Michihiro Naka, Tadashi Takayanagi and Tadaoki Uesugi- Boundary State Description of Tachyon...

8/3/2019 Michihiro Naka, Tadashi Takayanagi and Tadaoki Uesugi- Boundary State Description of Tachyon Condensation

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UT-890May, 2000

Boundary State Descriptionof Tachyon Condensation

Michihiro Naka, 1 Tadashi Takayanagi 2 and Tadaoki Uesugi 3

Department of Physics, Faculty of Science

University of Tokyo

Tokyo 113-0033, Japan

Abstract

We construct the explicit boundary state description of the vortex-type (codi-mension two) tachyon condensation in brane-antibrane systems generalizing theknown result of the kink-type (Frau et al. hep-th/9903123). In this description weshow how the RR-charge of the lower dimensional D-branes emerges. We also inves-tigate the tachyon condensation in T 4/ Z 2 orbifold and nd that the twisted sectorcan be treated almost in the same way as the untwisted sector from the viewpointof the boundary state. Further we discuss the higher codimension cases.

1 E-mail: [email protected] E-mail: [email protected]

E-mail: [email protected]


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Recently there have been tremendous progresses in understanding non-BPS congra-tions of D-branes and tachyon condensations in them, pioneered by Sen (for a review see[1]). In Superstring theory most of these systems are realized as either brane-antibranesystems [2, 3, 4, 5] or non-BPS D-branes [6, 5, 7, 8, 9, 10, 11, 12, 13]. The open stringsbetween a Dp-brane and an anti-Dp-brane are projected by the GSO projection oppositeto the usual cases and the tachyon survives. A non-BPS D-brane is dened as a D-branewithout any GSO projections and the tachyonic instability also occurs. Sen argued thatif the condensation of the constant tachyon eld stabilizes the system, then the system

will nally go down to the vacuum [4] and if the condensation has nontrivial congura-tions such as kinks or vortexes, then the nal object will be D-branes of correspondingcodimension [3, 5, 8, 14]. For example, the kink conguration in Dp Dp brane systemis identied as a non-BPS D( p1)-brane.

Three different approaches have been considered 4 to analyze those systems. The rstone is to use conformal eld theory descriptions of string world sheet with a boundary[5, 8, 9, 15, 14]. In this method, the tachyon condensation can be regarded as a marginaldeformation of the boundary conformal eld theory (BCFT) at a special radius. Thesecond is the K-theory approach by Witten. He argued that the topological charges of lower D-branes in a non-BPS conguration of D-branes are classied by the correspondingK-group [17, 12]. In other words we can say that the topological congurations of tachyonelds one-to-one correspond to the element of the K-group. The third one is the string eldtheory description [18]. The tachyon potential has been calculated in the case of a D-branein bosonic string [19] and a non-BPS D-brane in Superstring [20]. The numerical resultsare in good agreement with the Sens conjecture that the system at the minimum of thetachyon potential can be identied as the vacuum. The generations of lower dimensionalD-brane charges have been also discussed in this formalism [21].

In this paper we are interested in the rst approach. From the viewpoint of the openstrings the BCFT descriptions of tachyon kink condensations have been given in [5, 8, 9]

for a brane-antibrane system or a non-BPS D-brane in the presence of the orbifold andorientation projection. In order to discuss the generation of codimension two D-branes,the vortex line conguration of the tachyon eld is needed and is realized in [15] as a pairof the vortex and anti-vortex.

On the other hand we can use the boundary state formalism (for example see [22]),which can give more systematic CFT description of D-branes. In this formalism D-branesare constructed in the closed string Hilbert space. Therefore the couplings of D-branesto NSNS, RR-elds can be written down explicitly. The equivalence between the open

4 Quite recently a new approach which utilizes the noncommutative eld theory description of theworld volume theory has been considered in the presence of a large B-eld [16].

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In this section we review the descriptions of tachyon condensation from the viewpointof open strings. These results are needed when we compare the results with those gainedin the boundary state formalism. First we see the vortex-type tachyon condensation inthe D2D2 system following [15, 5] and investigate some details in a slightly generalizedsituation. Next we review several known facts about the brane-antibrane system in T 4/ Z 2orbifold and the tachyon condensation in that system. Such a system was rst discussedin [8, 11] and also considered in [9] using the T-dualized picture. In this paper we considertype II string theory only in the weak coupling region.

2.1 Tachyon condensation in a D2 D2 systemWe take a parallel D2-brane and an anti D2-brane in type IIA string theory along

x1, x2 and compactify these directions on a torus of radii 6 R1 = 1 , R2 = 1. Then weset a Z 2 Wilson line along each circle. There are four types of Chan-Paton factors forthe open strings in D2D2 system and are denoted by 1 , 1, 2, 3 using Pauli matrices.We use 1, 3 in order to represent the open strings with both ends on the same braneand the spectrum is determined by the conventional GSO projection. On the other hand1, 2 correspond to the open strings with two ends on two different branes and follow the

opposite GSO projection allowing the tachyon in the spectrum.We consider the condensation of the following two types 7 of the tachyon eld

T (1) (x1, x2) = ei12 (x

1 + x 2 ) ei12 (x

1 + x 2 ) , (2.1)

T (2) (x1, x2) = ie i12 (x

1

x2 ) + iei

12 (x

1

x2 ) . (2.2)

If we switch on only one of these, we get the tachyon kink conguration and a codimensionone D-brane or a non-BPS D1-brane will be generated. On the other hand if we condenseboth at the same time, the tachyon vortex line pair conguration will lead to a pair of codimension two D-branes or a D0 D0 system.

The corresponding open string vertex operators in (0)-picture are written asV T 1 = ( 1 + 2)(ei

12 (X

1 + X 2 ) + ei12 (X

1 + X 2 ) )1,V T 2 = ( 2 1)(ei

12 (X

1

X2 ) + ei

12 (X

1

X2 ))2, (2.3)

where X i = X iR + X iL , i = iR + iL (i = 1 , 2) denote the bosonic elds on the stringworld sheet in NS-R formalism and their superpartners.

6 In this paper we use = 1 unit.7 There are also other two marginal deformations which represent other tachyon condensations. But

these correspond to the shift of the vortex line center and the physical phenomena which occurs bysuch tachyon condensations do not change if we ignore these. Thus we only consider the tachyon elds

(2.1),(2.2) below.

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tex operators become marginal owing to the Wilson lines and the tachyon condensationcorresponding to such operators can be treated as the marginal deformation of CFT.

Now let us rotate the coordinates by 4

Y 1 =12(X

1 + X 2), Y 2 =12(X

1 X 2),1 =

12(

1 + 2), 2 =12(

1 2). (2.4)This procedure enables us to use the method of bosonization and fermionization as follows

ei2Y iR = 12(iR + i

iR ) i , e

i2Y iL = 12(iL + i

iL ) i ,

ei2 iR = 12(iR + i

iR ) i , e

i2 iL = 12(iL + i

iL ) i ,

ei2 iR = 12(iR + i

iR ) i , e

i2 iL = 12(iL + i

iL ) i ,

(2.5)

where i , i i (i = 1 , 2) are cocycle factors [15, 8] and we also assume iL,R , iL,R have thecocycle factor 3, 3. To be exact, other kinds of cocycle factors are needed in front of the exponential elds. The latter type of cocycle factors, which we will call second-typecocycle factors below, can not be ignored when we later discuss the bosonizations andfermionizations of boundary states. We leave the details in the appendix B.

The operator product expansions (OPE) among these elds are 8

Y iR (z)Y j

R (0)12

ij ln z , Y iL (z)Y j

L (0)12

ij ln z,

iR (z) jR (0)ij

iz

, iL (z) jL (0)ij

iz

,

iR (z) jR (0)ij

iz

, iL (z) jL (0)ij

iz

. (2.6)

Also the following identities are useful:

iR iR = i2Y iR , iL iL = i2Y iL ,

iR iR = i2 iR , iL iL = i2 iL . (2.7)

Now we can express the tachyon vertex operators (2.3) in the following convenient way

V T 1 = 2 i11 21 = 221 21,V T 2 = 2i22 12 = +2 22 12, (2.8)

8

Note that the factors i in the bosonic eld OPEs are due to second-type cocycle factors.

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is represented as the insertion of the following Wilson lines in terms of the eld

expi

22 1 21 + i 22 2 12 , (2.9)where denotes integration along the boundary and , mean parameters of tachyoncondensations. Notice that 21 commutes with 12 and the above Wilson line iswell dened without path ordering. The open string spectrum in the R sector does notchange in the presence of the Wilson line because for the R sector satises Neumannboundary condition at one end and Dirichlet boundary condition at the other end and

there is no zero mode for [5]. Therefore we will investigate only the NS sector.Now let us dene several projection operators in the following way

(1)F : |0 |0 , i i , (i , i ) (i , i ),h1 : (1, 1) (1, 1), (2, 2) (2, 2), i i ,

Y 1L,R Y 1L,R +2, Y

2L,R Y 2L,R ,

h2 : (1, 1) (1, 1), (2, 2) (2, 2), i i ,Y 1L,R Y 1L,R , Y 2L,R Y 2L,R +

2. (2.10)

As is clear from the above denition, ( 1)F is the fermion number on the world sheet andh1, h2 are the translation operators in the direction of Y 1, Y 2. We also dene (1)F

, h1 , h

2

similarly for .Since so far we have implicitly assumed the radius of circle in the direction of Y 1, Y 2

is 2, we should have a certain constraint in order to realize the physical periodicityX 1X

1 +2 , X 2X 2 +2 taking the effect of the Wilson line into consideration. Such

a constraint is given as(1)F h1h2 = ( 1)F

h1 h

2 = 1 , (2.11)

where we used eq.(2.10). There are eight sectors in NS sector which survive this projection

as follows

11, 1 3, 31, 3 3,1 1, 1 2, 2 1, 2 2. (2.12)

Four of these are insensitive to the tachyon condense or equally the insertion of the Wilsonlines. But the momenta of in the other four sectors are shifted in proportion to thedeformation parameters , . The details are shown in Table 1. Note that , haveperiodicity + 2 , + 2 by applying the same argument discussed in [5].

The main claim in [15] is that if the tachyon condensation develops into the point

= 1 , = 1, then the system is identied as the D0 D0 system where D0-brane6


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jection I 4 = 1 and below we only consider the case of + sign. Using the bosonizationsand fermionizations (2.5), it is easy to see V T

61 and the tachyon condensation

11

is described by the following Wilson line12

W = exp i

22 61 . (2.14)If we condense the tachyon into = 1, then the system is identied with the non-BPS

D1-brane stretching between the xed points. The justication of this statement will begiven later by constructing the boundary state. In [8] this non-BPS D1-brane is identiedwith a D2-brane wrapped on a non-supersymmetric cycle. Later we will also constructthe marginal deformation from D4 D4 to D0 D0 in this orbifold theory.

There is also a known interesting fact. If we consider a non-BPS D1-brane at thespecial radius R = 12 , then the vacuum amplitude of the system vanishes and the systemdevelops the bose-fermi degeneracy [26]. Later this phenomenon will be discussed in termsof the boundary state description.

3 Boundary state description of tachyon condensa-

tionIn this section, we construct the boundary state for a D2D2 system and condensate

a tachyon vortex pair. Mainly we follow the line of [24], where a tachyon kink wasconsidered. The crucial difference from that case is the emergence of nontrivial Chan-Paton factors in closed string sectors. After the condensation the nal object is identiedwith a D0 D0 system as expected. Next we also calculate the vacuum amplitude andinvestigate the consistency with open string picture. Finally we generalize these resultsinto the higher codimension cases. The denition and brief review of boundary states aregiven in appendix A.

11 Note that this marginal deformation is just the opposite to that considered in [8], where the defor-mation from non-BPS D1-brane to D 0 D 0 is considered.12 If we consider the case of the opposite twisted charge, then we get V T

6

1 and W =exp i 2 2 n 6 1 , where n denotes derivative in the normal direction.

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tion

First the boundary state for a D2-brane at xi (i = 39) which is extended to x1, x2

without any Wilson lines is given as follows :

|D2, x i =T p=2

2 |D2, xi

NSNS + |D2, x i RR ,

|D2, x i NSNS =12 dk2 7 eikx |D2, + , k iX NSNS |D2, , k iX NSNS ,

|D2, x i RR = 2 dk2

7

eikx |D2, + , k iX RR + |D2, , kiX RR , (3.1)where T p = 2 3 p

72 p is the normalization 13 of the Dp-brane boundary state and ki (i =

39) are the momenta in the direction of xi . The explicit forms of |D2, ,k i sector are

given below. The NSNS-sector is

|D2, ,k iX NSNS = exp

n =1

1n {( X )

0n ( X )

0n +

7

j =3( X ) jn ( X )

j

n}

exp

i

n =1 { 0

n + 12

0

n + 12

+7

j =3

j

n + 12

j

n + 12

} exp

n =1

2

i=1

1n

( X )in ( X )i

n exp + i

n =1

2

i=1 in +

12 in +

12

|D2, , k iX (0)NSNS , (3.2)and zero-mode

|D2, ,k iX (0)NSNS = wX

Z 2| 0, wX |kiX | (0)NSNS , (3.3)

where |(0)NSNS is the vacuum of world sheet theory and |nX , wX represents the zero

mode part of T 2 which has momenta nX = ( n1X

, n2X

) and windings wX = ( w1X

, w2X

). TheRR-sector is given as

|D2, , k iX RR = exp

n =1

1n {( X )

0n ( X )

0n +

7

j =3( X ) jn ( X )

j

n }

exp i

n =1 {0n

0n +

7

j =3 jn

j

n }

exp

n =1

2

i=1

1n

( X )in ( X )i

n exp + i

n =1

2

i=1 in

i

n

13

This can be determined by computing the cylinder amplitude.

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and zero-mode

|D2, , k iX (0)RR = wX

Z 2| 0, wX |kiX |, (0)RR , (3.5)

where |, (0)RR is the solution to

{ 0 i 0} |, (0)RR = { i0 + i i0} |, (0)RR = 0 ( = 0 , 1, 2). (3.6)The normalizations of the zero modes are dened as

|(0)NSNS = 1 , , |,

(0)RR = , ,

nX , wX | n X , w X = V n,n w,w , kiX |k iX = (2 )77(k k ), (3.7)where V is the volume of the time direction. Note that this is the solution to the condition(A.4) for a boundary state of a D2-brane. We use = 1 to indicate each choice of theopen string boundary condition (A.4). Also note that since we have used the light coneformalism [29, 13], the superscripts of oscillators run from 0 to 7, not to 9. We have dividedthe oscillator parts into X 0, X 3X

7 and X 1, X 2 ( are also divided) in eq(3.2),(3.4).This is because X 0, X 3X

7 part does not contribute to the later calculations of tachyoncondensation importantly. Therefore we abbreviate X 0, X 3

X 7, 0, 3

7 part andki from now on.

Next, we construct the boundary state for D2 D2 system where the position of theD2-brane and the D2-brane is xi = 0. This is given by the superposition of the boundarystates for a D2-brane as

|D2 D2, NSNS = |D2, NSNS + |D2 , NSNS ,|D2 D2, RR = |D2, RR |D2 , RR .

(3.8)

Here, we have two important points. One point is that we have switched on Z 2 Wilsonlines of the second D2-brane and we have expressed such a boundary state as that with aprime. Next point is the second D2-brane is the anti D-brane. Since an anti D-brane hasthe opposite RR charge to a D-brane, we have added a minus sign to the second boundarystate of RR sector.

From eq.(3.8), the oscillator parts of |D2 D2, > NSNS,RR are the same as eq.(3.2),eq.(3.4), and the zero mode parts are given by

|D2 D2, (0)NSNS = wX

Z 2| 0, wX (0)NSNS + ( 1)w

1X + w

2X | 0, wX (0)NSNS ,

|D2 D2, (0)RR = wX

Z 2| 0, wX , (0)RR (1)w

1X + w

2X | 0, wX , (0)RR , (3.9)

where the phase factors ( 1)w 1

X , (1)w 2

X are due to Z 2 Wilson lines [3, 24].

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,

by using the elds (Y 1, Y 2, 1, 2) as follows

|D2 D2, NSNS = exp

n =1

2

i=1

1n

(Y )in ( Y )i

n exp + i

n =1

2

i=1in +

12in +

12

2 wY

Z 2| 0, 2 wY (0)NSNS , (3.10)

|D2 D2, RR = exp

n =1

2

i=1

1n

(Y )in ( Y )i

n exp + i

n =1

2

i=1in

i

n

2 wY Z 2 | 0, 2 wY + 1,

(0)RR , (3.11)

where we dened 1 = (1 , 1).As explained in section 2, we have to change the base of the world sheet elds ( Y, )

into the bosonized ones ( , ) in order to describe the tachyon condensation. Then byusing (, ) modes how are the boundary states (3.10),(3.11) represented? Generalizingthe discussion in [24], we argue that the following boundary states are equivalent toeq.(3.10) and (3.11) respectively for = +1:

|D2

D2, + NSNS = 2 exp

n =1

2

i=1

1

ni

ni

nexp + i

n =1

2

i=1in +

1

2in +

1

2

w

Z 2| 0, 2 w (0)NSNS , (3.12)

|D2 D2, + RR = 2 exp

n =1

2

i=1

1n

in i

n exp + i

n =1

2

i=1in

i

n

w

Z 2| 0, 2 w + 1, + (0)RR . (3.13)

These are obtained simply by replacing ( Y )in , in , wiX in eq.(3.10),(3.11) with in , in , wi .In the appendix B we show they indeed satisfy the desirable boundary conditions if wetake the detailed (second-type) cocycle factors into consideration :

2Y i (w, w)| 2 =0 |D2 D2, + NSNS,RR = 0 , (3.14)(iR (w) i iL (w))| 2 =0 |D2 D2, + NSNS,RR = 0 . (3.15)

Note that eq.(3.14) and (3.15) are not enough 15 for the proof of the equivalence. Butfurther we can see that for several closed string states eq.(3.12) and (3.13) have the same

14 Note that here we have regarded the radii of Y 1 , Y 2 direction as R 1 = R 2 = 1 2 and we have usedthe relations w1Y = w

1X + w

2X , w

2Y = w

1X w

2X .

15 This is because these constraints do not determine the detailed structures of the zero modes such as

Wilson lines.

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functions, which we will see in the next subsection. We propose that these three evidencesare enough for the proof of the equivalence.

On the other hand, |D2D2, NSNS,RR is given by acting left-moving fermion numberoperator ( 1)F Y of (Y, ) system :

|D2 D2, NSNS,RR = ( 1)F Y |D2 D2, + NSNS,RR . (3.16)Note that the action of ( 1)F Y is given as

(

1)F Y : (Y )in

( Y )in , in

in ,

in in , in in , (3.17)and to zero mode

(1)F Y : |n , w (phase) | w

2, 2n , (3.18)

where phase comes from cocycle factors. Therefore for example, |D2 D2, NS isgiven by

|D2 D2, NSNS = 2 exp +

n =1

2

i=1

1

n i

n

i

n exp + i

n =1

2

i=1 i

n +12

i

n +12

w

Z 2(1)w

1 + w

2 | w , 0 (0)NSNS . (3.19)

Then it is straightforward to condense the tachyon by using the Wilson lines (2.9).Since closed strings usually do not have Chan-Paton factors, the tachyon condensation inthe closed string viewpoint corresponds to the insertion of the trace of (2.9) in front of the boundary state and the trace is given as

W 1(, )

cosw 1

2cos

w2

2. (3.20)

But, this is not sufficient. In [30] the authors argued that the Wess-Zumino terms inthe effective action of D2 D2 systems should possess the following coupling.

C 1dT dT , (3.21)where T, T are complex tachyon elds, and C 1 is the R-R 1-form which couples to a D0-brane. This implies that C 1 have Chan-Paton factor 3 since T, T have Chan-Paton factor1 i2 respectively. At rst sight you may think such an idea is not acceptable, but

16

This is almost the same calculation as that in appendix B of [24]. Thus we omit its detail.

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condition as we will see in the next subsection. Therefore we regard our results as anotherevidence of such an idea. Similar Chan-Paton factor for closed string vertex was dis-cussed [7, 1] in the case of a non-BPS D-brane17, where it was argued that the branch cutdue to a RR-vertex provides an extra Chan-Paton factor if its one end is on the non-BPSD-brane. We also argue that some of the states in NSNS sector have the Chan-Patonfactor 3. This fact can also be veried by the Cardys condition and will be required dueto the supersymmetry of the bulk theory. Therefore in these sectors we should insert 3in the trace.

Then the tachyon condensation switches not only (3.20) but also

W 3 (, )sinw 1

2sin

w22

, (3.22)

which is obtained by inserting 3 in the trace.Then by switching both contributions we obtain the following boundary state 18

|B(, ), + NSNS = 2 exp

n =1

2

i=1

1n

in in exp + i

n =1

2

i=1in +

12in +

12

w

Z 2cos(w 1)cos(w

2)

+sin( w 1) sin(w2) | 0, 2 w (0)NSNS , (3.23)

|B(, ), + RR = 2 exp

n =1

2

i=1

1n

in in exp + i

n =1

2

i=1in

i

n

w

Z 2cos{ (w1 +

12

)}cos{ (w2 +12

)}+sin { (w1 +

12

)}sin{ (w2 +12

)}| 0, 2 w + 1, + (0)RR .(3.24)

This satises |B(0, 0), + NSNS,RR = |D2 D2, + NSNS,RR .As explained in section 2.1 the point = = 1 is expected to be identied as a D0D0system [15] where a D0-brane and a D0-brane are produced at ( x1, x2) = (0 , 0) , (, )respectively. The boundary state of this D0 D0 system is

|D0 D0, + NSNS = 2 exp

n =1

2

i=1

1n

( Y )in ( Y )i

n exp i

n =1

2

i=1in +

12in +

12

17 For example, in the case of the non-BPS D2-brane the Wess-Zumino term is written as C 1 dT [31].18 Strictly speaking, from the above explanation we cant decide the relative normalization between the

rst and the second term. This is determined by the vacuum energy calculation in the next subsection.

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n Y Z 2

|D0 D0, + RR = 2 exp

n =1

2

i=1

1n

( Y )in ( Y )i

n exp i

n =1

2

i=1in

i

n

n Y

Z 2|nY +

12

, 0, + (0)RR . (3.26)

On the other hand, at this point the zero mode parts of eq.(3.23) and (3.24) becomerespectively

wZ 2 (1)

w 1

+ w 2 | 0, 2 w

(0)NSNS ,

wZ 2 (1)w

1 + w

2 | 0, 2 w + 1

(0)RR . (3.27)

Then just as we have veried that eq.(3.12) and (3.13) are equivalent to eq. (3.10),(3.11),we can also verify that |B(1, 1), + NSNS,RR is equivalent to eq.(3.25),(3.26) in the sameway. For example, eq.(3.25),(3.26) indeed satisfy the following equations which representthe boundary conditions of D0-branes :

1Y i (w)| 2 =0 |B(1, 1), + NSNS,RR = 0 , (3.28)iR (w) + i iL (w)

| 2 =0

|B(1, 1), + NSNS,RR = 0 , (i = 1 , 2). (3.29)

In other words the tachyon condensation from = = 0 to = = 1 changes theboundary conditions (3.14),(3.15) into (3.28),(3.29) and the crucial difference betweenthem is that the latter has the phase factor ( 1)w

1 + w

2 . At = = 0 only the rst

term of eq.(3.24) is nonzero and this corresponds to the RR charge of D2-brane. As thetachyon is condensed the second term also ceases to be zero and this means 19 that theRR charge of the D0-brane is generated. Finally at = = 1 only the second termis nonzero and this is the pure D0-brane RR charge. Note that if we ignored the factor(3.22) which corresponds to 3 sector, then the RR-sector boundary state would vanish at = = 1 and be inconsistent. In this way we see explicitly in the closed string formalismthat a tachyon kink on a brane-antibrane system produces a codimension two D-brane(see Figure 1).

Let us turn to the other points of , . It is easy to see that at ( , ) = (0 , 1), (1, 0)the RR-sector boundary state does vanish and each system corresponds to a non-BPS D1-brane stretching along the direction of Y 1 or Y 2 respectively (see Figure 1). Physicallythis can be interpreted as the statement that a tachyon kink produces a codimension one(non-BPS) D-brane. All of these identications will be veried further by the calculationof vacuum amplitudes including the detailed normalization.

19 It is easy to see that if , are small, then the second term is proportional to V T 1 V T 2

|D2 RR

|D 0 RR .

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D2+D2

non-BPD D1

D0

D0

x

x 1

2

2

2

(, ) = (0,0) (,) = (1,0) (,) = (1,1)

Figure 1: The tachyon condensation in D2 D2 system

3.2 Calculation of the vacuum amplitude

Here we calculate the vacuum amplitude of D2 D2 system for every value of , and translate it from the viewpoint of open string. As a result it will be shown that theboundary state have the correct normalizations or equally correct NSNS and RR-chargeneeded for the identication and that the additional NSNS and RR sector discussed beforeare indeed required in order to satisfy the Cardys condition.

First let us dene the propagator for closed string as

=12 0 ds esH c 1k2 + , (3.30)

where H c denotes the closed string Hamiltonian and its explicit form is given as

H c =i=1 ,2

(n i)2

2R2i+

12

(wi)2R2i +

i=1 ,2{n (i

n in +

i

n in ) +

r(ir

ir +

i

r ir )}

+9

i=3

12

(ki )2 +7

i=0 ,3{n (i

n in +

i

n in ) +

r( ir

ir +

i

r ir )}+ a, (3.31)

where a denotes the zero-energy for each sector and is given as a = 1 for NSNS-sectorand a = 0 for RR-sector.

Then the vacuum amplitudes for NSNS and RR sector are

Z NSNS =(T p=2 )2

16 0 ds dk2 7 dk2 7 {B(, ), + , k| |B(, ), + , k NSNS B(, ), + , k| |B(, ), , k NSNS },

=(T p=2 )2V D 2

4 0 1(2s ) 72 w 1

,w 2

{cos2(w 1)cos2(w2)

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s n w s n w q f 1(q)8 f 1(q)6f 2(q)2 ,Z RR = (T p=2 )2 0 ds dk2 7 dk2 7 B(, ), + , k| |B(, ), + , k RR ,

= (T p=2 )2V D 2

4 0 1(2s ) 72 w 1 ,w 2 {cos2( (w1 + 12)) cos2( (w2 + 12))+sin 2( (w1 +

12

)) sin 2( (w2 +12

))}q(w1 +

12 )

2 +( w 2 +12 )

2 f 2(q)8

f 1(q)8, (3.32)

where V D 2 = (2 )2V is the volume of D2-brane and we dened

f 1(q) = q1

12

n =1(1 q2n ) , f 2(q) = 2q

112

n =1

(1 + q2n ),

f 3(q) = q1

24

n =1(1 + q2n 1) , f 4(q) = q

124

n =1

(1 q2n 1), (3.33)with q = es .

Next let us perform modular transformations and interpret this as the open stringcylinder amplitude. We dene the modulus of the cylinder as t = s and introduceq = et . Then we get the following open string amplitude

Z openNS = 232 1V 0 dt t32 n 1 ,n 2 12(qn 21 + n 22 + q(n 1 )2 +( n 2 )2 )) f 3(q)

8

f 1(q)8

12

(1)n 1 + n 2 (qn21 + n

22 + q(n 1 )

2 +( n 2 )2

)f 4(q)8

f 1(q)8,

Z openR = 232 1V 0 dt t32 n 1 ,n 2 qn 21 + n 22 f 2(q)8f 1(q)8 , (3.34)

where we used the following identities

n

qn2

= f 1(q)f 3(q)2 ,n

(

1)n qn

2

= f 1(q)f 4(q)2,

nq(n

12 )

2

= f 1(q)f 2(q)2 , f 2(q)f 3(q)f 4(q) = 2, (3.35)and the modular properties

f 1(et ) = tf 1(et ) , f 2(et ) = f 4(et ),

f 3(et ) = f 3(et ) , f 4(e

t ) = f 2(et ). (3.36)

Now it is obvious that for each value of , the open string spectrum is well denedonly if we incorporate the additional sector of the boundary state dened in the previ-

ous subsection, otherwise the number of open string states for given n1, n2, H c would be

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Z = Z openNS + Z openR ,

= V D 2 0 dtt T rNS R 1 + (1)F h1 h

2

2q2H o , (3.37)

where T r is the trace over the open string Hilbert space including zero-modes and Chan-Paton sectors. Also H o means the open string Hamiltonian and is given as follows

H o = ( p0)2 +i=1 ,2

R i 2(wi)2 +

i=1 ,2{n i

n in +

rir

ir }

+7

i=0 ,3{n in in + r ir ir }+ a, (3.38)where a denotes the zero-energy and is given as a = 12 for NS-sector and a = 0 forR-sector.

This physically important constraint (3.37) is generally called Cardys condition [23].Also notice that the above open string spectrum is consistent with the momentum shiftshown in Table 1.

Finally let us verify the identication at particular , . In the case of (, ) = (1 , 0)or (0, 1) we get after the modular transformations

Z =1 , =0 = Z =0 , =1 = 232 1V 0 dt t

32

n 1 ,n 2qn

21 + n

22 f 3(q)8 f 2(q)8f 1(q)8 ,

= 22V 0 dtt T rNS R (q2H o ). (3.39)Therefore we can identify the system as a non-BPS D1-brane of which length is 2 2

as expected. Another case is ( , ) = (1 , 1) and the amplitude can be written as

Z =1 , =1 = 232 1V 0 dt t32 n 1 ,n 2 qn 21 + n 22 f 3(q)8 f 2(q)8f 1(q)8 (1)n 1 + n 2 qn 21 + n 22 f 4(q)8f 1(q)8 ,

= 2V

0

dtt

1

82t m 1 ,m 2 q2m 2

1 +2m 2

2

f 3(q)8

f 4(q)8

f 2(q)8

2f 1(q)8

+ q2(m 1 +12 )

2 +2( m 2 + 12 )2 f 3(q)8 + f 4(q)8 f 2(q)8

2f 1(q)8,

= 2V 0 dtt T rNS R 1 + (1)F 2 q2H c + 2 V 0 dtt T rNS R 1 (1)F 2 q2H c ,(3.40)

where we have dened m1 = n 1 + n 22 , m2 =n 1 n 22 . This shows explicitly that the system is

equivalent to a D0-brane and an anti D0-brane which are separated from each other by

x1 = x2 = .

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1

1

(1,1)

D2-D2

non-BPS D1 D0-D0

Figure 2: The moduli space of the non-supersymmetric D-brane congrations

3.3 Moduli space of non-supersymmetric D-branes

So far we have discussed the boundary states which describe various tachyon conden-sations in D2 D2 system at the critical radii. The tachyon condensations are param-eterized by , which have periodicity + 2 , + 2. At this particular radiusthe non-supersymmetric D-brane congrations for all values of , are realized in con-formal invariant manners. Figure 2 shows the moduli space of the non-supersymmetricD-brane congrations. In particular ( , ) = (0 , 0) (1, 0) (1, 1) can be regarded asa continuous version of the descent relation [9] (see Figure 1).

Realistically we are interested in the D2 D2 system at generic radii. If we shiftthe radius, tadpoles develop as can be seen by the method discussed in [5, 9, 15] or bycomputing one point functions using the boundary state we constructed. The tadpolesonly vanish at sin( ) = sin( ) = 0, which correspond to D2 D2, non-BPS D1-braneand D0 D0.

3.4 Tachyon condensation in general Dp Dp systemsIn this subsection, we generalize our construction to the higher codimension cases.

It is enough to consider the D8 D8 system (In odd codimension case, we have onlyto consider the decay to the D(1) D(1) system.). The different points from theanalysis of D2 D2 system are the slightly complicated choice of the gamma matricesand Chan-Paton factors. We mainly concentrate on these issues in the presentation here.

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. . . = R8 = 1) become 2 3 D0 D0 pairs via the tachyon condensation as the marginaldeformation. These 2 3 soliton - antisoliton pairs represent 2 3 D0D0 pairs, and a singleD0 brane is identied with the single codimension 8 soliton on the D8 D8 pair at thepoint where the tachyon condensation is maximum. Many other points are identied withthe bound states of D8D8, D 6D6, D 4D4, D 2D2, D 0D0 at critical radii. Weshow the emergence of the RR charges of the lower dimensional D-branes explicitly.

We switched on Z 2 Wilson line to the second anti-brane using the Wilson lines [15]

X 1X 2 direction 331 1, X 3X 4 direction 31 1 1,

X 5

X 6

direction 1 3

31, X

7

X 8

direction 1 1 3

3.(3.41)where we adopt the following representation of the SO (8) Clifford algebra

1 = 1 122, 2 = 3222, 3 = 1222,4 = 2222, 5 = 1 322, 6 = 1 1 12,7 = 1 1 32, 8 = 1 1 1 1. (3.42)

Above matrices for the Wilson lines have the properties such that the matrix in theX 2k+1 X 2k+2 direction anticommutes with 2k+1 , 2k+2 , and commutes with the other sixgamma matrices. Since we have seen that the oscillator parts dont contribute crucially

the tachyon condensation, we omit these parts. The zero mode part of the boundarystates of this D8 D8 system is given by|D8 D8, (0)NSNS =

wXZ 8

1 + (1)w1X + w

2X 1 + (1)w

3X + w

4X (3.43)

1 + (1)w5X + w

6X 1 + (1)w

7X + w

8X | 0, wX (0)NSNS ,

|D8 D8, (0)RR = wX

Z 81 (1)w

1X + w

2X 1 (1)w

3X + w

4X (3.44)

1 (1)w5X + w

6X 1 (1)w

7X + w

8X | 0, wX , (0)RR .

Now we change the variables as follows

Y 2k+1 = 12 X 2k+1 + X 2k+2 , Y 2k+2 = 12 X

2k+1 X 2k+2 ,2k+1R,L =

12

2k+1R,L +

2k+2R,L ,

2k+2R,L =

12

2k+1R,L 2k+2R,L ,

where k = 0 , 1, 2, 3. In terms of these variables, the zero mode parts of the boundarystate for D8 D8 system are rewritten as follows

|D8 D8, (0)NSNS = 16 wY

Z 8| 0, 2 wY (0)NSNS , (3.45)

|D8

D8, (0)RR = 16

wY

Z8

| 0, 2 wY + 1, (0)RR . (3.46)

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In order to describe the effect of the tachyon condensation, we change the variablesusing bosonization techniques. First, by fermionization Y i are represented by fermionsi , i and next, by bosonization we introduce free bosons i . In the following we neglectthe cocycle factors, but we can take their roles into account giving gamma matrix i toei2Y i , and 12345678 to i , i etc. Then the D8 D8 system at critical radii is describedwith the following projection as in section 2.1

(1)F h1 . . . h8 = ( 1)F h1 . . . h

8 = 1 . (3.47)

Using (, ) modes, we can write down the boundary states of D8

D8 system for

= +1 and their zero modes are given by

|D8 D8, + (0)NSNS = 16 w

Z 8| 0, 2 w (0)NSNS , (3.48)

|D8 D8, + (0)RR = 16 w

Z 8| 0, 2 w + 1, + (0)RR . (3.49)

The equivalence of two states written with different variables is provided by the fact thatthese states satisfy the denition equation of the boundary state for D8 D8 system.Also, the boundary states for = 1 is given as in the previous subsection.

Now we are ready to condense the tachyon. The tachyon condensation is representedas the insertion of the following Wilson line [15]

exp8

i=1

i i22 ii , (3.50)

where i represent the parameters of the condensation and have the periodicity i i +2.This represents the marginal deformation at the critical radius. These traces with theinsertion of the various Chan-Paton factors are given as the following ve types

W 1(

{

i}) :

8

i=1cos

iwi2

,

W ij ({ i}) :6

i=1cos

i wi2

sin 7w7

2sin

8w82

, 8C2 = 28 combinations ,

W ijk ({ i}) :4

i=1cos

i wi2

8

j =5sin

j w j2

, 8C4 = 70 combinations ,

W ijk mn ({ i}) : cos 1w1

2cos

2w22 8i=3 sin i w

i

2, 8C2 = 28 combinations ,

W 12345678 (

{ i

}) :

8

i=1

sin i wi

2. (3.51)

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tachyon condensation in the closed string sector.This can be understood from the following Wess-Zumino coupling [30]

D 8D 8 C RR STr eF , F = F + T T DT DT F TT

(3.52)

where D = d + A+ A and T, T represent the complex tachyon eld. The elds A+ , Adenote the gauge elds on the brane, anti-brane respectively, which is 0 in this case.Combining with the fact that the tachyon conguration is given by [17, 15]

for example T (x)i xi , at xi0, (3.53)

we can speculate that the RR elds C 2k+1 should have the following Chan-Paton factors

C 7 : ij , C 5 : ijk , C 3 : ijk mn , C 1 : 12345678 . (3.54)

For example, this can be understood as the following expression

D 8D 8 C D 0 dx1dx2. . .dx8 12345678 . (3.55)Thus the traces (3.51) correspond to the closed string sector that belongs to the eachChan-Paton factor. Thus the generation of the lower dimensional D-branes charges isinduced by the above Wess-Zumino terms which are characteristic of the brane-antibranesystems.

Switching on the above operators, we obtain the following boundary states

|B({ i}), + (0)NSNS = 16 w

Z 8K {m i}= {wi} | 0, 2 w (0)NSNS , (3.56)

|B({ i}), + (0)RR = 16 w

Z 8K {m i}= {wi +

12} | 0, 2 w + 1, +

(0)RR , (3.57)

where

K ({m i}) =8

i=1cos( i m i ) +

6

i=1cos( i m i ) sin( 7m7)sin( 8m8) + 27 terms

+4

i=1cos( im i )

8

j =5sin( j m j ) + 69 terms (3.58)

+ cos( 1m1)cos( 2m2) 8

i=3sin( i m i ) + 27 terms +

8

i=1sin( i m i ).

We have |B ({0}) , + NSNS,RR = |D8 D8, + NSNS,RR .21


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NSNS sector w

Z 8(1)w

1 + ... + w

8 | 0, 2 w (0)NSNS , (3.59)

RR sector w

Z 8(1)w

1 + ... + w

8 | 0, 2 w + 1, + (0)RR . (3.60)

The boundary state of this system is rewritten as follows

|D0 D0, + (0)NSNS = 16n Y

Z 8|nY , 0 (0)NSNS , (3.61)

|D0

D0, + (0)RR = 16

n Y Z 8 |

nY + 1/ 2, 0, + (0)RR . (3.62)

These boundary states satisfy the denition equation (the boundary conditions) for theD0 D0 system.

Again, the extra phase factor ( 1)w1 + ... + w

8 changes the boundary condition from

D8 D8 to D0 D0. This corresponds to the fact that D0-branes and D0-branes areproduced at each choices of

(x2k+1 , x2k+2 ) = (0 , 0) or (, ). (3.63)

Around the above points, there exists a soliton (anti-soliton) if the number of coordinate

pairs taking the value ( , ) is even (odd). Finally evaluating the vacuum amplitudesfor NSNS and RR sector with all the ghosts taking into account, it is easy to check theCardys constraint explicitly. Thus we have established in the closed string viewpointthat a tachyonic soliton on the D8D8 system produces a codimension eight D0-branes.

Next turn to the other points of i . The tadpole cancellation restricts the admissiblevalues to i = 0 , 1. We note the following basic observation.

= 0 = 1cos(w ) 1 (1)wsin(w ) 0 0

cos w +12 1 0

sin w + 12 0 (1)w(3.64)

Then in the case when 2 n of i is equal to 1, the system corresponds to the D(8 2n) D(8 2n) system. The number of such congurations corresponds to the possible choiceof the Chan-Paton factor in the closed string sector. For example, 28 ij corresponds tothe degree of freedom in order to set the direction of the codimension 2 among 8 directions.On the other hand, when the odd number of i is equal to 1, the boundary states in RRsector vanish and the system corresponds to a non-BPS D-branes.

Again we emphasize that the Chan-Paton factors in the closed string sector played

the crucial role in our analysis.

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tion in T 4/ Z 2 orbifold theory

In this section we construct the boundary state description of tachyon condensationsin the orbifold theory. We rst discuss the decay mode from a D0 D0 system to anon-BPS D1-brane in detail. Next we extend this result to the higher codimension cases.We also discuss the occurrence of the bose-fermi degeneracy [26, 27] in this formalism.

4.1 Construction of the boundary state

The boundary state which represents D0 D0 in T 4/ Z 2 orbifold at the radii R6 =2, R7 = R8 = R9 = R is given as follows

|B =T p=022(|U NSNS + |U RR ) +

N 22(|T NSNS + |T RR ), (4.1)

where |U , |T denote the untwisted, twisted part of the boundary state. The normaliza-tion for twisted sector is determined by comparing the closed string vacuum amplitudewith the open string one. The result is given by N = 2 3

32 .

The more detailed structure of each sector is written as

|U NSNS =12 dk2 5 n 6 ,n 7 ,n 8 ,n 9 1 + (1)n 6(22)(2R )3 |U, + , ki , n NSNS |U,, k i , n NSNS ,

|U RR = 2 dk2 5 n 6 ,n 7 ,n 8 ,n 9 1 (1)n 6(22)(2R )3 |U,+ , k i , n RR + |U,, k i , n RR ,|T NSNS = dk2 5 |T 1, + , ki NSNS + |T 1, , k i NSNS + |T 2, + , k i NSNS + |T 2, , k i NSNS ,

|T RR =

dk2

5

|T 1, + , ki RR +

|T 1,

, k i RR +

|T 2, + , k i RR +

|T 2,

, ki RR , (4.2)

where we set 1 i 5 and |T 1 , |T 2 represent the twisted sector boundary statescorresponding to two different xed points. As we will explain briey in the appendix A,

|U,,k i , n and |T 1, + , k i , |T 2, + , ki are dened by the conditions (A.4) expanding theelds (Y, ) in each Hilbert space.

Next we need to rewrite the above boundary state in terms of ( , ) in order to describethe tachyon condensation as discussed in the previous section. For the untwisted sectorthe procedure is almost the same and the result are as follows (we show below onlythe relevant modes which correspond to x6 direction and omit the superscript 6 in this

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|U,+ NSNS = exp( n

Z

1n

n n ir

Z + 12

r r )2 n Y |2nY , 0(0)NSNS

= exp(n

Z 1n

n n + ir

Z + 12

r r )2 w(1)w |0, w (0)NSNS ,

|U,+ RR = exp( n

Z

1n

n n i r

Z

r r )2 n Y |2nY + 1 , 0, +(0)RR

= exp(n

Z 1n

n n + ir

Z

r r )2w

(1)w |0, w +12

, + (0)RR ,

(4.3)

Next let us turn to the twisted sector. The twist operator which map the untwistedsector into twisted sector is needed [32]. A candidate for such an operator is given as

= ei1

2 (R L ) , (4.4)

which has the desired singular property as

(z)(0)

O(z12 ), X (z)(0)

O(z12 ), (4.5)

This operator leads to the correct boundary condition of twisted sector boundary state.Then we can rewrite the twisted sector boundary state as

|T, + NSNS = exp( n

Z

1n + 12

(n +12 ) (n +

12 ) i r

Z

r r ) {|T 1 (0)NSNS + |T 2 (0)NSNS }= exp(

n

Z 1n

n n + ir

Z + 12

r r ) w(1)w |w +

12

(0)NSNS ,

|T, + RR = exp( nZ1

n + 12 (n +1

2 ) (n +1

2 ) i rZ + 12 rr ) {|T 1

(0)RR + |T 2

(0)RR }

= exp(n

Z 1n

n n + ir

Z

r r ) w(1)w |w (0)RR .

(4.6)

Notice that this transformation or bosonization procedure can be veried by showingthe bosonized boundary state does indeed satisfy the boundary condition of the originalone as in section 3. It is also easy to see that the vacuum amplitude doesnt change bythe bosonization using the relations (3.35).

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Since we have constructed the boundary state of D0D0 system in terms of (, ), it isstraightforward to determine the boundary state which describe the tachyon condensationprocess in that system. The Wilson line corresponding to the tachyon condensationdiscussed in section 2 can be written as

W = Tr exp(i

22 1) = cos( w). (4.7)Then the effect of the tachyon condensation appears at the coefficients in front of the

zero mode parts as in section 3. If we consider the point = 1, which corresponds tothe maximal condensation, then the untwisted RR-sector and the twisted NSNS-sectorvanish. The untwisted NSNS-sector and the twisted RR-sector become as follows

|U,+ NSNS = exp( n

Z 1n

n n + ir

Z + 12

r r )2 w |w(0)NSNS

= exp(n

Z 1n

n n + ir

Z + 12

r r )2 wY |wY (0)NSNS ,

|T, + RR = exp( n

Z 1n

n n + ir

Z

r r ) w |w(0)RR

= exp(n

Z 1n + 12 (n +12 ) (n +

12 ) + i

r

Z + 12

r r ) {|T 1 (0)RR + |T 2 (0)RR },(4.8)

where we have rebosonized the expression using the basis ( Y, ). Note that the bound-ary condition is changed into that of D1-brane because of the extra phase cos( w) =(1)w .

Now it is obvious20 that the above boundary state is the same as that of a non-BPSD1-brane [8] stretching between the xed points.

In this way the tachyon condensation process from D0

D0 to a non-BPS D1-brane

(and also its reverse if we replace with 1 ) is explicitly shown by using boundarystate formalism. It would be an interesting fact that the twisted sector of ( Y, ) can beexpressed by using the untwisted sector of another eld basis( , ) and this is crucial inthe above discussion of the tachyon condensation in the orbifold theory. This fact willalso become important if we consider tachyon condensation processes in other orbifoldtheories.

20 If we start a D 0 D 0 which has the different relative twisted charge, then we can show by usingalmost the same procedure that the nal object is a non-BPS D1-brane with a Z 2 Wilson line after thetachyon condensation.

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Then we will be interested in the higher codimension cases. As we will show below,such generalizations are not so difficult in our boundary state formalism and the resultsremain almost the same as in section 3. Therefore the discussion is short.

To make things clear we consider the tachyon condensation that changes two D4D4pairs into two D0 D0 pairs (codimension four). Here D4 D4 system has appropriateZ 2 Wilson lines in the same sense of section 3. This process includes the decay modesinto two D2 D2 pairs. First let us dene the coordinates of T 4 as (x6, x7, x8, x9) andtheir 4 rotated coordinates as ( y

6, y7, y8, y9). We also take the radii of T 4 as R6 = R7 =

R8

= R9

= 1 in terms of the coordinates ( x6

, x7

, x8

, x9

) as in the previous discussionin at space. It is important to note that this D4 D4 system can be described interms of (y6, y7, y8, y9) as a D4 D4 system on T 4/ Z 2 of which radii are all 2 withthe projection ( 1)F h6h7h8h9 = 1 as in the case of at space. At this radius we canchange the basis ( Y i , i) into (i , i) by the bosonization procedures, which are trivialgeneralizations of eq.(4.3) and (4.6). Then we can describe the tachyon condensationprocesses and let us denote the corresponding four parameters as 1, 2, 3, 4. Noticethat in order to get four parameters 21 corresponding to the marginal deformation inthe four directions we should start with not one but two pieces of D4 D4. At thepoint 1 = 2 = 3 = 4 = 1 both the untwisted and the twisted boundary states

gain the same extra phase factor ( 1)w6 + w7 + w8 + w9 and the boundary conditions along

(y6, y7, y8, y9) are reversed. Then we get two D0-branes and two anti D0-branes which sitat ( x6, x7, x8, x9) = (0 , 0, 0, 0), ( , , , ) and (,, 0, 0), (0, 0, , ) respectively. In thisway we nd that the tachyon condensation of brane-antibrane system in T 4/ Z 2 orbifoldcan be treated almost in the same way as in at space except the treatment of the twistedsector.

4.4 Comments on bose-fermi degeneracy

Finally let us discuss the relation between the boundary state description in this sectionand the bose-fermi degeneracy [26]. First we compute the vacuum amplitude of the system(4.3),(4.6) with the insertion of the Wilson line (4.7) using ( , ) eld representation forx6 direction. The result is

B()| |B()=

V 16 ds( 12s ) 52 32

52

R3 {w ,n Y qw

2 +n 2Y

2 R 2 cos2(w )f 3(q)8

f 1(q)8 n Y qn 2Y

2 R 22f 3(q)f 4(q)7

f 2(q)f 1(q)7

21 If we started with one D 4 D4, then we would only get the decay modes into a D 2 D 2 and thecodimension four conguration is impossible.

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w ,n Y cos w 2 q2 2 R

f 1(q)8 .

+2 63{w cos2((w +

12

))q(w +12 )

2 f 3(q)5f 2(q)32f 1(q)5f 4(q)3 w cos2(w)qw

2

f 2(q)5f 3(q)32f 1(q)5f 4(q)3 },where nY = ( n7Y , n8Y , n9Y )Z

3 are momenta in the directions of ( y7, y8, y9). We can seethat this amplitude does vanish if R = 12 , = 1 and this phenomenon of non-BPS D1-brane is called bose-fermi degeneracy [26]. Below we would like to discuss this from theviewpoint of the boundary state.

The particular radii of torus R = 12 enable us to perform further bosonization proce-

dures in the direction of x7, x8, x9. The result is as follows (we only show the zero modesand oscillators which correspond to x6, x7, x8, x9)

|U,+ NSNS = exp n

Z

1n

( 6n 6n + i=7 ,8,9 i

n i

n )

ir

Z + 12

(6r 6r + i=7 ,8,9 i

r i

r ) 2w 6Y ,n Y

|w6Y , n 7Y , n 8Y , n 9Y (0)NSNS ,

= expn

Z

1n

(6n 6n i=7 ,8,9 i

n i

n )

ir

Z + 12

(6r 6r i=7 ,8,9 ir

ir ) 2 w

(1)w7 + w 8 + w 9 |w6 , 2w7 , 2w8 , 2w9 (0)NSNS ,

|T, + RR = exp n

Z

1n + 12

(6n 12 6

n 12

+i=7 ,8,9

in 12 in

12)

ir

Z + 12

(6r 6r + i=7 ,8,9 i

r i

r ) {|T 1 (0)RR + |T 2 (0)RR },

= expn

Z

1n

(6n 6n i=7 ,8,9 i

n i

n )

i r

Z(6r 6r i=7 ,8,9

ir

ir ) 2 w

(1)w7 + w

8 + w

9 |w6 , 2w7 , 2w8 , 2w9 , + (0)RR .

(4.10)

This expression shows that the twisted RR-sector in terms of ( Y, ) is rewritten tohave the same form as the untwisted RR-sector of D4-brane in terms of ( , )22. If we usethe basis ( Y, ) for NSNS-sector and ( , ) for RR-sector, each open string vacuum am-plitude of NS-sector and R-sector cancels each other and the occurrence of the bose-fermi

22 This expression also implies that the original non-BPS D1-brane in T 4 / Z 2 can be thought as a BPSD4-brane in terms of the eld ( , ) with a wrong GSO projection ( 1)F = I 4 (1)F

= 1 , though

the essence of this interpretation is not clear.

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critical radius are crucial in the bose-fermi degeneracy.

5 Conclusions

In this paper we have shown explicitly in the boundary state formalism that thetachyon condensation in 2 k1 pieces of D( p + 2 k) D( p + 2 k) at critical radii produces2k1 pieces of Dp Dp. Locally this means that a codimension 2 k soliton of the tachyoneld conguration corresponds to a Dp-brane (or Dp). We have also veried this resultsin T 4/ Z

2orbifold theory. Note that in these cases there are no gauge elds on the world

volume. But the generations of lower D-brane charges indeed occur due to the Wess-Zumino terms which are peculiar to brane-antibrane systems [30]. In the boundary statedescription we have succeeded to see these phenomena explicitly.

In the process of the explicit calculations we have found two remarkable facts. The rstis that the consistency with the open string picture (or Cardys condition) requires theclosed string sectors should have nontrivial Chan-Paton factors. This somewhat strangephenomenon only occurs if we discuss interactions of closed strings with brane-antibranesystems or non-BPS D-branes. These Chan-Paton factors also ensure the Wess-Zuminocoupling proposed in [30].

The second one is the fact in the case of T 4/ Z 2 orbifold we can treat the twistedsector boundary state in the same way as the untwisted one by changing the eld basis(or by bosonization procedure). This enables us to construct the boundary state whichdescribe the tachyon condensation in the orbifold theory. Another application of this factis the investigation of bose-fermi degeneracy [26, 27]. At the point where the degeneracyoccurs the boundary state of a non-BPS D1-brane becomes very much like that of a BPSD-brane by using the bosonization procedure. Naively it seems that a sort of a symmetryis enhanced at this particular moduli, but it is difficult to see this explicitly even inour formalism. We leave this as a future problem. So far the tachyon condensation infour dimensional orbifold theories other than T 4/ Z 2 have not been discussed. If one tryto construct the marginal deformations of BCFT in them, something like the previousbosonization procedures of the boundary state will be required.

Acknowledgments

T.T. would like to be grateful to Y. Matsuo for useful discussions and remarks. Thework of M.N. and T.T. is supported by JSPS Research Fellowships for Young Scientists.

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Here we give our CFT conventions and a short review of boundary states for generalDp-branes with or without the orbifold projection. Remember that we have used the lightcone formulation [29, 13] and ignored23 the non-zero modes of the elds (Y 8, Y 9, 8, 9)in the case of D2 D2 system. We use almost the same conventions as Sens except thedetailed normalizations.

A.1 CFT conventions

We dene z = ei 1 + 2

as the cylindrical coordinate of the world sheet and w = 1 + i2as its radial plane coordinate. First we list the mode expansions of ( Y, ) elds :

Y iR (z) = yiR i2 piY R ln z +

i2 n =0

1n

( Y )inzn

,

Y iL (z) = yiL i2 piY L ln z +

i2 n =0

1n

(Y )inzn

,

iR (z) = i12

r

Z +

irzr +

12

, iL (z) = i12

r

Z +

irzr +

12

, (A.1)

( piY R =

2( Y )

i

0, piY L =

2(Y )

i

0),where = 12 represents NS-sector and = 0 R-sector and we set i = 09.

Then the OPE relations (2.6) are equivalent to the following (anti)commutation rela-tions for modes :

[yiL , p jY L ] = [yiR , p jY R ] = iij ,[(Y )im , ( Y )

jn ] = [( Y )

im , ( Y )

jn ] = mm, n

ij ,

{ir , js }= {ir , js }= r, s ij , (A.2)where ij = ij for i = 1

9 and ij =

ij for i = 0. The vacuum of these modes

is dened as | (0)NSNS . If we compactify the coordinates yi on torus (radii R i ) then themomenta are quantized as follows

piY R =n iY R i

+ R i wiY , piY L =n iY R i R

i wiY , (A.3)

where n iY and wiY denote K.K. modes and winding modes.We have also used the elds ( X, ) and (, ) as another bases. The mode expansions

and commutation relations of these elds are dened in the same way.23 In the case of the orbifold theory discussed in section 4, we ignored the non-zero modes of

(X 4

, X 5

, 5

, 6

).

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A boundary state of Dp-brane is dened by the following boundary conditions 24 in theclosed string Hilbert space :

2Y (w)| 2 =0 |Dp, = 0 , ( = 0 p), 1Y i (w)| 2 =0 |Dp, = 0 , (i = p + 17),

(R (w) i L (w))| 2 =0 |Dp, = 0 , ( = 0 p),(iR (w) + i iL (w))| 2 =0 |Dp, = 0 , (i = p + 1 7), (A.4)

where = + , is the spin structure on the boundary and the GSO projection of theclosed string determines the correct linear combination of these spin structures. If weexpand the left-hand side of eq. (A.4), we get

(( Y )n + ( Y )

n )|Dp, = 0 , ( = 0 p),(( Y )in ( Y )in )|Dp, = 0 , (i = p + 1 7),

(r i r )|Dp, = 0 , ( = 0 p),(ir + i

i

r )|Dp, = 0 , (i = p + 17). (A.5)These conditions are easy to solve by using the commutation relations (A.1),(A.2).

Notice that for a BPS D-brane the boundary state consists of the NSNS-sector and RR-sector and the correct linear combination of them should be determined by comparing itscylinder amplitude with that of open string (see [22]). For example in the case of a (BPS)D2-brane the boundary state is given as eq.(3.1). Also note that for a non-BPS D-branethere is no RR-sector.

Finally let us see the orbifold case briey. In general the orbifold theories have twistedsectors in the closed string Hilbert space and therefore it is necessary to add twisted sectorboundary states |T to the untwisted one. The twisted sector boundary states are denedby the same equation (A.4), but the mode expansion is different from (A.1) because of the twisted boundary condition. In the case of T 4/ Z

2orbifold discussed in section 4, the

mode expansion of (Y i , i ) (i = 6 , 7, 8, 9) is shifted by half integer. For example, thetwisted sector boundary state of D0D0 is given as eq. (4.6), where we showed only themodes of (Y 6, 6). The correct linear combination of the twisted sector boundary statesand the untwisted one is also determined by the calculations of the cylinder amplitudeand this is called the Cardys condition [23].

24 Of course the conditions remain the same if we replace ( Y, ) with ( X, ), because this proceduredoes not mix the Neumann and Dirichlet conditions.

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version

In section 3, 4 we have used bosonized (and fermionized) descriptions of boundarystates at special radii. In [24] the authors calculate several one point functions in thecodimension one case and show that the results are the same as those before the bosoniza-tion. As a further evidence of the equivalence here we prove that the bosonized boundarystates discussed in section 3.1 satisfy the correct boundary conditions in the case of thetachyon condensation in D2D2 system. The other cases appeared in this paper can betreated almost in the same way.

B.1 Cocycle factors

In order to prove the correct boundary conditions the detailed cocyle factors should begiven explicitly. For example, the fermionization relations (2.5) are written incorporatingthe cocyle factors as

iC iY (2, 0) ei

2Y iR (z) =12(

iR iiR )(z),

i

C iY (0,

2) ei2Y iL (z)

=1

2(iL

iiL )( z), (B.1)

where i , C iY (kR , kL ) are both called cocycle factors . i (i = 1 , 2, 3) are 22 Pauli matrices.C iY (kR , kL ) are dened by (for example see [33])

C iY (kR , kL ) exp14

i (kiY R kiY L )( piY R + piY L ) (B.2)In bosonization procedure, they are needed to guarantee correct (anti)commutation rela-tions between various elds.

Next step is the rebosonization of two fermions,

12(iR i iR )(z) = iC i(2, 0) ei2

iR (z)

12(

iL i iL )( z) = iC i(0, 2) ei

2 iL (z) (B.3)

In this way we accomplished changing variables from ( Y i , i ) to (i , i )(see Figure 3).

B.2 Proof of the correct boundary conditions

Now let us prove the facts that the bosonized boundary states satisfy the correct

boundary conditions of the original ones and give a evidence that they are equivalent. We

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Y

F

B

F:fermionization

B:bosonization

Figure 3: Flow of the bosonization

take the example of D2D2 system discussed in section 3. Then two types of equivalenceshould be proved. The rst is that eq.(3.12) and (3.13) are equivalent to eq.(3.10) and(3.11) respectively. We can verify this by showing that eq.(3.12) and (3.13) satisfy theboundary conditions eq.(3.14),(3.15). The second case is that |B( = 1 , = 1) , + (seeeq.(3.23),(3.24)) is equivalent to the boundary state of D0 D0 system. We can alsoprove this in the same way by showing eq.(3.28),(3.29). Since these four equations can beproven in the same way, we show the proof of (3.14)below.

First let us note that eq.(3.12),(3.13) satisfy

(iR (w) iiL (w))| 2 =0 |D2 D2, + NSNS,RR = 0 , (B.4)and that we can replace 2Y i (w)| 2 =0 with (, ) variables. Then eq.(3.14) can be rewrit-ten as

2Y i (w)| 2 =0 |D2 D2, + NSNS,RR= i

z2i

iR (z) z C i(2, 0)ei2 iR (z) + C i(2, 0)ei2 iR (z)

+ z C i(0, 2)ei2 iL (z) + C i(0, 2)ei2 iL (z) |D2 D2, + NSNS,RR .

(B.5)The detail of the exponential is given as

: ei2 iR (z) : | 2 =0= exp i2R exp

i2 pR 1 exp

n =1

1n

n ein 1 exp

n =1

1n

n ein 1 .

(B.6)

If we note that eq.(3.12),(3.13) satisfy

(n + n )|D2 D2, + NSNS,RR = 0 , (B.7)32


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(B.5) cancel respectively. The proof is almost the same as in the case of |B( = 1 , =1), + except that the Z 2-phases (1)w

1 + w

2 of eq. (3.27) play an important role for

changing boundary conditions of Y 1, Y 2.

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