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Register a new userIntegrable Hamiltonian for Classical Strings on AdS_5 x S^5
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We find the Hamiltonian for physical excitations of the classical bosonic string propagating in the AdS_5 x S^5 space-time. The Hamiltonian is obtained in a so-called uniform gauge which is related to the static gauge by a 2d duality transformation. The Hamiltonian is of the Nambu type and depends on two parameters: a single S^5 angular momentum J and the string tension lambda. In the general case both parameters can be finite. The space of string states consists of short and long strings. In the sector of short strings the large J expansion with lambda=lambda/J^2 fixed recovers the plane-wave Hamiltonian and higher-order corrections recently studied in the literature. In the strong coupling limit lambdato infty, J fixed, the energy of short strings scales as sqrt[4]{lambda} while the energy of long strings scales as sqrt{lambda}. We further show that the gauge-fixed Hamiltonian is integrable by constructing the corresponding Lax representation. We discuss some general properties of the monodromy matrix, and verify that the asymptotic behavior of the quasi-momentum perfectly agrees with the one obtained earlier for some specific cases.
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We consider classical superstrings propagating on AdS_5 x S^5 space-time. We consistently truncate the superstring equations of motion to the so-called su(1|1) sector. By fixing the uniform gauge we show that physical excitations in this sector are described by two complex fermionic degrees of freedom and we obtain the corresponding Lagrangian. Remarkably, this Lagrangian can be cast in a two-dimensional Lorentz-invariant form. The kinetic part of the Lagrangian induces a non-trivial Poisson structure while the Hamiltonian is just the one of the massive Dirac fermion. We find a change of variables which brings the Poisson structure to the canonical form but makes the Hamiltonian nontrivial. The Hamiltonian is derived as an exact function of two parameters: the total S^5 angular momentum J and string tension lambda; it is a polynomial in 1/J and in sqrt{lambda} where lambda=frac{lambda}{J^2} is the effective BMN coupling. We identify the string states dual to the gauge theory operators from the closed su(1|1) sector of N=4 SYM and show that the corresponding near-plane wave energy shift computed from our Hamiltonian perfectly agrees with that recently found in the literature. Finally we show that the Hamiltonian is integrable by explicitly constructing the corresponding Lax representation.
We discuss some new simple closed bosonic string solutions in AdS_5 x S^5 that may be of interest in the context of AdS/CFT duality. In the first part of this work we consider solutions with two spins (S_1, S_2) in AdS_5. Starting from the flat-space solutions and using perturbation theory in the curvature of AdS_5 space, we construct leading terms in the small two-spin solution. We find corrections to the leading Regge term in the classical string energy and uncover a discontinuity in the spectrum for certain type of a solution. We then analyze the connection between small-spin and large-spin limits of string solutions in AdS_5. We show that the S_1 = S_2 solution in AdS_5 found in earlier papers admits both limits only in simplest cases of the folded and rigid circular strings. In the second part of the paper we construct a new class of chiral solutions in R_t x S^5 for which embedding coordinates of S^5 satisfy the linear Laplace equations. They generalize the previously studied rigid string solutions. We study in detail a simple nontrivial example.
We explore integrability properties of superstring equations of motion in AdS_5 x S^5. We impose light-cone kappa-symmetry and reparametrization gauges and construct a Lax representation for the corresponding Hamiltonian dynamics on subspace of physical superstring degrees of freedom. We present some explicit results for the corresponding conserved charges by consistently reducing the dynamics to AdS_3 x S^3 and AdS_3 x S^1 subsectors containing both bosonic and fermionic fields.
In this paper, considering the correspondence between spin chains and string sigma models, we explore the rotating string solutions over $ eta $ deformed $ AdS_5 times S^{5} $ in the so called fast spinning limit. In our analysis, we focus only on the bosonic part of the full superstring action and compute the relevant limits on both $(R times S^{3})_{eta} $ and $(R times S^{5})_{eta} $ models. The resulting system reveals that in the fast spinning limit, the sigma model on $ eta $ deformed $S^5$ could be $textit{approximately}$ thought of as the continuum limit of anisotropic $ SU(3) $ Heisenberg spin chain model. We compute the energy for a certain class of spinning strings in deformed $S^5$ and we show that this energy can be mapped to that of a similar spinning string in the purely imaginary $beta$ deformed background.
Using information from the marginality conditions of vertex operators for the AdS_5 x S^5 superstring, we determine the structure of the dependence of the energy of quantum string states on their conserved charges and the string tension proportional to lambda^(1/2). We consider states on the leading Regge trajectory in the flat space limit which carry one or two (equal) spins in AdS_5 or S^5 and an orbital momentum in S^5, with Konishi multiplet states being particular cases. We argue that the coefficients in the energy may be found by using a semiclassical expansion. By analyzing the examples of folded spinning strings in AdS_5 and S^5 as well as three cases of circular two-spin strings we demonstrate the universality of transcendental (zeta-function) parts of few leading coefficients. We also show the consistency with target space supersymmetry with different states belonging to the same multiplet having the same non-trivial part of the energy. We suggest, in particular, that a rational coefficient (found by Basso for the folded string using Bethe Ansatz considerations and which, in general, is yet to be determined by a direct two-loop string calculation) should, in fact, be universal.
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