### On String S-matrix, Bound States and TBA

, 2007

"... The study of finite J effects for the light-cone AdS5 × S5 superstring by means of the Thermodynamic Bethe Ansatz requires an understanding of a companion 2d theory which we call the mirror model. It is obtained from the original string model by the double Wick rotation. The S-matrices describing t ..."

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The study of finite J effects for the light-cone AdS5 × S5 superstring by means of the Thermodynamic Bethe Ansatz requires an understanding of a companion 2d theory which we call the mirror model. It is obtained from the original string model by the double Wick rotation. The S-matrices describing the scattering of physical excitations in the string and mirror models are related to each other by an analytic continuation. We show that the unitarity requirement for the mirror S-matrix fixes the S-matrices of both theories essentially uniquely. The resulting string S-matrix S(z1, z2) satisfies the generalized unitarity condition and, up to a scalar factor, is a meromorphic function on the elliptic curve associated to each variable z. The double Wick rotation is then accomplished by shifting the variables z by quarter of the imaginary period of the torus. We discuss the apparent bound states of the string and mirror models, and show that depending on a choice of the physical region there are one, two or 2M−1 solutions of the M-particle bound state equations sharing the same conserved charges. For very large but finite values of J, most of these solutions, however, exhibit various signs of pathological behavior. In particular, they

### The Dressing Factor and Crossing Equations Gleb Arutyunova∗ † b †

, 904

"... Abstract: We utilize the DHM integral representation for the BES dressing factor of the world-sheet S-matrix of the AdS5×S 5 light-cone string theory, and the crossing equations to fix the principal branch of the dressing factor on the rapidity torus. The results obtained are further used, in conjun ..."

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Abstract: We utilize the DHM integral representation for the BES dressing factor of the world-sheet S-matrix of the AdS5×S 5 light-cone string theory, and the crossing equations to fix the principal branch of the dressing factor on the rapidity torus. The results obtained are further used, in conjunction with the fusion procedure, to determine the bound state dressing factor of the mirror theory. We convincingly demonstrate that the mirror bound state S-matrix found in this way does not depend on the internal structure of a bound state solution employed in the fusion procedure. This welcome feature is in perfect parallel to string theory, where the corresponding bound state S-matrix has no bearing on bound state constituent particles as well. The mirror bound state S-matrix we found provides the final missing piece in setting up the TBA equations for the AdS5 × S 5 mirror theory.

### The S-matrix of String Bound States

, 803

"... Abstract: We find the S-matrix which describes the scattering of two-particle bound states of the light-cone string sigma model on AdS5 × S5. We realize the M-particle bound state representation of the centrally extended su(2|2) algebra on the space of homogeneous (super)symmetric polynomials of deg ..."

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Abstract: We find the S-matrix which describes the scattering of two-particle bound states of the light-cone string sigma model on AdS5 × S5. We realize the M-particle bound state representation of the centrally extended su(2|2) algebra on the space of homogeneous (super)symmetric polynomials of degree M depending on two bosonic and two fermionic variables. The scattering matrix SMN of M- and N-particle bound states is a differential operator of degree M + N acting on the product of the corresponding polynomials. We require this operator to obey the invariance condition and the Yang-Baxter equation, and we determine it for the two cases M = 1, N = 2 and M = N = 2. We show that the S-matrices found satisfy generalized physical unitarity, CPT invariance, parity transformation rule and crossing symmetry. Although the dressing factor as a function of four parameters x + 1, x−1, x+ 2, x−2 is universal for scattering of any bound states, it obeys a crossing symmetry equation which depends on M and N.

### Finite-Size Effects for Dyonic Giant Magnons

, 801

"... We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in ..."

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We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in which incoming particles are boundstates. By careful choice of poles in the su(2|2) 2-invariant S-matrix, we find agreement of the two results, which makes possible to predict the (leading) finite-size correction for dyonic giant magnons to all orders in the ’t Hooft coupling.

### Preprint typeset in JHEP style- PAPER VERSION Scattering of single spikes

, 710

"... Abstract: We apply the dressing method to a string solution given by a static string wrapped around the equator of a three-sphere and find that the result is the single spike solution recently discussed in the literature. Further application of the method allows the construction of solutions with mu ..."

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Abstract: We apply the dressing method to a string solution given by a static string wrapped around the equator of a three-sphere and find that the result is the single spike solution recently discussed in the literature. Further application of the method allows the construction of solutions with multiple spikes. In particular we construct the solution describing the scattering of two single spikes and compute the scattering phase shift. As a function of the dressing parameters, the result is exactly the same as the one for the giant magnon, up to non-logarithmic terms. This suggests that the single spikes should be described by an integrable spin chain closely related to the one associated to the giant magnons. The field theory interpretation of such spin chain however is still unclear. Keywords: Classical string solutions, AdS/CFT, spin chains, integrable systems. Contents

### D1-brane in β-Deformed Background

, 710

"... Abstract: We study various configurations of rotating and wound D1-brane in AdS5 ×S 5 background and in its β deformed version. We find giant magnon and spike solutions on the world-volume of D1-brane in AdS5 × S 5 background. We also analyse the equations of motion of D1-brane in β-deformed backgro ..."

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Abstract: We study various configurations of rotating and wound D1-brane in AdS5 ×S 5 background and in its β deformed version. We find giant magnon and spike solutions on the world-volume of D1-brane in AdS5 × S 5 background. We also analyse the equations of motion of D1-brane in β-deformed background. We show that in the limit of large electric flux on world-volume of D1-brane they reduce to the equations that describe collection of large number of fundamental strings. We also construct rotating and wound D1-brane solution that has two equal spins on S 5 γ.

### Finite-Size Effects for Dyonic Giant Magnons

, 801

"... We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in ..."

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We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in which incoming particles are boundstates. By careful choice of poles in the su(2|2) 2-invariant S-matrix, we find agreement of the two results, which makes possible to predict the (leading) finite-size correction for dyonic giant magnons to all orders in the ’t Hooft coupling.

### Finite-Size Effects for Dyonic Giant Magnons

, 801

"... We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in ..."

Abstract
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(Show Context)
We compute finite-size corrections to dyonic giant magnons in two ways. One is by examining the asymptotic behavior of helical strings of hep-th/0609026 as elliptic modulus k goes to unity, and the other is by applying the generalized Lüscher formula for µ-term of arXiv:0708.2208 to the situation in which incoming particles are boundstates. By careful choice of poles in the su(2|2) 2-invariant S-matrix, we find agreement of the two results, which makes possible to predict the (leading) finite-size correction for dyonic giant magnons to all orders in the ’t Hooft coupling.