>K9m 2014, tom 146, bbin. 1 (7), rap. 82 86
© 2014
OPEN STRING IN THE PRESENCE OF THE pp-WAVE, LINEAR DILATON, AND KALB-RAMOND BACKGROUNDS
M. Zoghi* D. Kamani**
Faculty of Physics, Amirkabir University of Technology (Tehran Polytechnic) P. O. Box
15875-4413, Tehran, Iran
Received January 10, 2014
We study open strings attached to a Dp-brane in the presence of the pp-wave background along with a constant antisymmetric B-field and the linear dilaton. The noncommutativity structure of this system is also investigated.
DOI: 10.7868/S0044451014070098
1. INTRODUCTION
String theory in various backgrounds has boon profoundly studied. Some of these backgrounds admit a solvable string theory. One of them is the />/>-wavo spacetimo fl], which is supported by a null, constant 5-forni flux and can be obtained from the AdS^ x S5 metric by taking the Penrose limit. The p/j-wave background is a maximal suporsymmetric space in which closed string theory is exactly solvable in the light-cone gauge [2, 3]. Another popular background is the constant antisymmetric U-field, which has boon extensively studied in the literature. It leads to nontrivial physics on the branes. The noncommutativity of the open string end points, which are attached to a D-brane [4], is a consequence of the mixed boundary condition in the U-field background. In addition, we have the linear dilaton field as a background, which is the simplest background for noncritical string theory [5]. Among the various conformal field theories (CFTs), the linear dilaton CFT has some interesting applications in string theory [6]. For example, the D-brane noncommutativity is investigated in various background fields such as the dilaton [7 10].
In this article, we consider all the three open-string background fields mentioned above and investigate on the solvability of the theory. Besides, it has been demonstrated that in the light-cone formulation of strings in the pjywave, the momentum space also becomes nonconiniutativc, which leads to a fully nonE-mail: zoghi'fflaut .ac .ir E-mail: kamani'fflaut .ac .ir
commutative phase space. This fact also motivated us to extend the problem by adding the above background fields and see whether any new kind of quantum geometry arises.
2. OPEN STRING IN A SET OF BACKGROUND FIELDS
The pjywave background consists of a plane wave metric, accompanied by a homogeneous R R 5-forni flux
d.s2 = - / 2 A'' X' (dX +)2+2 dX+dX ~ + dX1 dX1,
1= 1,2,... ,8, F5 = fdX+ A (dX1 A dX2 A dX3 A tIX4 + + dX5 A dX6 A dX7 A (IXs) .
We consider an open string attached to a D/>brane in the presence of the following background fields: the pjywave metric, a constant Ivalb Raniond tensor field Dtll,, and the dilaton field '!>. In the light-cone formalism, the coordinates are decomposed as
{.Y'..Y }{J{X'\1 1.2...../> - 1} U
[J{.Y'|/ /> — I.... ,9},
where X^ = (X° ± Xp)/\/2 and _Y+ = x+ + a'p+T. The string sigma-model action in the above backgrounds is
S =
Ait a'
(ta x
9u \y-hhabdaX1dbXJ + m2X1X + fabDIJdaXIdbXJ +
(2)
whoro S is the string worldshoot with the mot lie hab, and h = dot hab- Tho scalar curvaturo
RC2)
is const ructod from tho motric hab• Tho spacotimo motric is also gilv, which is givon by Eq. (1). Tho mass parameter in. i.o., tho IIlclSS of tho worldshoot fiolds X1, is defined as m := a'p+f.
In the conventional case, the dilaton is usually a general arbitrary function of the spacotimo coordinates, but considering only a linear dilaton gives rise to simplified equations. We suppose that the dilaton field has a linear form along the brane woiidvolume, i.e., $ = uaXa, where the parameters {aQ|a: = 0,1,... ,p} are constant. Regarding the diffeomorphism invariance of action (1), we are able to choose a conformally flat form for the worldshoot metric.
hab(*,T) =
Vab-
Because the dilaton field removes the Weyi symmetry, we are not allowed to set p(a,r) equal to zero. We finish our setup by setting u0 = up = 0 to avoid the presence of the coordinates X± in the action.
Equating the variation of the action to zero gives the equations of motion for the worldshoot fields X1 and p in the form
(a2 -tn2)X1 + -a'a1d2p = 0
md^X1 = o,
(3)
(4)
where ff2 = —dT2 + d,T2. In a noncritical string theory, i.e., for u2 = uju1 oc d — 26 0, these equations can be written as
(d2 -m^X1 +AljXJ = 0
d2p =
2m2 ai X'
a'u'
(5)
(6)
where the matrix is defined by Aij := m2ujuj/u2. The first equation reveals that the worldshoot fields X1 effectively feel the potential
l'(.Y) = \AijXiXj + I o,
where Vo is the potential at the origin of the coordinates. We observe that the presence of the linear dilaton and pjywave background simultaneously is the origin of this potential. However, the vanishing of the variation of the action also defines boundary conditions for the open string. For example, for the open string end at a = 0, we obtain the equations
(d^x1
B'jdr.xJ)\„=Q = o,
(7)
(tudrX1)^ = 0 (8)
for X1 and p.
It is not very easy to solve Eq. (5) in the general case. Therefore, we consider the situation where the only nonzero components of the vector uj are m and (i2, which gives $ = uiX1 + a-iX2. We also apply the
block diagonal form of the U-field /
D =
\
0 0 0
-V
0 \
0 V
0 /
(9)
where the nonzero elements are Bi-2 = b and U34 = b'. Now, rewriting Eq. (5), we obtain
Ai X1 + kX2 = 0.
A,.Y2 + kX1 = 0,
(10)
where the operators A{1;2} and the constant k are defined by
A
{1,2}
k : = m
•->2 2 = 0 — m
9 «1«2
'{2,1}
(H)
tr
By combining Eqs. (10), we obtain the equations A2Ai^Y1 — itX1 = 0, (12)
1
T
X2 =
(13)
The general solution of partial differential equation (12), which has rank four, with boundary condition (7) can be written as
A'1((T,r) = ( x1 cosuj0T+2a'p1 SmL°°T ) ch(uJob(r) —
\ <¿0 J
u;0
-2a'p2 cosier + x2uio sinu^r) sh(uioba)
oxp
7uvr X
n^O 1
x I 7-cos na H--b sm na
u;„. n )
83
6*
M. Zoghi, D. Kamani
>K3TO, tom 146, bbin. 1 (7), 2014
Thon Eq. (13) implies that
X'2(CT,T) = ——X1(a,T) u~2
Here, tho frequencies are
u;0 = ±-
m
(15)
(16)
u>„ = sign (n)\/m2 + n2. Wo noto that wo can write Eqs. (10) as A, Ai .V1' - k2X2 = 0,
which reveals that À'2 lias a solution similar to the one in Eq. (14), with the indices 1 and 2 interchanged. Comparing this solution for X2 and Eq. (14) leads to b = 0. In this case, the mixed boundary conditions for A'1 and A'2 reduce to the Neumann boundary conditions. Thus, the mode expansion for A'1 is
Xl{(t, t) = x1 cos hit + 2a'p1
Sill ll)T
m
a„
iV2a' N exp (—¿u>„r) — cosna.
(17)
and again X2(ct,t) = — (ui/u-2)X1(a,T).
Next with a different approach, we demonstrate that our setup is consistent only for b = 0. It is noteworthy that the dilaton term of the action cannot be treated just classically, but it has a quantum worldsheet correction that modifies the energy momentum tensor. String action (2) with the linear dilaton $ = a^X'1 defines a family of CFTs with the energy momentum tensor
T(z) = -1 : gtll,dA'W : +a,1d2X'\ T(z) = ~ : g^dX'+dX" : +«,J2A'J.
(18)
Recalling that A' is related to the other coordinates except A'+ and using thepjywave metric (1), we obtain
T(Z) = -1 : (±dXKdXK
trr
xhxK + dxldx,
+uKd2X'
(19)
T(Z) = -1 : (±8XKdXK + ^XKXK
dX'dXi) : +uK32X'
where
A' € {1,2,3,4}, / f {.->.... ./>— 1} I__J{/>— 1____,9}.
We note that we have assumed the only nontrivial dilaton coefficients to be a\ and «2- Because momentum does not flow at the boundary, we must have T(z) = = T(z) (at the boundary, i = !), hence we obtain the conditions
(20)
ukan!Bkl = 0, o'pjlh.i = 0, ukx0'Bkl = 0, M =1,2.
These equations imply that BV2 = b = 0.
3. QUANTIZATION
For the string coordinates A'3 and A'4, we can write the solutions as
X1 (<r, r) = A0' (<r, r) + X( (<r, r)
-Yq (<t, t) = ( x1 cosuot + 2a'p x c1i(u;o?/<t)
uiob'
ji SlIlUioT
u;0 B*j. x
x ^—x^ u>o sintJoT-\-2a' p'^ cosuJqtJ sh(ujob'tj), ^^ X( ((T, r ) = V2a' exp (—¿u>„r) x
x /—- cos na H---B j, sin no-
\ u;„. n
where A'q is the zero-mode part and X( is the oscillating part, and {/', J' = 3,4}. We note that both signs of tJo determine only one solution for X!l. According to our setup, we see that only the coordinates A'3 and A'4 contain the U-field elements. Therefore, only the A'3A'4 plane is nonconiniutativc, and we now investigate it.
The canonical momentum corresponding to the open string coordinate X1 is given by
Pl(a1 T) = {9tXI ~ BlJQ-XJ) ■ (22)
For the directions A'3 and A'4, the conjugate momenta also split into two pieces, the zero-mode part and the oscillating part:
P1' T) = PJ'((T,T) + P{'((T,T)
The matrix .1 /./' is inverse to M(„)//.//, and the matrix (DMis antisymmetric. The matrix M^ may be interpreted as a mode-dependent open string metric for the oscillators.
The above results allows calculating intrinsic commutation relations for the open string coordinates and their conjugate momenta:
PÎ (^r) =
2?ra'
M J, Ulo sillUioT +
+ 2a:'//' costJorj ch(u;ob'a) - j(BM)j, x x ^2a'// siriuioT + xJ u>ocosu>orj sh(^0b'cr)
(23)
Pi' (<7,T) =
n\/ ¿a'
7 J2 0XP ("'''
■lUJnT) X
x i a'l Mj'1 cos no- + i——<r(' Bj, sin no-
\ 'llUjn
The symmetric matrix M is given by
Mpji = (1 + b2)6iiji. (24)
Again, both signs of u>o specify one value for each of the momentum components P?' and P4.
It is known that in a D-brano with the U-field background, the spatial coordinates of the brano do not commute. We now investigate this in our setup. To quantize the open string theory, we use the symploctic form
/ 4 4 \
n= da[J2 Y.3I-J- dP1' A dx
(25)
K,I>= 3 ./'=3
This can be justified by analyzing the constraint structure of the theory (see, e.g., Rofs. [11,12]). With Eqs. (21) and (23), this differential form becomes
il =
F— !î .7"' —3 ^
/'=3 ./'=3
s1I2(ttu;O^) ~ 27m'?/2 2a:' sh2(TTu;o^) TTUio2^2
2iruob
(MB)/,./,d./' Ad/ -
(M B)r.r dp1' Ad//'
EK M n)I'J' ,/'„,/' -;-da„ A da „
«=1
(26)
where the symmetric matrices M(n)//j/ are defined by
2 7 2 '
M{n)VJ, = [ 1
Sr
i'j'-
(27)
T
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