D-branes and time dependent dynamics in two dimensional string theory

Fortschr. Phys. 52, No. 6 – 7, 531 – 539 (2004) / DOI 10.1002/prop.200310141

D-branes and time dependent dynamicsin two dimensional string theory

M. Gutperle∗

Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA

Received 15 December 2003, accepted 29 February 2004Published online 14 May 2004

We review recent progress in understanding unstable branes and their decays in two dimensionsional non-critical string theory using the dual formulation of the string theory as a matrix model.

c© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

Two dimensional string theories were studied intensely in the late eighties and early nineties, since theyadmit a nonperturbative definition in terms of a matrix model (see [1, 2] and references therein).

Two dimensional string theory was also the first place where nonperturbative effects of strength e− 1gs

were found [3]. Subsequently such effects were associated with D-branes [4] and D-instantons [5, 6].

Recently there has been a great deal of interest in time dependent backgrounds in string theory. It is im-portant to understand how string theory deals with spacelike singularities inside black holes or cosmologicalspacetimes. A somewhat simpler question is the description of the decay of unstable branes in open stringtheory. D-branes can be created as (spacelike) kinks of the open string tachyon field [7] and analogous timedependent tachyon lead to the picture of ‘S-branes’ or spacelike brane [8]. Sen found an exact boundaryconformal field theory realization of this idea by analytically continuation of an exact euclidean BCFT[9, 10]. There are many interesting questions associated with this system, e.g. what happened to the openstrings after the brane decays [11, 12] and what is the nature of the ‘tachyon matter’ [13]?

In some sense we have come a full circle in the last year: In [14] it was realized that the there is a simpledescription of unstable D-brane and their decays in two dimensional string theory using the matrix model.

It is also the case that the this system is a simple realization of the ideas of holography and open/closedstring duality [15]. The eigenvalues of the matrix in a 0+1 dimensional Matrix-quantum mechanics createan extra bulk dimension and the matrix model can be viewed as the worldvolume theory of the D0-branes.

In addition other matrix models have now been discussed using the perspective of branes, see e.g [16–20].It would also be interesting to readdress the question of the description of the two dimensional black holein the matrix model from the perspective of the unstable D-branes [21, 22, 24, 37].

Even though two dimensional string theory has far less rich dynamics than critical string theory and isat best a toy model. It seems sensible to expect that some of the general features and lessons of the twodimensional case carry over to the critical string theory.

∗ E-mail: [email protected]


532 M. Gutperle: D-branes and time dependent dynamics

2 Two dimensional string theory and the matrix model

Starting from an action for N ×N hermitian matrices,

S = βN

∫dt

{12tr(M)2 − trV (M)

}, (1)

in the singlet sector the dynamics reduce to the eigenvalues of the Hermitian matrix M . The dynamicsreduce to to a Hamiltonian describing nonrelativistic fermions moving in an potential. The matrix modelis related to the continuum string theory in the double scaling limit. In this case the details of the potentialdo not matter and the fermions move in an inverted harmonic oscillator potential.

H =∫dx

{12∂xψ

†∂xψ − x2

2ψ†ψ + µψ†ψ

}. (2)

Whereψ is a second quantized fermionic field obeying the anticommutation relations {ψ(x, t), ψ†(x′, t)} =δ(x− x′). The ground state is given by filling the energy levels in the inverse harmonic oscillator potentialup to µ = 1/g from the top.

The two dimensional bosonic string corresponds to filling the Fermi sea on one side of the potential. Thistheory suffers from nonperturbative instabilities and ambiguities due to the tunneling of fermions to theunfilled side of the potential. It was recently suggested [27, 28] that the theory with a symmetrically filledfermi sea corresponds to a noncritial type OB theory (with worldsheet but not spacetime supersymmetry),which does not suffer from instabilities. Note that in this case the regions x > 0, x < 0 do not correspondto two different asymptotic spatial regions but rather excitations which are even (odd) are identified withexcitations of the NS-NS tachyon (RR-scalar).

In the matrix model the basic dynamical process consists of fluctuations propagating in from x = −∞scatter off the potential and propagate back out to x = −∞. This is the matrix model version of the S-matrixfor tachyons. The tree level S-matrix is obtained by treating the fermions classically. In the classical limit,each fermion is described by a point in phase space. The Fermi sea is the region bounded by p±(x)0,where p±(x)0 = ±

√x2 − 2µ. It is convenient to express the fluctuations of the Fermi surface in terms of

a massless scalar field

p±(x, t) = ∓x± 1x

(±πS(q, t) − ∂qS(q, t)) (3)

where q = − ln(−x). For q → −∞, S becomes a free massless scalar field, admitting the mode expansion

S(q, t) =∫ ∞

−∞

dk√8π2k2

(ake

−i|k|t+ikq + a†ke

i|k|t−ikq)

(4)

Since the fermions are free it might be surprising that there is nontrivial scattering, however the S-matrixof string tachyons is defined only in terms of the bosonic field S, given by the bosonization of the relativisticfermion. The Fermions are free, but become nonrelativistic for small x, which leads to a nontrivial S-matrix.In [36] the relation between incoming and outgoing modes was worked out, here we use the notation of[31].

a†k = (

12µ)−ik

∞∑n=1

1n!

(i√2πµ

)n−1 Γ(1 − ik)Γ(2 − n− ik)

×∫ 0

−∞d1 . . . dkn(a†

k1− ak1) . . . (a

†kn

− akn)δ(±|k1| ± · · · ± |kn| − k) (5)

Here the signs in the delta functions are +|ki| for a†ki

and −|ki| for a aki . The bosonized modes ak

and the tachyonic modes in spacetime are related by a nonlocal ’leg-pole’ factor [ak]ws = −i(4µ)−ik(k2

×(Γ(ik)/Γ(−ik))ak


Fortschr. Phys. 52, No. 6 – 7 (2004) / www.fp-journal.org 533

3 D-brane decay in the matrix model

Besides the propagation of small ripples on the Fermi surface another class of excitations is given by movinga single fermion away from the Fermi-sea and onto the top of the harmonic oscillator potential.

It was conjectured in [14] that such a state corresponds to a D0-brane localized in the strong couplingregion. The corresponding boundary state in Liouville theory was constructed in [25]. In [26] a preciseidentification was given of the process in which the fermion rolls down the potential corresponds to therolling of the open string tachyon on the D0-brane. As t → ∞ single fermion comes asymptotically closeto the occupied by the Fermi sea. The state is best described in closed string language as an outgoing pulseof radiation. As shown [26], bosonization provides a dictionary between single fermion states and bosoniccollective field states [29, 30]. Starting from the ψ field appearing in (2), we change variables as

ψ(x, t) =1xe−iµt+ i

2 x2ψL(x, t) +

1xe−iµt− i

2 x2ψR(x, t) (6)

which amounts to removing the WKB part of the wavefunction for large negative x. Define q = − ln(−x)and substitute (6) into the Hamiltonian (2), keeping only terms which survive for large negative q. Oneobtains a Hamiltonian for a relativistic fermion

H =∫ ∞

−∞dq

(iψ†

R∂qψR − iψ†L∂qψL

). (7)

A relativistic Fermion can be bosonized in the following way

ψR/L = 1√2π

: exp(i√π

∫ q(πS ∓ ∂qS)dq′) : (8)

The bosonized field S is the same as the field which appeared earlier in (4).Single fermion states are obtained by acting with ψL,R on the vacuum |0〉. Of course, in our context the

relevant ground state consists of the filled Fermi sea. However, if we consider wavepackets which have verysmall overlap with states of the Fermi sea, we can effectively consider acting on the zero particle groundstate |0〉. Such states are

ψR(q, t)|0〉 = 1√2π

exp[−2i

√π

∫ ∞0

dk√8π2k2 a

†ke

−i(kq−|k|t)]|0〉 (9)

ψL(q, t)|0〉 = 1√2π

exp[2i

√π

∫ 0−∞

dk√8π2k2 a

†ke

−i(kq−|k|t)]|0〉 (10)

Classically, the fermions move along trajectories obeying the equations of motion. Trajectories with aturning point at t = 0 are

x(t) = −λ cosh t, λ = sinπλ. (11)

At early and late times, the trajectories become relativistic

t → ±∞ : q(t) = ∓t− lnλ

2. (12)

Therefore, at early and late times we can write

ψR = ψR(t− q), ψL = ψL(t+ q). (13)

Asymptotic states can be obtained by acting with either ψR or ψL in the region of large negative q. Wecan choose to work in terms of the ψR states, since the ψL states are related to these by reflecting off thepotential. It turns out to be convenient to write the incoming state as

ψR

(t− q + ln

µ

2

)|0〉, t− q = − ln

λ

2. (14)



According to the bosonization (8), the state (14) is√

2πψR(t− q + lnµ

2)|0〉 = exp

[−2i

√π

∫ ∞0

dk√8π2k2 a

†ke

ik(t−q+ln µ2 )

]|0〉 (15)

= exp[−2i

√π

∫ ∞0

dk√8π2k2 a

†kµ

ike−ik ln λ]|0〉 (16)

The state (16) can be viewed either as a single incoming fermion (the “open string interpretation"), or asan incoming pulse of tachyons (the “closed string interpretation".) We can now use our tree level S-matrixresult (5) to reexpress it in terms of outgoing states as

exp[2i

√π

∑∞n=1

1n!

∫ ∞0

dk√8π2k2

∫ 0−∞dk1 · · · dkne

iθ(k)(

i√2πµ

)n−1 Γ(1−ik)Γ(2−n−ik)

: (a†k1

− ak1) . . . (a†kn

− akn) : δ(±|k1| ± · · · ± |kn| − k)

]|0〉 (17)

where the phase factor is given by eiθ(k) = −2ike−ik ln λ. The state (17) represents a particular state ofoutgoing tachyons,

Note that the exponent of (17) contains both creation and annihilation operators for bosonic quanta. It isnatural to normal order this expression such that only creation operators remain. Note that such a proceduredecreases the number of creation operators with respect to the powers of g = 1/µ. It is therefore possibleto double expand the exponent of (17) in terms of g and the number creation operators a†

k.The leading contribution with a single creation operator is of order g0 and given by

exp

[2i

√π

∫ 0

−∞

dk1√8π2k2

1

eiθ(|k|)a†k1

]|0〉. (18)

As was shown in [26] agrees with a coherent state of outgoing closed string radiation which is given byexponentiating the disk one-point function. To get agreement, the time part of the CFT should include aboundary interaction λ cosh t, and the zero mode should be integrated using the Hartle-Hawking contourextending to t = +i∞. Alternatively, one can use the boundary interaction λet. We will think in terms ofthe λ cosh t interaction since it seems to correspond more naturally to our setup at early times. One featureof the Hartle-Hawking contour is that it restricts us to computing just the production of closed string statesand not their absorption, since convergence of the zero mode integral requires vertex operators to behaveas eiωt with ω > 0.

Similarly it was proposed in [31] to identify the other terms in (17) with other perturbative amplitudesin the rolling tachyon background. If we only keep terms with all creation operators we get

exp

[2i

√π

∞∑n=1

1n!

∫ 0

−∞dk1 · · · dkne

iθ(|k|)(

i√2πµ

)n−1 Γ(1 − ik)Γ(2 − n− ik)

a†k1

· · · a†kn

]|0〉. (19)

The counting of powers of the string coupling suggest that these terms arise from the sum of disk diagramswith any number of tachyon vertex operators. This result provides a conjecture for the Tachyon n-pontfunction on the disk for the boundary CFT defined as the product of a localized brane in the Liouvilledirection and a rolling tachyon CFT in the time direction. Note that the one-point function is the onlyamplitude which has been calculated, even for the two pointfuntion the relevant correlation function in therolling tachyon and boundary Liouville CFT are not known yet.

The remaining terms expansion of (17) involve normal ordered terms and hence a reduction of the numberof creation operators at a given order of g. In [31] it was conjectured that these terms correspond to stringdiagrams coming from worldsheets with more than one boundary, i.e. the annulus etc. Again if correct thisidentification provides us with a prediction for the n-point tachyon correlation functions on a sphere with



m holes. We should emphasize that the only approximation we have made was to treat the closed strings(the collective field) classically, which means that we are correctly including quantum effects due to openstrings. Of course, this is very familiar from the AdS/CFT correspondence, where we are used to sayingthat classical nonlinear closed string effects are dual to quantum open string effects.

4 Unstable branes in the matrix model

Classically, a fermion placed at the top of the inverted harmonic oscillator potential can stay there forever.In the two dimensional string theory this corresponds to the fact that one can construct a boundary statecorresponding to an eternal unstable D-brane. The boundary state is constructed by tensoring the (m,n) =(1, 1) boundary state of [25] with a Neumann boundary state for the free time directions X0.

| B〉 =| φ〉(m,n)=(1,1)⊗ | X0〉N (20)

Quantum mechanically, a localized wavefunction will spread and the unstable brane will have a finitelifetime. In the string theory the instability of the static unstable D0 brane manifests itself in the appearanceof an imaginary part in the annulus partition function [32, 33]. It is an interesting question whether such anunstable brane will modify the classical closed string scattering. From the worldsheet perspective one againexpects to find corrections from disk amplitudes. But on the matrix model side our preceding analysis doesnot directly apply since it heavily used the bosonization of the fermion in the asymptotic region, whereashere we want to keep the fermion at the origin.

4.1 Modified collective field theory

In the singlet sector the matrix model action (1) reduces to the following action for the eigenvalues

L =∑

i

1

2(∂tλi)2 − 1

2

∑j �=i

1(λi − λj)2

− V (λi)

(21)

In the action (21) the eigenvalues are treated as bosonic instead of fermionic. Note however that there is aa potential 1/(λi − λj)2 which repels eigenvalues from each other.

Das and Jevicki [34] introduced a collective field to describe the dynamics of the eigenvalues in the largeN limit ∂xφ(x, t) =

∑i δ(x− λi(t)) The dynamics of the collective field φ is described by the following

Lagrangian

L =∫dx

(12∂tφ∂tφ

∂xφ− π2

6(∂xφ)3 − (V (x) − µF )∂xφ

)(22)

In the double scaling limit the potential is given by V (x) = 12 (V0 − x2) and one takes N → ∞, µ =

V0 − µF → 0, keeping Nµ = µ fixed. The Lagrangian (22) becomes then

L =∫dx

(12∂tφ∂tφ

∂xφ− π2

6(∂xφ)3 +

(12x2 − µ

)∂xφ

)(23)

The string coupling constant is related to the height of the double scaled potential by µ = 1/g.A variation on the collective field theory of Das and Jevicki was developed in [35], where one splits

a single eigenvalue λ0(t) from the collective field φ(x, t) and treats it separately. This is justified foreigenvalue distributions where the single eigenvalue is away from the dense region of the Fermi sea, i.e.|λ| � 1

g . The dynamics of the filled Fermi sea is again described by the collective field φ(x, t). The actionfor the coupled system is

L =12(∂tλ0)2 +

12λ2

0 −∫dx

∂xφ

(x− λ0)2+

∫dx

(12∂tφ∂tφ

∂xφ− π2

6(∂xφ)3 +(

12x2 − 1

g)∂xφ

)(24)



In order to disentangle the dynamics of the single eigenvalue and the collective field it is useful to performthe following rescaling φ = g−1φ, x = g− 1

2 x and λ = g− 12 λ. The action (24) becomes

L =1g

(12(∂tλ0)2 +

12λ2

0

)−

∫dx

∂xφ

(x− λ0)2+

1g2

∫dx

(12∂tφ∂tφ

∂xφ− π2

6(∂xφ)3

+(

12x2 − 1

)∂xφ

)(25)

The form of (25) suggests an interpretation as an open-closed string field theory action. The part whichis of order 1/g can be interpreted as the action for the open string degree of freedom associated with theunstable D-brane. The part which is of order 1/g2 is the Das-Jevicki collective field action and correspondsto the action for the closed strings. The coupling between open and closed strings comes at order g0 and inthe limit g → 0 becomes unimportant. Furthermore the g → 0 limit corresponds to the limit where λ0 andφ can be treated as classical fields.

A simple solution of the decoupled equation is given by

λ0(t) = a1 cosh(t) + a2 sinh(t), ∂xφ =1π

√x2 − 2

g(26)

corresponding to a rolling eigenvalue and a static Fermi sea. It is interesting to analyze what happensif g is small but nonzero. When the eigenvalue λ is of order 1/g

12 the interaction term becomes important.

For the rolling tachyon (26) this happens at a time t = − log g12 . One might be tempted to argue that this

implies that there is a strong interaction between the eigenvalue and the Fermi sea, which starts at x2 = 2/g.However this is not clear, since the derivation of the action (24) assumed that the single eigenvalue is wellseparated from the Fermi sea, so the action (24) might not be a good description of the actual dynamics inthis case.

4.2 Scattering from a static D-brane

Instead of the rolling tachyon, in the classical limit one can consider a solution where the eigenvalue sits ontop of the inverse harmonic oscillator potential λ(t) = 0 for all t. The equation of motion for the collectivefield then becomes

∂t

(∂tφ

∂xφ

)− ∂x

{12

(∂tφ

∂xφ

)2

+π2

2(∂xφ)2 − 1

2x2 +

1g

+1x2

}= 0. (27)

The interaction term modifies the static solution

∂xφ =1π

√x2 − 2

g− 2x2 , (28)

which is valid for x2 > 2/g. Note that at large x the corrections to the standard solution of the static filledFermi-sea without any extra Fermion at the top of the potential are of order 1/x3 and subleading. A possibleinterpretation is that a localized D-brane near x = 0 only has a weak backreaction on the fields in the weakcoupling region at x = −∞.

The Das-Jevicki collective field is related to the description of the dynamics of the Fermi sea of Polchinskiby ∂xφ = 1

2π (p+ − p−). This implies that the equation of motion for p± get modified to

∂tp± = x+2x3 − p±∂xp± (29)



In the classical limit the motion of the fermions is described by an incompressible fluid moving in a potential.From (29) it is clear that the eigenvalue at λ0 = 0 produces a small modification of the Hamiltonian whichgoverns the motion of points in the phase space,

H =12p2 − 1

2x2 +

1x2 + µ. (30)

As closed string excitations are represented by small ripples in the Fermi sea whose turning point is atx2 ∼ 1/g, the extra term in (30) is only a small perturbation. The equations of motion following from (30)are hence

d2

dt2x(t) − x(t) − 2

x(t)3= 0. (31)

It is interesting that the equation of motion (4) is one of the few modifications of the (inverted) harmonicoscillator which can be solved exactly:

x(t) = a

√cosh2(t− σ) + b, b = −1

2(1 −

√1 − 8

a4 ). (32)

We have chosen the root for which (6) goes over to the solution for the inverted harmonic oscillator inthe limit g → 0. We now follow the arguments of Polchinski [36] to derive the classical scattering from thetime delay. The time delay for the motion from a given x and back is calculated using (6),

t′ − q = t+ q + ln(a2

4

), (33)

up to terms which vanish exponentially at late and early times respectively. Using the relation ε± =±(p± x)x one finds

ε−(t+ q) =a2

2

√1 − 8

a4 . (34)

The relation ε−(t+q) = ε+(t′ −q) can be reexpressed to produce a nonlinear relation between incomingand outgoing waves.

ε−(t+ q) = ε−

(t+ q + ln

(12

√ε−(t+ q)2 + 2

)). (35)

Expanding ε+(t− q) = 1g + δ+(t− q), ε−(t+ q) = 1

g + δ−(t+ q) and using the relation of δ± to thebosonic excitations of the tachyon one can calculate the S-matrix. It is easy to see that the formula for thetime delay (33) and hence the result for the S-matrix are modified at order g2.

At first sight this seems to be a puzzling feature since from string perturbation theory one would haveexpected that the corrections are of order g, coming from the disk versus sphere diagrams. This result canbe traced back to the fact that the interaction term in (25) comes at order g0 instead of order 1/g. It wouldbe interesting to understand this fact better from the second quantized fermion point of view. The basicpuzzle is that a fermion on the top of the potential has only an exponentially small overlap with states ofthe Fermi sea, and so would not seem to affect the perturbative scattering amplitudes.

A possible interpretation is that the backreaction on the closed string background caused by the presenceof the fermion on the top of the potential is weaker than one might have expected. Since the D0-brane islocalized in the strong coupling region in the Liouville direction the conventional intuition of gravitationalforces of branes in flat space might be misleading.

An indication that this is the case comes from considering the boundary state representing a static D-branein 2 dimensions. The vertex operator for on shell ’massless’ tachyon Vk is

Vk = e(2+ik)φ−i|k|t. (36)



The one point function on the disk of the Liouville primary Uk = e(2+ik)φ is given by [25]

〈e(2+ik)φ〉 =2√πi sinh(πk)µ−i k

2Γ(ik)

Γ(−ik) . (37)

Note that the Neumann boundary conditions on the X0 enforce k = 0 by momentum conservation. Hencein contrast to the rolling tachyon boundary state discussed in section five, the only physical state appearingin the boundary state is Vk with k = 0. It follows that the one point function of the tachyon Vertex operatoris zero because of the sinh(πk) factor in (37). This is in contrast to the rolling tachyon boundary state wherethe time component of the one pointfunction provides a 1/ sinh(πk) factor which leads to a finite result.

Our results further seem to imply the vanishing of all disk amplitudes with on-shell closed string vertexoperators. This conclusion is in fact consistent with the T-dual version of our setup, studied in [37]. Adirect calculation of these correlation functions to check this conjecture would be very interesting.

We should also note that the deformation of the inverted harmonic oscillator given by (30), has beendiscussed in two other contexts. Firstly in [38] a deformation of the matrix model potential was consideredand secondly in [39] it was argued that this Hamiltonian describes scattering in the matrix model for twodimensional type 0A string theory. It would be interesting whether there is any deeper relation betweenthese different theories.

5 Phase space density of the matrix model

The classical limit treats the fermions as an incompressible classical fluid, quantum mechanicaly howeverthe phase space is quantized and has a fundamental unit of order � = 1/g. An interesting formulation of thethe free fermion theory was given by Dhar et al. in [40, 41], which utilizes the W∞ symmetry underlyingthe free fermionic theory

The fundamental object is the bilocal fermion density operator given by

u(p, q, t) =∫ ∞

−∞dxψ†(q − �x/2, t)e−ipxψ(q + �x/2, t) (38)

where � can be identified with the string coupling g. The phase space density satisfies the followingconstraints ∫

dpdq

2πgu(p, q, t) = N (39)

cos(

�

2(∂q∂p′ − ∂q′∂p)

)u(p, q, t)u(p′, q′, t)|p=p′,q=q′ = u(p, q, t) (40)

u(p, q, t) satisfies the equation of motion

(∂t + p∂p + q∂q)u(p, q, t) = 0 (41)

Note that in the classical limit � → zero the constraints reduce to the condition that u becomes a projectoron regions of the Fermi sea which are filled with constant density. The above formulation does howeverincorporate the quantum effects and it would be very interesting to analyze the decay of unstable branesas well as other processes in the matrix model (like for example the cosmological solutions of [42]) inthis framework. For example in [41] a time dependent process corresponding to eigenvalue tunneling wasalready discussed.

Acknowledgements I would like to thank the organizers for a splendid conference and the warm hospitality. I wouldalso like to thank Per Kraus for an enjoyable collaboration. The work of the author is supported in part by NSF grant0245096. Any opinions, findings and conclusions are those of the author and not necessarily reflect the views of theNational Science Foundation.



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