aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION
Semiclassical Strings on Curved Branes
Sagar Biswas
Department of Physics and Meteorology,Indian Institute of Technology Kharagpur,Kharagpur-721 302, INDIA [email protected]
Kamal L Panigrahi
Department of Physics & Meteorology,Indian Institute of Technology Kharagpur,Kharagpur-721302, INDIA,andThe Abdus Salam International Centre for Theoretical Physics,Strada Costiera 11, Trieste, ITALY [email protected]
Abstract:
We study semiclassical strings in the near horizon geometry of certain curvedbranes. We investigate the rigidly rotating strings in the near horizon geometry of NS5-branes wrapped on
AdS × S and in the presence of background NS-NS flux. We studyseveral string solutions corresponding to giant magnon, single spike and more general foldedstrings for the fundamental string in this background. We comment that in the S-dualbackground the situation changes drastically. Keywords:
AdS-CFT correspondence, Bosonic Strings. ontents
1. Introduction 12. Rotating String on Curved NS5-branes 2
3. Folded String 12 θ = θ = 0 133.2 For ρ = θ = 0 143.3 For ρ = θ = 0 15
4. Discussion and Conclusion 15
1. Introduction
According to AdS/CFT duality [1],[2],[3] quantum closed string states in AdS should bedual to quantum Super Yang-Mills (SYM) states on the boundary. More precisely, thisduality implies the equality between the AdS energy E of quantum closed string states (asfunction of effective string tension T and other quantum numbers like the angular momenta J i on the sphere) and the dimension ∆ of the corresponding local SYM operators. Thoughthe state-operator matching is extremely difficult, but has been tractable in certain limits,such as the large angular momentum limit, on both sides of the duality [4],[5]. Further,it was observed that N = 4 SYM in planar limit can be described by an integrable spinchain model where the anomalous dimension of the gauge invariant operators were foundin [6], [7], [8], [9], [10], [11], [12]. In the dual picture, it was noticed that the stringtheory is integrable in the semiclassical limit as well, see for example [13], [14], [15], [16],[17], hence providing further insight into the AdS/CFT duality. However apart from few‘solvable’ examples of AdS/CFT, in many cases the exact nature of the boundary operatorsis not known, and hence it is interesting to make calculations in the gravity side and thenlook for possible operators on the boundary by invoking the duality map. The study ofrigidly rotating strings in semiclassical approximation in the gravity side has been one ofthe interesting areas of research in the last few years. In this connection a large numberof rotating and pulsating string solutions have been studied in AdS × S , AdS × CP ,orbifolded and in the near horizon geometry of certain nonlocal string theory backgrounds,see for example, [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31],[32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48],– 1 –49]. However, more recently, the integrability of the classical string motion in curved p -brane background has been explored in [50] in an attempt to see whether the full stringequations of motion is integrable. It is shown that moving away from the throat geometryor in other words switching on the brane charges actually destroyed the string integrability.Though the point like string equations are in complete agreement with the integrability,the equations describing an extended string in the complete D-brane background, theintegrable structure is lost. It is definitely interesting to look for string equations of motionin connection with integrability in various other situations.Further to understand AdS/CFT like dualities in more general backgrounds, arisingout of near horizon geometries of various branes in supergravities it is also interestingto look for classical string equations of motion in the gravity side and make statementsabout the integrability. It might help us in making some observations about the dualtheory which is apriori less understood. Branes solutions with curved worldvolumes havewidespread applications in string theory and black holes. In the past they have been usedto identify the non-perturbative states of strings in lower dimensions in various stringcompactifications. The curved brane solutions can be constructed from the elementarysolutions of the NS-NS sector which are associated with conformally invariant sigma model[51],[52]. Indeed a large class of solutions have been constructed in [53] by using variousdualities in string theory. We would like to study semiclassical strings in some of thesebackgrounds. Specifically we wish to study rigidly rotating strings in the near horizongeometry of stack of NS5-brane with AdS × S worldvolume. We study the most generalform of the string equation of motion and solve for the giant magnon and spiky like strings.We further study a few general pulsating strings in this background. Finally, we make somecomments regarding F-string in the S-dual background, namely the nature of the solutionsto the F-string equations of motion on a D5-brane wrapped on AdS × S . We remark thatthe possible non appearance of the giant magnon or spike like solution is perhaps due to thenon integrability of the classical string equations of motion in the D5-brane background.The rest of the paper is organized as follows. In section-2, we study rigidly rotatingstrings on NS5-brane wrapped on AdS × S space. We find two limiting cases correspond-ing to giant magnon and single spike solutions for the string in this background. We presentthe regularized dispersion relations among various conserved charges corresponding to thestring motion. Section-3 is devoted to the study of pulsating strings in this background.In section 4, we make some remarks about the string motion in the D5-brane wrapped on AdS × S and conclude.
2. Rotating String on Curved NS5-branes
We start with the solutions presented in [53] that correspond to intersecting
N S − N S ′ − N S − N S ′ branes in supergravity. The details of this background is given by the followingform of the metric, 2-form Neveu-Schwarz (NS) field strength and dilaton [53] ds = g − ( x, y )( − dt + dz ) + H ( x ) dx n dx n + H ′ ( y ) dy m dy m , (2.1) dB = dg − ∧ dt ∧ dz + ⋆dH + ⋆dH ′ , – 2 – φ = H ( x ) H ( y ) g ( x, y ) , where [ H ′ ( y ) ∂ x + H ( x ) ∂ y ] g ( x, y ) = 0 . (2.2)A particular solution is given by g ( x, y ) = H ( x ) H ′ ( y ) , (2.3)with H , = 1 + Q , x , and H ′ , = 1 + Q ′ , y , where Q , etc correspond to the charges ofF1 and NS5-brane respectively. The above solution corresponds to the so called ”dyonicstring” generalization of the 5 NS + 5 NS solution in supergravity.To continue further let us choose H ′ = 1, in which the solution is just a direct productof N S
N S
N S − brane. Further, in the near horizon limitas x →
0, the metric becomes, ds = x Q ( − dt + dz ) + Q dx x + Q d Ω + H ′ ( y )( dy + y d Ω ′ ) (2.4)where we have set for simplicity H = H . The resulting sigma model describes a curvedNS5-brane wrapped on AdS × S and defines an exact CFT. We are interested in studyingrigidly rotating string on this NS5-brane in the near horizon geometry. In the near horizonlimit, y →
0, we get the following form of the metric and NS-NS B-field. ds = Q ′ ( − cosh ρdt + dρ + sinh ρdϕ + d Ω ) + Q ′ ( dy y + d Ω ′ ) , (2.5)with d Ω = dθ + sin θ dφ + cos θ dψ , and d Ω ′ = dθ + sin θ dφ + cos θ dψ . The above metric is further supported by a NS-NS two form field given by B = 2 Q ′ sin θ dφ ∧ dψ . Note that for convenience we have set Q = Q ′ . This background is also associated withan appropriate form of the dilaton whose explicit form will not be needed here. To proceedfurther we make the following change of variables χ = ln y . The final form of metric andbackground field now takes the form ds = Q ′ ( − cosh ρdt + dρ + sinh ρdϕ + dθ + sin θ dφ + cos θ dψ + dχ + dθ + sin θ dφ + cos θ dψ ) , B φ ψ = 2 Q ′ sin θ . (2.6)Note that we have used a different parameter to represent the AdS × S subspace. Westart by writing down the Polyakov action of the F-string in the above background, S = − √ λ π Z dσdτ [ √− γγ αβ g MN ∂ α X M ∂ β X N − e αβ ∂ α X M ∂ β X N b MN ] , (2.7)– 3 –here the ’t Hooft coupling √ λ = Q ′ , γ αβ is the worldsheet metric and e αβ is the antisym-metric tensor defined as e τσ = − e στ = 1. Under conformal gauge (i.e. √− γγ αβ = η αβ )with η ττ = − η σσ = 1 and η τσ = η στ = 0, the Polyakov action in the above backgroundtakes the form, S = − √ λ π Z dσdτ h − cosh ρ ( t ′ − ˙ t ) + ρ ′ − ˙ ρ + sinh ρ ( ϕ ′ − ˙ ϕ ) + θ ′ − ˙ θ + sin θ ( φ ′ − ˙ φ ) + cos θ ( ψ ′ − ˙ ψ ) + χ ′ − ˙ χ + θ ′ − ˙ θ + sin θ ( φ ′ − ˙ φ )+ cos θ ( ψ ′ − ˙ ψ ) − θ ( ˙ φ ψ ′ − ˙ ψ φ ′ ) i , (2.8)where ‘dots’ and ‘primes’ denote the derivative with respect to τ and σ respectively. Forstudying the rigidly rotating strings we choose the following ansatz, ρ = ρ ( y ) , t = τ + h ( y ) , ϕ = µ ( τ + h ( y )) ,θ = θ ( y ) , φ = ν ( τ + g ( y )) , ψ = ω ( τ + f ( y )) ,θ = θ ( y ) , φ = ν ( τ + g ( y )) , ψ = ω ( τ + f ( y )) , χ = κτ . (2.9)where y = σ − vτ . Variation of the action with respect to X M gives us the followingequation of motion2 ∂ α ( η αβ ∂ β X N g KN ) − η αβ ∂ α X M ∂ β X N ∂ K g MN − ∂ α ( e αβ ∂ β X N b KN )+ e αβ ∂ α X M ∂ β X N ∂ K b MN = 0 , (2.10)and variation with respect to the metric gives the two Virasoro constraints, g MN ( ∂ τ X M ∂ τ X N + ∂ σ X M ∂ σ X N ) = 0 ,g MN ( ∂ τ X M ∂ σ X N ) = 0 . (2.11)Next we have to solve these equations by the ansatz we have proposed above in eqn. (2.9).Solving for t, ϕ we get, ∂h ∂y = 11 − v [ c cosh ρ − v ] , ∂h ∂y = 11 − v [ c sinh ρ − v ] . (2.12)Substituting these for ρ equation we get,(1 − v ) ∂ ρ∂y = sinh ρ cosh ρ [(1 − c cosh ρ ) − µ (1 − c sinh ρ )] , (1 − v ) ( ∂ρ∂y ) = (1 − µ ) sinh ρ + c cosh ρ − µ c sinh ρ + c , (2.13)where c , c and c are integration constants as well. Similarly solving for φ and ψ equations we get, ∂g ∂y = 11 − v [ c sin θ − v ] , ∂f ∂y = 11 − v [ c cos θ − v ] . (2.14)– 4 –ubstituting these for θ equation we get,(1 − v ) (cid:18) ∂θ ∂y (cid:19) = ( ω − ν ) sin θ − ν c sin θ − ω c cos θ + c , (2.15)where c , c and c are integration constants. Again solving for φ and ψ equations weget, ∂g ∂y = 11 − v [ c ν sin θ − ω ν − v ] , ∂f ∂y = 11 − v [ c ω cos θ − ν ω − v ] . Substituting these in θ equation we get,(1 − v ) (cid:18) ∂θ ∂y (cid:19) = 3( ν − ω ) sin θ − c sin θ − c cos θ + c , (2.16)where c , c and c are integration constants as well. Now the Virasoro constraint T τσ = 0gives (1 − v ) h (cid:18) ∂ρ∂y (cid:19) + (cid:18) ∂θ ∂y (cid:19) + (cid:18) ∂θ ∂y (cid:19) i = cosh ρ − µ sinh ρ − ν sin θ − ω cos θ − ν − ω − ν cos θ − ω sin θ + c cosh ρ − µ c sinh ρ − ν c sin θ − ω c cos θ − c sin θ − c cos θ + 4( ω c + ν c )+ 1 + v v ( − c + µ c + ν c + ω c + ν c + ω c − ν ω ) (2.17)Further, the Virasoro constraint T ττ + T σσ = 0 gives(1 − v ) h (cid:18) ∂ρ∂y (cid:19) + (cid:18) ∂θ ∂y (cid:19) + (cid:18) ∂θ ∂y (cid:19) i = cosh ρ − µ sinh ρ − ν sin θ − ω cos θ − ν − ω − ν cos θ − ω sin θ + c cosh ρ − µ c sinh ρ − ν c sin θ − ω c cos θ − c sin θ − c cos θ + 4( ω c + ν c )+ 4 v v ( − c + µ c + ν c + ω c + ν c + ω c − ν ω ) − (1 − v ) κ v . (2.18)Subtracting the above two equations we get the following relation among various parame-ters, − c + µ c + ν c + ω c + ν c + ω c − ν ω + κ v = 0 . (2.19)In what follows we will look at the two limiting cases corresponding to giant magnon andsingle spike solutions for the string in the curved NS5-brane near horizon background. Recall, we have from (2.15) (cid:18) ∂θ ∂y (cid:19) = 1(1 − v ) (cid:20) ( ω − ν ) sin θ − ν c sin θ − ω c cos θ + c (cid:21) – 5 – θ ∂y → θ → π implies c = 0 and c = ν c + ν − ω , substituting this in the aboveequation we get, ∂θ ∂y = p ν − ω − v cot θ q sin θ − α , (2.20)where α = ν c ν − ω . Further, we have from (2.16) (cid:18) ∂θ ∂y (cid:19) = 1(1 − v ) (cid:20) ν − ω ) sin θ − c sin θ − c cos θ + c (cid:21) . Similarly, ∂θ ∂y → θ → π implies c = 0 and c = 3( ω − ν ) + c . Substituting this inthe above equation we get ∂θ ∂y = p ω − ν )1 − v cot θ q sin θ − α , (2.21)where α = c ω − ν ) . Substituting the values of ∂θ ∂y and ∂θ ∂y with c = c = 0 in firstVirasoro constraint (2.17) we get,(1 − v ) (cid:18) ∂ρ∂y (cid:19) = 1 + (1 − µ ) sinh ρ + c cosh ρ − µ c sinh ρ − α , (2.22)where α = ν (1 + c ) + 1 ν { c − µ c − ν c − κ v } − { κ (1 + v ) ++ κ v (1 + 4 ω ν ) + ν − ν ω } (2.23)In limit ∂ρ∂y → ρ → c = 0 and c = α −
1. Hence ∂ρ∂y = p − µ − v tanh ρ q cosh ρ + α , (2.24)where α = − α − µ . Looking at the symmetry of the background of the near horizon ofNS5-branes, a number of conserved charges can be constructed as follows E = − Z ∂ L ∂ ˙ t dσ = √ λ π − v Z (cosh ρ − c v ) dσ ,S = Z ∂ L ∂ ˙ ϕ dσ = √ λ π µ − v Z sinh ρdσ ,J = Z ∂ L ∂ ˙ φ dσ = √ λ π ν − v Z (sin θ − c v ) dσ ,J = Z ∂ L ∂ ˙ ψ dσ = √ λ π ω − v Z cos θ dσ ,K = Z ∂ L ∂ ˙ φ dσ = √ λ π − v Z (3 ν cos θ − ν − c v ) dσ , – 6 – = Z ∂ L ∂ ˙ ψ dσ = √ λ π − v Z (4 ω + 2 ν v − c − ω cos θ ) dσ ,P = Z ∂ L ∂ ˙ χ dσ = √ λ π κ Z dσ . (2.25)Also we have the following relation among various integration constants c = 1 ν [ c − ν c − κ v + 2 ν ω ] (2.26) Let us look at various solutions to the string equations of motion derived in the last sectionwith appropriate choice of integration constant. First, we choose c = c = v . Now theconserved quantities become, E = √ λ π − v Z (cosh ρ − v ) dσ, (2.27) S = √ λ π µ − v Z sinh ρdσ,J = √ λ π ν − v Z (sin θ − v ) dσ,J = √ λ π ω − v Z cos θ dσ,K = √ λ π − v Z (3 ν cos θ − ν − c v ) dσ,K = √ λ π − v Z (4 ω + 2 ν v − c − ω cos θ ) dσP = √ λ π κ Z dσ . Also the relation among the integration constants now becomes c = 1 ν [ v (1 − ν − κ ) + 2 ν ω ] . (2.28)It is straightforward to see that the among various conserved charges we get the followingrelations, E − Sµ = J ν + J ω (2.29)and K ν + K ω = 11 − v κ (cid:20) ω + 2 ν vω − ( ω v + 2 ν )(2 ν ω + v (1 − ν − κ ) ν ω (cid:21) P . (2.30)– 7 –o find the explicit relation among various conserved charges which looks like the spikystring, we now write the explicit expression of the conserved charges. Now J = √ λπ ν p ν − ω [(1 − v ) Z arcsin( α ) π sin θ dθ cos θ q sin θ − α − Z arcsin( α ) π sin θ cos θ dθ q sin θ − α ] . (2.31) J diverges, but on regularization we get,( J ) reg = √ λπ ν p ν − ω q − α . (2.32)On the other hand J is finite and is written as J = − √ λπ ω p ν − ω q − α . (2.33)Similarly, K and K both diverge, however the regularized expressions are given by( K ) reg = − √ λπ ν p ω − ν ) q − α , (2.34)and ( K ) reg = √ λπ ω p ω − ν ) q − α . (2.35)Now the angle difference between the end points of the string is given by∆ φ = ν Z ∞−∞ dy ∂g ∂y = − α ) , (2.36)which implies α = cos ∆ φ . However, ∆ φ = ν R ∞−∞ dy ∂g ∂y diverge, but the regularizedexpression is given by (∆ φ ) reg = − α ) , (2.37)which implies α = cos (∆ φ ) reg . In terms of ∆ φ and (∆ φ ) reg we can express,( J ) reg = √ λπ ν p ν − ω sin ∆ φ , J = − √ λπ ω p ν − ω sin ∆ φ , (2.38)and they satisfy the relation,( J ) reg = s J + (cid:18) λπ (cid:19) sin ∆ φ . (2.39)This relation looks precisely like the single spike dispersion relation with two spins on R × S [33]. Now,( K ) reg = − √ λπ ν p ω − ν ) sin (∆ φ ) reg , ( K ) reg = √ λπ ω p ω − ν ) sin (∆ φ ) reg , (2.40)– 8 –nd they satisfy the relation,( K ) reg = s ( K ) reg + 3 (cid:18) λπ (cid:19) sin (∆ φ ) reg . (2.41)We wish to mention that due to the presence of the background B -field in the metric whichis essentially the volume form of the three sphere in the transverse space, we get a factorof 3 in the dispersion relation in (2.41) as compared to (2.39). We also have energy E andspin S of AdS space as conserved quantities, which are diverging. However the regularizedexpressions are given by E reg = (cid:18) Sµ (cid:19) reg = − √ λπ p α p − µ . (2.42)So they satisfy E reg − (cid:18) Sµ (cid:19) reg = 0 . (2.43)The regularized spin can be rewritten as, S reg µ = r S + λπ (1 + α ) . (2.44)The time difference ∆ t between the end point of the string can be defined as,∆ t = Z ∞−∞ ∂h ∂y dy = − v p − µ [ Z ∞ sinh ρdρ cosh ρ q cosh ρ + α ] , (2.45)which is finite and is given by,(∆ t ) = − v p α − p α − p − µ ! , (2.46)which implies p α − p − µ = − sin( ∆ t p α − v ) . In terms of ∆ t we can express, E reg = S reg µ = vuut S + (cid:18) λπ (cid:19) cos ∆ t p α − v ! . (2.47)– 9 – .3 Magnon Case In this case, let us choose c = c = v . Then the conserved quantities become, E = √ λ π − v Z sinh ρdσ ,S = √ λ π µ − v Z sinh ρdσ ,J = − √ λ π ν − v Z cos θ dσ ,J = √ λ π ω − v Z cos θ dσ ,K = √ λ π − v Z (3 ν cos θ − ν − c v ) dσ ,K = √ λ π − v Z (4 ω + 2 ν v − c − ω cos θ ) dσ ,P = √ λ π κ Z dσ . (2.48)Also the relation among the integration constants now become c = 1 ν (cid:20) v (1 − ν ) − κ v + 2 ν ω (cid:21) . (2.49)Among the conserved quantities we get the following relations, E − Sµ = J ν + J ω = 0 , (2.50)and K ν + K ω = 11 − v κ " ω + 2 ν vω − ( ω v + 2 ν )(2 ν ω − κ v + v (1 − ν )) ν ω P . (2.51)The explicit expression of spin S associated with AdS is diverging, but the regularizedform is, S reg µ = − √ λπ p α p − µ (2.52)This can be rewritten as, S reg µ = r S + λπ (1 + α ) (2.53)The time difference ∆ t between the end point of the string can be defined as,∆ t = Z ∞−∞ ∂h ∂y dy = 2 p − µ [( 1 v − v ) Z ∞ cosh ρdρ sinh ρ q cosh ρ + α − v Z ∞ sinh ρdρ cosh ρ q cosh ρ + α ] , (2.54)– 10 –hich diverges, however the regularized (∆ t ) reg is ,(∆ t ) reg = − v p α − p α − p − µ ! , (2.55)which implies p α − p − µ = − sin( ∆ t reg v p α −
12 ) . In terms of (∆ t ) reg we can express, S reg µ = s S + (cid:18) λπ (cid:19) cos ( ∆ t reg v p α −
12 ) . (2.56)Again the angle difference ∆ φ is defined as,∆ φ = ν Z ∞−∞ ∂g ∂y dy = 2 ν p ν − ω v Z π arcsin( α ) cos θ dθ sin θ q sin θ − α + (cid:18) v − v (cid:19) Z π arcsin( α ) sin θ dθ cos θ q sin θ − α , (2.57)diverges. After excluding the divergence part, we get the regularized ∆ φ ,(∆ φ ) reg = − α ) , (2.58)which implies α = − sin (∆ φ ) reg . The angular momentum J is given by, J = − √ λπ ω p ν − ω cos (∆ φ ) reg , (2.59)which can be rewritten as, J ν ω = r J + λπ cos (∆ φ ) reg J ) reg ω . (2.60)Therefore we can write the giant magnon dispersion relation as,( E − J ) reg = S reg µ + ( J ) reg ω = s S + λπ cos ( ∆ t reg v p α −
12 ) + r J + λπ cos (∆ φ ) reg . (2.61)One may like to find further relations among the other charges such as K , K and ∆ φ which are confined to the transverse space of the NS-brane.– 11 – . Folded String In this section we wish to study some string solutions which are pulsating in the backgroundof the near horizon geometry of the curved NS5-branes and also contain some extra angularmomentum. To study folded strings on this background we choose the following ansatz, ρ ( σ ) = ρ ( σ + 2 π ) , t = κτ, ϕ = µ τ, χ = µ τ, (3.1) θ ( σ ) = θ ( σ + 2 π ) , φ ( σ ) = φ ( σ + 2 π ) , ψ = ω τ,θ ( σ ) = θ ( σ + 2 π ) , φ ( σ ) = φ ( σ + 2 π ) , ψ = ω τ . The Polyakov action of the string, in the conformal gauge and with these ansatz, becomes, S = − √ λ π Z dσdτ h cosh ρ ˙ t + ρ ′ − sinh ρ ˙ ϕ + θ ′ + sin θ φ ′ − cos θ ˙ ψ − ˙ χ + θ ′ + sin θ φ ′ − cos θ ˙ ψ + 4 sin θ ˙ ψ φ ′ i . (3.2)Solving for ρ , θ and θ equations we get, ρ ′′ = sinh ρ cosh ρ ( κ − µ ) ,θ ′′ = sin θ cos θ ( φ ′ + ω ) ,θ ′′ = sin θ cos θ ( φ ′ + ω + 4 ω φ ′ ) , (3.3)Again solving for φ and φ equations, we get, ddσ ( φ ′ sin θ ) = 0 , ddσ ( φ ′ sin θ ) + 4 ω sin θ cos θ dθ dσ = 0 . (3.4)Integrating these two, we get φ ′ = c sin θ , φ ′ = c sin θ − ω , (3.5)where c and c are integration constants. Substituting the values of φ ′ and φ ′ in (3.3)equations of motion and integrating them we get, ρ ′ = ( κ − µ ) sinh ρ + c ,θ ′ = − c sin θ + ω sin θ + c ,θ ′ = − c sin θ − ω sin θ + c , (3.6)where c , c and c are integration constants as well. Now from the Virasoro constraints,we get the following relation among various integration constants, − κ − µ + ω − ω + 4 c ω + c + c − c = 0 . (3.7)– 12 –he conserved quantities in these case are given by, E = √ λκ π Z π dσ cosh ρ ,S = √ λµ π Z π dσ sinh ρJ = √ λω π Z π dσ cos θ ,K = √ λ π Z π dσ (4 ω − ω cos θ − c ) ,P = √ λµ . (3.8)We can choose c = c = 0, so that we can express our result in terms of elliptic functionsas is the usual practice. In what follows, we wish to study few subset of pulsating solutions. θ = θ = 0In this section we wish to studying the string which pulsates in the AdS subspace andwhich contains extra charges due to the transverse motion of the string along the radialdirection. We define the energy and spin density as E = E √ λ = κ π Z π dσ cosh ρ , S = S √ λ = µ π Z π dσ sinh ρ . (3.9)Hence they satisfy the relation E κ − S µ = 1or, E = κ + κµ S (3.10)Also we can define, J = J √ λ = ω , K = K √ λ = ω , P = P √ λ = µ . (3.11)Now we have, ρ ′ = dρdσ = q c + ( κ − µ ) sinh ρ Z π dσ = 4 Z ρ dρ q c + ( κ − µ ) sinh ρ (3.12)where ρ corresponds to maximum value sinh ρ . Solving this we get, q µ − κ = 2 π K ( q ) , (3.13)– 13 –here K ( q ) is the elliptic function of first kind with argument q = q c κ − µ . Also we have, E = κ π Z π dσ cosh ρ = 4 κ π Z ρ dρ cosh ρ q c + ( κ − µ ) sinh ρ . (3.14)Solving this integration we get, E = 2 κπ E ( q ) p µ − κ , (3.15)where E ( q ) is the elliptic function of second kind. Combining the two equations (3.13) and(3.15) we get κ = (cid:18) K ( q ) E ( q ) E (cid:19) ,µ = (cid:18) K ( q ) E ( q ) E (cid:19) + 4 π ( K ( q )) . (3.16)Similar solutions were found in [54]. ρ = θ = 0In this section we wish to study strings that pulsate in one of the S and at the same timehave extra charges due to the transverse motion of the string in the radial direction ofthe NS5-brane. Note that the extra charges appear because of the translational symmetryalong the ξ direction of the original background. For this case we have, E = κ , S = 0 , K = ω , P = µ . (3.17)We also have, dθ dσ = q c + ω sin θ , (3.18)which implies ω = − π K ( r ) , (3.19)where K ( r ) is the elliptic function of first kind with the argument r = √ c ω and J = ω π Z π dσ cos θ , (3.20)which implies J = − π E ( r ) , (3.21)where E ( r ) is the elliptic function of second kind. Combining the two equations (3.19) and(3.21) we get, ω = (cid:18) K ( r ) E ( r ) J (cid:19) . (3.22)Similar solutions have been found in [54]. – 14 – .3 For ρ = θ = 0This is an interesting case, where not only the string pulsates in one of the S , it alsohas extra charges due to the transverse motion of the string in the radial direction andfurthermore there is a non-zero B-which contributes the equations of motion. Hence thefundamental string would know the presence of such field through the energy-spin relation-ship. For this case we have, E = κ , S = 0 , J = ω , P = µ . (3.23)We also have, dθ dσ = q c − ω sin θ , (3.24)which implies ω = 2 √ π K ( s ) (3.25)where K ( s ) is the elliptic function of first kind with the argument s = √ c √ ω and K = ω π Z π dσ (4 − θ ) (3.26)= 4 ω π h Z π dθ q c − ω sin θ − Z π dθ cos θ q c − ω sin θ i , which implies K = 2 √ π [4 K ( s ) − E ( s )] , (3.27)where E ( s ) is the elliptic function of second kind. Combining these two equations we canwrite, K = (cid:20) − E ( s ) K ( s ) (cid:21) ω , or, ω = K h − E ( s ) K ( s ) i . (3.28)
4. Discussion and Conclusion
In this paper we have studied semiclassical strings in the near horizon geometry of curvedNS5-branes, namely on the NS5-branes with
AdS × S worldvolume. We have foundthe most general solutions of the equations of motion of the probe fundamental stringin this background and found out solutions corresponding to giant magnon, single spikeand furthermore the pulsating strings. We have found out the dispersion relation amongvarious conserved charges and compare them with the existing ones. The novelty of thesesolutions is that they contain the information about the background NS-NS field. Further,the presence of the charge, P , in the dispersion relation reflects the fact that the motion of– 15 –he string in the radial direction ξ in the near horizon geometry of NS5-branes is free. In thespirit of the non-integrability of the classical strings in the generic p -brane background, onecan try to investigate the fundamental string equations of motion in the S-dual background,i.e. the D5-brane background wrapped on AdS × S , which is presented in this paper. Thedetails of the background is given by [53]. By looking at the classical integrability of thestring solutions presented in this paper in the NS5-brane background, one might be temptedto believe that similar solution would appear because of the AdS × S × S structure ofthe parent background (in the absence of any brane charges). But we notice that whilesolving for the equations of motion in the D5-brane background, it is not possible to findsimple or similar solution corresponding to the usual giant magnon and single spike strings.This is perhaps a hint to believe that F-string equations of motion are non-integrable inthe D5-brane background. However, we wish to remark that the background solutionsfor the NS5-brane and D5-brane are similar being related by S-duality, but this S-dualitydoes not act on classical string solutions in these backgrounds. Therefore, classical stringsolutions would indeed be very different. Hence in D5-brane background case there is noreason to expect integrability of probe fundamental string equations. It would perhaps beinteresting to study D1-brane equations of motion in the D5-brane background and lookfor exact solutions in the context of integrability. We wish to come back to this issue infuture. Acknowledgements:
We would like to thank A. Tseytlin for some comments. KLP wouldlike to thank the Abdus Salam I.C.T.P, Trieste for hospitality under Associate Scheme,where a part of this work was completed.
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