Open-String T-Duality and D-Branes

Closed-string T-duality exchanges momentum and winding. Open-string T-duality does something even more geometric: it changes the boundary condition of the string endpoint.

A coordinate with Neumann boundary conditions becomes, after T-duality, a coordinate with Dirichlet boundary conditions. Thus an open string whose endpoint was free to move in a compact direction is reinterpreted as a string ending on a hypersurface in the dual geometry. That hypersurface is a D-brane.

This page explains the mechanism and the first physical consequences: Wilson lines become brane positions, separated branes give massive stretched strings, coincident branes give enhanced gauge symmetry, and transverse open-string polarizations become scalar fields describing brane motion.

Open strings on a compact Neumann circle

Let $Y\sim Y+2\pi R$ be a compact spatial direction with Neumann boundary conditions at the open-string endpoints:

$$\partial_\sigma Y\big|_{\sigma=0,\pi}=0.$$

For $0\leq\sigma\leq\pi$, the standard open-string mode expansion is

$$Y(\tau,\sigma) = y+2\alpha' p_Y\tau +i\sqrt{2\alpha'}\sum_{m\neq 0}\frac{\alpha_m^Y}{m}e^{-im\tau}\cos(m\sigma),$$

with quantized momentum

$$p_Y=\frac{n}{R}.$$

Unlike a closed string, a single open string with free endpoints has no conserved winding number in a Neumann direction. The string can unwind through its endpoints. Nevertheless, the left/right decomposition still exists locally, and we can define the dual coordinate

$$\widetilde Y=Y_L-Y_R.$$

The local T-duality relations are, up to a sign convention,

$$\partial_\tau\widetilde Y=\partial_\sigma Y, \qquad \partial_\sigma\widetilde Y=\partial_\tau Y.$$

Neumann becomes Dirichlet

Apply the first relation at the boundary. If $Y$ obeys Neumann boundary conditions,

$$\partial_\sigma Y\big|_{\partial\Sigma}=0,$$

then

$$\partial_\tau\widetilde Y\big|_{\partial\Sigma}=0.$$

The endpoint value of $\widetilde Y$ is independent of worldsheet time. That is a Dirichlet condition:

$$\boxed{ \text{Neumann for }Y \quad\Longleftrightarrow\quad \text{Dirichlet for }\widetilde Y. }$$

Open-string T-duality changes boundary conditions. A free endpoint in the original coordinate $Y$ becomes a fixed endpoint in the dual coordinate $\widetilde Y$.

This is the first appearance of D-branes. Start with an open bosonic string whose endpoints are free in all $25$ spatial directions; this is a space-filling D25-brane. T-dualizing one spatial direction produces endpoints fixed in the dual direction, so the string ends on a D24-brane. More generally, T-dualizing $q$ independent Neumann directions lowers the brane dimension by $q$.

For Type II superstrings, the corresponding statement starts from a space-filling D9-brane and gives

$$\text{D}9\xrightarrow{\text{T-dualize }q\text{ directions}}\text{D}(9-q).$$

The dual open-string expansion

Using

$$\partial_\sigma\widetilde Y=\partial_\tau Y,$$

and the Neumann expansion for $Y$, the dual coordinate has a term linear in $\sigma$:

$$\widetilde Y(\tau,\sigma) = \widetilde y+2\alpha' p_Y\sigma +\sqrt{2\alpha'}\sum_{m\neq 0}\frac{\alpha_m^Y}{m}e^{-im\tau}\sin(m\sigma),$$

up to an irrelevant constant and a possible overall sign. Since $p_Y=n/R$,

$$\widetilde Y(\tau,\pi)-\widetilde Y(\tau,0) =2\pi\alpha'\frac{n}{R} =2\pi nR', \qquad R'=\frac{\alpha'}{R}.$$

Thus the original open-string momentum quantum number becomes winding in the dual circle. But it is not winding of a closed string; rather, it says that the open string may stretch between the same D-brane after going around the dual circle $n$ times.

The open-string momentum $n/R$ becomes endpoint separation $2\pi nR'$ in the dual coordinate.

D-branes as hypersurfaces

A D$p$-brane is a $(p+1)$-dimensional hypersurface in spacetime, including time. Open strings can end on it. The name means that the transverse coordinates of the string endpoint obey Dirichlet boundary conditions.

For a flat static D$p$-brane in $D$ spacetime dimensions,

$$\begin{array}{ll} \text{Neumann:} & X^a,\quad a=0,\ldots,p,\\ \text{Dirichlet:} & X^i,\quad i=p+1,\ldots,D-1. \end{array}$$

At the open-string boundary,

$$\partial_\sigma X^a=0, \qquad \delta X^i=0 \quad\text{or equivalently}\quad \partial_\tau X^i=0.$$

A D$p$-brane has Neumann boundary conditions along its worldvolume and Dirichlet boundary conditions in the transverse directions.

This language may sound like D-branes are fixed external objects. They are not. The open strings ending on a D-brane contain massless modes that become dynamical fields on the brane. We will make this more precise near the end of this page and then derive the nonlinear effective action on the next page.

Wilson lines become brane positions

Now add Chan-Paton labels. Suppose the compact direction is Neumann before duality, and the open string endpoints carry labels $i,j=1,\ldots,N$. A constant gauge field $A_Y$ around the compact circle has zero field strength, but its Wilson line is physical:

$$\exp\left(i\oint A_Y\,dY\right)=e^{i\theta}.$$

For $U(N)$, choose a diagonal Wilson line,

$$\theta=\operatorname{diag}(\theta_1,\ldots,\theta_N).$$

An open string beginning on label $i$ and ending on label $j$ has shifted compact momentum

$$\boxed{ p_Y=\frac{1}{R}\left(n+\frac{\theta_i-\theta_j}{2\pi}\right). }$$

Therefore the bosonic open-string mass formula becomes

$$M^2= \frac{1}{R^2}\left(n+\frac{\theta_i-\theta_j}{2\pi}\right)^2 +\frac{1}{\alpha'}(N_{\rm osc}-1).$$

The same momentum shift appears in the superstring, with the intercept replaced by the appropriate NS or R value.

Under T-duality,

$$R'=\frac{\alpha'}{R},$$

and the Wilson-line eigenvalues become positions on the dual circle:

$$\boxed{ \widetilde Y_i=R'\theta_i \qquad\text{mod }2\pi R'. }$$

The mass formula becomes

$$\boxed{ M^2= \left( \frac{\widetilde Y_i-\widetilde Y_j+2\pi nR'}{2\pi\alpha'} \right)^2 +\frac{1}{\alpha'}(N_{\rm osc}-1). }$$

The first term is the classical mass contribution from stretching a string of length

$$\ell_{ij}^{(n)}= \widetilde Y_i-\widetilde Y_j+2\pi nR'$$

between two branes, allowing the string to wrap the dual circle $n$ times before ending.

A diagonal Wilson line in the original description becomes a set of D-brane positions in the dual description.

Gauge symmetry from coincident branes

The Wilson-line interpretation immediately explains symmetry breaking and enhancement.

If the eigenvalues $\theta_i$ are all distinct, the dual branes sit at different points. Strings stretching between different branes have nonzero length and are massive. Only strings beginning and ending on the same brane remain massless, so the gauge group is broken to the subgroup preserving each stack.

For example,

$$U(N)\longrightarrow U(N_1)\times U(N_2)\times\cdots$$

if the branes split into stacks of multiplicities $N_1,N_2,\ldots$.

If all branes coincide, then strings with arbitrary Chan-Paton labels can be massless. The off-diagonal open strings restore the nonabelian gauge symmetry.

Separating branes Higgses the gauge symmetry. Coincident branes make off-diagonal strings massless and restore the nonabelian gauge group.

This is one of the most useful pictures in string theory: nonabelian gauge symmetry is carried by open strings whose endpoints have Chan-Paton labels, and the Higgs mechanism is literally the geometric separation of branes.

Worldvolume fields from open-string polarizations

For a D$p$-brane, the massless open-string state splits according to whether its polarization is tangent or transverse to the brane.

The tangent polarizations

$$\alpha_{-1}^{a}|0;k\rangle, \qquad a=0,\ldots,p,$$

produce a gauge field on the brane:

$$A_a(x^0,\ldots,x^p).$$

The transverse polarizations

$$\alpha_{-1}^{i}|0;k\rangle, \qquad i=p+1,\ldots,D-1,$$

produce scalar fields on the brane:

$$\Phi^i(x^0,\ldots,x^p).$$

Geometrically, these scalars describe transverse fluctuations of the D-brane:

$$X^i(x^a)=x_0^i+2\pi\alpha'\Phi^i(x^a),$$

up to a conventional normalization.

A massless open-string polarization tangent to the brane gives a worldvolume gauge field $A_a$; a transverse polarization gives a scalar $\Phi^i$ describing brane motion.

Thus D-branes are not merely places where strings end. They are dynamical objects with their own low-energy field theory. The next page turns this into the Dirac-Born-Infeld action and explains why D-branes carry R-R charges.

Exercises

Exercise 1. Neumann-to-Dirichlet boundary conditions

Use

$$\partial_\tau\widetilde Y=\partial_\sigma Y, \qquad \partial_\sigma\widetilde Y=\partial_\tau Y$$

to show that Neumann boundary conditions for $Y$ imply Dirichlet boundary conditions for $\widetilde Y$.

Solution

At an open-string boundary, Neumann boundary conditions mean

$$\partial_\sigma Y\big|_{\partial\Sigma}=0.$$

Using the first T-duality relation,

$$\partial_\tau\widetilde Y\big|_{\partial\Sigma} = \partial_\sigma Y\big|_{\partial\Sigma}=0.$$

Therefore $\widetilde Y$ is constant along the boundary as a function of $\tau$. This is a Dirichlet boundary condition: the endpoint is fixed in the dual coordinate.

Exercise 2. Endpoint separation in the dual circle

Starting from the Neumann expansion

$$Y=y+2\alpha'\frac{n}{R}\tau+\cdots,$$

show that the dual coordinate satisfies

$$\widetilde Y(\pi)-\widetilde Y(0)=2\pi n\frac{\alpha'}{R}.$$

Solution

The T-duality relation gives

$$\partial_\sigma\widetilde Y=\partial_\tau Y.$$

The zero-mode part of $\partial_\tau Y$ is

$$\partial_\tau Y=2\alpha'\frac{n}{R}.$$

Therefore

$$\widetilde Y(\pi)-\widetilde Y(0) =\int_0^\pi d\sigma\,\partial_\sigma\widetilde Y =\int_0^\pi d\sigma\,2\alpha'\frac{n}{R} =2\pi n\frac{\alpha'}{R}.$$

Since $R'=\alpha'/R$, this is $2\pi nR'$.

Exercise 3. Wilson-line masses as stretched-string masses

An open string with endpoint labels $i,j$ has shifted compact momentum

$$p_Y=\frac{1}{R}\left(n+\frac{\theta_i-\theta_j}{2\pi}\right).$$

Show that after T-duality this equals the stretched-string mass contribution for branes at $\widetilde Y_i=R'\theta_i$, with $R'=\alpha'/R$.

Solution

The dual separation, including $n$ windings around the dual circle, is

$$\Delta\widetilde Y_{ij}^{(n)} = \widetilde Y_i-\widetilde Y_j+2\pi nR' =R'(\theta_i-\theta_j+2\pi n).$$

The mass contribution of a stretched string is

$$M_{\rm stretch}^2 = \left(\frac{\Delta\widetilde Y_{ij}^{(n)}}{2\pi\alpha'}\right)^2.$$

Using $R'=\alpha'/R$ gives

$$M_{\rm stretch}^2 = \left[\frac{(\alpha'/R)(\theta_i-\theta_j+2\pi n)}{2\pi\alpha'}\right]^2 = \frac{1}{R^2}\left(n+\frac{\theta_i-\theta_j}{2\pi}\right)^2.$$

This is exactly the shifted momentum contribution.

Exercise 4. Gauge enhancement from coincident branes

Explain why $N$ coincident D-branes give a nonabelian gauge group, whereas separated branes give only the gauge group of the individual stacks.

Solution

Open strings carry Chan-Paton labels $|i,j\rangle$. If the branes are separated, a string with $i\neq j$ has nonzero length and therefore a positive stretching contribution to $M^2$. These off-diagonal states are massive. Only strings with both endpoints on the same stack remain massless, so the gauge group is the product of the groups associated with the stacks.

If all branes coincide, the stretching length for every pair $i,j$ can vanish. Then the off-diagonal states also become massless. Together with the diagonal states they fill out the adjoint representation of the nonabelian gauge group.

Exercise 5. Why transverse polarizations are scalar fields

For a D$p$-brane, why does a transverse massless polarization produce a scalar field on the brane rather than a vector field?

Solution

The low-energy open-string field propagates only along the brane coordinates $x^a$, $a=0,\ldots,p$. A tangent polarization carries an index $a$, so it transforms as a worldvolume vector $A_a(x)$.

A transverse polarization carries an index $i=p+1,\ldots,D-1$. This is not a vector index under Lorentz transformations along the brane. From the brane worldvolume viewpoint it is a scalar label telling us which transverse direction is being displaced. Thus the field is $\Phi^i(x)$, a scalar on the brane and a vector under rotations of the transverse space.