SU(2|2) for Theories with Sixteen Supercharges at Weak and Strong Coupling

We consider the dimensional reductions of N=4 Supersymmetric Yang-Mills theory on R x S^3 to the three-dimensional theory on R x S^2, the orbifolded theory on R x S^3/Z_k, and the plane-wave matrix model. With explicit emphasis on the three-dimensional theory, we demonstrate the realization of the SU(2|3) algebra in a radial Hamiltonian framework. Using this structure we constrain the form of the spin chains, their S-matrices, and the corresponding one- and two-loop Hamiltonian of the three dimensional theory and find putative signs of integrability up to the two-loop order. The string duals of these theories admit the IIA plane-wave geometry as their Penrose limit. Using known results for strings quantized on this background, we explicitly construct the strong-coupling dual extended SU(2|2) algebra and discuss its implications for the gauge theories.Comment: 37 pages, 1 figure. v2 some minor improvements to the text, version to appear in Phys.Rev.

Review of AdS/CFT Integrability, Chapter I.3: Long-range spin chains

In this contribution we briefly review recent developments in the theory of long-range integrable spin chains. These spin chains constitute a natural generalisation of the well-studied integrable nearest-neighbour chains and are of particular relevance to the integrability in the AdS/CFT correspondence since the dilatation operator in the asymptotic region is conjectured to be a Hamiltonian of an integrable long-range psu spin chain.Comment: 17 pages, see also overview article arXiv:1012.3982, v2: references to other chapters updated, v3: minor typos corrected, references adde

Matrix Reduction and the su(2|2) superalgebra in AdS/CFT

We study the supersymmetry generators Q, S on the 1-loop vectorless sector of N=4 Super Yang-Mills, by reduction to the plane-wave matrix model. Using a coherent basis in the su(2|2) sector, a comparison with the algebra given by Beisert in nlin/0610017 is presented, and some parameters (up to one-loop) are determined. We make a final comparison of these supercharges with the results that can be obtained from the string action by working in the light-cone-gauge and discretizing the string.Comment: 20 pages, no figures v2: Typos corrected, references adde

The Complete One-Loop Dilation Operator of N=2 SuperConformal QCD

We evaluate the full planar one-loop dilation operator of N=2 SuperConformal QCD, the SU(N_c) super Yang-Mills theory with N_f = 2 N_c fundamental hypermultiplets, in the flavor-singlet sector. Remarkably, the spin-chain Hamiltonian turns out to be completely fixed by superconformal symmetry, as in N=4 SYM. We present a more general calculation, for the superconformal quiver theory with SU(N_c)X SU(N_c) gauge group, which interpolates between N=2 SCQCD and the Z_2 orbifold of N=4 SYM; here symmetry fixes the Hamiltonian up to a single parameter, corresponding to the ratio of the two marginal gauge couplings.Comment: v2: typo corrected, cosmetic changes. JHEP versio

Higher-Loop Integrability in N=4 Gauge Theory

The dilatation operator measures scaling dimensions of local operator in a conformal field theory. Algebraic methods of constructing the dilatation operator in four-dimensional N=4 gauge theory are reviewed. These led to the discovery of novel integrable spin chain models in the planar limit. Making use of Bethe ansaetze, a superficial discrepancy in the AdS/CFT correspondence was found, we discuss this issue and give a possible resolution

Fluctuations and Energy Shifts in the Bethe Ansatz

We study fluctuations and finite size corrections for the ferromagnetic thermodynamic limit in the Bethe ansatz for the Heisenberg XXX1/2 spin chain, which is the AdS/CFT dual of semiclassical spinning strings. For this system we derive the standard quantum mechanical formula which expresses the energy shift as a sum over fluctuation energies. As an example we apply our results to the simplest, one-cut solution of this system and derive its spectrum of fluctuations.Comment: 8 pages, 1 figure, v2: comparison to string theory improved, reference adde

Long-Range GL(n) Integrable Spin Chains and Plane-Wave Matrix Theory

Quantum spin chains arise naturally from perturbative large-N field theories and matrix models. The Hamiltonian of such a model is a long-range deformation of nearest-neighbor type interactions. Here, we study the most general long-range integrable spin chain with spins transforming in the fundamental representation of gl(n). We derive the Hamiltonian and the corresponding asymptotic Bethe ansatz at the leading four perturbative orders with several free parameters. Furthermore, we propose Bethe equations for all orders and identify the moduli of the integrable system. We finally apply our results to plane-wave matrix theory and show that the Hamiltonian in a closed sector is not of this form and therefore not integrable beyond the first perturbative order. This also implies that the complete model is not integrable.Comment: 22 pages, v2: reference adde

The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure

In this paper we investigate the algebraic structure of AdS/CFT in the strong-coupling limit. We propose an expression for the classical r-matrix with (deformed) u(2|2) symmetry, which leads to a quasi-triangular Lie bialgebra as the underlying symmetry algebra. On the fundamental representation our r-matrix coincides with the classical limit of the quantum R-matrix.Comment: 31 pages, v2: added comment on classical double structure in 4.5. new section 5 on relation to other algebras (from old appendix and new results). fixed typos and mathematical inaccuracies, added references, v3: improved mathematical presentation, to appear in CM

On Quantum Corrections to Spinning Strings and Bethe Equations

Recently, it was demonstrated that one-loop energy shifts of spinning superstrings on AdS5xS5 agree with certain Bethe equations for quantum strings at small effective coupling. However, the string result required artificial regularization by zeta-function. Here we show that this matching is indeed correct up to fourth order in effective coupling; beyond, we find new contributions at odd powers. We show that these are reproduced by quantum corrections within the Bethe ansatz. They might also identify the "three-loop discrepancy" between string and gauge theory as an order-of-limits effect.Comment: 12 pages, v2, v3: minor corrections, footnotes and references added, v3: to appear in Phys. Lett.

Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence

We define Landau-Lifshitz sigma models on general coset space $G/H$, with $H$ a maximal stability sub-group of $G$. These are non-relativistic models that have $G$-valued N\"other charges, local $H$ invariance and are classically integrable. Using this definition, we construct the $PSU(2,2|4)/PS(U(2|2)^2)$ Landau-Lifshitz sigma-model. This sigma model describes the thermodynamic limit of the spin-chain Hamiltonian obtained from the complete one-loop dilatation operator of the N=4 super Yang-Mills (SYM) theory. In the second part of the paper, we identify a number of consistent truncations of the Type IIB Green-Schwarz action on $AdS_5\times S^5$ whose field content consists of two real bosons and 4,8 or 16 real fermions. We show that $\kappa$-symmetry acts trivially in these sub-sectors. In the context of the large spin limit of the AdS/CFT correspondence, we map the Lagrangians of these sub-sectors to corresponding truncations of the $PSU(2,2|4)/PS(U(2|2)^2)$ Landau-Lifshitz sigma-model.Comment: 42 page