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Semi-classical strings in AdS_4*CP^3

In this paper, we study the semi-classical strings in AdS_4*CP^3 spacetime. We construct various kinds of string solutions, including the point-like, circular, folded and pulsating strings. For the circular and folded strings, we figure out their field theory dual operators. In particular, we discuss the anomalous dimensions of the corresponding operators from decoupled SU(2)*SU(2) spin chain. We find that in the large angular momentum limit, the field theory and string theory results are in very good agreement up to an interpolating function of coupling constant.


Introduction
The semi-classical strings has played an important role in studying AdS 5 /SY M 4 correspondence. In a remarkable paper [1], it was found that the fluctuations around the point-like string captured the right string spectrum of BMN plane-wave background [2]. Moreover, the study of the multi-spin string solutions reveals that even when the configuration is far from BPS, the energy of the string solution with large angular momentum could be in perfect agreement with the ones calculated in dual gauge theory. The agreement beyond BPS limit relies on the fact that on string side the quantum corrections of strings was suppressed by the large quantum number, while on the field theory side the dual field operators to these string solutions are the composite operators, whose anomalous dimension matrix (ADM) could be related to the Hamiltonian of integrable spin chain. Moreover this agreement suggest that both sides of AdS 4 /SY M 4 correspondence are integrable. There are lots of study on this topic. Please see [3] for a nice review and references.
Very recently, inspired by the study of Bagger-Lambert-Gustavson theory on N membranes [4,5,6], in [7] Aharony et.al. proposed a N = 6 Chern-Simons theory coupled with bi-fundamental matter, describing N membrane on S 7 /Z k . In particular, it was pointed out that in the limit with k ≪ N ≪ k 5 , this theory is dual to IIA string theory in AdS 4 × CP 3 . Many aspects of the BPS sector of this new AdS 4 /CF T 3 correspondence have been studied in [7]. Quite recently, the near-BPS sector are also studied by using the Penrose limit of this IIA string theory background [8,9,10]. Other relevant work could be found in [12]- [25]. In particular, in [11,9] it was pointed out there exist integrable spin-chain structure in the field theory. It would be interesting to go beyond the near BPS sector and investigate the semiclassical configurations and their implications in AdS 4 /CF T 3 correspondence.
In this paper, we will study the semi-classical string solutions in AdS 4 × CP 3 . We will mainly focus on the spinning solutions in CP 3 . We will calculate their energy and angular momenta and compare with their field theory duals. This is possible due to the recent study of 2-loop integrable structure in the field theory [11,9]. Our solutions include a point-like one which is BPS and corresponding to the ground states of the string theory in the IIA plan wave background obtained by taking Penrose limit. We also study the circular and folded string configurations. We find that some of our solutions are far from BPS, but still has a field theory dual. For the circular string and folded string, we discuss the energy of the corresponding operator from algebraic Bethe Ansatz equation(ABAE). The agreement between string theory and field theory results could be perfect if an interpolating function of coupling constant is introduced.
The paper will be organized as follows. In section 2, we will set up our system and have a general discussion of the semi-classical solutions. In section 3, we will present several kinds of spinning solutions. In section 4, we will discuss the field dual to some of these spinnning string solutions. We will end with conclusion and discussion in section 5.

Action and equations of motion
The three-dimensional N = 6 superconformal theory proposed by ABJM in [7] is a Chern-Simons theory with gauge group SU (N ) × SU (N ) with bifundamental superfields A 1 , A 2 and anti-bifundamental superfields B 1 , B 2 . The action has a pure Chern-Simons part: (2) ), (2.1) and the superpotential Note that the sign before the Chern-Simons terms are opposite and the superpotential has actually an SU (2) × SU (2) global symmetry, which acting on the A's and the B's separately. The level of the Chern-Simons theory for the two components of the gauge group are k and −k, respectively. When k and N satisfy k ≪ N ≪ k 5 , this field theory is dual to IIA superstring theory on AdS 4 × CP 3 with constant dilaton, RR two-form and four-form fluxes. The constant dilaton reads: Since we are interested in semiclassical fundamental string solution in this background, the RR fluxes play no roles here. Let us start from the metric of AdS 4 × CP 3 , ds 2 = 1 4 R 2 (− cosh 2 ρdt 2 + dρ 2 + sinh 2 ρ(dθ 2 + sin 2 θdφ 2 )) + R 2 (dξ 2 + cos 2 ξ sin 2 ξ(dψ + 1 2 cos θ 1 dϕ 1 − 1 2 cos θ 2 dϕ 2 ) 2 + 1 4 cos 2 ξ(dθ 2 1 + sin 2 θ 1 dϕ 2 1 ) + 1 4 sin 2 ξ(dθ 2 2 + sin 2 θ 2 dϕ 2 2 )). (2.4) The radius R here is 1 R = 2 5/4 π 1/2 λ 1/4 , (2.5) where λ ≡ N/k is the 't Hooft coupling constant of this three dimensional superconformal theory. In the above metric, the CP 3 part is the standard Fubini-Study metric which could be obtained in the following way. We first consider a unit S 7 embedded in C 4 . Then we parameterize the complex coordinates Z i 's as follows: (2.10) Here 0 ≤ ξ < π 2 , 0 ≤ y < 2π, −2π < ψ < 2π and (θ i , ϕ i ) are coordinates of two S 2 's. Now the induced metric on S 7 can be written as a U (1) fiber over CP 3 : In this way, we get the metric on CP 3 as above (here A is a one-form). For most of our solutions, we assume that t, φ, ψ, ϕ 1 , ϕ 2 are function of τ only and ρ, ξ, θ 1 , θ 2 are only the periodic functions of σ. The bosonic part of the string Lagrangian is where G M N is the background metric above and g ab is the worldsheet metric. We can choose the conformal gauge, in which g ab = e γ Diag(−1, 1). Then we have the action: The relation betweenλ and the 't Hooft coupling λ is: (2.14) From the metric, we know that the background has at least five Killing vectors corresponding to the translations along t, φ, ψ, ϕ 1 , ϕ 2 . The Killing vector along t gives the conserved energy and the other four Killing vectors give the conserved angular momenta. Let us make ansatz: With this setup, we have the conserved quantities: The equations of motion for t, φ, ψ, ϕ 1 , ϕ 2 are just the conservation of E, S, J i 's. Now we turn to the equations of motion for ρ, θ, ξ, θ 1 , θ 2 . The equations of motion are: The Virasoro constraint gives

Various semi-classical string solution
In this section, we would like to discuss the semi-classical string solutions in AdS 4 × CP 3 , with emphasis on the spinning string solutions in CP 3 . For the string solutions in AdS 4 , the discussion is very similar to the case in AdS 5 . The only difference lies at the fact that there is only one spin quantum number in AdS 4 while there are two spins in AdS 5 . The rotation string and pulsating string could be constructed easily.
For the semi-classical string in CP 3 , the construction is different. In this case, ρ is constant and t = κτ , , θ, φ is constant, then the equation of motion for θ is satisfied and from the equation of motion for ρ, we get Compared to the similar construction of multi-spin solutions in AdS 5 × S 5 [26], the construction of spinning solutions in AdS 4 × CP 3 is more tricky and restrictive, whose dual field operators are also not transparent. We manage to find the dual operators for point-like string, a class of circular string and a class of folded string, by matching the global charges.

Point-like solution
Consider the point-like solution in which there is no dependence on σ. 2 Let θ 1 = θ 2 = 0, then the equations of motion for θ 1 and θ 2 are satisfied. And from the equation of motion for ξ, we get So for generic ω i 's, ξ can only take π/4. The Virasoro constraints give Then we get Recall that J i 's is the quantum number corresponding to the Killing vector The essential fact is that following [8], we can identify X 1 , X 2 , X 3 , X 4 in (2.7)-(2.10) as the scalar fields A 1 , A 2 ,B 1 ,B 2 . Then we can write down the charges of the scalar fields as Then the chiral operator Tr(A 1 B 1 ) J has J 1 = J, J 2 = J/2, J 3 = −J/2, E = J, in perfect match with the relation (3.4). In fact, as the case studied in [1], the point-like solution is dual to the chiral primary operators, which is BPS and can be identified as the ground state of the IIA string in the plane-wave background. The identification of Tr(A 1 B 1 ) J as the ground state has also been pointed out in [8]. Similar to the IIB string in AdS 5 × S 5 , it could be expected that the fluctuations around this point-like solution is actually the IIA string spectrum in the plane-wave background [1,28,8].

Folded string I
Let us try the following ansatz: θ 1 = θ 2 = 0, then we have Since we consider the folded string here, ξ will take its maximal value at some ξ 0 . When ξ = ξ 0 , we have ξ ′ = 0 and so κ 2 = sin 2 2ξ 0ω 2 . Then we get ξ ′2 =ω 2 4 (sin 2 2ξ 0 − sin 2 2ξ), (3.12) which leads to 2π = 4 The angular momenta in this case is just J 1 = λ dσ 2π cos 2 ξ sin 2 ξω (3.14) The Eqs. (3.13) and (3.14) give us where q = sin 2 2ξ 0 and E(q), K(q) are the elliptic integrals of first kind and second kind, respectively. The energy E is This string configuration is a folded string. The relation between different angular momenta suggest that the solution has only one angular momentum. It is reminiscent of the folded string solution spinning in S 5 discussed in [1]. It is interesting to consider the large J limit, which corresponds to ξ 0 → π/4. In this limit, one has E → ∞, J 1 → ∞ and 2π . This looks similar to the relation in the giant magnon case with p = 1 in [9], but actually folded string solution is very different from the semiclassical string of giant magnon explicitly constructed in [10,22].

Circular strings
In this subsection, we will fix ξ = ξ 0 being a constant. First let us further assume that ω 3 = 0, θ 1 = 0, the equation of motion for θ 1 is satisfied. From the equation of motion for θ 2 we have θ ′′ 2 = 0, (3.20) so θ 2 = nσ, (we choose n = 0). The equation of motion for ξ gives: (3.21) For this solution, we have: This is a string solution with one independent angular momentum. For n = 0, we just come back to the point-like solution discussed before. We can also make ansatz that ω 2 = ω 3 = 0, then the equations of motion for θ 1 , θ 2 give which gives θ i = n i σ. If we choose n 1 and n 2 to be nonzero, then J 2 = J 3 = 0, The equation of motion for ξ gives: The relation between E and J 1 is 4 sin 2 ξ 0 cos 2 ξ 0 +λ 16 (cos 2 ξ 0 n 2 1 + sin 2 ξ 0 n 2 2 ). In this case, the circular string has a field theory dual. Let us consider the composite operator in field theory Tr((A 1 B 1 ) J (A 2 B 2 ) J ). It has J 1 = 2J, J 2 = J 3 = 0 and at the classical level, E = 2J. This is in consistent with the relations that the circular string respect to. We will study this operator in the next section.

Folded string II
The study of the circular string solutions suggest that one may have to fix ξ = π/4 in order to find their field theory dual operator chains. In this case, from the equation of motion for ξ, it is quite natural to require θ 1 = ±θ 2 . Actually, this is the only possible way to have nontrivial solution. Then from the equations for θ 1 and θ 2 , we find that this is only possible for ω 2 = −ω 3 and θ ′′ 1 = sin θ 1 ω 1 ω 2 .

(3.28)
This equation is in consistency with the Virasoro constraint, which has Let us first consider two special case. When we take ω 2 = 0, this reduce to the circular string we studied before.
However, the case with ω 1 = 0 is also interesting. In this case, we have This is another circular string solution, quite similar to the first one, but the field theory dual is very different. To respect the relation between quantum numbers, we are led to considering the following operators: The energy of string in the large J 2 limit is Classically the dual operator has dimension ∆ = 2J 2 , in consistent with the zero order string energy. It should be keep in mind that the first order correction in the string energy is of orderλ/J 2 2 , similar to the circular string we discussed before. In general, the solution is a folded string configuration. Since θ 1 is periodic of σ, so we have −θ 1 (0) ≤ θ 1 (σ) ≤ θ 1 (0), κ 2 = ω 2 1 + ω 2 2 + 2ω 1 ω 2 cos θ 1 (0). (3.32) If ω 1 ω 2 > 0, θ is varying around π. On the contrary, if ω 1 ω 2 < 0, the folded string is centered at θ 1 = 0. Without losing generality, we will assume ω 1 > 0, ω 2 < 0 such that We have the following relation In the case that ω 1 = −ω 2 , we have J 1 = −2J 2 . It is convenient to introduce the following quantities: In terms of these quantities, we have , (3.42) and the following key relations (3.44) We will show that in the next section, due to the above relation the folded string could be in perfect match with the dual field theory operators up to an interpolating function. The dual operators in the field theory is somehow subtle. To match the above angular momenta, we propose the following identification: However, if we take ω 1 = −ω 2 , then the above operators reduce to Tr(A 2 B 2 ) J 1 , which is a BPS primary and has ∆ = J 1 without quantum correction. But from the string calculation we know that there do exist higher order corrections.

Pulsating string
Before ending this section, let us discuss pulsating string, another kind of semi-classical string solution. The pulsating string purely in AdS 4 is quite similar to the one in AdS 5 [27]. So we focus on the circular pulsating string expanding and contracting on CP 3 . To simplify the discussion, we let θ, φ, θ 1 , θ 2 , ϕ 1 , ϕ 2 be fixed to zero, t = κτ, ψ = nσ and ρ, ξ be the function of τ . Then the Green-Schwarz action is which leads to the equations of motion On the other hand, the Virasoro constraint is now 1 4 (−κ 2 cosh 2 ρ +ρ 2 ) +ξ 2 + cos 2 ξ sin 2 ξn 2 = 0 (3.49) To be consistent with the equations of motion, we notice thatρ has to be vanishing. So we fix ρ = 0. Then we have only one equation to solve. Let η = 2ξ, we havė This looks nice a one-dimensional pendulum. From eq. (3.50), we know that when η take the maximal value, η 0 ,η = 0. Then κ 2 = n 2 sin 2 η 0 . Sȯ η 2 = n 2 (sin 2 η 0 − sin 2 η).

Field theory dual operators
The integrable spin chain in ABJM theory has been discussed in [11,9]. It was pointed out in [11], at two-loop order, the anomalous dimension matrix (ADM) of the composite operators constructed from the scalar fields could be identified with an integrable Hamiltonian of an SU (4) spin chain. Including the fermions in the operators, the spin chain is extended to SU (2|2) in [9]. More precisely, following the notation in [11] let us consider the following gauge invariant operators of the form In general, the leading order ADM of this class of composite operators can be identified with the Hamiltonian of an SU (4) spin chain with sites alternating between the fundamental and anti-fundamental representations. We will not review the relevant discussions here.
There are three sets of Bethe roots, satisfying the coupled Bethe equations. Let us consider SU (2) × SU (2) subsector of the SU (4) spin chain, in which Y A take only A 1 , A 2 and Y † A take only B 1 , B 2 . Obviously the operator dual to the circular string and the folded string (ω 1 + ω 2 > 0 case) belong to this subsector. In this case, we have two decoupled SU (2) chains. The middle Bethe roots r j in [11] will not appear in this case. And the Bethe equations are: The trace condition is The energy is Here we have L = 2J and M u = M v = L/2. One choice to satisfy the trace condition is to satisfy and separately. Then these two chains are totally unrelated. For one chain in the large J limit, the Bethe roots may distribute in two different ways, as discussed in [29]. For the circular string, the Bethe roots distribute along the imaginary axis. It is natural to expect that for the circular string in this paper, the Bethe roots distribution follow the same way. Then use the results in [29], we get that in the large N and large J limit, the anomalous dimension of this operator: So the dimension of this operator obtained from the field theory side at the weak coupling is: (4.10) Comparing this result with the string energy (3.27), we find that at large J limit, the energy is proportional to J 1 at the leading order and the first order correction is always 1/J 2 1 . However, on string side the first order contribution is linear inλ, while on the field theory side, it is quadratic inλ. This is very similar to what happens in the point-like string case, where the string and field theory result on fluctuation spectrum has a mismatch of factor λ.
The folded string case is more impressive. Let us just consider the dual operators Tr((A 1 B 1 ) , which belong to the decoupled subsector. For a single subsector, the operator looks like the same one corresponding to the IIB folded string in AdS 5 × S 5 with two angular momenta, namely the operators of the form TrZ J 1 Φ J 2 + · · ·. In IIB case, the match between the string and the field theory result is in a highly nontrivial way [30]. Without getting into details, we can show that our folded string is also in good match with the field theory result, up to an interpolating functin of the 't Hooft coupling constant. Notice that the relation (3.43) is the same as the relation (2.2) in [30], after identifying the angular momenta properly. And since we have the similar integrable structure, the discussion in [30] could be applied to our case straightforwardly 3 . In this way, we show that for the folded string, we have very good first leading order match in AdS 4 /CF T 3 correspondence.
For other circular string and folded string, the study of the dual operators is quite similar. the dual operators should belong to the decoupled SU (2) × SU (2) subsector. And the relation between energy and the angular momenta in these cases is consisent with this statement.

Conclusion
In this paper, we studied the semi-classical string configurations in AdS 4 × CP 3 and their possible field theory dual. We constructed point-like, circular, folded and pulsating strings in CP 3 and calculate their energy and angular momenta. For the circular strings and one class of folded string configurations, we figured out their field theory dual operators. For one class of circular string, the dual operators fall into the SU (2) × SU (2) subsector of an integrable SU (4) spin chain. In this subsector, the fact that the two spin chains decouple allow us to calculate the eigenvalues of ADM from Bethe equations. On the field side, the ADM get correction only at two-loop order, which is proportional to λ 2 . This suggest that in the large angular momentum limit, an effective expansion parameter could be λ 2 /J 2 . On the other hand, from the string calculation, we learn that the first order correction is linear in the expansion parameter λ/J 2 . As suggested in [9,8,10], there should exist a interpolating function f (λ) which approaches λ at weak coupling and √ λ at strong coupling. Combining the result we found in this paper, we suggest that in the study of the spinning strings, the effective expansion parameter is f 2 (λ)/J 2 . The nontrivial functional match in the first leading order in the folded string case gives very strong support to this suggestion.