Adam Rej

Review of AdS/CFT Integrability: An Overview

This is the introductory chapter of a review collection on integrability in the context of the AdS/CFT correspondence. In the collection, we present an overview of the achievements and the status of this subject as of the year 2010.

Dressing and Wrapping

We prove that the validity of the recently proposed dressed, asymptotic Bethe ansatz for the planar AdS/CFT system is indeed limited at weak coupling by operator wrapping effects. This is done by comparing the Bethe ansatz predictions for the four-loop anomalous dimension of finite-spin twist-two operators to BFKL constraints from high-energy scattering amplitudes in gauge theory. We find disagreement, which means that the ansatz breaks down for length-two operators at four-loop order. Our method supplies precision tools for multiple all-loop tests of the veracity of any yet-to-be constructed set of exact spectral equations. Finally we present a conjecture for the exact four-loop anomalous dimension of the family of twist-two operators, which includes the Konishi field.

Planar N=4 Gauge Theory and the Hubbard Model

Recently it was established that a certain integrable long-range spin chain describes the dilatation operator of N = 4 gauge theory in the su(2) sector to at least three-loop order, while exhibiting BMN scaling to all orders in perturbation theory. Here we identify this spin chain as an approximation to an integrable short-ranged model of strongly correlated electrons: The Hubbard model.

Five-loop anomalous dimension of twist-two operators

Abstract In this article we calculate the five-loop anomalous dimension of twist-two operators in the planar N = 4 SYM theory. Firstly, using reciprocity, we derive the contribution of the asymptotic Bethe ansatz. Subsequently, we employ the first finite-size correction for the AdS 5 × S 5 sigma model to determine the wrapping correction. The anomalous dimension found in this way passes all known tests provided by the NLO BFKL equation and double-logarithmic constraints. This result thus furnishes an infinite number of experimental data for testing the veracity of the recently proposed spectral equations for planar AdS/CFT correspondence.

A Generalized Scaling Function for AdS/CFT

We study a refined large spin limit for twist operators in the sl(2) sector of AdS/CFT. We derive a novel non-perturbative equation for the generalized two-parameter scaling function associated with this limit, and analyze it for weak coupling. It is expected to smoothly interpolate between weakly coupled gauge theory and string theory for strong coupling.

Analytic three-loop Solutions for N = 4 SYM Twist Operators

We introduce a method to obtain the analytic solution of the higher-order Baxter equation for twist-two and twist-three operators of planar N = 4 SYM. Our result proofs the conjectured formula for the threeloop anomalous dimension of twist-two operators. As such we derive the maximally transcendental part of the corresponding three-loop QCD result from the maximal supersymmetric gauge theory in four dimension purely by methods of integrability.

Review of AdS/CFT Integrability, Chapter I.3: Long-Range Spin Chains

In this contribution, we review long-range integrable spin chains that originate from the recently discovered integrability in the planar AdS/CFT correspondence. We also briefly summarise the theory of generic integrable perturbatively long-range spin chains.

Integrability and the AdS/CFT correspondence

In this paper, we review the recently discovered asymptotic integrability in the planar SYM theory and discuss its breakdown beyond the asymptotic region due to the wrapping interactions. We also discuss novel dynamical tests of the AdS/CFT correspondence one can perform in the special cases when the wrapping interactions may be neglected.

Konishi at strong coupling from ABE

In this communication, we derive analytically the scaling dimension of the Konishi operator in the planar gauge theory at strong coupling from the asymptotic Bethe equations. The first two leading terms agree with the recent string computation and numerical analysis of TBA equations. At the third order we find a spurious logarithm of the coupling constant, which should be absent in the anomalous dimension of any finite operator of the planar SYM theory. Showing the cancellation of this term would provide an important test at strong coupling of the recently proposed sets of TBA equations for the planar AdS/CFT correspondence.

The Yangian of \mathfrak{s}\mathfrak{l}\left( {n\left| m \right.} \right) and its quantum R-matrices

In this paper we study Yangians of