1 Introduction

Some significant and important steps have been made to introduce the D-branes as essential objects in the string theory [1,2,3,4]. One of the main problem of the D-branes is their stability. The fate of an unstable D-brane can be investigated via the dynamics of the open string tachyon, i.e., the tachyon condensation process [5,6,7,8,9,10,11,12]. An unstable D-brane usually decays to another lower dimensional unstable D-brane as an intermediate state [13,14,15,16]. This intercurrent state eventually collapses to the closed string vacuum or decays to a lower dimensional stable configuration. There are various trusty approaches for studying these concepts, e.g.,: string field theory [17,18,19,20], the first quantized string theory [5,6,7,8,9,10,11,12, 21, 22], the renormalization group flow method [23,24,25], and the boundary string field theory [14, 17,18,19,20, 26, 27].

On the other hand, we have the boundary state formalism for describing the D-branes [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67]. A boundary state prominently encodes all properties of its corresponding D-brane, and is a source for emitting all closed string states. Thus, this adequate state can be used to study the time evolution of the brane during the tachyon condensation process [46,47,48,49,50,51]. Note that the rolling tachyon has a boundary state description which is valid during the finite time. Therefore, after elapsing this time the energy of the system will be completely dissipated into the bulk [47, 48].

Among the various D-branes the dynamical-dressed branes motivated us to examine their behaviors under the tachyon condensation experience. This stimulation is due to the background fields and dynamics of such branes. Thus, in this paper we shall consider a single Dp-brane with a transverse rotation, which has been dressed with the Kalb–Ramond field, a U(1) gauge potential and an open string tachyon field. The boundary state, associated with this Dp-brane, enables us to study the response of it in conflicting with the tachyon condensation phenomenon. We shall observe that the rotation of the brane and its field-dressing do not induce a resistance to protect it against the collapse. That is, the dimensional reduction of the brane will drastically occur.

This paper is organized as follows. In Sect. 2, the boundary state, corresponding to a rotating Dp-brane with the foregoing background fields, will be constructed. In Sect. 3, evolution of this Dp-brane under the condensation of the open string tachyon will be investigated. Section 4 is devoted to the conclusions.

2 The boundary state corresponding to our dynamical-dressed Dp-brane

We begin with the following closed string action

$$\begin{aligned} S= & {} - \frac{1}{4\pi \alpha '} \int _\Sigma {\mathrm{d}}^{2}\sigma \left( \sqrt{-h}h^{ab}g_{\mu \nu }\partial _a X^{\mu }\partial _b X^{\nu } \right. \nonumber \\&\left. + \epsilon ^{ab} B_{\mu \nu } \partial _a X^{\mu }\partial _b X^{\nu } \right) \nonumber \\&+ \frac{1}{2\pi \alpha '} \int _{\partial \Sigma } {\mathrm{d}}\sigma \left( A_\alpha \partial _{\sigma }X^{\alpha } + T^2(X^\alpha ) \right) , \end{aligned}$$
(2.1)

where \(\Sigma \) is the worldsheet of a closed string which is emitted by a static Dp-brane, and \(\partial \Sigma \) is the boundary of it. The coordinates \(\{x^\alpha |\alpha =0, 1, \ldots ,p \}\) specify the directions which are along the worldvolume of this brane, and the set \(\{x^i| i= p+1, \ldots ,d-1\}\) will be used for the perpendicular directions to it. The field \(A_\alpha \) is a U(1) gauge potential which lives in the worldvolume of the brane, and \(T^2(X^\alpha )\) is the open string tachyon field.

In fact, the states of our tachyon field and the gauge potential belong to the open string spectrum. Thus, their corresponding fields obviously appear in the surface terms of the string action. This implies that these fields do not have any coupling with the worldsheet curvature. That is, the action (2.1) has the Weyl symmetry. Therefore, for the metric of the worldsheet we can choose the flat gauge \(h_{ab} =\eta _{ab}=\mathrm{diag} (-1,1)\). Beside, the spacetime metric is chosen as \(g_{\mu \nu }=\eta _{\mu \nu }= \mathrm{diag} (-1,1,\ldots ,1)\), and for the gauge field we select the gauge \(A_{\alpha }=-\frac{1}{2}F_{\alpha \beta }X^{\beta }\) with the constant field strength \(F_{\alpha \beta }\). In addition, we apply the tachyon profile \(T^2=-2\pi i \alpha ' U_{\alpha \beta }X^{\alpha }X^{\beta }\) where the tachyon matrix \(U_{\alpha \beta }\) is constant and symmetric.

The Kalb–Ramond field \(B_{\mu \nu }\) also will be considered constant. This implies that the second term of the action is total derivative, i.e., it is a surface term. Since the equation of motion originates from the bulk part of the action, the constant B-field obviously does not contribute to the equation of motion.

Variation of the action defines the following equations for the boundary state, associated with the static Dp-brane with the background fields

$$\begin{aligned}&\left( {\partial }_{\tau }X^{\alpha } +{\mathcal {F}}^\alpha _{\;\;\;\beta }\partial _\sigma X^\beta -B^\alpha _{\;\;\;i} \partial _\sigma X^i \right. \nonumber \\&\quad \left. -4\pi i \alpha ' U^\alpha _{\;\;\;\beta } X^{\beta }\right) _{\tau =0} |B \rangle _{\mathrm{(static)}}=0 , \nonumber \\&(X^i -y^i)_{\tau =0}|B \rangle _{\mathrm{(static)}}=0 , \end{aligned}$$
(2.2)

where the transverse vector \(y^i\) indicates the brane location, and \({\mathcal {F}}_{\alpha \beta } = F_{\alpha \beta } -B_{\alpha \beta }\) exhibits the total field strength. Making use of the second equation, the third term of the first equation vanishes.

Before rotating the brane we should remind the following facts. We know that string theory was started with the string action in the flat Minkowski background \({{\mathbb {R}}}^{1,25}\), and the flat hyperplanes (the D-branes) were discovered. However, in the string spectrum there exists a massless closed string state which prominently is corresponding to the fluctuations of the geometry [1]. Therefore, since the D-branes are sources of energy–momentum tensors, the flatness of the background spacetime in the presence of them is a reliable approximation which has been widely applied to the various subjects of the string theory and branes. In our setup we considered the flat spacetime, accompanied by the constant Kalb–Ramond field and a zero dilaton field. These imply that our static Dp-brane represents a trivial solution of the supergravity equations. That is, in the first approximation this brane does not induce a curvature to the spacetime. Besides, it does not live in a non-flat background.

In fact, the rotation of the perpendicular coordinates to a brane worldvolume, which describes a spinning brane, deforms the metric of the background spacetime. However, imposing some other motions to the brane does not change the metric. For example, in the flat spacetime see the D-branes with transverse velocities [2, 35, 62,63,64,65], the rotated D-branes [63], the D-branes with tangential rotations [66, 67], and so on. In fact, in these examples the first approximation of the background metric has been manifestly applied. Our D-brane will rotate in a transverse plane to itself, thus, for a small angular velocity we have a quasi-static D-brane. Hence, similar to the foregoing examples, at least for such small rotations we can apply the first approximation of the metric. This elaborates that the equations of the boundary state, corresponding to the rotating brane with the transverse rotation, and also the equation of motion of an emitted closed string from the brane will be reliably written in the initial flat spacetime.

Here we compare effects of the different dynamics of the branes on the background metric. In a spinning brane for each rotating perpendicular plane to the brane there exists one rotational parameter. These adequate variables induce a deformation to the spacetime metric. For a boosted brane the velocity components are introduced, which do not deform the metric. In our system the only rotational parameter is the angular velocity. By employing the foregoing first approximation of the metric, accompanied by the quasi-static rotation of the brane, one can see that the flat metric under the transformations (2.3) remains invariant.

Now we impose a transverse rotation to the brane. Let \(x^{i_0}\) be the horizontal axis and \(x^{\alpha _0}\) (with \({\alpha _0} \ne 0\)) be the vertical one. At the time \(t=0\) the direction \(x^{\alpha _0}\) is along the brane, and the direction \(x^{i_0}\) is perpendicular to it. The brane is rotating, e.g. counterclockwise, with the constant angular velocity “\(\omega \)”. The axis of the rotation is one of the normal directions to the plane \(x^{i_0}x^{\alpha _0}\). The coordinate system \(\{x'^{\mu }\}\) is stuck to the brane such that at each moment the planes \(x^{i_0}x^{\alpha _0}\) and \(x'^{i_0}x'^{\alpha _0}\) have common origin and they are coincident. Thus, we receive the following coordinate transformations

$$\begin{aligned} x'^{i_0}= & {} x^{i_0} \cos (\omega t) +x^{\alpha _0} \sin (\omega t), \nonumber \\ x'^{\alpha _0}= & {} -x^{i_0} \sin (\omega t) +x^{\alpha _0}\cos (\omega t), \nonumber \\ x'^{{\bar{\alpha }}}= & {} x^{{\bar{\alpha }}}, \nonumber \\ x'^{{\bar{i}}}= & {} x^{{\bar{i}}}, \end{aligned}$$
(2.3)

where the new indices \({{\bar{\alpha }}}\) and \({{\bar{i}}}\) belong to the sets

$$\begin{aligned} {{\bar{\alpha }}}\in & {} \{0,1,\ldots , p\}-\{\alpha _0\}, \\ {{\bar{i}}}\in & {} \{p+1,\ldots , d-1\}-\{i_0\}. \end{aligned}$$

The equation of motion, extracted from the action (2.1) for the flat spacetime and worldsheet, clearly takes the form \((\partial ^2_\tau - \partial ^2_\sigma ) X^\mu (\sigma , \tau )=0\). By applying the transformations (2.3) and the quasi-static approximation for the brane dynamics we receive \((\partial ^2_\tau - \partial ^2_\sigma ) X'^\mu (\sigma , \tau )=0\). Besides, the quasi-static rotating Dp-brane possesses the following boundary state equations

$$\begin{aligned}&\bigg [ {\partial }_{\tau }X^{{{\bar{\alpha }}}} +{\mathcal {F}}^{{{\bar{\alpha }}}}_{\;\;\;{{\bar{\beta }}}} \partial _\sigma X^{{{\bar{\beta }}}}\nonumber \\&\quad +{\mathcal {F}}^{{{\bar{\alpha }}}}_{\;\;\;\alpha _0}\cos (\omega t) \left( - \sin (\omega t)\partial _\sigma X^{i_0} +\cos (\omega t)\partial _\sigma X^{\alpha _0}\right) \nonumber \\&-4\pi i \alpha ' U^{{{\bar{\alpha }}}}_{\;\;\;{{\bar{\beta }}}} X^{{{\bar{\beta }}}} -4\pi i \alpha ' U^{{{\bar{\alpha }}}}_{\;\;\;\alpha _0} \cos (\omega t) \left( - X^{i_0}\sin (\omega t)\right. \nonumber \\&\quad \left. +X^{\alpha _0}\cos (\omega t)\right) \bigg ]_{\tau =0}|B(t)\rangle =0 , \nonumber \\&\bigg [ \cos (\omega t){\partial }_{\tau }X^{\alpha _0} -\sin (\omega t) \partial _\tau X^{i_0} +{\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}}\cos (\omega t) \partial _\sigma X^{{{\bar{\beta }}}} \nonumber \\&\quad -4\pi i \alpha ' U^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos (\omega t) X^{{{\bar{\beta }}}} \nonumber \\&-4\pi i \alpha ' U^{\alpha _0}_{\;\;\;\alpha _0}\cos ^2 (\omega t) \left( - X^{i_0}\sin (\omega t) \right. \nonumber \\&\quad \left. + X^{\alpha _0}\cos (\omega t) \right) \bigg ]_{\tau =0}|B(t)\rangle =0 , \nonumber \\&[ X^{i_0}\cos (\omega t) + X^{\alpha _0}\sin (\omega t) ]_{\tau =0}|B(t)\rangle =0 , \nonumber \\&(X^{{\bar{i}}} - y^{{\bar{i}}})_{\tau =0}|B(t)\rangle =0 . \end{aligned}$$
(2.4)

Note that the time variable “t” is the center-of-mass part of the emitted closed string coordinate \(X^0 (\sigma , \tau )\), i.e. \(t=x^0\). Therefore, the argument of the sine and cosine is “\(\omega t\)” but not “\(\omega X^0 \)”. If we use \(\omega ={\mathrm{d}} \theta / {\mathrm{d}} X^0 (\sigma , \tau )\), instead of \(\omega ={\mathrm{d}} \theta / {\mathrm{d}} x^0\), we obtain a non-constant angular velocity \(\omega (\sigma , \tau )\). In this case each point of the emitted closed string from the rotating brane possesses its own angular velocity, which is not consistent with the assumption of the constant angular velocity of the rotating brane.

Equation (2.4) can be rewritten in terms of the zero modes and oscillators of the closed string coordinates

$$\begin{aligned}&\bigg [ p^{{\bar{\alpha }}} -2\pi i U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}}x^{{\bar{\beta }}} -2\pi i U^{{\bar{\alpha }}}_{\;\;\;{\alpha _0}} \cos (\omega t) \left( -x^{i_0}\sin (\omega t) \right. \nonumber \\&\quad \left. + x^{\alpha _0}\cos (\omega t)\right) \bigg ] {|B(t)\rangle }^{\left( 0\right) }\ =0, \nonumber \\&\bigg [ p^{\alpha _0}\cos (\omega t) - p^{i_0}\sin (\omega t) -2\pi i U^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos (\omega t)x^{{\bar{\beta }}} \nonumber \\&-2\pi i U^{\alpha _0}_{\;\;\;{\alpha _0}} \cos ^2(\omega t) \left( -x^{i_0} \sin (\omega t)\right. \nonumber \\&\quad \left. + x^{\alpha _0} \cos (\omega t)\right) \bigg ] {|B(t)\rangle }^{\left( 0\right) }\ =0, \nonumber \\&[ x^{i_0} \cos (\omega t)+ x^{\alpha _0} \sin (\omega t)] {|B(t)\rangle }^{\left( 0\right) }\ =0, \nonumber \\&(x^{{\bar{i}}}-y^{{\bar{i}}}){|B(t)\rangle }^{\left( 0\right) }\ =0, \end{aligned}$$
(2.5)

for the zero-mode part, and we have

$$\begin{aligned}&\bigg [ \alpha ^{{\bar{\alpha }}}_m +{{\tilde{\alpha }}}^{{\bar{\alpha }}}_{-m} -\left( {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}}-\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}}\right) \left( \alpha ^{{\bar{\beta }}}_m -{{\tilde{\alpha }}}^{{\bar{\beta }}}_{-m} \right) \nonumber \\&\quad +\left( {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;\alpha _0} -\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;\alpha _0}\right) \cos (\omega t) [\left( \alpha ^{i_0}_m -{{\tilde{\alpha }}}^{i_0}_{-m}\right) \sin (\omega t) \nonumber \\&\quad - \left( \alpha ^{\alpha _0}_m -{{\tilde{\alpha }}}^{\alpha _0}_{-m} \right) \cos (\omega t)]\bigg ]{|B(t)\rangle }^{(\mathrm{osc})}=0, \nonumber \\&\bigg [\left( \alpha ^{\alpha _0}_m + {{\tilde{\alpha }}}^{\alpha _0}_{-m} \right) \cos (\omega t) - \left( \alpha ^{i_0}_m + {{\tilde{\alpha }}}^{i_0}_{-m}\right) \sin (\omega t) \nonumber \\&\quad -\left( {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} -\frac{2\pi \alpha '}{m}U^{\alpha _0}_{\;\;\;{{\bar{\beta }}}}\right) \cos (\omega t) \left( \alpha ^{{\bar{\beta }}}_m -{{\tilde{\alpha }}}^{{\bar{\beta }}}_{-m}\right) \nonumber \\&\quad -\frac{2\pi \alpha '}{m}U^{\alpha _0}_{\;\;\; \alpha _0} \cos ^2 (\omega t)\left[ \left( \alpha ^{i_0}_m -{{\tilde{\alpha }}}^{i_0}_{-m}\right) \sin (\omega t) \right. \nonumber \\&\quad \left. -\left( \alpha ^{\alpha _0}_m -{{\tilde{\alpha }}}^{\alpha _0}_{-m}\right) \cos (\omega t) \right] \bigg ]{|B(t)\rangle }^{(\mathrm{osc})}=0, \nonumber \\&\left[ \left( \alpha ^{i_0}_m - {{\tilde{\alpha }}}^{i_0}_{-m}\right) \cos (\omega t) +\left( \alpha ^{\alpha _0}_m - {{\tilde{\alpha }}}^{\alpha _0}_{-m} \right) \sin (\omega t) \right] {|B(t)\rangle }^{(\mathrm{osc})}\nonumber \\&\quad =0, \nonumber \\&(\alpha ^{{\bar{i}}}_m-{{\tilde{\alpha }}}^{{\bar{i}}}_{-m}) {|B(t)\rangle }^{(\mathrm{osc})}\ =0, \end{aligned}$$
(2.6)

for the oscillating part, with \(m \in {\mathbb {Z}}-\{0\}\). Note that we decomposed the boundary state to the zero-mode portion and the oscillating part, i.e., \(|B(t)\rangle ={|B(t)\rangle }^{\left( 0\right) } \otimes {|B(t)\rangle }^{\left( \mathrm{osc}\right) }\).

In fact, solving Eqs. (2.5) and (2.6) is very difficult. For simplification we impose the restriction \(U_{\alpha \alpha _0}=0\), or equivalently \(U_{{{\bar{\alpha }}} \alpha _0}=U_{\alpha _0 \alpha _0}=0\). Therefore, the solution of the zero-mode part of the boundary state is given by

$$\begin{aligned} {\mathrm{|}B(t)\rangle }^{\left( 0\right) }= & {} \frac{1}{\sqrt{\det {{\tilde{U}}}}}\int ^{\infty }_{\mathrm{-}\infty } \exp \bigg [-\frac{1}{4\pi }\sum _{{\bar{\alpha }}} {\left( {{\tilde{U}}}^{-1}\right) }_{{{\bar{\alpha }}} {{\bar{\alpha }}}} {\left( p^{{\bar{\alpha }}}\right) }^2 \nonumber \\&- \frac{1}{2\pi }\sum _{{{\bar{\alpha }}} \ne {{\bar{\beta }}}} \left( {{\tilde{U}}}^{-1} \right) _{{{\bar{\alpha }}}{{\bar{\beta }}}} p^{{\bar{\alpha }}}p^{{\bar{\beta }}}\bigg ] \left( \prod _{{\bar{\alpha }}}\mathrm{|} p^{{\bar{\alpha }}}\rangle {\mathrm{d}}p^{{\bar{\alpha }}}\right) \nonumber \\&\times \delta \left[ x^{i_0} \cos (\omega t) +x^{\alpha _0} \sin (\omega t) \right] \nonumber \\&\times \prod _{{\bar{i}}}\delta (x^{{\bar{i}}}-y^{{\bar{i}}}) \mathrm{|}p^{{\bar{i}}} = 0 \rangle \otimes | p^{i_0} =0 \rangle \otimes | p^{\alpha _0}\nonumber \\&= 0 \rangle , \end{aligned}$$
(2.7)

where, according to the condition \(U_{\alpha \alpha _0}=0\), the \(p \times p\) symmetric matrix \({{\tilde{U}}}\) is defined by eliminating the \({\alpha _0}\)th column and \({\alpha _0}\)th row of the \((p+1) \times (p+1)\) tachyon matrix U. From the disk partition function we deduce the prefactor \(1/\sqrt{\det {{\tilde{U}}}}\) [52, 53]. The exponential part of \({\mathrm{|}B(t)\rangle }^{\left( 0\right) }\), which is absent for the conventional boundary states, clearly is an effect of the tachyon field. We observe that the zero-mode part of the boundary state is independent of the total field strength and the parameter \(\alpha '\). This is due to the fact that we considered a non-compact brane. The compact case extremely contains these factors [36,37,38].

For solving Eq. (2.6) we define the new oscillators

$$\begin{aligned} A_m= & {} \alpha ^{i_0}_m\cos (\omega t) + \alpha ^{\alpha _0}_m\sin (\omega t), \nonumber \\ {{\tilde{A}}}_m= & {} {{\tilde{\alpha }}}^{i_0}_m\cos (\omega t) + {{\tilde{\alpha }}}^{\alpha _0}_m\sin (\omega t), \nonumber \\ B_m= & {} \alpha ^{\alpha _0}_m\cos (\omega t) - \alpha ^{i_0}_m\sin (\omega t), \nonumber \\ {{\tilde{B}}}_m= & {} {{\tilde{\alpha }}}^{\alpha _0}_m\cos (\omega t) - {{\tilde{\alpha }}}^{i_0}_m\sin (\omega t). \end{aligned}$$
(2.8)

These oscillators possess the following nonzero commutators

$$\begin{aligned}{}[ A_m , A_n] = [{{\tilde{A}}}_m , {{\tilde{A}}}_n] =[ B_m , B_n] = [{{\tilde{B}}}_m , {{\tilde{B}}}_n] = m \delta _{m+n,0},\nonumber \\ \end{aligned}$$
(2.9)

and all other commutators among them vanish.

By applying the coherent state method, and after some heavy calculations, the oscillating part of the boundary state finds the feature

$$\begin{aligned}&| B(t) \rangle ^{\mathrm{(osc)}} \nonumber \\&\quad = \frac{T_p}{g_s}\prod ^{\infty }_{n=1} \left[ \det \left( \mathbf{1} -{{{\mathcal {F}}}'(t)} +\frac{2\pi \alpha '}{n}{{\bar{U}}} \right) \right] ^{-1} \nonumber \\&\qquad \times \exp \bigg \{-\sum ^{\infty }_{m=1} {\frac{1}{m}\bigg [{\alpha }^{{\bar{\alpha }}}_{-m} Q_{(m){{\bar{\alpha }}}{{\bar{\beta }}}}{\tilde{\alpha }}^{{\bar{\beta }}}_{-m}} -\alpha ^{{\bar{i}}}_{-m} {{\tilde{\alpha }}}^{{\bar{i}}}_{-m} \nonumber \\&\qquad -A_{-m}{{\tilde{A}}}_{-m} \nonumber \\&\qquad + \left( 1 + 2 \left( M^{-1}_m\right) ^{{\bar{\alpha }}} _{\;\;\;{{\bar{\beta }}}} {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}} {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\alpha }}}} \cos ^2 (\omega t)\right) B_{-m}{{\tilde{B}}}_{-m} \nonumber \\&\qquad + 2 {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}} \cos (\omega t) \left( \left( M^{-1}_m\right) _{{{\bar{\alpha }}} {{\bar{\beta }}}} \alpha ^{{\bar{\alpha }}}_{-m}{{\tilde{B}}}_{-m} \right. \nonumber \\&\qquad \left. -\left( M^{-1}_m\right) _{{{\bar{\beta }}} {{\bar{\alpha }}}} B_{-m}{{\tilde{\alpha }}}^{{\bar{\alpha }}}_{-m} \right) \bigg ]\bigg \} {|0\rangle }, \end{aligned}$$
(2.10)

where by putting the \({\alpha _0}\)th column and \({\alpha _0}\)th row of the tachyon matrix U to zero the \((p+1) \times (p+1)\) matrix \({{\bar{U}}}\) is obtained. By multiplying the \({\alpha _0}\)th column and \({\alpha _0}\)th row of the field strength matrix \({{\mathcal {F}}}\) with \(\cos (\omega t)\) we receive the matrix \({{{\mathcal {F}}}'}(t)\). The other matrices are defined by

$$\begin{aligned}&Q_{(m){{\bar{\alpha }}}{{\bar{\beta }}}} = \left( M^{-1}_{m} N_{m}\right) _{{{\bar{\alpha }}}{{\bar{\beta }}}}\;, \nonumber \\&M^{{\bar{\alpha }}}_{(m){{\bar{\beta }}}} = \delta ^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} - {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} +\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} -{\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;\alpha _0} {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos ^2 (\omega t), \nonumber \\&N^{{\bar{\alpha }}}_{(m){{\bar{\beta }}}} = \delta ^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} + {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} -\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} +{\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;\alpha _0} {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos ^2 (\omega t).\nonumber \\ \end{aligned}$$
(2.11)

As we see these matrices depend on the mode number “m” which is induced by the tachyon matrix. The normalization factor, i.e., the infinite product in the first line of Eq. (2.10), is originated by the disk partition function. The state (2.10) specifies that \(A_{m}\) and \({{\tilde{A}}}_{m}\) are Dirichlet oscillators. Similarly, in the case \({\mathcal {F}}_{\alpha _0 {{\bar{\alpha }}}}=0\) the variables \(B_{m}\) and \({{\tilde{B}}}_{m}\) became Neumann oscillators.

In Eq. (2.6) one can express the right-moving annihilation oscillators in terms of the left-moving creation oscillators. This obviously eventuates to the boundary state (2.10). However, in these equations it is possible to express the left-moving annihilation oscillators in terms of the right-moving creation oscillators. In this case, applying the coherent state method leads to another form for the boundary state of the oscillating part. Equality of these boundary states elaborates the following conditions

$$\begin{aligned} M_m M'^T_m= & {} N_m N'^T_m , \nonumber \\ 2 \left( M^{-1}_m\right) _{{{\bar{\alpha }}} {{\bar{\beta }}}} {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}}= & {} -{\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \left( Q'_m +\mathbf{1} \right) ^{{\bar{\beta }}}_{\;\;\;{{\bar{\alpha }}}}, \nonumber \\ 2 \left( N'^{-1}_m\right) _{{{\bar{\alpha }}} {{\bar{\beta }}}} {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}}= & {} -{\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \left( Q_m +\mathbf{1} \right) ^{{\bar{\beta }}}_{\;\;\;{{\bar{\alpha }}}}, \nonumber \\ \left( M^{-1}_m\right) _{{{\bar{\alpha }}} {{\bar{\beta }}}} {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}}= & {} \left( N'^{-1}_m\right) _{{{\bar{\alpha }}} {{\bar{\beta }}}} {\mathcal {F}}^{{\bar{\beta }}}_{\;\;\;{\alpha _0}}, \end{aligned}$$
(2.12)

where the new matrices have the definitions

$$\begin{aligned} {Q'}_{(m){{\bar{\alpha }}}{{\bar{\beta }}}}= & {} \left( N'^{-1}_{m} M'_{m}\right) _{{{\bar{\alpha }}}{{\bar{\beta }}}}, \nonumber \\ M'^{{\bar{\alpha }}}_{(m){{\bar{\beta }}}}= & {} \delta ^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} - {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} -\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} +{\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;\alpha _0} {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos ^2 (\omega t), \nonumber \\ N'^{{\bar{\alpha }}}_{(m){{\bar{\beta }}}}= & {} \delta ^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} + {\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} +\frac{2\pi \alpha '}{m}U^{{\bar{\alpha }}}_{\;\;\;{{\bar{\beta }}}} -{\mathcal {F}}^{{\bar{\alpha }}}_{\;\;\;\alpha _0} {\mathcal {F}}^{\alpha _0}_{\;\;\;{{\bar{\beta }}}} \cos ^2 (\omega t).\nonumber \\ \end{aligned}$$
(2.13)

In fact, by substituting the explicit forms of the matrices from Eqs. (2.11) and (2.13) into Eq. (2.12) we see that the first, the second and the third equations of (2.12) are trivial identities. That is, they do not impose any relation among the parameters of our setup. For the odd values of the brane dimension “p” the fourth equation is an identity, and for the even values of “p” it only gives rise to the condition \(\det \left( {{\mathcal {F}}}_{{{\bar{\alpha }}}{{\bar{\beta }}}}\right) =0\).

Note that the total boundary state includes a part which comes from the conformal ghosts. This portion manifestly is independent of the background fields and the brane rotation. Thus, it obviously is null under the tachyon condensation process. Hence, we shall not consider it.

3 Effect of the tachyon condensation on our Dp-brane

According to the Sen’s papers [5,6,7,8,9,10,11,12], in the presence of the open string tachyonic field our knowledge about the vacua of the string theories, the fate of the D-branes, their instability, and so on, was improved. During the process of the tachyon condensation the brane drastically collapses, and finally we receive a collection of the closed strings. These imply that decadence of unstable objects is very important phenomenon. For example, these objects specify an approach to achieve the background independent formulation of string theory.

Since the boundary state is a source for emitting all quantum states of closed string, and accurately describes all properties of the corresponding brane, and comprises a specific normalization factor, it is a favorable and convenient tool for finding the treatment and behavior of a single D-brane under the experience of the tachyon condensation. Hence, in this section we shall use this adequate formalism.

For imposing the condensation on the tachyon field, some of the matrix elements of the tachyon matrix should be infinite. For this purpose let the system tend to the infrared fixed point via the limit \(U_{pp} \rightarrow \infty \). This defines the tachyon condensation along the \(x^p\)-direction, where we assume \(x^{\alpha _0} \ne x^p\). Now we should take the limit of the total boundary state to acquire the behavior of our dynamical-dressed Dp-brane under the tachyon condensation process.

At first we obtain the behavior of the zero-mode part of the boundary state, i.e., Eq. (2.7). Under the limit \(U_{pp} \rightarrow \infty \) its prefactor transforms to

$$\begin{aligned} \frac{1}{\sqrt{ U_{pp} \det {\tilde{{\tilde{U}}}}}}, \end{aligned}$$

where the \((p-1) \times (p-1)\) symmetric matrix \({\tilde{{\tilde{U}}}}\) is defined by eliminating the last and \(\alpha _0\)th columns and also the last and \(\alpha _0\)th rows of the tachyon matrix U. At the IR fixed point limit we have

$$\begin{aligned} {\mathop {\lim }_{U_{pp}\rightarrow \infty } {{\tilde{U}}}^{-1}} =\left( \begin{array}{cc} {\tilde{{\tilde{U}}}}^{-1} &{} \mathbf{0}_{(p-1)\times 1} \\ \mathbf{0}_{1 \times (p-1)} &{} 0 \end{array} \right) . \end{aligned}$$
(3.1)

Adding all these together we receive the limit

$$\begin{aligned}&| {{\mathcal {B}}}(t) \rangle ^{(0)}\nonumber \\&\quad = \frac{2\pi }{\sqrt{ U_{pp}\det {\tilde{{\tilde{U}}}}}} \int ^\infty _{-\infty }\exp \bigg [-\frac{1}{4\pi }\sum _a {\left( {\tilde{{\tilde{U}}}}^{-1}\right) }_{aa} {\left( p^a\right) }^2 \nonumber \\&\qquad - \frac{1}{2\pi }\sum _{a \ne b} \left( {\tilde{{\tilde{U}}}}^{-1} \right) _{ab}p^ap^b\bigg ]\left( \prod _a\mathrm{|} p^a\rangle {\mathrm{d}}p^a\right) \nonumber \\&\qquad \times \delta \left[ x^{i_0} \cos (\omega t) +x^{\alpha _0} \sin (\omega t) \right] \nonumber \\&\qquad \times \prod _{{\bar{i}}} \delta (x^{{\bar{i}}}-y^{{\bar{i}}}) \mathrm{|}p^{{\bar{i}}} = 0 \rangle \otimes \delta (x^p)|p^p=0 \rangle \nonumber \\&\qquad \otimes | p^{i_0} =0 \rangle \otimes | p^{\alpha _0}= 0 \rangle , \end{aligned}$$
(3.2)

where \(a,b \in \{0,1,\ldots ,p \}-\{\alpha _0 ,p\}\). Since the exponential factor has lost the momentum component \(p^p\) we obtained the state \(\sqrt{2\pi }|x^p=0 \rangle = 2\pi \delta (x^p)|p^p=0 \rangle \). However, up to the factor \(2\pi /\sqrt{ U_{pp}}\) the Eq. (3.2) accurately represents the zero-mode boundary state of a D\((p-1)\)-brane which is rotating inside the \(x^{i_0}x^{\alpha _0}\)-plane with the angular velocity \(\omega \).

Now look at the oscillating part of the boundary state. By the method of the zeta function regularization we have \(\prod ^\infty _{n=1}(n\lambda ) \rightarrow \sqrt{2\pi /\lambda }\), and accordingly the prefactor of Eq. (2.10) possesses the limit

$$\begin{aligned} \frac{T_{p-1}\sqrt{U_{pp}}}{g_s}\prod ^{\infty }_{n=1} \left[ \det \left( \mathbf{1} -{\tilde{{{\mathcal {F}}}'}} +\frac{2\pi \alpha '}{n}{\tilde{{\bar{U}}}} \right) _{p\times p}\right] ^{-1}, \end{aligned}$$
(3.3)

where the profitable relation \(2\pi \sqrt{\alpha '}\;T_p =T_{p-1}\) was used. For the forms of the \(p\times p\) matrices \({\tilde{{{\mathcal {F}}}'}}\) and \({\tilde{{\bar{U}}}}\), eliminate the last rows and last columns of the \((p+1)\times (p+1)\) matrices \({{{\mathcal {F}}}'}\) and \({{\bar{U}}}\), respectively.

For calculating the limit of \(M^{-1}_m\) we use

$$\begin{aligned} {\mathop {\lim }_{U_{pp}\rightarrow \infty } \det M_m}=\frac{2\pi \alpha '}{m}U_{pp}\det M^{(p-1)}_m , \end{aligned}$$
(3.4)

where by eliminating the last row and last column of \(M_m\) the \((p-1)\times (p-1)\) matrix \(M^{(p-1)}_m\) is acquired. Since the last row and last column of \(M^{-1}_m\) contain the factor \(1/U_{pp}\), we receive

$$\begin{aligned} {\mathop {\lim }_{U_{pp}\rightarrow \infty } M^{-1}_m}=\left( \begin{array}{cc} \left( M^{(p-1)}_m \right) ^{-1} &{} \mathbf{0}_{(p-1)\times 1} \\ \mathbf{0}_{1 \times (p-1)} &{} 0 \end{array} \right) . \end{aligned}$$
(3.5)

Beside, the limit of \(Q_m=M^{-1}_m N_m\) finds the feature

$$\begin{aligned} {\mathop {\lim }_{U_{pp}\rightarrow \infty } Q_m}=\left( \begin{array}{cc} \left( M^{(p-1)}_m \right) ^{-1} N^{(p-1)}_m &{} \;\;{} \mathbf{0}_{(p-1)\times 1} \\ \mathbf{0}_{1 \times (p-1)} &{} \;\;-1 \end{array} \right) . \end{aligned}$$
(3.6)

Note that the limit of the matrix \(Q_m\) is not the product of the limits of \(M^{-1}_m\) and \(N_m\). After performing the product \(M^{-1}_m N_m\) we have taken the limit of \(Q_m\). The structures of the matrices \(M^{(p-1)}_m\), \(N^{(p-1)}_m\) and \(Q^{(p-1)}_m =\left( M^{(p-1)}_m \right) ^{-1} N^{(p-1)}_m\) are similar to the matrices of Eq. (2.11) in which \({{\bar{\alpha }}}\) and \({{\bar{\beta }}}\) must be replaced with the indices \(a,b \in \{0,1, \ldots p \}-\{\alpha _0 , p\}\).

Adding all these together, the effect of the tachyon condensation on the oscillating part of the boundary state is given by

$$\begin{aligned}&| {{\mathcal {B}}}(t) \rangle ^{\mathrm{(osc)}} \nonumber \\&\quad = \frac{T_{p-1}\sqrt{U_{pp}}}{g_s}\prod ^{\infty }_{n=1} \left[ \det \left( \mathbf{1} -{\tilde{{{\mathcal {F}}}'}} +\frac{2\pi \alpha '}{n}{\tilde{{\bar{U}}}} \right) _{p \times p}\right] ^{-1} \nonumber \\&\qquad \times \exp \bigg \{-\sum ^{\infty }_{m=1} {\frac{1}{m}\bigg [{\alpha }^a_{-m} \left( Q^{(p-1)}_{(m)}\right) _{ab}{\tilde{\alpha }}^b_{-m}} \nonumber \\&\qquad -\alpha ^p_{-m} {{\tilde{\alpha }}}^p_{-m} -\alpha ^{{\bar{i}}}_{-m} {{\tilde{\alpha }}}^{{\bar{i}}}_{-m} - A_{-m}{{\tilde{A}}}_{-m} \nonumber \\&\qquad + 2 {\mathcal {F}}^b_{\;\;\;{\alpha _0}} \cos (\omega t) \left( \left[ \left( M^{(p-1)}_m \right) ^{-1}\right] _{ab} \alpha ^a_{-m}{{\tilde{B}}}_{-m} \right. \nonumber \\&\qquad \left. -\left[ \left( M^{(p-1)}_m \right) ^{-1}\right] _{ba} B_{-m}{{\tilde{\alpha }}}^a_{-m} \right) \nonumber \\&\qquad + \left( 1 + 2\left[ \left( M^{(p-1)}_m \right) ^{-1}\right] ^a_{\;\;\;b} {\mathcal {F}}^b_{\;\;\;{\alpha _0}} {\mathcal {F}}^{\alpha _0}_{\;\;\;\;a} \cos ^2 (\omega t)\right) B_{-m}{{\tilde{B}}}_{-m} \bigg ]\bigg \} {|0\rangle }.\nonumber \\ \end{aligned}$$
(3.7)

As expected, the sign of the operator \(\alpha ^p_{-m} {{\tilde{\alpha }}}^p_{-m}\) has changed, i.e., the previous Neumann direction \(x^p\) has been transformed to a Dirichlet direction. By comparing this equation with Eq. (2.10) we observe that, up to the factor \(\sqrt{U_{pp}}\;\), Eq. (3.7) manifestly describes the oscillating part of the boundary state which is corresponding to the D\((p-1)\)-brane.

For the total boundary state at the IR fixed point the extra factors \(1/\sqrt{U_{pp}}\) and \(\sqrt{U_{pp}}\) of Eqs. (3.2) and (3.7) exactly cancel each other. Similar cancellation between the zero-mode portion and the oscillating part also occurs in the D-\({\bar{D}}\) systems [20, 26, 27, 68]. However, according to the product of the states (3.2) and (3.7) we have proved that, during the tachyon condensation process, the transverse rotation and background fields cannot protect the brane against the collapse. That is, the unstable Dp-brane lost its \(x^p\)-direction and conveniently reduced to a D\((p-1)\)-brane. The resulted brane is rotating inside the \(x^{i_0}x^{\alpha _0}\)-plane with the same frequency “\(\omega \)”. The delta functions of Eq. (3.2) prominently clarify that this D\((p-1)\)-brane has been localized at the position \(x^p=0 \;,\; x^{{\bar{i}}}=y^{{\bar{i}}}\), and its configuration at the times \(t \in \{\frac{2\pi n}{\omega }|n\in {\mathbb {Z}}\}\) is along the directions \(\{x^1, x^2, \ldots , x^{p-1}\}\).

4 Conclusions

In the framework of the bosonic string theory we constructed a profitable boundary state, associated with a dynamical Dp-brane with a transverse rotation, in the presence of the anti-symmetric tensor field \(B_{\mu \nu }\), a U(1) internal gauge potential and a tachyonic field of the open string spectrum. Though we imposed a uniform rotation to the brane but the time dependence of the corresponding boundary state is very intricate. Besides, the rotational dynamics induced the deformed versions of the tachyon matrix and total field strength to the boundary state.

We investigated the effects of the tachyon condensation on the foregoing Dp-brane through its boundary state. We demonstrated that at the infrared fixed point the background fields, accompanied by the transverse rotation of the brane, cannot prevent the unstable brane against the collapse. Therefore, the tachyon condensation was eventually terminated by the dimensional reduction of the brane. The resulted D\((p-1)\)-brane possessed the same angular frequency as the previous one. Presence of the remaining tachyon field implies that the subsequent brane also is an unstable object, and at the IR fixed point will be collapsed.