Neumann-Rosochatius system for strings in ABJ Model
PPrepared for submission to JHEP
Neumann-Rosochatius system for stringsin ABJ Model
Adrita Chakraborty a Kamal L. Panigrahi ba Centre For Theoretical Studies, Indian Institute of Technology Kharagpur-721302, India b Department of Physics, Indian Institute of Technology Kharagpur-721302, India
E-mail: [email protected] , [email protected] Abstract:
Neumann-Rosochatius system is a well known one dimensional inte-grable system. We study the rotating and pulsating string in
AdS × CP with a B NS holonomy turned on over CP ⊂ CP , the so called Aharony-Bergman-Jafferis(ABJ) background. We observe that the string equations of motion in both casesare integrable and the Lagrangians reduce to a form similar to that of a deformedNeumann-Rosochatius system. We find out the scaling relations among various con-served charges and comment on the finite size effect for the dyonic giant magnons on R t × CP with two angular momenta. For the pulsating string we derive the energyas function of oscillation number and angular momenta along CP . Keywords:
AdS/CFT correspondence, Semiclassical string a r X i v : . [ h e p - t h ] N ov ontents AdS × CP with two form NS-NS flux 3 R t × CP with flux 10 Planar integrability of both gauge theory as well as string theory has played a vitalrole in understanding the celebrated AdS/CFT duality conjecture[1][2][3]in a betterway. In this context, it was first observed by Minahan and Zarembo[4] that theone-loop dilatation operators of the SU (2) sector of N = 4 Supersymmetric YangMills (SYM) theory can be identified with the Hamiltonian of the Heisenberg spinchain [5][6]. As proving the duality for all values of coupling is extremely hard, thesemiclassical string states in the gravity side have been used to look for suitablegauge theory operators on the boundary. The usual
AdS /CF T duality has beengeneralized to AdS /CF T in the presence of mixed NS-NS and R-R flux as well. Thesigma model for the string in AdS × S × T in this mixed flux background has beenproved to be classically integrable [7]. The background solution has further beenshown to satisfy the type IIB supergravity field equations, provided the parametersassociated to field strengths of NS-NS flux (say q ) and R-R flux (say ˆ q ) are related bythe constraint q + ˆ q = 1 . This AdS × S × T background with ’mixed’ three-formfluxes has been an interesting testing laboratory for proving AdS /CF T duality inthe presence of fluxes. This background is conjectured to be originated from the nearhorizon geometry of the intersecting ( F − N S − D − D branes in supergravity,although an explicit construction is yet to be found.In adding further examples of the AdS/CFT duality, ABJM theory [8] has been– 1 –onjectured to be dual to the M -theory on AdS × S /Z k with N units of four-formflux, which for k << N << k can be compactified down to a 10 dimensional type IIAstring theory on AdS × CP , with k being the level of Chern-Simon (CS) theory withgauge group U ( N ) × U ( N ) . The ABJ model [9] is an interesting extension of the abovewith the gauge group U ( M ) k × U ( N ) − k and the amount of maximal supersymmetryremaining fixed. The corresponding string dual in the limit of k (cid:28) N (cid:28) k isconjectured to be type IIA superstring theory on AdS × CP background with aNS-NS two form holonomy over CP ⊂ CP . Using the integrability property ofthe classical string-sigma model, it is relevant to find generic string solutions andtheir corresponding field theory duals. It has been proved that the ABJ theory isintegrable both in nonplanar and planar limits [10–13] similar to its gravity dual.In understanding the string geodesics better, several classes of rigidly rotating andpulsating string solutions in the background of AdS × CP with and without fluxhas been studied, e.g. in [14–19].To this end, [20–22] provided a novel way to find out a large class of simple rotat-ing string solutions by solving a very renowned one dimensional integrable system,known as Neumann model that describes an oscillator on a sphere. The Neumann-Rosochatius (NR) system on the other hand depicts a particle on a sphere with anadditional centrifugal potential (proportional to r ). Reduction of the string-sigmamodel on AdS × S , AdS × CP , etc. to the NR system has been quite usefulin unravelling new relationships between the integrable structures of the two sidesof the AdS/CFT duality[18, 23–27]. Such an approach has previously been usedto study the finite size effect to the giant magnon (GM) solutions in AdS × S background[28]. It is worth emphasizing that the NR integrable system is very ef-fective in dealing with the classical strings in R t × CP . Indeed in [28], the stringdynamics on AdS × CP has been studied by using the NR integrable system. Thedispersion relation and subsequently the finite size effects for the giant magnon andsingle spike solutions for the string on R t × CP with two angular momenta have beenstudied in detail. In this article, we wish to extend the analysis presented in [28]to the case of spinning closed strings in AdS × CP in the presence of B NS holonomy.Among various classes of semiclassical strings, the pulsating strings have muchbetter stability than the non-pulsating ones [29]. The pulsating string concept wasfirst introduced in [30], where it was shown that they correspond to certain highlyexcited sigma model operators. However unlike rotating strings, pulsating strings areless explored. These solutions were first introduced in [31] and further generalizedin [32], [33], [34]. They have also been explored in AdS × S , for e.g. in [35],[20],in AdS × CP , for e.g. in [36], [37]. We wish to show that the pulsating string in AdS × CP with B NS holonomy can also be reduced to a NR system. We derivethe energy of such pulsating strings as a function of the oscillation number and theangular momenta along CP . – 2 –he rest of the paper is organised as follows. In section 2 we study classicalstring action in AdS × CP background in the presence of B NS holonomy and thecorresponding NR system. In section 3 we study of rigidly rotating strings in R t × CP and compute finite size effect for the dyonic giant magnon solution with two angularmomenta. Section 4 is devoted to the study of pulsating string in R t × CP . Weconclude in section 5 with a brief discussion of our results. AdS × CP with two form NS-NS flux We start by writing down the Polyakov action for the bosonic string in the form S = − T (cid:90) dτ dσ √− γγ αβ G αβ − T (cid:90) dσdτ (cid:15) αβ B MN ∂ α X M ∂ β X N , (2.1)where G αβ = G MN ∂ α X M ∂ β X N , X N ( τ, σ ) , M, N = 0 , . . . , are embedding coordi-nates of the string. Further, ∂ α ≡ ∂∂σ α , σ = τ, σ = σ and T is the string tension.The string is embedded into ten dimensional background with the metric G MN andNS-NS two form field B MN . Finally γ αβ is two dimensional world-sheet metric whoseequations of motion have the form T αβ = − √− γ δSδγ αβ = − T γ αβ γ γδ G γδ + T G αβ = 0 . (2.2)The supergravity dual background of so called ABJ theory is AdS × CP backgroundwith B NS flux turned on CP ⊂ CP . The metric is given as ds = G MN dx M dx N = R (cid:18) ds AdS + ds CP (cid:19) , (2.3)which in terms of the background coordinates assumes the following form ds = R − cosh ρdt + dρ + sinh ρ ( dη + sin ηdχ )]+ R [ dξ + cos ξ sin ξ ( dψ + 12 cos θ dφ −
12 cos θ dφ ) ++ 14 cos ξ ( dθ + sin θ dφ ) + 14 sin ξ ( dθ + sin θ dφ )] , accompanied by the following NS-NS B -field as B NS = − b (cid:16) sin 2 ξdξ ∧ (2 dψ + cos θ dφ − cos θ dφ ) + cos ξ sin θ dθ ∧ dφ + sin ξ sin θ dθ ∧ dφ (cid:17) . (2.4)In addition to the above form of metric and NS-NS flux, there is a dilaton field andRamond-Ramond two form and four form flux respectively, whose detailed forms arenot needed in what follows. When taking α (cid:48) = 1 , the curvature radius R is given by R = 2 / πλ / , which is precisely the same as that of ABJM theory.– 3 – .1 Constraints and NR integrable system It is convenient to describe string moving in this background with the help of theembedding coordinates Y i for AdS and X i for CP , where the embedding coordinatesdescribing the background must satisfy the following constraints (see for example[14]). For AdS part, the constraint is (cid:88) i,j =0 η ij Y i Y j + R , (2.5)while for CP , we have the following constraints (cid:88) i =0 X i − R = 0 , (cid:88) i =1 , , , ( X i ∂ α X i +1 − X i +1 ∂ α X i ) = 0 . (2.6)In case of AdS , the embedding coordinates are related to the global ones by Y + iY = R ρe it , Y + iY = R ρ sin ηe iχ ,Y = R ρ cos η, Y + i (cid:113) ( Y + Y ) = R ρe iη (2.7)while in case of CP we have X + iX = R √ ξe iθ , X + iX = R √ ξe iθ ,X + iX = R √ ψe iφ , X + iX = R √ ψe iφ . (2.8)For simplicity, we are interested in the string dynamics on R t × CP subspace, whichcan be achieved by putting Y = Y = Y = 0 . The metric (2.4) in this case, takesthe following form ds = R − dt ] + R [ dξ + cos ξ sin ξ ( dψ + 12 cos θ dφ −
12 cos θ dφ ) +14 cos ξ ( dθ + sin θ dφ ) + 14 sin ξ ( dθ + sin θ dφ )] . (2.9)Now let us define the X i ’s in terms of the polar coordinates as follows : W = X + iX = Rr e i Φ , W = X + iX = Rr e i Φ (2.10) W = X + iX = Rr e i Φ , W = X + iX = Rr e i Φ (2.11)Therefore the embedding of the string in R t × CP may be reduced as z = Z ( τ, σ ) = R e it ( τ,σ ) , w a = W a ( τ, σ ) = Rr a ( τ, σ ) e i Φ a ( τ,σ ) . (2.12)– 4 –n the case of embedding in CP , it must be : (cid:88) a =1 W a ¯ W a = R , (cid:88) a =1 ( W a ∂ α ¯ W a − ¯ W a ∂ α W a ) = 0 (2.13)In terms of the embedding coordinates, the CP constraints become (cid:88) a =1 r a ( τ, σ ) = 0 , (cid:88) a =1 r a ( τ, σ ) ∂ α Φ a ( τ, σ ) = 0 (2.14)For this embedding, the metric induced on the string worldsheet G αβ and the B αβ isgiven by, G αβ = − ∂ ( α Z β ) ¯ Z + (cid:88) a =1 ∂ ( α W a ∂ β ) ¯ W a , (2.15) B αβ = B MN ∂ α X M ∂ β X N = b (cid:15) αβ (cid:34) ∂ ( α Z∂ β ) ¯ Z + (cid:88) a =1 ∂ ( α W a ∂ β ) ¯ W a (cid:35) . (2.16)Putting the expressions of Z and W a we get G αβ = − R ∂ α t∂ β t ) + R (cid:88) a =1 ( ∂ α r a ∂ β r a + r a ∂ α Φ a ∂ β Φ a ) , (2.17) B αβ = B MN ∂ α X M ∂ β X N = bR (cid:88) a =1 r a [ ∂ σ r a ∂ τ Φ a − ∂ τ r a ∂ σ Φ a ] . (2.18)The corresponding Lagrangian in target space now becomes L = −√ λ (cid:8) ( ∂ τ t ) − ( ∂ σ t ) (cid:88) a =1 ( ∂ σ r a ) − (cid:88) a =1 ( ∂ τ r a ) − (cid:88) a =1 r a [( ∂ τ Φ a ) − ( ∂ σ Φ a ) ] (cid:9) − b √ λ (cid:88) a =1 r a ( ∂ σ r a ∂ τ Φ a − ∂ τ r a ∂ σ Φ a )+ √ λ Λ( (cid:88) a =1 r a −
1) + √ λ Λ (cid:88) a =1 ( r a ∂ τ Φ a ) + √ λ Λ (cid:88) a =1 ( r a ∂ σ Φ a ) . (2.19)Here M, N = τ, σ and Λ , Λ , Λ are suitable Lagrange multipliers corresponding tothe constraints. NR system, being an integrable modification of the first proposed Neumann inte-grable model, illustrates the constrained motion of a harmonic oscillator of unit mass– 5 –n a ( N − dimensional unit sphere under another centrifugal potential barrier. TheLagrangian for such a system is given by L = 12 N (cid:88) i =1 (cid:104) x (cid:48) i + x i (cid:16) K (cid:48) i − ω i (cid:17)(cid:105) − Λ2 (cid:32) N (cid:88) i =1 x i − (cid:33) , (2.20)where K (cid:48) i = v i x i , with v i being a constant and Λ is a suitable Lagrange multiplier todeal with the spherical geometry. The corresponding equation of motion is x (cid:48)(cid:48) i = (cid:16) K (cid:48) i − ω i + Λ (cid:17) x i . (2.21)The Hamiltonian for such a system may be written as H = 12 (cid:88) i =1 (cid:104) x (cid:48) i − x i (cid:16) K (cid:48) i − ω i (cid:17)(cid:105) , (2.22)where (cid:80) Ni =1 x i = 1 . We wish to study the spinning string in R t × CP backgroundin the presence of B NS holonomy. We use the following parametrization: t = κτ, r a ( τ, σ ) = r a ( ζ ) , Φ a ( τ, σ ) = ω a τ + f a ( ζ ) , ζ = ασ + βτ , (2.23)where κ, ω a , α, β are constants. Using the ansatz (2.23) for the string rotating in R t × CP , the Lagrangian (2.19) becomes, L = −√ λ (cid:34) κ α − β ) (cid:88) a =1 (cid:18) r a (cid:48) + r a ( f (cid:48) a − βω a α − β ) − α ω a r a ( α − β ) (cid:19)(cid:35) − b √ λ (cid:88) a =1 r (cid:48) a r a αω a + √ λ Λ( (cid:88) a =1 r a −
1) + √ λ Λ (cid:88) a =1 ( r a ω a ) + √ λ Λ (cid:88) a =1 ( r a f (cid:48) a ) . (2.24)Equation of motion for f a is f (cid:48) a = 1 α − β (cid:20) C a r a + βω a + Λ (cid:21) , (2.25)where C a ’s are proper integration constants. Putting this expression of f a in theLagrangian (2.24)we get L = − √ λ (cid:34) κ α − β ) (cid:88) a =1 (cid:18) r (cid:48) a + 1( α − β ) ( C a r a + 2 C a Λ + Λ r a ) − α ω a r a ( α − β ) (cid:19)(cid:35) − B √ λ (cid:88) a =1 αω a r (cid:48) a r a + √ λ Λ( (cid:88) a =1 r a −
1) + √ λ Λ (cid:88) a =1 ( r a ω a )+ √ λ Λ (cid:88) a =1 α − β ) ( C a + βω a r a + Λ r a ) . (2.26)– 6 –rom (2.26), we calculate the equation of motion for r a as ( α − β ) r (cid:48)(cid:48) a − C a ( α − β ) r a + [2(Λ + Λ ω a ) + ω a + (Λ + βω a )( α − β ) ] r a = 0 . (2.27)We note that, this equation can also be derived from the following Lagrangian: L = (cid:88) a =1 [( α − β ) r (cid:48) a − α − β ) C a r a − ω a r a ] + (cid:88) a =1 αω a r a r (cid:48) a − (cid:88) a =1 r a − − (cid:88) a =1 ( r a ω a ) + (cid:88) a =1 α − β ) ( βω a r a + Λ r a ) . (2.28)Here it is quite obvious from equation (2.28) that the Lagrangian is that for the NRsystem only with an extra term added due to the presence of B NS two-form holonomythrough CP and two additional constraints (2.14) to deal with the geometry of CP .Now the equation of motion for Λ gives (cid:80) a =1 ( βω a + Λ ) = 0 ,so that the term (cid:80) a =1 1( α − β ) ( βω a r a + Λ r a ) in the Lagrangian becomes zero. Nowfrom the constraints of AdS × CP , we get the C a ’s as Λ = − (cid:88) a =1 C a = 0 , C a = β ω a r a . (2.29)The Hamiltonian for such system is given as H NR = ( α − β ) (cid:88) a =1 (cid:20) r (cid:48) a + 1( α − β ) ( C a r a + α ω a r a ) (cid:21) = α + β α − β κ , (2.30)whose form is exactly the same as equation (2.22), thereby supporting the NR ap-proach of studying the gravity dual of ABJ theory in planar limit. We note thatin deriving the above relations, we use the Virasoro constraints, G ττ + G σσ = 0 and G τσ = 0 which gives the conserved Hamiltonian H NR and the relation be-tween the embedding coordinates and the arbitrary constants C a . Another relation (cid:80) a =1 ω a C a + βκ = 0 along with the Hamiltonian helps to satisfy both the Vira-soro constraints simultaneusly. For closed strings, r a and f a satisfy the periodicityconditions as, r a ( ζ + 2 πα ) = r a ( ζ ) , f a ( ζ + 2 πα ) = f a ( ζ ) + 2 πn a , (2.31)where n a is the integer winding number.– 7 – .3 Integrals of motion Integrability of any system requires the existence of infinite number of conservedquantities, also known as the integrals of motion, to be in involution. K. Uhlenbackfist introduced the integrals of motion for Neumann model which states that theremust be N number of integrals of motion I i such that [38] { I i , I j } = 0 ∀ i, j ∈ { , , ......, N } , (2.32)so that the integrable features of NR system exists. For any arbitrary values of theconstants v i , the integrals of motion for the NR system assume the following form I i = x i + (cid:88) j (cid:54) = i ω i − ω j (cid:20)(cid:16) x i x (cid:48) j − x j x (cid:48) i (cid:17) + v i x j x i + v j x i x j (cid:21) . (2.33)To construct the integrals of motion for the string moving in ABJ background weuse the parametrization (2.12). With this, the Lagrangian in the desired backgroundreads as L = − ( ∂ τ t ) + ( ∂ σ t ) + (cid:88) a (cid:0) ∂ τ W a ∂ τ ¯ W a − ∂ σ W a ∂ σ ¯ W a (cid:1) + i (cid:88) a (cid:0) − ∂ τ W a ∂ σ ¯ W a + ∂ τ ¯ W a ∂ σ W a (cid:1) − Λ (cid:32)(cid:88) a W a ¯ W a − (cid:33) − Λ (cid:88) a (cid:0) W a ∂ τ ¯ W a − ¯ W a ∂ τ W a (cid:1) − Λ (cid:88) a (cid:0) W a ∂ σ ¯ W a − ¯ W a ∂ σ W a (cid:1) . (2.34)For convenience, let us use the following ansatz[22] W a = x a ( ζ ) e iω a τ , (2.35)where ζ = ασ + βτ and x a ( ζ ) = r a ( ζ ) e if a ( ζ ) .The equation of motion for x a is given by ( α − β ) x (cid:48)(cid:48) a − (2 iβω a − β − α ) x (cid:48) a − ( ω a + 2 i Λ ω a − Λ) x a = 0 . (2.36)It may be noted that this equation of motion can also be derived from the followingLagrangian L = (cid:34)(cid:88) a ( α − β ) x (cid:48) a ¯ x (cid:48) a + iβ (cid:88) a ω a ( x (cid:48) a ¯ x a − ¯ x (cid:48) a x a ) − (cid:88) a ω a x a ¯ x a (cid:35) + 12 (cid:88) a αω a ( x a ¯ x (cid:48) a + ¯ x a x (cid:48) a ) + Λ (cid:32)(cid:88) a x a ¯ x a − (cid:33) + Λ β (cid:88) a (cid:16) x a ¯ x (cid:48) a − ¯ x a x (cid:48) a (cid:17) − i Λ (cid:88) a ω a x a ¯ x a + Λ α (cid:88) a (cid:16) x a ¯ x (cid:48) a − ¯ x a x (cid:48) a (cid:17) . (2.37) here we consider b = 1 for simplicity – 8 –his Lagrangian is equivalent to (2.28). The momenta p a conjugate to x a can bederived as p a = ∂ L ∂ ¯ x (cid:48) a = ( α − β ) x (cid:48) a − iβω a x a + α ω a x a + Λ βx a + Λ αx a , (2.38)Therefore, (¯ x b p a − x a ¯ p b ) = ( α − β )( x (cid:48) a ¯ x b − x a ¯ x (cid:48) b ) − iβ ( ω a + ω b ) x a ¯ x b + α ω a + ω b ) x a ¯ x b . (2.39)Now, we know that the form of the integral of motion for any NR system is F α = α x a x (cid:48) a + (cid:88) b (cid:54) = a | ¯ x b p a − x a ¯ p b | ( ω a − ω b ) . (2.40)Using equation (2.35), we get, x (cid:48) a = ( r (cid:48) a + ir a f (cid:48) a ) e if a . (2.41)Again, from the equations of motion we get i ( α − β )( x (cid:48) a ¯ x a − ¯ x (cid:48) a x a ) (cid:48) = [ − βω a − i (Λ β + Λ α )] x a ¯ x a . (2.42)Now using equations (2.39-2.42) we get, α x a ¯ x (cid:48) a = α r a , | ¯ x b p a − x a ¯ p b | = (¯ x b p a − x a ¯ p b )( x b ¯ p a − ¯ x a p b )= ( α − β )( r (cid:48) a r b − r a r (cid:48) b ) + ( C a r b r a + C b r a r b ) +2 β (cid:2) C a r b ω a + C a r b ω b + C b r a ω a + C b r a ω b (cid:3) + α ω a + ω b ) r a r b . This finally yields F α = α r a + (cid:88) b (cid:54) = a ( r (cid:48) a r b − r a r (cid:48) b ) ( ω a − ω b ) + (cid:88) b (cid:54) = a ω a − ω b ) (cid:18) C a r b r a + C b r a r b (cid:19) + α (cid:88) b (cid:54) = a (cid:18) ω a + ω b ω a − ω b (cid:19) r a r b . (2.43)These are the Uhlenback integrals of motion for the closed rotating string in R t × CP in the presence of the B NS holonomy. – 9 – Dyonic magnon on R t × CP with flux Let us assume that the string is spinning in the given background with two indepen-dent angular momenta ω = − ω and ω = − ω . For such a system we consider theembedding coordinates as r = r = √ sin θ and r = r = √ cos θ. The equation ofmotion (2.27) then reduces to θ (cid:48) ( ζ ) = 1( α − β ) [( α + β ) κ − C + C )(sin θ ) − C + C )(cos θ ) − α ( ω (sin θ ) + ω (cos θ ) )] , (3.1)where we have used the Hamiltonian(2.30). We wish to study the giant magnonsolution. For this we put C = C = 0 so that (cid:80) a =1 C a ω a = − βκ yields C = − C = − βκ ω . Thus the expression for (3.1) reduces to (cos θ ) (cid:48) = ∓ α − β ) [( α + β ) κ − ( α + β ) κ θ − β κ ω − α ω + 2 α ω cos θ − α ω cos θ + α ( ω − ω ) cos θ ] , (3.2)with a solution cos θ = z + dn ( Cζ | m ) , (3.3)where z ± = 12(1 − ω ω ) (cid:34) y + y − ω ω ± (cid:115) ( y − y ) − [2( y + y − y y ) − ω ω ] ω ω (cid:35) . (3.4)In the above exprerssion, y = 1 − κ ω and y = 1 − β κ α ω ; C = ∓ α √ ( ω − ω )( α − β ) z + and m = 1 − z − z .For f a = α − β (cid:82) dζ ( C a r a + βω a ) , one has f = − f = ∓ βαz + (cid:113) − ω ω (cid:20) K (1 − z − z ) − κ ω (1 − z ) (cid:8) Π( am ( Cζ ) , − ( z − z − )(1 − z ) | m ) (cid:9)(cid:21) , (3.5) f = − f = ∓ βω αω z + K (1 − z − z ) (cid:113) − ω ω . (3.6)The full string solution in this case using NR integrable system is the same as theones obtained in [18] for the string in AdS × CP without the B NS flux. As the Lagrangian does not depend on t and φ a we have the conserved charges as E = − (cid:90) dσ ∂ L ∂ ( ∂ τ t ) , J a = (cid:90) dσ ∂ L ∂ ( ∂ τ φ a ) , (3.7)– 10 –ith a = 1 , . Therefore, E S = κ √ λ α (cid:90) dζ, J a = 2 √ λ ( α − β ) (cid:90) dζ ( βα C a + α ω a r a ) − b π (cid:90) r a r (cid:48) a dζ (3.8)In what follows we will be looking at the conserved charges for α > β . E and J a become E = E s √ λ = κ α (cid:90) dζ = 2 κ (1 − β α ) ω z + (cid:113) − ω ω K (1 − z − z ) . (3.9) J = J √ λ = 2 α − β (cid:90) ( βα C a + αω r ) dζ − b π √ λ (cid:90) r r (cid:48) dζ (3.10) J = 2 α − β (cid:90) αω r dζ + b π √ λ (cid:90) r r (cid:48) dζ (3.11)Therefore, the final expression of the currents after doing the integration are asfollows: J = 2 z (cid:113) − ω ω z + (1 − β κ α ω ) K (1 − z − z ) − z (cid:113) − ω ω E (1 − z − z ) − b (cid:0) z − z − (cid:1) , (3.12) J = 21 − ω ω ω ω z + E (cid:18) − z − z (cid:19) + b (cid:0) z − z − (cid:1) , (3.13) J = −J − b (cid:0) z − z − (cid:1) , (3.14) J = −J + b (cid:0) z − z − (cid:1) . (3.15)Therefore it is quite obvious that the constraints of CP geometry support (cid:88) a =1 J a = 0 (3.16)by using the NR approach. The computation of ∆Φ gives p =∆Φ = 2 (cid:90) θ max θ min dθθ (cid:48) f (cid:48) = 2 βαz + (cid:113) − ω ω (cid:20) K (1 − z − z ) − κ ω (1 − z ) (cid:8) Π ( am ( Cζ ) , − ( z − z − )(1 − z ) | m ) (cid:9)(cid:21) . (3.17)Here, let us assume, u ≡ ω ω , v ≡ βα , (cid:15) ≡ z − z , where u and v are also functions of (cid:15) .Considering z − z = (cid:15) → the leading order value of z + is z + = (cid:114) J + 4 sin p , (3.18)– 11 –nd hence (cid:15) = 16 exp − J + J + (cid:113) J + 4 sin p ) J + 4 sin p (cid:114) J + 4 sin p p . (3.19)Putting all the expansions of the elliptic integrals and their coefficients we get thedispersion relation as E − J = (cid:114) J + 4 sin p −
32 sin p (cid:113) J + 4 sin p exp − J + J + (cid:113) J + 4 sin p ) J + 4 sin p (cid:114) J + 4 sin p p − b (cid:16) J + 4 sin p (cid:17) exp − J + J + (cid:113) J + 4 sin p ) J + 4 sin p (cid:114) J + 4 sin p p − . (3.20)This result resembles with the dispersion relation for dyonic giant magnon in the R t × S [39, 40] with an extra B dependent term. For the string rotating in R t × CP subspace there exists a similar dispersion relation between E , J and J with a formexactly the same as that of (3.20). Again with one angular momentum J to be zeroit reduces to the giant magnon dispersion relation in R t × S subspace [19]. In this section, we analyze the case of closed pulsating string in R t × CP with B NS flux. It will be worth showing that the pulsating strings with two-form NS-NS fluxesin R t × CP can also be reduced to a deformed NR system. The embedding ansatzfor closed pulsating string in R t × CP is [26]: Z ( τ ) = Y + iY = R z ( τ ) e ih ( τ ) (4.1a) W ( τ, σ ) = X + iX = Rr ( τ ) e i ( f ( τ )+ m σ ) (4.1b) W ( τ, σ ) = X + iX = Rr ( τ ) e i ( f ( τ )+ m σ ) (4.1c) W ( τ, σ ) = X + iX = Rr ( τ ) e i ( f ( τ )+ m σ ) (4.1d) W ( τ, σ ) = X + iX = Rr ( τ ) e i ( f ( τ )+ m σ ) (4.1e)– 12 –here, z = z ( τ ) and r a = r a ( τ ) , with a = 1 , , , , . The winding numbers m a are kept only along the σ direction to make the time direction single-valued. Takingsuch an ansatz and considering all the constraints, the Lagrangian for the pulsatingstring may be derived as: L = −√ λ (cid:34) ˙ z + z ˙ h (cid:35) + √ λ (cid:88) a =1 (cid:104) ˙ r a + r a ˙ f a − r a m a (cid:105) + b √ λ (cid:88) a =1 r a ˙ r a m a − Λ2 √ λ (cid:32) (cid:88) a =1 r a − (cid:33) − ˜Λ2 √ λ (cid:0) z + 1 (cid:1) −√ λ Λ (cid:88) a =1 r a ˙ f a − √
8Λ Λ (cid:88) a =1 r a m a , (4.2)where the derivative with respect to τ is denoted by dots. Λ , ˜Λ , Λ and Λ are suitableLagrange multipliers. The equations of motion for z and f a are given by ¨ z − C z + 4 ˜Λ z = 0 , (4.3) ˙ f a = C a r a + Λ (4.4)Substituting the expression of ˙ f a in the equation(4.2) we get, L = −√ λ (cid:34) ˙ z + z ˙ h (cid:35) + √ λ (cid:88) a =1 (cid:20) ˙ r a + C a r a + C a Λ + Λ r a − r a m a (cid:21) + b √ λ (cid:88) a =1 r a ˙ r a m a − Λ2 √ λ (cid:32) (cid:88) a =1 r a − (cid:33) − ˜Λ2 √ λ (cid:0) z + 1 (cid:1) −√ λ Λ (cid:88) a =1 (cid:18) C a + Λ r a (cid:19) − √
8Λ Λ (cid:88) a =1 r a m a . (4.5)This eventually yields the equation of motion for r a as: ¨ r a + C a r a + Λ (cid:0) Λ + Λ m a + m a (cid:1) r a = 0 . (4.6)One can show that the equations of motion (4.3) and (4.6) can also be obtained froma Lagrangian L NR = ˙ z − C z − ˜Λ (cid:0) z + 1 (cid:1) + (cid:88) a =1 (cid:18) ˙ r a + C a r a (cid:19) + B (cid:88) a =1 r a ˙ r a m a + (cid:88) a =1 (cid:0) r a − (cid:1) (cid:32) (cid:88) a =1 C a (cid:33) − (cid:88) a =1 m a r a + Λ (cid:32) (cid:88) a =1 r a − (cid:33) + Λ (cid:88) a =1 m a r a . (4.7)– 13 –t is obvious that this Lagrangian is of the form of a NR system with z , z , r a and r a type of terms. Here we have used Λ = (cid:0) − (cid:80) a =1 C a (cid:1) . The Hamiltonianformulation yields H NR = ˙ z C z + (cid:88) a =1 (cid:18) ˙ r a − C a r a (cid:19) + (cid:88) a =1 m a r a . (4.8)The corresponding Virasoro constraints may be written as: (cid:88) a =1 (cid:104) ˙ r a + r a ˙ f a + r a m a (cid:105) = (cid:18) ˙ z C z (cid:19) , (4.9) (cid:88) a =1 r a ˙ f a m a = 0 . (4.10)The Uhlenback integrals of motion involved in the motion of the pulsating string in R t × CP in the presence of flux may be obtained by using the similar procedure asdescribed in section (2.3) and those are F = z + r a (cid:88) b (cid:54) = a ( ˙ r a r b − r a ˙ r b ) m a − m b + (cid:88) b (cid:54) = a m a − m b (cid:18) C a r b r a + C b r a r b (cid:19) +2 (cid:88) b (cid:54) = a ( r a ˙ r a r b − r b ˙ r b r a ) m a + m b + 14 (cid:88) b (cid:54) = a (cid:18) m a − m b m a + m b (cid:19) r a r b . (4.11) To find the expression of r a ( τ ) , firstly we take r = r = √ sin θ , r = r = √ cos θ , m = − m and m = − m , so that the constraints become (cid:88) a =1 r a = 1 , (cid:88) a =1 r a m a = 0 . (4.12)Now if we consider z ( τ ) = 1 and h ( τ ) = τ = t , from the Virasoro constraint(4.9)we get ˙ θ = 14 − C sin θ − (cid:0) m sin θ + m cos θ (cid:1) , (4.13)with C = C =0 and C = − C . Equation of motion(4.13) finally yields ∂∂τ (cos θ ) = ˙ r = ∓ (cid:113) m − m (cid:113) (cos θ − z − ) ( z − cos θ ) (4.14)which gives a solution for all r a ’s, a = 1 , , , as r = r = 1 √ (cid:113) − z dn ( Aτ | m ); r = r = z + √ dn ( Aτ | m ) , (4.15) here also we consider b = 1 for simplicity. – 14 –here A ≡ ∓ z + τ (cid:113) m − m , m ≡ − z − z , (4.16) z ± = 12 ( m − m ) (cid:18) − m + m (cid:19) ± (cid:115)(cid:18) − m + m (cid:19) − m − m ) (cid:18) m + 2 C − (cid:19) . (4.17)Hence the string solutions may be written as Z = R e iτ , W = R √ (cid:113) − z dn ( Aτ | m ) e i ( m σ + f ) , (4.18a) W = R √ z + dn ( Aτ | m ) e i ( m σ + f ) , W = R √ (cid:113) − z dn ( Aτ | m ) e − i ( m σ + f ) , (4.18b) W = R √ z + dn ( Aτ | m ) e − i ( m σ + f ) . (4.18c)Now, the conserved charges for the pulsating string in such a background may befound from the target space Lagrangian as E s = ∂ L ∂ ( ∂ τ t ) = − √ λ , J a = ∂ L ∂ ( ∂ τ φ a ) = −√ λr a ˙ f a (4.19)It is obvious that the presence of flux in the background does not affect the conservedcharges in the case of a pulsating string. Now expressing the Virasoro constraint(4.9)in terms of E = E √ λ and J a = J a √ λ , the oscillation number can be written as [41] N = N √ λ = (cid:73) r ˙ r dr = √ λ (cid:90) √ R dr r (cid:113) r E − J − m r . (4.20)where √ R = r max = m (cid:104) E ± (cid:112) E − J m (cid:105) . From the above we get, ∂ N ∂m = − m (cid:90) √ R r (cid:114)(cid:16) r − a − m (cid:17) (cid:16) a + m − r (cid:17) dr , (4.21)where a ± = E ± (cid:112) E − J m and √ R = (cid:113) a + m . Expressing it in terms of standardelliptic integrals and dropping some terms for sake of simplicity we get, ∂ N ∂m = 2 √ m a + (cid:20) K (cid:18) a − a + (cid:19) − E (cid:18) a − a + (cid:19)(cid:21) . (4.22)Now we use the following standard expansions of the elliptic integrals of first andsecond kinds, K ( (cid:15) ) = π π(cid:15) π(cid:15)
128 + 25 π(cid:15)
512 + 1225 π(cid:15) O [ (cid:15) ] , (4.23)– 15 – ( (cid:15) ) = π − π(cid:15) − π(cid:15) − π(cid:15) − π(cid:15) O [ (cid:15) ] , (4.24)where (cid:15) = a − a + . Substituting the expressions for a + and a − in the above expansionand expanding for small E and J , we get ∂ N ∂m a = (cid:18) π J a m a − π J a m a + 1752048 π J a m a + O [ J a ] (cid:19) − (cid:18) π m a + 117 π J a m a − π J a m a + O [ J a ] (cid:19) E + (cid:18) − πm a + 21128 π J a m a − π J a m a + O [ J a ] (cid:19) E − (cid:18) m − π J a m a + 5252048 π J a m a + O [ J a ] E (cid:19) + O [ E ] . (4.25)Integrating the equation(4.25) with respect to m and then inverting the series weget the energy as a function of oscillation number, winding number and conservedangular momenta as, E = M + K ( m a ) 5 πm a J a + O [ J a ] , (4.26)where, M = π √ m a + √N ,K ( m a ) = m a − N (cid:16) m a + π (cid:17) . (4.27)When we take m a → and J a → it yields the first order term of the small energylimit expansion of the energy for the string pulsating in one plane[42, 43]. We have shown, in this paper, that the string motion in
AdS × CP in the presenceof B NS holonomy is integrable by reducing the Lagrangian of such a system into theform of a NR model with an additional term proportional to the flux. We have elu-cidated the Lagrangian and Hamiltonian formulation and computed the integrals ofmotion in our pursuit to show it to be integrable. We have then turned out attentionfor studying the rotating string with two angular momenta and have found out therelevant scaling relation among various charges corresponding to the giant magnonsolution of string rotating in this background. We have also computed the leadingorder finite size correction of such dispersion relation. For the pulsating string wehave performed the Lagrangian and Hamiltonian formulation and integrals of motionto show that it reduces to a NR system. We have derived the integrable equations of– 16 –otion, the pulsating string profile and the short string energy as function of oscilla-tion number and angular momenta. It would be certainly interesting to generalize theconstruction to the more generic rotating and pulsating string solutions and checkthe integrability. The presence of B NS field does not destroy the integrability of thebackground and hence it will be interesting to check for general rotating strings inthe ABJ background. It would also be interesting to look at the D1-string equationof motion in this background and check the integrability via the NR system. Wewish to come back to some of these issues in future. It will also be interesting tolook for finite-size corrections by using the Lüscher correction formulation based onexact S-matrix. References [1] J. M. Maldacena, “The Large N limit of superconformal field theories andsupergravity,” Int. J. Theor. Phys. , 1113 (1999), [Adv. Theor. Math. Phys. , 231(1998)], [hep-th/9711200].[2] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “Gauge theory correlators fromnoncritical string theory,” Phys. Lett. B , 105 (1998), [hep-th/9802109].[3] E. Witten, “Anti-de Sitter space and holography,” Adv. Theor. Math. Phys. , 253(1998), [hep-th/9802150].[4] J. A. Minahan and K. Zarembo, “The Bethe ansatz for N=4 superYang-Mills,” JHEP , 013 (2003), [hep-th/0212208].[5] N. Beisert, “The complete one loop dilatation operator of N=4 superYang-Millstheory,” Nucl. Phys. B , 3 (2004), [hep-th/0307015].[6] N. Beisert and M. Staudacher, “The N=4 SYM integrable super spin chain,” Nucl.Phys. B , 439 (2003), [hep-th/0307042].[7] A. Cagnazzo and K. Zarembo, “B-field in AdS(3)/CFT(2) Correspondence andIntegrability,” JHEP , 133 (2012) Erratum: [JHEP , 003 (2013)][arXiv:1209.4049 [hep-th]].[8] O. Aharony, O. Bergman, D. L. Jafferis and J. Maldacena, “N=6 superconformalChern-Simons-matter theories, M2-branes and their gravity duals,” JHEP , 091(2008), [arXiv:0806.1218 [hep-th]].[9] O. Aharony, O. Bergman and D. L. Jafferis, “Fractional M2-branes,” JHEP ,043 (2008), [arXiv:0807.4924 [hep-th]].[10] D. Bak, D. Gang and S. J. Rey, “Integrable Spin Chain of Superconformal U(M) xanti-U(N) Chern-Simons Theory,” JHEP , 038 (2008), [arXiv:0808.0170[hep-th]]. – 17 –
11] J. A. Minahan, W. Schulgin and K. Zarembo, “Two loop integrability forChern-Simons theories with N=6 supersymmetry,” JHEP , 057 (2009),[arXiv:0901.1142 [hep-th]].[12] R. de Mello Koch, B. A. E. Mohammed, J. Murugan and A. Prinsloo, “Beyond thePlanar Limit in ABJM,” JHEP , 037 (2012), [arXiv:1202.4925 [hep-th]].[13] B. A. E. Mohammed, “Nonplanar Integrability and Parity in ABJ Theory,” Int. J.Mod. Phys. A , 1350043 (2013), [arXiv:1207.6948 [hep-th]].[14] G. Grignani, T. Harmark, M. Orselli and G. W. Semenoff, “Finite size GiantMagnons in the string dual of N=6 superconformal Chern-Simons theory,” JHEP , 008 (2008), [arXiv:0807.0205 [hep-th]].[15] D. Astolfi, V. G. M. Puletti, G. Grignani, T. Harmark and M. Orselli, “Finite-sizecorrections in the SU(2) x SU(2) sector of type IIA string theory on AdS(4) xCP**3,” Nucl. Phys. B , 150 (2009), [arXiv:0807.1527 [hep-th]].[16] C. Ahn and P. Bozhilov, “Finite-size Giant Magnons on AdS xCP γ ,” Phys. Lett. B , 186 (2011), [arXiv:1106.3686 [hep-th]].[17] C. Ahn and P. Bozhilov, “Finite-size Effect of the Dyonic Giant Magnons in N=6super Chern-Simons Theory,” Phys. Rev. D , 046008 (2009), [arXiv:0810.2079[hep-th]].[18] C. Ahn, P. Bozhilov and R. C. Rashkov, “Neumann-Rosochatius integrable systemfor strings on AdS(4) x CP**3,” JHEP , 017 (2008), [arXiv:0807.3134 [hep-th]].[19] S. Jain and K. L. Panigrahi, “Spiky Strings in AdS(4) x CP**3 with Neveu-SchwarzFlux,” JHEP , 064 (2008), [arXiv:0810.3516 [hep-th]].[20] G. Arutyunov, J. Russo and A. A. Tseytlin, “Spinning strings in AdS(5) x S**5: Newintegrable system relations,”, Phys. Rev. D , 086009 (2004) [hep-th/0311004].[21] G. Arutyunov, S. Frolov, J. Russo and A. A. Tseytlin, “Spinning strings in AdS(5) xS**5 and integrable systems,” Nucl. Phys. B , 3 (2003), [hep-th/0307191].[22] M. Kruczenski, J. Russo and A. A. Tseytlin, “Spiky strings and giant magnons onS**5,” JHEP , 002 (2006), [hep-th/0607044].[23] R. Hernandez and J. M. Nieto, “Spinning strings in the η -deformedNeumann-Rosochatius system,” Phys. Rev. D , no. 8, 086010 (2017),[arXiv:1707.08032 [hep-th]].[24] R. Hernandez and J. M. Nieto, “Spinning strings in AdS × S with NS-NS flux,”Nucl. Phys. B , 236 (2014) Erratum: [Nucl. Phys. B , 303 (2015)],[arXiv:1407.7475 [hep-th]].[25] R. Hernandez and J. M. Nieto, “Elliptic solutions in the Neumann-Rosochatiussystem with mixed flux,” Phys. Rev. D , no. 12, 126006 (2015), [arXiv:1502.05203[hep-th]]. – 18 –
26] R. Hernandez, J. M. Nieto and R. Ruiz, “Pulsating strings with mixed three-formflux,” JHEP , 078 (2018), [arXiv:1803.03078 [hep-th]].[27] G. Arutyunov, M. Heinze and D. Medina-Rincon, “Integrability of the η -deformedNeumann-Rosochatius model,” J. Phys. A , no. 3, 035401 (2017),[arXiv:1607.05190 [hep-th]].[28] C. Ahn and P. Bozhilov, “Finite-size effects of Membranes on AdS(4) x S**7,” JHEP , 054 (2008), [arXiv:0807.0566 [hep-th]].[29] A. Khan and A. L. Larsen, “Improved stability for pulsating multi-spin stringsolitons,” Int. J. Mod. Phys. A , 133 (2006), [hep-th/0502063].[30] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, “A Semiclassical limit of the gauge/ string correspondence,” Nucl. Phys. B , 99 (2002), [hep-th/0204051].[31] J. A. Minahan, Nucl. Phys. B , 203 (2003) doi:10.1016/S0550-3213(02)00966-5[hep-th/0209047].[32] J. Engquist, J. A. Minahan and K. Zarembo, “Yang-Mills duals for semiclassicalstrings on AdS(5) x S(5),” JHEP , 063 (2003), [hep-th/0310188].[33] H. Dimov and R. C. Rashkov, “Generalized pulsating strings,” JHEP , 068(2004), [hep-th/0404012].[34] M. Smedback, “Pulsating strings on AdS(5) x S**5,” JHEP , 004 (2004),[hep-th/0405102].[35] A. Khan and A. L. Larsen, “Spinning pulsating string solitons in AdS(5) x S**5,”Phys. Rev. D , 026001 (2004), [hep-th/0310019].[36] B. Chen and J. B. Wu, “Semi-classical strings in AdS(4) x CP**3,” JHEP , 096(2008), [arXiv:0807.0802 [hep-th]].[37] H. Dimov and R. C. Rashkov, “On the pulsating strings in AdS(4) x CP**3,” Adv.High Energy Phys. , 953987 (2009), [arXiv:0908.2218 [hep-th]].[38] K. Uhlenbeck, “Equivariant harmonic maps into spheres”, Lect. Notes Math. , 024(2006), [hep-th/0605155].[40] Y. Hatsuda and R. Suzuki, “Finite-Size Effects for Dyonic Giant Magnons,” Nucl.Phys. B , 349 (2008), [arXiv:0801.0747 [hep-th]].[41] P. M. Pradhan and K. L. Panigrahi, “Pulsating Strings With Angular Momenta,”Phys. Rev. D , no. 8, 086005 (2013), [arXiv:1306.0457 [hep-th]].[42] I. Y. Park, A. Tirziu and A. A. Tseytlin, “Semiclassical circular strings in AdS(5) and’long’ gauge field strength operators,” Phys. Rev. D , 126008 (2005),[hep-th/0505130].[43] C. Cardona, “Pulsating strings from two dimensional CFT on ( T ) N /S ( N ) ,” Nucl.Phys. B , 512 (2015), [arXiv:1408.5035 [hep-th]]., 512 (2015), [arXiv:1408.5035 [hep-th]].