Finite-size effects for single spike

Abstract

We use the reduction of the string dynamics on R t × S 3 to the Neumann-Rosochatius integrable system to map all string solutions described by this dynamical system onto solutions of the complex sine-Gordon integrable model. This mapping relates the parameters in the solutions on both sides of the correspondence. In the framework of this approach, we find finite-size string solutions, their images in the (complex) sine-Gordon system, and the leading finite-size effects of the single spike "E − ∆ϕ" relation for both R t × S 2 and R t × S 3 cases.Recent developments in AdS/CFT correspondence between type IIB strings on AdS 5 ×S 5 and its dual N = 4 super Yang-Mills (SYM) theories [1] are mainly based on the integrabilities discovered in both theories. Integrability of the SYM side appears in the calculations of conformal dimensions which are related to the string energies according to the AdS/CFT correspondence. A remarkable observation by Minahan and Zarembo [2] is that the conformal dimension of an operator composed of scalar fields in the N = 4 SYM in the planar limit can be computed by diagonalizing the Hamiltonian of one-dimensional integrable spin chain model. This task can be done by solving a set of coupled algebraic equations called Bethe ansatz equations. It has been shown that explicit calculation of the eigenvalues for various SYM operators agree with those computed from the SYM perturbation theory. This result has been further extended to the full P SU(2, 2|4) sector [3] and the Bethe ansatz equations which are supposed to hold for all loops are conjectured [4,5].The string side of the correspondence is mostly studied at the classical level due to the lack of full quantization. The type IIB string theory on AdS 5 × S 5 is described by a nonlinear sigma model with P SU(2, 2|4) symmetry [6]. This sigma model has been shown to have an infinite number of local and nonlocal conserved currents [7] and some of the conserved charges such as energy and angular momentum are computed explicitly from the classical integrability (see, for example, [8] and the references therein). These results based on the classical integrability provide valuable information on the AdS/CFT duality in the domain of large t'Hooft coupling constant. A new direction to quantizing the string theory is to find exact S-matrix between the fundamental spectrum of the theory on the world sheet. It has been shown that the S-matrix along with the exact particle spectrum can be determined by the underlying symmetry and integrability in the theory [9,10]. The overall scalar factor of the S-matrix, sometimes called as dressing phase, which can not be determined by the symmetry alone, has been computed exactly [11,12] from the crossing relation [13]. In this process, the explicit expression of the dressing phase in the classical limit, which was determined by the classical integrability, was essential [14].Various classical solutions play an important role in testing and understanding the correspondence. The classical giant magnon (GM) state [15] ...

Neumann-Rosochatius integrable system for strings onAdS4× ℂℙ3

“…It coincides with the string result for R t × S 3 found in [30]. As in the GM case, the difference is in the identification (3.6).…”

“…While the string dynamics are quite different, computations are identical to the cases in the sphere geometries. Therefore, we will provide only the results here, referring technical details to [30,18].…”

Large-spin and large-winding expansions of giant magnons and single spikes

“…by adding extra angular momenta. Finite momentum effects for single spikes have been considered in [39]. The following result,…”

“…As we have already noted, only the first six classical leading terms (A n0 , 1 ≤ n ≤ 6) in the dispersion relation of giant magnons have been computed in [19], along with one subleading and one next-to-subleading term (A 21 , A 20 ) that were found in [17]. For single spikes, the leading finite-size correction 10 was computed in [39]. Many more classical terms can be obtained with Mathematica (cf.…”

Finite-size giant magnons on AdS4×CPγ3

“…Integrating (2.6) and (2.7), one can find string solutions with very different properties. All of them are related to solutions of the complex sine-Gordon integrable model in an explicit way [10].…”