Open Strings, T-Duality, and the Emergence of D-Branes

Closed-string T-duality says that a circle of radius $R$ is physically equivalent to a circle of radius $R'=\alpha'/R$, with momentum and winding exchanged. For open strings the same operation has a more dramatic interpretation. An open string with Neumann boundary conditions has quantized momentum along a compact circle, but it has no closed-string winding number. After T-duality this momentum becomes a physical separation of endpoints in the dual coordinate. The endpoints are no longer free to move in that direction: they lie on a hypersurface. That hypersurface is a D-brane.

The core statement is

$$\text{T-duality along an open-string Neumann direction} \quad\Longrightarrow\quad \text{Dirichlet boundary condition in the dual direction}.$$

This is the shortest conceptual path from perturbative open strings to D-branes. The letter $D$ stands for Dirichlet. We will first phrase the discussion in the bosonic string, where the boundary-condition argument is completely transparent. The same kinematics applies in superstring theory; the GSO projection, spacetime fermions, Ramond—Ramond charges, and type IIA/IIB chirality will become important later.

Closed-string reminder

Let $Y$ be a compact target-space coordinate,

$$Y\sim Y+2\pi R.$$

For a closed string, the left- and right-moving momenta are

$$p_L={n\over R}+{wR\over \alpha'}, \qquad p_R={n\over R}-{wR\over \alpha'}, \qquad n,w\in\mathbb Z.$$

Writing

$$Y(z,\bar z)=Y_L(z)+Y_R(\bar z),$$

T-duality is the operation

$$Y_L\longrightarrow Y_L, \qquad Y_R\longrightarrow -Y_R.$$

Equivalently, the dual coordinate is

$$Y'=Y_L-Y_R, \qquad R'={\alpha'\over R},$$

and the quantum numbers are exchanged:

$$n\longleftrightarrow w.$$

In Lorentzian worldsheet coordinates $(\tau,\sigma)$, the local T-duality equations are

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

These equations are the most useful form for open strings, because open strings are defined by boundary conditions.

Neumann boundary conditions become Dirichlet boundary conditions

Take an open string with $0\leq \sigma\leq \pi$ and Neumann boundary conditions along $Y$:

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

Using the first T-duality equation, the boundary condition becomes

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

Thus the endpoint value of $Y'$ is independent of worldsheet time. In other words,

$$Y'(\tau,0)=\text{constant}, \qquad Y'(\tau,\pi)=\text{constant}.$$

That is a Dirichlet boundary condition. Conversely, a Dirichlet condition in $Y$ becomes a Neumann condition in $Y'$. Therefore

$$\boxed{\text{T-duality along a brane direction maps }Dp\to D(p-1)},$$

while

$$\boxed{\text{T-duality along a transverse direction maps }Dp\to D(p+1)}.$$

For open strings, T-duality is visible directly at the boundary. A Neumann endpoint, free to move around the original circle, becomes an endpoint fixed at a position in the T-dual circle.

This should already feel surprising. Closed-string T-duality changes the radius of a compact dimension. Open-string T-duality changes the dimension of the locus on which endpoints can move.

The fixed locus is not an externally added wall. Perturbative open strings contain the degrees of freedom that move it. In later lectures these collective coordinates will become scalar fields on the brane worldvolume, and their nonlinear dynamics will be described by the Dirac—Born—Infeld action.

Open strings on a compact Neumann circle

For an open string with Neumann boundary conditions along $Y$, the mode expansion is

$$Y(\tau,\sigma)=y+2\alpha' p\,\tau +i\sqrt{2\alpha'}\sum_{m\neq0}{\alpha_m\over m}e^{-im\tau}\cos m\sigma.$$

Compactness quantizes the center-of-mass momentum:

$$p={n\over R}, \qquad n\in\mathbb Z.$$

A dual coordinate satisfying

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

is

$$Y'(\tau,\sigma)=y'+2\alpha' p\,\sigma +\sqrt{2\alpha'}\sum_{m\neq0}{\alpha_m\over m}e^{-im\tau}\sin m\sigma.$$

At the endpoints the sine term vanishes. Hence

$$Y'(\tau,0)=y',$$

and

$$Y'(\tau,\pi)=y'+2\pi\alpha' p =y'+2\pi n{\alpha'\over R} =y'+2\pi nR'.$$

Therefore

$$\Delta Y'=Y'(\tau,\pi)-Y'(\tau,0)=2\pi nR'.$$

The original open-string momentum quantum number $n$ becomes the number of times the dual string stretches to an image brane on the dual circle. The phrase “momentum becomes winding” is still correct, but for open strings it means winding-image separation rather than closed-string winding around a loop.

The small-radius limit

Let the original radius shrink, $R\to0$. In the original description, nonzero momentum modes have masses of order $|n|/R$ and decouple. The low-energy open-string spectrum seems to forget the compact direction.

The dual description explains what happened geometrically. The dual radius grows,

$$R'={\alpha'\over R}\to\infty,$$

but the endpoints are fixed in $Y'$. The open strings propagate only along the directions parallel to the fixed hypersurface. In the bosonic string, T-dualizing one compact Neumann direction of a space-filling D25-brane gives a D24-brane localized in the dual coordinate. T-dualizing $k$ such directions gives a D$(25-k)$-brane.

Thus open-string T-duality does not merely say that a tiny circle is equivalent to a large circle. It says that the open strings on the large dual circle are confined to a lower-dimensional brane.

Wilson lines become brane positions

A constant gauge field along a noncompact direction is locally pure gauge. A constant gauge field along a compact circle can be physical, because its holonomy is gauge invariant. For a $U(1)$ gauge field along $Y$,

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

A charged wavefunction obeys

$$\psi(Y+2\pi R)=e^{i\theta}\psi(Y),$$

so its momentum is shifted:

$$p={1\over R}\left(n+{\theta\over 2\pi}\right).$$

After T-duality, this shift is a displacement in the dual coordinate:

$$\Delta Y'=2\pi\alpha'p=(2\pi n+\theta)R'.$$

Modulo the circumference $2\pi R'$, the Wilson-line phase corresponds to the brane position

$$Y'_0=\theta R'.$$

For $U(N)$ Chan—Paton factors, choose a diagonal Wilson line

$$W=\operatorname{diag}\left(e^{i\theta_1},e^{i\theta_2},\ldots,e^{i\theta_N}\right).$$

An open string in the Chan—Paton sector $|ij\rangle$ has charge $+1$ under the $i$-th diagonal $U(1)$ and charge $-1$ under the $j$-th diagonal $U(1)$. Its wavefunction picks up the phase

$$e^{i(\theta_i-\theta_j)}.$$

Therefore its momentum is

$$p_{ij}={2\pi n+\theta_i-\theta_j\over 2\pi R}.$$

The T-dual separation is

$$\Delta Y'_{ij}=2\pi\alpha' p_{ij} =\left(2\pi n+\theta_i-\theta_j\right)R'.$$

The individual brane positions are

$$Y'_i=\theta_iR' \qquad \text{mod }2\pi R'.$$

The eigenvalues of a Wilson line are angular positions on the T-dual circle. A Chan—Paton sector $|ij\rangle$ becomes a string stretched from brane $i$ to brane $j$, possibly after going around the dual circle.

This dictionary is worth remembering:

$$\boxed{\text{Wilson-line eigenvalue }\theta_i \quad\longleftrightarrow\quad \text{D-brane position }Y'_i=\theta_iR'.}$$

It also gives a clean geometric picture of gauge-symmetry breaking. If all $N$ branes coincide, the full $U(N)$ gauge symmetry is present. If the branes sit at generic distinct positions, only the diagonal gauge fields remain massless and the low-energy gauge symmetry is

$$U(1)^N.$$

If the branes form coincident stacks of multiplicities $N_1,N_2,\ldots,N_s$, the unbroken group is

$$U(N_1)\times U(N_2)\times\cdots\times U(N_s).$$

In field-theory language this is Higgsing. In string language the W-bosons are open strings stretched between separated branes.

Stretched strings and the mass formula

Let two parallel D-branes be separated by a transverse vector $\Delta\vec y$. An open string stretching from one brane to the other has classical length $|\Delta\vec y|$, so its classical energy is

$$E_{\rm stretch}=T_{\rm F1}|\Delta\vec y| ={|\Delta\vec y|\over 2\pi\alpha'}.$$

The full mass formula contains this classical contribution plus the usual oscillator contribution:

$$M^2=\left({|\Delta\vec y|\over 2\pi\alpha'}\right)^2 +{1\over\alpha'}(N-a).$$

For the bosonic open string, $a=1$. In the RNS superstring, $a=1/2$ in the NS sector and $a=0$ in the R sector before the GSO projection.

For branes on the dual circle, the separation for a string in the sector $|ij\rangle$ is

$$\Delta Y'_{ij,n}=\left(2\pi n+\theta_i-\theta_j\right)R'.$$

Its compact contribution to the mass is

$$M^2_{\rm compact} =\left({\Delta Y'_{ij,n}\over 2\pi\alpha'}\right)^2 =\left({2\pi n+\theta_i-\theta_j\over 2\pi R}\right)^2.$$

This is precisely the shifted Kaluza—Klein momentum before T-duality.

Strings with both endpoints on the same brane give diagonal gauge fields. Strings stretched between distinct branes give off-diagonal states whose masses are proportional to the brane separation.

When branes coincide, $\Delta\vec y=0$, the off-diagonal strings can become massless. These are the extra gauge bosons needed to enhance

$$U(1)^N\longrightarrow U(N).$$

The matrix indices of Yang—Mills theory are therefore not abstract labels from the open-string point of view. They are labels of string endpoints.

Boundary conditions for a D$p$-brane

A flat D$p$-brane has Neumann boundary conditions along its $p+1$ worldvolume directions and Dirichlet boundary conditions in the transverse directions. Let

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

Then at $\sigma=0,\pi$,

$$\partial_\sigma X^\mu=0, \qquad X^a=y^a.$$

For an open string stretched from brane $i$ to brane $j$,

$$X^a(\tau,0)=y_i^a, \qquad X^a(\tau,\pi)=y_j^a.$$

The classical solution contains a linear piece,

$$X^a(\tau,\sigma)=y_i^a+{\sigma\over\pi}(y_j^a-y_i^a)+\text{oscillators}.$$

That linear term is the worldsheet origin of the stretching contribution to the mass.

Worldvolume fields

The massless open-string states on a D$p$-brane split according to whether the oscillator polarization lies along the brane or transverse to it.

Along Neumann directions,

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

gives gauge fields $A_\mu(\xi)$ on the brane worldvolume. The corresponding boundary vertex operator is schematically

$$V_A\sim \epsilon_\mu\,\partial_t X^\mu e^{ik\cdot X},$$

where $\partial_t$ is tangent to the worldsheet boundary.

Along Dirichlet directions,

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

gives scalar fields $\Phi^a(\xi)$ on the brane. The corresponding vertex operator is schematically

$$V_\Phi\sim \epsilon_a\,\partial_n X^a e^{ik\cdot X},$$

where $\partial_n$ is normal to the boundary.

For one brane, these scalars describe transverse motion:

$$y^a(\xi)=2\pi\alpha'\Phi^a(\xi).$$

For $N$ coincident branes, $\Phi^a$ is matrix-valued. The diagonal entries are brane positions; the off-diagonal entries are open strings connecting different branes. This is the first appearance of a powerful idea: the transverse geometry of coincident D-branes is encoded in matrices.

Low-energy gauge theory

At energies far below the string scale, the massless open strings on $N$ coincident D$p$-branes are described by a $U(N)$ gauge theory in $p+1$ dimensions. In a supersymmetric setting the bosonic terms have the schematic form

$$S_{\rm wv}=-{1\over g_{\rm YM}^2}\int d^{p+1}\xi\,\operatorname{Tr}\left( {1\over4}F_{\mu\nu}F^{\mu\nu} +{1\over2}D_\mu\Phi^aD^\mu\Phi^a -{1\over4}[\Phi^a,\Phi^b]^2 +\cdots \right).$$

The commutator potential vanishes when the scalar matrices are mutually diagonal, which is the moduli space of separated parallel branes. Off-diagonal fluctuations become massive when the branes separate, exactly as predicted by the stretched-string mass formula.

For a single abelian brane, the Maxwell action is the leading term in the Dirac—Born—Infeld action,

$$S_{\rm BI}=-T_p\int d^{p+1}\xi\, \sqrt{-\det\left(\eta_{\mu\nu}+2\pi\alpha'F_{\mu\nu}+\partial_\mu y^a\partial_\nu y^a\right)}.$$

The relation between gauge fields and geometry is already visible from T-duality:

$$A_Y\quad\longleftrightarrow\quad {Y'\over 2\pi\alpha'}.$$

A gauge component along a compact direction becomes a transverse scalar after T-duality. This is why the worldvolume gauge field and the brane-position fields sit in the same open-string multiplet.

Summary

Open-string T-duality turns a statement about compact momentum into a statement about boundary geometry. The essential dictionary is

$$\begin{array}{ccl} \text{Neumann along }Y &\longleftrightarrow& \text{Dirichlet along }Y',\\ R &\longleftrightarrow& \alpha'/R,\\ \text{momentum }n/R &\longleftrightarrow& \text{endpoint separation }2\pi nR',\\ \text{Wilson-line eigenvalue }\theta_i &\longleftrightarrow& \text{brane position }Y'_i=\theta_iR',\\ \text{Chan--Paton labels }i,j &\longleftrightarrow& \text{open-string endpoints on branes }i,j. \end{array}$$

D-branes are therefore not optional additions to open-string theory. They are forced by T-duality. Once they appear, gauge symmetry, brane geometry, and string endpoint labels become different languages for the same structure.

Exercises

Exercise 1: Boundary conditions under T-duality

Let $Y'$ be defined by

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

Show that Neumann boundary conditions for $Y$ imply Dirichlet boundary conditions for $Y'$.

Solution

At the boundary,

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

Using $\partial_\tau Y'=\partial_\sigma Y$, this becomes

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

Hence the endpoint value of $Y'$ is independent of $\tau$:

$$Y'(\tau,0)=\text{constant}, \qquad Y'(\tau,\pi)=\text{constant}.$$

That is the Dirichlet condition.

Exercise 2: The dual open-string coordinate

Starting from

$$Y(\tau,\sigma)=y+2\alpha'p\tau +i\sqrt{2\alpha'}\sum_{m\neq0}{\alpha_m\over m}e^{-im\tau}\cos m\sigma,$$

verify that

$$Y'(\tau,\sigma)=y'+2\alpha'p\sigma +\sqrt{2\alpha'}\sum_{m\neq0}{\alpha_m\over m}e^{-im\tau}\sin m\sigma$$

satisfies the T-duality equations, and compute $Y'(\pi)-Y'(0)$.

Solution

Differentiating gives

$$\partial_\tau Y'= -i\sqrt{2\alpha'}\sum_{m\neq0}\alpha_m e^{-im\tau}\sin m\sigma,$$

which equals $\partial_\sigma Y$. Also,

$$\partial_\sigma Y'=2\alpha'p+\sqrt{2\alpha'}\sum_{m\neq0}\alpha_m e^{-im\tau}\cos m\sigma,$$

which equals $\partial_\tau Y$. Since the sine term vanishes at both endpoints,

$$Y'(\tau,\pi)-Y'(\tau,0)=2\pi\alpha'p.$$

With $p=n/R$ and $R'=\alpha'/R$,

$$Y'(\tau,\pi)-Y'(\tau,0)=2\pi nR'.$$

Exercise 3: Wilson lines and shifted momenta

For a diagonal Wilson line

$$W=\operatorname{diag}(e^{i\theta_1},\ldots,e^{i\theta_N}),$$

show that an open string in the $|ij\rangle$ sector has momentum

$$p_{ij}={2\pi n+\theta_i-\theta_j\over 2\pi R}.$$

Solution

The endpoint at $i$ contributes the phase $e^{i\theta_i}$ and the endpoint at $j$ contributes the conjugate phase $e^{-i\theta_j}$. Therefore a trip around the circle gives

$$e^{i(\theta_i-\theta_j)}.$$

A momentum eigenfunction changes by $e^{2\pi iRp}$. Thus

$$e^{2\pi iRp}=e^{i(\theta_i-\theta_j)}$$

up to the integer Fourier mode $n$. Hence

$$2\pi Rp=2\pi n+\theta_i-\theta_j,$$

which gives the desired result.

Exercise 4: Mass of a stretched string

An open string stretches between two parallel D-branes separated by distance $L$. Derive the stretching contribution to the mass and write the bosonic open-string mass formula.

Solution

The string tension is

$$T_{\rm F1}={1\over 2\pi\alpha'}.$$

A static string of length $L$ has energy

$$E=T_{\rm F1}L={L\over 2\pi\alpha'}.$$

This contributes $E^2$ to $M^2$. For the bosonic open string the oscillator contribution is $(N-1)/\alpha'$, so

$$M^2=\left({L\over 2\pi\alpha'}\right)^2+{N-1\over\alpha'}.$$

Exercise 5: Gauge enhancement

Explain why $N$ separated branes have low-energy gauge group $U(1)^N$, while $N$ coincident branes have $U(N)$.

Solution

For separated branes, strings beginning and ending on the same brane give $N$ diagonal massless gauge fields, hence $U(1)^N$. Strings stretching between different branes carry off-diagonal Chan—Paton labels $|ij\rangle$ with $i\neq j$, but they are massive because their length is nonzero.

When the branes coincide, the stretching length of the off-diagonal strings vanishes. The off-diagonal vector states become massless and combine with the diagonal ones into the adjoint representation of $U(N)$. Thus the gauge group is enhanced from $U(1)^N$ to $U(N)$.

Exercise 6: Scalars on a D$p$-brane

In bosonic string theory in $D=26$, how many massless scalar fields live on a single D$p$-brane, and what do they represent?

Solution

A D$p$-brane has $p+1$ Neumann directions, including time. Therefore the number of transverse Dirichlet directions is

$$26-(p+1)=25-p.$$

There is one massless scalar for each transverse direction, so the brane has $25-p$ massless scalar fields. They describe transverse fluctuations of the brane position.