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Preprint typeset in JHEP style. - HYPER VERSION hep-th/0005176
A Note on the Path Integral Representation
of the Boundary State of D-brane
Yong Zhang∗
Institute of Theoretical Physics, Academia Sinica,
PO Box 2735, Beijing 100080, P. R. China
Abstract: In this paper we construct path integral representations of the boundary
states in some special backgrounds such as the U(1) gauge field background, the linear
dilaton background and the open string tachyon background. The initial purpose of this
paper is to construct a general solution of the boundary conformal field theory with the
analytical approach, mainly for the constraint equations(Ln− L˜−n)|B〉 = 0 are difficult
to be solved to obtain the solution represented by string modes from the pure algebraic
approach. However in the path integral representation it is easy transforming those
algebraic equations into the differential equations which can be solved. Another purpose
of this paper is to try to explore an open question. we do not know how to construct
an exact theory of D-branes in the general background until now. However many
recent researches show the boundary state description indeed seizes some fundamental
features of D-branes in the rather special backgrounds. Since the general background
field effects can be easily introduced in the path integral representation, we argue that
path integral representation of the boundary state should provide an exact description
of D branes in the general backgrounds.
∗zhangyo@itp.ac.cn

Contents
1. Introduction 1
2. The path integral representation of the boundary state in the U(1)
gauge field background 3
2.1 The path integral representation of the boundary state 3
2.2 T-duality of D-brane 5
2.3 The path integral representation of the boundary state in the U(1) gauge
field background 7
2.4 Some notes of the path integral representation of the boundary state 10
3. The path integral representation of the boundary state in other spe-
cial backgrounds 11
3.1 The path integral representation of the boundary state in the linear
dilaton background 11
3.2 The path integral representation of the boundary state in the open string
tachyon background 14
4. Discussions 15
A. The calculation of the normalization factor N(F ) 17
B. The conformal path integral representation of the boundary state in
the linear dilaton background 18
1. Introduction
It is well known that D-branes in[1] play very important roles in the study of nonper-
turbative string theory dynamics. However, we do not know how to construct an exact
theory of D-brane in the general background until now. So we have to limit ourselves
within a very small circle, namely in some special backgrounds. This kind of study
may have some heuristic implications in seeking a good theory describing D-brane. On
the other hand many recent researches show the boundary state description indeed
1

seizes some basic features of D-brane. In fact the concept of the boundary state is
introduced into string theory before that of D-brane. The BPS object can be repre-
sented by boundary state, which satisfies the constraint (Q+ Q˜)|B〉 = 0. We can read
off the tension and R-R charge from the tree diagram of the boundary state. These
are the most important proofs supporting the conjecture that D-brane is the classical
soliton solution with R-R charge, namely p-brane. Many relevant literatures have been
included in [2].
The concept of the boundary state is fundamental. Firstly, the tadpole can be
represented with the boundary state which is the source of closed string. Naively,
it seems reasonable that the boundary state should be seen as quantum state of an
independent dynamic object arising from string vacuum. D-brane is just defined as
that object where open string end points end. Secondly, the boundary state can be
defined according to the idea that one loop amplitude of open string can be regarded
as tree amplitude of closed string. From this physical picture, the holographic principle
[3] can be argued because non-abelian gauge field theory appear from the low energy
limits of open string theory and the low energy limit of closed string theory can lead to
effective theory of gravity. Thirdly in [4] Cardy generalizes the boundary state concept
so as to get an exact description of the boundary conformal field theory. Now the
boundary conformal field theory of Gepner model, which is important in the heterotic
string theory, has been proposed in [5]. This generalizes D-brane concept, since the
original D-brane only comes from the system of Type I string, Type IIA string and Type
IIB string. Fourthly, in [6], the path integral representation of boundary state provides
a simple proof supporting the fundamental argument in Matrix theory that physics
of Dp brane is equivalent to that of infinitely many D(p−2r) branes. Fifthly, the path
integral formalism has been incorporated into recent development of Noncommutative
geometry in string theory in [7]. Finally, in [2] the classical p-brane solution can be
constructed from one point amplitude of boundary state with closed string states. This
is also an important proof that D-brane is the classical soliton solution with R-R charge.
Here we only give some examples of the path integral representation of the con-
formal invariant boundary state 1. Since the boundary conformal field theory is a
candidate for D-brane theory, we must find solutions of the boundary conformal field
theory. In the case of path integral representation, algebraic constraint equation can
be transformed into differential equation which is easily solved to obtain the solution
represented by string modes. Although the conformal invariant boundary state satis-
1The Feynman path integral is one way to represent a quantum theory, and it is a very natural
method for describing interactions in string theory. Since we have had a systematic study of string
theory using the Polyakov path integral, we may try to explore the path integral representation of
D-brane.
2

fying Cardy condition can be constructed with Ishibashi state, this formalism is not
often used in the practical calculation. For example in those review articles [2] the
mode representation of the boundary state is used calculating various amplitudes 2.
The organization of this paper is as follows. In section 2, only from conformal
invariance we deduce the path integral representation of boundary state in constant
U(1) gauge field background 3. and show some applications of this formalism in T-
duality and Matrix theory. In section 3, we construct the conformal invariant boundary
state in the linear dilaton background with the path integral representation, and at the
same time construct the path integral representation of the boundary state in open
string tachyon background. Section 4 is devoted to discussions.
2. The path integral representation of the boundary state in
the U(1) gauge field background
The boundary state must be conformal invariant, which is the requirement of the
reparametrization invariance on the world sheet. So the conformal invariance can be
directly used to construct the path integral representation of boundary state in the
general background. In this section, firstly this procedure will be given in detail. Sec-
ondly T-duality of D-brane to construct Dp brane from D−1 brane can be carried out in
the path integral formalism of the boundary state. Then the simplest solution of those
differential equations is found right to be the path integral representation of boundary
state in constant U(1) gauge field background.
2.1 The path integral representation of the boundary state
In this paper only consider the bosonic string case. In fact the generalization to super-
symmetrical case is straightforward. The holomorphic stress tensor reads
T = −1
2
: (∂X)2 :, (2.1)
2It is necessary to notice the path integral representation of boundary state is easily connected
with divergence. Although the path integral representation of the boundary state in non-constant U(1)
gauge field background in [8] can be obtained, those boundary states contain bad divergences in [8] and
so need regularization and renormalization. Another example is the path integral representation of the
boundary state in the linear dilaton background whose procedure of regularization and renormalization
will be discussed in Appendix B.
3Although the path integral representation of boundary state in constant U(1) gauge field back-
ground in [9] had been argued so as to determine the normalization factor, here the familiar form of
path integral representation in [9] can be deduced directly from the conformal invariance.
3

and a similar anti-holomorphic counterpart. It is necessary to work with mode expan-
sions
X(τ, σ) = q + (α0 + α˜0)τ + i
∑
n 6=0
1
n
(
αne−in(τ−σ) + α˜ne−in(τ+σ)
)
, (2.2)
X(z, z¯) = q − i(α0lnz + α˜0lnz¯)− i
∑
n 6=0
1
n (α−nz
n + α˜−nz¯n) , (2.3)
Ln =
1
2
∑
m
: αm+nα−m : (2.4)
in which we have taken the Regge slope α′ = 2. The commutators are [αm, αn] =
mδm+n,0, and similarly for the right-moving modes. Let Kn = Ln−L˜−n. The boundary
condition is entirely encoded in the boundary state |B〉, and the conformal invariant
condition is Kn|B〉 = 0. To solve equations Kn|B, p〉 = 0, it is essential to adopt the
coherent state technique introduced in [9]. Introduce the following coherent state which
satisfies
(αn − α˜−n − xn)|x, p〉 = 0, (2.5)
where n can be either positive or negative. Further the conjugate coherent state should
satisfy
〈x, p|(αn − α˜−n − xn) = 0. (2.6)
If requiring 〈x, p| = (|x, p〉)†, then x−n = (xn)†. Solve this equation (2.5), the corre-
spondent solution is
|x, p〉 = exp
( ∞
∑
n=1
1
n [−
1
2
xnx−n + α−nα˜−n + xnα−n − x−nα˜−n]
)
|p〉. (2.7)
This set of states form a complete orthogonal basis as can be checked. The formulas
eA+B = eAeBe−12 [A,B], and eAeB = eBeAe[A,B] when [[A,B], A] = [[A,B], B] = 0 are very
helpful, and will be often used in the involved calculation in this paper. Observing this
coherent state, we can find each oscillation mode can be traced in this state! Basing
on this point, our goal may be realized that transforming the algebraic equations into
the differential equations. So the detailed operation is in essence a kind of substitution
α−n|x, p〉 =
(
n ∂∂xn
+
1
2
x−n
)
|x, p〉,
α˜−n|x, p〉 =
(
−n ∂∂x−n
− 1
2
xn
)
|x, p〉,
(2.8)
4

with the requirement of n 6= 0. We postulate that the boundary state with the boundary
interaction is of the form
|B, p〉 =
∫
[dx]|x, p〉 exp (S(x)), (2.9)
in which S(x) represents the special boundary interaction. In fact the general form of
the boundary interaction should admit string zero mode qˆ or differential operator ∂∂x .
To solve Kn|B, p〉 = 0, one first computes
Kn|x, p〉 =
(
pxn +
∞
∑
m6=0,−n
mxm+n
∂
∂xm
)
|x, p = p¯〉, (2.10)
in which n 6= 0 is necessary. Substituting the above relations into Kn|B, p〉 = 0 and
integrating by parts, the conformal invariance condition for S(x) is transformed into
(
−pxn +
∞
∑
m6=0,−n
mxm+n
∂
∂xm
)
S(x) = 0. (2.11)
However for n = 0 case the thing is not similar. Applying the above procedure to
K0|B, p〉 = 0, (2.12)
we obtain the equation
∞
∑
m6=0
mxm
∂
∂xm
S(x) = 0. (2.13)
To our special ansatz of the boundary interaction and the special choice of vacuum
state, the solution of the equation (2.13) and the equation (2.11) corresponds to a
special conformal invariant boundary state. In order to obtain one boundary state, we
may take various ansatz of vacuum state and boundary interaction. For example, if we
take vacuum state |p = −p¯〉, then in the n 6= 0 case the equation is the form,
( ∞
∑
m6=0
mxm+n
∂
∂xm
)
S(x) = 0. (2.14)
2.2 T-duality of D-brane
Since T-duality interchanges Neumann and Dirichlet boundary conditions, a further
T-duality in a direction tanget to a Dp-brane reduces it to a D(p−1) brane, while a
T-duality in an orthogonal direction turns it into a D(p+1) brane. In the path integral
5

representation of the boundary state of D-brane, T-duality of Dp brane can be carried
out.
The coherent state which satisfies
(αn − α˜−n − xn)|x, p = −p¯〉 = 0 (2.15)
has to satisfy the following constraint
(αn + α˜−n)|x, p = −p¯〉 = −2n
∂
∂x−n
|x, p = −p¯〉. (2.16)
Therefore the boundary state
|B 〉N =
∫
[dx]|x, p = −p¯〉 (2.17)
satisfies Neumann condition ∂τX(σ)|B 〉N = 0. By carrying out the path integral, its
detailed formalism is
|B 〉N = A exp
( ∞
∑
n=1
1
n [−α−nα˜−n]
)
|p = −p¯〉, (2.18)
where A is a proportional constant. Further, another coherent state has to be also
defined in order to realize our goal in this subsection. Firstly, some formulas should be
given by
X(σ) = q + i
√
α′
2
∑
n 6=0
1
n (αn − α˜−n) e
inσ, (2.19)
∂τX(σ) = α′P +
√
α′
2
∑
n 6=0
(αn + α˜−n) einσ, (2.20)
where P is the total string momentum. Their commutator is [X(σ), P (σ′)] = iδ(σ −
σ′), in which the string momentum P (σ) at the point labeled by the σ is defined as
1
2piα′∂τX(σ). The Fourier expansion of δ(σ−σ′) takes the form 12pi
∑
m exp(−im(σ−σ′)).
Now define a new coherent state
X(σ)|x〉 = x(σ)|x〉. (2.21)
The difference between the coherent state (2.5) and (2.21) only lies in that the latter
is the position eigenstate qˆ |x〉 = q|x〉 and the former is the momentum eigenstate
Pˆ |x, p〉 = P |x, p〉. Naturally, they may be connected by the transform
∫
Dx(σ)|x〉 =
∫
[dx]|x, p = −p¯〉. (2.22)
6

And the coherent state (2.21) can be constructed with the boundary state of instanton
namely D−1 brane,
|x〉 = exp
(
−i
∫
dσP (σ) · x(σ)
)
|B〉−1, (2.23)
where the boundary state of instanton satisfies the constraint X(σ)|B〉−1 = 0. We
conclude this subsection by constructing Dp brane boundary state with D−1 brane
boundary state
|B〉p =
∫ p
∏
i=0
Dxi(σ) exp
(
−i
∫
dσP i(σ) · xi(σ)
)
|B〉−1 (2.24)
with i is limited to i = 0, 1, · · ·, p 4. It can be checked such Dp brane boundary state
will satisfy the boundary conditions,
Xµ(σ)|B〉p = 0, µ = p + 1, · · ·, D − 1; (2.25)
P i(σ)|B〉p = 0, i = 0, 1, · · ·, p. (2.26)
On the other hand only from the above expression of (2.24), T-duality of boundary
state seems to be Fourier transform in the configuration space involved with T-duality.
2.3 The path integral representation of the boundary state in the U(1) gauge
field background
We now turn to solve the differential equation (2.13) and the differential equation
(2.14). It is easy to find one simple solution
S(x) = 1
4
Fµν
∑
m6=0
xmµx−mν
m (2.27)
where x0 = 0 and Fµν = −Fνµ are necessary and the factor 14 has been fixed due to the
following calculation of the normalization factor. The correspondent boundary state is
the form
|B, p = 0〉 =
∫
[dx] exp
(
1
4
Fµν
∑
m6=0
xmµx−mν
m
)
|x, p = −p¯ = 0〉. (2.28)
4The idea used in [10] where the path integral representation was used to support open string T-
duality has been generalized. Here as the result of a T-duality in an orthogonal direction the boundary
state of Dp brane can be constructed from the boundary state of D−1 brane. In (2.24) a further T-
duality in a direction tanget to a Dp-brane is equivalent to changing the measure
∏p
i=0 Dxi(σ) to
∏p−1
i=0 Dxi(σ).
7

Firstly this above solution 5 can be integrated out to show the final result is right
the boundary state in the constant U(1) gauge field background in [2]. The final result
of (2.28) in the Euclidean spacetime is
|B, p = 0〉 = N(F ) exp
(
−
∞
∑
n=1
1
nα−n
(
1− F
1 + F
)
α˜−n
)
|p = −p¯ = 0〉, (2.29)
with the normalization factor N(F ) = (det(1 + F ))
1
2 . In bosonic string case the nor-
malization factor is just the effective action of gauge potential in[11] 6.
The purpose of arguing the path integral representation in [9] is to obtain the
normalization factor. Here shows this approach to get normalization factor is rather
natural in contrast with the other methods. The method in [2] is to compare the one
point amplitude of the boundary state with that from Dirac-Born-Infeld action. In
addition the detailed form of the normalization factor in [8] is in the requirement of
gauge invariance of D-brane source term contained in the action of closed string field
theory.
With the form (2.19) and the following
∂σX(σ) = −
∑
n 6=0
(αn − α˜−n) einσ, (2.30)
the path integral representation of (2.28) may be written with another form
|B, p = 0〉 =
∫
[dx] exp
( i
8πFµν
∫
0
2pi
dσ xµ(σ) · ∂σxν
)
|x, p = −p¯ = 0〉 (2.31)
which shows the boundary interaction of U(1) constant gauge field background.
With the help of (2.23), the above boundary state can be changed into the form
|B, p = 0〉 =
∫
Dx(σ) exp
( i
8πFµν
∫
0
2pi
dσ xµ(σ) · ∂σxν − i
∫
0
2pi
dσPµ · xµ
)
|B〉−1
(2.32)
5It is nontrivial to point out that in the presence of a constant U(1) gauge field background Virasoro
generators are not modified.
6We would like to thank Professor A.A.Tseytlin for bringing our attention to the paper[11]. In fact,
our purpose which is different from of [11] is to give an interesting example to show potential value
of path integral representation. The familiar boundary state in U(1) constant gauge background can
be only determined by conformal invariance. The boundary state can be used to determine boundary
condition from which the action can be constructed. Naturally this example also shows the effective
action or the normalization factor can be determined directly from the conformal invariance.
8

from which the boundary condition determining such above boundary state may be
found. Indeed, one can show that the following identity holds:
0 =
∫
Dx(σ) δδxµ(σ) exp
( i
8πFµν
∫
0
2pi
dσ xµ(σ) · ∂σxν − i
∫
0
2pi
dσPµ · xµ
)
|B〉−1
= [
i
4πFµν∂σX
ν − iPµ(σ)]|B, p = 0〉, (2.33)
where the boundary condition
(∂τXµ − Fµν∂σXν) |B, p = 0〉 = 0 (2.34)
can be extracted. Here it is both the boundary condition and the conformal invariance
which require
p = −p¯ = 0. (2.35)
Such the boundary condition just corresponds to the open string theory
S =
∫
dτ
∫ 2pi
0
dσ
{
1
4πα′ [(∂τX)
2 − (∂σX)2]− [δ(σ)− δ(σ − 2π)]X˙µAµ(X)
}
. (2.36)
The following is a simple review to [6] where the path integral representation of
the boundary state in the U(1) constant gauge field background can be also argued
from the point of Matrix theory. Notice that their conventions are a little different
with ours since the definition of X(τ, σ) and two order antisymmetry tensor Fµν are
not completely fixed.
The configuration of infinitely many D-instantons can be expressed by ∞ × ∞
hermitian matrices XM (M = 0, · · · , D − 1). The one we consider is
X i = Qˆi, (i = 0, · · · , p)
XM = 0 (M = p + 1, · · · , D − 1), (2.37)
where Qˆi (i = 0 · · · , p) satisfy
[Qˆi, Qˆj] = iθij . (2.38)
Here (p + 1)× (p + 1) matrix θ is assumed to be invertible.
In Matrix theory this configuration of D-instantons is equivalent to a Dp-brane. A
quick way to see the equivalence is to look at the boundary states. The boundary state
|B〉 corresponding to the configuration eq. (2.37) can be written as follows:
|B〉 = TrXe−i
∫ 2pi
0 dσPi·Xˆi|B〉−1. (2.39)
9

|B〉−1 includes also ghost part which is not relevant to the discussion here. The
factor in front of |B〉−1 is an analogue of Wilson loop and corresponds to the background
eq. (2.37). Eq. (2.39) can be rewritten with the path integral as
|B, p = 0〉 =
∫
Dx(σ) exp
( i
2
∫
dσ xi · ∂σxjFij − i
∫
dσPi · xi
)
|B〉−1, (2.40)
where Fij = (θ−1)ij .
It is straightforward to perform the path integral in eq. (2.40). In conclusion |B〉
is equivalent to the boundary state for a Dp-brane with U(1) gauge field strength Fij
on the worldvolume.
2.4 Some notes of the path integral representation of the boundary state
To construct a good theory of Dp brane in the general background, we can argue
|B〉Np =
∫ p
∏
i=0
Dxi(σ) exp
(
S(qˆ, xn,
∂
∂xn
, · · ·)
)
|x(σ)〉, (2.41)
where |B〉Np is the Neumann part of the boundary state |B〉p which is defined as
|B〉p = |B〉Np · |B〉Dp (2.42)
with |B〉Dp =
∏D−1
µ=p+1 |B〉
µ
−1. In essence (2.41) may be seen as Fourier transform between
the boundary state |B〉Np with boundary interaction and the coherent state |x(σ)〉 7.
For the path integral representation of the boundary state
|B, p〉 =
∫
[dx]|x, p = −p¯〉 exp (S(x)) (2.43)
we can check those equations
( ∞
∑
m6=0
mxm+n
∂
∂xm
)
S(x) = 0, n 6= 0,
∞
∑
m6=0
mxm
∂
∂xm
S(x) = 0,
x0 = 0 (2.44)
7From the above expression, it can be argued that physics of the world volume of Dp brane, which
is represented by |B〉Np , may be equivalent to the complete physics of (p + 1) particles in space time,
which is represented by quantum state
∏p
i=0 |x(σ) 〉i. The involved motion of these quantum states is
with the weight
(
S(qˆ, xn, ∂∂xn , · · ·)
)
which is determined by the respective paths.
10

from the conformal invariance are consistent with the following equations
2n ∂∂xµn
S(x) = Fµνxν−n, n 6= 0,
P = 0 (2.45)
from the mixed boundary condition determining the boundary state in the constant
U(1) gauge field background. Therefore when we take the ansatz S(x), we will know
such path integral representation shows the physics of D brane in constant U(1) gauge
field background before the detailed path integral calculation in Appendix A. In fact it
is important to firstly estimate the form of S with some physical consideration before
determining the detailed form of S by the conformal invariance.
In this paper some ansatz can be changed. Define the coherent state
(αn + α˜−n − pn)|pp〉 = 0 (2.46)
then the boundary state with the boundary interaction can be expanded into the sum
of the above coherent state,
|B,P 〉 =
∫
[dp] exp
(
S(qˆ, pn,
∂
∂pn
, · · ·)
)
|pp〉 = 0. (2.47)
So all the calculation in this paper can be carried out from the new starting point.
3. The path integral representation of the boundary state in
other special backgrounds
Besides the U(1) gauge field background, the linear dilaton background and the tachyon
background are important in recent research. Especially, it seems attracting how to
construct the boundary state of D-brane in Type 0 string under the closed string
tachyon background and make it consistent with the argument that D-brane is the
classical soliton solution with R-R charge.
3.1 The path integral representation of the boundary state in the linear
dilaton background
We will construct the conformal invariant boundary state in the linear dilaton back-
ground with path integral representation 8.
8Professor Miao Li introduced me his paper [12] and advised me to read this paper carefully. In [12]
the conformal invariance was directly used to construct the path integral representation of boundary
state in the linear dilaton background. However the path integral representation of the boundary state
in [12] can be found not to be exactly conformal invariant. In the following the conformal invariant
boundary state will be constructed.
11

Now the starting point is still the stress tensor
T = −1
2
(∂φ)2 + Q∂2φ (3.1)
and a similar anti-holomorphic counterpart. The central charge of this free scalar is
c = 1+ 12Q2, and Q =
√
2 in two dimensional string theory. Consider a unit disk, the
conformal invariance condition on the boundary means no net energy-momentum flow
out of the boundary. It is convenient to work with mode expansions
φ = ϕ0 − i(plnz + p¯lnz¯)− i
∑
n 6=0
1
n (α−nz
n + α˜−nz¯n) ,
Ln = [α0 + iQ(n + 1)]αn +
1
2
∑
m6=0,−n
αm+nα−m, n 6= 0,
L0 = [
α0
2
+ iQ]α0 +
1
2
∑
m6=0
αmα−m. (3.2)
The similar formula for L˜n. The commutators are [αm, αn] = mδm+n,0, and similarly
for the right-moving modes. Let Kn = Ln − L˜−n. The boundary condition is entirely
encoded in the boundary state |B〉, and the conformal invariance condition is Kn|B〉 =
0.
The usual Neumann boundary condition is given by ∂rφ = 0 on the boundary of
the unit disk. In terms of the boundary state, it states that
P |B〉N = (αn + α˜−n)|B〉N = 0.
Due to the existence of the background charge Q, one has to modify the boundary
condition a bit: p = −iQ. So there must be a net momentum flow out of the boundary
(in view of spacetime φ). One way to see this is to consider the commutators
[Km, αn + α˜−n] = 2iQmδm+n,0 − n (αm+n + α˜−m−n) . (3.3)
When the case n + m = 0 occurs, the above commutator can be showed in the clearer
form
[Km, αn + α˜−n] = m(2iQ + α0 + α˜0). (3.4)
Actually, we have taken the following conventions,
2p = α0 + α˜0, α0 = α˜0 = p. (3.5)
So when p = −iQ, the center term disappears, and it is possible to impose both the
conformal invariance condition and Neumann boundary condition. To be as close to
12

the ordinary Dirichlet condition as possible, one requires that a net momentum transfer
is possible if one scatters string states against the object described by the boundary
state. So |B, p〉 is an eigen-state of p with arbitrary number p. To solve equations
Kn|B, p〉 = 0, we may construct the path integral representation of the conformal
invariant boundary state. However the ansatz S(x, qˆ, ˆ˜q) is one little special form with
the contribution of zero mode,
S(x, qˆ, ˆ˜q) = V ncoup exp
(
qˆ + ˆ˜q
2Q
)
Φ(x), (3.6)
where the operator qˆ and ˆ˜q satisfy the commutative relation [qˆ, ˆ˜q] = 0, [α0, qˆ] = −i and
[α˜0, ˆ˜q] = −i. And V ncoup is the coupling constant before renormalization procedure. The
boundary state has been changed into the form
|B〉 =
∫
[dx] exp
(
V ncoup exp
(
qˆ + ˆ˜q
2Q
)
Φ(x)
)
|x, p = p¯ = −iQ〉. (3.7)
So the equation Kn|B, p〉 = 0 now can be changed into the following two equations,
−i
2QxnΦ(x) + 2iQn
2 ∂
∂x−n
Φ(x) =
∞
∑
m6=0,−n
mxm+n
∂
∂xm
Φ(x), (3.8)
∞
∑
m6=0
mxm
∂
∂xm
Φ(x) = 0. (3.9)
The special solution of the two equations could be found as follow,
Φ(x) =
∮ dz
z exp
(
∑
m6=0
ixm
2Qmzm
)
. (3.10)
As we have claimed, such above solution contains the divergence. Naturally the renor-
malized boundary state which will be given in Appendix B is conformal invariant only
in some specific cases.
Actually the conformal invariant boundary state in the linear dilaton background
can be constructed without considering the path integral representation. Define the
screening charge
QQ =
∮
dz : exp (ik ·X(z)) : (3.11)
with k must satisfy the following equation
k2 + 2iQ · k = 2. (3.12)
13

Since the commutative relation [Ln, QQ] = 0, the conformal invariant boundary state
9 is
|B〉 = Fun(QQ)|B, p = p¯ = −iQ〉N (3.13)
in which Fun is function of QQ. The boundary state |B, p = p¯ = −iQ〉N is the
Neumanm boundary state with the momemtum which satisfies
(αn + α˜−n)|B, p = p¯ = −iQ〉N = 0, n 6= 0,
(α0 + α˜0)|B, p = p¯ = −iQ〉N = 2p|B, p = p¯ = −iQ〉N ,
(Ln − L˜−n)|B, p = p¯ = −iQ〉N = 0. (3.14)
Here the boundary state (3.13)is still rather specifically. In principle the number of
conformal invariant boundary state in the linear dilaton background is infinitely many.
3.2 The path integral representation of the boundary state in the open string
tachyon background
The path integral representation of the boundary state in the open string tachyon
background will be constructed, which provides some clues to construct the conformal
invariant boundary state of D-brane from Type 0 string in the tachyon background 10.
The conformal invariant boundary state in the open string tachyon background 11
is the form
|B〉 = exp
(
∮
dz : exp (ik ·X(z)) :
)
|B, p = −p¯〉N , (3.15)
with the convention k2 = 2. So the path integral representation of the boundary state
in the open string tachyon background will have one form
|B〉 =
∫
[dx]|x, p = −p¯〉+ (3.16)
∫
[dx]
∞
∑
l=1
Φl|x, p = −p¯〉. (3.17)
9This conformal invariant boundary state is useful in Appendix B.
10It is necessary to point out that tachyon appearing in Type 0 string theory is closed string state
but here tachyon is open string state. Since the boundary state is the closed string source, it is possible
to construct the boundary state of D-brane in Type 0 string. On the other hand in order to support
the argument the path integral representation of boundary state can provide an exact description of
D-brane in general background, we have to give a nontrivial example. So the work to ensure the
boundary state representing the boundary state of D-brane in Type 0 string is just nontrivial.
11The boundary state in the open string tachyon background has been researched in [13].
14

The differential equations from the conformal invariance constraint are
Φl
(
−nl ∂∂x−n
+
1
2
lxn
)
+ Φl
(
∑
m6=0
mxm+n
∂
∂xm
)
= 0, n 6= 0, (3.18)
and for n = 0 case the differential equation is
Φl
(
1
2
l2k2 + lk · p
)
+ Φl
(
∑
m6=0
mxm
∂
∂xm
)
= 0. (3.19)
Finally the ansatz of Φl, namely Φl
(
qˆ, p, xn, x−n, ∂∂xn
)
is solved as
Φl =
l
∏
i=1
∮
dzi expik·qˆ zk·pi z
(i−1)k2
i
exp
(
∑
n=1
k · xn
−2nzni
)
exp
(
∑
n 6=0
kzni ·
∂
∂xn
)
exp
(
∑
n=1
k · x−nzni
2n
)
, (3.20)
with the definition of
∏l
i=1
∮
[dzi] ≡
∮
[dz1]
∮
[dz2] · · ·
∮
[dzl]. In essence this solution is
relevant with the form of (3.15). The factor z(i−1)k
2
i appearing in the (3.20) is from the
noncommutative relation between zero modes, for example l = 2 case,
∮
dz2 expik·qˆ z2k·pˆ
∮
dz1 expik·qˆ z1k·p|x, p = −p¯〉 (3.21)
=
∮
dz1 expik·qˆ z1k·p
∮
dz2 expik·qˆ z2k·pz2k
2|x, p = −p¯〉. (3.22)
Although such above calculation is not easy, it shows the path integral representation
of the boundary state is very useful if we can take some tricks in the given background.
4. Discussions
In this paper some examples of the path integral representation of the boundary state
are given in some special backgrounds such as the U(1) gauge field background, the
linear dilaton background and the open string tachyon background. The path integral
representation of the boundary state in the U(1) gauge field background contains much
essential information, especially the normalization factor which is the effective action
of gauge field in the bosonic string theory. The application in Matrix theory and
Noncommutative geometry hints that Path integral representation of the boundary
state could be rather fundamental in describing D-brane in the general background.
15

The initial purpose of this paper is to construct a general solution of the boundary
conformal field theory with analytical approach, mainly for the constraint equations
(Ln− L˜−n)|B〉 = 0 are difficult to be solved to obtain the solution represented by string
modes from the pure algebraic approach. However in the path integral representation it
is easy transforming those algebraic equations into the differential formalism which can
be solved. In addition, Cardy condition which ensures the existence of open string the-
ory in the boundary conformal field theory is vital to this view point that the boundary
conformal field theory is an exact description of D-brane in the general background. We
will consider Cardy condition realization in the path integral representation in the fu-
ture work. Finally, it is also our wish that the path integral representation of boundary
state should be used to support or interpret the known ansatz in [3] 12,
〈
exp
∫
Sd
φ0O
〉
CFT
= ZS(φ0). (4.1)
It is possible to supersymmetrize all the results in this paper. There are still a lot of
work to be done. The path integral representation of constant gauge field configuration
should be generalized to nonconstant background in the conformal invariant require-
ment. The path integral representation of the boundary state in the linear dilaton
background might be useful to construct the boundary state under AdS3 background,
especially in the light cone gauge in [17]. Most important, the construction of the
boundary state in tachyon background is potentially useful for the generalization of
AdS/CFT in non-super symmetrical case. The recent research on D-brane in Type 0
string shows we must face the tachyon problem in arriving at the purpose. However,
it seems that the usual boundary state is not reasonable which is constructed from
the general procedure without considering the tachyon background effect, because the
classical p-brane from the usual boundary state [18] is not the solution in [19]. This can
be the result of the introduction of the tachyon background, which makes the equa-
tions corresponding to p-brane solution become nonlinear. But the usual boundary
state is only the linear combination of closed string states. Since nonsupersymmetrical
generalization of AdS/CFT in Type 0 string case seems more reasonable than in that
12It has been recently argued that holographic principle should be deduced from noncommutative
geometry, such as in [14]. And it seems that two fundamental principles are intrinsic consistent in
[15]. In fact they can be both argued from the point of string/M theory. So it is possible to deduce
AdS/CFT duality from string/M theory. Naturally we wish that this procedure may be depend on
the path integral representation of boundary state. From string amplitude with D brane, which may
be represented by saturating string state with boundary state in[2], much useful information about
supergravity can be obtained in[16]. However those cases are in flat spacetime, but now we have to
face curved spacetime such as AdS. We wish we could continue this research because this will provide
a truely nontrivial example of path integral representation of boundary state of D-brane.
16

approach of supersymmetrical breaking case used in [20], it is possible to construct the
boundary state with the tachyon background in Type 0 string case.
Acknowledgments
We would like to thank Miao Li for helpful comments on the manuscript and Yi-Hong
Gao for a helpful discussion.
A. The calculation of the normalization factor N(F )
In the following the calculation of the normalization factor N(F ) is given. The special
solution of the differential equation (2.13) and the differential equation (2.14) is
S(x) = 1
4
Fµν
∑
m6=0
xmµx−mν
m ,
=
−1
2
Fµν
∞
∑
m=1
x−mµxmν
m . (A.1)
The entire calculation of the path integral of (2.28) has to be put in the Euclidean
spacetime. The calculation of the (2.28) is as follow,
|B, p = 0〉 =
∫
[dx] exp
(
1
4
Fµν
∑
m6=0
xmµx−mν
m
)
|x, p = −p¯ = 0〉,
=
∫
[dx] exp
(
−1
2
∞
∑
m=1
1
m(x−m − 2α−m
1
1 + F )(1 + F )(xm + 2
1
1 + F α˜−m)
)
exp
(
−
∞
∑
m=1
1
mα−m
(
1− F
1 + F
)
α˜−m
)
|p = −p¯ = 0〉. (A.2)
The integral measure in the above formula has to be chosen as
∫
[dx] ≡
∫ ∞
∏
n=1
dx−ndxn
n , (A.3)
∫ dx−ndxn
n ≡
∫ dadb
(2π)D
, (A.4)
with the definition of xn =
√n(a+ ib) and x−n =
√n(a− ib). The choice of (A.4) aims
at using the following formula in our calculation,
∫ dadb
(2π)D
exp
(−1
2
(a− ib)A(a + ib)
)
=
1
AD . (A.5)
17

So the final result can be given,
|B, p = 0〉 =
∫
[dx] exp
(
1
4
Fµν
∑
m6=0
xmµx−mν
m
)
|x, p = −p¯ = 0〉,
= (Det(1 + F ))−1 exp
(
−
∞
∑
m=1
1
mα−m
(
1− F
1 + F
)
α˜−m
)
|p = −p¯ = 0〉. (A.6)
In conclusion, the normalization factor should take the form
N(F ) = (Det(1 + F ))−1. (A.7)
Since
Det(1 + F ) =
∞
∏
m=1
det(1 + F ),
∞
∑
m=1
1 = ζ(0) = −1
2
, (A.8)
we obtain
N(F ) = (det(1 + F ))
1
2 . (A.9)
B. The conformal path integral representation of the boundary
state in the linear dilaton background
The path integral representation of the boundary state in the linear dilaton background
is the form,
|B〉 =
∫
[dx] exp
(
V ncoup exp
(
qˆ + ˆ˜q
2Q
)
Φ(x)
)
|x, p = p¯ = −iQ〉. (B.1)
The conformal invariant constraint of the boundary state, (Ln − L˜−n)|B〉 = 0, can be
expressed by
(Ln − L˜−n)|B〉 =
∫
[dx]
[
α0αn − α˜0α˜−n, expS(x, qˆ, ˆ˜q)
]
|x, p = p¯ = −iQ〉 +
∫
[dx] expS(x, qˆ, ˆ˜q)
(
−2iQn2 ∂∂x−n
+
∞
∑
m6=0,−n
mxm+n
∂
∂xm
)
|x, p = p¯ = −iQ〉 = 0. (B.2)
18

So the equation Kn|B, p〉 = 0 now can be changed into the following two equations,
2iQn2 ∂∂x−n
Φ(x) = i
2QxnΦ(x) +
∞
∑
m6=0,−n
mxm+n
∂
∂xm
Φ(x), n 6= 0, (B.3)
∞
∑
m6=0
mxm
∂
∂xm
Φ(x) = 0, (B.4)
with the commutative relation
[
α0αn − α˜0α˜−n, expS(x, qˆ, ˆ˜q)
]
=
−i
2Q(αn − α˜−n)S(x, qˆ,
ˆ˜q) expS(x, qˆ, ˆ˜q). (B.5)
The special solution of the two equations could be found,
Φ(x) =
∮ dz
z exp
(
∑
m6=0
ixm
2Qmzm
)
. (B.6)
Now we come to check the solution. The right part of the (B.4) is
i
2QxnΦ(x) +
∞
∑
m6=0,−n
mxm+n
∂
∂xm
Φ(x)
=
∮
dz (−zn) ddz exp
(
∑
m6=0
ixm
2Qmzm
)
; (B.7)
and the left part of the (B.4) is
2iQn2 ∂∂x−n
Φ(x) =
∮
dz nzn−1 exp
(
∑
m6=0
ixm
2Qmzm
)
; (B.8)
so the difference between the above expressions is zero integral 13. With (αn − α˜−n −
xn)|x, p = p¯ = −iQ〉 = 0, and qˆ|x, p = p¯ = −iQ〉 = ˆ˜q|x, p = p¯ = −iQ〉, the boundary
state is transformed into
|B〉 =
∫
[dx] exp
(
V ncoup exp
(
qˆ + ˆ˜q
2Q
)
Φ(x)
)
|x, p = p¯ = −iQ〉
= exp
(
V ncoup exp
(
qˆ + ˆ˜q
2Q
)
∮ dz
z exp
(
∑
m6=0
i(αm − α˜−m)
2Qmzm
))
|B, p = p¯ = −iQ〉N
= exp
(
V ncoup exp(
qˆ
Q)
∮ dz
z exp
(
∑
m6=0
iαm
Qmzm
))
|B, p = p¯ = −iQ〉N (B.9)
13if x0 6= 0 then the conclusion of the zero integral is not correct.
19

which contains the divergence, so needs normalizing and renormalization. With the
commutator [αn, αm] = n(1 − ǫ)|n|δn+m,0, the expression exp
(
∑
m6=0
iαm
Qmzm
)
may be
represented by
exp
( ∞
∑
m=1
iα−mzm
−Qm
)
exp
( ∞
∑
m=1
iαm
Qmzm
)
(ǫ−12 Q2). (B.10)
With the renormalized coupling constant V rcoup, the boundary state in the path integral
representation has the final form
|B〉 = exp
(
V rcoup exp(
qˆ
Q)
∮ dz
z exp
( ∞
∑
m=1
α−mzm
iQm
)
exp
( ∞
∑
m=1
αm
−iQmzm
))
|B, p = p¯ = −iQ〉N .
(B.11)
Since the boundary state |B, p = p¯ = −iQ〉N is conformal invariant, we have to verify
that the former part of |B, p = p¯ = −iQ〉N should commute with Virasoro generator Ln
to ensure the conformal invariance of |B〉 . In fact that part is similar to the screening
charge (3.11),
QQ =
∮
dz : exp (ik ·X(z)) :
=
∮
dz exp(ik · qˆ) zk·pˆ exp
( ∞
∑
m=1
k · α−mzm
m
)
exp
( ∞
∑
m=1
k · αm
−mzm
)
|B, p = p¯ = −iQ〉N , (B.12)
with k must satisfy the following equation,
k2 + 2iQ · k = 2. (B.13)
With k = 1iQ , the screening charge is just the former factor of the first order term
of coupling constant of the boundary state. In addition, the solutions of equation
(B.13) may be labeled by 1a+ or
1
a−
. The a+ and a− are respectively 12(Q +
√
Q2 − 2)
and 12(Q −
√
Q2 − 2). So the solution k = 1/(iQ) is good in the large Q limit with
the corrections suppressed by 1/Q2 etc. to k. Such a limit will change the boundary
state|B〉 into |B, p = p¯ = −iQ〉N which is conformal invariant. Therefore it is rea-
sonable that the renormalized boundary state should be conformal invariant in some
special backgrounds.
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22