Charlotte Kristjansen

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.

Wrapping interactions and a new source of corrections to the spin-chain/string duality

Abstract Assuming that the world-sheet sigma-model in the AdS/CFT correspondence is an integrable quantum field theory, we deduce that there might be new corrections to the spin-chain/string Bethe ansatz paradigm. These come from virtual particles propagating around the circumference of the cylinder and render Bethe ansatz quantization conditions only approximate. We determine the nature of these corrections both at weak and at strong coupling in the near-BMN limit, and find that the first corrections behave qualitatively as wrapping interactions at weak coupling.

Matrix model calculations beyond the spherical limit

We propose an improved iterative scheme for calculating higher genus contributions to the multi-loop (or multi-point) correlators and the partition function of the hermitian one matrix model. We present explicit results up to genus two. We develop a version which gives directly the result in the double scaling limit and present explicit results up to genus four. Using the latter version we prove that the hermitian and the complex matrix model are equivalent in the double scaling limit and that in this limit they are both equivalent to the Kontsevich model. We discuss how our results away from the double scaling limit are related to the structure of moduli space.

The strong coupling limit of the scaling function from the quantum string Bethe ansatz

Abstract Using the quantum string Bethe ansatz we derive the one-loop energy of a folded string rotating with angular momenta ( S , J ) in AdS 3 × S 1 ⊂ AdS 5 × S 5 in the limit 1 ≪ J ≪ S , z = λ log ( S / J ) / ( π J ) fixed. The one-loop energy is a sum of two contributions, one originating from the Hernandez–Lopez phase and another one being due to spin chain finite size effects. We find a result which at the functional level exactly matches the result of a string theory computation. Expanding the result for large z we obtain the strong coupling limit of the scaling function for low twist, high spin operators of the SL ( 2 ) sector of N = 4 SYM. In particular, we recover the famous − 3 log ( 2 ) π . Its appearance is a result of non-trivial cancellations between the finite size effects and the Hernandez–Lopez correction.

A Universality test of the quantum string Bethe ansatz.

We show that the quantum corrected string Bethe ansatz passes an important universality test by demonstrating that it correctly incorporates the non-analytical terms in the string sigma model one-loop correction for rational three-spin strings with two out of the three spins identical. Subsequently, we use the quantum corrected string Bethe ansatz to predict the exact form of the non-analytic terms for the generic rational three-spin string.

Holographic three-point functions of giant gravitons

Working within the AdS/CFT correspondence we calculate the three-point function of two giant gravitons and one pointlike graviton using methods of semi classical string theory and considering both the case where the giant gravitons wrap an S3 ⊂ S5 and the case where the giant gravitons wrap an S3 ⊂ AdS5. We likewise calculate the correlation function in

Non-planar ABJM theory and integrability

\mathcal{N} = 4

Integrable 2D Lorentzian gravity and random walks

SYM using two Schur polynomials and a single trace chiral primary. We find that the gauge and string theory results have structural similarities but do not match perfectly, and interpret this in terms of the Schur polynomials’ inability to interpolate between dual giant and pointlike gravitons.

The relation between Euclidean and Lorentzian 2D quantum gravity

Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2) × SU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length operators as well as to two different types of BMN operators. As in = 4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity — a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account the degeneracies between parity pairs disappear hinting the absence of higher conserved charges. The analysis of the BMN operators resembles that of = 4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

Observing 4d baby universes in quantum gravity

Abstract We introduce and solve a family of discrete models of 2D Lorentzian gravity with higher curvature weight, which possess mutually commuting transfer matrices, and whose spectral parameter interpolates between flat and curved space-times. We further establish a one-to-one correspondence between Lorentzian triangulations and directed random walks. This gives a simple explanation why the Lorentzian triangulations have fractal dimension 2 and why the curvature model lies in the universality class of pure Lorentzian gravity. We also study integrable generalizations of the curvature model with arbitrary polygonal tiles. All of them are found to lie in the same universality class.