Open String Field Theory and Tachyon Condensation

Perturbative string theory is usually presented as a first-quantized theory: choose a worldsheet conformal field theory, insert vertex operators, integrate over moduli space, and obtain an $S$-matrix. This is beautiful, but it hides one thing that ordinary quantum field theory makes manifest: an off-shell space of fields and an action whose classical solutions represent different vacua.

String field theory restores that viewpoint. Instead of describing only on-shell vertex operators, we assemble the entire open-string Fock space into one huge spacetime field. The basic variable is not a scalar, vector, or spinor, but a string field

$$|\Psi\rangle\in \mathcal H_{\rm open},$$

where $\mathcal H_{\rm open}$ is the state space of the boundary conformal field theory describing open strings on a chosen D-brane background. The payoff is enormous: unstable D-branes, tachyon condensation, and D-brane descent become classical solutions of a single action.

On this page we focus on bosonic open string field theory, where the structure is cleanest. Superstring field theory exists, but picture number, Ramond-sector constraints, and contact terms make the construction more delicate. The conceptual lessons are already visible in Witten’s cubic bosonic theory.

Backgrounds, boundary conditions, and why open SFT is useful

A perturbative string background is specified by a worldsheet CFT. For closed strings, the bulk CFT encodes the spacetime metric, $B$-field, dilaton, and other closed-string background data. For open strings, one must also specify boundary conditions. These boundary conditions are the worldsheet avatar of D-branes.

Worldsheet object	Spacetime interpretation
Bulk CFT	closed-string background
Boundary CFT	D-brane or collection of D-branes
Open-string Hilbert space $\mathcal H$	fluctuations living on that D-brane background
BRST cohomology	physical open-string spectrum
Classical SFT solution	a new boundary condition in the same closed-string background

Thus open string field theory is not initially a theory of all possible closed-string backgrounds. It starts with a fixed closed-string CFT and a chosen boundary CFT. Within that setting it gives an off-shell theory whose classical solutions can describe the disappearance, recombination, or descent of D-branes.

This is exactly what one needs for the open-string tachyon. A tachyon is not a small nuisance in the spectrum; it is a signal that the boundary condition is unstable. Ordinary on-shell amplitudes do not tell us where the system goes. String field theory does.

The open string field

Let $\mathcal H$ be the full matter-plus-ghost state space of the open-string BCFT. The open string field is a ghost-number-one state,

$$\operatorname{gh}(\Psi)=1.$$

For a space-filling bosonic D25-brane, a useful schematic expansion is

$$|\Psi\rangle = \int {d^{26}k\over (2\pi)^{26}} \left[ T(k)c_1|k\rangle +A_\mu(k)\alpha_{-1}^{\mu}c_1|k\rangle +\cdots \right].$$

The coefficient $T(k)$ is the open-string tachyon field. The coefficient $A_\mu(k)$ is the gauge field. The ellipsis contains the infinite tower of massive open-string fields, together with auxiliary and gauge degrees of freedom. For a D$p$-brane one restricts the continuous momentum to the $p+1$ Neumann directions; transverse scalar fields appear among the massless coefficients.

The phrase “string field” is therefore literal but dangerous if taken too naively. $\Psi$ is a spacetime field with infinitely many components, one for each basis state of the open-string Fock space. The field theory is local on the worldsheet, not local in the ordinary spacetime sense.

Open string field theory is built from three pieces of worldsheet data: the BRST differential $Q_B$, Witten’s star product $*$, and the BPZ inner product $\langle\,,\,\rangle$. Together they define a gauge-invariant cubic action.

The physical open-string spectrum is recovered by linearizing around the perturbative vacuum $\Psi=0$. At the free level the equation is

$$Q_B\Psi=0,$$

and the gauge redundancy is

$$\delta\Psi=Q_B\Lambda, \qquad \operatorname{gh}(\Lambda)=0.$$

Thus the free spectrum is BRST cohomology at ghost number one, exactly as in the first-quantized description. The nontrivial content of string field theory is the nonlinear completion of this equation.

BPZ inner product and surface states

The BPZ inner product is the two-string vertex. If $A$ and $B$ are open-string states, then $\langle A,B\rangle$ is computed as a disk or upper-half-plane correlation function after applying the BPZ conformal map to one of the states.

Ghost number is important. For the bosonic open string, a nonzero disk correlator must soak up three $c$-ghost zero modes. Therefore

$$\langle A,B\rangle\neq0 \quad\Longrightarrow\quad \operatorname{gh}(A)+\operatorname{gh}(B)=3.$$

Since $\Psi$ has ghost number one and $Q_B$ raises ghost number by one, the kinetic term $\langle \Psi,Q_B\Psi\rangle$ has total ghost number three. Likewise, because the star product adds ghost number, $\langle\Psi,\Psi*\Psi\rangle$ is also allowed.

A particularly useful language is that of surface states. Given a Riemann surface $\Sigma$ with one open-string puncture and a local coordinate map $f$ around that puncture, one defines a state $|\Sigma\rangle$ by

$$\langle \phi|\Sigma\rangle = \big\langle f\circ\phi(0)\big\rangle_\Sigma.$$

Here $\phi$ is any test state, represented by a local operator inserted at the puncture. This turns a worldsheet surface into a vector in the open-string Hilbert space. Surface states are the natural bridge between the CFT picture and the algebraic star-product picture.

Witten’s cubic action

Witten’s open bosonic string field theory has the action

$$S[\Psi] =-{1\over g_o^2}\left( {1\over2}\langle\Psi,Q_B\Psi\rangle +{1\over3}\langle\Psi,\Psi*\Psi\rangle \right).$$

Here $g_o$ is the open-string coupling. The string field $\Psi$ is Grassmann odd and has ghost number one. The BRST operator $Q_B$ has ghost number one. The star product is an associative product on open-string states:

$$(A*B)*C=A*(B*C).$$

Varying the action gives

$$\delta S =-{1\over g_o^2}\langle\delta\Psi,Q_B\Psi+\Psi*\Psi\rangle,$$

so the classical equation of motion is

$$\boxed{Q_B\Psi+\Psi*\Psi=0.}$$

The nonlinear gauge transformation is

$$\boxed{\delta_\Lambda\Psi=Q_B\Lambda+\Psi*\Lambda-\Lambda*\Psi,}$$

where $\Lambda$ is a ghost-number-zero gauge parameter. This is the stringy analogue of

$$\delta A=d\lambda+[A,\lambda], \qquad F=dA+A\wedge A.$$

The analogy is not cosmetic. The equation

$$Q_B\Psi+\Psi*\Psi=0$$

is a Maurer—Cartan equation in a differential graded algebra. The BRST operator acts like a differential, and the star product acts like a noncommutative multiplication.

The algebraic identities needed for gauge invariance are

$$Q_B^2=0,$$ $$Q_B(A*B)=(Q_BA)*B+(-1)^{|A|}A*(Q_BB),$$

and cyclicity,

$$\langle A,B*C\rangle=\langle A*B,C\rangle,$$

together with the BPZ property of $Q_B$,

$$\langle Q_BA,B\rangle=-(-1)^{|A|}\langle A,Q_BB\rangle.$$

Here $|A|$ denotes the Grassmann parity of $A$. These identities are not added by hand. They are consequences of BRST invariance and sewing of worldsheet surfaces.

Geometry of the star product

The star product $A*B$ is defined geometrically. Think of an open string as an interval. Split each string into left and right halves. Then glue the right half of the first string to the left half of the second string. The output is a new open string made from the left half of the first string and the right half of the second string.

Witten’s star product glues the right half of one open string to the left half of another. Equivalently, $\langle C,A*B\rangle$ is a disk correlator with three open-string insertions.

This half-string gluing is the reason the action is cubic. Higher open-string tree amplitudes are generated by cubic vertices connected by propagators, just as in ordinary cubic field theory. The difference is that the propagator includes an integral over a strip length. In Siegel gauge,

$$b_0\Psi=0,$$

the propagator is formally

$${b_0\over L_0},$$

which can be written as an integral over worldsheet proper time,

$${b_0\over L_0}=b_0\int_0^\infty ds\, e^{-sL_0}.$$

The parameter $s$ is precisely the length of the strip glued between cubic vertices. In this way Feynman diagrams of open SFT cover the moduli spaces of bordered Riemann surfaces. This is the worldsheet origin of the spacetime Feynman rules.

There are subtleties at the open-string midpoint, especially in superstring field theory and in singular analytic solutions. But for the basic bosonic theory the geometric message is robust: the product $*$ is worldsheet sewing.

Wedge states and the sliver

A simple and powerful class of surface states is formed by wedge states. Let $W_\alpha$ denote a strip-like surface of width $\alpha$, with the convention

$$W_0=I,$$

where $I$ is the identity string field. The star product simply adds widths:

$$\boxed{W_\alpha*W_\beta=W_{\alpha+\beta}.}$$

In this convention $W_1$ is the $SL(2,\mathbb R)$-invariant vacuum state, and the limit

$$W_\infty=\lim_{\alpha\to\infty}W_\alpha$$

is the sliver state. Because adding an infinite strip to an infinite strip still gives an infinite strip, the sliver is a projector:

$$W_\infty*W_\infty=W_\infty.$$

Projectors are important because they behave like classical solitons in noncommutative field theory. Indeed, before the full analytic tachyon-vacuum solution was found, projector-like surface states already gave strong evidence that D-brane decay and D-brane descent could be described within the star algebra.

It is also useful to package wedge states as

$$W_\alpha=e^{-\alpha K},$$

where $K$ is the generator of translations in strip width. Then

$${d\over d\alpha}W_\alpha=-KW_\alpha.$$

This notation leads naturally to the $K,B,c$ subalgebra.

The $K,B,c$ algebra

Many analytic solutions of open SFT live in a small subalgebra generated by three string fields:

$$K, \qquad B, \qquad c.$$

Geometrically, $K$ is a line integral of the stress tensor along the strip, $B$ is a corresponding line integral of the $b$ ghost, and $c$ is a boundary $c$-ghost insertion. Their algebra is

$$[K,B]=0, \qquad B^2=0, \qquad c^2=0, \qquad \{B,c\}=1.$$

The BRST operator acts as

$$Q_BK=0, \qquad Q_BB=K, \qquad Q_Bc=cKc.$$

This is the string-field-theoretic analogue of choosing a clever coordinate system on the space of gauge fields. The full string field is infinite-dimensional, but the tachyon vacuum can be written in terms of wedge states and a small number of ghost insertions.

A typical building block has the schematic form

$$\psi_n\sim c\,B\,W_n\,c,$$

where $W_n$ is a wedge state. The derivative $\psi'_n=\partial_n\psi_n$ measures how the state changes as the strip width changes. Schnabl’s analytic tachyon-vacuum solution can be written schematically as

$$\Psi_{\rm Sch} = \lim_{N\to\infty}\left(\sum_{n=0}^{N}\psi'_n-\psi_N\right).$$

The final term is often called a boundary or phantom term. It is essential: without it, the limiting expression has the right formal shape but the wrong gauge-invariant observables. This is a recurring theme in string field theory: states that look harmless algebraically may encode singular worldsheet limits.

Tachyon condensation

The bosonic open string on a D-brane has a tachyon. In the first-quantized spectrum this is simply a state with

$$m^2=-{1\over \alpha'}.$$

In spacetime language it means that the perturbative vacuum $\Psi=0$ is unstable. The central question is: what is the endpoint?

Sen’s answer is one of the most important insights into D-brane physics:

1. The open-string tachyon condenses to a classical solution $\Psi_{\rm tv}$ whose energy cancels the D-brane tension.
2. Around this endpoint there are no perturbative open-string excitations.
3. Lower-dimensional D-branes appear as solitonic lump solutions of the tachyon field.

In open SFT the energy density of a static classical solution is

$$E[\Psi]=-{S[\Psi]\over V_{p+1}},$$

where $V_{p+1}$ is the worldvolume volume. The perturbative D-brane is normalized to have $E[0]=0$ in the open-string field theory action. Sen’s first conjecture says

$$\boxed{E[\Psi_{\rm tv}]=-T_p.}$$

Thus the total energy of the original D$p$-brane plus the tachyon condensate is zero. The brane has disappeared, leaving the closed-string vacuum of the same bulk background.

For the bosonic space-filling D25-brane, in the conventional normalization $\alpha'=1$,

$$T_{25}={1\over 2\pi^2g_o^2},$$

and the exact analytic solution gives

$$E[\Psi_{\rm tv}]=-{1\over 2\pi^2g_o^2}.$$

This is an exact cancellation of the D-brane tension. It is not merely a qualitative picture.

The perturbative open-string vacuum describes an unstable D-brane. At the tachyon vacuum the energy shift is $-T_p$, canceling the brane tension. Localized tachyon profiles represent lower-dimensional D-branes.

No open strings at the tachyon vacuum

Let $\Psi_0$ be any classical solution,

$$Q_B\Psi_0+\Psi_0*\Psi_0=0.$$

Write fluctuations around it as

$$\Psi=\Psi_0+\Phi.$$

The quadratic part of the shifted action is governed by a new BRST operator

$$Q_{\Psi_0}\Phi =Q_B\Phi+\Psi_0*\Phi-(-1)^{|\Phi|}\Phi*\Psi_0.$$

The equation of motion for $\Psi_0$ implies

$$Q_{\Psi_0}^2=0.$$

So the physical open-string spectrum around the new solution is the cohomology of $Q_{\Psi_0}$.

For the tachyon vacuum $\Psi_{\rm tv}$, Sen’s second conjecture states that this cohomology is empty. In analytic solutions this can be shown by constructing a homotopy operator $A$ satisfying

$$Q_{\Psi_{\rm tv}}A=I.$$

If $Q_{\Psi_{\rm tv}}\chi=0$, then

$$\chi=I*\chi=(Q_{\Psi_{\rm tv}}A)*\chi =Q_{\Psi_{\rm tv}}(A*\chi),$$

up to the standard graded signs. Therefore every closed state is exact, and the cohomology vanishes. This is the precise mathematical version of the slogan: after the D-brane decays, no open strings remain.

Closed strings are not gone. The tachyon vacuum is the closed-string background without the original D-brane. Open-string degrees of freedom were tied to the brane; once the brane disappears, their perturbative spectrum disappears with it.

D-brane descent as tachyon lumps

Sen’s third conjecture says that lower-dimensional D-branes arise as localized solitons of the tachyon field. For example, on an unstable space-filling brane one can look for a tachyon profile depending on one transverse coordinate $x$ such that

$$T(x)\to T_{\rm tv} \quad\text{as}\quad x\to\pm\infty,$$

but with a localized defect near $x=0$. This defect carries the tension and open-string spectrum of a D-brane of one lower dimension.

The expected descent relation is

$$T_{p-1}=2\pi\sqrt{\alpha'}\,T_p.$$

More generally, a codimension-$q$ tachyon lump on a space-filling brane represents a D$(p-q)$-brane. The lump is not an extra object inserted by hand; it is a classical solution in the original open string field theory.

This is conceptually striking. A theory formulated using the open strings of one D-brane contains, nonperturbatively, other D-branes as solitons. In modern language, D-brane charge is encoded in the topology of the open-string tachyon configuration, closely related to K-theory.

Level truncation and the meaning of exactness

Before analytic solutions were known, tachyon condensation was tested by level truncation. One truncates the string field to oscillator level $L$ and keeps interaction terms up to some maximum level. At the crudest level one keeps only the tachyon mode,

$$|\Psi\rangle\approx t\,c_1|0\rangle,$$

and obtains a cubic potential with an unstable critical point. Adding more and more massive fields rapidly improves the approximation to

$$E[\Psi_{\rm tv}]=-T_p.$$

The convergence was one of the early triumphs of open SFT: the infinite tower of string modes is not decorative. Massive fields are essential for quantitatively correct off-shell physics.

The analytic tachyon-vacuum solution made the story exact. It showed that Witten’s cubic action, despite its compact form, contains the full nonperturbative physics of D-brane annihilation in a fixed closed-string background.

What open SFT does and does not claim

Open string field theory around a given BCFT is a theory of open strings on that boundary condition. Classical solutions can represent new boundary conditions, including the absence of the original brane. But the closed-string bulk CFT is held fixed in the simplest formulation.

This distinction matters. If one wants a fully dynamical theory in which closed-string backgrounds themselves change, one needs closed string field theory or open-closed string field theory. These are more complicated because the moduli spaces of closed Riemann surfaces have more intricate decomposition properties.

Still, open SFT is one of the sharpest laboratories for nonperturbative string theory. It gives an honest action, exact classical solutions, gauge-invariant observables, and quantitative control of D-brane decay.

Exercises

Exercise 1: Varying Witten’s cubic action

Starting from

$$S[\Psi] =-{1\over g_o^2}\left( {1\over2}\langle\Psi,Q_B\Psi\rangle +{1\over3}\langle\Psi,\Psi*\Psi\rangle \right),$$

show that the equation of motion is

$$Q_B\Psi+\Psi*\Psi=0.$$

Solution

Use the BPZ property of $Q_B$ and cyclicity of the cubic vertex. The variation of the kinetic term is

$$\delta\left({1\over2}\langle\Psi,Q_B\Psi\rangle\right) ={1\over2}\langle\delta\Psi,Q_B\Psi\rangle +{1\over2}\langle\Psi,Q_B\delta\Psi\rangle.$$

Because $\Psi$ is Grassmann odd and $Q_B$ is BPZ odd, the second term equals the first. Thus

$$\delta S_2=-{1\over g_o^2}\langle\delta\Psi,Q_B\Psi\rangle.$$

For the cubic term, cyclicity gives three equal contributions:

$$\delta\left({1\over3}\langle\Psi,\Psi*\Psi\rangle\right) = \langle\delta\Psi,\Psi*\Psi\rangle.$$

Therefore

$$\delta S=-{1\over g_o^2}\langle\delta\Psi,Q_B\Psi+\Psi*\Psi\rangle.$$

Since $\delta\Psi$ is arbitrary, the equation of motion is

$$Q_B\Psi+\Psi*\Psi=0.$$

Exercise 2: Ghost number bookkeeping

Show that the kinetic and cubic terms in the open bosonic SFT action have the correct total ghost number on the disk.

Solution

The bosonic open-string field has ghost number one:

$$\operatorname{gh}(\Psi)=1.$$

The BRST charge has ghost number one, so

$$\operatorname{gh}(Q_B\Psi)=2.$$

Thus the kinetic pairing has total ghost number

$$\operatorname{gh}(\Psi)+\operatorname{gh}(Q_B\Psi)=1+2=3.$$

For the cubic term, the star product adds ghost number:

$$\operatorname{gh}(\Psi*\Psi)=2.$$

Then

$$\operatorname{gh}(\Psi)+\operatorname{gh}(\Psi*\Psi)=1+2=3.$$

This is exactly the required ghost number for a nonzero disk correlator, because three $c$-ghost zero modes must be saturated.

Exercise 3: Linearized spectrum as BRST cohomology

Linearize the open SFT equation of motion around $\Psi=0$ and show that the physical free spectrum is BRST cohomology at ghost number one.

Solution

The full equation is

$$Q_B\Psi+\Psi*\Psi=0.$$

Near $\Psi=0$, the quadratic term is negligible, so the free equation is

$$Q_B\Psi=0.$$

The gauge transformation is

$$\delta\Psi=Q_B\Lambda+O(\Psi),$$

so at the linearized level

$$\delta\Psi=Q_B\Lambda.$$

Thus physical free fluctuations are BRST-closed states modulo BRST-exact states:

$$\mathcal H_{\rm phys}={\ker Q_B\over \operatorname{im}Q_B}$$

at ghost number one. This is precisely the first-quantized open-string physical-state condition.

Exercise 4: Wedge-state semigroup and the sliver projector

Assume wedge states obey

$$W_\alpha*W_\beta=W_{\alpha+\beta}.$$

Show that $W_0$ is the identity and that $W_\infty$ is a projector.

Solution

Set $\alpha=0$. Then

$$W_0*W_\beta=W_\beta, \qquad W_\beta*W_0=W_\beta.$$

So $W_0$ acts as the identity string field.

Now define

$$W_\infty=\lim_{\alpha\to\infty}W_\alpha.$$

Then

$$W_\infty*W_\infty = \lim_{\alpha,\beta\to\infty}W_{\alpha+\beta} =W_\infty.$$

Therefore $W_\infty$ is a projector. This infinite-width wedge state is called the sliver.

Exercise 5: The shifted BRST operator squares to zero

Let $\Psi_0$ satisfy the classical equation

$$Q_B\Psi_0+\Psi_0*\Psi_0=0.$$

Define

$$Q_{\Psi_0}\Phi =Q_B\Phi+\Psi_0*\Phi-(-1)^{|\Phi|}\Phi*\Psi_0.$$

Show that $Q_{\Psi_0}^2=0$.

Solution

The operator $Q_{\Psi_0}$ is the covariantized BRST operator. A direct computation using the derivation property of $Q_B$ and associativity of $*$ gives

$$Q_{\Psi_0}^2\Phi = (Q_B\Psi_0+\Psi_0*\Psi_0)*\Phi - \Phi*(Q_B\Psi_0+\Psi_0*\Psi_0),$$

with the appropriate graded signs included in the second term.

Since $\Psi_0$ solves

$$Q_B\Psi_0+\Psi_0*\Psi_0=0,$$

both terms vanish. Hence

$$Q_{\Psi_0}^2\Phi=0$$

for every fluctuation $\Phi$.

Exercise 6: Trivial cohomology from a homotopy operator

Suppose there exists a string field $A$ such that

$$Q_{\Psi_{\rm tv}}A=I.$$

Show that the cohomology of $Q_{\Psi_{\rm tv}}$ is trivial.

Solution

Let $\chi$ be a closed state:

$$Q_{\Psi_{\rm tv}}\chi=0.$$

Since $I$ is the identity string field,

$$\chi=I*\chi.$$

Using $I=Q_{\Psi_{\rm tv}}A$, we get

$$\chi=(Q_{\Psi_{\rm tv}}A)*\chi.$$

Because $Q_{\Psi_{\rm tv}}$ is a derivation and $Q_{\Psi_{\rm tv}}\chi=0$, this can be written as

$$\chi=Q_{\Psi_{\rm tv}}(A*\chi),$$

up to the standard sign determined by the Grassmann parity of $A$. Hence every closed state is exact. The cohomology is empty.

Exercise 7: D-brane descent and tension ratios

Using the descent relation

$$T_{p-1}=2\pi\sqrt{\alpha'}\,T_p,$$

show that a codimension-$q$ tachyon lump on a D$p$-brane has the expected tension of a D$(p-q)$-brane.

Solution

Apply the descent relation repeatedly. For one codimension,

$$T_{p-1}=2\pi\sqrt{\alpha'}\,T_p.$$

For two codimensions,

$$T_{p-2}=2\pi\sqrt{\alpha'}\,T_{p-1} =(2\pi\sqrt{\alpha'})^2T_p.$$

Continuing $q$ times gives

$$T_{p-q}=(2\pi\sqrt{\alpha'})^qT_p.$$

Therefore a codimension-$q$ tachyon lump has exactly the tension expected for a D$(p-q)$-brane, provided the SFT solution has the correct descent normalization.