Superstring disk amplitudes in a rolling tachyon background

PHYSICAL REVIEW D 73, 046007 (2006)

Superstring disk amplitudes in a rolling tachyon background

Niko Jokela,1,* Esko Keski-Vakkuri,2,† and Jaydeep Majumder1,‡

1Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FIN-00014, Finland2Department of Physical Sciences, University of Helsinki, P.O. Box 64, FIN-00014, Finland

(Received 21 November 2005; published 24 February 2006)

*Electronic†Electronic‡Electronic

1550-7998=20

We study the tree level scattering or emission of n closed superstrings from a decaying non-BPS branein Type II superstring theory. We attempt to calculate generic n-point superstring disk amplitudes in therolling tachyon background. We show that these can be written as infinite power series of Toeplitzdeterminants, related to expectation values of a periodic function in Circular Unitary Ensembles. Furtheranalytical progress is possible in the special case of bulk-boundary disk amplitudes. These are interpretedas probability amplitudes for emission of a closed string with initial conditions perturbed by the additionof an open string vertex operator. This calculation has been performed previously in bosonic string theory,here we extend the analysis for superstrings. We obtain a result for the average energy of closedsuperstrings produced in the perturbed background.

DOI: 10.1103/PhysRevD.73.046007 PACS numbers: 11.25.Uv, 11.25.Db, 11.25.Hf, 11.80.�m

I. INTRODUCTION

One of the basic open questions in string theory isunderstanding the decay of unstable branes. Sen has pro-posed a conformal field theory (CFT) description for spa-tially homogenous decay by deforming the open stringworld sheet theory by exactly marginal rolling tachyonbackgrounds [1–3]. This process can be interpreted as aspacelike brane localized in time (full S-brane) [4]. Analternative, rescaled rolling tachyon background [5] corre-sponds to decay starting from past infinity (half S-brane).One can also consider brane decay on a space-time orbifoldwith a semi-infinite time direction, to obtain a model wherethe unstable brane is prepared at origin of time and thendecays [6]. Basic questions such as computing amplitudesfor scattering or emission of strings from decaying braneshave turned out to lead into quite complicated calculationsrendering it difficult to draw out lessons of physics interest.Several different approaches to this problem have beenexplored, such as timelike boundary Liouville theory [7]and matrix integrals [8]. In particular, for full S-branes, aprescription based upon analytic continuation to imaginarytime where the full S-brane corresponds to an array ofsmeared D-branes, was proposed in [9]. Further referencesinclude [10–31], and the recent reviews [32,33]. Recently,for half S-branes, this problem was elaborated and mappedinto the study of random matrices [34]. In this paper weextend this approach to a study of superstring scatteringfrom unstable branes in superstring theory. It would beinteresting to compare the random matrix approach withthat of [9].

The general setup is also interesting from the point ofview of cosmology. Recently, there has been progress inconstructing string theoretic models of inflation. Of par-

address: [email protected]: [email protected]: [email protected]

06=73(4)=046007(8)$23.00 046007

ticular motivational interest here are models based in TypeIIB superstring theory, where inflation arises from inter-actions of branes in (single or multiple) warped throats[35–39]. In these models, it has been proposed [36–39]that reheating after inflation is associated with Kaluza-Klein (KK) modes of gravitons that are produced copiouslyas end decay products of massive closed strings emittedfrom decaying D �D-systems at the throats. However, theemission of massive closed strings is at present undercalculational control only for production of single strings,see [14,18,19,27]. In this paper we aim for progress incalculating closed string n-point disk amplitudes in therolling tachyon background in superstring theory, thatcould be interpreted as probability amplitudes for multi-string emission. This is a very complicated problem, andwe are able to make only partial progress.

One technique to organize these calculations is to mapthem to a computation in the language of random matrices:the amplitudes turn out to involve power series of expec-tation values of periodic functions in Circular UnitaryEnsembles (CUEs) of U�N� matrices of increasing rank.This was found in [34] in the context of two-point diskamplitudes in bosonic string theory; in this paper we gen-eralize the observation for generic n-point disk amplitudesin superstring theory. Such expectation value calculationsare a basic question in the theory of random matrices.However, for the particular periodic functions that arisein the calculations, the expectation values are only known1

as Toeplitz determinants of Fourier coefficients of thefunction. Further progress, needed for extracting physicslessons from the amplitudes, is then associated with newprogress in the field of random matrix theory and mathe-matical analysis.

In the special case of bulk-boundary disk amplitudes,two-point functions of one bulk and one boundary vertex

1As far as we are aware of.

-1 © 2006 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.73.046007

NIKO JOKELA, ESKO KESKI-VAKKURI, AND JAYDEEP MAJUMDER PHYSICAL REVIEW D 73, 046007 (2006)

operator, it is known that the calculations can be carried outto the point of actually finding corrections to the one-pointamplitude in an analytic form. These results have beenderived in bosonic string theory [34], and also in [17,24]using Liouville theory methods. In this paper we willextend the calculations and results to the case ofsuperstrings.

This paper is organized as follows. In Section II, weconsider generic n-point superstring disk amplitudes andshow how they are related to infinite power series ofexpectation values in CUEs, or Toeplitz determinants ofincreasing rank. In Section III, we calculate the bulk-boundary disk amplitudes in superstring theory. InSection IV, we interpret the open string vertex operatoras an additional initial perturbation on the decaying brane,and calculate how it corrects the average energy of theemitted closed strings in the decay. Finally, Section V is abrief summary.

II. GENERIC CLOSED STRING DISKAMPLITUDES AND RANDOM MATRICES

We begin by attempting to compute NS-NS and R-Rdisk amplitudes in the background of a decaying brane.Depending on the external momentum assignments, thesecould be interpreted as scattering or emission probabilityamplitudes. As in [27,34], we focus on the 1

2 S-brane orrolling tachyon background, which for the non-BPS braneof Type II superstring corresponds to the exactly marginaldeformation

�SB � ��2p��

Z dt2�

0eX0=��2p

� �1; (1)

where 0 is the time component of the world sheet fermionfield and �1 is a Chan-Paton factor associated with theboundary tachyon, which can be related to the one-dimensional boundary fermion � [5,27,40,41]; see theappendix for an elaboration on this point. For the bulkclosed string vertex operators Vs, one can adopt convenientgauge choices [14,42] (see also [27,43]), where the depen-dence on the time component X0 of the bosonic field takesa simple form:

Vs � ei!cX0V?s �Xi; i; ~ i; . . .� (2)

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in the NS-NS sector, and

Vs � ei!cX0�s0

~�~s0V?s �Xi; i; ~ i; . . .� (3)

in the R-R sector, with the spin fields �s0� eis0H0

in thebosonized form. (The ellipsis refers to ghosts and super-conformal ghosts.) Thus, for a generic n-point closed stringamplitude, the nontrivial part of the computation due to thepresence of the rolling tachyon amounts to the expectationvalue

An�!1; . . . ;!n��

�Yna�1

ei!aX0�za;�za��

deformed

�

�e��2p��1

R�

��dt=2�� 0e�X

0=��2p�Yna�1

ei!aX0�za;�za��

(4)

for vertex operators in the NS-NS sector, and

An�!1; . . . ; !n� �

�Yna�1

ei!aX0�za;�za��a�s0~��a�~s0

�deformed

(5)

in the R-R sector. Consider, for example, the NS-NS sectordisk amplitude in more detail. Bosonizing the fermionicsuperpartner 0 and expanding, we obtain (odd termsvanish, since Tr��n1� � 0, for n � odd)

An�!1; . . . ; !n� �Z 1�1

dx0eix0P

na�1

!aX1N�0

��e�x0=��2p��2N

�2N�!

O�!1; . . . ; !n�; (6)

where

O�!1; . . . ; !n� �Z �

��

Y2Ni�1

dti2�

��eiH�ti� � e�iH�ti��

eX0 �ti��

2pYna�1

ei!aX0�za;�za��; (7)

and we have separated out the zero mode from the fluctu-ating part, X0 � x0 X00, and further dropped the super-script 0. The Wick contractions in (7) are easily calculatedand we obtain

O�!1; . . . ; !n� �Xf�ig��

Z 1�1

dhY2Ni�1

��iei�ih�Z �

��

Y2Ni�1

dti2�

Y1�i<j�2N

jeiti � eitj j1�i�j

Y2Ni�1

Yna�1

j1� zae�iti ji

��2p!a

Y1�a<b�n

jza � zbj�!a!b

Yna;b�1

j1� za �zbj��!a!b�=2; (8)

where we have separated out the zero mode h from H. The integral over it enforces a constraint

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SUPERSTRING DISK AMPLITUDES IN A ROLLING . . . PHYSICAL REVIEW D 73, 046007 (2006)

X2Ni�1

�i � 0: (9)

In the sum over �i � �, all the combinations subject to theconstraint contribute equally to (8), as can be seen by anappropriate relabeling of the ti’s. Thus, we can choose

�1; . . . ; �N � 1; �N1; . . . ; �2N � �1 (10)

and count the number of all equivalent terms. This is arandom walk problem, there are �2N�!=�N!�2 such terms.The remaining integrals then factorize and we can write

O�!1; . . . ; !n� � ��1�N�2N�!Y

1�a<b�n

jza � zbj�!a!b

Yna;b�1

j1� za �zbj�1

2!a!bI2N�!1; . . . ; !n�;

(11)

where IN is the integral

IN�!1; . . . ; !n� �1

N!

Z �

��

YNi�1

dti2�

Y1�i<j�N

jeiti � eitj j2YNi�1

�Yna�1

j1� zae�iti ji

��2p!a

�: (12)

The study of this type of integrals is a central question inthe theory of random matrices [44]. We can recognize it asthe expectation value of a periodic function with respect tothe Circular Unitary Ensemble of U�N� matrices,

IN � EU�N�

�YNi�1

f�ti��; (13)

where the periodic function is

f�t� �Yna�1

j1� zae�itji��2p!a : (14)

It contains the information about the locations (modularparameters) za of the closed string vertex operators and theon-shell energies !a. Alternatively, because of the facto-rization, we could have written the result as a U�N� U�N� integral as in [27],

I2N � EU�N�

�YNi�1

f�ti�� EU�N�

�YNi�1

f�ti��

� EU�N�U�N�

�Y2Ni�1

f�ti��: (15)

The integrals (13) can then be evaluated by Heine’sidentity [34,45] and rewritten as Toeplitz determinants ofthe Fourier coefficients2 of f,

2For example, in the case of a 2-point function the Fouriercoefficients turn out to be related to hypergeometric functions,see [34].

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IN � det�f�k�l��1�k;l�N � DN�f�; (16)

where

f �k�l� �Z dt

2�f�t�ei�k�l�t: (17)

Thus the amplitude becomes a Fourier transform of aninfinite series of Toeplitz determinants,

An�!1; . . . ; !n� �Y

1�a<b�n

jza � zbj�!a!b

Yna;b�1

j1� za �zbj��1=2�!a!b

Z 1�1

dx0eix0P

na�1

!aF�x0;!1; . . . ; !n�;

(18)

where

F�x0;!1; . . . ; !n� �X1N�0

��2�2e��2px0�N�DN�f��2: (19)

Unfortunately, the Toeplitz determinants are in generalquite complicated so a more detailed analysis of the infiniteseries is extremely difficult. For example, the radius ofconvergence of (19) is difficult to determine. By physicsreasons, we expect the infinite series to converge at leastfor sufficiently early times (as then the amplitude ap-proaches that for scattering from a stable D-brane). Itmight also be possible to gain some further insight intothe behavior of the series from numerical methods.However, in order to perform the Fourier transform in(18), one would need an analytic expression for (19) andthen analytically continue beyond its expected conver-gence radius, a much harder task.

For R-R sector the story is a bit modified. The amplitude(5) becomes

An�!1; . . . ; !n� �Z 1�1

dx0eix0P

na�1

!aX1N�0

��e�x0=��2p��N

N!

O�!1; . . . ; !n�Tr��1�Nn; (20)

where

O�!1; . . . ; !n� �Z �

��

YNi�1

dti2�

��eiH�ti� � e�iH�ti��e�X

0�ti��=��2p

Yna�1

ei!aX0�za;�za�eisaH�za�ei~sa ~H� �za��: (21)

In the R-R sector one has to explain why the amplitudewith a single R-R vertex operator in the bulk is nonvanish-ing for an odd number of insertions of boundary tachyonvertex operators [46–48]. From (1), it is clear that thevertex operator of the tachyon contains the Chan-Patonmatrix �1. The Chan-Paton Hilbert space is two dimen-

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sional, even for a single non-BPS D-brane, since a non-BPS Dp-brane of Type IIA(B) theory can be thought of as abound state of a Dp-Dp -pair of Type IIB(A) theory. For anodd number of tachyon vertex operator insertions on theboundary, naively we expect that the amplitude vanishesbecause of the presence of the factor Tr��2n1

1 � � 0, n �ve integer. However, this is not the full story, at least forbulk-boundary amplitudes involving R-R sector (whichwill be considered in Section III).

For concreteness, let us suppose we are considering anon-BPS Dp-brane in Type IIA theory (so p is odd). It isobtained by taking a Dp-Dp -brane pair in Type IIB andmodding it out by ��1�FL , where FL is the left-movingspace-time fermion number. The R-R and R-NS sectors ofType IIA can be thought of as ‘‘twisted sector‘‘ states under��1�FL orbifold in Type IIB theory. For diagrams involvingR-R operators it is easier if we stick to Type IIB orbifoldrather than Type IIA language. The operator ��1�FL doesnot act on the matter or ghost part of any open string vertexoperator, but it has an action on the 2 2 CP Hilbertspace. Since under its action a BPS Dp-brane gets ex-changed with a Dp-brane, the representation of ��1�FL

in the CP Hilbert space is �1. So a Type IIA disk diagramwith some N number of boundary tachyon vertex operatorsand a R-R vertex operator inserted in the bulk, from TypeIIB orbifold perspective, is equivalent to a disk diagramwith a cut, associated with the ��1�FL operator, ending onthe boundary. Because of above representation of ��1�FL

in the CP Hilbert space, the trace part in the full amplitudegets another factor of �1, where the cut hits the boundary,i.e., now the trace from the CP sector is Tr��N1

1 �. This isnonvanishing only when N � odd. It is straightforward toextend this procedure for n insertions of R-R vertex op-erators in the bulk. The amplitude will then be nonvanish-ing if �N n� � even. Thus, if N and n are even integersseparately, the amplitude is still nonvanishing.3

For a correlation function in (21) with n number of bulkR-R and N of boundary tachyon operator insertions, thezero mode integral from the temporal part imposes a con-straint

XNi�1

�i � �Xna�1

�s�a�0 ~s�a�0 � � k 2 Z: (22)

By inspection one can see thatN, n, and k all have the same(even or odd) parity.

Similar considerations as for the NS-NS amplitudesshow that the constraint can be satisfied in ( N

N�k2

) equivalent

ways. Omitting contractions which are not relevant for ourdiscussion, the source-dependent part of the amplitudeagain leads to a series of expectation values of periodic

3The whole analysis can be done in terms of the Gliozzi-Scherk-Olive (GSO) operator ��1�F instead of ��1�FL , where Fis the left-moving world sheet fermion number. This is a bitinvolved; interested readers may consult Ref. [47].

046007

functions in CUE ensembles,

O�!1; . . . ; !n� �EU��N�k�=2�

� Y�N�k�=2

i�1

f��ti��

EU��Nk�=2�

� Y�Nk�=2

i�1

f�ti��; (23)

where the periodic functions f��t� resemble (14) but differin their exponents. The underlyingU�N�k2 � U�

Nk2 � struc-

ture was found in [27] in the case of generic 1-pointamplitudes. By Heine’s identity, the source-dependentpart of the amplitude can again be rewritten as an infiniteseries of (products of) Toeplitz determinants of increasingrank. This structure generalizes also to generic (n1 n2)-point disk amplitudes, where n1 (n2) counts NS-NS (R-R)bulk insertions.

Further progress on disk amplitude calculations dependson new techniques, and we hope to return to this problem inthe future. However, it is known that there are some specialamplitudes, where an analytic solution can be found—thebulk-boundary amplitudes. They were computed in thebosonic case in [34], and we will next extend these resultsto superstrings.

III. BULK-BOUNDARY DISK AMPLITUDES

Let us consider the case n � 2, and place4 the othervertex operator into the boundary of the disk, thus renam-ing !1 � !c, !2 � !o. We consider the operator in theboundary to represent an additional open string. In otherwords, we will consider the amplitudes

ANSNS;NS�!c;!o� � hei!cX0�z;�z�ei!oX0�t�ideformed (24)

and

ARR;NS�!c;!o� � hei!cX0�z;�z��s0

~�~s0ei!oX0�t�ideformed:

(25)

We can choose the bulk vertex operator to be inserted at theorigin of the disk, z � �z � 0, while the location t of theboundary vertex operator remains a free modular parame-ter to be integrated over in the end.

A. NS-NS bulk vertex operator

Consider first the case with a NS-NS bulk vertex opera-tor. The amplitude becomes

A2�!c;!o� �Z 1�1

dx0eix0�!o!c�

X1N�0

��2�2e��2px0�N

�IN�!o��2; (26)

4The vertex operator cannot be mapped into the boundary by aconformal transformation.

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where IN�!o� is the integral

IN�!o� �1

N!

Z �

��

YNi�1

dti2�

Y1�i<j�N

jeiti � eitj j2

YNi�1

j1� eite�iti ji��2p!o: (27)

We have removed an apparent divergence resulting fromthe self-contractions on the boundary, by an appropriate

5Apart from an irrelevant overall phase factor �i.

046007

normal ordering [5]. The multiple integrals over ti do notdepend on t, hence we can set t � 0. As noted in [34], theintegral IN can be evaluated using Selberg’s integral for-mula. After some algebra, we can then evaluate the ampli-tude in a closed form in terms of known functions. Defininga ‘‘chemical potential’’ � � � log��2�2e

��2px0�, care-

fully following the calculational strategy in [34], and car-rying out the x0 integral using the real contour of [14] weobtain

A2�!c;!o� �Z 1�1

dx0eix0�!o!c�

X1N�0

e�N��YNj�1

��j��j i��2p!o�

��j i!o=��2p��2

�2

�Z 1�1

dx0eix0�!o!c�

X1N�0

e�N�e2R1

0dtH�t;!o=

��2p��e�Nt�1�

��i��

2p

��i��2p�!o!c�

sinh��!o !c�=��2p�

exp�2 G

�!c��

2p ;

!o��2p

�; (28)

where

G�!c��

2p ;

!o��2p

�Z 1

0dtH

�t;!o��

2p

�ei�!o!c�t=

��2p

� 1�; (29)

with

H�t; !o� ��1� e�i!ot�2

2t�1� cosht�; (30)

similar to the result in [34]. As a simple consistency checkwe can verify that the result reduces5 to the answer in [27]in the absence of the initial open string perturbation, !o �0. This follows easily since H�t; !o� vanishes in the limit.

B. R-R bulk vertex operator

Consider then the R-R closed string vertex operator [27]

�s0~�~s0ei!cX0�zc;�zc�; (31)

and bosonize the spin fields

�s0� eis0H

0; ~�~s0

� ei~s0~H0: (32)

Note that in the series expansion of the amplitude the termswith N � even vanish. The relevant Wick contractionsnow give

�Yi

�eiH�ti� � e�iH�ti��eis0H�0�ei~s0~H�0�ei!cX0�0;0�ei!oX0�t�eX

0�ti�=��2p�

� �2s0�s0;~s0

��1�N�2N 1�!

N!�N 1�!

� Y1�i<j�N

jeiti � eitj j2�� Y

N1�i<j�2N1

jeiti � eitj j2�

�Yi

j1� eite�iti ji��2p!o

� �irrelevant terms�: (33)

Note that we have already integrated out the zero modes ofH and ~H. We have suppressed the details of terms that willultimately not contribute because the bulk vertex operatorhas been placed at the origin. The amplitude becomes

A2�!c;!o� � �2s0�s0;~s0

Z 1�1

dx0eix0�!o!c�

X1N�0

��1�N

��ex0=��2p

�2N1IN IN1; (34)

where IN , IN1 are the same Selberg integrals as before,giving

IN �YNj�1

��j��j i��2p!o�

��j i!o=��2p��2: (35)

Proceeding as before, we get

A2�!c;!o� � �2s0�s0;~s0

Z 1�1

dx0eix0�!o!c�

X1N�0

��1�N

��ex0=��2p

�2N1 exp�Z 1

0dtH�t; !o=

��2p�

�e�Nt�1 e�t� � 2��; (36)

and, after some algebra, finally

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A2�!c;!o� � �2s0��s0;~s0��

2p

��i��2p�!o!c�

cosh��!o !c�=��2p�

exp�G�!c��

2p �

i2;!o��

2p

G

�!c��

2p

i2;!o��

2p

�:

(37)

Note that this again meets the result in [27] as !o ! 0.We would like to add a few comments on the delta

function present in the Eq. (37). It implies that the left-and right-movers in the R-R field are such that s0 � ~s0. Itresults from the correlation function of the spin field alongthe temporal direction of the R-R vertex operator Vs givenin Eq. (3). Apparently, it does not contain any informationabout the nature of the theory, i.e., whether this result holdsin either Type IIA or Type IIB or both. Certainly, suchinformation cannot come from the temporal part of thecorrelation function. These informations are contained inother parts of the vertex operators, which we have sup-pressed since they do not take part in the physics of therolling tachyon. First, the full R-R vertex operator has aspatial part V?s , as defined in (3), which depends on thespin fields along spatial directions. Second, the R-R vertexoperator also has a piece with R-R field strength given by

F� � F�1 �k��1 ��k��; (38)

where �, are spinor indices (�; � 1; . . . ; 32) and wesuppressed the normalization constants which are not soimportant for our purpose. Finally, there is another piecewhich also contributes Gamma matrices. This comes fromthe standard doubling trick procedure for such bulk-boundary correlation function computations, where weextend the definition of the holomorphic field from upperhalf-plane (UHP) to lower half-plane (LHP) by equating itto its antiholomorphic partner:

X�z� ��X�z�; for z 2 UHP

� eX�z�; for z 2 LHP; (39)

where the � sign is for Neumann (Dirichlet) directions.For a spin field, it gives

~S��z� � ��0 �p��S��z�; (40)

where p is the dimensionality of the Dp-brane, and for anon-BPS brane p � odd�even� for Type IIA (IIB). Oncewe take all this into consideration, the restriction on thesets fs0, sig and f~s0, ~sig turns out to be6

For Type IIA: s0 ~s0 � 0) s0 � �~s0;X4

i�1

�si ~si� � �1; (41)

6For our purpose we choose the chirality of left- and right-moving R sector spinors in such a way that for Type IIA :

Pisi �

even andPi~si � odd, whereas for Type IIB it is:

Pisi �

Pi~si �

even.

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For Type IIB: s0 ~s0 � �1) s0 � ~s0 � �12 ;X4

i�1

�si ~si� � 0; (42)

so that the Type IIA (IIB) spinor chiralities can be satisfiedcorrectly.

IV. ENERGY EMISSION

We are ultimately interested in computing the expecta-tion value of total emitted energy from the decaying brane.For the unperturbed initial state of the D-brane (spatiallyhomogeneous decay), it was found in [27] that the totalenergy of closed strings emitted was divergent for aDp-brane with p � 2. We now shortly examine how thisis modified when the initial state is perturbed by addition ofthe boundary tachyon vertex operator, thus extending thediscussion in [34] to superstring.

So the relevant question is, how does the inclusion of anopen string perturbation change the asymptotics of branedecay into closed strings? For this we need the asymptoticsof G�!c;!o� for !c � !o. Using a method described in[49] we find for large n that

eG�inis=2;�is=2� �Ynj�1

��j��js�

��js=2��2

�n�s=2�2

e�s=2�2�1�P1

j�3��s�j��2j�1�1�=�2j�1��j�1�:

(43)

So upon analytic continuation7 we get the asymptotics withlarge !c � !o,

2G�!c��

2p ;

!o��2p

��2!2

o log�!c

!o

(44)

for NS-NS bulk amplitude, and

G�!c��

2p �

i2;!o��

2p

G

�!c��

2p

i2;!o��

2p

��2!2

o log�j!c i=

��2pj

!o

��2!2

o log�!c

!o

(45)

for R-R bulk amplitude. The total emitted energy is calcu-lated by summing over all emitted closed string energies[14,34]

EVp�Xs

1

2jA2�!c;!o�j

2 �1

�2��pZd!c!

�p=2�2!2o

c ;

(46)

showing that the result is in close analogy to bosonic case.

7Assuming that the ratio eG�inis=2;�is=2�=n�s=2�2 is analyticaround n � 1.

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The lesson is that, without the perturbation !o � 0,emitted energy diverges for p < 3 and is finite for p � 3.But morally speaking, we expect divergencies to indicatethat the unstable brane decays completely into closedstrings, whereas for finite emitted energy into a lower

dimensional brane. The extra factor!�2!2o

c is a suppression(enhancement) if !o is real (imaginary), depending on thedimension of the Dp-brane. However, for perturbationswith imaginary !o, decay into closed strings is enhanced,so we expect a complete decay to closed strings for all p.See [34] for additional discussion.

V. SUMMARY

We have investigated superstring disk amplitudes in therolling tachyon background corresponding to an eternallydecaying non-BPS brane. Such computations address verybasic questions about how these branes decay. We haveshown here that the general structure of the amplitudes is aFourier transform of a power series in the target space-timecoordinate, where the coefficients are Toeplitz determi-nants arising from expectation values of a periodic functionin Circular Unitary Ensembles of increasing rank. Theperiodic function encodes the essential information aboutthe amplitude. The determinants of increasing rank Ncompute disk amplitudes with N open string tachyon ver-tex operators from the rolling tachyon background. Furtherprogress is related to advance in solving mathematicalproblems in the context of random matrices, in particular,there is a need to investigate grand canonical ensembles,where the rank of the ensemble (corresponding to thenumber of open string tachyon insertions) can vary. Sofar, the calculations can be carried out fully only in thespecial case of bulk-boundary amplitudes. Apart from themore difficult mathematical problems, one tractable direc-tion to pursue could be to study the field theory limit of thepower series of the individual terms, and compare it withresults computed from the effective Dirac-Born-Infeldfield theory, in the spirit of earlier such investigations(see, e.g., [50–53]).

ACKNOWLEDGMENTS

We are grateful to Vijay Balasubramanian, Per Kraus,and Asad Naqvi for discussions. N. J. was supported in part

046007

by Magnus Ehrnrooth foundation. E. K-V. was supported inpart by the Academy of Finland.

APPENDIX: RELATIONS BETWEEN BOUNDARYFERMIONS AND PAULI MATRICES �i

Recall that �, �� are in fact Grassmann variables:

f�; ��g � 1; (A1)

�2 � ��2 � 0: (A2)

Their spinorial representation on the two-dimensionalHilbert space can be easily worked out, and is given by

� �0 1

0 0

!; ��

0 0

1 0

!;

� �� 1 0

0 0

!; ��

0 0

0 1

!:

(A3)

Defining �� 12 ��1 � i�2�, we find

� � �; �� ; �3 � ��; ��;

�1 � � ��; �2 � �i�� :(A4)

The relevant part of the supersymmetric boundary action interms of �, �� on a brane-antibrane pair is

�SB � i

��2

�

s Z@�dt� �� D�T � ��D�T��t�: (A5)

On a brane-antibrane pair, the tachyon T is a complex field.Substituting T � U iV in the above, we get

�SB � i

��2

�

s Z@�dt�� D�U i� �� D�V�:

(A6)

Next, to obtain a non-BPS D-brane from a brane-antibranepair, we choose the ��1�FL projection in such a way thatonly the 2nd term in the above equation gets projected in.Choosing V �

��2�p

�eX0=��2p

, we arrive at (1). The boundaryfermion � used in [27] is actually �� 1 in ournotation.

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Superstring disk amplitudes in a rolling tachyon background