870 research outputs found
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
Anomalous dimensions at twist-3 in the sl(2) sector of N=4 SYM
We consider twist-3 operators in the sl(2) sector of N=4 SYM built out of
three scalar fields with derivatives. We extract from the Bethe Ansatz
equations of this sector the exact lowest anomalous dimension gamma(s) of
scaling fields for several values of the operator spin s. We propose compact
closed expressions for the spin dependence of gamma(s) up to the four loop
level and show that they obey a simple new twist-3 transcendentality principle.
As a check, we reproduce the four loop universal cusp anomalous dimension
governing the logarithmic large spin limit of gamma(s).Comment: 26 pages, JHEP styl
The Ising model and planar N=4 Yang-Mills
The scattering-matrix for planar Yang-Mills with N=4 supersymmetry relies on
the assumption that integrability holds to all orders in perturbation theory.
In this note we define a map from the spectral variables x^{\pm},
parameterizing the long-range magnon momenta, to couplings in a two-dimensional
Ising model. Under this map integrability of planar N=4 Yang-Mills becomes
equivalent to the Yang-Baxter equation for the two-dimensional Ising model, and
the long-range variables x^{\pm} translate into the entries of the Ising
transfer matrices. We explore the Ising correlation length which equals the
inverse magnon momentum in the small momentum limit. The critical regime is
thus reached for vanishing magnon momentum. We also discuss the meaning of the
Kramers-Wannier duality transformation on the gauge theory, together with that
of the Ising model critical points.Comment: 24 pages. v2: References added and minor typos correcte
Landau-Lifshitz sigma-models, fermions and the AdS/CFT correspondence
We define Landau-Lifshitz sigma models on general coset space , with
a maximal stability sub-group of . These are non-relativistic models that
have -valued N\"other charges, local invariance and are classically
integrable. Using this definition, we construct the
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 whose field content consists of two
real bosons and 4,8 or 16 real fermions. We show that -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 Landau-Lifshitz
sigma-model.Comment: 42 page
The Bound State S-Matrix for AdS5 x S5 Superstring
We determine the S-matrix that describes scattering of arbitrary bound states
in the light-cone string theory in AdS5 x S5. The corresponding construction
relies on the Yangian symmetry and the superspace formalism for the bound state
representations. The basic analytic structure supporting the S-matrix entries
turns out to be the hypergeometric function 4F3. We show that for particular
bound state numbers it reproduces all the scattering matrices previously
obtained in the literature. Our findings should be relevant for the TBA and
Luescher approaches to the finite-size spectral problem. They also shed some
light on the construction of the universal R-matrix for the centrally-extended
psu(2|2) superalgebra.Comment: 37 pages, 2 figures, v2: typos correcte
Three loop anomalous dimensions of twist-3 gauge operators in N=4 SYM
We propose a closed expression for the three loop anomalous dimension of a
class of twist-3 operators built with gauge fields and covariant derivatives.
To this aim, we solve the long-range Bethe Ansatz equations at finite spin and
provide a consistent analytical formula obtained assuming maximal
transcendentality violation as suggested by the known one-loop anomalous
dimension. The final result reproduces the universal cusp anomalous dimension
and obeys recursion relations inspired by the principle of reciprocity
invariance.Comment: 20 pages, JHEP styl
Bound States of the q-Deformed AdS5 x S5 Superstring S-matrix
The investigation of the q deformation of the S-matrix for excitations on the
string world sheet in AdS5 x S5 is continued. We argue that due to the lack of
Lorentz invariance the situation is more subtle than in a relativistic theory
in that the nature of bound states depends on their momentum. At low enough
momentum |p|<E the bound states transform in the anti-symmetric representation
of the super-algebra symmetry and become the solitons of the Pohlmeyer reduced
theory in the relativistic limit. At a critical momentum |p|=E they become
marginally unstable, and at higher momenta the stable bound states are in the
symmetric representation and become the familiar magnons in the string limit as
q->1. This subtlety fixes a problem involving the consistency of crossing
symmetry with the relativistic limit found in earlier work. With mirror
kinematics, obtained after a double Wick rotation, the bound state structure is
simpler and there are no marginally unstable bound states.Comment: 25 page
Twist-three at five loops, Bethe Ansatz and wrapping
We present a formula for the five-loop anomalous dimension of N=4 SYM
twist-three operators in the sl(2) sector. We obtain its asymptotic part from
the Bethe Ansatz and finite volume corrections from the generalized Luescher
formalism, considering scattering processes of spin chain magnons with virtual
particles that travel along the cylinder. The complete result respects the
expected large spin scaling properties and passes non-trivial tests including
reciprocity constraints. We analyze the pole structure and find agreement with
a conjectured resummation formula. In analogy with the twist-two anomalous
dimension at four-loops, wrapping effects are of order log^2 M/M^2 for large
values of the spin.Comment: 19 page
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