$SL(2, Z)$ invariant rotating $(m,n)$ strings in $AdS_3\times S^3$ with mixed flux

We study rigidly rotating and pulsating $(m,n)$ strings in $AdS_3 \times S^3$ with mixed three form flux. The $AdS_3 \times S^3$ background with mixed three form flux is obtained in the near horizon limit of $SL(2,Z)$-transformed solution, corresponding to the bound state of NS5-branes and fundamental strings. We study the probe $(m,n)-$string in this background by solving the manifest $SL(2,Z)-$covariant form of the action. We find out the dyonic giant magnon and single spike solutions corresponding to the equations of motion of a probe string in this background and find out various relationships among the conserved charges. We further study a class of pulsating $(m,n)$ string in $AdS_3$ with mixed three form flux.


Introduction
Integrability in string theory has been proved to be one of the most useful techniques in studying string spectrum in various semisymmetric superspaces [1] 1 . The appearance of integrability on both sides of the AdS/CFT correspondence [3,4,5] has added tremendous amount of progress in the study of string theory. In this context, type IIB superstring theory on AdS 5 × S 5 has been shown to be described as supercoset sigma model [6]. The appearance of integrability via appearance of hidden charges was first exploited in [7]. With the realization that the counting of gauge invariant operators from gauge theory side can be elegantly formulated in terms of an integrable spin chain, it has been established that integrability played an important role on both sides of the duality, since the dual string theory is integrable in the semiclassical limit. In this connection, a special limit was put forth using in which both sides of the duality were analyzed in great detail. In particular, the spectrum on the field theory side was shown to consist of elementary excitations, the so called magnons which carry momentum p along the finitely or infinitely long spin chain. On the string theory side, the dual string state derived from the rigidly rotating string in the R × S 3 appears to give the same dispersion relation between the string energy (E) and the angular momentum (J) in the large 't Hooft limit and is known as the giant magnon [8]. A more general kind of rotating strings, known as spiky strings, are dual to higher twist operators also presented in [9]. It was further argued that they both fall into the category outlook.

Rotating (m, n)−string in AdS 3 × S 3 with mixed flux
We begin this section with a review of the construction of AdS 3 × S 3 background with mixed three form fluxes that was performed recently in [30]. The starting point is the AdS 3 × S 3 background with NS-NS two form 3 where ds 2

AdS 3
is the line element of AdS 3 space expressed in dimensionless variables. It is well known that the solution given in (2.1) is a solution of type IIB supergravity equations of motion. On the other hand, we also know that type IIB superstring theory possesses SL(2, Z)−duality transformation that leaves the metric in the Einstein frame unchanged. In case of two forms, it is convenient to introduce the vector B defined as where B and C (2) are NSNS and RR two forms respectively. The vector B transforms under SL(2, Z) transformation asB and where a, b, c, d are integers. Type IIB theory also has two scalar fields χ and Φ, where the dilaton Φ is in the NS-NS sector while χ belongs to the RR sector. It is convenient to combine these fields into a complex field τ = χ + ie −Φ and introduce the following matrix that transforms under SL(2, Z) transformation aŝ where Λ is given in (2.4). 3 We ignore the part of the metric corresponding to four torus T 4 with the volume V4.
Then in order to find AdS 3 × S 3 background with mixed three form fluxes, we perform SL(2, Z) transformation of the ansatz (2.1) and we obtain the line element in the form [30] where ds 2 T = dx 2 6 + · · · + dx 2 9 . We see that the new solution has the curvature radius Further, there are the following NSNS and RR three formsH and dilaton and zero form RR field as Our goal is to study the dynamics of the probe (m, n)−string in this background.
To do this, we introduce the action for (m, n)-string that has the form 4 where m, n count the number of fundamental string (m) and D1-branes (n) and hence they have to be integers. It is important that the action (2.10) is manifestly invariant under SL(2, Z) transformation when B and M transform as in (2.2) and (2.6) and when m transforms aŝ m = Λm . (2.12) Note that this action is expressed using Einstein frame metric g M N which is related to the string frame metric G M N by the relation g M N = e −Φ/2 G M N where the Einstein frame metric is invariant under SL(2, Z)-transformations. Then it was shown in [30] that (m, n)−string in mixed AdS 3 ×S 3 background with mixed three form fluxes can be mapped to (m ′ , n ′ )−string in AdS 3 × S 3 background with NSNS three form flux. Explicitly, using the manifest covariance of the action (2.10) we obtain where G M N , B M N and Φ N S correspond to the background (2.1) and we used the fact that We see that we reduced the problem of the dynamics of (m, n)−string in mixed AdS 3 × S 3 background to the much simpler analysis of (m ′ , n ′ )−string in pure NSNS AdS 3 × S 3 background where the action is given in (2.13). On the other hand, this action is nonlinear due to the presence of the square root of the determinant that makes the analysis of equations of motion rather awkward. For that reason it is useful to rewrite this action into Polyakov-like form when we introduce an auxiliary metric γ αβ and write the action S (m,n) into the form is the charge vector of (d, −b)−flux background. Note that we have also used the fact that Φ N S is constant for the background (2.1). To see an equivalence between (2.16) and Nambu-Goto form of the action, note that the equations of motion for γ αβ have the form that has clearly a solution γ αβ = G αβ . Inserting this solution into (2.16) we obtain the original action. In the following we use the Polyakov form of the action due to the manifest linearity of the theory. The equations of motion with respect to γ have been already determined in (2.18) while the equations of motion with respect to x M can be easily determined where Our goal is to find solutions of the equations of motion derived above that correspond to giant magnon or single spike configurations. For that reason it is convenient to use the following explicit form of the background metric (2.1) where we used ordinary symbols for coordinates instead of symbols with tilde used in (2.1) keeping in mind that all coordinates are dimensionless. Note that due to the fact that the action does not depend explicitly on φ, φ 1 , φ 2 and t, the action is invariant under constant shifts where all ǫ ′ s are constants. With the help of the standard Noether theorem we derive the following conserved currents where ǫ τ σ = −ǫ στ = 1. Note that these currents obey the relations Using these relations we derive the following conserved charges (2.25) Now we try to solve the equations of motion explicitly when we consider the following ansatz where y is a function of world sheet coordinates y = ασ + βτ , together with ρ = 0 and φ = 0. At the same time we impose the conformal gauge when γ τ τ = −1 , γ σσ = 1 , γ τ σ = 0. In this case the components of the stress energy tensor have the form The equation of motion for φ 1 implies where Φ 1 = const. and g ′ 1 = ∂g 1 ∂y In the same way the equation of motion for φ 2 implies (2.30) From these two equations we can see one important point that the case of the single angular momentum, i.e., ω 2 = 0 , g 2 = 0 is possible only when q (m,n) = 0 as follows from the equation of motion for φ 2 .
In order to find the equation of motion for θ we use the constraint T τ τ = 0 and we obtain This equation simplifies considerably when we impose the boundary condition that for θ → π 2 , θ ′ = 0. Since lim θ→ π 2 cos θ = 0, we have to demand that Φ 2 = 0 and the previous equation implies that can be solved for γ. Let us now consider the constraint T τ σ = 0 that implies that for θ = π/2, θ ′ (π/2) = 0 implies and we have to analyze under which condition this equation is obeyed. The first possibility is that g ′ 1 | θ=π/2 = 0 and using (2.29) we find that this is possible when The second possibility how to obey (2.33) is to demand that ω 1 + βg ′ 1 | θ=π/2 = 0 which implies These two values of the constants Φ I 1 , Φ II 2 determine whether we have giant spike or giant magnon solution. Before we proceed to the discussion of the general case with two angular momenta, we consider the simpler case of single angular momentum.

Single Angular Momentum
Let us now consider the case when g ′ 2 = 0 and ω 2 = 0. As we argued previously this is possible on condition when q (m,n) = 0 too. In this case we have while the constraint T τ τ = 0 takes the form If we impose again the condition that for θ = π 2 , θ ′ = 0 we obtain the equation that can be solved for γ. Further, the constraint T τ σ = 0 has the form that for θ = π/2, θ ′ (π/2) = 0 implies that can be solved for two values of Φ I,II and We begin with the first case.

Spike Solution
In the second case, we have Φ II 1 = − α 2 β ω 1 . Equation (3.3) gives γ 2 = α 2 β 2 ω 2 1 while the constraint T τ τ = 0 implies following differential equation for θ It is easy to see that the energy is equal to which is divergent. On the other hand note that J φ 1 is now finite and is equal to In order to find finite dispersion relation, let us determine the angle difference that is divergent. Then it is easy to find following dispersion relation E + κτ (m,n) L 2 △φ 1 = 2κτ (m,n) L 2 π 2 − θ 1 .

(3.17)
This is the dispersion relation corresponding to the single spike solution of the string.

Two Angular Momenta
In this section we consider a more interesting case of two non-zero angular momenta. Recall that in section (2) we determined that these solutions are characterized by condition Φ 2 = 0 and two values of Φ 1 given in (2.35) and (2.36). Let us begin with the first case.
The explicit form of the conserved charge E is which is divergent. As for the remaining conserved charges J φ 1 and J φ 2 , we get and and the angle difference  Collecting these results together we obtain the following dispersion relation (4.6) Using previous integral we evaluate the right side of the equation above and we obtain and hence we derive the final form of the dispersion relation This dispersion relation is the generalization of the dispersion relations derived in [18] and also in [21] to the case of (m, n)−string in (d, −b)−mixed flux background. We see that this dispersion relation is linear in the △φ 1 that is identified with the world-sheet momentum p which spoils periodicity of this solution. On the other hand it is clear that this dispersion relation reduces to the usual giant magnon dispersion relation when m T q = 0 and also n ′ = 1 that corresponds to (b, d)−string in (d, −b)−flux background and the dispersion relation has the form that again corresponds to SL(2, Z)−rotation of the dispersion relation of D1-brane in pure NSNS flux background. On the other hand it is interesting to analyze dispersion relation when n ′ = 0 that implies q (m,n) = τ (m,n) that corresponds to m = an c . If we again consider the case when m ′ = 1 we find that this corresponds to (a, c)−string and we find that the dispersion relation has the form (4.10) We interpret this solution as the bound state of J φ 2 elementary magnons so that for J φ 2 = 1 we obtain massless dispersion relation that has nice physical interpretation. The (a, c)−string in (d, −b)−flux background is defined by SL(2, Z)−rotation of fundamental string in pure NSNS flux background, where the matrix Λ is given in (2.4). On the other hand we know that fundamental string in AdS 3 × S 3 with NSNS flux has exact WZW conformal field theory description with massless dispersion relation [18].

Second Limiting Case
Now, we consider the second case with Φ II τ (m,n) αω 2 ). From (2.32) we again find γ 2 = α 2 β 2 ω 2 1 Using (2.31) we find that the differential equation for θ has the form where Using (4.12) we determine the following conserved charges together with the angle difference dθ sin θ cos θ sin 2 θ − sin 2 θ 0 (4.14) From (4.13) and (4.14) we see that E and △φ 1 are both divergent but their combination is finite and is equal to Further, the dispersion relation between angular momenta can be written as where (△φ 1 ) reg = −2 cos −1 (sin θ 0 ). Note that (4.16) could be rewritten using the original variables m and M and we could discuss the various properties of this relation with dependence on the values of vector m exactly in the same way as in previous section but we will not repeat it here since the discussion would be exactly the same.

Pulsating (m, n)−string in AdS 3 with mixed flux
In this section we will analyze the pulsating (m, n)−string in mixed three form flux background. Recall that such a string has an action 2) and the equations of motion for γ αβ In order to find pulsating (m, n)−string in AdS 3 , we consider the following ansatz Then it is easy to find the form of the induced metric G τ τ = L 2 (− cosh 2 ρṫ 2 +ρ 2 ) , G σσ = L 2 κ 2 sinh 2 ρ , G τ σ = G στ = 0 .
On the other hand the equation of motion for ρ is more complicated and is equal to  Now using (5.7) we find that P t is equal to (5.10) The second important quantity is the oscillation number that is associated with string motion along the ρ direction where Π ρ is the canonical momentum conjugate to ρ Π ρ = τ (m,n) L 2ρ .

(5.12)
Now using the equation (5.6) and (5.7) we obtain differential equation for ρ in the forṁ and hence the oscillation number N is equal to Changing the variable x = sinh ρ, we get where R + and R − are the roots of the quadratic equation in the square root with .

Conclusion
In this paper, we have studied the rotating and pulsating (m, n)−type string in AdS 3 × S 3 background with mixed fluxes which has been obtained by taking the SL(2, Z) transformations of the usual (F 1 − N S5) bound state followed by a near horizon geometry. We have applied SL(2, Z) transformation on the (m, n)−probe string action and generated (m ′ , n ′ )−string action, where the m ′ and n ′ are the SL(2, Z) invariants. The giant magnon and its dyonic counterpart solutions have been obtained by solving relevant equations of motion of the probe string in the above background in the presence of NS-NS fluxes. We have shown various regularized dispersion relations among different conserved charges that the background admits. We have also checked that in the absence of probe D-string charges, the relations among various charges do match exactly with the F-string result. Furthermore, we have looked at an oscillating (m, n)−string in the background of AdS 3 with NS-NS flux. In short string limit, we have obtained the energy of such a string in terms of the oscillation number. The work done in this paper can be extended in several ways. One of the interesting problems to consider is to study the pulsating and circular string solutions of (m, n)−type string in the R × S 3 with the NS-NS flux turned on. A point to note, however, is that the pulsating and oscillating strings in R × S 3 is qualitatively different from the AdS 3 case. It is left as a further example for future work. In the context of obtaining the mixed flux background, one of the backgrounds which one might look for is the AdS 3 × S 3 × S 3 × S 1 with mixed flux. A way to do so would be to start with the N S1 − N S1 ′ − N S5 − N S5 ′ intersecting brane solution of [32] and then apply the SL(2, Z) transformation followed by a rotation and check the commutativity of the these two operations. At present, it appears to be a nice idea to pursue further.