\dot X-X')^2=0.$$

In light-cone worldsheet coordinates this is especially simple:

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

The normalization of $T_{\pm\pm}$ varies slightly between classical and CFT conventions; the constraint $T_{\pm\pm}=0$ is convention-independent.

In conformal gauge the residual left- and right-moving directions are $\sigma^+=\tau+\sigma$ and $\sigma^-=\tau-\sigma$. The metric equation of motion imposes $T_{++}=T_{--}=0$.

These constraints remove the unphysical longitudinal oscillations. Classically, together with residual conformal transformations, they leave $D-2$ transverse physical degrees of freedom. Quantum mechanically, their mode expansion becomes the Virasoro constraints on string states.

Summary

The Polyakov formulation is powerful because it converts the square-root Nambu-Goto action into a free field theory plus constraints:

$$S_{\rm conf} = \frac{1}{4\pi\alpha'} \int d\tau d\sigma\, (\dot X^2-X^{\prime 2}),$$ $$(\partial_\tau^2-\partial_\sigma^2)X^\mu=0,$$ $$T_{++}=T_{--}=0.$$

For open strings the variational principle also requires boundary conditions. Neumann directions describe directions along which the endpoint is free to move; Dirichlet directions describe directions transverse to a D-brane.

The next lecture note uses these equations and constraints to analyze explicit classical string solutions, especially the rotating open string that gives the classical Regge relation between spin and mass.

Exercises

Exercise 1. Weyl invariance is special to two dimensions

Consider the generalized action in $d$ worldsheet dimensions,

$$S_d=-\frac{T}{2}\int d^d\sigma\sqrt{-h}\,h^{\alpha\beta} \partial_\alpha X\cdot\partial_\beta X.$$

Under $h_{\alpha\beta}\mapsto e^{2\omega}h_{\alpha\beta}$, determine how the integrand scales. For what value of $d$ is the action Weyl invariant?

Solution

In $d$ dimensions,

$$\sqrt{-h}\mapsto e^{d\omega}\sqrt{-h}, \qquad h^{\alpha\beta}\mapsto e^{-2\omega}h^{\alpha\beta}.$$

Thus

$$\sqrt{-h}\,h^{\alpha\beta} \partial_\alpha X\cdot\partial_\beta X \mapsto e^{(d-2)\omega} \sqrt{-h}\,h^{\alpha\beta} \partial_\alpha X\cdot\partial_\beta X.$$

The action is Weyl invariant only when

$$d-2=0,$$

so $d=2$.

Exercise 2. Derive the open-string boundary term

Starting from

$$S_{\rm conf}= \frac{1}{4\pi\alpha'}\int d\tau d\sigma\, (\dot X^2-X^{\prime 2}),$$

show that the boundary term at $\sigma=0,\pi$ is proportional to

$$[\delta X\cdot X']_{\sigma=0}^{\sigma=\pi}.$$

Explain why this leads to Neumann or Dirichlet boundary conditions.

Solution

Varying the action gives

$$\delta S_{\rm conf} =\frac{1}{2\pi\alpha'} \int d\tau d\sigma\, (\dot X\cdot\delta\dot X-X'\cdot\delta X').$$

Integrating by parts in $\tau$ and $\sigma$ gives

$$\delta S_{\rm conf} =-\frac{1}{2\pi\alpha'} \int d\tau d\sigma\, \delta X\cdot(\partial_\tau^2-\partial_\sigma^2)X - \frac{1}{2\pi\alpha'} \int d\tau\,[\delta X\cdot X']_{0}^{\pi},$$

where endpoint terms in $\tau$ vanish when the initial and final configurations are fixed.

For the boundary term to vanish, one may impose

$$X'=0$$

at the boundary, which is Neumann, or

$$\delta X=0,$$

which is Dirichlet. These choices may be made independently in each target-space direction.

Exercise 3. Residual conformal transformations

Show that the transformations

$$\sigma^+\mapsto f(\sigma^+), \qquad \sigma^-\mapsto g(\sigma^-)$$

preserve conformal gauge up to a Weyl rescaling.

Solution

In conformal gauge,

$$ds^2=-d\sigma^+d\sigma^-.$$

After the transformation,

$$d\sigma^+\mapsto f'(\sigma^+)d\sigma^+, \qquad d\sigma^-\mapsto g'(\sigma^-)d\sigma^-.$$

Therefore

$$ds^2\mapsto -f'(\sigma^+)g'(\sigma^-)d\sigma^+d\sigma^-.$$

This differs from the original metric only by the local factor

$$e^{2\omega}=f'(\sigma^+)g'(\sigma^-).$$

A Weyl transformation removes this factor, so the combined transformation preserves conformal gauge.

Exercise 4. Equivalent forms of the constraints

Prove that

$$\dot X^2+X^{\prime 2}=0, \qquad \dot X\cdot X'=0$$

are equivalent to

$$(\dot X+X')^2=0, \qquad (\dot X-X')^2=0.$$

Solution

Compute

$$(\dot X+X')^2 =\dot X^2+X^{\prime 2}+2\dot X\cdot X',$$

and

$$(\dot X-X')^2 =\dot X^2+X^{\prime 2}-2\dot X\cdot X'.$$

If

$$\dot X^2+X^{\prime 2}=0, \qquad \dot X\cdot X'=0,$$

then both squared expressions vanish.

Conversely, if both squared expressions vanish, adding them gives

$$2(\dot X^2+X^{\prime 2})=0,$$

and subtracting them gives

$$4\dot X\cdot X'=0.$$

Thus the two forms are equivalent.

Exercise 5. D$p$-brane dimensions

In $D$-dimensional flat spacetime, an open string has Neumann boundary conditions in $X^0,X^1,\ldots,X^p$ and Dirichlet boundary conditions in the remaining directions. How many spatial dimensions does the brane have, and how many transverse scalar fluctuations should one expect on its worldvolume?

Solution

The Neumann directions include time $X^0$ and $p$ spatial directions $X^1,\ldots,X^p$. Hence the brane is a D$p$-brane: it has $p$ spatial dimensions and a $(p+1)$-dimensional worldvolume.

The Dirichlet directions are

$$X^{p+1},\ldots,X^{D-1},$$

so their number is

$$D-1-p=D-p-1.$$

When the D-brane becomes dynamical, its transverse position can fluctuate in each of these directions. Thus one expects $D-p-1$ scalar fields on the brane worldvolume.