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.Comment: 13 pages, Talk given at Strings 2004, Paris, 28 June - 2 July, v2: reference adde

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

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.

A large spin limit of strings on AdS_5 x S^5 in a non-compact sector

We study the scaling law of the energy spectrum of classical strings on AdS_5 x S^5, in particular, in the SL(2) sector for large S (AdS spin) and fixed J (S^1 \subset S^5 spin). For any finite gap solution, we identify the limit in which the energy exhibits the logarithmic scaling in S, characteristic to the anomalous dimension of low-twist gauge theory operators. Our result therefore shows that the log S scaling, first observed by Gubser, Klebanov and Polyakov for the folded string, is universal also on the string side, suggesting another interesting window to explore the AdS/CFT correspondence as in the BMN/Frolov-Tseytlin limit.Comment: 11 pages, 1 figure; (v2) a reference added; (v3) comments added, typos correcte

N-Impurity Superstring Spectra Near the pp-Wave Limit

The complicated non-linear sigma model that characterizes the first finite-radius curvature correction to the pp-wave limit of IIB superstring theory on AdS_5 x S^5 has been shown to generate energy spectra that perfectly match, to two loops in the modified 't Hooft parameter lambda', finite R-charge corrections to anomalous dimension spectra of large-R N=4 super Yang-Mills theory in the planar limit. This test of the AdS/CFT correspondence has been carried out for the specific cases of two and three string excitations, which are dual to gauge theory R-charge impurities. We generalize this analysis on the string side by directly computing string energy eigenvalues in certain protected sectors of the theory for an arbitrary number of worldsheet excitations with arbitrary mode-number assignments. While our results match all existing gauge theory predictions to two-loop order in lambda', we again observe a mismatch at three loops between string and gauge theory. We find remarkable agreement to all loops in lambda', however, with the near pp-wave limit of a recently proposed Bethe ansatz for the quantized string Hamiltonian in the su(2) sector. Based on earlier two- and three-impurity results, we also infer the full multiplet decomposition of the N-impurity superstring theory with distinct mode excitations to two loops in lambda'.Comment: LaTeX, 24 page

Spin Chain for Quantum Strings

We review and compare the integrable structures in N=4 gauge theory and string theory on AdS5xS5. Recently, Bethe ansaetze for gauge theory/weak coupling and string theory/strong coupling were proposed to describe scaling dimensions in the su(2) subsector. Here we investigate the Bethe equations for quantum string theory, naively extrapolated to weak coupling. Excitingly, we find a spin chain Hamiltonian similar, but not equal, to the gauge theory dilatation operator.Comment: 10 pages, Contribution to RTN and EXT Workshop, Kolymbari, Crete, Greece, 5-10 September 2004, v2: numerical comparision with spinning strings added, coefficient of H4 in Tab. 3 correcte

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.

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

Higher Impurity AdS/CFT Correspondence in the Near-BMN Limit

The pp-wave/BMN limit of the AdS/CFT correspondence has exposed the Maldacena conjecture to a new regimen of direct tests. In one line of pursuit, finite-radius curvature corrections to the Penrose limit (which appear in inverse powers of the string angular momentum J) have been found to induce a complicated system of interaction perturbations to string theory on the pp-wave; these have been successfully matched to corresponding corrections to the BMN dimensions of N=4 super Yang-Mills (SYM) operators to two loops in the 't Hooft coupling lambda. This result is tempered by a well-established breakdown in the correspondence at three loops. Notwithstanding the third-order mismatch, we proceed with this line of investigation by subjecting the string and gauge theories to new and significantly more rigorous tests. Specifically, we extend our earlier results at O(1/J) in the curvature expansion to include string states and SYM operators with three worldsheet or R-charge impurities. In accordance with the two-impurity problem, we find a perfect and intricate agreement between both sides of the correspondence to two-loop order in lambda and, once again, the string and gauge theory predictions fail to agree at third order.Comment: 45 pages, LaTeX; notation and references correcte

Spin bits at two loops

We consider the Super Yang--Mills/spin system map to construct the SU(2) spin bit model at the level of two loops in Yang--Mills perturbation theory. The model describes a spin system with chaining interaction. In the large $N$ limit the model is shown to be reduced to the two loop planar integrable spin chain.Comment: 10 pages, 3 figures, References and Acknowledgements adde