Finite-size Effect for Dyonic Giant Magnons in $CP^3_{\beta}$

We studied the finite-size giant magnons in $\text{AdS}_4\times\text{CP}^3_{\beta}$ background using the classical spectral curve constructed in this paper. We computed the finite-size corrections to the dispersion relations for the $RP^3$ giant magnons using our twisted algebraic curve based on the method proposed in arXiv:0810.1246, in which the authors computed the finite-size corrections of giant magnons in $\text{AdS}_4\times\text{CP}^3$ by introducing a finite-size resolvent $G_{\text{finite}}(x)$. We obtained exactly the same result as in arXiv:1106.3686, where a totally different approach was used.


Introduction
The gauge/gravity duality conjecture proposed in [3] is a striking and thought-inspiring idea, which relates a large N gauge theory to a certain string theory in such a non-trivial way that makes it very hard to prove. Since integrable structures were found in both sides in AdS 5 /CFT 4 and AdS 4 /CFT 3 dual pairs [4], [5], [6], [7], integrability techniques can help us to understand and to test this conjecture qualitatively. See [8] for a review.
Investigating such dualities without or with less supersymmetry is very important. One approach is to make marginal deformations in the field theory side. The β-deformation of AdS 5 /CFT 4 is such an example, which in the field theory it corresponds to an exact marginal deformation [9], while the deformed background in the string theory can be generated by TsT transformations [10]. People have also investigated the three-parameter deformation which is also called γ-deformation as a generalization of β-deformation [11], [12]. Another interesting example is the duality between type IIA string theory on AdS 4 × CP 3 γ and γ-deformed ABJM theory [13]. This is a three-parameter deformation which breaks supersymmetry completely, while here we are interested in the β-deformed theory with γ 1 = γ 2 = 0, γ 3 = β which preserves N = 2 supersymmetry. 1 In our recent paper [15], the integrable structure of γ-deformed ABJM theory was investigated in detail. The twist matrices were obtained in various bases. In this paper, we will try to calculate the finite-size effect for giant magnons in AdS 4 × CP 3 β using classical spectral curve. In order to compare our results with [2], we had better to use different charges from our previous choice in [15]. These charges are listed in Table 1. These two choices can be easily related through non-singular linear transformations, so they are equivalent. The twist matrix still has the form (in the su 2 grading) , (1.1) 1 We always assume that the deformation parameters are real, since otherwise generically integrability will be broken [14]. but with the relation between δ i 's and γ i 's different, We make natural assumption that the type IIA string theory on AdS 4 × CP 3 γ is integrable based on previous results in the field theory side. Its classical integrability can be proven similarly to the studies in [17]. Although we did not make explicit calculations, we expect that the AdS 4 × CP 3 type IIA string in the γ-deformed background with boundary condition ϕ i (2π) − ϕ i (0) = 2πm i 2 is equivalent to string with twisted boundary conditionsφ i (2π) −φ i (0) = 2π(m i − γ j ǫ ijk J k ) in the undeformed theory based on a general analysis on TsT transformation, where ϕ i 's are the three U (1) directions ϕ 1 , ϕ 2 , ψ listed in Table 1, and J i 's are the corresponding conserved angular momentum.
The dispersion relation for RP 3 giant magnons in AdS 4 × CP 3 β has been calculated in [18], and the finite-size corrections were obtained in [2] by searching for the needed classical string solution directly. In this paper, we use the classical spectral curve method to calculate these quantities and to compare our results to the ones of [2]. Finally, we find they match perfectly.
2 γ-deformed ABA at strong coupling In this section we briefly review the γ-deformed AdS 4 /CFT 3 asymptotic Bethe ansatz(ABA) equations and introduce some notations which will be useful in the following sections. In this section, we use the same notation as our previous paper [15]. Let's first define some useful functions as and the functions with no subscript mean a product of type-4 and type-4 ones: We also introduce the excitation number vector K = (L|K 1 , K 2 , K 3 , K 4 , K4) as before, then the γ-deformed ABA can be written as (in the su 2 grading) where the x ± are Zhukowsky variables, and we have already used the general notation The spectrum of all conserved charges is given by the momentum carrying roots u 4 and u4 alone from, The conserved momentum reads p =p + 2π(AK) 0 , (2.7) At weak coupling, h(λ) ≃ λ, and at strong coupling, h(λ) ≃ λ/2. The BES dressing phase σ BES behaves like: In the scaling limit u a,j ≃ L ≃ √ λ ≃ K a ≫ 1, we find and g is defined through g ≡ λ/8. It is convenient to introduce the resolvents: We can recover the conserved charges Q n from these resolvents: The classical spectral curve of γ-deformed AdS 4 /CFT 3 In this section, we try to obtain the classical spectral curve from the generating functional [19], [20].
In the context of γ-deformation, the twisted generating functional is needed. The twisted generating functional for γ-deformed AdS 4 /CFT 3 has been constructed in [15] and reads (we have used a different gauge from there) In the scaling limit, we have the following expansions At strong coupling, the generating functional becomes where λ a = e −iqa , with q a being the so called quasi-momenta. We have refined the formal expansion parameter in the following way The AdS 4 × CP 3 γ classical string algebraic curve is a ten sheets Riemann surface parameterized by {e iqa(x) , e −iqa(x) }. After some computations, we find where φ 3 = −i log τ 1 + 2π(AK) 0 ,φ 4 = i log τ 1 ,φ 1 = φ 2 = 0, and q 11−i = −q i , i = 1, 2, · · · , 5. As discussed in [15], we must also introduce the twist phases The relations between the conserved angular momentum and the excitation numbers read where we denote the SU (4) Dynkin labels as [p 1 , q, p 2 ]. See Appendix B for more information. Using the above relations, we can write the twists AK appearing in the ABA equations as With the help of the formulas given in Appendix A, it's very easy to show that the twisted ABA eqs. (2.2) are equivalent to the following equations in the scaling limit. In this section and the following, we will work with the β-deformation of AdS 4 /CFT 3 , where we take γ 1 = γ 2 = 0, γ 3 ≡ β, and we label the Dynkin nodes as (r, u, v) instead of (3,4,4). We use the ansatz for solutions mostly in CP 3 β [1] 3 , where the twists φ i and ∆φ i adding here are to be determined. Let's make some comments on these two kind of twists. The giant magnons are open string solutions with endpoints located on different places. If we want to treat them as closed strings, we have to consider Z M orbifold of deformed ABJM theory. The ∆φ i 's incorporate this effect. While the existence of φ i are due to that we are considering the β-deformed theory. The Dynkin labels of SU (4) are related to the excitation numbers by While in this sector, In the last section, we have calculate the twists φ 1 , . . . , φ 5 , with the help of eq. (3.9), we can write them in terms of the angular momentum. For the case at hand, we have

Fix the twists from orbifolding
In this section we will mainly discuss the RP 3 giant magnon, which is the dyonic generalization of RP 2 giant magnon. We closely follow the treatment of [21] (early work on treating giant magnon as closed string on orbifold includes [22]- [24]). Let us temporarily put aside the deformation. As mentioned above, giant magnons are open string solutions with non-periodic boundary conditions. For RP 3 magnon, we have Formally identifying this open string as closed string leads us to consider Z M orbifolding of ABJM theory. This theory appears when we consider low energy effective theory of N M 2 branes put at C 4 /(Z M × Z M k ) orbifold singularity [25]- [28] with Z M acting on C 4 as Notice that this Z M is inside SU (4) R-symmetry group of ABJM theory. It is easy to see that we should identify p/2 with 2πm/M . Then we have to set the charges of Y I 's as From the analysis in [29], we have where t I 's are parameters of the orbifold. Comparing these two results, we find t 1 = t 2 = t 3 = m.
Then the charges appearing in the su 2 -grading orbifolding ABA equations are  This leads to ∆φ 1 = ∆φ 2 = ∆φ 5 = 0, ∆φ 3 = ∆φ 4 = −p/2, (4.12) exactly the same as the results given in [30] (see also [31]). Thus the asymptotic behaviour of the quasi-momentum when Comparing our ansatz eq. (4.1) with the asymptotic behaviors eq. (4.13), we conclude α = ∆/2g and S = 0. The total momentum condition from the Bethe ansatz now reads p = 2πn − 2π(AK) 0 + p. (4.14) Here n ∈ Z, p is the momentum of giant magnon andp is defined as For β-deformed case, we have And the above result is consistent with the results from inversion symmetry For our convenience, we can set n = 0. In fact, any even n gives the correct result. This require comes from the ambiguity of the definition ofp. Let's make this point more clear.p is defined up to some integer multiply 2π. For 'small giant magnon', any n gives the same result. But for RP 3 magnon, this integer must be a even number, because we have two copies of 'small giant magnon'.

Infinite size dispersion relation
Studying giant magnons using the classical spectral curve method first appears in [32], where the authors show that giant magnons correspond to logarithmic cuts in the algebraic curve language. The giant magnons in CP 3 β have various forms. But essentially they belong to two different classes: the 'small' and 'big' giant magnons and their dyonic generalizations. They can all be constructed by setting some of the resolvents in ansatz eq. (4.1) to be where (X + ) * = X − . As a warm-up exercise, let's first consider the 'small giant magnon' with The charges can be read from the asymptotic behavior of this curvẽ Then the dispersion relation can be find as Now we consider another kind of dyonic giant magnons in CP 3 β , which are also called pair of small giant magnons or RP 3 magnons. They can be constructed by putting a 'small giant magnon' in each sector, G u (x) = G v (x) = G mag (x) and with G r (x) = 0. From this setting, we obtain the charges as Thus we find the dispersion relation wherep = p − 2πβJ 1 .

Finite-size corrections
The finite-size effects for giant magnons were usually computed through Lüscher formula, which basically contain two different terms coming from two types of spacetime interpretation: the µ-term and the F-term, which correspond to the leading classical corrections and the first quantum corrections respectively. The F-term is easily compute from classical spectral curve. In [1], the authors proposed a method to compute the µ-term from the algebraic curve, see also [33]. As we don't not attend to construct the Drinfeld-Reshetikhin twist [34] of the AdS 4 /CFT 3 S-matrix in this short note, we will try to compute the finite-size corrections using the twisted classical spectral curve proposed above. The basic idea is that at finite size we can think the giant magnon solutions obtain a small square root cut tail in each ends of the logarithmic cut. Considering this, we use the finite-size resolvent [1] G finite (x) = 2i log where Y ± are points shift by some small amount away from X ± : As a simple check, when taking δ = 0, this finite-size resolvent goes back to the infinite-size resolvent (4.18). Now we will study the finite-size RP 3 dyonic giant magnons using the method mentioned above. For this purpose, we set G u (x) = G v (x) = G finite (x) and with G r (x) = 0. We write The momentum of the giant magnon is The expansion ofp in small δ is The non-trivial large x asymptotic of this curve is from which we solve Then we also expand these two charges in δ up to O(δ 3 ) as The coefficients have already been compute in [35], and we report the results here. The result for J 1 is Then we find the corrections to E = ∆ − J 2 begin at order δ 2 We now need to find out δ by solving the constraint where C 47 is the square-root branch cut connecting the 4th and 7th sheets as we are considering the RP 3 magnons, for which [K r , K u , K v ] = [0, J 1 , J 1 ]. We are interested in the leading finite-size corrections, therefore we can evaluate at x = X + as (5.14) Solving this equation, we can fix δ as To ensure the energy corrections being a real number, we must impose δ to be real (for generic case the contributions at the O(δ) order may be nonvanishing). Then the real part of δ 2 is ℜ(δ 2 ) = |δ 2 | = 64 sin 2 (p/4) exp − ∆r(r 2 + 1) sin(p/4) 2g((r 2 − 1) 2 + 4r 2 sin 2 (p/4)) = 64 sin 2 (p/4) exp The condition ℑ(δ) = 0 gives the relation of the phase φ and the conserved charges 2φ = πn ′ 1 − πβ + J 1 sin(p/2) J 2 1 + 64g 2 sin 4 (p/4) J 2 − J 1 sin(p/2) J 2 1 + 64g 2 sin 2 (p/4) J 2 1 + 64g 2 sin 4 (p/4) , (5.18) where n ′ 1 = n 1 for plus sign in (5.15) while n ′ 1 = n 1 − 1 for minus sign in (5.15). Here we still havẽ p = p − 2πβJ 1 as before.

Conclusion
In this note, we have constructed the classical spectral curve of γ-deformed AdS 4 /CFT 3 , which is consistent with the twisted ABA equations at strong coupling. The γ-deformed algebraic curve is not very different from the original one as we only adding some appropriate phases which can be easily obtained from the twisted generating functional and some other natural requirement. To check our proposal, we compute the finite-size corrections of the dyonic giant magnons in AdS 4 × CP 3 β using our twisted algebraic curve. Our result is the same as the one in [2]. We hope this can be further verified by construct the twisted S-matrix and appropriate twisted boundary as in [34] for AdS 5 /CFT 4 , and using the generalized Lüscher formula [36] to compute the corrections of the dispersion relation.
Recently, the quantum spectral curve for γ-deformed AdS 5 /CFT 4 has been construct in [37], it's also very interesting to find out it for AdS 4 /CFT 3 with γ-deformation.