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Register a new userReview of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry
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J. M. Drummond
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Scattering amplitudes in planar N=4 super Yang-Mills theory reveal a remarkable symmetry structure. In addition to the superconformal symmetry of the Lagrangian of the theory, the planar amplitudes exhibit a dual superconformal symmetry. The presence of this additional symmetry imposes strong restrictions on the amplitudes and is connected to a duality relating scattering amplitudes to Wilson loops defined on polygonal light-like contours. The combination of the superconformal and dual superconformal symmetries gives rise to a Yangian, an algebraic structure which is known to be related to the appearance of integrability in other regimes of the theory. We discuss two dual formulations of the symmetry and address the classification of its invariants.
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With a view to understanding extended-BMS symmetries in the framework of the $AdS_4/CFT_3$ correspondence, asymptotically AdS geometries are constructed with null impulsive shockwaves involving a discontinuity in superrotation parameters. The holographic dual is proposed to be a two-dimensional Euclidean defect conformal field localized on a particular timeslice in a three-dimensional conformal field theory on de Sitter spacetime. The defect conformal field theory generates a natural action of the Virasoro algebra. The large radius of curvature limit $elltoinfty$ yields spacetimes with nontrivial extended-BMS charges.
Using the thermodynamical Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in arXiv:0901.3753 for the spectrum of all operators in planar N=4 SYM theory follows from these equations. In particular, we present the integral equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analogue of DHM formula for the dressing kernel in the mirror kinematics.
We study, using the dual AdS description, the vacua of field theories where some of the gauge symmetry is broken by expectation values of scalar fields. In such vacua, operators built out of the scalar fields acquire expectation values, and we show how to calculate them from the behavior of perturbations to the AdS background near the boundary. Specific examples include the ${cal N}=4$ SYM theory, and theories on D3 branes placed on orbifolds and conifolds. We also clarify some subtleties of the AdS/CFT correspondence that arise in this analysis. In particular, we explain how scalar fields in AdS space of sufficiently negative mass-squared can be associated with CFT operators of {it two} possible dimensions. All dimensions are bounded from below by $(d-2)/2$; this is the unitarity bound for scalar operators in $d$-dimensional field theory. We further argue that the generating functional for correlators in the theory with one choice of operator dimension is a Legendre transform of the generating functional in the theory with the other choice.
We study, using ADHM construction, instanton effects in an ${CN}=2$ superconformal $Sp(N)$ gauge theory, arising as effective field theory on a system of $N$ D-3-branes near an orientifold 7-plane and 8 D-7-branes in type I string theory. We work out the measure for the collective coordinates of multi-instantons in the gauge theory and compare with the measure for the collective coordinates of $(-1)$-branes in the presence of 3- and 7-branes in type I theory. We analyse the large-N limit of the measure and find that it admits two classes of saddle points: In the first class the space of collective coordinates has the geometry of $AdS_5times S^3$ which on the string theory side has the interpretation of the D-instantons being stuck on the 7-branes and therefore the resulting moduli space being $AdS_5times S^3$, In the second class the geometry is $AdS_5times S^5/Z_2$ and on the string theory side it means that the D-instantons are free to move in the 10-dimensional bulk. We discuss in detail a correlator of four O(8) flavour currents on the Yang-Mills side, which receives contributions from the first type of saddle points only, and show that it matches with the correlator obtained from $F^4$ coupling on the string theory side, which receives contribution from D-instantons, in perfect accord with the AdS/CFT correspondence. In particular we observe that the sectors with odd number of instantons give contribution to an O(8)-odd invariant coupling, thereby breaking O(8) down to SO(8) in type I string theory. We finally discuss correlators related to $R^4$, which receive contributions from both saddle points.
We parametrize the (2+1)-dimensional AdS space and the BTZ black hole with Fefferman-Graham coordinates starting from the AdS boundary. We consider various boundary metrics: Rindler, static de Sitter and FRW. In each case, we compute the holographic stress-energy tensor of the dual CFT and confirm that it has the correct form, including the effects of the conformal anomaly. We find that the Fefferman-Graham parametrization also spans a second copy of the AdS space, including a second boundary. For the boundary metrics we consider, the Fefferman-Graham coordinates do not cover the whole AdS space. We propose that the length of the line delimiting the excluded region at a given time can be identified with the entropy of the dual CFT on a background determined by the boundary metric. For Rindler and de Sitter backgrounds our proposal reproduces the expected entropy. For a FRW background it produces a generalization of the Cardy formula that takes into account the vacuum energy related to the expansion.
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