Gleb Arutyunov

Bethe ansatz for quantum strings

We propose Bethe equations for the diagonalization of the Hamiltonian of quantum strings on AdS5 ×S 5 at large string tension and restricted to certain large charge states from a closed su(2) subsector. The ansatz differs from the recently proposed all-loop gauge theory asymptotic Bethe ansatz by additional factorized scattering terms for the local excitations. We also show that our ansatz quantitatively reproduces everything that is currently known about the string spectrum of these states. Firstly, by construction, we recover the integral Bethe equations describing semiclassical spinning strings. Secondly, we explain how to derive the 1/J energy corrections of M-impurity BMN states, provide explicit, general formulae for both distinct and confluent mode numbers, and compare to asymptotic gauge theory. In the special cases M = 2,3 we reproduce the results of direct quantization of Callan et al. Lastly, at large string tension and relatively small charge we recover the famous 2 4 √ n 2 � asymptotics of massive string modes at level n. Remarkably, this behavior is entirely determined by the novel scattering terms. This is qualitatively consistent with the conjecture that these terms occur due to wrapping effects in gauge theory. Our finding does not in itself cure the disagreements between gauge and string theory, but leads us to speculate about the structure of an interpolating Bethe ansatz for the AdS/CFT system at finite coupling and charge.

Spinning strings in AdS5 x S5 and integrable systems

We show that solitonic solutions of the classical string action on the AdS5 × S 5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous dimensions of the corresponding composite operators in N = 4 super YM theory. We obtain a closed system of equations that determines this subleading coefficient and, therefore, the 1-loop anomalous dimensions of the dual SYM operators. We expect that an equivalent system of equations should follow from the thermodynamic limit of the algebraic Bethe ansatz for the SO(6) spin chain derived from SYM theory. We also identify a particular string solution whose classical energy exactly reproduces the one-loop anomalous dimension of a certain set of SYM operators with two independent R charges J1, J2.

Thermodynamic Bethe Ansatz for the AdS_5 x S^5 Mirror Model

We use the string hypothesis for the mirror theory to derive the Thermodynamic Bethe Ansatz equations for the AdS5 × S5 mirror model. We further demonstrate how these equations can be used to construct the associated Y-system recently discussed in the literature, putting particular emphasis on the assumptions and the range of validity of the corresponding construction.

Foundations of the AdS5 × S5 superstring: I

We review the recent advances towards finding the spectrum of the AdS5 ? S5 superstring. We thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem. In certain cases our exposition is original and cannot be found in the existing literature. The present part I deals with foundations of classical string theory in AdS5 ? S5, light-cone perturbative quantization and the derivation of the exact light-cone world-sheet scattering matrix.

Spinning strings in AdS(5) x S**5: New integrable system relations

A general class of rotating closed string solutions in AdS5 × S 5 is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional “centrifugal” potential. We expect that the reduction of the AdS5 × S 5 sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS5 and three S 5 spins. As in other recently discussed examples, the leading large-spin correction to the classical energy turns out to be proportional to the square of the string tension or the ’t Hooft coupling λ, suggesting that it can be matched onto the one-loop anomalous dimensions of the corresponding “long” operators on the SYM side of the AdS/CFT duality.

Finite-size effects from giant magnons

In order to analyze finite-size effects for the gauge-fixed string sigma model on AdS5 × S 5 , we construct one-soliton solutions carrying finite angular momentum J . In the infinite J limit the solutions reduce to the recently constructed one-magnon configuration of Hofman and Maldacena. The solutions do not satisfy the level-matching condition and hence exhibit a dependence on the gauge choice, which however disappears as the size J is taken to infinity. Interestingly, the solutions do not conserve all the global charges of the psu(2, 2|4) algebra of the sigma model, implying that the symmetry algebra of the gauge-fixed string sigma model is different from psu(2, 2|4) for finite J , once one gives up the level-matching condition. The magnon dispersion relation exhibits exponential corrections with respect to the infinite J solution. We also find a generalisation of our one-magnon configuration to a solution carrying two charges on the sphere. We comment on the possible implications of our findings for the existence of the Bethe ansatz describing the spectrum of strings carrying finite charges.

The Zamolodchikov-Faddeev Algebra for AdS5 x S5 Superstring

We discuss the Zamolodchikov-Faddeev algebra for the superstring sigma-model on AdS5 ×S 5 . We find the canonical su(2|2) 2 invariant S-matrix satisfy- ing the standard Yang-Baxter and crossing symmetry equations. Its near-plane-wave expansion matches exactly the leading order term recently obtained by the direct per- turbative computation. We also show that the S-matrix obtained by Beisert in the gauge theory framework does not satisfy the standard Yang-Baxter equation, and, as a consequence, the corresponding ZF algebra is twisted. The S-matrices in gauge and string theories however are physically equivalent and related by a non-local transformation of the basis states which is explicitly constructed.

On String S-matrix, Bound States and TBA

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 might receive a finite J correction to their energy which is complex, or the energy correction might exceed corrections arising due to finite J modifications of the Bethe equations thus making the asymptotic Bethe ansatz inapplicable.

Matching Higher Conserved Charges for Strings and Spins

We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N = 4 gauge theory and energies of spinning strings on AdS5 × S 5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.

Integrable Hamiltonian for Classical Strings on AdS5 x S5

We flnd the hamiltonian for physical excitations of the classical bosonic string propagating in the AdS5£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 ‚. In the general case both parameters can be flnite. The space of string states consists of short and long strings. In the sector of short strings the large J expansion with ‚ 0 = ‚=J 2 flxed recovers the plane-wave hamiltonian and higher-order corrections recently studied in the literature. In the strong coupling limit ‚ ! 1, J flxed, the energy of short strings scales as 4 p ‚ while the energy of long strings scales as p ‚. We further show that the gauge-flxed 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 speciflc cases.