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 prompt optical/near-infrared flare of GRB 050904: the most luminous transient ever detected

With a redshift of z=6.295, GRB 050904 is the most distant gamma-ray burst ever discovered. It was an energetic event at all wavelengths and the afterglow was observed in detail in the near-infrared bands. We gathered all available optical and NIR afterglow photometry of this GRB to construct a composite NIR light curve spanning several decades in time and flux density. Transforming the NIR light curve into the optical, we find that the afterglow of GRB 050904 was more luminous at early times than any other GRB afterglow in the pre-\emph{Swift} era, making it at these wavelengths the most luminous transient ever detected. Given the intrinsic properties of GRB 050904 and its afterglow, we discuss if this burst is markedly different from other GRBs at lower redshifts.Comment: The Astronomical Journal, in press; revised version, including the comments of the referee (one figure added, text restructured, all conclusions unchanged), 7 pages, 3 figure

World-sheet scattering in AdS_5 x S^5 at two loops

We study the AdS_5 x S^5 sigma-model truncated to the near-flat-space limit to two-loops in perturbation theory. In addition to extending previously known one-loop results to the full SU(2|2)^2 S-matrix we calculate the two-loop correction to the dispersion relation and then compute the complete two-loop S-matrix. The result of the perturbative calculation can be compared with the appropriate limit of the conjectured S-matrix for the full theory and complete agreement is found.Comment: 26pages, 3 figure

On the Integrability of large N Plane-Wave Matrix Theory

We show the three-loop integrability of large N plane-wave matrix theory in a subsector of states comprised of two complex light scalar fields. This is done by diagonalizing the theory's Hamiltonian in perturbation theory and taking the large N limit. At one-loop level the result is known to be equal to the Heisenberg spin-1/2 chain, which is a well-known integrable system. Here, integrability implies the existence of hidden conserved charges and results in a degeneracy of parity pairs in the spectrum. In order to confirm integrability at higher loops, we show that this degeneracy is not lifted and that (corrected) conserved charges exist. Plane-wave matrix theory is intricately connected to N=4 Super Yang-Mills, as it arises as a consistent reduction of the gauge theory on a three-sphere. We find that after appropriately renormalizing the mass parameter of the plane-wave matrix theory the effective Hamiltonian is identical to the dilatation operator of N=4 Super Yang-Mills theory in the considered subsector. Our results therefore represent a strong support for the conjectured three-loop integrability of planar N=4 SYM and are in disagreement with a recent dual string theory finding. Finally, we study the stability of the large N integrability against nonsupersymmetric deformations of the model

Magnon Bound-state Scattering in Gauge and String Theory

It has been shown that, in the infinite length limit, the magnons of the gauge theory spin chain can form bound states carrying one finite and one strictly infinite R-charge. These bound states have been argued to be associated to simple poles of the multi-particle scattering matrix and to world sheet solitons carrying the same charges. Classically, they can be mapped to the solitons of the complex sine-Gordon theory. Under relatively general assumptions we derive the condition that simple poles of the two-particle scattering matrix correspond to physical bound states and construct higher bound states ``one magnon at a time''. We construct the scattering matrix of the bound states of the BDS and the AFS S-matrices. The bound state S-matrix exhibits simple and double poles and thus its analytic structure is much richer than that of the elementary magnon S-matrix. We also discuss the bound states appearing in larger sectors and their S-matrices. The large 't Hooft coupling limit of the scattering phase of the bound states in the SU(2) sector is found to agree with the semiclassical scattering of world sheet solitons. Intriguingly, the contribution of the dressing phase has an independent world sheet interpretation as the soliton-antisoliton scattering phase shift. The small momentum limit provides independent tests of these identifications.Comment: 25 pages, Latex V2: clarifying comments added to footnote 1 and footnote 10; references added V3: typos correcte

Higher charges and regularized quantum trace identities in su(1,1) Landau-Lifshitz model

We solve the operator ordering problem for the quantum continuous integrable su(1,1) Landau-Lifshitz model, and give a prescription to obtain the quantum trace identities, and the spectrum for the higher-order local charges. We also show that this method, based on operator regularization and renormalization, which guarantees quantum integrability, as well as the construction of self-adjoint extensions, can be used as an alternative to the discretization procedure, and unlike the latter, is based only on integrable representations.Comment: 27 pages; misprints corrected, references adde

The S-matrix of the Faddeev-Reshetikhin Model, Diagonalizability and PT Symmetry

We study the question of diagonalizability of the Hamiltonian for the Faddeev-Reshetikhin (FR) model in the two particle sector. Although the two particle S-matrix element for the FR model, which may be relevant for the quantization of strings on $AdS_{5}\times S^{5}$, has been calculated recently using field theoretic methods, we find that the Hamiltonian for the system in this sector is not diagonalizable. We trace the difficulty to the fact that the interaction term in the Hamiltonian violating Lorentz invariance leads to discontinuity conditions (matching conditions) that cannot be satisfied. We determine the most general quartic interaction Hamiltonian that can be diagonalized. This includes the bosonic Thirring model as well as the bosonic chiral Gross-Neveu model which we find share the same S-matrix. We explain this by showing, through a Fierz transformation, that these two models are in fact equivalent. In addition, we find a general quartic interaction Hamiltonian, violating Lorentz invariance, that can be diagonalized with the same two particle S-matrix element as calculated by Klose and Zarembo for the FR model. This family of generalized interaction Hamiltonians is not Hermitian, but is $PT$ symmetric. We show that the wave functions for this system are also $PT$ symmetric. Thus, the theory is in a $PT$ unbroken phase which guarantees the reality of the energy spectrum as well as the unitarity of the S-matrix.Comment: 32 pages, 1 figure; references added, version published in JHE

On the breakdown of perturbative integrability in large N matrix models

We study the perturbative integrability of the planar sector of a massive SU(N) matrix quantum mechanical theory with global SO(6) invariance and Yang-Mills-like interaction. This model arises as a consistent truncation of maximally supersymmetric Yang-Mills theory on a three-sphere to the lowest modes of the scalar fields. In fact, our studies mimic the current investigations concerning the integrability properties of this gauge theory. Like in the field theory we can prove the planar integrability of the SO(6) model at first perturbative order. At higher orders we restrict ourselves to the widely studied SU(2) subsector spanned by two complexified scalar fields of the theory. We show that our toy model satisfies all commonly studied integrability requirements such as degeneracies in the spectrum, existence of conserved charges and factorized scattering up to third perturbative order. These are the same qualitative features as the ones found in super Yang-Mills theory, which were enough to conjecture the all-loop integrability of that theory. For the SO(6) model, however, we show that these properties are not sufficient to predict higher loop integrability. In fact, we explicitly demonstrate the breakdown of perturbative integrability at fourth order.Comment: 27 page