Vector profile and gauge invariant observables of string field theory solutions for constant magnetic field background
Accepted: May
Vector pro le and gauge invariant observables of string eld theory solutions for constant magnetic
Nobuyuki Ishibashi 1 5
Isao Kishimoto 1 3
Toru Masuda 0 1 4
Tomohiko Takahashi 1 2
0 CORE of STEM, Nara Women's University
1 University of Tsukuba , Tsukuba, Ibaraki 3058571 , Japan
2 Department of Physics, Nara Women's University
3 Faculty of Education, Niigata University
4 CEICO, Institute of Physics of the Czech Academy of Sciences
5 Tomonaga Center for the History of the Universe
We study pro les and gauge invariant observables of classical solutions corresponding to a constant magnetic eld on a torus in open string eld theory. We numerically nd that the pro le is not discontinuous on the torus, although the solution describes topologically nontrivial con gurations in the context of low energy e ective theory. From the gauge invariant observables, we show that the solution provide correct couplings of closed strings to a Dbrane with constant magnetic eld.
Bosonic Strings; String Field Theory; Dbranes; Tachyon Condensation

1 Introduction
2 Classical solutions for constant magnetic eld background
3 Pro les of the classical solution
3.1
3.2
3.3
3.4
4.1
4.2
4.3
Pro les and dual states
Quasiperiodicity of the solution
Tachyon pro le
Vector pro le
Calculation of gauge invariant observables
Comparison with DiracBornInfeld action
4 Gauge invariant observables for the classical solution
5 Concluding remarks
A Conformal transformation of the vector vertex operator
B Calculation of correlation functions in the vector pro le
moduli space of string theory. In bosonic open string eld theory, Erler and Maccaferri [1]
proposed a way to construct classical solutions representing any timeindependent open
string background by use of boundary condition changing (BCC) operators. Following
their method, a solution corresponding to constant magnetic
eld background has been
constructed by some of the present authors in [
2
]. It was found that the classical action
of the solution calculated from the operator product expansions (OPEs) of BCC operators
agrees with the DiracBornInfeld action.
This magnetic solution has several new features compared with the solutions discovered
de ned as follows. Using the Fock space expression, a solution j i of the open string eld
theory can be expanded as
j i =
X tp jTpi + X Ap Vp
p
p
+
;
where the lower mass states, corresponding to the tachyon and the massless vector eld
are expressed as
jTpi = c1 jpi = ceip X (0) j0i ;
Vp
= c1
1 jpi = i
r 2
0
The position representation of the component elds is given by the Fourier transform of
tp; Ap;
:
to ask whether the magnetic solution needs multiple patches in string
eld theory. This
question may be examined by evaluating the vector pro le A (x) corresponding to the
U(1) gauge eld. If A (x) has discontinuities, we need to divide the torus by coordinate
patches to represent the con guration by smooth gauge elds.
Secondly, we will evaluate gauge invariant observables for the magnetic solution. In
conventional eld theory, topologically nontrivial con gurations are characterized by some
gauge invariant quantities which take discrete values. Such quantities have not been found
in string eld theory. Instead, we have the gauge invariant observable which is associated
with onshell closed string vertex operators [3]:
OV ( )
hIj c(i)c( i)V (i; i) j i
;
(1.5)
where I is the identity string eld, V (z; z) denotes an onshell closed string vertex operator
and z = +i; z =
i in the complex plane correspond to the midpoint
=
=2 of the
open string. If
is a classical solution, the observable represents a coupling of the closed
string vertex operator V to the Dbrane to which the solution corresponds. Accordingly,
we expect that the observable for a massless antisymmetric tensor vertex has a nontrivial
value since the corresponding Dbrane has constant background magnetic eld. In order to
con rm the existence of background magnetic eld and nd a clue for topological invariants
in string eld theory, we calculate the gauge invariant observables with massless and
zeromomentum closed string vertex operators.
This paper is organized as follows. In section 2, we brie y review the constant magnetic
solution on torus. In section 3, we study tachyon and vector pro les of the solution. As
{ 2 {
(1.1)
(1.2)
(1.3)
(1.4)
a byproduct, we show periodic and quasiperiodic properties of the solutions. In section
4, we evaluate the gauge invariant observables of the solution. By comparing the resulting
observables with the DiracBornInfeld (DBI) action, we
nd that the solution indeed
corresponds to constant magnetic eld background. In section 5, we will give concluding
remarks. In the appendices, details of calculations are exhibited.
2
Classical solutions for constant magnetic
eld background
We would like to consider the con guration with a constant background F . We
concentrate on the spatial directions X1 and X2, since a real antisymmetric tensor F
can be
transformed into a block diagonal form with 2
2 blocks. Let us consider the bosonic open
string eld theory in which these spatial directions are toroidally compacti ed with radii
R1 and R2, and the Neumann boundary conditions are imposed on the variables X1; X2.
The time direction X0 is required to be noncompact in order to construct the solution
following ErlerMaccaferri's method [1], but other directions are unspeci ed here.
To nd the classical solution corresponding to a constant F12 background, we need to
prepare the BCC operators which changes the open string boundary conditions for X1; X2
from the Neumann boundary condition to the one with F12 and vice versa. Such operators
correspond to the open strings with one edge with the free boundary conditions and the
other coupled to the constant magnetic eld. The zero mode coordinates x1; x2 of these
open strings become noncommutative and we need to introduce the following operators
jN j pairs of BCC operators:
conformal weight
The zero mode algebra has a jN j dimensional representation. Correspondingly, we can nd
k (k = 1;
; jN j) [
2
], which are primary elds with
h =
; ( tan
= 2
0F12 ):
Following ErlerMaccaferri's method, we multiply k
; k by the vertex operators e iphX0
k
;
k. They are primary elds with conformal weight zero and satisfy the OPEs
and appropriate normalization factors and construct jN j pairs of modi ed BCC operators
k
;
(1
2
x
1
R1
)
{ 3 {
k(s) l(0)
k;l;
l(s) k(0)
l;k
j cos
j
=
l;k
for small positive s.
Having these BCC operators, the classical solution corresponding to the constant
magnetic eld background [
2
] are given as follows:
where
tv is the ErlerSchnabl solution for the tachyon vacuum [4],
:
1 + K
; (2.8)
(2.9)
It should be noted that 1k;l and 2k;l independently satisfy the string eld reality condition,1
X Ak;l 1
k;l
k;l z
=
X Ak;l 1k;l;
k;l
X Ak;lQB 2
k;l
k;l z
=
X Ak;lQB 2k;l:
k;l
(2.10)
These are useful in checking the correctness of the pro le calculation presented later.
A = diag( 1;
; 1; 0;
; 0 );
 {z }  jN{jzM }
M
then the solution can be regarded as describing M Dbranes with magnetic eld
condensation. Using the OPEs (2.3), it is easy to show that the energy of the solution is given by
M times that of a single Dbrane with magnetic eld condensation.
For later discussion, it is convenient to rewrite the solution (2.4) in the following way.
By using the algebra of K, B, c and k
, l, the solution
k;l can be decomposed to three
parts:
and Ak;l is a hermitian jN j
jN j matrix satisfying A2 = A. The second term is a solution
to the equations of motion around the tachyon vacuum. Suppose that A is given by
where 1k;l and 2k;l are de ned as
Pro les of the classical solution
Pro les and dual states
Now let us study the pro les of the magnetic solution
0. In order to extract momentum
space pro les from the solution expanded as (1.1), we de ne the states dual to the tachyon
1We follow the de nition of the conjugate z given in [1]. K, B and c are selfconjugate and z = . In
(2.4)
(2.5)
(2.6)
(2.7)
and massless vector states, jTpi and Vp , by
~
Tp =
~
V
p
=
1
1
;
1
i
and they are orthogonal to other higher massive states. The momentum space pro le is
derived from the inner product of the dual state and
0
:
HJEP05(218)4
tp = T~p; 0 ;
Ap = V~p ; 0 :
Using (2.7), we can see that the pro les (3.4) are decomposed to three parts. The
calculations are simpli ed by using the following identities:
QB T~p
QB V~p
Consequently, for example, we only need to deal with
the tachyon pro le.
3.2
Quasiperiodicity of the solution
Before starting the calculation of the pro les, we would like to point out that the pro les
satisfy quasiperiodic relations. Here we deal with the solution corresponding to a single
Dbrane with constant magnetic eld F12 6= 0, namely M = 1 in (2.6). We can construct
jN j independent solutions
0k =
tv +
k;k
(k = 1;
; jN j):
To derive the tachyon pro le from
0k, we have to calculate the inner products T~p; tv
and T~p; 1k;k as seen from (3.5). The former has been known to be a constant [4].2 The
latter can be calculated by rewriting in terms of the correlation function including the
tachyon vertex and the BCC operators, which has been derived in [
2
]:
2
D
e ip X ( ) k( 1) l ( 2)
E =
(
( 1
Cl;kp = ! n12n2 n2l
2) 0p2 Cl;kp;
k l; n1 (modN)
0p2
2 ;
T~p; tv =
Z 1
0
dt
Z 1
0
ds
e s t
4 2pts
(s + t + 1)2 1 + cos s + t + 1
(t s)
= 0:509038
:
{ 5 {
tv and
1k;l for the calculation of
(3.1)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
where ! is an N th root of unity, ! = exp(2 i=N ), and ni are momentum quantum
numbers, pi = ni=Ri (i = 1; 2), and ln
( )
(x). Here, it is important to notice that the correlation function depends on the
parameters k and l only through the normalization factor, Ck;pl. Given this, we nd that
the inner product T~p; 1k;k
depends on k as follows,
T~p; 1k;k
= Ck;pk
= e i 2 RN2 k p2
:
Therefore, it turns out that the tachyon pro le tk(x) of
0k satis es the quasiperiodic
Moreover, since the factor Ck;pk includes 0; n1 (modN) we nd
t
k
x1 +
2 R1 ; x2
N
= tk x1; x2 :
Similarly to the tachyon case, other pro les can be calculated also by using a 3point
functions of the BCC operators. The matter vertex operators are of the form @nX
eipX .
Since the operators U; V are not included in the derivatives of X1; X2 [
2
], the 3point
function depends on the parameters k and l only through Ck;pl. Consequently, we nd that
all pro les of the solution satisfy the quasiperiodic relations similar to (3.11) and (3.12),
namely, the space representation of the solution satis es
k
0
k
0
x1 +
x1; x2 +
2 R1 ; x2
N
Notice that the quasiperiodicity relations of these forms arise because we have taken
the BCC operators corresponding to the eigenstates of V = exp(ix2=R2) [
2
]. If we take
the BCC operators corresponding to the eigenstates of U = exp(ix1=R1), we have a set
of classical solutions which is periodic in the x
1 direction and quasiperiodic in the x
2
direction. In addition, we should comment that, due to the translational symmetry of the
they correspond to degenerate states in magnetic elds, socalled Landau level.3
theory, we can generate other set of solutions from
torus. In our case, we simply just choose
0k by arbitrary displacement of the
0k as jN j independent solutions in the sense that
3In a naive expectation, the jN j independent solutions 0k (k = 1;
; jN j) are physically equivalent
as a result of the translational symmetry and the relation (3.14). However, it is di cult to connect the
translation is represented as a gauge transformation in open string eld theory.
{ 6 {
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
Now let us calculate the tachyon pro le. The inner product T~p; 1k;l is rewritten in terms
of correlation functions with the help of KBc algebra:
T~p; 1k;l
=
dt2
L
L
0
Z 1 dt1 e t1 t2 t3
1
2
1
2
iL denotes a correlation function on the in nite cylinder
of circumference L. The correlation functions which appear in (3.15) can be obtained by
conformally transforming (3.8) and the ghost correlation functions which are de ned on
the complex plane. We eventually get the tachyon pro le for the jN jth solution
0
jNj as
where t0 = T~p; tv . The function Gt(u) is de ned by
g( 0u
1) exp
( )
(1
)
2 log 4) ; (3.17)
Z 1
0
ds
Z 1
s+t+1
dL
sin (s+t+1)
L
L
2L sin (s+1=2) sin (t+1=2)
L
!z
:
The tachyon pro le for the kth solution can be derived by using (3.11).
Now, we carry out numerical evaluation of the tachyon pro le. In order to do so,
we need to evaluate the triple integration on the right hand side of (3.18). However, this
expression is inappropriate for numerical integration, because the rate of convergence is
very slow due to the in nite integration region and it gives an inaccurate value even by
use of Monte Carlo method. An expression of g (z) which is better suited than (3.18) for
numerical integration can be obtained by a change of variables:
g(z) =
of the pro le depends on the behavior of g (z) when z is very large.
4 z
The Fourier coe cients of (3.16) are evaluated by using the numerical results of g(z)
and the tachyon pro le can be obtained numerically by summing up the Fourier series. We
nd that the Fourier coe cient approaches zero fast enough as n; m ! 1 so that we can
approximate the series by a
nite summation over lower modes. The plot of a result is
depicted in gure 2.4
3.4
Vector pro le
Next, we consider the vector pro le of the magnetic solution. Since the vector pro le of
tv vanishes, the momentum space vector pro le of
0k is given by
Ap
= V~p ; 1k;k
QBV~p ; 2k;k :
(3.20)
As in the case of the tachyon pro le, each inner product can be rewritten in terms of
correlation functions on in nite cylinder of circumference L = t1 + t2 + t3 + 1:
V~p ; 1k;l
=
2
0p2
1
4The reason why the x1 and x2 dependence are di erent from each other is that the solution is constructed
by using the BCC operators corresponding to the eigenstates of V = exp(ix2=R2).
{ 8 {
D
L
2
0p2
(2 )2R1R2
1
2
1
2
1 Z 1 dt3 Z 1
c(t1)
E
L
c(t1 + t2)B
L
E
L
; (3.21)
Bc(t1)
; (3.22)
2
0
where we have used the expressions for the dual vector vertex operator, (3.2) and (3.6).
The computation of the correlation functions on the right hand side of (3.21) is not so
simple because the vector vertex operator in (3.21) is not a primary
eld. As derived in
appendix A, under the conformal transformation z = f ( ), the vector vertex transforms as
QBV~p ; 2k;l
follows;
E
L
9
>
=
>
;
d
dz
0p2+1 ><
8
>
:
i 0p
2
d
2
d
dz
2
dz2 e ip X ( ) :
(3.23)
Thus, we need correlation functions for vector and tachyon vertex operators to derive
the vector pro le.
A detailed derivation of these correlation functions is presented in
appendix B. Using the results, it turns out that only the term (3.21) contributes and we
nd:
Ap
=
the position space representation of the vector pro le is obtained as follows:
A1(x) =
A2(x) =
1
p
where the function G(u) is de ned in terms of g (z) in (3.18) as
G(u) = ( 0u + 1) g( 0u) exp
2p3. Here, we take 0 = 1.
2p3. Here, we take 0 = 1.
As in the tachyon pro le, we can numerically calculate the vector pro le by using the
numerical results of g(z) and summing up the Fourier series. The di erence of the two
pro les is in the fact that the Fourier coe cients of the vector pro le include momentum
factors, N m=R1 or n=R2. Hence the asymptotic behavior of the pro le may change in the
ultraviolet region and discontinuities could be found for the vector pro le.
However, we observe that the Fourier coe cients for the vector pro le rapidly converges
to zero as n; m !
1 and the position space representation of the pro le seems to be
absolutely convergent. Consequently, we get the plot shown in gures 3 and 4, and we
conclude that the vector pro le has no discontinuities.5
Here, we would like to comment on pro les for other massive modes. The reason why
the tachyon and vector pro les are not discontinuous is because the momentum space pro le
becomes zero rapidly for large momenta. This behavior is due to the exponential factor
5From the vector pro le, we can calculate the quantity F~12
not as a constant but as a smooth function, and the space average of F~12 over the torus becomes zero. Since
F~12 is not invariant under gauge transformations in string
eld theory, it is no wonder that F~12 does not
correspond to the constant magnetic eld.
0p2=2 in the normalization factor of the 3 point function (3.9).6 Since this exponential
factor always appears in the expression of other pro les, we expect that other pro les also
do not have any discontinuities.
Gauge invariant observables for the classical solution
Calculation of gauge invariant observables
Let us consider the gauge invariant observable (1.5) for the kth solution (3.7) with the
following closed string vertex operators at zero momentum:
HJEP05(218)4
B(z; z)
G(z; z)
where X = (X1 + iX2)=p2 and X~ = (X1
iX2)=p2. These correspond to the
antisymmetric tensor eld B12 and the sum of graviton eld G11 + G22 in the spatial directions X1
and X2. Substituting the kth solution (3.7) into (1.5) and expressing it in terms of the
correlation functions on the in nite cylinder, the observables can be rewritten as
OV ( tv)
1
j cos
Z 1 dt1 Z 1
0
c(s + t1 + i1)c(s + t1
i1)c(s)
E
CL
;
E
CL
where L = s + t1 + t2 . For B(z; z) and G(z; z), OV ( tv) has been calculated as7
OB( tv) = 0;
OG( tv) =
0
2 i
(2 )2R1R2:
Here we normalize the observable by dividing it by the volume of the directions other than
X1 and X2.
be derived from the correlator
For the vertex operators (4.1) and (4.2), the matter correlation function in (4.3) can
0
2
z 1
w
+ (1
(z
w)2
) wz
k l ;
6It gives exponential factor in (3.17) and (3.27), where 2 (1)
( )
(1
) 2 log 4
0 for 0
7As in [
7
], if V (z; z) is decomposed by the matter primary eld Vn(z) as V (z; z) = P
m;n mnVm(z)Vn(z),
and the OPE of the primary elds is Vm(z)Vn(z0)
vmn=(z
z0)2, the observable for the tachyon vacuum
solutions is given by
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
< 1.
The matter 4point function in (4.3) on the in nite cylinder can be obtained from (4.6)
by a conformal transformation. To make our calculation wellde ned, we regularize the
correlators by replacing
i1 by
iM and taking the limit M ! 1. As a result, we nd
iM ) k(x1) k(x2) CL
!
0
2
L
2
( 4)e L 4M n ei 2L (1 )(x1 x2) + (1
)e i 2L (x1 x2)o
Multiplying the ghost correlator, the 4point function in (4.3) is given as
c(s + t1 + i1)c(s + t1
i1)c(s) CL
where L = s + t1 + t2. Similarly its conjugate is given by
c(s + t1 + i1)c(s + t1
i1)c(s) CL
which is also obtained by acting the X2 parity transformation
$ 1
on (4.9) [
2
].
Having calculated the correlation functions, the observables can be obtained by
intek l =
j cos
) ne i 2L s
ei 2L (1 )so
) ne i 2L (1 )s
ei 2L so
2
0F12
j cos
j cos
j
j
;
;
j cos
j
and its conjugate. Here
gration of (4.3):
(2 )2R1R2
Comparison with DiracBornInfeld action
Let us check if the gauge invariant observables (4.11) and (4.12) are consistent with what
is known about the magnetic background. Here we consider a D2brane extended in the
X1 and X2 directions for simplicity. The coupling of the D2brane to massless modes of
closed strings is described by the DiracBornInfeld (DBI) action:8
S =
T2e
(2 )2R1R2pdet(Gab + Bab + 2
0Fab);
(4.13)
8Hereafter, we adopt the static gauge and normalize the action by dividing it by the volume of the time
direction.
where T2 is the D2brane tension, and
, Gab and Bab denote the induced
elds on
the brane.
Fixing to the static gauge, we nd the variation of the DBI action under an in nitesimal
variation of B12 around the at background, namely hGabi = ab and hBabi = 0:
The observable (4.11) can be expressed by using the DBI action as
T2e
(
S
B12 F126=0
B12 F12=0
0F12)2
S
B12:
)
nd the variation of the DBI action under an in nitesimal variation
G12 does not appear in this result and the graviton
included implicitly in the dilaton
eld as
=
24 + (1=4) ln det Gab when the spacetime
is of the form M 24
T 2 [
8
]. From (4.16) we get the relation
T2e
(
S
G11
+
S
G22
These results (4.15) and (4.17) show that the observables correctly re ect the coupling
of the D2brane with the constant magnetic eld to the closed string modes. Consequently,
it is explicitly found that (3.7) can be regarded as the classical solution corresponding to
magnetic eld condensation.
4.3
Tdual of a D2brane with F12 6= 0
We would like to examine if the couplings (4.11) and (4.12) are consistent with Tduality.
It is well known that the D2brane is transformed into a Dstring tilted in the dual torus,
which is extended along the line [
8
]
where X01 denotes the coordinate dual to X1. From the Dirac quantization condition (2.1),
it follows that the Dstring winds N times around the X2 direction.
The DBI action for the Dstring is given by
(4.14)
(4.15)
(4.16)
eld is
(4.17)
(4.18)
(4.19)
on the Dstring to coincide with X2. Then, the
In this gauge, the induced metric is calculated as
0F12)2G011 + 4
0F12G012 + G022:
Substituting this into the DBI action (4.19), we can calculate the variation of S under an
HJEP05(218)4
in nitesimal variation of G0ab:
S =
T1e
0
+
S can be rewritten as
T2e
where we have used the relation 0 = 0
24
(1=4) ln det G0ab as in the D2brane case.
By using the relations between the parameters [
8
]
0 = p 0
R1 e
;
T1 = 2
p
0T2;
Combining with the results (4.11) and (4.12), we nd that
G012 F126=0
G012 F12=0
G012
0F12)2 +
0F12)2 +
S
S
G022
)
;
These relations implies that the gauge invariant observables reproduce correct couplings of
the Dsting to the closed string modes in the dual space, because, in the dual space, the
vertex operators (4:1) and (4.2) correspond to G012 and a
G011 + G022 respectively.
1
1
2 p1 + (2
2 p1 + (2
1
1
0F12)2
0F12)2
!
!
(
G011)
)
G022 :
S
( G011)
+
S
G022
F12=0
(4.20)
(4.21)
(4.23)
(4.24)
(4.25)
)
: (4.26)
Concluding remarks
In this paper, we have studied and explicitly calculated the tachyon and vector pro les for
the constant magnetic eld solution on a torus constructed by following ErlerMaccaferri's
method. In addition, we have calculated gauge invariant observables for the solution and
found that the solution reproduces correct couplings of the Dbrane with a constant
magnetic eld to the closed string modes.
A remarkable feature of the resulting pro les is that they have no discontinuity on the
torus. This result does not seem to be consistent with the fact that the solution corresponds
to a topologically nontrivial con guration. In low energy eld theory, the gauge eld will
have discontinuities if one tries to describe the con guration without dividing the torus into
patches. We expect that the same thing happens in string eld theory, but we have found
that pro les for any states do not have discontinuities. One possibility is that pro les for
normalized states are not the right quantities to be chosen in observing such phenomena.
Instead, we might have to consider a sum of in nitely many pro les or even a nonlinear
functional of the string eld. Another possibility is that our results could be an indication
of nonlocality of string
eld theory. Since the star product is a nonlocal operation from
the target space viewpoint, even if a string
eld is de ned in a coordinate patch, it can
spread beyond the boundary after a gauge transformation. Therefore, the nonexistence of
discontinuity itself may be a natural consequence of the nonlocality of string eld theory.
Another important question about the solution is how we can de ne the topological
invariant characterizing the constant magnetic
eld solution in the framework of string
eld theory. The magnetic eld is proportional to an integer due to the Dirac quantization
condition, which is derived based on low energy theory. This quantization condition should
be derived from the string eld theory itself. Unfortunately, the gauge invariant observables
calculated in this paper do not provide any clue about such a quantization condition,
although it is interesting to notice that the observable with the antisymmetric tensor mode
becomes nonzero as a result of the magnetic eld background. In order to capture the
\topological" nature, we may need new insights from noncommutative geometry, matrix
theory and so on [9{12].
There are many interesting possibilities for future exploration related to the constant
magnetic
eld solution. Suppose that we consider the con guration which is Tdual to
the magnetic eld solution along both X1 and X2 directions. The resulting con guration
may correspond to a con guration of multiple Dparticles on the torus. Such a system
was studied in the case of a noncompact space [13]. In this case, the coordinates of
the Dparticles become noncommutative and the observable (4.11) represents the coupling
of the Dparticle to a symplectic form characterizing the noncommutativity. Therefore,
Dparticles with noncommutative coordinates may be described in terms of the string
eld theory in this dual background. Such a string
eld theory may be considered as
another version of Matrix theories [14, 15] although it has no supersymmetry. It is also
possible to make the relation between noncommutative geometry and constant magnetic
eld background manifest by a similarity transformation for string
elds [16, 17]. It will
be interesting to nd such a similarity transformation in the background of the magnetic
eld solution.
Acknowledgments
The authors would like to thank T. Erler, M. Kudrna, C. Maccaferri, M. Schnabl, Y. Kaneko
and S. Watamura for helpful comments. We also thank the organizers of \SFT@HIT" at
Holon, especially M. Kroyter, for hospitality. T. M. also would like to thank the
organizers of \String Field Theory and String Phenomenology" at HarishChandra Research
Institute, Allahabad, for their hospitality and for providing him a stimulating
environment. This work was supported in part by JSPS GrantinAid for Scienti c Research
(C) (JP25400242), JSPS GrantinAid for Young Scientists (B) (JP25800134), and JSPS
GrantinAid for Scienti c Research (C) (JP15K05056), and Nara Women's University
Intramural Grant for Project Research. The research of T.M. was supported by the Grant
Agency of the Czech Republic under the grant 1722899S. The numerical computation in
this work was partly carried out at the Yukawa Institute Computer Facility.
A
Conformal transformation of the vector vertex operator
In this appendix, we would like to derive the conformal transformation (3.23) of the vector
vertex operator. The OPE of the vertex operator with the energy momentum tensor is
(
i 0p
)3
eip X ( ) +
(
0p2 + 1
)2
1
The right hand side includes the tachyon vertex operator which is a primary eld of weight
given by
transformed as
T ( ) eip X ( )
(
0p2
)2
1
eip X ( ) +
Let us consider a conformal transformation
! z = f ( ), under which a eld ( ) is
( ) ! f
( ) = Uf ( )Uf 1:
Here, the operator Uf is given in terms of the energymomentum tensor as follows:
Since the tachyon vertex is primary, it is transformed as
I d
2 i
Uf = eT (v);
T (v) =
v( )T ( );
Uf eip X ( ) Uf 1 =
dz
d
0p2
eip X (z):
From the OPEs (A.1) and (A.2), we can derive the commutation relations of the vertex
operators with T (v):
i
2
(A.2)
(A.3)
(A.4)
(A.5)
(A.7)
Let us consider the transformation of the vector vertex operator under the following one
parameter family of conformal transformations:
By using the commutation relations (A.6) and (A.7), we can see that the result should be
expressed as
Yt( ) @X eip X ( ) + Zt ( ) eip X ( );
dYt( )
dZt ( )
dt
dt
= ( 0p2 + 1)v0 ft( ) Yt( );
= ( 0p2)v0 ft( ) Zt ( )
Yt( ) = ft0( )
where Yt( ) and Zt( ) are some functions of t and . Di erentiating these with respect to
t and using (A.6) and (A.7), we obtain the equations
where ft( ) is de ned by ft( ) = exp(t v( )@) . Integrating these with the initial conditions
Yt=0 = 1 and Zt=0 = 0, we nd
2
i 0p v00 ft( ) Yt( );
i
2
p
ft00( )
ft0( )
ft0( )
e ip X ( ) k( 1) l ( 2) + regular terms;
(
E
!
By setting t = 1 and p
p , we obtain the formula (3.23).
B
Calculation of correlation functions in the vector pro le
In this appendix, we will show how to calculate the correlation functions which appear
in (3.21) and (3.22).
Here we begin with the calculation of the following 4point functions on the complex
The rst two are derived directly from boundary conditions of X. Moreover, in the limit
, we nd that the 4point function behaves as
plane:
behaves as
D
E
Here X = (X1 + iX2)=p2, and , 1 and 2 are taken to be real and satisfy 2 < 1 < , so
that the operator e ip X is on the boundary with the Neumann boundary condition. The
correlation function is a 1form on the complex plane with respect to the variable
and it
E
E
which xes the normalization of the four point function. These conditions determine the
4point function as
E = i 0p
(
where the 3point function on the right hand side is given by (3.8) [
2
]. By taking the limit
in (B.4), we obtain a correlation function involving the vector vertex operator on
the complex plane:
D
e ip X ( ) k( 1) l ( 2) ;
E
(B.4)
+ (1
(
1)(
) 1 +
2)
(
Taking the normalization of the modi ed BCC operators [
2
] into account, we obtain
= i 0p Cl;kp (2 )2R1R2
f
j cos
j
+ (1
) 1 +
2g (
( 1
The correlation function for the conjugate vertex operator @X~ e ip X (X~ = (X1 iX2)=p2)
can be calculated in a similar way and we nd
E
= i 0p~Cl;kp (2 )2R1R2
f
j cos
j
+
1 + (1
) 2g (
( 1
Eq. (B.7) can also be obtained by using the fact that the X2 parity transformation X2
X2 corresponds to the following transformations of the parameters:[
2
]
Having found the 3point functions on the complex plane, we can nd the one on the
in nite cylinder with circumference L by the conformal transformation:
By using the transformation law (3.23), we obtain the following 3point functions on the
cylinder from (B.6) and (B.7):
D
=
L
h Cl;kp (2 )2R1R2
i 0p
2
1
2
j cos
0
sin
j
p sin
(z1
L
E
L
(z2
L
z2) cos
where h = 0p2 + 1 and 12 =
21 = 1.
p ! p~;
! 1
:
L
z =
arctan :
L
z3)
sin (z2 z3) h 1
L
sin (z1 z2) sin (z1 z3) h
L
!
(B.8)
(z1
z3)
L
+ sin
(z1
L
z3) cos
(z1
z2)
L
; (B.9)
The matter correlation functions that appear in (3.21) are calculated by using (B.9) as
(B.12)
(B.13)
1) cos t2 sin
1) sin t2 cos
p h
sin
h+1
sin
1) cos t2 cos
t3+ 21 sin
2
t1+ 21
j
cos
j
cos t2 sin
sin
h 2
t1+ 21 sin
t2
h+1
p h
sin
h
sin
t1+ 21 sin
t2
h+1
(h
j
cos
cos t2 sin
sin
h 2
j
t2
h
sin
h+1
+(h
(h
t1+ 21 sin
t3+ 21
1) cos t2 sin
1) sin t2 cos
t1+ 21 + sin t2 cos
t1+ 21
t3+ 21 sin
t3+ 21 sin
t1+ 21 cos
t1+ 21 cos
t1+ 21
t1+ 21
D
ip X
and
D
ip X
=
L
i
2
L
i
2
1
2
C
p
sin
h
+(h
(h
1
2
C
p
L
h+1
0p
L
h+1
0p
1
2
1
2
L
E
1
2
n
2
E
n
L
1
2
+h sin t2 sin
t3+ 21 cos
2
t1+ 21 + sin t2 sin
t3+ 21 sin
2
t1+ 21
o
#
; (B.11)
where s is de ned by
s =
s=L. The ghost correlation functions which appear in (3.21)
are given as
Combining these results, we nd that the integrand in (3.21) turns out to be
D
ip X
D
ip X
k
l
(t1 + t2)@ (t1)
L
c(t1 + t2)
1
2
E
E
L
L
=
=
E D
L
L
L
2
2
sin
sin
2
2
1
2
L
t3+ 21 ;
t1+ 21 :
1
2
L
h 1 Cl;kp (2 )2R1R2 "
j cos
j
2
cos t2 sin2
sinh
+
1
2
t1+ 12 sinh
t3+ 12
t1+ 12 + sin2
o
1) cos t2 sin t1+ 12 sin t3+ 12 sin t1+ 12 cos t1+ 12
The second term in the bracket on the right hand side proportional to p is antisymmetric
under the interchange of t1 and t3. Since the integration measure in (3.21) is symmetric
with respect to it, the second term does not contribute to the vector pro le.
Next let us turn to (3.22). The correlation function which appears in the integrand
in (3.22) is given as
L
j cos
j sinh
sinh 2 t2
t1+ 12 sinh
t3+ 12
n
sin t2 (1 + 2t1) cos t1+ 12 sin t3+ 12
(1 + 2t3) cos t3+ 12 sin t1+ 12
o
2 (t1
L
t3) cos t2 sin t1+ 12 sin t3+ 12
2 sin t2 sin t1+ 12 sin t3+ 12 cos2
t1+ 12
cos2
t3+ 12
2 cos t2 cos t1+ 12 sin2
t1+ 12 sin t3+ 12 cos t3+ 12 sin2
t3+ 12 sin t1+ 12
: (B.15)
Since (B.15) is antisymmetric under the interchange of t1 and t3, we nd that the right
hand side of (3.22) vanishes.
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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