Tristan McLoughlin

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.

Quantizing string theory in AdS5×S5: beyond the pp-wave

In a certain kinematic limit, where the effects of spacetime curvature (and other background fields) greatly simplify, the light-cone gauge world-sheet action for a type IIB superstring on AdS5 × S 5 reduces to that of a free field theory. It has been conjectured by Berenstein, Maldacena, and Nastase that the energy spectrum of this string theory matches the dimensions of operators in the appropriately defined large R-charge large-Nc sector of N = 4 supersymmetric Yang–Mills theory in four dimensions. This holographic equivalence is thought to be exact, independent of any simplifying kinematic limits. As a step toward verifying this larger conjecture, we have computed the complete set of first curvature corrections to the spectrum of light-cone gauge string theory that arises in the expansion of AdS5 × S 5 about the plane-wave limit. The resulting spectrum has the complete dependence on λ = g 2 Y M Nc; corresponding results in the gauge theory are known only to second order in λ. We find precise agreement to this order, including the N = 4 extended supermultiplet structure. In the process, we demonstrate that the complicated schemes put forward in recent years for defining the Green–Schwarz superstring action in background Ramond-Ramond fields can be reduced to a practical (and correct) method for quantizing the string.

Holography beyond the Penrose limit

The flat pp-wave background geometry has been realized as a particular Penrose limit of AdS5 × S 5 . It describes a string that has been infinitely boosted along an equatorial null geodesic in the S 5 subspace. The string worldsheet Hamiltonian in this background is free. Finite boosts lead to curvature corrections that induce interacting perturbations of the string worldsheet Hamiltonian. We develop a systematic light-cone gauge quantization of the interacting worldsheet string theory and use it to obtain the interacting spectrum of the so-called ‘two-impurity’ states of the string. The quantization is technically rather intricate and we provide a detailed account of the methods we use to extract explicit results. We give a systematic treatment of the fermionic states and are able to show that the spectrum possesses the proper extended supermultiplet structure (a non-trivial fact since half the supersymmetry is nonlinearly realized). We test holography by comparing the string energy spectrum with the scaling dimensions of corresponding gauge theory operators. We confirm earlier results that agreement obtains in low orders of perturbation theory, but breaks down at third order. The methods presented here can be used to explore these issues in a wider context than is specifically dealt with in this paper.

Spinning strings at one-loop in AdS(4) x P**3

We analyze the folded spinning string in AdS4 × P 3 with spin S in AdS4 and angular momentum J in P 3 . We calculate the one-loop correction to its energy in the scaling limit of both ln S and J large with their ratio kept fixed. This result should correspond to the first subleading strong coupling correction to the anomalous dimension of operators of the type T r(D S (Y † Y ) J ) in the dual N = 6 Chern-Simons-matter theory. Our result appears to depart from the predictions for the generalized scaling function found from the all-loop Bethe equations conjectured for this AdS4/CFT3 duality. We comment on the possible origin of this difference.

Exacting N=4 Superconformal Symmetry

Tree level scattering amplitudes in N = 4 super Yang-Mills theory are al- most, but not exactly invariant under the free action of the N = 4 superconformal algebra. What causes the non-invariance is the holomorphic anomaly at poles where external par- ticles become collinear. In this paper we propose a deformation of the free superconformal representation by contributions which change the number of external legs. This modified classical representation not only makes tree amplitudes fully invariant, but it also leads to additional constraints from symmetry alone mediating between hitherto unrelated ampli- tudes. Moreover, in a constructive approach it appears to fully constrain all tree amplitudes when combined with dual superconformal alias Yangian symmetry.

One-Loop Superconformal and Yangian Symmetries of Scattering Amplitudes in N=4 Super Yang-Mills

Recently it has been argued that tree-level scattering amplitudes in

Scattering in Mass-Deformed N>=4 Chern-Simons Models

\mathcal{N} = 4

New relations for three-dimensional supersymmetric scattering amplitudes.

Yang-Mills theory are uniquely determined by a careful study of their superconformal and Yangian symmetries. However, at one-loop order these symmetries are known to become anomalous due to infrared divergences. We compute these one-loop anomalies for amplitudes defined through dimensional regularisation by studying the tree-level symmetry transformations of the unitarity branch cuts, keeping track of the crucial collinear terms arising from the holomorphic anomaly. We extract the superconformal anomalies and show that they may be cancelled through a universal one-loop deformation of the treelevel symmetry generators which involves only tree-level data. Specialising to the planar theory we also obtain the analogous deformation for the level-one Yangian generator of momentum. Explicit checks of our one-loop deformation are performed for MHV and the 6-point NMHV amplitudes.