Finite-size effect of \eta-deformed AdS_5 x S^5 at strong coupling

We compute Luscher corrections for a giant magnon in the \eta-deformed (AdS_5\times S^5)_{\eta} using the su(2|2)_q-invariant S-matrix at strong coupling and compare with the finite-size effect of the corresponding string state, derived previously. We find that these two results match and confirm that the su(2|2)_q-invariant S-matrix is describing world-sheet excitations of the \eta-deformed background.


Introduction
AdS/CF T duality [1], a correspondence between string theories in AdS background with certain supersymmetric and conformal Yang-Mills theories on the boundary space-time of the AdS space, has been a hot topic for theoretical researches and produced many important quantitative results and applications (for overview see [2]). In these developments, integrability has played a crucial role on both sides of the correspondence. Two-dimensional world-sheet actions for the string theory moving in the background are described by nonlinear sigma models on coset group manifolds which are classically integrable. Aspects of quantum integrable structure of supersymmetric Yang-Mills theories appear in Bethe ansatz equations and related exact integrable machineries which can determine conformal dimensions of the CFTs. Quantum S-matrices of the world-sheet actions provide integrable framework which interpolate from the strong to weak coupling limits.
An important direction of research is to find new AdS/CFT pairs which show novel integrability structures. One such string theory, which has been studied recently, is the type IIB superstring theory in the η-deformed targe space (AdS 5 × S 5 ) η for a real parameter η [3]. The classical integrability of nonlinear sigma model is provided by solutions of the classical Yang-Baxter equation [4]. (See [5,6,7] for related issues.) It has been conjectured in [3] that full quantum S-matrix of the deformed sigma model is given by the R-matrix of the q-deformed Hubbard model which has been proposed much earlier in [8]. When q is a complex phase, the dressing phase of the S-matrix and bound-states have been analyzed in [9]. Scattering amplitudes of bosonic exitations for small values of the world-sheet momentum have been computed and shown to agree with the q-deformed S-matrix in the large string tension (strong coupling) limit for real q with explicit relation with η [10]. Based on the exact S-matrix, thermodynamic Bethe ansatz equations for ground states and dressing phase for real q have been studied in [11].
A pertinent issue which should be mentioned is that the deformed sigma model is not a fully consistent string theory at quantum level. It has been found that this η-deformed sigma model does not solve the type IIB supergravity equations of motion [12], but rather a generalization of them [13]. This generalized ones allow only scale invariance but not full Weyl invariance at one-loop [14]. The Weyl invarince can be restored if the deformation is generalized by some modified solutions of the Yang-Baxter equation [15]. This suggests that one should pay attention to treat the η-deformed theory at quantum level.
In this letter, we provide another evidence for the q-deformed S-matrix to describe the string theory on the η-deformed geometry. For this purpose, we consider finite-size effects of a giant magnon state, a classical string configuration living on a subspace of the (AdS 5 ×S 5 ) η [16]. These corrections have been computed for the undeformed AdS 5 ×S 5 in [17,18] and for the γ-deformed AdS 5 ×S 5 in [19,20] from both directions of string solutions and world-sheet S-matrices. For the η-deformed case, this effect has been studied from only string theory side in [21], which will be reviewed in sect.2. Exact q-deformed S-matrix and related formula will be presented in sect.3. We present our computation of the Lüscher corrections for a giant magnon based on q-deformed S-matrix in sect.4 along with a conjecture on the deformed dressing phase in sect.5. In sect.6, we conclude with a short summary and comments.
2 Finite-size effect of a giant magnon in (AdS 5 × S 5 ) η In this section, we give a brief review on computing the energy of a giant magnon using Neumann-Rosochatius ansatz following [21]. The giant magnon is defined in the R t × S 3 η subspace of (AdS 5 × S 5 ) η , where backgound metric and B-field are given by Deformation parameterη is related to original parameter η byη = 2η/(1 − η 2 ).
One can solve the giant magnon configuration using an ansatz for the dynamics of the target space coordinates where τ and σ are the string world-sheet coordinates and the Virasoro constraints. If we restrict further to S 2 by setting the isometry angle φ 2 to zero, conserved charges E s , J 1 corrsponding to other isometric coordinates t, φ 1 are given by complete elliptic integrals of first and third kinds (W = κ 2 /ω 2 1 ): 3) where the parameters are satisfying The momentum of a giant magnon, which is related to the deficit angle by ∆φ 1 = p, satisfies Eqs.(2.3) and (2.4) generate the dispersion relation of a giant magnon at finite J 1 .
In the limit of J 1 ≫ g ≫ 1 one can solve the parameter relations in terms of small ǫ-expansions to determine the energy and angular momentum (2.5) The first term is the energy dispersion relation of a giant magnon in the infinite volume and the second one is the small finite-size (or finite J 1 ) correction. In next sections, we are going to reproduce this result from the su(2|2) q S-matrix.

q-deformed S-matrix
The quantum-deformed S-matrix can be written as a graded tensor product of su(2|2) qinvarint matrix as follows: The overall scalar factor S su(2) is given by [ with q-deformed dressing phase σ. The su(2|2) q -invarint S-matrix has 16 × 16 elements, . (3.9) The parameters x ± satisfy a shortening relation and related to energy E and momentum p by The constant ξ is related to the string tension g and deformation parameter q by It is claimed that the quantum group parameter q is related toη by q = e −ν/g with ν =η 1 +η 2 . (3.13) General energy-momentum dispersion relation follows from this (3.14) At strong coupling limit g ≫ 1, Eqs. where x ± 0 (p) = e ±ip/2 1 +η 2 sin 2 p 2 ∓η sin p 2 , (3.17) .

(3.18)
Also the dispersion relation in Eq.(3.14) becomes which is consistent with that of giant magnon string state given in the first term of Eq.(2.5).

Lüscher corrections
Leading finite-size corrections in the strong coupling limit are the µ-term Lüscher corrections which arise from residues of S-matrix in the contour integrals of the F -term formula. Explicit µ-term Lüscher formula for one su(2) giant magnon state with su(2|2) index (11) is given by [22,18], whereq is the location of S-matrix the poles. The physical giant magnon has momentum p and energy given by (3.19), while the momentum q ⋆ of the virtual particle satisfies the following on-shell relation We also use a short notationq ⋆ = q ⋆ (q).
We start with locating the poles of the S-matrix. The overall scalar factor S su(2) (p, q ⋆ ) in (3.7) have both s-channel pole at x − (q ⋆ ) = x + (p) and t-channel pole at x − (q ⋆ ) = 1/x + (p).
We have checked that the t-channel gives exactly same results as the s-channel. We will present a detailed computation for the s-channel here and multiply a factor 2 at the end.

q-deformed Dressing phase
The dressing phase has been proposed first in terms of q-deformed Gamma function for q a complex phase, [9] , (5.32) where a = ν/g for g ≫ 1 and u(z) is defined by An integral representation for Γ q 2 given in [9] can be analytically continued for real q to get strong coupling limit [10] log where ψ −2 is the poly-gamma function. The integrals over two unit circles in (5.32) may develop a branch cut for ν ≥ 1/2 but can be handled with proper care as pointed out in [11].
For computing σ(p,q ⋆ ) at strong coupling, we combine the χ functions with arguments x ± (p), x ± (q ⋆ ) given in Eqs. (3.16) and (4.22) to get with a short notation x ± 0 = x ± 0 (p) given in (3.17). Due to complicated branch cuts appearing in the contour integrals, we could not evaluate this integral analytically. However, we have found numerically that the integration depends onη very insensitively within available accuracy. This leads to our conjecture that the dressing phase with given arguments in the strong coupling limit is which is the result for the undeformed case, computed from the AFS phase [23] in [18].

Conclusion
Compared with finite-size giant magnon computation (2.5), the strong coupling Lüscher correction match quite well except 1 +η 2 in the overall factor. We think this factor should be modified in the string theory computation. Apart from this minor discrepancy, both coefficient and exponent of the exponential factor show correct dependence on the momentum and deformation parameter. Our check is valid for the su(2) sector with generic value of p and supports that the q-deformed S-matrix should describe the string theory in the η-deformed AdS background. It will be interesting to further elaborate the q-deformed dressing phase to check (5.36) both numerically and analytically. Another interesting but less studied domain is the weak coupling limit of the S-matrix, which could be related to certain q-deformed spin-chain.