# (*m*, *n*)-String in (*p*, *q*)-string and (*p*, *q*)-five-brane background

## Abstract

We study dynamics of (*m*, *n*)-string in (*p*, *q*)-five-brane and (*p*, *q*)-string background. We determine world-volume stress energy tensor and we analyze the dependence of the string’s dynamics on the values of the charges (*m*, *n*) and the value of the angular momentum.

## 1 Introduction and summary

Low energy effective actions of superstring theories have reached a spectrum of solutions that preserve some fractions of supersymmetry; for a review see for example [1, 2, 3, 4]. These objects have the property that they are sources of various form fields that are presented in supergravity theories. Further, fundamental string, D-brane, and NS5-brane solutions preserve one half of the space-time supersymmetries and can be considered as the building block of other solutions. For example, taking the intersection of these configurations we get backgrounds that preserve some fractions of supersymmetry [5]. Another possibility is to generate new solutions using the U-duality symmetry of M-theory (for a review see for example [6]), which is basically the symmetry of M-theory on its maximally supersymmetric toroidal compactifications. For example, M-theory compactified on a two torus possesses the *SL*(2, *Z*) symmetry, which turns out to be the non-perturbative *SL*(2, *Z*) duality of type IIB theory. More precisely, it is well known that the low effective action of type IIB supergravity written in an Einstein frame is invariant under *SL*(2, *R*) duality. A special case of *SL*(2, *R*) transformation is the S-duality transformation that roughly speaking transforms the theory at weak coupling to strong coupling. The fact that the type IIB supergravity action is invariant under this symmetry suggests the possibility to generate new supergravity solutions when we apply a *SL*(2, *R*) rotation on known supergravity solutions, as for example fundamental string or NS5-brane backgrounds. Such a procedure was first used in a famous paper [7] where the manifestly *SL*(2, *R*) covariant supergravity solution corresponding to a (*p*, *q*)-string was found. The extension of this analysis to the case of an NS5-brane was performed in [8] when the *SL*(2, *Z*) covariant expression for supergravity solutions corresponding to the (*p*, *q*)-five brane was derived.^{1} These backgrounds are very interesting and certainly deserve to be studied further. In particular, it is well known that the continuous classical symmetry group *SL*(2, *R*) of type IIB supergravity cannot be a symmetry of the full string theory when non-perturbative effects break it to a discrete subgroup *SL*(2, *Z*). To see this more clearly, note that the fundamental string carries one unit of NSNS two-form charge and hence this charge has to be quantized in integer units. On the other hand *SL*(2, *R*) transformations map a fundamental string into a string with *d* units of this charge where *d* is an entry of the *SL*(2, *R*) matrix. From this result we conclude that *d* has to be integer. In a similar way we can argue that the *SL*(2, *R*) symmetry of the low energy effective action has to be broken to its *SL*(2, *Z*) subgroup when a fundamental string is mapped under this duality to a (*p*, *q*)-string that carries charge *p* of NSNS two-form and charge *q* of the Ramond–Ramond two-form [9]. It was also shown in [9] that the type IIB string effective action together with the (*p*, *q*)-string action is covariant under *SL*(2, *R*) transformations. However, the fact that the (*p*, *q*) string has to map to another \((p',q')\)-string where \(p',q'\) are integers suggests that the full symmetry group of the combined action breaks to *SL*(2, *Z*). On the other hand, solutions found in [7, 8] were determined using the *SL*(2, *R*) matrices so that it is interesting to analyze the problem of an (*m*, *n*)-string probe in such a background and this is precisely the aim of this paper.

We begin with the D1-brane action that we rewrite into a manifestly covariant *SL*(2, *Z*) form; for a related analysis see [10] and for a very elegant formulation of the manifestly *SL*(2, *Z*) covariant superstring, see [11, 12]. Now using the fact that (*p*, *q*)-five and fundamental string solutions were derived using *SL*(2, *R*) transformations we can map the problem of the dynamics of the (*m*, *n*)-string in this background to the problem of the analysis of the \((m',n')\)-string in the original NS5-brane and fundamental string background with the crucial exception that the harmonic functions that define these solutions have constant factors that differ from the factors that define NS5-brane and fundamental string solutions. It is also important to stress that now \((m',n')\) are not integers but depend on *p*, *q* and also on asymptotic values of the dilaton and Ramond–Ramond zero-form. We think that this is not a quite satisfactory resort and one can ask the questions whether it would be possible to find (*p*, *q*)-string and five-brane backgrounds that are derived from the NS5-brane and fundamental string background through manifest *SL*(2, *Z*) transformations when the probe (*m*, *n*)-string will transform in an appropriate way. This problem is currently under study and we return to it in the near future. We rather focus on the dynamics of the probe (*m*, *n*)-string in the backgrounds [7, 8], following the very nice analysis introduced in [14]. Using a manifest *SL*(2, *Z*) covariant formulation of a probe (*m*, *n*)-string we can analyze the time evolution of the homogeneous time-dependent string in a given background. We determine the components of the world-sheet stress energy tensor and study its time evolution. The properties of this stress energy tensor and the dynamics of the probe depend on the values of *m*, *n* and hence our results can be considered as a generalization of the analysis performed in [14].

As the next step we analyze the dynamics of the probe (*m*, *n*)-string in the background of (*p*, *q*)-macroscopic string. Thanks to the form of the solution [7] we formulate this problem as the analysis of the dynamics of \((m',n')\)-string in the background of fundamental string. This problem was studied previously in [15] but we focus on a different aspect of the dynamics of the probe. Explicitly we will be interested in the behavior of the probe where the difference between its energy and the rest energy is small. We find that the potential is flat, which is in agreement with the fact that the string probe in the fundamental string background can form a marginal bound state with the strings that are sources of this background. We also analyze the situation with a non-zero angular momentum and we find that there is a potential barrier that does not allow the probe string to move towards to the horizon. These results are in agreement with the analysis performed in [15].

The organization of this paper is as follows. In the next section (Sect. 2) we review *SL*(2, *R*) duality of the type IIB low energy effective action. We also introduce a manifestly *SL*(2, *R*) covariant action for (*m*, *n*)-string. In Sect. 3 we study the dynamics of this string in the background of a (*p*, *q*)-five brane. Finally in Sect. 4 we study the dynamics of the (*m*, *n*)-string in the background of a (*p*, *q*)-string.

## 2 *SL*(2, *R*)-Covariance of type IIB low energy effective action

*H*corresponds to the NSNS three-form, while

*F*belongs to the RR sector and does not couple to the usual string world-sheet. Type IIB theory has also two scalar fields, which can be combined into a complex field \(\tau = \chi +\mathrm{ie}^{-\Phi }\). The dilaton \(\Phi \) is in the NSNS sector, while \(\chi \) belongs to the RR sector. The other Bose fields are the metric \(g_{\mu \nu }\) and the self-dual five-form field strength \(F_5\), which we set zero in this paper. Then it is possible to write down a covariant form of the bosonic part of type IIB effective action,where \(\tilde{\kappa }^2_{10}=\frac{1}{4\pi }(4\pi ^2\alpha ')^4\) and where we have combined \(B,C^{(2)}\) into

*SL*(2,

*R*) transformation

*SL*(2,

*R*) covariance of type IIB effective action it is possible to derive solutions corresponding to the (

*p*,

*q*)-five brane [8] and fundamental string [7]. It will be certainly interesting to analyze the properties of a given background with the help of the appropriate probe, which will be a probe (

*m*,

*n*)-string. For that reason we introduce a manifestly covariant form of the (

*m*,

*n*)-string action.

### 2.1 (*m*, *n*)-String action

In this section we formulate the action for the (*m*, *n*)-string. Even if such a formulation is well known [9, 10, 11, 12, 13] we derive this action in a slightly different way with the help of the Hamiltonian formalism which will also be useful for the analysis of the dynamics of the probe (*m*, *n*)-string in (*p*, *q*)-five and (*p*, *q*)-string background.

*n*coincident D1-branes in a general background,

*m*,

*n*)-string is derived when we fix the gauge generated by \(\mathcal {G}\) with the gauge fixing function \(A_\sigma =\mathrm {const}\). Then the fixing of the gauge implies that \(\pi ^\sigma =f(\tau )\), but the equation of motion for \(\pi ^\sigma \) implies that \(\partial _\tau \pi ^\sigma =0\) and hence \(\pi ^\sigma =m\), where

*m*is an integer that counts the number of fundamental strings bound to

*n*D1-branes. After this partial gauge fixing the Hamiltonian density has the form

*m*,

*n*)-string action we derive the Lagrangian density corresponding to the Hamiltonian (15). Explicitly, from (15) we obtain equations of motion for \(x^M\)

*SL*(2,

*R*) form,

*SL*(2,

*R*) covariant. On the other hand, since

*m*,

*n*count the number of fundamental strings and D1-branes and hence have to be integers, we find that the non-perturbative duality group of type IIB superstring theory is

*SL*(2,

*Z*), which will have an important consequence for the analysis of the dynamics of (

*m*,

*n*)-string in (

*p*,

*q*)-five-brane and (

*p*,

*q*)-fundamental string background.

## 3 (*m*, *n*)-string in the background of (*p*, *q*)-five brane

*m*,

*n*)-string in the background of a (

*p*,

*q*)-five brane that has the form [8]

*p*,

*q*)-five brane. Further \(A_{(p,q)}\) is defined as

*m*,

*n*)-string action (19) in a given background. The analysis of this problem simplifies considerably when we realize how the solution (22) was determined. Following [8] and [7] we introduce the

*SL*(2,

*R*) matrix

*m*,

*n*equal to

*m*,

*n*)-string in a (

*p*,

*q*)-five-brane background is equivalent to the action of \((m',n')\)-string in an NS5-brane background with the important exception that the harmonic function has the factor \(Q_{(p,q)}\) (23) instead of the standard one, which corresponds to the number of NS5-branes. Note also that \(m',n'\) depend on

*m*,

*n*,

*p*,

*q*and the moduli \(\Phi _0\) and \(\chi _0\) as follows from (30). The crucial point, however, is that \(m',n'\) are not integers, which suggests inconsistency with the (

*p*,

*q*)-five-brane background. We come to this important observation later.

*m*,

*n*)-string in this background. It is convenient to impose the static gauge

*p*,

*q*)-five brane. Then we define the components of the two dimensional stress energy tensor as

### 3.1 Gauge fixing in Hamiltonian formalism

*m*,

*n*)-string in (

*p*,

*q*)-five-brane background. It turns out that it is useful to perform this analysis in the canonical approach when we impose the static gauge using the two gauge fixing functions

*p*,

*q*)-five branes. Since they are conserved we restrict ourselves to the case when \(p_\alpha =0\). At the same time we find that \(p_\phi \) is conserved as well and we denote this constant as \(p_\phi =L\). On the other hand the equations of motion for \(\theta ,p_\theta \) have the form

*r*. The equation of motion for

*r*gives

*E*. Then we can solve \(\mathcal {H}_\mathrm{fix}=E\) for \(p_r\) as

### 3.2 The Case \(L=0\)

*SL*(2,

*Z*) transformation these two objects transform differently. Explicitly, since the NS5-brane is a magnetically charged object with respect to the NSNS two-form it transforms in the same way as in (4). Then the (

*p*,

*q*)-five brane arises from the NS5-brane through the following

*SL*(2,

*Z*) transformation:

*SL*(2,

*Z*) transformation as in (21). Then we find that for \(\Lambda \) given in (61) we obtain an (

*a*,

*c*)-string where \((ap+qc=1)\). Since NS5-brane and fundamental string form a marginal bound state the previous arguments suggest that such a bound state exists also for the (

*p*,

*q*)-five brane and the (

*a*,

*c*)-fundamental string. Then the condition given in (59) is not in agreement with this claim. In other words the condition (59) says that there should exist a marginal bound state between (

*p*,

*q*)-five brane and (

*p*,

*q*)-fundamental string, which is not consistent with the

*SL*(2,

*Z*) duality of the type IIB string theory as was argued above. This is fact suggests that the (

*p*,

*q*)-five-brane background is not consistent from the probe point of view. Stated differently, the method of the constructions of (

*p*,

*q*)-five-brane and fundamental string backgrounds that was used in [7, 8] in fact does not lead to the correct form of the background from the string probe point of view. We mean that the resolution of this paradox can be found when we construct the background (

*p*,

*q*)-five-brane solution with the help of an

*SL*(2,

*Z*) transformation rather than the procedure used in [8], which was based on the

*SL*(2,

*R*) transformation. This question is now under active investigation and we hope to report our results soon.

### 3.3 The case \(L\ne 0\)

*L*. Following [14] we rewrite the equation of motion for (49) in the form

*r*. For small

*r*we obtain

*r*we have

*r*we have to require that the potential approaches zero from below, which implies

*r*. Further, the equation of motion for \(\phi \) implies

*r*.

*m*,

*n*)-string initially at a large distance from the (

*p*,

*q*)-five brane. The probe moves towards the (

*p*,

*q*)-five brane until it reaches the point when the effective potential vanishes, that is, at

*m*,

*n*)-string from the collection of (

*p*,

*q*)-five branes. Since the analysis is completely the same as in [14] we will not repeat it here.

## 4 (*m*, *n*)-String in (*p*, *q*)-string background

*m*,

*n*)-string in the macroscopic (

*p*,

*q*)-string background [7],

*p*,

*q*)-string solution was derived through an

*SL*(2,

*R*) transformation from the fundamental string background. Then clearly a fundamental string in a macroscopic string background maps to the same object under an

*SL*(2,

*Z*) transformation. On the other hand from (77) we see that this is not exactly true, since the probe string does not carry integer charge. We again leave the resolution of this paradox for future research.

*m*,

*n*)-string in static gauge. As in the previous section we temporarily replace the fixed two dimensional metric \(\eta _{\alpha \beta }\) with the two dimensional metric \(\gamma _{\alpha \beta }\) and write the gauge fixed action in the form

*r*and angular variable \(\phi \). As a result the Hamiltonian density (75) simplifies considerably:

*m*,

*n*)-string.

*E*is conserved we can express \(p_r\) as

### 4.1 The case \(L=0\)

*r*we obtain

*r*. Then the probe string cannot escape to infinity and it moves in the bounded region around the (

*p*,

*q*)-string background. On the other hand for \(E^2>T_{D1}^2(m'^2+ n^2)\) the potential is negative for all values and hence the probe string can move to infinity. Let us first consider the case when \(n'=0\). This case corresponds to the situation of the motion of fundamental string in the background of a collection of fundamental strings. We can expect that it is possible to form a marginal bound state of \(N+m'\) fundamental strings. In fact, for \(E-T_{D1}m'=\epsilon \ll 1\) we find that the effective potential has the form

*N*fundamental strings. Note that this approximation is valid on condition that

### 4.2 The case \(L\ne 0\)

*p*,

*q*)-string then it moves towards it, however, it cannot cross the horizon. Rather it approaches the distance given by \(r_+\) and then it is deflected. On the other hand the \(m'\)-string that is initially in the region below \(r_-\) will spirally move towards the the horizon. This situation is similar to the case of the (

*m*, 1)-string studied in [15] and we will not repeat it here.

## Footnotes

## Notes

### Acknowledgments

This work was supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028.

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