15 November 2001
Physics Letters B 520 (2001) 385–390www.elsevier.com/locate/npe
Tachyon condensation and open string field theory
Taejin LeeDepartment of Physics, Kangwon National University, Chuncheon 200-701, South Korea
Received 20 August 2001; accepted 9 September 2001Editor: T. Yanagida
Abstract
We perform canonical quantization of open string on a unstable D-brane in the background of tachyon condensation.Evaluating the Polyakov path-integral on a strip, we obtain a field theoretical propagator in the open string theory. Ascondensation occurs the string field theory is continuously deformed. At the infrared fixed point of condensation, the openstring field on the unstable D-brane transmutes to that on the lower-dimensional D-brane with the correct D-brane tension. 2001 Elsevier Science B.V. All rights reserved.
PACS: 11.25.Sq; 11.25.-w; 04.60.Ds
1. Introduction
Tachyon condensation is a noble phenomenon instring theory, which determines the ultimate fates ofthe unstable D-branes and the D–�D-brane pairs. Theunstable systems in string theory are expected to re-duce to stable lower-dimensional D-brane systems ordisappear into vacuum, leaving only the closed stringspectrum behind. Since the celebrated Sen’s conjec-ture [1] on the tachyon condensation many importantaspects of this noble phenomena have been exploredby numerous authors. Since tachyon condensation isan off-shell phenomenon, the theoretical framework todeal with it should be the second quantized string the-ory. The main tools to discuss the tachyon condensa-tion are the open string field theory with the level trun-cation [2] and the boundary string field theory [3,4].The former one, which is based on the Witten’s cubicopen string field theory [5], has been a useful practi-
E-mail address: [email protected] (T. Lee).
cal tool to describe the decay of the unstable D-branesto the bosonic string vacuum. The latter one, whichis based on the background independent string fieldtheory, has been useful to obtain the effective tachyonpotential. These two approaches are considered to becomplementary to each other.
In a recent paper [6] we discuss the tachyon con-densation in a single D-brane, using the boundary stateformulation [7,8], which is closely related to the latterone. As we point out, the boundary state formulationcontains the boundary string field theory, since the nor-malization factor of the boundary state corresponds tothe disk partition function, which is the main ingredi-ent of the latter approach. Moreover, it provides an ex-plicit form of the quantum state of the system in termsof the closed string wavefunction. Thus, we may finda direct connection between the boundary state for-mulation and the former approach based on the stringfield theory if we appropriately utilize the open–closedstring duality. It suggests that the succinct boundarystate formulation of the tachyon condensation maybe transcribed into the open string field theory. The
0370-2693/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0370-2693(01)01132-7
386 T. Lee / Physics Letters B 520 (2001) 385–390
purpose of this Letter is to construct the open stringfield theory in the background of the tachyon conden-sation and to show that the descent relations amongthe D-branes is also well described in the frameworkof the open string field theory. To this end we per-form canonical quantization [9] of the open string ona unstable D-brane in the background of the tachyoncondensation. Then we evaluate the Polyakov stringpath-integral on a strip to obtain the field theoreticalpropagator of the open string in the background of thetachyon condensation. At the infrared fixed point, theopen string field on the unstable D-brane transmutesto that on the lower dimensional D-brane with the cor-rection D-brane tension.
2. Canonical quantization
It is well known that the field theoretical stringpropagator is obtained from the first quantized stringtheory, by evaluating the Polyakov path-integral overa strip, which is the world-sheet of the open string inthis case. Following the same steps, we will constructthe field theoretical open string propagator in thebackground of the tachyon condensation. To this endwe perform canonical quantization of the open stringattached on D-brane in the tachyon background. Thenintegration over the proper time yields the stringpropagator, therefore the kinetic part of the secondquantized string theory. As we vary the parameter ofthe tachyon profile, the field theoretical action for theopen string on a D-brane is continuously deformedand eventually reduced to that on a lower dimensionalD-brane. For the sake of simplicity we consider thebosonic string on a single D-brane. Extension to moregeneral cases is straightforward.
The action for the open string in the background ofthe tachyon condensation is given as1
S = SM + ST
= − 1
4πα′
∫M
dτ dσ√−hhαβgµν∂αXµ∂βX
ν
1 Note that in the boundary state formulation we takeST =i∫dσ T (X). The boundary state formulation is related to the
canonical formulation of the open string theory under discussion bythe double Wick rotation.
(1)+∫∂M
dτ NT (X),
where we consider a simple tachyon profile,T (X) =uijX
iXj . HereN is an einbein on the world-line ofthe end points of the open string and its relation to theworld-sheet metric is given by
(2)√−hhαβ = 1
N
(−1 00 N2
).
That is,N is the lapse function of the world-sheetmetric. The string action is manifestly invariant underthe reparametrization. We may fix this reparametriza-tion invariance by choosing the proper-time gauge,dN/dτ = 0, equivalentlyN = T , constant. Hereafterwe confine our discussion to the proper-time gauge.The string propagator is defined as a Polyakov path-integral over a strip [10]
(3)G[Xf ;Xi] =∫D[N]D[X]exp(iSM + iST ),
where the path integral is subject to the boundaryconditionXµ(τf , σ )=X
µf (σ), X
µ(τi, σ )=Xµi (σ ).
In order to understand the structure of the openstring propagator on a D-brane, let us first consider aflat D-brane whereST = 0. Introducing the canonicalmomentaPµ, we find that the propagator is written as
(4)
G[Xf ;Xi] =∞∫
0
dT
∫D[X,P ] ei
∫ T0
( ∫PµX
µdσ−H )dτ .
If we expand the canonical variables in terms ofnormal modes
Xµ(σ)=∑n
Xµn e
inσ ,
Pµ(σ)=∑n
Pµne−inσ ,
we find that the Hamiltonian is given as
(5)
H = 1
2
∑n
gµν
((2πα′)Pµ
n Pν−n + n2
(2πα′)Xµn X
ν−n).
For the open string on a Dp-brane ind dimensions,we need to impose the Neumann boundary conditionfor Xi and the Dirichlet boundary condition forXa ;∂σX
i |∂M = 0, Xa |∂M = 0, where i = 0,1, . . . , p,
T. Lee / Physics Letters B 520 (2001) 385–390 387
a = p + 1, . . . , d − 1. These boundary conditionsresult in the following constraints:
(6)Xin −Xi−n = 0, P i
n − P i−n = 0,
(7)Xan +Xa−n = 0, P a
n + Pa−n = 0,
xa = 0, pa = 0,
for n = 1,2, . . . . Thus, the canonical variables arewritten as
Xi = xi + √2∑n=1
Y in cosnσ,
P i = pi + √2∑n=1
Kin cosnσ,
Xa = √2∑n=1
�Y an sinnσ, P a = √2∑n=1
�Kan sinnσ,
where(Yn,Kn) and(�Yn, �Kn) form canonical pairs,
Yµn = 1√2
(Xµn +X
µ−n
), �Yµn = i√
2
(Xµn −X
µ−n
),
Kµn = 1√
2
(Pµn + P
µ−n
),
�Kµn = − i√
2
(Pµn − P
µ−n
).
The procedure given above is equivalent to reducing afree closed string to an open string on the Dp-braneby imposing an orbifold condition:
Xi(σ)=Xi(−σ), Xa(σ )= −Xa(−σ),P i(σ )= P i(−σ), P a(σ )= −Pa(−σ).If these constraints are imposed, the Hamiltonian isread as
H = 1
2gij (2πα′)pipj
+ 1
2
∑n=1
gij
{(2πα′)Ki
nKjn + n2
(2πα′)Y inY
jn
}
(8)
+ 1
2
∑n=1
gab
{(2πα′)�Ka
n�Kbn + n2
(2πα′)�Y an �Y bn
}.
Now the field theoretical propagator follows fromintegrating over the proper-time
G[Xf ;Xi] =∞∫
0
dT 〈Xf |e−iT H |Xi〉
= i
∫D[Φ]Φ[Xf ]Φ[Xi]
(9)× exp
(−i
∫D[X]Φ[X]KΦ[X]
),
whereΦ[X] = Φ[xi, Y i,�Y a] andK = H . Hence, theHamiltonian in the first quantized theory correspondsto the kinetic operator for string field in the secondquantized theory.
3. Background of tachyon condensation
The background of the tachyon condensation altersthe boundary conditions for the open string on theD-brane. In order to have consistent equations ofmotion from the action Eq. (1) we need to impose thefollowing boundary conditions on∂M
(10)
(− 1
2πα′ gij ∂σXj + 2uijXj
)∣∣∣∣σ=π
= 0,
(11)
(1
2πα′ gij ∂σXj + 2uijXj
)∣∣∣∣σ=0
= 0.
If we rewrite these boundary conditions in terms ofnormal modes, we get
(12)∑n
nXin + i(2πα′)2
(g−1u
)ij
∑n
Xjn = 0,∑
n
nXin(−1)n
(13)− i(2πα′)2(g−1u
)ij
∑n
Xjn(−1)n = 0.
In the framework of the canonical quantization wetreat them as primary constraints. Let us denote thefirst constraint Eq. (12) as a primary constraintΦi
0
(14)Φi0 =
∑n
(nI + i(2πα′)2g−1u
)ij X
jn = 0.
Then the commutator of the primary constraint withthe Hamiltonian generates a secondary constraintΨi0,which is conjugate to the primary constraintΦi
0
(15)Ψi0 =∑n
(nI − 2i(2πα′)ug−1)
ijPjn = 0.
The Dirac procedure requires further{H,Ψi0} = 0and it generates yet another constraintΦi
1. We maycontinue this procedure until it does not generates
388 T. Lee / Physics Letters B 520 (2001) 385–390
additional new constraints. Repeating it we obtain acomplete set of constraints
(16)
Φim =
∑n
(n2m+1I + 2i(2πα′)n2mg−1u
)ij X
jn = 0,
(17)
Ψim =∑n
(n2m+1I − 2i(2πα′)n2mug−1)
ij Pjn = 0,
wherem= 0,1,2, . . . . All these constraints belong tothe second class. We may apply the same procedure tothe primary constraint Eq. (13), but we only get a setof constraints equivalent to the set we already have.Thus, they are redundant. It is quite useful to rearrangethese set of constraints. From the constraints Eq. (16)it follows that∑m=0
Φim
(iσ )2m
(2m)!=
∑n
(nI + 2i(2πα′)g−1u
)ij cosnσ Xj
n = 0.
If we make use of the following simple algebra,
2π∫0
dσ
πcosnσ cosmσ = δ(n−m)+ δ(n+m),
we find that the set of constraints{Φim = 0, m =
0,1,2, . . .} is equivalent to{xi = 0, �Y im = 2
m(2πα′)
(g−1u
)ij Y
jm,
(18)m= 1,2, . . .
}.
By a similar algebra, we conclude that the set of theconstraints{�Kim = 0, m = 0,1,2, . . .} is equivalentto the following set of constraints{pi = 0, �Kim = 2
m(2πα′)
(ug−1)
ijKjm,
(19)m= 1,2, . . .
}.
If the tachyon condensation does not occur,u= 0, theconstraints reduce to{�Y im = �Kim = 0, m = 1,2, . . .},i.e., the open string is attached to a flat Dp-brane.As one of the parameters of the tachyon profile,uppis turned on and reaches the infrared fixed point,upp → ∞, the constraints for the canonical variables
in the direction ofp turn into the Dirichlet constraints,{Ypm = Kpm = 0, m = 1,2, . . .}. Therefore, we findthat the open string is now attached to a D(p − 1)-brane.
If we exploit the explicit solution of the constraints,we easily see how the Hamiltonian is deformed as thecondensation develops. Let us suppose that we turn onsome of the tachyon profile parameters. Then the partof the Hamiltonian, which governs the dynamics of thecanonical variables in the directions where the profileparameters are turned on, may be written as
H = (2πα′)2
∑n=1
(�Kng
−1 �Kn
+(n
2
)2 1
(2πα′)2�Knu
−1gu−1 �Kn
)+ 1
2
1
(2πα′)∑n=1
n2(
�Yng�Yn
(20)+(n
2
)2 1
(2πα′)2�Yngu−1gu−1g�Yn
).
Thus, asu → ∞, it becomes the kinetic term for theopen string variables along the Dirichlet directions
(21)
H = (2πα′)2
∑n=1
�Kng−1 �Kn + 1
2
1
(2πα′)∑n=1
n2�Yng�Yn.
4. Open string field theory
As the parameter of the tachyon profile is turned on,the string field and the Hamiltonian, equivalently thekinetic operator in string field theory are deformed aswe expect. Now let us examine what effect the tachyoncondensation background makes on the string fieldaction. Evaluating the Polyakov path-integral over astrip we obtain the kinetic part of the string field action
(22)S = Tp
∫D[X] 1
2Φ[X]KΦ[X],
where Tp is the tension of the Dp-brane. Let ussuppose that we turn on only one of the tachyon profileparametersupp = u. Then, taking the constraints Eqs.(18), (19) into account, we may write the measure
T. Lee / Physics Letters B 520 (2001) 385–390 389
D[X] as
D[X] =D[Xp
]×
∏i=0,...,p−1
D[Y i
] ∏a=p+1,...,d−1
D[ �Y a],
D[Xp
] = dxp√g
∏n=1
dYpn d�Ypn δ
(23)×(
�Ypn − 2
n(2πα′)g−1uY
pn
),
whereg = gpp and the metricgij is diagonal. If thetachyon condensation does not occur,
∫D[Xp] →∫
D[Yp]. As the system reaches the infrared fixedpoint of the condensation, whereu → ∞, the mea-sure for the canonical variables in thepth directionbecomes∫D
[Xp
] →∫dxp
√g
∏n=1
d�Ypn(
n
2πα′
)(g
2u
)
(24)= 2π√α′
√2u
g
∫dxp
√g
∏n=1
d�Ypn ,
where we make use of the zeta function regularization.As the tachyon condensation is turned on the
string becomes inactive in the corresponding direction.Hence, it is appropriate to integrate outxp. In orderto find the dependence of the string fieldΦ on xp
we should be careful in defining the string propagatorEq. (9). Since the tachyon background term alsocontributes to the path-integral through the space-likeboundary, the string propagator may be written as
G[Xf ;Xi] =∞∫
0
dT 〈Xf |e−iT H |Xi〉
(25)× e−πux2p(T )−πux2
p(0).
This expression is consistent with the analysis ofthe disk diagram in the boundary state formulation[6], which is related to the open string field theoryby the open–closed string duality. Thanks to theconstraints the propagator depends only on the zeromodex through the boundary action on the space-like
boundarye−πux2p(T )−πux2
p(0). It implies that the stringfieldΦ[X] can be factorized as
(26)Φ[X] = e−πux2pΦ
[Y,�Y ]
in the infrared fixed limit. Hence, the string field actionbecomes in the infrared fixed limit
S = 2π√α′ Tp
√2u
∫dx e−2πux2
×∫D
[Y,�Y ]
Φ[Y,�Y ]
KΦ[Y,�Y ]
(27)= 2π√α′ Tp
∫D
[Y,�Y ]
Φ[Y,�Y ]
KΦ[Y,�Y ]
,
where
D[Y,�Y ] =
∏i=0,...,p−1
D[Y i
] ∏a=p,...,d−1
D[ �Y a].
Therefore, we find that the string field action forthe open string on a Dp-brane turns into that forthe open string on a D(p − 1)-brane as the tachyoncondensation develops. We also confirm the well-known relationship between the D-brane tensionsfrom Eq. (27)Tp−1 = 2π
√α′ Tp . Defining the string
propagator we may construct the interacting openstring field theory by gluing strings together. After thetachyon condensation occurs, the strings may be gluedas usual. Depending how the strings are glued, weget the Witten’s cubic open string [5] or the covariantstring field theory [11].
5. Conclusions
We conclude this paper with a few remarks. Asthe tachyon condensation develops, the Dp-braneturns into a lower-dimensional D-brane. We find thatthe tachyon condensation background enters into theopen string field theory through the constraints to beimposed on the string variables. As the tachyon profileparameter varies, the string field on the Dp-branetransmutes into that on a lower dimensional D-brane.In this transmutation process it is also pointed out thatthe measure plays an important role in determining thetension of the lower dimensional D-brane.
The effect of the tachyon condensation on the openstring may be seen more clearly as we evaluate thedistance between two ends of the open string on theD-brane, which is given by
(28)
∣∣X(0)−X(π)∣∣2 =
∑n,m=1
2(2n− 1)(2m− 1)
(2πα′)2
× �Y in(gu−1gu−1g
)ij
�Y jm.
390 T. Lee / Physics Letters B 520 (2001) 385–390
Since it is order of 1/u2, it vanishes in the infraredfixed limit. If the tachyon condensation takes place inevery direction on the world-surface of the D-brane,the two ends of the open string approach to eachother. Eventually in the infrared fixed point limit theycoincide and the open string turns into a closed string.
In the present work we discuss the canonical quan-tization of the open string in the background of thetachyon condensation. Although some important as-pects of the tachyon condensation can be understooddirectly in the open string field theory, there remainsmuch room for improvement to explore the full dy-namical aspects of the tachyon condensation, includ-ing the quantum corrections [12].
Acknowledgement
This work was supported by grant No. 2000-2-11100-002-5 from the Basic Research Program of theKorea Science and Engineering Foundation. Part ofthis work was done during the author’s visit to APCTP(Korea) and KIAS (Korea).
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