We construct three dimensional Chern-Simons-matter theories with gauge groups U(N) × U(N) and SU(N) × SU(N) which have explicit = 6 superconformal symmetry. Using brane constructions we argue that the U(N) × U(N) theory at level k describes the low energy limit of N M2-branes probing a C4/Zk singularity. At large N the theory is then dual to M-theory on AdS4 × S7/Zk. The theory also has a 't Hooft limit (of large N with a fixed ratio N/k) which is dual to type IIA string theory on AdS4 × CP3. For k = 1 the theory is conjectured to describe N M2-branes in flat space, although our construction realizes explicitly only six of the eight supersymmetries. We give some evidence for this conjecture, which is similar to the evidence for mirror symmetry in d = 3 gauge theories. When the gauge group is SU(2) × SU(2) our theory has extra symmetries and becomes identical to the Bagger-Lambert theory.

N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals

Superstring theoryをめぐって(放談室)

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.

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Review of AdS/CFT Integrability: An Overview

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

Attempts to solve Yang-Mills theory must eventually face the problem of analyzing the theory at intermediate values of the coupling constant. In this regime neither perturbation theory nor the gravity dual are adequate, and one must consider the full string theory in the appropriate background. We suggest that in some nontrivial cases the world sheet theory may be exactly solvable. For the Green-Schwarz superstring on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ we find an infinite set of nonlocal classically conserved charges of the type that exist in integrable field theories.

Hidden symmetries of the AdS(5) x S**5 superstring

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.

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Dressing and Wrapping

We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N = 4 gauge theory and energies of spinning strings on AdS5 × S 5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.

Matching Higher Conserved Charges for Strings and Spins

We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N=4 gauge theory and energies of spinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating function, containing information to arbitrary loop order, is constructed by solving exactly the Backlund equations of the integrable classical string sigma model. Our finding should be an important step towards matching the integrable structures on the string and gauge side of the AdS/CFT correspondence.

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.

https://iopscience.iop.org/article/10.1088/1742-5468/2008/07/P07015/pdf

A Generalized Scaling Function for AdS/CFT

https://openresearch-repository.anu.edu.au/bitstream/1885/68789/2/01_Bazhanov_Baxter_q-operators_and_2011.pdf

Matthias Staudacher

Papers

Dressing and Wrapping

Matching Higher Conserved Charges for Strings and Spins

Matching Higher Conserved Charges for Strings and Spins

A Generalized Scaling Function for AdS/CFT

Baxter Q-operators and representations of Yangians