Finite-size giant magnons on \eta-deformed AdS_5 x S^5

We consider strings moving in the R_t x S^3_\eta subspace of the \eta-deformed AdS_5 x S^5 and obtain a class of solutions depending on several parameters. They are characterized by the string energy and two angular momenta. Finite-size dyonic giant magnon belongs to this class of solutions. Further on, we restrict ourselves to the case of giant magnon with one nonzero angular momentum, and obtain the leading finite-size correction to the dispersion relation.


Introduction
In the recent years important progress has been made in the field of AdS/CF T duality [1] (for overview see [2]). The main achievements are due to the discovery of integrable structures on both sides of the correspondence.
The most developed case is the correspondence between strings moving in AdS 5 × S 5 and N = 4 SYM in four dimensions. The so-called γ-deformation of AdS 5 × S 5 has been proposed in [3]. It was shown in [4] that this deformation is still integrable for real γ (known as β-or T sT -deformation).
A new integrable deformation of the type IIB AdS 5 × S 5 superstring action, depending on one real parameter η, has been found recently in [5]. The bosonic part of the superstring sigma model Lagrangian on this η-deformed background was determined in [6]. Then the authors of [6] used it to compute the perturbative S-matrix of bosonic particles in the model.
Interesting new developments were made in [7]. There the spectrum of a string moving on η-deformed AdS 5 × S 5 is considered. This is done by treating the corresponding worldsheet theory as integrable field theory. In particular, it was found that the dispersion relation for the infinite-size giant magnons [8] on this background, in the large string tension limit g → ∞ is given by whereη is related to the deformation parameter η according tõ Here, we are going to extend the result (1.1) to the case of finite-size giant magnons.
The paper is organized as follows. In Sec.2 we give the bosonic part of the string Lagrangian on η-deformed AdS 5 × S 5 found in [6] and extract from it the background fields. Then in Sec.3, we obtain the exact solutions for the finite-size dyonic giant magnon coordinates, the corresponding conserved charges and the angular difference along one of the isometric coordinates on the deformed sphere S 3 η 1 . In Sec.4 we find the dispersion relation for the giant magnons with one nonzero angular momentum, including the leading finite-size effect on it. Sec.5 is devoted to our concluding remarks.

String Lagrangian and background fields
The bosonic part of the string Lagrangian L on the η-deformed AdS 5 × S 5 found in [6] is given by a sum of the Lagrangians L a and L s , for the AdS and sphere subspaces. Since there is nonzero B-field on both subspaces, which leads to the appearance of Wess-Zumino terms, these Lagrangians can be further decomposed as where the superscript "g" is related to the dependence on the background metric. The explicit expressions for the Lagrangians in (2.1) are as follows [6] L g a = − where we introduced the notation Comparing (2.2)-(2.5) with the Polyakov string Lagrangian, one can extract the components of the background fields. They are given by , g ζζ = ρ 2 1 +η 2 ρ 4 sin 2 ζ (2.7) , g ξξ = r 2 1 +η 2 r 4 sin 2 ξ (2.8) Since we are going to consider giant magnon solutions, we restrict ourselves to the R t × S 3 η subspace, which corresponds to the following choice in AdS η On S 5 η we first introduce the angleθ in the following way r = sinθ, which leads to Now, to go to S 3 η , we can safely set φ = 0,θ = π 2 (we also exchange φ 1 and φ 2 and replace ξ with θ). Thus, the background seen by the string moving in the R t × S 3 η subspace can be written as

Exact results
Here and further on we will work in conformal gauge when γ αβ = diag(−1, 1) and the string Lagrangian and Virasoro constraints have the form are the fields induced on the string worldsheet. For the case under consideration The corresponding nonzero components of g M N and b M N are given in (2.9). Now, we impose the following ansatz for the string embedding where τ and σ are the string world-sheet coordinates, F i (ξ), θ(ξ) are arbitrary functions of ξ, and κ, ω i , α, β are parameters.
Replacing (3.3) into (3.1) one finds the following solutions of the equations of motion for φ i (τ, σ) (we introduced the notation χ ≡ cos 2 θ) where C 1 , C 2 are integration constants.
By using (3.4), (3.5), one can show that the Virasoro constraints (3.2) take the form dχ dξ , Next, we solve (3.7) with respect to C 1 and replace the solution into (3.6). The result is where The solution ξ(χ) of (3.8) is where F is the incomplete elliptic integral of first kind and χ η > χ p > χ > χ m > χ n .
Inverting ξ(χ) to χ(ξ), one finds where DN(x, m) is one of the Jacobi elliptic functions and By using (3.8) we can find the explicit solutions for the isometric angles φ 1 , φ 2 . They are given by where Π is the incomplete elliptic integral of third kind.
Now, let us go to the computations of the conserved charges Q µ , i.e. the string energy E s and the two angular momenta J 1 , J 2 . Starting with and applying the ansatz (3.3), one finds We will need also the expression for the angular difference ∆φ 1 . The computations give the following result .

Solving the integrals in (3.17)-(3.20) and introducing the notations
we finally obtain where K and Π are the complete elliptic integrals of first and third kind.

Small ǫ-expansions and dispersion relation
In this section we restrict ourselves to the simpler case of giant magnons with one nonzero angular momentum. To this end, we set the second isometric angle φ 2 = 0. From the solution (3.16) it is clear that φ 2 is zero when or equivalently (see (3.21)) Then it follows from (3.12) that χ n = 0 because χ η > χ p > χ m > 0 for the finite-size case.
In addition, we express χ m through the other parameters in correspondence with (3.21) As a cosequence (3.9)-(3.11) take the form In order to obtain the leading finite-size effect on the dispersion relation, we consider the limit ǫ → 0 in (4.1)-(4.3) first. We will use the following small ǫ-expansions for the remaining parameters χ η = χ η0 + (χ η1 + χ η2 log ǫ)ǫ (4.4) Replacing (4.4) into (4.1)-(4.3) and expanding in ǫ one finds the following solution of the resulting equations
For ǫ = 0, (4.9) reduces to the dispersion relation for the infinite-size giant magnon obtained in [7] for the large g case. In the limitη → 0, (4.9) gives the correct result for the undeformed case found in [9].

Concluding Remarks
Here we dealt with strings moving in the R t × S 3 η subspace of the η-deformed AdS 5 × S 5 . The finite-size dyonic giant magnon solution is contained in the string configurations we considered.
We derived the explicit exact solutions for the string coordinates and the corresponding conserved charges. Then we restricted ourselves to the case of giant magnons with one nonzero angular momentum and obtained the dispersion relation for them including the leading finite-size effect on it.
It will be interesting to extend the result (4.9), (4.10) to the case of dyonic giant magnons. We will report on this soon.
Another possible direction of further investigation is to show that (4.9), (4.10) can be reproduced by using the Lüscher formula for the finite-size effects on the dispersion relation and we are going to do that in the near future.