Open/Closed Duality and D-Brane Scattering

A D-brane has two equally important descriptions. From the viewpoint of open strings it is the hypersurface on which endpoints live. From the viewpoint of closed strings it is a heavy coherent source for the massless fields of the bulk: the metric, dilaton, antisymmetric tensor in the presence of flux, and, in the superstring, Ramond—Ramond potentials. The most economical way to see that these are the same statement is open/closed duality.

The basic worldsheet is the annulus. If time runs around the annulus, it is a one-loop vacuum diagram of open strings. If time runs across the annulus, it is a tree-level closed-string propagator between two boundary states. This is not an analogy; it is the same Riemann surface with a different choice of Hamiltonian evolution.

The annulus can be quantized as an open-string one-loop diagram with modulus $t$, or after the modular transformation $t=1/\ell$ as closed-string propagation for proper time $\ell$ between two D-brane boundary states.

This simple fact is one of the central structural observations behind D-branes. It tells us that a perturbative open-string loop already knows about tree-level gravity, and it lets us determine the normalization of D-brane sources by comparing two descriptions of the same amplitude.

The annulus in the open-string channel

Consider two parallel D$p$-branes separated by a transverse vector $y^i$, with

$$y^2=y^iy^i, \qquad i=p+1,\ldots,9.$$

An open string stretched between them has Neumann boundary conditions along the common worldvolume directions and Dirichlet boundary conditions in the transverse directions. The separation contributes a classical stretching energy. In the NS sector of the superstring, schematically,

$$\alpha' M^2={y^2\over 4\pi^2\alpha'}+N-{1\over 2},$$

while in the R sector,

$$\alpha' M^2={y^2\over 4\pi^2\alpha'}+N.$$

For the moment we suppress the detailed spin-structure sum and write the open-channel annulus amplitude in the form

$$\mathcal A_{\text{open}}(y) =V_{p+1}\int_0^\infty {dt\over 2t} \int {d^{p+1}k\over (2\pi)^{p+1}} \operatorname{Tr}_{\text{osc}} \exp\left[-2\pi t\left(\alpha' k^2+{y^2\over 4\pi^2\alpha'}+N-a\right)\right].$$

Here $V_{p+1}$ is the regulated brane worldvolume, $t$ is the open-string proper time, and $a$ is the normal-ordering constant appropriate to the sector. The Gaussian integral over momenta tangent to the brane gives

$$\int {d^{p+1}k\over (2\pi)^{p+1}} \exp(-2\pi\alpha't k^2) =(8\pi^2\alpha't)^{-(p+1)/2}.$$

Thus the amplitude has the universal structure

$$\mathcal A_{\text{open}}(y) ={V_{p+1}\over 2}\int_0^\infty {dt\over t} (8\pi^2\alpha't)^{-(p+1)/2} \exp\left[-{y^2t\over 2\pi\alpha'}\right] Z_{\text{osc}}(t),$$

where $Z_{\text{osc}}(t)$ is the oscillator partition function, including the GSO projection and possible Chan—Paton factors. The details of $Z_{\text{osc}}$ distinguish the bosonic string, type II BPS branes, brane—antibrane systems, and non-BPS branes. The kinematic lesson is already visible in the exponential: a long stretched string is expensive in the open-string channel.

The endpoints of the annulus are boundaries, so the amplitude is weighted as an open-string one-loop effect. In string perturbation theory this diagram is order $g_s^0$. That fact will match the closed-string interpretation: two disk one-point couplings to a D-brane are each proportional to the brane tension $T_p\sim 1/g_s$, while the closed-string propagator carries $\kappa_{10}^2\sim g_s^2$.

The same annulus in the closed-string channel

Now set

$$\ell={1\over t}.$$

The same annulus can be viewed as a closed string emitted from one boundary and absorbed by the other. Formally,

$$\mathcal A_{\text{closed}}(y) =\langle B_p,y_1|\,\Delta\,|B_p,y_2\rangle,$$

where $|B_p,y\rangle$ is the boundary state describing a D$p$-brane localized at transverse position $y$, and

$$\Delta=\int_0^\infty d\ell\, \exp\left[-\pi\ell\left(L_0+\widetilde L_0-2a_{\text{closed}}\right)\right]$$

is the closed-string propagator. The boundary state imposes the D-brane boundary conditions as operator equations. For a flat D$p$-brane,

$$(\alpha_n^a+\widetilde\alpha_{-n}^a)|B_p\rangle=0, \qquad a=0,\ldots,p,$$

and

$$(\alpha_n^i-\widetilde\alpha_{-n}^i)|B_p\rangle=0, \qquad i=p+1,\ldots,9.$$

The plus sign in the Neumann equation says that the normal derivative vanishes at the boundary; the minus sign in the Dirichlet equation says that the coordinate is fixed. The zero-mode part of the Dirichlet condition localizes the brane:

$$(X^i-y^i)|B_p,y\rangle=0.$$

After the modular transformation, the separation appears as the heat kernel for propagation in the transverse space:

$$\exp\left[-{y^2t\over 2\pi\alpha'}\right] \quad\longrightarrow\quad \exp\left[-{y^2\over 2\pi\alpha'\ell}\right].$$

Large $\ell$ means that the closed string propagates for a long proper time. Hence the closed-channel infrared limit is controlled by the lightest closed-string states.

This is the crucial UV/IR exchange:

$$t\to 0 \quad\Longleftrightarrow\quad \ell\to\infty.$$

What looks like the ultraviolet region of an open-string loop is the infrared region of closed-string propagation. Therefore an open-channel divergence at small $t$ is not interpreted as an ordinary short-distance UV catastrophe. It is a closed-string tadpole, massless exchange effect, or tachyon instability, depending on the spectrum.

Long-distance force from massless closed strings

At brane separation

$$y\gg \sqrt{\alpha'},$$

massive closed strings are exponentially suppressed. The leading interaction comes from massless exchange. In the transverse space of dimension

$$d=9-p,$$

the massless scalar Green function is

$$G_d(y)=\int {d^dk\over (2\pi)^d}{e^{ik\cdot y}\over k^2}.$$

For $d>2$,

$$G_d(y)= {\Gamma\left({d-2\over2}\right)\over 4\pi^{d/2}} {1\over y^{d-2}}.$$

Thus the long-distance annulus amplitude per unit brane volume has the general form

$${ \mathcal A_{\text{long}}(y)\over V_{p+1}} \sim \left(2\kappa_{10}^2T_p^2\right)G_{9-p}(y) \times(\text{polarization and charge factor}).$$

The phrase “polarization and charge factor” hides the tensor algebra of the exchanged fields. NS—NS exchange includes the graviton and dilaton, and in more general backgrounds the $B$-field. In type II theory, RR exchange must also be included for BPS D-branes. For two identical supersymmetric D-branes, the NS—NS attraction cancels the RR repulsion. For a brane—antibrane pair, the RR sign reverses and the forces add. We will analyze this cancellation and its BPS meaning in later pages; here the important point is the open/closed dictionary itself.

At large separation the annulus is dominated by the lightest closed strings. The amplitude reduces to field-theory exchange in the transverse space, with potential proportional to $G_{9-p}(y)$.

This is the spacetime meaning of the annulus: it is the one-loop vacuum energy of stretched open strings, but it is also the classical interaction energy generated by closed-string fields sourced by the branes.

D-brane scattering of closed strings

A second way to see the same physics is to scatter a closed string from a D-brane. The worldsheet is now a disk with two closed-string vertex operators inserted in the interior. The boundary imposes D-brane boundary conditions.

Because the brane is translationally invariant along its worldvolume, parallel momentum is conserved:

$$p_1^a+p_2^a=0.$$

Transverse momentum need not be conserved by the closed string alone, since the brane can absorb recoil. Define the momentum transfer

$$q^i=p_1^i+p_2^i,$$

and the invariant

$$t=-q^2.$$

A convenient second invariant is

$$s=-p_{1a}p_1^a,$$

which measures the energy flowing through possible open-string states on the brane.

A closed string hitting a D-brane can transfer transverse momentum to it. The disk two-point amplitude contains closed-string poles in the momentum-transfer channel and open-string poles in the worldvolume channel.

The boundary can be encoded in a reflection matrix

$$D^M{}_{N} = \begin{cases} +\delta^a{}_b, & \text{Neumann directions},\\ -\delta^i{}_j, & \text{Dirichlet directions}. \end{cases}$$

In the doubling trick, right-movers are reflected into left-movers by

$$\widetilde X^M(\bar z)\longrightarrow D^M{}_{N}X^N(\bar z).$$

A closed-string vertex on the disk therefore behaves like a product of two holomorphic insertions with momenta related by $D$. This is why a two-closed-string disk amplitude has a beta-function structure.

For massless external closed strings the amplitude takes the schematic form

$$\mathcal A_{\text{disk}} =\mathcal C_p\,K(1,2)\, {\Gamma(-\alpha's)\Gamma(-\alpha't/4) \over \Gamma(1-\alpha's-\alpha't/4)}.$$

Here $K(1,2)$ is a polarization-dependent kinematic factor, and $\mathcal C_p$ is proportional to the gravitational coupling times the D-brane tension. The exact form of $K$ depends on whether the external states are gravitons, dilatons, or $B$-fields. The analytic structure, however, is universal.

The gamma functions display two kinds of poles:

$$\alpha's=0,1,2,\ldots \qquad\text{open-string poles},$$

and

$${\alpha't\over4}=0,1,2,\ldots \qquad\text{closed-string poles}.$$

The leading $t=0$ pole is massless closed-string exchange between the probe and the brane. The leading $s=0$ pole is a massless open-string excitation on the brane. At low energy, the amplitude decomposes into

$$\mathcal A_{\text{disk}} = \mathcal A_{\text{closed pole}} + \mathcal A_{\text{open pole}} + \mathcal A_{\text{contact}} + O(\alpha').$$

This is exactly what an effective action should reproduce: bulk supergravity exchange, brane worldvolume exchange, and local DBI-type contact interactions.

Linear couplings from the DBI action

Let us see the $t$-channel pole from field theory. The string-frame DBI action for a brane with $F=0$ is

$$S_{\text{DBI}} =-\tau_p\int d^{p+1}\xi\,e^{-\Phi} \sqrt{-\det P[G]_{ab}}.$$

Around the weakly coupled flat background,

$$e^{\Phi_0}=g_s, \qquad G_{MN}=\eta_{MN}+h_{MN}, \qquad \Phi=\Phi_0+\varphi,$$

this becomes

$$S_{\text{DBI}} =-T_p\int d^{p+1}\xi \left[ 1+{1\over2}h^a{}_{a}-\varphi+\cdots \right].$$

The constant term is the brane tension. The term linear in $h_{ab}$ is the coupling to the graviton. The term linear in $\varphi$ is the coupling to the dilaton. The antisymmetric field $B_{ab}$ has no linear term for a flat brane with $F=0$ because

$$\operatorname{tr} B=0.$$

The coupling to $B$ appears through the gauge-invariant combination

$$P[B]+2\pi\alpha'F,$$

and becomes important once a background $B$-field or worldvolume flux is turned on.

In field theory, a canonically normalized graviton couples as

$$S_{\text{int}}={\kappa_{10}\over2} \int d^{10}x\,h_{MN}T^{MN}.$$

For a static flat brane at $x^i=y^i$,

$$T^{MN}(x) \propto T_p\,\delta^{(9-p)}(x_\perp-y)\, \delta^M_a\delta^N_b\eta^{ab}.$$

The propagator of a massless field carrying transverse momentum $q$ is $1/q^2=-1/t$. Therefore the disk amplitude must contain a long-distance singularity of the form

$$\mathcal A_{\text{disk}}\big|_{t\to0} \sim {\kappa_{10}T_p\over t} K_{\text{massless}}(1,2).$$

Matching the coefficient of this pole to the exact disk amplitude fixes the normalization of the D-brane source. This is one of the clean worldsheet derivations of

$$T_p={1\over g_s(2\pi)^p(\alpha')^{(p+1)/2}}.$$

The $1/g_s$ scaling is the key physical point. A D-brane is heavy and nonperturbative from the closed-string viewpoint, but open strings ending on it are perfectly perturbative.

Reading effective actions from poles

Tree-level string amplitudes contain more information than just scattering probabilities. Their singularities tell us the spectrum and the couplings of the low-energy effective theory.

For the disk two-point function:

• The $t$-channel closed-string pole is reproduced by bulk supergravity fields coupling to the brane stress tensor and dilaton source.
• The $s$-channel open-string pole is reproduced by gauge and scalar fields propagating on the brane.
• The non-singular terms are local contact terms on the brane, organized by the DBI action and higher-derivative corrections.

For the annulus:

• The open-channel trace describes quantum fluctuations of open strings stretched between branes.
• The closed-channel cylinder describes tree-level exchange of closed strings.
• The $t\to0$ open UV region is the $\ell\to\infty$ closed IR region, so divergences are interpreted as closed-string tadpoles or long-range massless exchange.

This is why D-branes are such precise probes of string theory. A single amplitude simultaneously constrains open-string gauge theory, closed-string gravity, and their coupling.

A compact dictionary

The core dictionary for this page is:

Open-string channel	Closed-string channel
annulus one-loop trace	cylinder tree exchange
modulus $t$	modulus $\ell=1/t$
$t\to\infty$ open IR	$\ell\to0$ closed UV
$t\to0$ open UV	$\ell\to\infty$ closed IR
stretched open-string mass $y/(2\pi\alpha')$	closed-string propagation across distance $y$
Chan—Paton and boundary conditions	boundary-state normalization and gluing
open-string poles in $s$	brane worldvolume excitations
closed-string poles in $t$	bulk closed-string fields

The punchline is worth repeating: D-branes are simultaneously hypersurfaces for open strings and coherent sources for closed strings. The consistency of the two descriptions is not optional; it is enforced by the modular geometry of the annulus and by factorization of disk amplitudes.

Exercises

Exercise 1. Boundary-state gluing conditions

Show that the closed-string boundary state of a flat D$p$-brane satisfies

$$(\alpha_n^a+\widetilde\alpha_{-n}^a)|B_p\rangle=0, \qquad (\alpha_n^i-\widetilde\alpha_{-n}^i)|B_p\rangle=0.$$

Explain the relative signs.

Solution

In the closed-string channel, the boundary is placed at fixed worldsheet time $\tau=0$. Neumann boundary conditions become

$$\partial_\tau X^a(\tau=0,\sigma)|B_p\rangle=0.$$

Using the closed-string mode expansion, the oscillator part of $\partial_\tau X^a$ is proportional to

$$\sum_n(\alpha_n^a+\widetilde\alpha_{-n}^a)e^{-in\sigma}.$$

Therefore Neumann directions obey

$$(\alpha_n^a+\widetilde\alpha_{-n}^a)|B_p\rangle=0.$$

For Dirichlet directions, the coordinate itself is fixed:

$$X^i(\tau=0,\sigma)|B_p,y\rangle=y^i|B_p,y\rangle.$$

The oscillator part of this condition gives

$$(\alpha_n^i-\widetilde\alpha_{-n}^i)|B_p,y\rangle=0,$$

while the zero-mode part gives $(\hat x^i-y^i)|B_p,y\rangle=0$. The sign difference is exactly the sign of the reflection matrix: $D=+1$ along Neumann directions and $D=-1$ along Dirichlet directions.

Exercise 2. The stretched-string Gaussian

Starting from

$$M_{\text{stretch}}^2={y^2\over(2\pi\alpha')^2},$$

show that the open-string Schwinger factor $e^{-2\pi t\alpha'M^2}$ produces

$$\exp\left[-{y^2t\over2\pi\alpha'}\right].$$

Solution

Substitute the stretching mass:

$$e^{-2\pi t\alpha'M_{\text{stretch}}^2} = \exp\left[ -2\pi t\alpha'{y^2\over(2\pi\alpha')^2} \right].$$

Since

$$2\pi\alpha'\,{1\over(2\pi\alpha')^2} ={1\over2\pi\alpha'},$$

we get

$$e^{-2\pi t\alpha'M_{\text{stretch}}^2} = \exp\left[-{y^2t\over2\pi\alpha'}\right].$$

This is the separation-dependent Gaussian in the open-channel annulus.

Exercise 3. Open and closed poles in the disk amplitude

Consider

$$B(s,t)= {\Gamma(-\alpha's)\Gamma(-\alpha't/4) \over \Gamma(1-\alpha's-\alpha't/4)}.$$

Identify the open-string and closed-string poles.

Solution

The gamma function has simple poles at nonpositive integers:

$$\Gamma(z)\quad\text{has poles at}\quad z=0,-1,-2,\ldots.$$

The factor $\Gamma(-\alpha's)$ therefore has poles when

$$-\alpha's=-n, \qquad n=0,1,2,\ldots,$$

or

$$\alpha's=n.$$

These are open-string poles because $s$ is the invariant energy flowing along the brane worldvolume.

The factor $\Gamma(-\alpha't/4)$ has poles when

$$-\alpha't/4=-n,$$

or

$${\alpha't\over4}=n.$$

These are closed-string poles in the momentum-transfer channel. The leading $n=0$ pole is the massless closed-string pole.

Exercise 4. The transverse Green function

For $d>2$, evaluate

$$G_d(y)=\int {d^dk\over(2\pi)^d}{e^{ik\cdot y}\over k^2}.$$

Solution

Use the Schwinger representation

$${1\over k^2}=\int_0^\infty d\lambda\,e^{-\lambda k^2}.$$

Then

$$G_d(y)=\int_0^\infty d\lambda \int {d^dk\over(2\pi)^d} e^{-\lambda k^2+ik\cdot y}.$$

The Gaussian integral is

$$\int {d^dk\over(2\pi)^d} e^{-\lambda k^2+ik\cdot y} = {1\over(4\pi\lambda)^{d/2}} e^{-y^2/(4\lambda)}.$$

Thus

$$G_d(y) ={1\over(4\pi)^{d/2}} \int_0^\infty d\lambda\,\lambda^{-d/2} e^{-y^2/(4\lambda)}.$$

Set $u=y^2/(4\lambda)$. Then

$$G_d(y) ={1\over(4\pi)^{d/2}} \left({4\over y^2}\right)^{d/2-1} \int_0^\infty du\,u^{d/2-2}e^{-u}.$$

The integral is $\Gamma(d/2-1)$, so

$$G_d(y)= {\Gamma\left({d-2\over2}\right)\over4\pi^{d/2}} {1\over y^{d-2}}.$$

Exercise 5. Linearized DBI coupling to the graviton

Starting from

$$S=-T_p\int d^{p+1}\xi \sqrt{-\det(\eta_{ab}+h_{ab})},$$

derive the leading coupling

$$S_{\text{lin}} =-{T_p\over2}\int d^{p+1}\xi\,h^a{}_{a}.$$

Solution

For a small matrix $M$,

$$\sqrt{\det(1+M)} =1+{1\over2}\operatorname{tr}M+O(M^2).$$

Here

$$M^a{}_b=\eta^{ac}h_{cb},$$

so

$$\operatorname{tr}M=h^a{}_{a}.$$

Substituting into the action gives

$$S=-T_p\int d^{p+1}\xi \left[ 1+{1\over2}h^a{}_{a}+O(h^2) \right].$$

The first term is the tension. The linear term is

$$S_{\text{lin}} =-{T_p\over2}\int d^{p+1}\xi\,h^a{}_{a}.$$

This coupling is responsible for the graviton contribution to the massless $t$-channel pole.

Exercise 6. Open UV equals closed IR

Explain why the region $t\to0$ of the open-channel annulus is the infrared region of closed-string propagation.

Solution

In the open-string channel, the annulus is

$$\int_0^\infty {dt\over t} \operatorname{Tr}_{\text{open}} \left(e^{-2\pi t(L_0-a)}\right).$$

The parameter $t$ is the proper time of the open-string loop. Therefore $t\to0$ is a short-proper-time region, which is ultraviolet from the open-string perspective.

In the closed-string channel,

$$\ell={1\over t}.$$

The same amplitude becomes

$$\int_0^\infty d\ell\, \langle B_1| e^{-\pi\ell(L_0+\widetilde L_0-2a)} |B_2\rangle.$$

Thus $t\to0$ corresponds to $\ell\to\infty$. Long closed-string proper time projects onto the lowest-energy closed-string states. Therefore the open ultraviolet corner is the closed infrared corner. Divergences in this region signal closed-string tachyons, massless tadpoles, or long-range massless exchange.