Higher loops, integrability and the near BMN limit

In this note we consider higher-loop contributions to the planar dilatation operator of N=4 SYM in the su(2) subsector of two complex scalar fields. We investigate the constraints on the form of this object due to interactions of two excitations in the BMN limit. We then consider two scenarios to uniquely fix some higher-loop contributions: (i) Higher-loop integrability fixes the dilatation generator up to at least four-loops. Among other results, this allows to conjecture an all-loop expression for the energy in the near BMN limit. (ii) The near plane-wave limit of string theory and the BMN correspondence fix the dilatation generator up to three-loops. We comment on the difference between both scenarios.Comment: 6 page

The su(2|3) Dynamic Spin Chain

The complete one-loop, planar dilatation operator of the N=4 superconformal gauge theory was recently derived and shown to be integrable. Here, we present further compelling evidence for a generalisation of this integrable structure to higher orders of the coupling constant. For that we consider the su(2|3) subsector and investigate the restrictions imposed on the spin chain Hamiltonian by the symmetry algebra. This allows us to uniquely fix the energy shifts up to the three-loop level and thus prove the correctness of a conjecture in hep-th/0303060. A novel aspect of this spin chain model is that the higher-loop Hamiltonian, as for N=4 SYM in general, does not preserve the number of spin sites. Yet this dynamic spin chain appears to be integrable.Comment: 34 pages, 5 figures, v2: additional coefficient at three loops explained, discussion of integrability enhanced, figures adde

On Yangian Symmetry in Planar N=4 SYM

Planar N=4 supersymmetric Yang-Mills theory appears to be perturbatively integrable. This work reviews integrability in terms of a Yangian algebra and compares the application to the problems of anomalous dimensions and scattering amplitudes.Comment: 24 pages, To Lev Lipatov on the occasion of his 70th birthday, v2: references and typos corrected, v3: no changes, references updated, published in "Subtleties in Quantum Field Theory", pp. 175, ed: D. Diakonov and "Gribov-80 Memorial Volume: Quantum Chromodynamics and Beyond", pp. 413, ed: Yu. Dokshitzer, P. L\'evai, J. Ny\'ir

The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2) Symmetry

We investigate the integrable structure of spin chain models with centrally extended su(2|2) and psu(2,2|4) symmetry. These chains have their origin in the planar AdS/CFT correspondence, but they also contain the one-dimensional Hubbard model as a special case. We begin with an overview of the representation theory of centrally extended su(2|2). These results are applied in the construction and investigation of an interesting S-matrix with su(2|2) symmetry. In particular, they enable a remarkably simple proof of the Yang-Baxter relation. We also show the equivalence of the S-matrix to Shastry's R-matrix and thus uncover a hidden supersymmetry in the integrable structure of the Hubbard model. We then construct eigenvalues of the corresponding transfer matrix in order to formulate an analytic Bethe ansatz. Finally, the form of transfer matrix eigenvalues for models with psu(2,2|4) symmetry is sketched.Comment: 66 pages, v2: minor changes, references added, to appear in JSTA

Construction of Lax Connections by Exponentiation

We propose a method for constructing the Lax connection of two-dimensional relativistic integrable sigma models on coset spaces by means of exponentiation of a suitable operator. We derive a simple quadratic relation that this operator must satisfy for an entire one-parameter family of connections to be flat.Comment: 17 page

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

Bonus Yangian Symmetry for the Planar S-Matrix of N=4 Super Yang-Mills

Recent developments in the determination of the planar S-matrix of N=4 Super Yang-Mills are closely related to its Yangian symmetry. Here we provide evidence for a yet unobserved additional symmetry: the Yangian level-one helicity operator.Comment: 8 pages, v2: minor change

Maximally extended sl(2|2), q-deformed d(2,1;epsilon) and 3D kappa-Poincar\'e

We show that the maximal extension sl(2) times psl(2|2) times C3 of the sl(2|2) superalgebra can be obtained as a contraction limit of the semi-simple superalgebra d(2,1;epsilon) times sl(2). We reproduce earlier results on the corresponding q-deformed Hopf algebra and its universal R-matrix by means of contraction. We make the curious observation that the above algebra is related to kappa-Poincar\'e symmetry. When dropping the graded part psl(2|2) we find a novel one-parameter deformation of the 3D kappa-Poincar\'e algebra. Our construction also provides a concise exact expression for its universal R-matrix.Comment: 25 page