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

We describe how to compute planar gluon scattering amplitudes at strong coupling in = 4 super Yang Mills by using the gauge/string duality. The computation boils down to finding a certain classical string configuration whose boundary conditions are determined by the gluon momenta. The results are infrared divergent. We introduce the gravity version of dimensional regularization to define finite quantities. The leading and subleading IR divergencies are characterized by two functions of the coupling that we compute at strong coupling. We compute also the full finite form for the four point amplitude and we find agreement with a recent ansatz by Bern, Dixon and Smirnov.

Gluon scattering amplitudes at strong coupling

We use localization techniques to compute the expectation values of supersymmetric Wilson loops in Chern-Simons theories with matter. We find the path-integral reduces to a non-Gaussian matrix model. The Wilson loops we consider preserve a single complex supersymmetry, and exist in any N = 2 theory, though the localization requires superconformal symmetry. We present explicit results for the cases of pure Chern-Simons theory with gauge group U(N), showing agreement with the known results, and ABJM, showing agreement with perturbative calculations. Our method applies to other theories, such as Gaiotto-Witten theories, BLG, and their variants.

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Exact results for Wilson loops in superconformal Chern-Simons theories with matter

For 3-dimensional field theories with $ \mathcal{N} = 2 $
 supersymmetry the Euclidean path integrals on the three-sphere can be calculated using the method of localization; they reduce to certain matrix integrals that depend on the R-charges of the matter fields. We solve a number ofsuch large N matrix models and calculate the free energy F as a function of the trial R-charges consistent with the marginality of the super potential. In all our $ \mathcal{N} = 2 $
 superconformal examples, the local maximization of F yields answers that scale as N
 3/2 and agree with the dual M-theory backgrounds AdS
 4 × Y, where Y are 7-dimensional Sasaki-Einstein spaces. We also find in toric examples that local F-maximization is equivalent to the minimization of the volume of Y over the space of Sasakian metrics, a procedure also referred to as Z-minimization. Moreover, we find that the functions F and Z are related for any trial R-charges. In the models we study F is positive and decreases along RG flows. We therefore propose the “F-theorem” that we hope applies to all 3-d field theories: the finite part of the free energy on the three-sphere decreases along RG trajectories and is stationary at RG fixed points. We also show that in an infinite class of Chern-Simons-matter gauge theories where the Chern-Simons levels do not sum to zero, the free energy grows as N
 5/3 at large N. This non-trivial scaling matches that of the free energy of the gravity duals in type IIA string theory with Romans mass.

Towards the F-theorem: $ \mathcal{N} = 2 $ field theories on the three-sphere

Hexagon Wilson loop = six-gluon MHV amplitude

We discuss the N=2 superspace formulation of the N=8 superconformal Bagger-Lambert-Gustavsson theory, and of the N=6 superconformal Aharony-Bergman-Jafferis-Maldacena U(N)xU(N) Chern-Simons theory. In particular, we prove the full SU(4) R-symmetry of the ABJM theory. We then consider orbifold projections of this theory that give non-chiral and chiral (U(N)xU(N))^n superconformal quiver gauge theories. We argue that these theories are dual to certain AdS_4 x S^7/(Z_n x Z_k) backgrounds of M-theory. We also study a SU(3) invariant mass term in the superpotential that makes the N=8 theory flow to a N=2 superconformal gauge theory with a sextic superpotential. We conjecture that this gauge theory is dual to the U(1)_R x SU(3) invariant extremum of the N=8 gauged supergravity, which was discovered by N. Warner 25 years ago and whose uplifting to 11 dimensions was found more recently.

Superconformal Chern-Simons Theories and AdS_4/CFT_3 Correspondence

We discuss the N = 2 superspace formulation of the N = 8 superconformal Bagger-Lambert-Gustavsson theory, and of the N = 6 superconformal Aharony-Bergman-Jafferis-Maldacena U(N) x U(N) Chern-Simons theory. In particular, we prove the full SU(4)R-symmetry of the ABJM theory. We then consider orbifold projections of this theory that give non-chiral and chiral (U( N) x U(N))(n) superconformal quiver gauge theories. We argue that these theories are dual to certain AdS(4) x S-7/(Z(n) x Z (k$) over tilde) backgrounds of M-theory. We also study a SU(3) invariant mass term in the superpotential that makes the N = 8 theory flow to a N = 2 superconformal gauge theory with a sextic superpotential. We conjecture that this gauge theory is dual to the U(1)(R) x SU(3) invariant extremum of the N = 8 gauged supergravity, which was discovered by N. Warner 25 years ago and whose uplifting to 11 dimensions was found more recently.

Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence

In two remarkable recent papers the planar perturbative expansion was proposed for the universal function of the coupling appearing in the dimensions of high-spin operators of the N=4 super Yang-Mills theory. We study numerically the integral equation derived by Beisert, Eden, and Staudacher, which resumes the perturbative series. In a confirmation of the anti-de Sitter-space/conformal-field-theory (AdS/CFT) correspondence, we find a smooth function whose two leading terms at strong coupling match the results obtained for the semiclassical folded string spinning in AdS5. We also make a numerical prediction for the third term in the strong coupling series.

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Test of the anti-de Sitter-space/conformal-field-theory correspondence using high-spin operators.

We calculate the non-abelian R-charges of BPS monopole operators in three-dimensional gauge theories with $ \mathcal{N} = 3 $ supersymmetry. This class of models includes ABJM theory, the proposed gauge theory dual of M-theory on $ {{Ad{S_4} \times {S^7}} \mathord{\left/{\vphantom {{Ad{S_4} \times {S^7}} {{\mathbb{Z}_k}}}} \right.} {{\mathbb{Z}_k}}} $. In the UV limit of the $ \mathcal{N} = 3 $ theories the Yang-Mills coupling becomes weak and the monopole operators are described by classical backgrounds. This allows us to find their SU(2)R charges in a one-loop computation which by virtue of the non-renormalization of non-abelian R-charges yields the exact result for any value of the coupling. The spectrum of SU(2)R charges is found by quantizing the SU(2)/U(1) collective coordinate of the BPS background, whose dynamics is that of a charged particle on a sphere with a Wess-Zumino term representing a magnetic monopole at its center. If the Wess-Zumino coefficient is h, then the smallest possible SU(2)R representation for BPS monopole operators has spin |h|/2. We find, in agreement with earlier proposals, that h is proportional to the sum of the U(1)R charges of all the fermion fields weighted by the effective monopole charges determined by their gauge representations. The field content of ABJM theory is such that h = 0. This proves for any Chern-Simons level k the existence of monopole operators which are singlets under all global symmetries and have vanishing scaling dimensions. These operators are essential for matching the spectrum of the ABJM theory with supergravity and for the supersymmetry enhancement to $ \mathcal{N} = 8 $.

Charges of monopole operators in Chern-Simons Yang-Mills theory

We give an exact analytic solution of the strong coupling limit of the integral equation which was recently proposed to describe the universal scaling function of high spin operators in = 4 gauge theory. The solution agrees with the prediction from string theory, confirms the earlier numerical analysis and provides a basis for developing a systematic perturbation theory around strong coupling.

Marcus K. Benna

Papers

Superconformal Chern-Simons Theories and AdS_4/CFT_3 Correspondence

Superconformal Chern-Simons Theories and AdS(4)/CFT(3) Correspondence

Test of the anti-de Sitter-space/conformal-field-theory correspondence using high-spin operators.

Charges of monopole operators in Chern-Simons Yang-Mills theory

On the strong coupling scaling dimension of high spin operators