Interaction of Dynamical Fractional Branes with Background Fields: Superstring Calculations

TL;DR

This paper calculates the boundary state for fractional Dp-branes with background fields and motion in a superstring framework, analyzing their interactions and behavior at large distances.

Contribution

It introduces a method to compute boundary states for fractional D-branes with background fields and motion in superstring theory, enabling analysis of their interactions.

Findings

Derived the boundary state for fractional D-branes with background fields.

Calculated the interaction amplitude between two moving fractional D-branes.

Analyzed the large-distance behavior of the interaction amplitude.

Abstract

We compute the boundary state corresponding to a fractional D$p$-brane with transverse motion and internal background fields: Kalb-Ramond and a $U(1)$ gauge field. The spacetime has the orbifold structure $\mathbb{R}^{1,5} \times\mathbb{C}^{2}/\mathbb{Z}_{2}$. The calculations are in the superstring theory. Using this boundary state we shall obtain the interaction amplitude between two parallel moving fractional D$p$-branes. We shall extract behavior of the interaction amplitude for large distances of the branes.

Equations

$$S=$$

$$[\gamma {\partial }_{\tau }(X^0-v^i{X}^i)+\gamma {\mathcal{F}}_{\bar{\alpha }}^0{\partial }_{\sigma }{X}^{\bar{\alpha }}{]}_{\tau =0}|B_X⟩=0,$$

$${\partial }_{+}{X}^{\mu }(\sigma ,\tau )$$

$${\partial }_{-}{X}^{\mu }(\sigma ,\tau )$$

$$X^{\mu }(\sigma ,\tau )\to \sum _t\frac{1}{2t}\left(i{\psi }_t^{\mu }{e}^{-2it(\tau -\sigma )}+\eta {\tilde{\psi }}_t^{\mu }{e}^{-2it(\tau +\sigma )}\right).$$

$$|B⟩={\mathcal{N}}^{\mathrm{U}}\left(|B{⟩}_{\mathrm{NS}}^{\mathrm{U}}+{ϵ}_1|B{⟩}_R^{\mathrm{U}}\right)+{\mathcal{N}}^{\mathrm{T}}\left(|B{⟩}_{\mathrm{NS}}^{\mathrm{T}}+{ϵ}_2|B{⟩}_{\mathrm{R}}^{\mathrm{T}}\right),$$

$${\psi }_{+}^{\mu }=\sum _t{\tilde{\psi }}_t^{\mu }{e}^{-2it(\tau +\sigma )},{\psi }_{-}^{\mu }=\sum _t{\psi }_t^{\mu }{e}^{-2it(\tau -\sigma )},$$

$${\psi }_t^{\lambda }\text{and}{\tilde{\psi }}_t^{\lambda },t\in \mathbb{Z},$$

$${\psi }_t^{\lambda }\text{and}{\tilde{\psi }}_t^{\lambda },t\in \mathbb{Z}+\frac{1}{2},$$

$$\displaystyle\bigg{[}[\gamma(\delta_{\lambda}^{0}-v^{i}\delta_{\lambda}^{i})-\gamma\mathcal{F}^{0}_{~{}~{}\bar{\alpha}}\delta_{\lambda}^{\bar{\alpha}}]\psi^{\lambda}_{t}-$$

$$\left({\psi }_t^{\lambda }-i\eta S_{(t){\lambda }^{\prime }}^{\lambda }{\tilde{\psi }}_{-t}^{{\lambda }^{\prime }}\right)|B_{\psi },\eta {⟩}^{\mathrm{osc},\mathrm{T}}=0,$$

$$M_{(t)\lambda }^0$$

$$M_{(t)\lambda }^{\bar{\alpha }}$$

$$M_{(t)\lambda }^i$$

$$N_{(t)\lambda }^0$$

$$N_{(t)\lambda }^{\bar{\alpha }}$$

$$N_{(t)\lambda }^i$$

$$\left({\psi }_r^a+i\eta {\tilde{\psi }}_{-r}^a\right)|B_{\psi },\eta {⟩}^{\mathrm{osc},\mathrm{T}}=0,$$

$$\left({\psi }_0^a+i\eta {\tilde{\psi }}_0^a\right)|B_{\psi },\eta {⟩}_{\mathrm{NS}}^{(0)\mathrm{T}}=0.$$

$$\left({\psi }_0^{\lambda }-i\eta S_{{\lambda }^{\prime }}^{\lambda }{\tilde{\psi }}_0^{{\lambda }^{\prime }}\right)|B_{\psi },\eta {⟩}_{\mathrm{R}}^{(0)\mathrm{T}}=0.$$

$$|B_X{⟩}^{\mathrm{osc}}$$

$$|B_X{⟩}^{(0)}=\frac{T_p}{2}{\int }_{-\infty }^{\infty }\prod _{\lambda =0}^{5}d{p}^{\lambda }\prod _i\left[\delta \left({\hat{x}}^i-v^i{\hat{x}}^0-y^i\right)|p^i⟩\right]\prod _{\alpha }|p^{\alpha }⟩.$$

$$|B_{\psi },\eta {⟩}_{\mathrm{NS}}^{\mathrm{T}}$$

$$|B_{\psi },\eta {⟩}_{\mathrm{NS}}^{(0)\mathrm{T}}={\left(\bar{C}\frac{1+i\eta \bar{\Gamma }}{1+i\eta }\right)}_{LM}|L⟩\otimes |\tilde{M}⟩,$$

$$|B_{\psi },\eta {⟩}_{\mathrm{R}}^{\mathrm{T}}$$

$$|B_{\psi },\eta {⟩}_{\mathrm{R}}^{(0)\mathrm{T}}={\left(\tilde{C}{\Gamma }^0\dots {\Gamma }^p\frac{1+i\eta \tilde{\Gamma }}{1+i\eta }G\right)}_{AB}|A⟩\otimes |\tilde{B}⟩,$$

$${\Gamma }^{\lambda }G=S_{{\lambda }^{\prime }}^{\lambda }G{\Gamma }^{{\lambda }^{\prime }}.$$

$$G=;\exp \left(\frac{1}{2}{\bar{\Phi }}_{\lambda {\lambda }^{\prime }}{\Gamma }^{\lambda }{\Gamma }^{{\lambda }^{\prime }}\right);,$$

$$S=(\Lambda -\Phi {)}^{-1}({\Lambda }^{\prime }+\Phi ),$$

$${\Lambda }_{\lambda }^0=\gamma ({\delta }_{\lambda }^0-v^i{\delta }_{\lambda }^i)$$

$${\Lambda }_{\lambda }^{\bar{\alpha }}={\delta }_{\lambda }^{\bar{\alpha }}$$

$${\Lambda }_{\lambda }^i={\delta }_{\lambda }^i$$

$${\Phi }_{\lambda }^0$$

$${\Phi }_{\lambda }^{\bar{\alpha }}$$

$${\Phi }_{\lambda }^i$$

Full text

PACS numbers: 11.25.-w; 11.25.Uv

Keywords: Boundary state; Moving fractional branes; Background fields; Interaction.

1 Introduction

The D-branes are known as objects which have critical roles in the string theory [1]. They can be studied in various ways, for example, they may be known as hypersurfaces that boundaries of the string worldsheets can sit on them. In addition, the boundary states in the closed string channel elaborate all properties of the D-branes, such as their dynamics [2], [3], [4], [5], [6], [7]. Precisely, the boundary state formalism can be used to investigate the interactions between the D-branes. So far this adequate procedure for some configurations of the D-branes, in the presence of the background fields, for the bosonic and superstring theories has been extensively utilized [8], [9], [10], [11], [12], [13], [14], [15]. Besides, the setups of the regular branes and systems of the fractional branes, such as BPS or stable non-BPS, have attracted the attention of the researchers [16], [17], [18], [19], [20], [21].

Previously we have studied a system of two fractional branes in the bosonic string theory [22]. In this paper we extend the calculations for a generalized setup in the superstring theory. Thus, we consider two parallel fractional D$p$-branes with the Kalb-Ramond field $B_{\mu\nu}$ as the background field and two $U(1)$ gauge potentials as the internal fields on the worldvolumes of the branes. The branes have velocities which are perpendicular to their volumes. We apply a $10$-dimensional factorizable spacetime with the orbifold structure, i.e. $\mathbb{R}^{1,5}\times\mathbb{C}^{2}/\mathbb{Z}_{2}$. The interaction amplitude of the branes and its long-range behavior will be extracted. We shall see that our results considerably are novel and interesting. In fact, due to the fermionic degrees of freedom in the above-mentioned generalized setup, calculations are not straightforward, and drastically we need some novel techniques.

This paper is organized as follows. In Sec. 2, the boundary state of a closed superstring, associated with a moving fractional D$p$-brane with various background fields, will be constructed. In Sec. 3, we shall calculate the interaction amplitude and the related long-range force concerning two parallel branes in the twisted NS-NS and R-R sectors of the superstring theory. Section 4 is devoted to the conclusions.

2 The boundary state

A fractional brane of our system is stuck at the fixed-points of the noncompact orbifold $\mathbb{C}^{2}/\mathbb{Z}_{2}$, where the complex coordinates of $\mathbb{C}^{2}$ are constructed from $\{x^{a}|a=6,7,8,9\}$, which the $\mathbb{Z}_{2}$-group acts on them. The fixed-points are located at the hyperplane $x^{a}=0$.

In this section we extract the boundary state of a moving fractional D$p$-brane with background fields through the sigma-model action of closed string. Let us have a glance on the bosonic part of the string sigma-model that we use

[TABLE]

where the set $\{x^{\alpha}|\alpha=0,1,\cdot\cdot\cdot,p\}$ shows the brane directions, $\Sigma$ indicates the worldsheet of a closed string and $\partial\Sigma$ is its boundary. The variables $h_{ab}$ and $G_{\mu\nu}$ are the metrics of the string worldsheet and the $10$-dimensional spacetime, which will be considered flat metrics, e.g. $G_{\mu\nu}=\eta_{\mu\nu}={\rm diag}(-1,1,\cdot\cdot\cdot,1)$. For the $U(1)$ gauge potential we consider the gauge $A_{\alpha}=-\frac{1}{2}F_{\alpha\beta}X^{\beta}$, where the field strength $F_{\alpha\beta}$ is constant. In addition, for the next purposes, we consider the constant Kalb-Ramond field $B_{\mu\nu}$.

2.1 The boundary state equations

2.1.1 The bosonic part

Variation of the action with respect to $X^{\mu}(\sigma,\tau)$ gives the equations of motion and the boundary state equations. Imposing a boost on the resulted boundary state equations, along the non-orbifoldy perpendicular directions to the brane, we obtain the boundary state equations of a moving fractional brane [22],

[TABLE]

where $\gamma=\sqrt{1-v^{i}v^{i}}$, the set $\{x^{\bar{\alpha}}|{\bar{\alpha}}=1,\cdot\cdot\cdot,p\}$ shows the spatial directions of the brane, and $\{x^{i}|i=p+1,\ldots,5\}$ indicates the directions perpendicular to its worldvolume, except the orbifoldy directions. The parameters $\{y^{i}|i=p+1,\ldots,5\}\bigcup\{y^{a}|a=6,7,8,9\}$ represent the initial location of the brane. Since the branes are stuck at the orbifold fixed-points we have $\{y^{a}=0|a=6,7,8,9\}$. For simplifying the equations we assumed that the mixed elements $B^{\alpha}_{~{}~{}i}$ and $B^{\alpha}_{~{}~{}a}$ to be zero. The total field strength is defined by $\mathcal{F}_{\alpha\beta}=F_{\alpha\beta}-B_{\alpha\beta}$.

Note that the dimension of the orbifold crucially puts a bound on the dimension of the brane. In order to have a velocity perpendicular to the brane, the maximum spatial dimension of the brane is 4, i.e. $p\leq 4$.

2.1.2 The fermionic part

To find the boundary state equations for the worldsheet fermions, we accurately perform the worldsheet supersymmetry on the bosonic boundary equations (2.2). That is, we use following replacements

[TABLE]

where $\eta=\pm 1$ is introduced for the GSO projection of the boundary state. To obtain the replacement for $X^{\mu}$ in terms of the fermionic coordinates, Eq. (2.3) induces

[TABLE]

It is well known that the spectrum of the string in the presence of the orbifoldy directions is a linear combination of two sectors: the states which are invariant under the orbifold projection, which are called the untwisted sector, and the states of the twisted sector which are living at the orbifold fixed-points [19]. Therefore, in general, the boundary state of a fractional brane in an orbifold theory can be written as

[TABLE]

where ‘U’ and ‘T’ indicate the untwisted and twisted sectors, $\mathcal{N}^{\rm U}$ and $\mathcal{N}^{\rm T}$ are normalization constants, $\epsilon_{1}$ and $\epsilon_{2}$ are two R-R charges ($\epsilon=+1,-1$ stands for brane and anti-brane, respectively). Eq. (2.5) has been written in this form to save the generality, which admits one to consider various configurations of the branes (brane-antibrane) in the both twisted and untwisted sectors.

We are not going to perform calculations for the complete boundary state (2.5). This is due to the fact that the untwisted states $|B\rangle^{\rm U}_{\rm NS,R}$ represent the usual boundary states with the standard GSO-projection which have been vastly studied for various setups [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Thus, we shall not repeat the calculations for this sector, instead we shall just concentrate on the twisted states $|B\rangle^{\rm T}_{\rm NS,R}$ in the next sections.

For the boundary state equations in the twisted part, we have to use the fermionic mode expansions of the twisted sector, which are given by

[TABLE]

where in the twisted R-R sector there are

[TABLE]

where $\{\lambda|\lambda=0,1,2,3,4,5\}$ stand for the non-orbifoldy directions and $\{a|a=6,7,8,9\}$ represent the orbifoldy directions. In the twisted NS-NS sector we have

[TABLE]

By replacing Eqs. (2.3) and (2.4) into Eq. (2.2), and applying the above mode expansions for the worldsheet fermions, we acquire

[TABLE]

The boundary state equations of the usual (non-orbifoldy) directions, in the both twisted R-R and NS-NS sectors (for the nonzero-mode numbers of the R-R sector) possess the following feature

[TABLE]

where $t\in\mathbb{Z}-\{0\}\;(\mathbb{Z}+1/2)$ for the R-R (NS-NS) sector. The mode-dependent matrix $S^{\lambda}_{(t)\lambda^{\prime}}$ is defined by $S_{(t)}=M_{(t)}^{-1}N_{(t)}$ with

[TABLE]

The boundary state equations of the orbifoldy directions, in the both twisted NS-NS and R-R sectors (for the nonzero-mode numbers of the NS-NS sector) find the following form

[TABLE]

where $r\in\mathbb{Z}+1/2\;(\mathbb{Z}-\{0\})$ for the R-R (NS-NS) sector.

There are zero-mode parts in the both twisted NS-NS and R-R sectors. For the twisted NS-NS sector the zero-mode boundary state equation finds the feature

[TABLE]

Similarly, the zero-mode boundary state equation for the twisted R-R sector has the form

[TABLE]

2.2 The solutions of the boundary state equations

2.2.1 The bosonic part

We have calculated the bosonic part of the boundary state in the Ref. [22], hence we adduse the final feature of it. Its oscillating part is

[TABLE]

where the matrix $S_{(m)}$ is the same as one that was exhibited in the fermionic part. The zero-mode portion of the bosonic boundary state is

[TABLE]

Note that since there is no motion along the orbifoldy directions, because the brane is stuck at the orbifold fixed-points $(x^{a}=0)$, Eq. (2.14) does not include bosonic zero modes for these directions.

2.2.2 The NS-NS sector

Using the coherent state method, the oscillating part of the boundary state in the twisted NS-NS sector possesses the form

[TABLE]

where the zero-mode part of this sector comes from the orbifoldy directions, and according to Eq. (2.11) it is independent of the background fields and velocity. Therefore, it has the structure [19], [20],

[TABLE]

where $\bar{\Gamma}={\Gamma}^{6}{\Gamma}^{7}{\Gamma}^{8}{\Gamma}^{9}$, $\bar{C}$ is the charge conjugate matrix of $SO(4)$, and $|L\rangle$ and $|\tilde{M}\rangle$ are the spinors of $SO(4)$.

2.2.3 The R-R sector

Applying the same procedure of the bosonic part and the NS-NS sector, we acquire the following boundary state in the twisted R-R sector

[TABLE]

in which ${|B_{\psi},\eta\rangle^{(0){\rm T}}_{\rm R}}$ is the solution of Eq. (2.12). Its trusty form is given by

[TABLE]

where $\tilde{\Gamma}={\Gamma}^{0}{\Gamma}^{1}\ldots{\Gamma}^{5}$, $|A\rangle$ and $|\tilde{B}\rangle$ are spinors of $SO(1,5)$, $\tilde{C}$ is the charge conjugate matrix of $SO(1,5)$, and the matrix $G$ is specified by the equation

[TABLE]

The matrix $G$ has the conventional solution

[TABLE]

where the symbol $;~{};$ means that we have to expand the exponential with the convention that all $\Gamma$-matrices anticommute. This implies that there are a finite number of terms in the expansion. Also $\bar{\Phi}=(\Phi-\Phi^{T})/2$, and the matrices $\Phi$ and $S$ have the relation

[TABLE]

in which

[TABLE]

Thus, the matrix elements of $\Phi$ are given by

[TABLE]

Therefore, we find the following expressions for the matrix elements of $\bar{\Phi}$,

[TABLE]

As an example, the explicit form of the matrix $\bar{\Phi}$ for a fractional D2-brane that moves along the $x^{3}$-direction is

[TABLE]

where $\mathcal{F}^{\bar{\alpha}}_{~{}~{}0}=\mathcal{F}^{0}_{~{}~{}{\bar{\alpha}}}$ and $\mathcal{F}^{\bar{\alpha}}_{~{}~{}{\bar{\beta}}}=-\mathcal{F}^{\bar{\beta}}_{~{}~{}{\bar{\alpha}}}$.

3 Interaction of the D$p$-branes

The interaction between two D-branes can be described in two different but equivalent ways, i.e. the open and closed string approaches. In the open string method the interaction amplitude is elaborated by the one-loop diagram of an open string which is stretched between two D-branes. In the closed string approach, the interaction of two D-branes is precisely recasted by the tree-level exchange of a closed string that is emitted from one brane and is absorbed by another brane. We apply the second approach.

Now we are ready to calculate the interaction amplitude between two parallel fractional D$p$-branes with the velocities $v^{i}_{1}$ and $v^{i}_{2}$ in the twisted sector. For achieving this, we apply the interaction amplitude $\mathcal{A}_{\rm NS,R}^{\rm T}={}_{\rm NS,R}^{\rm T}\langle B_{1}|D|B_{2}\rangle_{\rm NS,R}^{\rm T}$, where $D$ is the closed string propagator

[TABLE]

The closed string Hamiltonian $H_{\rm closed}$ is sum of the Hamiltonians of the matter part and ghost part. The matter part of the Hamiltonian, in each of the NS-NS and R-R sectors, is sum of the Hamiltonians of the bosonic part and the fermionic part, i.e.,

[TABLE]

$|B\rangle^{\rm T}_{\rm NS,R}$ is the total projected boundary state associated with the fractional D$p$-brane in the twisted sector

[TABLE]

where $|B_{\rm gh}\rangle$ and $|B_{\rm sgh},\eta\rangle_{\rm NS,R}$ are the ghost and superghost boundary states, respectively. The orbifold projection, the brane velocity and the background fields do not change the (super)ghost boundary state.

It is noticeable that a physical boundary state should be invariant under the GSO-projection. Thus, it is a linear combination of two states corresponding to $\eta=\pm 1$. In the twisted sector, invariance of the boundary state under the action of the orbifold projection, in addition to the GSO-projection, changes the conventional form of the combination of the states, i.e. in the both twisted NS-NS and R-R sectors there is a plus sign in the combination of the states [23], [24],

[TABLE]

3.1 The interaction amplitude in the NS-NS sector

Using the total projected boundary states, the interaction amplitude in the twisted NS-NS sector finds the following feature

[TABLE]

where $q=e^{-2t}$, and $V$ is the common volume of the branes. The damping exponential factor depends on the distance of the branes. The last line is induced by the oscillators of the matter part, conformal ghosts and super conformal ghosts. Since the ghost portion of the boundary state is not influenced by the velocities, background fields and orbifoldization, the contribution of the ghosts has been introduced by manipulation of the amplitude.

As it was mentioned, the orbifoldy directions induce the zero-mode boundary state in the twisted NS-NS sector. This brings in the factor ‘4’ in the last line of the amplitude for $\delta_{\eta_{1}\eta_{2},1}$, while its contribution to $\delta_{\eta_{1}\eta_{2},-1}$ is zero. Vanishing of this part of the interaction has significant effects on the amplitude of the distant branes, which will be discussed later.

3.2 The interaction amplitude in the R-R sector

Calculation of the interaction amplitude in the twisted R-R sector drastically needs more attention. This is due to the difficulties of the computations concerning to the divergent contribution of the zero-mode boundary state in this sector. Thus, we should use a suitable regularization to evaluate it. Similar to the Ref. [25], we apply the regulator $x^{2(F_{0}+G_{0})}$ as follows

[TABLE]

where $(-1)^{F_{0}}=i\tilde{\Gamma}$ and $G_{0}=-\gamma_{0}\beta_{0}$. The calculations of the superghost part eventuate to the same result of the untwisted sector

[TABLE]

For the fermionic part we obtain

[TABLE]

Substituting the explicit forms of $G_{1}$ and $G_{2}$ from Eq. (2.20), after a rigorous calculation, we receive the following regular factor for the interaction amplitude

[TABLE]

where $\Psi_{\lambda_{1}\lambda_{2}}=\frac{1}{2}(\bar{\Phi}_{(2)\lambda_{1}\lambda_{2}}-\bar{\Phi}_{(1)\lambda_{1}\lambda_{2}})$ and $\epsilon^{\lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}}$ is the Levi-Civita symbol. The factor $(-7)^{p+1}$ originates from the anticommutation relations between the $\Gamma$-matrices which appeared in the product $\Gamma^{0}\Gamma^{1}\ldots\Gamma^{p}$ and in the expansion of the exponential functions $G_{1}$ and $G_{2}$. In the absence of the background fields the matrices $G_{1}$ and $G_{2}$ become trivial, and hence the conventional results of the literature will be accurately reproduced. For acquiring Eq. (3.6) we have used the operators [21],

[TABLE]

which imply that the regulator of the fermionic zero modes satisfies the relations

[TABLE]

Finally, the interaction amplitude in the twisted R-R sector is

[TABLE]

The total amplitude in the twisted sector is

[TABLE]

where $\mathcal{A}^{\rm T}(\eta_{1},\eta_{2})=\mathcal{A}_{\rm NS-NS}^{\rm T}(\eta_{1},\eta_{2})+\mathcal{A}_{\rm R-R}^{\rm T}(\eta_{1},\eta_{2})$. Comparing the resulted amplitudes (3.3) and (3.8) with the conventional forms of the interaction amplitudes reveals that the presence of the orbifoldy directions induces significant effects on the interaction.

Note that the complete form of the theory contains both the utwisted and twisted sectors. That is, the total interaction amplitude is sum of the amplitude of the twisted sector, i.e. Eq. (3.9), and the amplitude in the untwisted sector. In fact, since for the various setups the untwisted sector has been deeply studied we only investigated the twisted sector of our setup.

3.3 Interaction of the distant branes

For obtaining the long-range forces of the theory, the behavior of the interaction amplitude for distant branes should be evaluated. Hence, we discuss the evolution of the interaction amplitude under the limit $t\rightarrow\infty$ for the oscillating part of the amplitudes in the both twisted sectors. The limit is not applied on the first and second lines of Eqs. (3.3) and (3.8), because these factors originate from the bosonic zero modes, and the oscillators do not have any contribution to them. We shall see that the orbifold projection, accompanied by the background fields, imposes some new effects on the long-range forces.

The interaction amplitude for the twisted NS-NS sector, after passing a long enough time, is

[TABLE]

This amplitude contains three terms. We observe that the sum of the second and third terms, which is induced by the massless states (i.e. graviton, dilaton and Kalb-Ramond), ghosts and superghosts, vanishes. This is due to the fact that the orbifold projection deformed the zero-point energy of the closed string Hamiltonian, therefore, this peculiar result was appeared. However, even under this condition, the first term, i.e. “1” in the last line, remains. Note that in the untwisted sector the first term, i.e. “1” disappears, and the second and third terms, without the damping factor $e^{-2t}$, withstand.

The twisted R-R sector possesses the following contribution to the amplitude of the distant branes

[TABLE]

where the last line originates from the fermionic zero-mode part.

The total amplitude in the twisted sector, due to the exchange of the massless states, is

[TABLE]

where ${\tilde{\mathcal{A}}}^{\rm T}(\eta_{1},\eta_{2})={\tilde{\mathcal{A}}}_{\rm NS-NS}^{\rm T}(\eta_{1},\eta_{2})+{\tilde{\mathcal{A}}}_{\rm R-R}^{\rm T}(\eta_{1},\eta_{2})$.

Again note that the complete form of the theory includes both the twisted and untwisted sectors. Therefore, the total amplitude of the long-range force is sum of the amplitude of the twisted sector, i.e. Eq. (3.12), and that one of the untwisted sector.

4 Conclusions

As a first result, for a fractional moving D$p$-brane with internal background fields which lives in an orbifoldized spacetime $\mathbb{R}^{1,5}\times\mathbb{C}^{2}/\mathbb{Z}_{2}$, we constructed a corresponding boundary state. The background fields are Kalb-Ramond tensor and a $U(1)$ gauge potential. The orbifoldy directions divide the boundary state into two different parts: the twisted and untwisted sectors [19]. Unlike the untwisted sector, the boundary state of the twisted NS-NS sector includes a zero-mode portion. Our setup imposed a modified zero-mode part for the twisted R-R sector.

As another result, the interaction amplitude of two parallel moving fractional D$p$-branes, or a system of one fractional brane and one fractional anti-brane with the background fields, for the both twisted NS-NS and R-R sectors, was acquired. The orbifoldy directions introduced some significant effects on the interaction. For example, in the amplitude of the twisted NS-NS sector, the contribution of the GSO-projected sectors with different spin structures vanishes. However, various parameters in the amplitude, i.e. the parameters of the background fields, the velocities, the branes dimensions, accompanied by the orbifoldy directions, elegantly give a generalized feature to this interaction. Thus, the strength of the interaction accurately is adjustable.

Obtaining the interaction amplitude for distant branes is the other conclusion. We observed that in the both twisted sectors contribution of the massless states, unlike the conventional case, prominently vanishes. In addition, in the twisted NS-NS sector we received an extra term. These are effects of the orbifoldization.

Added Notes:

By setting all background fields and the velocities to zero our results reduce to the simple setups of the D-branes, e.g. see the Refs. [21, 25, 26].

According to the paper [27] and the following references therein [5, 6, 7, 65], the orbifold conformal field theory is obtained with a non-trivial background $B$-field. This implies that the Cardy’s condition for our setup, i.e., that in the presence of the background fields and the velocities which have resemblance with the electric components of the anti-symmetric $B$-field, is satisfied.