D-brane on Deformed AdS_3 times S^3

We study D1-brane in AdS_3 times S^3 kappa-deformed background with non-trivial dilaton and Ramond-Ramond fields. We consider purely time-dependent and spatially-dependent ansatz where we study the solutions of the equations of motion for D1-brane in given background. We find that the behavior of these solutions crucially depends on the value of the parameter a that was introduced in [arXiv:1411.1066 [hep-th]].


Introduction
The recent developments in the field of higher-dimensional extended objects have led to the deep understanding of the superstrings and supergravity theories. D-branes have been by now well understood both from the conformal field theory (CFT) and from the geometric, target-space viewpoint 1 . Such hyperplanes are dynamical rather than rigid and they are defined by the property that open strings can end on them [3]. The incorporation of such D-branes permits to argue that the different types of string theories are different states of a single theory, which also contain states with arbitrary configurations of D-branes. The dynamics of Dp-brane is governed by the action where S DBI is Dirac-Born-Infeld action and S W Z is Wess-Zumino action of the form where σ µ , µ = 0, . . . , p label world-volume of Dp-brane, Φ(x) is the dilaton and g αβ , b αβ given in (1.2) are the pull-backs of the target space metric and the NS-NS two form field to the world-volume of Dp-brane where x M (σ) are embedding coordinates of D1-brane. Finally, F αβ = ∂ α A β − ∂ β A α is the field strength for the world-volume gauge field A α . The coupling of Dp-brane to the Ramond-Ramond fields is expressed through the Wess-Zumino term (1.3) where it is understood that expressions given there are forms and the multiplications between them have the form of the wedge product. Motivated by the recent surge of interest in finding out the dualities in D-branes and fundamental strings bound states in anti-de Sitter space and gaining more insight into CFT, we study in this paper the dynamics of D1-branes described by Dirac-Born-Infeld action and Wess-Zumino terms. The corresponding target-space geometries are threedimensional κ-deformed AdS 3 × S 3 space-time. Very interesting class of deformations of target space-time have been introduced in [22,23] that preserve the integrability of the twodimensional quantum field theory on the world sheet 2 . In the κ-deformed anti-de Sitter background model, the metric is a direct sum of the deformed AdS n and S n parts and could be truncated from the ten-dimensional metric to κ-deformed AdS 3 ×S 3 for example [4]. The presence of the deformation parameter κ introduces new interesting results that reproduce the ordinary undeformed case in the limit κ → 0 as in [6]. Hence, it is interesting to study the dynamics of D1-brane in given background as well. In fact, recently a one-parameter model of the κ-deformed background AdS 3 × S 3 with non-trivial Ramond-Ramond (RR) forms and dilaton was proposed in [7]. A remarkable property of given background is that it depends on parameter a where it is presumed that a is a particular function of κ while the full solutions were constructed for the special values a = 0 and a = 1 only. In our present work, we will use these one-parameter backgrounds to analyze static and time-dependent solutions of D1-brane equations of motion in the background with non-trivial dilaton and with RR fields. Our analysis will reveal subtle features. Specifically, for the κ deformed AdS 3 background with a = 0 we will show that D1-brane does not see the presence of the singularity of the κ-deformed background and can reach ρ → ∞ limit. We also find that the static solutions are very simple deformations of the static solution known as AdS D1brane in AdS 3 background with RR flux. Since such a solution has not been found in the global coordinates before we present this result in the Appendix A. Moreover, Appendix B exhibits static solutions of pD1-branes in AdS 3 space-time with non-trivial B N S field. The case of (1, q) string was analyzed previously in [18], but we provide an extended analysis here for the (p, q) string in order to see the S-duality between given solution and the static D1-brane solution in undeformed AdS 3 space-time with Ramond-Ramond fields.
We also consider static and time-dependent solutions of D1-brane equations of motion for the κ-deformed background when the value of the parameter a is equal to 1. In this case we find D1-brane cannot cross the singularity ρ c = L κ and we also find that the static solutions is more complicated than in case a = 0.
The plan of this paper is as follows: In section (2) we will consider static D1-brane in the κ-deformed background AdS 3 × S 3 [7]. We study the solutions of the corresponding equations of motions and discuss the possibility of the D1-brane to reach the ρ → ∞ limit of the deformed AdS 3 × S 3 space. In section (3) we consider a time-dependent ansatz. In conclusion (4) we present summary of our results and their possible extension. Finally, some details of the calculations are summarized in the appendices. In appendix (A) we find static D1-brane solutions in non-deformed AdS 3 × S 3 background with non-trivial RR fields. In appendix (B), we find static solutions of pD1-branes in non-deformed AdS 3 space-time with non-trivial B N S field in global coordinates which is simple generalization of the solution found in [18].

D1-brane in κ-Deformed Background
In this section, we will study time-independent solutions of the equations of motion that follow from D-brane actions (1.2) and (1.3) in case when D1-brane is embedded in κdeformed AdS 3 × S 3 background [7] that has the form with non-trivial dilaton and Ramond-Ramond fields where L is the inverse curvature scale. Consider D1-brane in given background whose dynamics is governed by the action Generally, the equations of motion for x M that follow from given action have the form where ǫ τ σ = −ǫ στ = 1. On the other hand the equation of motion for A α implies where Π is constant that counts the number of fundamental strings. Using this result, we express F τ σ as Then the equations of motion (2.4) simplify as where T F 1 = 1 2πα ′ . This is the form of the equation of motion for D1-brane that we will be our starting point. If we now return to the specific background given in (2.2) , we see that since the background depends on r through its square we find that the equation of motion for constant r has the form δL that has solution r = 0. In the same way we can show that the equations of motions for ϕ and ψ have the solutions ϕ = ψ = 0. In other words we will not consider solutions with non-trivial behavior on deformed S (3) .

Static Solutions
Let us now consider the static D1-brane solution when we assume the following ansatz For this ansatz the equation of motion for x 0 = t is obeyed automatically. In order to solve for ρ it is more convenient to consider the equation of motion for χ since the background fields do not depend on χ explicitly. Then, we obtain (2.11) From given equation we derive the differential equation for ρ in the form (2.12) In the following, we will solve this equation for different background fields by considering two a-families.

The case a=0
Let us begin with the case a = 0 so that we have the following background fields Then (2.12) gives (2.14) This equation can be integrated at least in principle. However, when we choose C = L the given equation simplifies considerably that has a solution .
We choose the integration constant by requiring that D1-brane approaches ρ → ∞ for χ → 0. Hence, the final result is Surprisingly, we find that there is only mirror modification of the static solution of D1brane in AdS 3 background with non-trivial RR fields that is presented in appendix (A), where this modification is given by the presence of the deformation parameter κ. Further, we also see that D1-brane can be stretched through the singularity ρ c = L κ and can reach ρ → ∞. This is very remarkable result especially in the light of the solution that we find in case a = 1.

The case a=1
In this case, the relevant components of the background fields at r = 0 are We simplify this equation by choosing C = L √ 1+κ 2 . Hence, the previous equation has the form where Solving this equation, we get We choose the integration constant χ 0 in such a way that for ρ → ρ min , χ → π 2 . Then we obtain (2.23) From (2.22), we see that D1-brane does not reach the maximum value ρ max for χ in the interval χ ∈ (0, 2π). More precisely, D1-brane reaches the maximum value at ρ max for χ → −∞ that implies that D1-brane has to wrap compact χ direction infinitely many times. We also see that now D1-brane does not cross the singularity at ρ c = L κ . In summary, we see qualitative different behaviors of these two solutions corresponding to the cases a = 0 and a = 1. We will also see this difference in case of pure time-dependent solutions that will be analyzed in the next section.

Time-Dependent Solution
In order to find time-dependent solution, we consider an ansatz For such ansatz, we find that the equation of motion for χ is automatically obeyed. On the other hand, the equation of motion for t gives that implies following differential equation forρ In the following we will consider two different one-parameter a-families of background fields.

The case a=0
Substituting the fields of (2.13) in (3.4), we obtaiṅ Let us impose the condition C = L that simplifies the given equation considerably. For this condition, the turning point at whichρ = 0 will be at which is less than L κ . Hence, we see that in this case the D1-brane does not cross the singularity at ρ c = L κ . On the other hand let us consider the case when Π = 0. From (3.5), we obtaiṅ It seems interesting that now the expressions containing the deformation parameter κ disappear. The turning point is at this implies that in order to have real solution we have to demand that C < L 2 . For C = L 2 , we realize that the D1-brane reaches ρ → ∞ asymptotically. More explicitly, in such case we can easily integrate the differential equation with the result and we see that for t → ∞ D1-brane approaches (ρ = ∞).
Finally, for C > L 2 the D1-brane reaches ρ → ∞ since the expression under the square root is then always positive without any restrictions on the radial coordinate ρ. In other words, D1-brane with zero electric field can probe the space-time beyond the singularity ρ c = L κ as well.

The case a=1
In this case, substituting the fields of (2.18) in (3.4), the differential equation has the forṁ We are interested in the special case when Π = 0. In this case, we find the turning point at After the analysis of the expression under the square root (the discriminant), we realize that it is always positive. Then we have to consider two cases.
In the first case, A 2 − 1 L 2 < 0 then to have an overall positive quantity, we require a condition In the second case we have A 2 − 1 L 2 > 0 that gives In this case however we have also to demand that Now, however we find that the second condition is always obeyed since the second interval is included in the first. Therefore, there always exist real roots corresponding to the turning points ρ r.t. . Let us try to determine the value of the turning point for large C L ≫ 1 (large energy limit). In this case, we can write A ≈ − C L 2 κ 2 and we obtain so that when we restrict to the terms linear in L C ≪ 1 we obtain two roots In other words we find two situations. In the first case D1-brane is in the region below the singularity ρ 2 c = L 2 κ 2 and can reach its turning point at ρ 2 min and then it returns back. In the second case, D1-brane is in the region ρ 2 > ρ 2 max i.e. beyond the singularity. However, it is important that in both of these cases, D1-brane cannot go through the singularity.
Finally, we compare this result with the analysis of the time-dependent solution of the fundamental string in κ-deformed background. Recall that the fundamental string is described by the Nambu-Gotto action (3.17) The equation of motion for t for the time dependent ansatz again implies g tt g τ τ −g χχ (g tt + g ρρρ 2 ) = C , C = constant . (3.18) Solving given equation forρ we obtaiṅ Note that for large C, ρ − has the form We see that the allowed regions for the propagation of the string is (0, ρ 2 − ) and ( L 2 κ 2 , ∞). In other words, strings cannot cross the singularity at ρ 2 c = L 2 κ 2 when it is originally confined in the region around the point ρ = 0. On the other hand, for C L ≪ 1 we obtain ρ 2 − ≈ C 2 ≪ L 2 and the string is confined in the region around ρ = 0 there is no sign of the deformation of the target space-time.

Conclusion
In this paper we have studied the dynamics of D1-brane in κ-deformed AdS 3 × S 3 background with non-trivial dilaton and Ramond-Ramond fields [7]. We have found that the background with a = 0 possesses many interesting properties. We have shown that the static solution of D1-brane in the presence of RR-charges can reach ρ → ∞ limit of the deformed AdS 3 × S 3 space-time and that given solution is a slight modification from the AdS D1-brane solution in undeformed AdS 3 × S 3 background with Ramond-Ramond flux that is found in Appendix (A). Moreover, it is also very interesting that the time dependent solution does not see the presence of the singularity at ρ c = L κ . In other words, D1-brane in deformed AdS 3 × S 3 space-time can cross given singularity and reach ρ → ∞. Hence, D1-brane can be considered as natural probe of given space-time.
These results are in sharp contrast with the case a = 1 where we have shown that the D1-brane does not reach the singularity. Explicitly, it was shown that in such conditions the D1-brane can move in the region beyond the singularity or in a region below the singularity, but it can not cross the singularity in both situations. The latter result was confirmed by analyzing the dynamics of the fundamental string in given background. Again, it was demonstrated that a string originally confined in the region around ρ = 0 can not cross the singularity.
We have further examined the static gauge ansatz of pD1-branes bound to q fundamental strings with non-trivial NS-NS flux in global coordinates. After solving the equations of motion, we were able to generalize Bachas result [18] for the constant C that determines the radius of AdS. We have shown that C is proportional to the number of fundamental strings in the bound state and inversely proportional to the number of D1-branes. Finally, we considered the time-dependent solution of pD1-branes bound to q fundamental strings in the same background. We were able to show that in the limit T (p,q) → qT F the fundamental string can indeed reach ρ → ∞.
The present analysis can be extended in various directions. First of all it would be very interesting and challenging to study the dynamics of pD1-branes in the κ-deformed AdS 3 × S 3 with complex deformation parameter. Further one can try to find the complete solution with arbitrary parameter a then proceed with a similar analysis as we did in this paper. It would be also interesting to perform analysis of D1-brane and fundamental string configurations that could describe Wilson loops in dual field theory. We hope to return to these problems in future.

Acknowledgement:
This work was supported by the Grant agency of the Czech republic under the grant P201/12/G028.

A. D1-brane as Probe of AdS 3 × S 3 Background with Ramond-Ramond Background
In this appendix we consider the static solution of D1-brane equations of motions in the non deformed AdS 3 × S 3 background with non-zero Ramond Ramond field. Let us be more explicit and consider the case of the near horizon limit of D1-D5 brane system that in global coordinates has the form 3 φχ = Q 5 sin 2 θ , (A.1) 3 We follow the conventions used in [19,20].
where ϕ, φ, χ, y m ∈ [0, 2π] and θ ∈ [0, π] and where The equations of motion for D1-brane in given background have the form while the equation of motion for A α again implies Let us now presume an ansatz where the D1-brane is wrapping τ and ϕ directions and where ρ = ρ(σ), where we use the notation σ 0 = τ, σ 1 = σ keeping in mind that σ is dimensionless. For the ansatz (A.5), the equation of motion (A.3) for M = 0 is automatically satisfied while that of ϕ implies that can be solved for ρ ′ as and where now ρ ′ ≡ dρ dϕ . Let us now choose the constant C ′ in such a way that when the equation above simplifies considerably and hence we find the solution We again choose the integration constant that for ϕ → 0, the system approaches ρ → ∞ so that This is the solution corresponding to AdS D1-brane in AdS 3 background with non-trivial RR fields. In the next appendix we show that given configuration is S-dual to the specific bound state of D1-branes and fundamental strings in AdS 3 background with non-trivial B N S two form.
B. pD1-branes in AdS 3 with B N S field in global coordinates Let us now consider a collection of pD1-branes in AdS 3 background with the metric Since we are interested in the collective dynamics of the bound state of p D1-branes it is clear that given action is the standard DBI action multiplied with the number p so that Note that the equations of motion for x M that follow from given action have the form The equation of motion for A α implies where q is the number of fundamental strings bound to p D1-branes. Note that using the previous result we can express b τ σ + (2πα ′ )F τ σ as Let us now choose the following ansatz t = τ, χ = σ , ρ = ρ(σ) .

(B.7)
In this case we find that the equation of motion for t is automatically obeyed while the equation of motion for χ implies where C is a constant. From given equation we obtain differential equation for ρ ′ If we choose the integration constant C to be equal to we find simple differential equation for ρ which is the generalization of the solution found in [18] to the case of the bound state of p D1-branes and q fundamental strings. Note that the special case when we have p = Π D1-branes and q = 1 fundamental strings is S-dual to the solution found in the previous section which is the bound state of single D1-brane and Π fundamental strings. It is also instructive to consider time-dependent solution corresponding to the motion of the bound state of pD1-branes and q fundamental strings in given background when we consider an ansatz x 0 = τ , ρ = ρ(τ ) , χ = σ . (B.13) Then the equation of motion for t implies p 2 T 2 D1 + q 2 T 2 F 1 g tt g τ τ −detg αβ − qT F 1 b tχ = C (B.14) and hence we obtainρ = √ −g tt √ g ρρ 1 + (p 2 T 2 D1 + q 2 T 2 F 1 )g tt g χχ (C − qT F 1 b tχ ) 2 .
(B.15) -14 -For the background given in (B.1) and (B.2) we obtain that there is a turning point at Note that there is a special formal case when p = 0 when the turning point occurs at .

(B.17)
We see that given turning point is real when C < 2 qT F 1 L . We also see from (B.17) that the string can reach ρ → ∞ when In fact, it is easy to see that for C > C cr , the expression under the square root in (B.15) is always positive for all ρ. Hence, for C > C cr and for p the given configuration can always reach ρ → ∞. Finally, we would like to compare the given result with the analysis of the motion of fundamental strings in the given background. Recall that the dynamics of the classical string is governed by the Nambu-Gotto action Consider the time-dependent ansatz as in case of D1-brane x 0 = τ , χ = σ , g σσ = g χχ , g τ τ = g tt + g ρρ (ρ) 2 .

(B.21)
Then the equation of motion for χ is obeyed automatically while that for t implieṡ Now the expression on the right has turning point at Integrating both sides we obtain and we see that given string reaches ρ = ∞ in the limit t → ∞. It is also easy to see that for C > L/2 there is no turning point and fundamental string always reaches ρ → ∞ which is well known fact [18].