(m, n)-String in (p, q)-string and (p, q)-five-brane background
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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.
KeywordsFundamental String Supergravity Solution Primary Constraint Stress Energy Tensor String Background
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 . Another possibility is to generate new solutions using the U-duality symmetry of M-theory (for a review see for example ), 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  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  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 . It was also shown in  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  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 . 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 .
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  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  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 .
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
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.
3 (m, n)-string in the background of (p, q)-five brane
3.1 Gauge fixing in Hamiltonian formalism
3.2 The Case \(L=0\)
3.3 The case \(L\ne 0\)
4 (m, n)-String in (p, q)-string background
4.1 The case \(L=0\)
4.2 The case \(L\ne 0\)
This work was supported by the Grant Agency of the Czech Republic under the Grant P201/12/G028.
- 3.K.S. Stelle, Lectures on supergravity p-branes, in Trieste 1996, High energy physics and cosmology, pp. 287–339. arXiv:hep-th/9701088
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- 14.D. Kutasov, D-brane dynamics near NS5-branes. arXiv:hep-th/0405058
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