The Compactified D-Brane Cylinder Amplitude and T Duality

Abstract

In this paper, we address how to implement T-duality to the closed string tree cylinder amplitude between a Dp brane and a Dp′ brane with $p-p^{\prime }=2n$. To achieve this, we compute the closed string tree cylinder amplitude, for the first time, between these two D-branes with common longitudinal and transverse circle compactifications. We then show explicitly how to perform a T-duality for this amplitude along either a longitudinal or a transverse compactified direction to both branes. At the decompactification limit, we show that either the compactified cylinder amplitude or the T dual compactified cylinder gives the known non-compactified one as expected.

1. Introduction

In general, we know that a Dp brane will become a D(p − 1) brane if a T-duality is performed along its longitudinal direction and a D(p + 1) brane if the T-duality is conducted along a direction transverse to it. However, it is not clear, to the best of our knowledge, how to implement this T-duality in a closed string tree cylinder amplitude between a Dp and a Dp′, placed parallel at a separation, with $p-p^{\prime }=2n$ and $2n\le p\le 8$ (without loss of generality, we assume from now on $p>p^{\prime }$) in Type II superstring theories (It is known that when $p-p^{\prime }=2n$ and $n=0,2$, this amplitude vanishes due to the underlying systems preserving 1/2 and 1/4 spacetime supersymmetry, respectively. For $n=4$, we have only one system consisting of a D8 and a D0, and the corresponding amplitude also vanishes. However, it is known that this system has various subtle issues [1,2,3,4], and for this reason we will not consider them in this paper for simplicity. Therefore, in this paper, we limit ourselves to $n=0,1,2,3$ in Type II theories).

We try to address this in Type II theories in the present paper. As will become clear, we can perform a T-duality along a direction transverse to both branes for such systems if we periodically place them along this direction in order to generate an effective circle compactification with a radius $R\to 0$ and then perform the T-duality to create a system of D(p + 1) and D(p′ + 1) without compactification. We will show that this T-duality works only in the $R\to 0$ limit. However, this same process will not work if a T-duality is performed along a longitudinal direction common to both branes. We will explain the reasoning behind this.

In order to properly perform a T-duality on the amplitude in general, we need to compute the corresponding amplitude with at least one compactified direction which is either longitudinal or transverse to both branes, and the T-duality is performed along this direction.

To achieve this, we compute in this paper the general closed string tree cylinder amplitude, for the first time, between a Dp and a Dp′, as specified above, with $k\le p^{\prime }$ spatial longitudinal circle compactifications of radii $r_1,r_2,\cdots r_k$ and $l\le 9-p$ transverse circle compactifications of radii $R_1,R_2,\cdots R_l$. This amplitude will serve as a starting point for us to perform a T-duality along either a longitudinal or a transverse direction.

This paper is arranged as follows: In the following section, we will give a brief review of the Dp brane boundary state representation as given, for example, in the nice review in Reference [5]. We will use this representation to demonstrate in this section how each sector of the matter part in the D-brane boundary state transforms under a T-duality along either a longitudinal or transverse direction to the brane. This shows that the only part which we need to take into consideration when one such direction is compactified is the bosonic matter zero mode to the boundary state. The corresponding compactified bosonic matter zero mode boundary state is already given in [6] when the compactified circles have the same radius. We generalize this boundary state to one with $k\le p-2n$ spatial longitudinal compactifications with the respective radii $r_1$ and $\cdots r_k$ and $l\le 9-p$ transverse compactifications with the respective radii $R_1$ and $\cdots R_l$, as specified earlier. With such compactified boundary states, we compute the corresponding closed string cylinder amplitude between a Dp and a Dp′ with $p-p^{\prime }=2n$. In Section 4, we address how to implement a T-duality along either the longitudinal or the transverse direction to the cylinder amplitude computed in the previous section and check the consistency, along with other discussions. We conclude this paper in Section 5.

2. The D-Brane Boundary State and Its T-Duality

D-branes are sort of non-perturbative with respect to the fundamental string or F-string. We specify our discussion here on Type-II superstring theories in 10 dimensions. For a weak string coupling, D-branes can be viewed as rigid at least up to the string scale $l_s=\sqrt{2{\alpha }^{\prime }}$. In this case, a Dp-brane can be represented as a coherent state of closed string excitations, called the Dp-brane boundary state. For such a description, there are two sectors, namely the NS-NS and R-R sectors, due to the closed string fermionic boundary conditions obtained via a special conformal transformation from the open string ones defining the D-brane and being consistent with T-duality (see detail on this in the nice review in Reference [5]). In each sector, we have two implementations for the boundary state of a Dp-brane, resulting in two boundary states $|B,\eta \rangle$ with $\eta =\pm$. However, only the combinations

$$\begin{array}{ccccc}\hfill & & {|B\rangle }_{\mathrm{NSNS}}={\displaystyle \frac{1}{2}}\left[{|B,+\rangle }_{\mathrm{NSNS}}-{|B,-\rangle }_{\mathrm{NSNS}}\right],\hfill & & \\ & & {|B\rangle }_{\mathrm{RR}}={\displaystyle \frac{1}{2}}\left[{|B,+\rangle }_{\mathrm{RR}}+{|B,-\rangle }_{\mathrm{RR}}\right],\hfill \end{array}$$

(1)

are selected by the Gliozzi–Scherk–Olive (GSO) projection in the NS-NS and R-R sectors, respectively (As is well-known, in the RNS formulation of, say, Type-II superstrings, the GSO projection is a must to guarantee the surviving states selected by this projection to be spacetime supersymmetric. In the present context, a Dp-brane is a vacuum-like configuration, preserving 1/2 of the $N=2$ spacetime supersymmetries, and only its GSO projected boundary state in each sector can guarantee this 1/2 SUSY property. This can be checked explicitly by computing the total stringy interaction between two such Dp-branes placed parallel at a separation. This interaction has two contributions: one from the GSO-projected boundary state in the NS-NS sector, which is attractive due to the brane tension or mass, and the other from the GSO projected boundary state in the R-R sector, which is repulsive due to the brane RR charge. It turns out that the former contribution cancels precisely the latter, and the total interaction is null, a direct check and also a consequence of 1/2 BPS property. This is the case for $n=0$. For $n=2$, there is no contribution from the R-R sector (different RR charges do not interact), but, for this case, the contribution from the NS-NS sector also vanishes. This is also a consequence of this system preserving 1/4 of the underlying spacetime supersymmetry. Once again, one has to use the GSO projected state to guarantee this. In other words, the states projected out by GSO are not physical and should not be used in computing amplitudes. All these were explicitly checked before; for example, see Equation (9.290) in the nice review from Reference [5]). The boundary state can be expressed as the product of a matter part and a ghost part, due to its invariance under BRST, as

$$|B,\eta \rangle =|B_{\mathrm{mat}},\eta \rangle |B_{\mathrm{g}},\eta \rangle ,$$

(2)

where

$$|B_{\mathrm{mat}},\eta \rangle =|B_X\rangle |B_{\psi },\eta \rangle ,|B_{\mathrm{g}},\eta \rangle =|B_{\mathrm{gh}}\rangle |B_{\mathrm{sgh}},\eta \rangle ,$$

(3)

with here $\eta =\pm$. The bosonic boundary state and the fermionic one in each sector can be determined by their respective overlap conditions (Classically, as mentioned earlier, we obtain the closed string boundary conditions from the respective open string ones defining the D-brane. Quantum mechanically, such obtained closed string boundary conditions become operators, and, in general, they cannot be set to vanish. However, they can act on the respective boundary state to vanish, and this defines the respective closed string boundary state. Such defining boundary state equations are called the overlap conditions) (Again, see [5] for details). They are

$$|B_X\rangle =\exp (-\sum _{n=1}^{\infty }{\displaystyle \frac{1}{n}}{\alpha }_{-n}·S·{\tilde{\alpha }}_{-n}){|B_X\rangle }_0,$$

(4)

and

$$|B_{\psi }{,\eta \rangle }_{\mathrm{NS}}=-\mathrm{i}\exp (i\eta \sum _{m=1/2}^{\infty }{\psi }_{-m}·S·{\tilde{\psi }}_{-m})|0\rangle ,$$

(5)

for the NS-NS sector and

$$|B_{\psi }{,\eta \rangle }_{\mathrm{R}}=-\exp (i\eta \sum _{m=1}^{\infty }{\psi }_{-m}·S·{\tilde{\psi }}_{-m}){|B,\eta \rangle }_{0\mathrm{R}},$$

(6)

for the R-R sector. The ghost boundary states are the standard ones as given in [3], and they are irrelevant to the T-dualities we are interested in this paper. For this reason, we will not list them here. The matrix S in the absence of D-brane worldvolume fluxes is simply

$$S_p=\left({\eta }_{\alpha \beta },-{\delta }_{ij}\right)$$

(7)

and the zero-mode boundary states are, respectively,

$$|B_X{\rangle }_0={\displaystyle \frac{c_p}{2}}{\delta }^{(9-p)}(q^i-y^i)\prod _{\mu =0}^9|k^{\mu }=0\rangle ,$$

(8)

for the bosonic sector with the overall normalization $c_p=\sqrt{\pi }{\left(2\pi \sqrt{{\alpha }^{\prime }}\right)}^{3-p}$, and

$$|B_{\psi }{,\eta \rangle }_{0\mathrm{R}}={\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{B}\rangle ,$$

(9)

for the R-R sector.

In the above, the Greek indices $\alpha ,\beta ,\cdots$ label the worldvolume directions $0,1,\cdots ,p$ along which the Dp brane extends, while the Latin indices $i,j,\cdots$ label the directions transverse to the brane, i.e., $p+1,\cdots ,9$. We have also denoted by $y^i$ the positions of the D-brane along the transverse directions and by C the charge conjugation matrix. $|A\rangle |\tilde{B}\rangle$ stands for the spinor vacuum of the R-R sector. Note that the $\eta$ in the above denotes either sign ± or the worldvolume Minkowski flat metric and should be clear from the content.

The GSO projected boundary states given in (1) are the ones we are going to use in computing the closed string cylinder amplitude between a Dp brane and a Dp′ brane with $p-p^{\prime }=2n$, as specified earlier.

Our purpose is to determine how to perform a T-duality on the amplitude. As mentioned earlier, we need to consider the corresponding compactified D-brane boundary states. In the following, we will show that the only thing indeed relevant is the bosonic matter zero-mode boundary state. To this end, we pose here to recall briefly the simplest T-duality acting on a closed string in Type-II superstring theories.

To achieve this, we compactify a spatial direction of the 10-dimensional spacetime, say X, to obtain a circle of a radius R in one theory (say IIA) with its closed string wrapping this circle W times and having a quantized momentum K, and to obtain the other circle of $\tilde{R}$ in the other theory (say IIB) with its closed string having the winding $\tilde{W}$ and momentum $\tilde{K}$. If the radius R is related to the other radius $\tilde{R}$ via $R\tilde{R}={\alpha }^{\prime }$, and the winding and momentum modes are exchanged via $K\leftrightarrow \tilde{W}$ and $W\leftrightarrow \tilde{K}$, then we have the equivalent relation $\mathrm{IIA}\leftrightarrow \mathrm{IIB}$ via this T-duality. The above simple picture of T-duality can be generalized to all modes for both bosonic and fermionic modes in the following way:

$$\left\{\begin{array}{c}{X}_R(\tau -\sigma )\to -X_R(\tau -\sigma );{\psi }_R(\tau -\sigma )\to -{\psi }_R(\tau -\sigma ),\hfill \\ ({\alpha }_n\to -{\alpha }_n,x\leftrightarrow \widehat{x};{\psi }_t\to -{\psi }_t)\hfill \end{array}\right.$$

(10)

for the right mover, while for the left mover

$$\left\{\begin{array}{c}{X}_L(\tau +\sigma )\to X_L(\tau +\sigma );{\psi }_L(\tau +\sigma )\to {\psi }_L(\tau +\sigma ),\hfill \\ ({\tilde{\alpha }}_n\to {\tilde{\alpha }}_n,x\leftrightarrow \widehat{x};{\tilde{\psi }}_t\to {\tilde{\psi }}_t).\hfill \end{array}\right.$$

(11)

where the index n is an integer, while the index t is an integer in the R sector and a half-integer in the NS sector. Here, x and $\widehat{x}$ are the respective initial positions of X in the original and T dual theories. Note that

$${\alpha }_0=\sqrt{{\displaystyle \frac{{\alpha }^{\prime }}{2}}}\left({\displaystyle \frac{K}{R}}-{\displaystyle \frac{WR}{{\alpha }^{\prime }}}\right),{\tilde{\alpha }}_0=\sqrt{{\displaystyle \frac{{\alpha }^{\prime }}{2}}}\left({\displaystyle \frac{K}{R}}+{\displaystyle \frac{WR}{{\alpha }^{\prime }}}\right).$$

(12)

Under T-duality, the generators of Virasoro algebra of the matter part for either the left or the right mover remain invariant. We can see this easily with the following zero modes as examples:

$$\begin{array}{ccccc}\hfill L_0^{\mathrm{matter}}& =& {\displaystyle \frac{{\alpha }^{\prime }}{4}}{\left({\displaystyle \frac{K}{R}}-{\displaystyle \frac{WR}{{\alpha }^{\prime }}}\right)}^2+{\displaystyle \frac{{\alpha }^{\prime }}{4}}{\widehat{p}}^2+\sum _{n=1}^{\infty }{\alpha }_{-n}·{\alpha }_n+\sum _{t>0}^{\infty }{\psi }_{-t}·{\psi }_t,\hfill & & \\ \hfill {\tilde{L}}_0^{\mathrm{matter}}& =& {\displaystyle \frac{{\alpha }^{\prime }}{4}}{\left({\displaystyle \frac{K}{R}}+{\displaystyle \frac{WR}{{\alpha }^{\prime }}}\right)}^2+{\displaystyle \frac{{\alpha }^{\prime }}{4}}{\widehat{p}}^2+\sum _{n=1}^{\infty }{\tilde{\alpha }}_{-n}·{\tilde{\alpha }}_n+\sum _{t>0}^{\infty }{\tilde{\psi }}_{-t}·{\tilde{\psi }}_t,\hfill \end{array}$$

(13)

where $\widehat{p}$ labels the momentum along the non-compactified directions. Note that in the R sector, we have fermionic zero modes ${\psi }_0$ and ${\tilde{\psi }}_0$, and they transform under the T-duality along the compactified direction as

$${\psi }_0\to -{\psi }_0,{\tilde{\psi }}_0\to {\tilde{\psi }}_0.$$

(14)

Note also that ${\psi }_0\sim \Gamma$, with $\Gamma$ being the Dirac matrix along the compactified direction (We adopt here that the 10-dimensional Dirac matrices are all real with the spatial ones symmetric and the temporal one antisymmetric. Their explicit representations are given, for example, in the Appendix in [6]. The charge conjugate C is also given explicitly, which remains invariant under T-duality). This implies that $\Gamma \to -\Gamma$ under T-duality. This further implies that the chiral operator ${\Gamma }_{11}\to -{\Gamma }_{11}$. Therefore, under the T-duality, the chirality of one of the two chiral spinors in Type II flips, which gives

$$\mathrm{IIA}\leftrightarrow \mathrm{IIB}.$$

(15)

With this brief introduction to T-duality, we now examine how the boundary state given above transforms under it. To this end, let us first examine how each of the exponential factors, due to either the bosonic or the fermionic oscillators, transforms under the T-duality in the respective boundary states of matter. Given (10) and (11), it is easy to see that the change to each of these factors under the T-duality amounts to a change in the matrix $S_p$ (7), as follows:

$$S_p=\left({\eta }_{\alpha \beta },-{\delta }_{ij}\right)\to S_{p-1}=\left({\eta }_{{\alpha }^{\prime }{\beta }^{\prime }},-{\delta }_{i^{\prime }{j}^{\prime }}\right),$$

(16)

where, without loss of generality (for simplicity and for convenience), we have chosen the compactification $X=X^p$, i.e., along the brane direction, with ${\alpha }^{\prime },{\beta }^{\prime }=0,1,\cdots p-1$ and $i^{\prime },j^{\prime }=p,p+1,\cdots 9$ or

$$S_p=\left({\eta }_{\alpha \beta },-{\delta }_{ij}\right)\to S_{p+1}=\left({\eta }_{{\alpha }^{\prime }{\beta }^{\prime }},-{\delta }_{i^{\prime }{j}^{\prime }}\right),$$

(17)

where we have chosen the compactification $X=X^{p+1}$, i.e., along the direction transverse to the brane, with now ${\alpha }^{\prime },{\beta }^{\prime }=0,1,\cdots ,p+1$ and $i^{\prime },j^{\prime }=p+2,p+3,\cdots 9$. Either of these is consistent with T-duality. We will now examine how the fermionic zero-mode boundary state transforms under T-duality. If we choose $X=X^p$ for simplicity, we have, under a T-duality along this direction, ${\Gamma }^p\to -{\Gamma }^p$ and ${\Gamma }_{11}\to -{\Gamma }_{11}$. We then have from (9)

$$\begin{array}{ccc}\hfill |B_{\psi }^p{,\eta \rangle }_{0\mathrm{R}}& =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{B}\rangle \hfill \\ & \to & -{\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1-\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{B}\rangle \hfill \\ & =& -{\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^{p-1}{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AD}|A\rangle {\left({\Gamma }^p\right)}_{DB}|\tilde{B}\rangle ,\hfill \end{array}$$

(18)

where we have used $C\to C$ under the T-duality as mentioned earlier. Note also that we can choose ${\left({\Gamma }_{11}\right)}_{AB}|B\rangle =|A\rangle$ and ${\left({\Gamma }_{11}\right)}_{AB}|\tilde{B}\rangle =\pm |\tilde{A}\rangle$, where the sign depends on which theory (IIA or IIB) we start with. For the $‘-$’ sign, we start with IIA, while for the $‘+$’ sign, we start with IIB. Denoting $|\tilde{\mathcal{B}}\rangle =-{\left({\Gamma }^p\right)}_{BD}|\tilde{D}\rangle$, we have ${\left({\Gamma }_{11}\right)}_{AB}|\tilde{\mathcal{B}}\rangle =\mp |\tilde{\mathcal{A}}\rangle$. In other words, after the T-duality, we transform $|\tilde{B}\rangle$ to $|\tilde{\mathcal{B}}\rangle$ with an opposite chirality (Given that ${\Gamma }_{11}$ is a symmetric matrix, we can also apply this to $|A\rangle$.), i.e., $\mathrm{IIA}\leftrightarrow \mathrm{IIB}$, as expected. Given all this, we have from (18) under the T-duality

$$\begin{array}{ccc}\hfill |B_{\psi }^p{,\eta \rangle }_{0\mathrm{R}}& \to & -{\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^{p-1}{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AD}|A\rangle {\left({\Gamma }^p\right)}_{DB}|\tilde{B}\rangle \hfill \\ & =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^{p-1}{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{\mathcal{B}}\rangle \hfill \\ & =& |B_{\psi }^{p-1}{,\eta \rangle }_{0\mathrm{R}}.\hfill \end{array}$$

(19)

Therefore, under the T-duality along a longitudinal direction of the Dp brane, we have $|B_{\psi }^p{,\eta \rangle }_{0\mathrm{R}}\to {|B_{\psi }^{p-1},\eta \rangle }_{0\mathrm{R}}$ ($\mathrm{IIA}\leftrightarrow \mathrm{IIB}$), and the R sector zero-mode boundary state transforms as expected. If we choose instead to perform a T-duality along a direction, say $X=X^{p+1}$, transverse to the p-brane, we still have ${\Gamma }_{11}\to -{\Gamma }_{11}$ due to ${\Gamma }^{p+1}\to -{\Gamma }^{p+1}$. Since ${\left({\Gamma }^{p+1}\right)}^2={\mathbf{I}}_{32\times 32}$ with ${\mathbf{I}}_{N\times N}$ the $N\times N$ unit matrix, we then have, from (9),

$$\begin{array}{ccc}\hfill |B_{\psi }^p{,\eta \rangle }_{0\mathrm{R}}& =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{B}\rangle \hfill \\ & \to & {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1-\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{B}\rangle \hfill \\ & =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\displaystyle \frac{1-\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}{\left({\Gamma }^{p+1}\right)}^2\right)}_{AB}|A\rangle |\tilde{B}\rangle \hfill \\ & =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\Gamma }^{p+1}{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AD}|A\rangle {\left({\Gamma }^{p+1}\right)}_{DB}|\tilde{B}\rangle \hfill \\ & =& {\left(C{\Gamma }^0{\Gamma }^1\cdots {\Gamma }^p{\Gamma }^{p+1}{\displaystyle \frac{1+\mathrm{i}\eta {\Gamma }_{11}}{1+\mathrm{i}\eta }}\right)}_{AB}|A\rangle |\tilde{\mathcal{B}}\rangle \hfill \\ & =& |B_{\psi }^{p+1}{,\eta \rangle }_{0\mathrm{R}},\hfill \end{array}$$

(20)

where $|\tilde{\mathcal{B}}\rangle \equiv {\left({\Gamma }^{p+1}\right)}_{BD}|\tilde{D}\rangle$. Therefore, under the T-duality along a direction transverse to the Dp brane, by a similar token, we have $|B_{\psi }^p{,\eta \rangle }_{0\mathrm{R}}\to {|B_{\psi }^{p+1},\eta \rangle }_{0\mathrm{R}}$ and $\mathrm{IIA}\leftrightarrow \mathrm{IIB}$, also as expected. The only thing left is to determine how the bosonic zero mode boundary state transforms under the T-duality either along a longitudinal or transverse direction.

Given that the bosonic zero modes indeed change from the non-compactified case to the compactified one, as shown earlier, we expect that the bosonic zero mode boundary state also changes, as already stressed and considered in [6]. We list the bosonic zero-mode boundary state for a Dp-brane with k longitudinal compactified directions of radii $r_{{\alpha }_i}$ ($i=1,\cdots k$) and l transverse compactified directions of radii $R_{j_m}$ ($m=1,\cdots ,l$), generalized to different radii from that given in [6], as

$$\begin{array}{ccccc}\hfill |{\Omega }_p\rangle & =& {\mathcal{N}}_p\prod _{i=1}^k\left[\sum _{{\omega }_{{\alpha }_i}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}|n_{{\alpha }_i}=0,{\omega }_{{\alpha }_i}\rangle \right]|k^0=0,{\widehat{k}}^{\Vert }=0\rangle \hfill & & \\ & & \times \prod _{m=1}^l\left[\sum _{n_{j_m}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}|n_{j_m},{\omega }_{j_m}=0\rangle \right]{\widehat{\delta }}^{(\perp )}\left({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp }\right)|{\widehat{k}}^{\perp }=0\rangle ,\hfill \end{array}$$

(21)

where we use ‘$^$’ to denote those non-compactified directions, either longitudinal or transverse. In the above, the normalization factor

$${\mathcal{N}}_p={\displaystyle \frac{c_p}{2}}\prod _{i=1}^k{\left({\displaystyle \frac{2\pi r_{{\alpha }_i}}{{\Phi }_{{\alpha }_i}}}\right)}^{1/2}\prod _{m=1}^l{\left({\displaystyle \frac{1}{2\pi R_{j_m}{\Phi }_{j_m}}}\right)}^{1/2}.$$

(22)

Following [6], we introduce, for convenience, the ‘position’ and ‘momentum’ operators, respectively, for the momentum and winding degrees of freedom, as

$$\left[q_{\omega }^{\mu },p_{\omega }^{\nu }\right]=i{\delta }^{\mu \nu },\left[q_n^{\mu },p_n^{\nu }\right]=i{\delta }^{\mu \nu },$$

(23)

where $\mu ,\nu$ are along either the longitudinal or the transverse compactified spatial directions. We then have

$$p_n^{\alpha }|{\Omega }_p\rangle =0,p_{\omega }^j|{\Omega }_p\rangle =0,$$

(24)

where $\alpha$ represents one of ${\alpha }_i$, and j represents one of $j_m$. By denoting the eigenstate $|n_{\nu },{\omega }_{\nu }\rangle$ of the respective ‘momentum’ operators, we have

$$p_n^{\mu }|n_{\mu },{\omega }_{\mu }\rangle ={\displaystyle \frac{n_{\mu }}{a_{\mu }}}|n_{\mu },{\omega }_{\mu }\rangle ,p_{\omega }^{\mu }|n_{\mu },{\omega }_{\mu }\rangle ={\displaystyle \frac{{\omega }_{\mu }{a}_{\mu }}{{\alpha }^{\prime }}}|n_{\mu },{\omega }_{\mu }\rangle ,$$

(25)

where $a_{\mu }$ is the radius of the compactified direction, which can be either one of $r_{{\alpha }_i}$ or one of $R_{j_m}$, as mentioned earlier. Given the above, it is easy to write down the state as

$$|n_{\mu },{\omega }_{\mu }\rangle \equiv e^{i{q}_n^{\mu }{n}_{\mu }/a_{\mu }}{e}^{i{q}_{\omega }^{\mu }{\omega }_{\mu }{a}_{\mu }/{\alpha }^{\prime }}|0,0\rangle ,$$

(26)

with $|0,0\rangle$ denoting the zero-momentum and zero-winding state. The normalization of this state is given as

$$\langle n_{\mu }^{\prime },{\omega }_{\mu }^{\prime }|n_{\mu },{\omega }_{\mu }\rangle ={\Phi }_{\mu }{\delta }_{n_{\mu }^{\prime },n_{\mu }}{\delta }_{{\omega }_{\mu }^{\prime },{\omega }_{\mu }},$$

(27)

where ${\Phi }_{\mu }$ is the so called ‘self-dual’ volume [6], which has the following properties:

$${\Phi }_{\mu }=2\pi a_{\mu }\mathrm{if}{a}_{\mu }\to \infty ;{\Phi }_{\mu }={\displaystyle \frac{2\pi {\alpha }^{\prime }}{a_{\mu }}}\mathrm{if}{a}_{\mu }\to 0.$$

(28)

Note also that for the decompactification limit, we have that when $r_{\alpha }\to \infty$

$$\sum _{{\omega }_{\alpha }}{e}^{i\left(q_{\omega }^{\alpha }-y^{\alpha }\right){\omega }_{\alpha }{r}_{\alpha }/{\alpha }^{\prime }}|0,0\rangle \to |0,0\rangle$$

(29)

where $\alpha$ is one of ${\alpha }_i$, and when $R_j\to \infty$

$$\begin{array}{ccccc}\hfill & & \sum _{n_j}{e}^{i\left(q_n^j-y^j\right)n_j/R_j}|0,0\rangle \to R_j\int d{k}^j{e}^{i\left(q^j-y^j\right)k^j}|0,0\rangle =\hfill & & \\ & & 2\pi R_j\int {\displaystyle \frac{d{k}^j}{2\pi }}{e}^{i\left(q^j-y^j\right)k^j}|0,0\rangle =2\pi R_j\delta \left(q^j-y^j\right)|0,0\rangle ,\hfill \end{array}$$

(30)

where j is one of $j_m$. In the above, when taking $r_{\alpha }\to \infty$, only the ${\omega }_{\alpha }=0$ term survives in the sum in (29), while in (30), we have replaced the sum with an integral of k given by $k^j=n_j/R_j$ when $R_j\to \infty$.

With the above preparation, we now come to performing a T-duality along either a longitudinal or a transverse direction to the zero-mode state $|{\Omega }_p\rangle$ to see if it is consistent with our expectations. Let us begin with a T-duality along a longitudinal direction first. Without loss of generality, let us perform this T-duality along the ${\alpha }_k$-direction. We then need to send $r_{{\alpha }_k}\to {\alpha }^{\prime }/r_{{\alpha }_k}$ and $n_{{\alpha }_k}\leftrightarrow {\omega }_{{\alpha }_k}$. With this, we have

$$\begin{array}{ccc}\hfill |{\Omega }_p\rangle & \to & {\tilde{\mathcal{N}}}_p\prod _{i=1}^{k-1}\left[\sum _{{\omega }_{{\alpha }_i}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}|n_{{\alpha }_i}=0,{\omega }_{{\alpha }_i}\rangle \right]|k^0=0,{\widehat{k}}^{\Vert }=0\rangle \hfill \\ & & \times \left(\sum _{n_{{\alpha }_k}}{e}^{-i{y}^{{\alpha }_k}{n}_{{\alpha }_k}/r_{{\alpha }_k}}|n_{{\alpha }_k},{\omega }_{{\alpha }_k}=0\rangle \right)\hfill \\ & & \times \prod _{m=1}^l\left[\sum _{n_{j_m}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}|n_{j_m},{\omega }_{j_m}=0\rangle \right]{\widehat{\delta }}^{(\perp )}\left({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp }\right)|{\tilde{k}}^{\perp }=0\rangle \hfill \\ & =& {\tilde{\mathcal{N}}}_p\prod _{i=1}^{k-1}\left[\sum _{{\omega }_{{\alpha }_i}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}|n_{{\alpha }_i}=0,{\omega }_{{\alpha }_i}\rangle \right]|k^0=0,{\widehat{k}}^{\Vert }=0\rangle \hfill \\ & & \times \prod _{m=1}^{l+1}\left[\sum _{n_{j_m}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}|n_{j_m},{\omega }_{j_m}=0\rangle \right]{\widehat{\delta }}^{(\perp )}\left({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp }\right)|{\tilde{k}}^{\perp }=0\rangle ,\hfill \end{array}$$

(31)

where in the last equality, we have taken $n_{j_{l+1}}=n_{{\alpha }_k}$, $j_{l+1}={\alpha }_k$ and $R_{j_{l+1}}=r_{{\alpha }_k}$. Note also that under this T-duality, we have

$$\begin{array}{ccc}\hfill {\mathcal{N}}_p\to {\tilde{\mathcal{N}}}_p& =& {\displaystyle \frac{c_p}{2}}{\left({\displaystyle \frac{2\pi {\alpha }^{\prime }}{r_{{\alpha }_k}{\Phi }_{{\alpha }_k}}}\right)}^{1/2}\prod _{i=1}^{k-1}{\left({\displaystyle \frac{2\pi r_{{\alpha }_i}}{{\Phi }_{{\alpha }_i}}}\right)}^{1/2}\prod _{m=1}^l{\left({\displaystyle \frac{1}{2\pi R_{j_m}{\Phi }_{j_m}}}\right)}^{1/2}\hfill \\ & =& {\displaystyle \frac{c_p}{2}}\left(2\pi \sqrt{{\alpha }^{\prime }}\right){\left({\displaystyle \frac{1}{2\pi r_{{\alpha }_k}{\Phi }_{{\alpha }_k}}}\right)}^{1/2}\prod _{i=1}^{k-1}{\left({\displaystyle \frac{2\pi r_{{\alpha }_i}}{{\Phi }_{{\alpha }_i}}}\right)}^{1/2}\prod _{m=1}^l{\left({\displaystyle \frac{1}{2\pi R_{j_m}{\Phi }_{j_m}}}\right)}^{1/2}\hfill \\ & =& {\displaystyle \frac{c_{p-1}}{2}}\prod _{i=1}^{k-1}{\left({\displaystyle \frac{2\pi r_{{\alpha }_i}}{{\Phi }_{{\alpha }_i}}}\right)}^{1/2}\prod _{m=1}^{l+1}{\left({\displaystyle \frac{1}{2\pi R_{j_m}{\Phi }_{j_m}}}\right)}^{1/2}={\mathcal{N}}_{p-1}.\hfill \end{array}$$

(32)

With the above, we have, under this T-duality, the expected transformation

$${|\Omega \rangle }_p\to {|\Omega \rangle }_{p-1}.$$

(33)

By a similar token, one can also show that when a T-duality is performed along a transverse compactified direction,

$${|\Omega \rangle }_p\to {|\Omega \rangle }_{p+1},$$

(34)

which is also expected.

3. The Compactified D-Brane Cylinder Amplitude

With the preparation given in the previous section, we are now ready to compute the closed string tree cylinder amplitude between a Dp and a Dp′ with $p-p^{\prime }=2n$. The two D-branes are placed parallel at a separation along their common non-compactified transverse directions. Here, the compactifications are made with respect to the longitudinal and transverse directions common to both of the D-branes. In other words, for the compactified boundary state given in the previous section, we need to restrict $k\le p^{\prime }$ and $l\le 9-p$, as discussed in the Introduction.

The cylinder amplitude between a Dp-brane and a Dp′-brane, placed parallel at a separation y, can be calculated via

$${\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}=\langle B^{p^{\prime }}\left|D\right|B^p\rangle ,$$

(35)

where D is the closed string propagator defined as

$$D={\displaystyle \frac{{\alpha }^{\prime }}{4\pi }}{\int }_{\left|z\right|\le 1}{\displaystyle \frac{d^{2}z}{{\left|z\right|}^2}}{z}^{L_0}{\overline{z}}^{{\tilde{L}}_0}.$$

(36)

This amplitude represents the tree closed string interaction amplitude between the two D-branes, which can be viewed as that the Dp′-brane, represented by its boundary state $\langle B^{p^{\prime }}|$, emits a virtual closed string at some instant, and the emitted virtual closed string, represented here by the closed string propagator D above, propagates for some time and is finally absorbed by the Dp-brane, represented here by the boundary state $|B^p\rangle$, just like the Coulomb interaction between two point of charge or the Newtonian interaction between two points of mass, which can be viewed as one charge (or one mass), emits a virtual photon (or a virtual graviton), and the virtual photon (or the virtual graviton) propagates for some time and is finally absorbed by the other charge (or the other mass). In the present context, if one draws the corresponding diagram for the closed string, the amplitude looks like a cylinder, which is therefore usually called cylinder amplitude.

In the above, $L_0$ and ${\tilde{L}}_0$ are the respective left and right mover total zero-mode Virasoro generators of matter fields, ghosts, and superghosts. For example, $L_0=L_0^X+L_0^{\psi }+L_0^{\mathrm{gh}}+L_0^{\mathrm{sgh}}$ where $L_0^X,L_0^{\psi },L_0^{\mathrm{gh}}$ and $L_0^{\mathrm{sgh}}$ represent contributions from matter fields $X^{\mu }$, matter fields ${\psi }^{\mu }$, ghosts b and c, and superghosts $\beta$ and $\gamma$, respectively. The matter parts $L^{X_0}+L_0^{\psi }$ (also ${\tilde{L}}_0^X+{\tilde{L}}_0^{\psi }$), except for their corresponding zero-mode parts, which will be given later, are already given in (13). The explicit expressions for the ghost part have nothing to do with T-duality and can be found in any standard discussion of superstring theories, for example in [5]; therefore, they will not be presented here. The boundary states $|B_p\rangle$ and $|B_{p^{\prime }}\rangle$ used above are the respective GSO-projected ones given in (1).

In general, we have two contributions to the total amplitude given in (35), one from the NS-NS sector and the other from the R-R sector. In other words, we have

$${\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}={\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}^{\mathrm{NS}-\mathrm{NS}}+{\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}^{\mathrm{R}-\mathrm{R}}.$$

(37)

Computing each of these is boiled down to the following equation in each sector

$${\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}({\eta }^{\prime },\eta )=\langle B^{p^{\prime }},{\eta }^{\prime }\left|D\right|B^p,\eta \rangle ,$$

(38)

with the respective boundary state given by (2), for which we also take the compactified case into consideration. As discussed in detail in [7,8], we have ${\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}({\eta }^{\prime },\eta )={\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}\left({\eta }^{\prime }\eta \right)$, and this amplitude can be factorized as

$${\Gamma }_{{\mathrm{Dp}}^{\prime }|\mathrm{Dp}}\left({\eta }^{\prime }\eta \right)={\displaystyle \frac{{\alpha }^{\prime }}{4\pi }}{\int }_{\left|z\right|\le 1}{\displaystyle \frac{d^{2}z}{{\left|z\right|}^2}}{A}^X{A}^{\mathrm{bc}}{A}^{\psi }\left({\eta }^{\prime }\eta \right)A^{\beta \gamma }\left({\eta }^{\prime }\eta \right).$$

(39)

In the above, we have

$$\begin{array}{ccccc}\hfill & & A^X=\langle B_X^{p^{\prime }}|z^{L_0^X}{\overline{z}}^{{\tilde{L}}_0^X}|B_X^p\rangle ,A^{\psi }\left({\eta }^{\prime }\eta \right)=\langle B_{\psi }^{p^{\prime }},{\eta }^{\prime }{\left|\right|z|}^{2L_0^{\psi }}|B_{\psi }^p,\eta \rangle ,\hfill & & \\ & & A^{\mathrm{bc}}=\langle B_{\mathrm{gh}}{\left|\right|z|}^{2L_0^{\mathrm{gh}}}|B_{\mathrm{gh}}\rangle ,A^{\beta \gamma }\left({\eta }^{\prime }\eta \right)=\langle B_{\mathrm{sgh}},{\eta }^{\prime }{\left|\right|z|}^{2L_0^{\mathrm{sgh}}}|B_{\mathrm{sgh}},\eta \rangle .\hfill \end{array}$$

(40)

As stressed earlier, the total amplitude has a contribution from the R-R sector only when $p=p^{\prime }$, for which this amplitude vanishes due to the cancellation between the contribution from the NS-NS sector and that from the R-R sector because of the 1/2 BPS nature of this system. This certainly still holds true when a T-duality is performed. We can understand this easily as follows: The bosonic zero-mode contribution to the boundary state remains the same to both sectors. The oscillator contributions to the amplitude from all sectors remain the same before and after the T-duality because the only quantity relevant to the T-duality is the matrix $S{S}^{\prime T}$ [8], with the respective S and $S^{\prime }$ given by (7) for the Dp-brane and Dp′-brane, which remains invariant under the T-duality. As demonstrated in the previous section, the fermionic zero-mode boundary state in R-R sector will have a sign change in front of ${\Gamma }_{11}$—for example, see the second line in (18) or (20)—under the T-duality. We can absorb this ‘-’ sign by defining $\tilde{\eta }=-\eta$ and ${\tilde{\eta }}^{\prime }=-{\eta }^{\prime }$. Since this zero-mode contribution to the amplitude depends only on the product $\tilde{\eta }{\tilde{\eta }}^{\prime }=\eta {\eta }^{\prime }$, not individual $\tilde{\eta }$ and ${\tilde{\eta }}^{\prime }$, this contribution will also remain invariant under the T-duality.

When $p\ne p^{\prime }$ for which the total amplitude is non-vanishing, the only contribution to the amplitude now comes from the NS-NS sector. As discussed before, only the bosonic zero-mode contribution to the amplitude will change under the T-duality. Therefore, this is our focus in computing the non-vanishing closed string tree cylinder amplitude for $p-p^{\prime }=2,6$. The non-compactified cylinder amplitude for either $p-p^{\prime }=2$ or $p-p^{\prime }=6$ can be obtained from the general one given in [8] by setting the worldvolume flux vanishing (The non-compactified amplitude for the $p-p^{\prime }=2n$ case in the absence of general worldvolume fluxes was computed earlier in [5], but the identities of various $\theta$ functions have not been used to simplify the integrand of the amplitude as we did in [8]). For $p-p^{\prime }=2$, it is for $2\le p\le 8$

$${\Gamma }_{\mathrm{D}(\mathrm{p}-2)|\mathrm{Dp}}\left(y\right)={\displaystyle \frac{2V_{p-1}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{p-1}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p}{2}}}}{e}^{-{\displaystyle \frac{y^2}{2\pi {\alpha }^{\prime }t}}}\prod _{n=1}^{\infty }{\displaystyle \frac{{\left(1+{\left|z\right|}^{4n}\right)}^4}{{\left(1-{\left|z\right|}^{2n}\right)}^6{\left(1+{\left|z\right|}^{2n}\right)}^2}},$$

(41)

where $\left|z\right|=e^{-\pi t}$ and y is the brane separation along the transverse directions common to both of the branes. For $p-p^{\prime }=6$, we have the amplitude for $6\le p\le 8$ as

$${\Gamma }_{\mathrm{D}(\mathrm{p}-6)|\mathrm{Dp}}\left(y\right)=-{\displaystyle \frac{V_{p-5}}{2{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{p-5}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p}{2}}}}{e}^{-{\displaystyle \frac{y^2}{2\pi {\alpha }^{\prime }t}}}\prod _{n=1}^{\infty }{\displaystyle \frac{{(1+|z|}^{4n}{)}^4}{{(1-|z|}^{2n}{)}^2{(1+|z|}^{2n}{)}^6}},$$

(42)

where again $\left|z\right|=e^{-\pi t}$ and y the brane separation in the transverse directions common to both of the branes. These two amplitudes will provide consistent checks for the corresponding compactified ones which will be computed in the following when the respective decompactification limits are taken. Note that the infinite product in the integrand of the amplitude either (41) or (42) comes from the contribution of various oscillator modes and this will remain the same in the corresponding compactified case. Therefore, in the compactified case, we need only to compute the bosonic zero-mode contribution to the amplitude. This boils down to computing the bosonic zero-mode contribution to the matrix element of $A^X$ given in (40). In other words, we need to compute

$$A_0^X={⠀}_0\langle B_X^{p^{\prime }}|z^{{\displaystyle \frac{1}{2}}{\alpha }_{0R}^2+{\displaystyle \frac{1}{2}}{\widehat{\alpha }}_0^2}{\overline{z}}^{{\displaystyle \frac{1}{2}}{\alpha }_{0L}^2+{\displaystyle \frac{1}{2}}{\widehat{\alpha }}_0^2}{|B_X^p\rangle }_0,$$

(43)

where ${\alpha }_{0R},{\alpha }_{0L}$ denote the right-mover and left-mover bosonic zero-modes, respectively, along the compactified directions, while ${\widehat{\alpha }}_0={\widehat{\tilde{\alpha }}}_0$ denote the respective bosonic zero-modes along the non-compactified directions. We will use this $A_0^X$ to replace the corresponding one in the non-compactified case in the amplitude (39) while the rest, due to contributions from various oscillators and the possible fermionic zero-mode in the R-R sector, remains the same as in the non-compactified case. Note that in the compactified case $|B_X^p{\rangle }_0=|{\Omega }_p\rangle$, $|B_X^{p^{\prime }}{\rangle }_0=|{\Omega }_{p^{\prime }}\rangle$, and also the following:

$${\alpha }_{0R}={\displaystyle \frac{l_s}{2}}{p}_R={\displaystyle \frac{l_s}{2}}\left(p_n-p_{\omega }\right),{\alpha }_{0L}={\displaystyle \frac{l_s}{2}}{p}_L={\displaystyle \frac{l_s}{2}}\left(p_n+p_{\omega }\right),$$

(44)

and

$${\widehat{\alpha }}_0={\widehat{\tilde{\alpha }}}_0={\displaystyle \frac{l_s}{2}}\widehat{p},$$

(45)

with $l_s=\sqrt{2{\alpha }^{\prime }}$. This means that we have

$$\begin{array}{ccc}\hfill A_0^X& =& \langle {\Omega }_{p^{\prime }}|z^{{\displaystyle \frac{1}{2}}{\alpha }_{0R}^2+{\displaystyle \frac{1}{2}}{\widehat{\alpha }}_0^2}{\overline{z}}^{{\displaystyle \frac{1}{2}}{\alpha }_{0L}^2+{\displaystyle \frac{1}{2}}{\widehat{\alpha }}_0^2}|{\Omega }_p\rangle \hfill \\ & =& \langle {\Omega }_{p^{\prime }}|z^{{\displaystyle \frac{{\alpha }^{\prime }}{4}}\left[{\sum }_{i=1}^k{\left(p_n^{{\alpha }_i}-p_{\omega }^{{\alpha }_i}\right)}^2+{\sum }_{m=1}^l{\left(p_n^{j_m}-p_{\omega }^{j_m}\right)}^2+{\widehat{p}}^2\right]}{\overline{z}}^{{\displaystyle \frac{{\alpha }^{\prime }}{4}}\left[{\sum }_{i=1}^k{\left(p_n^{{\alpha }_i}+p_{\omega }^{{\alpha }_i}\right)}^2+{\sum }_{m=1}^l{\left(p_n^{j_m}+p_{\omega }^{j_m}\right)}^2+{\widehat{p}}^2\right]}|{\Omega }_p\rangle \hfill \\ & =& \langle {\Omega }_{p^{\prime }}|z^{{\displaystyle \frac{{\alpha }^{\prime }}{4}}\left[{\sum }_{i=1}^k{\left(p_{\omega }^{{\alpha }_i}\right)}^2+{\sum }_{m=1}^l{\left(p_n^{j_m}\right)}^2+{\widehat{p}}^2\right]}{\overline{z}}^{{\displaystyle \frac{{\alpha }^{\prime }}{4}}\left[{\sum }_{i=1}^k{\left(p_{\omega }^{{\alpha }_i}\right)}^2+{\sum }_{m=1}^l{\left(p_n^{j_m}\right)}^2+{\widehat{p}}^2\right]}|{\Omega }_p\rangle \hfill \\ & =& \langle {\Omega }_{p^{\prime }}||z{|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}\left[{\sum }_{i=1}^k{\left(p_{\omega }^{{\alpha }_i}\right)}^2+{\sum }_{m=1}^l{\left(p_n^{j_m}\right)}^2+{\widehat{p}}^2\right]}|{\Omega }_p\rangle \hfill \\ & =& {\mathcal{N}}_{p^{\prime }}{\mathcal{N}}_p\langle k^0=0,{\widehat{k}}^{\prime \Vert }=0,{\widehat{k}}^{\prime \perp }=0|{\widehat{\delta }}^{(9+2n-p-l)}({\widehat{q}}^{\prime \perp })\hfill \\ & & \times \prod _{i=1}^k\left[\left(\sum _{{\omega }_{{\alpha }_i}^{\prime }}\langle {\omega }_{{\alpha }_i}^{\prime }|\right){\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\left(p_{\omega }^{{\alpha }_i}\right)}^2}\left(\sum _{{\omega }_{{\alpha }_i}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}|{\omega }_{{\alpha }_i}\rangle \right)\right]\hfill \\ & & \times \prod _{m=1}^l\left[\left(\sum _{n_{j_m}^{\prime }}\langle n_{j_m}^{\prime }|\right){\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\left(p_n^{j_m}\right)}^2}\left(\sum _{n_{j_m}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}|n_{j_m}\rangle \right)\right]\hfill \\ & & \times {\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\widehat{p}}^2}{\widehat{\delta }}^{(9-p-l)}({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp })|k^0=0,{\widehat{k}}^{\Vert }=0,{\widehat{k}}^{\perp }=0\rangle ,\hfill \end{array}$$

(46)

where in the third line, we have used $p_n^{{\alpha }_i}|{\Omega }_p\rangle =0,p_{\omega }^{j_m}|{\Omega }_p\rangle =0$ as given in (24), and in the last equality, we have used the explicit expression for $|{\Omega }_p\rangle$ given in (21). Note also that $|y^{j_m}|$ is the brane separation along the respective compactified direction. This can be further expressed as

$$\begin{array}{cccc}\hfill A_0^X& =& {\mathcal{N}}_{p^{\prime }}{\mathcal{N}}_p\prod _{i=1}^k{\Phi }_{{\alpha }_i}\left(\sum _{{\omega }_{{\alpha }_i}}{\left|z\right|}^{{\displaystyle \frac{{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}\right)\prod _{m=1}^l{\Phi }_{j_m}\left(\sum _{n_{j_m}}{\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}\right)\hfill & \\ & & \times \langle k^0=0,{\widehat{k}}^{\prime \Vert }=0,{\widehat{k}}^{\prime \perp }=0|{\widehat{\delta }}^{(9+2n-p-l)}\left({\widehat{q}}^{\prime \perp }\right)|z{|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\widehat{p}}^2}{\widehat{\delta }}^{(9-p-l)}({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp })\hfill & \\ & & \times |k^0=0,{\widehat{k}}^{\Vert }=0,{\widehat{k}}^{\perp }=0\rangle .\hfill \end{array}$$

(47)

In the following, we move to compute the factor in the last two lines above. To this end, note that

$$V_{1+p-2n-k}=\langle k^0=0,{\widehat{k}}^{\prime \Vert }=0|k^0=0,{\widehat{k}}^{\prime \Vert }=0\rangle ,$$

(48)

we have then

$$\begin{array}{ccc}& & \langle k^0=0,{\widehat{k}}^{\prime \Vert }=0,{\widehat{k}}^{\prime \perp }=0|{\widehat{\delta }}^{(9+2n-p-l)}({\widehat{q}}^{\prime \perp })|z{|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\widehat{p}}^2}{\widehat{\delta }}^{(9-p-l)}({\widehat{q}}^{\perp }-{\widehat{y}}^{\perp })|k^0=0,{\widehat{k}}^{\Vert }=0,{\widehat{k}}^{\perp }=0\rangle \hfill \\ & =& \int {\displaystyle \frac{d^{2n+9-p-l}{Q}^{\prime }}{{\left(2\pi \right)}^{2n+9-p-l}}}\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}\langle k^0=0,{\widehat{k}}^{\prime \Vert }=0,{\widehat{k}}^{\prime \perp }=-Q^{\prime }{\left|\right|z|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{\widehat{p}}^2}{e}^{-iQ·{\widehat{y}}^{\perp }}\hfill \\ & & \times |k^0=0,{\widehat{k}}^{\Vert }=0,{\widehat{k}}^{\perp }=Q\rangle \hfill \\ & =& V_{1+2p-2n-k}\int d^{2n+9-p-l}{Q}^{\prime }\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}{\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{Q}^2}{e}^{-iQ·{\widehat{y}}^{\perp }}{\delta }^{\left(2n\right)}\left(Q^{\prime }\right){\delta }^{(9-p-l)}(Q+Q^{\prime })\hfill \\ & =& V_{1+2p-2n-k}\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}{\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{Q}^2}{e}^{-iQ·{\widehat{y}}^{\perp }}.\hfill \end{array}$$

(49)

This means that we have

$$\begin{array}{ccc}& & A_0^X={\mathcal{N}}_{p^{\prime }}{\mathcal{N}}_p\prod _{i=1}^k{\Phi }_{{\alpha }_i}\left(\sum _{{\omega }_{{\alpha }_i}}{\left|z\right|}^{{\displaystyle \frac{{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}}{e}^{-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}\right)\prod _{m=1}^l{\Phi }_{j_m}\left(\sum _{n_{j_m}}{\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}}{e}^{-i{y}^{j_m}{n}_{j_m}/R_{j_m}}\right)\hfill \\ & & \times V_{1+2p-2n-k}\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}{\left|z\right|}^{{\displaystyle \frac{{\alpha }^{\prime }}{2}}{Q}^2}{e}^{-iQ·{\widehat{y}}^{\perp }}\hfill \\ & & ={\mathcal{N}}_{p^{\prime }}{\mathcal{N}}_p\prod _{i=1}^k\prod _{m=1}^l{\Phi }_{{\alpha }_i}{\Phi }_{j_m}\left(\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{y}^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}/{\alpha }^{\prime }}\right)\left(\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{y}^{j_m}{n}_{j_m}/R_{j_m}}\right)\hfill \\ & & \times V_{1+2p-2n-k}\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }}{2}}{Q}^2-iQ·{\widehat{y}}^{\perp }},\hfill \end{array}$$

(50)

where we have set $\left|z\right|=e^{-\pi t}$. Note that

$$\begin{array}{ccccc}\hfill {\mathcal{N}}_{p^{\prime }}{\mathcal{N}}_p\prod _{i=1}^k\prod _{m=1}^l{\Phi }_{{\alpha }_i}{\Phi }_{j_m}& =& {\displaystyle \frac{c_{p-2n}{c}_p}{4}}\prod _{i=1}^k{\displaystyle \frac{2\pi r_{{\alpha }_i}}{{\Phi }_{{\alpha }_i}}}\prod _{m=1}^l{\displaystyle \frac{1}{2\pi R_{j_m}{\Phi }_{j_m}}}\prod _{i=1}^k\prod _{m=1}^l{\Phi }_{{\alpha }_i}{\Phi }_{j_m}\hfill & & \\ & =& {\displaystyle \frac{c_{p-2n}{c}_p}{4}}\prod _{i=1}^{k}2\pi r_{{\alpha }_i}\prod _{m=1}^l{\displaystyle \frac{1}{2\pi R_{j_m}}},\hfill \end{array}$$

(51)

where we have used (22) for ${\mathcal{N}}_p$ and ${\mathcal{N}}_{p^{\prime }}$, and

$$\int {\displaystyle \frac{d^{9-p-l}Q}{{\left(2\pi \right)}^{9-p-l}}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }}{2}}{Q}^2-iQ·{\widehat{y}}^{\perp }}={\left(2{\pi }^2{\alpha }^{\prime }t\right)}^{-{\displaystyle \frac{9-p-l}{2}}}{e}^{-{\displaystyle \frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}}.$$

(52)

With the above, we have

$$\begin{array}{ccccc}\hfill A_0^X& =& {\displaystyle \frac{c_{p-2n}{c}_p}{4{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{\frac{9-p-l}{2}}}}{V}_{1+p-2n-k}\prod _{i=1}^k\left(2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right)\hfill & & \\ & & \times \prod _{m=1}^l\left({\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right){\displaystyle \frac{e^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}}{t^{\frac{9-p-l}{2}}}}.\hfill \end{array}$$

(53)

Let us make a consistent check of the above to see if it yields the decompactification limit when we take $r_{{\alpha }_i}\to \infty$ and $R_{j_m}\to \infty$. When we take $r_{{\alpha }_i}\to \infty$, it is clear that only the winding ${\omega }_{{\alpha }_i}=0$ term survives. When $R_{j_m}\to \infty$ is taken, we set $k_{j_m}=n_{j_m}/R_{j_m}\to d{k}_{j_m}=1/R_{j_m}$ and pass the summation to an integration as follows:

$$\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}=R_{j_m}\int d{k}_{j_m}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{k}_{j_m}^2}{2}}-i{y}^{j_m}{k}_{j_m}}=2\pi R_{j_m}{\left(2{\pi }^2{\alpha }^{\prime }t\right)}^{-1/2}{e}^{-{\displaystyle \frac{{(y^{j_m})}^2}{2\pi {\alpha }^{\prime }t}}}.$$

(54)

We then have

$$\begin{array}{ccccc}\hfill A_0^X& \to & {\displaystyle \frac{c_{p-2n}{c}_p}{4{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{\frac{9-p-l}{2}}}}{V}_{1+p-2n-k}\left(\prod _{i=1}^{k}2\pi r_{{\alpha }_i}\right)\left(\prod _{m=1}^l{\displaystyle \frac{e^{-\frac{{(y^{j_m})}^2}{2\pi {\alpha }^{\prime }t}}}{\sqrt{2{\pi }^2{\alpha }^{\prime }t}}}\right){\displaystyle \frac{e^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}}{t^{\frac{9-p-l}{2}}}}\hfill & & \\ & =& {\displaystyle \frac{c_{p-2n}{c}_p{V}_{1+p-2n}}{4{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{\frac{9-p}{2}}}}{\displaystyle \frac{e^{-\frac{{({\widehat{y}}^{\perp })}^2+{\sum }_{m=1}^l{\left(y^{j_m}\right)}^2}{2\pi {\alpha }^{\prime }t}}}{t^{\frac{9-p}{2}}}},\hfill \end{array}$$

(55)

where we have set $V_{1+p-2n}=V_{1+p+2n-k}{\prod }_{i=1}^{k}2\pi r_{{\alpha }_i}$ with $r_{{\alpha }_i}\to \infty$. Note also that $y^2={\left({\widehat{y}}^{\perp }\right)}^2+{\sum }_{m=1}^l{\left(y^{j_m}\right)}^2$ is precisely the brane separation along the transverse directions to both branes when the decompactification limit is taken. Therefore,

$$A_0^X\to {\displaystyle \frac{c_{p-2n}{c}_p{V}_{1+p-2n}}{4{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{\frac{9-p}{2}}}}{\displaystyle \frac{e^{-\frac{y^2}{2\pi {\alpha }^{\prime }t}}}{t^{\frac{9-p}{2}}}},$$

(56)

giving precisely the corresponding decompactification limit.

With the above, we have the compactified cylinder amplitude between a D(p − 2n)-brane and a Dp-brane as

$$\begin{array}{ccccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}& =& {\displaystyle \frac{c_{p-2n}{c}_p}{4}}{\displaystyle \frac{{\alpha }^{\prime }}{4\pi }}2{\pi }^2{\displaystyle \frac{2^2{V}_{1+p-2n-k}}{{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{\frac{9-p-l}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p-l}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\hfill & & \\ & \times & \prod _{i=1}^k\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^l\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right],\hfill \end{array}$$

(57)

where we have used

$${\int }_{\left|z\right|\le 1}{\displaystyle \frac{d{z}^2}{{\left|z\right|}^2}}=2{\pi }^2{\int }_0^{\infty }dt.$$

(58)

As pointed out earlier, the various oscillator contributions plus a possible fermionic zero-mode contribution in the R-R sector to the amplitude remain inert with respect to the compactifications, and we denote these contributions by $2^2{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}$ in the above amplitude. Note that $Z_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)=0$ for $n=0,2$ due to the BPS nature of the underlying systems, for $n=1$ from (41)

$$Z_{(\mathrm{p}-2)|\mathrm{p}}\left(t\right)={\displaystyle \frac{{\left(1+{\left|z\right|}^{4n}\right)}^4}{{\left(1-{\left|z\right|}^{2n}\right)}^6{\left(1+{\left|z\right|}^{2n}\right)}^2}},$$

(59)

and for $n=3$ from (42)

$$Z_{(\mathrm{p}-6)|\mathrm{p}}\left(t\right)=\prod _{n=1}^{\infty }\frac{{(1+|z|}^{4n}{)}^4}{{(1-|z|}^{2n}{)}^2{(1+|z|}^{2n}{)}^6}.$$

(60)

Note that

$$c_p=\sqrt{\pi }{\left(2\pi \sqrt{{\alpha }^{\prime }}\right)}^{3-p},c_{p-2n}=\sqrt{\pi }{\left(2\pi \sqrt{{\alpha }^{\prime }}\right)}^{3+2n-p},$$

(61)

we have

$$c_p{c}_{p-2n}{\displaystyle \frac{{\alpha }^{\prime }}{4\pi }}2{\pi }^2={\displaystyle \frac{2^n}{2^{p-1}}}{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{4+n-p}.$$

(62)

As a result, we have the compactified cylinder amplitude (57) as

$$\begin{array}{ccccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\left(y\right)& =& {\displaystyle \frac{2^{2-n-l}{V}_{1+p-2n-k}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-l}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p-l}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\hfill & & \\ & \times & \prod _{i=1}^k\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^l\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right].\hfill \end{array}$$

(63)

Taking the decompactification limit, as discussed earlier, we have

$$\begin{array}{ccccc}\hfill & & \prod _{i=1}^k\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\to \prod _{i=1}^{k}2\pi r_{{\alpha }_i},\hfill & & \\ & & \prod _{m=1}^l\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right]\to \prod _{m=1}^l\frac{e^{-\frac{{(y^{j_m})}^2}{2\pi {\alpha }^{\prime }t}}}{\sqrt{2{\pi }^2{\alpha }^{\prime }t}}={\displaystyle \frac{e^{-{\sum }_{m=1}^l\frac{{(y^{j_m})}^2}{2\pi {\alpha }^{\prime }t}}}{{\left(2{\pi }^2{\alpha }^{\prime }t\right)}^{\frac{l}{2}}}}.\hfill \end{array}$$

(64)

We then have the decompactified cylinder amplitude from (63) as

$${\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\left(y\right)={\displaystyle \frac{2^{2-n}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p}{2}}}}{e}^{-{\displaystyle \frac{y^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right),$$

(65)

where, as before, $V_{1+p-2n}=V_{1+p-2n-k}{\prod }_{i=1}^{k}2\pi r_{{\alpha }_i}$ for $r_{{\alpha }_i}\to \infty$ and $y^2={\left({\widehat{y}}^{\perp }\right)}^2+{\sum }_{m=1}^l{\left(y^{j_m}\right)}^2$. This gives precisely the one in (41) for $n=1$ and the one in (42) for $n=3$.

In summary, we have in this section computed the compactified cylinder amplitude (63) between a D(p - 2n) and a Dp with $2n\le p\le 8$. This amplitude has $k\le p-2n$ compactified longitudinal directions and $l\le 9-p$ compactified transverse directions common to both the branes. As specified earlier, the respective compactified radii are $r_{{\alpha }_i}$ and $R_{j_m}$, with $i=1,2,\cdots k$ and $m=1,2,\cdots l$. These two D-branes are placed parallel along the non-compactified transverse directions at a separation of ${\widehat{y}}^{\perp }$ and along each of the compactified transverse directions at $|y^{j_m}|$. $y^{{\alpha }_i}$ are the Wilson lines turned on along the respective compactified worldvolume directions of the Dp-brane while keeping the D(p − 2n)-brane absent of these.

4. T-Duality

Given the cylinder amplitude (63), performing a T-duality along either a compactified longitudinal direction or a compactified transverse direction becomes easy.

Without loss of generality, let us first perform this T-duality specifically along the longitudinal ${\alpha }_k$ direction. We expect to obtain the corresponding compactified amplitude via ${\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\to {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n}-1|\mathrm{D}(\mathrm{p}-1)}$. Let us check if this is indeed true. To this end, we send $r_{{\alpha }_k}\to R_{j_{l+1}}={\alpha }^{\prime }/r_{{\alpha }_k}$, ${\omega }_{{\alpha }_k}\to n_{j_{l+1}}={\omega }_{{\alpha }_k}$, and $y^{{\alpha }_k}\to y^{j_{l+1}}=y^{{\alpha }_k}$ to the cylinder amplitude (63). We then have

$$2\pi r_{{\alpha }_k}\sum _{{\omega }_{{\alpha }_k}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_k}^2{r}_{{\alpha }_k}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_k}{\omega }_{{\alpha }_k}{r}_{{\alpha }_k}}{{\alpha }^{\prime }}}}\to {\displaystyle \frac{8{\pi }^2{\alpha }^{\prime }}{2}}{\displaystyle \frac{1}{2\pi R_{j_{l+1}}}}\sum _{n_{j_{l+1}}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_{l+1}}^2}{2R_{j_{l+1}}^2}}-i{\displaystyle \frac{y^{j_{l+1}}{n}_{j_{l+1}}}{R_{j_{l+1}}}}}.$$

(66)

We then have from (63)

$$\begin{array}{ccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}& \to & {\displaystyle \frac{2^{2-n-l}{V}_{1+p-2n-k}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-l}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p-l}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}-2\mathrm{n}-1)\left|\right(\mathrm{p}-1)}\left(t\right)\hfill \\ & \times & \prod _{i=1}^{k-1}\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^l\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right]\hfill \\ & \times & {\displaystyle \frac{8{\pi }^2{\alpha }^{\prime }}{2}}{\displaystyle \frac{1}{2\pi R_{j_{l+1}}}}\sum _{n_{j_{l+1}}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_{l+1}}^2}{2R_{j_{l+1}}^2}}-i{\displaystyle \frac{y^{j_{l+1}}{n}_{j_{l+1}}}{R_{j_{l+1}}}}}\hfill \\ & =& {\displaystyle \frac{2^{2-n-(l+1)}{V}_{1+(p-1)-2n-(k-1)}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+(p-1)-2n-(l+1)}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-(p-1)-(l+1)}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}-2\mathrm{n}-1)\left|\right(\mathrm{p}-1)}\left(t\right)\hfill \\ & \times & \prod _{i=1}^{k-1}\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^{l+1}\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right]\hfill \\ & =& {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n}-1)\left|\mathrm{D}\right(\mathrm{p}-1)},\hfill \end{array}$$

(67)

where in the first line, we have sent $Z_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\to Z_{(\mathrm{p}-2\mathrm{n}-1)\left|\right(\mathrm{p}-1)}\left(t\right)=Z_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)$ due to the inert of various oscillator contributions to the amplitude under T-duality, and ${\widehat{y}}^{\perp }$ remains the same, so this indeed goes as expected.

We now move to discuss performing a T-duality along a compactified transverse direction, say, along $j_l$, to the amplitude (63). We then expect to have ${\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\to {\Gamma }_{\mathrm{D}(\mathrm{p}+1-2\mathrm{n})\left|\mathrm{D}\right(\mathrm{p}+1)}$. To this end, we set $R_{j_l}\to r_{{\alpha }_{k+1}}={\alpha }^{\prime }/R_{j_l}$, $n_{j_l}\to {\omega }_{{\alpha }_{k+1}}=n_{j_l}$ and $y^{j_l}\to y^{{\alpha }_{k+1}}=y^{j_l}$. We have

$${\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\to {\displaystyle \frac{2}{8{\pi }^2{\alpha }^{\prime }}}2\pi r_{{\alpha }_{k+1}}\sum _{{\omega }_{{\alpha }_{k+1}}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_{k+1}}^2{r}_{{\alpha }_{k+1}}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_{k+1}}{\omega }_{{\alpha }_{k+1}}{r}_{{\alpha }_{k+1}}}{{\alpha }^{\prime }}}}.$$

(68)

We then have from (63)

$$\begin{array}{ccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}& \to & {\displaystyle \frac{2^{2-n-l}{V}_{1+p-2n-k}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-l}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p-l}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)\hfill \\ & \times & \prod _{i=1}^k\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^{l-1}\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right]\hfill \\ & \times & {\displaystyle \frac{2}{8{\pi }^2{\alpha }^{\prime }}}2\pi r_{{\alpha }_{k+1}}\sum _{{\omega }_{{\alpha }_{k+1}}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_{k+1}}^2{r}_{{\alpha }_{k+1}}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_{k+1}}{\omega }_{{\alpha }_{k+1}}{r}_{{\alpha }_{k+1}}}{{\alpha }^{\prime }}}}\hfill \\ & =& {\displaystyle \frac{2^{2-n-(l-1)}{V}_{1+(p+1)-2n-(k+1)}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+(p+1)-2n-(l-1)}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-(p+1)-(l-1)}{2}}}}{e}^{-\frac{{({\widehat{y}}^{\perp })}^2}{2\pi {\alpha }^{\prime }t}}{Z}_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)\hfill \\ & \times & \prod _{i=1}^{k+1}\left[2\pi r_{{\alpha }_i}\sum _{{\omega }_{{\alpha }_i}}{e}^{-{\displaystyle \frac{\pi t{\omega }_{{\alpha }_i}^2{r}_{{\alpha }_i}^2}{2{\alpha }^{\prime }}}-i{\displaystyle \frac{y^{{\alpha }_i}{\omega }_{{\alpha }_i}{r}_{{\alpha }_i}}{{\alpha }^{\prime }}}}\right]\prod _{m=1}^{l-1}\left[{\displaystyle \frac{1}{2\pi R_{j_m}}}\sum _{n_{j_m}}{e}^{-{\displaystyle \frac{\pi t{\alpha }^{\prime }{n}_{j_m}^2}{2R_{j_m}^2}}-i{\displaystyle \frac{y^{j_m}{n}_{j_m}}{R_{j_m}}}}\right]\hfill \\ & =& {\Gamma }_{\mathrm{D}(\mathrm{p}+1-2\mathrm{n})\left|\mathrm{D}\right(\mathrm{p}+1)},\hfill \end{array}$$

(69)

where in the first line, we have also set $Z_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)=Z_{(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\left(t\right)$ for the same reason as explained earlier. In the end, we obtain the expected result.

As demonstrated already in the previous section, the decompactification limit will give the respective known cylinder amplitude after the T-dualities discussed above.

In the following, we will explain, as promised in the Introduction, that the usual low-energy approach does work for a T-duality along a transverse direction common to the two D-branes for the corresponding decompactified cylinder amplitude but does not work for a T-duality along a longitudinal direction.

The simple reason is that to obtain the decompactified result after the T-duality, we need to send the initial compactified radius to zero. To perform a low-energy T-duality along a transverse direction, we first need to generate an isometry along this direction by periodically placing this system along this direction and by sending the radius to vanishing. In other words, we do not consider the momentum modes along this effective compactification with a zero radius; therefore, we do not have a winding in the T-dual system, which is consistent with the fact that the T-dual radius becomes infinitely large, and there is no winding allowed. However, this is not the case if we apply a low-energy T-dual along the longitudinal direction. The system already has an isometry along the longitudinal direction, since the decompactified amplitude is independent of the longitudinal directions. If we simply assume that one of them is compactified with a vanishing radius, then it is clear that an infinite number of windings is not accounted for, which is the opposite to what we did in (63). This implies that in the T-dual system with an infinite radius, the contribution from the corresponding infinite number of momentum modes, which produces a continuous momentum in the infinitely large radius limit, to the amplitude has not been accounted for (This contribution gives rise to one of the factors given in the second equation in (64)). As a result, this kind of T-duality does not work for the decompactified amplitude if it is performed along a longitudinal direction.

Here, we provide an explicit demonstration of the T-duality acting on the non-compactified or the decompactified cylinder amplitude (65) along a transverse direction to both the branes. To achieve this, we first write $y^2=y_{p+1}^2+y_{p+2}^2+\cdots y_9^2=y_{p+1}^2+{\overline{y}}^2$. We will perform a T-duality along the $y_{p+1}$-direction. As such, we first need to place the $\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}$ system periodically along this direction for an effective compactification along this direction. This can be achieved by setting D(p − 2n) and Dp as

$${\overrightarrow{y}}_{\mathrm{D}(\mathrm{p}-2\mathrm{n})}=2\pi ka{\widehat{e}}_{p+1},k=0,\pm 1,\pm 2,\cdots ,$$

(70)

and

$${\overrightarrow{y}}_{\mathrm{Dp}}=\overrightarrow{\overline{y}}+(y_{p+1}+2\pi la){\widehat{e}}_{p+1},l=0,\pm 1,\pm 2,\cdots ,$$

(71)

where ${\widehat{e}}_{p+1}$ is the unit vector along $y_{p+1}$, and a is the radius of compactification with $a\to 0$. We then have

$${\overrightarrow{y}}_{k,l}={\overrightarrow{y}}_{\mathrm{Dp}}-{\overrightarrow{y}}_{\mathrm{D}(\mathrm{p}-2\mathrm{n})}=\overrightarrow{\overline{y}}+\left[y_{p+1}+2\pi a(l-k)\right]{\widehat{e}}_{p+1}.$$

(72)

As a result, we now have

$$y_{k,l}^2={\overline{y}}^2+{\left[y_{p+1}+2\pi a(l-k)\right]}^2.$$

(73)

Then, the total cylinder amplitude between the periodic D(p − 2n) and the periodic Dp with ${\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}^{(k,l)}$ given by (64) is

$$\begin{array}{ccccc}\hfill \Gamma & =& \sum _{k,l}{\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}^{(k,l)}\left(y_{k,l}\right)\hfill & & \\ & =& {\displaystyle \frac{2^{2-n}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{9-p}{2}}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\sum _{k,l}{e}^{-{\displaystyle \frac{y_{k,l}^2}{2\pi {\alpha }^{\prime }t}}}.\hfill \end{array}$$

(74)

As stressed earlier, the current approach is valid only when we take the compactified radius $a\to 0$, since the momentum modes along the compactified circle have not been taken into consideration. Note that in the above integrand, we have the sum

$$\sum _{k,l}{e}^{-{\displaystyle \frac{y_{k,l}^2}{2\pi {\alpha }^{\prime }t}}}=\sum _{k,l}{e}^{-{\displaystyle \frac{{\overline{y}}^2+{\left[y_{p+1}+2\pi a(l-k)\right]}^2}{2\pi {\alpha }^{\prime }t}}}=e^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}\sum _{k,l}{e}^{-{\displaystyle \frac{{\left[y_{p+1}+2\pi a(l-k)\right]}^2}{2\pi {\alpha }^{\prime }t}}}.$$

(75)

If we set $m=l-k$, then the above sum can be expressed as

$$\sum _{k,l}{e}^{-{\displaystyle \frac{y_{k,l}^2}{2\pi {\alpha }^{\prime }t}}}=N_{\infty }{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}\sum _m{e}^{-{\displaystyle \frac{{\left(y_{p+1}+2\pi am\right)}^2}{2\pi {\alpha }^{\prime }t}}},$$

(76)

where $N_{\infty }={\sum }_k$. We now set $z=y_{p+1}+2\pi am$ and $dz=2\pi a\to 0$ if $a\to 0$, so we can pass

$$\sum _m{e}^{-{\displaystyle \frac{{\left(y_{p+1}+2\pi am\right)}^2}{2\pi {\alpha }^{\prime }t}}}={\int }_{-\infty }^{\infty }{\displaystyle \frac{dz}{2\pi a}}{e}^{-{\displaystyle \frac{z^2}{2\pi {\alpha }^{\prime }t}}}={\displaystyle \frac{\sqrt{2{\pi }^2{\alpha }^{\prime }t}}{2\pi a}}.$$

(77)

We then have the total amplitude

$$\Gamma =N_{\infty }{\displaystyle \frac{{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{1/2}}{2\pi a}}{\displaystyle \frac{2^{2-n}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right).$$

(78)

Therefore, the cylinder amplitude between a D(p − 2n) and a Dp with such a transverse compactification with $a\to 0$ is given as

$$\begin{array}{ccccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}\left(\overline{y}\right)& =& {\displaystyle \frac{\Gamma }{N_{\infty }}}={\displaystyle \frac{{\left(2{\pi }^2{\alpha }^{\prime }\right)}^{1/2}}{2\pi a}}{\displaystyle \frac{2^{2-n}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\hfill & & \\ & =& {\displaystyle \frac{1}{2\pi a}}{\displaystyle \frac{2^{2-n-1}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-1}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right),\hfill \end{array}$$

(79)

which corresponds precisely to the $l=1,k=0$ case, when $a\to 0$, given in (63) in the previous section. This also clearly shows that the low-energy T-duality only works when $a\to 0$ (otherwise, the momentum modes must be taken into consideration). T-duality along this direction amounts to setting $a\to \overline{a}={\alpha }^{\prime }/a\to \infty$ in the above, for which we have

$$\begin{array}{ccc}\hfill {\Gamma }_{\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}}& \to & {\displaystyle \frac{\overline{a}}{2\pi {\alpha }^{\prime }}}{\displaystyle \frac{2^{2-n-1}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-1}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}-2\mathrm{n})|\mathrm{p}}\left(t\right)\hfill \\ & =\hfill & {\displaystyle \frac{2\pi \overline{a}}{4{\pi }^2{\alpha }^{\prime }}}{\displaystyle \frac{2^{2-n-1}{V}_{1+p-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+p-2n-1}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)\\ & =& {\displaystyle \frac{2^{2-n}{V}_{1+(p+1)-2n}}{{\left(8{\pi }^2{\alpha }^{\prime }\right)}^{\frac{1+(p+1)-2n}{2}}}}{\int }_0^{\infty }{\displaystyle \frac{dt}{t^{\frac{8-p}{2}}}}{e}^{-{\displaystyle \frac{{\overline{y}}^2}{2\pi {\alpha }^{\prime }t}}}{Z}_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)\hfill \\ & =& {\Gamma }_{\mathrm{D}(\mathrm{p}+1-2\mathrm{n})\left|\mathrm{D}\right(\mathrm{p}+1)}\left(\overline{y}\right),\hfill \end{array}$$

(80)

where, in the last equality, we have set $V_{1+(p+1)-2n}=V_{1+p-2n}2\pi \overline{a}$ with $\overline{a}\to \infty$ and also $Z_{(\mathrm{p}+1-2\mathrm{n})\left|\right(\mathrm{p}+1)}\left(t\right)=Z_{(p-2n)|p}\left(t\right)$. This result is the expected one. In other words, we can apply this approach to the cylinder amplitude for the initial $D0\left|D\right(2n)$ system with a T-duality once a time along one of the directions transverse to both the branes in order to obtain the general cylinder amplitude for a non-compactified $\mathrm{D}(\mathrm{p}-2\mathrm{n})|\mathrm{Dp}$ system with $2n\le p\le 8$. However, this does not work in the reverse direction, i.e., starting from the cylinder amplitude for the $\mathrm{D}(8-2\mathrm{n})|\mathrm{D}8$ system.

5. Conclusions and Discussion

In this paper, we compute, for the first time, the closed string tree cylinder amplitude between a D(p − 2 n)-brane and a Dp-brane with $n=0,1,2,3$ and $2n\le p\le 8$ for which there are $k\le p-2n$ longitudinal circle compactifications common to both branes with the respective radii $r_{{\alpha }_1},r_{{\alpha }_2},\cdots r_{{\alpha }_k}$ and $l\le 9-p$ transverse circle compactifications with the respective radii $R_{j_1},R_{j_2},\cdots R_{j_l}$. These two branes are placed parallel at a separation of ${\widehat{y}}^{\perp }$ along the non-compactified directions transverse to both branes and are also along each of the compactified transverse circles with a separation of $|y^{j_m}|$. Note that there are Wilson lines $y^{{\alpha }_i}$ turned on along the Dp-brane worldvolume, but there are no such lines on the D(p − 2n)-brane for the sake of simplicity.

With this amplitude, when the decompactification limits are taken, it gives the corresponding known non-compactified amplitude. We explicitly check T-duality along either a longitudinal or a transverse direction to both of these branes, and the results meet our expectations. We explain the validity of the usual low-energy T-duality, which is usually employed, for example, in obtaining different supergravity D-brane configurations from a given one and in obtaining the non-compactified cylinder amplitude when it is applied to a transverse direction to both of the branes while taking the compactified radius to vanishing. The rationale behind this is that the low-energy T-duality does not take the momentum and winding modes into consideration, and this can only be good if the compactification circle is taken along a transverse direction, and the radius of this circle is taken to vanish, as explained in the previous section.