ROTATING MOVING D-BRANES WITH BACKGROUND FIELDS
IN SUPERSTRING THEORY
F. Safarzadeh-Maleki* D. Kamani**
Department of Physics, Amirkabir University of Technology (Tehran Polytechnic) 15875-4413, Tehran, Iran
Received February 14, 2014
Using the boundary state formalism, we study rotating and moving Dp-branes in the presence of the Kalb-Ra-mond field, U{1) gauge potential and tachyon background fields. The rotation and motion are in the brane volumes. The interaction amplitude of two Dp-branes is studied, and especially the contribution of the super-string massless modes is segregated. Because of the tachyon fields, rotations and velocities of the branes, the behavior of the interaction amplitude reveals obvious differences from what is conventional.
DOI: 10.7868/S0044451014100125
1. INTRODUCTION
D-branos, as essential ingredients of superstring theory fl], have important applications in different aspects of theoretical physics. These objects are classical solutions of the low-energy string effective action and hence can be described in terms of closed strings. Besides, D-branos with nonzero background internal fields have shown several interesting properties [2 7]. For example, these fields affect the emitted closed strings of the branes and therefore modify the brane interactions.
On the other hand, we have the boundary state formalism for describing the D-branos [2,8 14], which is a useful tool in many complicated situations. This is because the boundary state encodes all relevant properties of the D-branos, and that is why the formalism has boon widely used recently in studying properties of D-branos in string theory. A boundary state can describe creation of a closed string from the vacuum, or equivalently it can be interpreted as a source for a closed string, emitted by a D-brane. Among achievements in this formalism is its extension to superstring theory and the analysis of the contribution of conformai and super-conformal ghosts. The overlap of two boundary states corresponding to two D-branos, via a closed-string propagator, gives the amplitude of interaction of the branes. So far, this method has boon
* E-mail: f.safarzadeh'öaut .ac .ir
E-mail: kamani'&aut .ac .ir
suitably applied to various configurations in the presence of different background fields, including stationary branes, moving branes with constant velocities, angled branes [15 19], various configurations in a compact spacetimo [15], in the presence of the tachyon field [19,20], a bound state of two D-branos [13], and so on.
Previously, we studied a general configuration of rotating and moving D/>branes in bosonic string theory in the presence of the following background fields: the Kalb Ramond field, U( 1) gauge potentials that live in the D-branes world volumes, and tachyon fields [20]. In this paper, the same setup is considered in superstring theory. The novelty of the results is considerable. Our procedure is as follows. For this setup, we obtain the boundary state associated with the brane and then compute the interaction between two such D/>branes as a closed superstring tree-level diagram in the covari-ant formalism. The generality of the setup strongly recasts the feature of boundary states and interaction of the branes. We observe that the interaction amplitude and its long-range part, which occurs between distant branes, exhibit some appealing behaviors.
We note that we consider rotation of each brane in its volume and its motion along the brane directions. Due to the various fields inside the brane, there are preferred directions, which indicates the breaking of the Lorentz symmetry, and hence such rotation and motion are meaningful.
This paper is organized as follows. In Sec. 2, the boundary state of a closed superstring, correspond-
7 >K9T<£>, iibiii.4(10)
769
ing to a rotating moving D/>branc with various background fields is constructed. In See. 3, interaction of two D/>brancs in the NS NS and R R sectors of the superstring is calculated. In Sec. 4, the long-range force of the interaction is extracted. Section 5 is devoted to the conclusions.
2. BOUNDARY STATE ASSOCIATED WITH A ROTATING MOVING D-BRANE WITH BACKGROUND FIELDS
We use the following signia-niodel action for a closed string to describe a rotating and moving D/>bra-ne in the presence of the Ivalb Ramond, photonic, and tachyonic fields:
5 =
G^daX^dhX1
£ahBlu,da x»dhx'
associated with the 9 — p scalar fields, from the world-volume point of view. These scalars represent coordinates of the brane. We keep them fixed, that is, assume that the branes do not execute transverse motion. For the gauge field Aa, we have chosen the special gauge in the third equation in (2).
In the literature, the tachyon field is usually nonzero just in one dimension and its effects are studied on a space-filling brane, while we here consider a D/>branc with an arbitrary value of p. Besides, the square form of the tachyon profile is used to produce a Gaussian integral. Hence, the tachyon field has components along all directions of the brane worldvolume. The gauge and tachyon fields are in the spectrum of open strings attached to the D/>branc.
In fact, in the presence of an antisymmetric field and a local gauge field, there are preferred alignments in the brane, and hence the rotation and motion of the brane in its volume is sensible.
j^-j j da(AadcrXa + oJafj.J^ + T(Xa)), (1) 2.1. Bosonic part of the boundary state
fE Til the closed-stririff onerator formalism, the D-l
where E is the worldsheet of a closed string emitted (absorbed) by the brane, and is the boundary of the worldsheet. Besides, "a:" and "/?" are indices along the brane worldvolume and "/" is used for the directions perpendicular to it. In addition, the background fields GllLl,, Btll,, Aa, T and the antisymmetric variables u)aij and are respectively the spacetime metric, a Ivalb Ramond (antisymmetric tensor) field, a gauge field, a tachyon field, the angular velocity, and the angular momentum density of the brane. We consider the following forms for these variables:
G,lv = = diag(^l, 1,... ,1), BllLl, = const, Bai = 0, 1
An = --Fai;iXf , Fni-i = const,
In the closed-string operator formalism, the D-bra-nes of the type-IIA and type-IIB theories can be described by boundary states. These are closed-string states that insert a boundary on the closed-string worldsheet and enforce appropriate boundary conditions on it. We now extract the corresponding boundary state in our setup. Requiring the vanishing of the variation of the action with respect to the closed string coordinates A''J(<r, r), we obtain the boundary-state equations
[(''laß + ^aß)dTXß + .Fnr;i),TXi
+ UnfiX%=0\Bhos) = 0,
(ó"A'¿)T=0|i?6os) = o,
(3)
aß
1,
T(X) = -Cnr,.\n X\ Uaß = Ußa = const,
Uai = Uij = 0, J^ = 2<^jafjXndTXfi.
(2)
Map ■The last equation indicates the rotation and motion of the brane. The components {u;oa|<5 = 1,2,... ,p} denote the velocity of the brane, while the elements {<jj&p\a,d = 1,2,... ,p} represent its rotation.
We note that in the presence of a D/>branc, the 10-dimensional U( 1) gauge field Atl is decomposed into a longitudinal U( 1) gauge field which lives in the worldvolume of the D/>branc, and a transverse part A;
where Taß = 9aAß — 8ßAa — Baß is the total field strength.
It is worthwhile to show that the Lorentz symmetry is broken along the worldvolume of the brane. Equations (3) leads to
'hosWbos) —
= J da o
(A^1U)a X^X1 - (A-1Uf1XaX'
I Bhos), (4)
where Aaij = 'tfap + 4u;Q/j. We observe that for restoring the Lorentz in variance, all elements of the tachyon
matrix Uai:i and the total field strength J"aii must vanish. We demonstrated this for the bosonic part of the boundary state. This procedure can also be applied to the total boundary state, which includes the bosonic and formionic parts, to prove the breakdown of the Lorontz invarianco along the worldvolume of the brano.
Introducing the closed-string mode expansion
A''J((T, r) = x'1 + 2a'p'1T +
/u \—* 1
in Eq. (3) gives
Vaß + i'^atí - faß H Vaß + aß + Taß
ali(--2in{T-<r)
2 m
-U,
■ à'1 e~'2in{T+,T)
(K
aß
-Uaß 'I
[^'(Vaß
(«m -
2 m
x IBhos)io^ = 0, ^aß )PÍ + Uaß.^ 1 IBhoEy0) = 0, 11 Bhos){
(5)
\(osc) = 0
=0,
where the set {yl\i = p+1,... ,9} indicates the position of the brano. Besides, for the boundary state
IBhos) = |Dhos){a) O |Dhos){osc),
the components |Uf,os)i0^ and |i?6OS)iosc') respectively represent boundary states for the zero modes and oscillating modes.
The solution of the oscillating part, which can be found by the coherent state method, is given by
\Bhos)iosc) = n^Çoor1 X
«=1
x exp
(m)i-t
in I
(6)
whore the matrices are defined as
Q(m)aß = Vaß + l'-^aß ~ Faß +
Si
A
(m)
A -
= (Q(m)Ar(m))a/ï,
*(m.)aß
N(m)aß = Vaß + ^aß + Faß
2m
aß
Uaß ■
(7)
Because the mode-dependent matrix A(m) is not orthogonal in general, the matrix (A' also appears in the definition of S(m)fll,. In Eq. (C), the
normalization factor n«=i Q(n.)]_ can deduced from the disk partition function.
The boundary state for the zero modes is given by
I
IBhoK)i0] = j exp <| ia'
a=0
¿ (C/_1 A + A TIJ-
1 1 r>a ) aß1
jP
a,ß=Q,a=£ß
x n \pa)tlpa IP^' - y^tf = °>-
\ a / i
The integration over the momenta indicates that the effects of all values of the momentum components have boon taken into account. As we see, unlike the oscillating part, the total field strength does not enter Eq. (8).
We note that for calculating the interaction amplitude, the contribution of the conformal ghosts b, c, b and c in the bosonic boundary state must also be taken into account.
2.2. Fermionic part of the boundary state
The supersymmotric version of action (1) is invariant under the global worldsheet supersymmotry
6X'1 = «/•'-%
Stf* = -iffdaX^e, u G {T,<T},
where e is an infinitesimal constant anticommuting spinor. Because we need the explicit forms of the components of the worldsheet fermions
i* =
4''i „h'1
wo write the
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.