Amplitudes at Weak Coupling as Polytopes in AdS_5

We show that one-loop scalar box functions can be interpreted as volumes of geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal space-time as boundary. When the tetrahedron is space-like, it lies in a totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on the boundary. It is a classical result that the volume of such a tetrahedron is given by the Bloch-Wigner dilogarithm and this agrees with the standard physics formulae for such box functions. The combinations of box functions that arise in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to the volume of a three-dimensional polytope without boundary, all of whose vertices are attached to a null polygon (which in other formulations is interpreted as a Wilson loop) at infinity.Comment: 16 pages, 5 figure

Universal structure of subleading infrared poles at strong coupling

Recently a concise expression for the subleading infrared singularity of dimensional-regularized gauge theories has been proposed. For conformal theories, such relation involves a universal eikonal contribution plus a non-eikonal contribution, related to the subleading term in the anomalous dimension of twist two operators with large spin. In this note we make use of the AdS/CFT correspondence in order to check such conjecture at strong coupling for the case of N=4 SYM.Comment: 13 page

One-loop corrections to AdS_5 x S^5 superstring partition function via Pohlmeyer reduction

We discuss semiclassical expansions around a class of classical string configurations lying in AdS_3 x S^1 using the Pohlmeyer-reduced from of the AdS_5 x S^5 superstring theory. The Pohlmeyer reduction of the AdS_5 x S^5 superstring theory is a gauged Wess-Zumino-Witten model with an integrable potential and two-dimensional fermionic fields. It was recently conjectured that the quantum string partition function is equal to the quantum reduced theory partition function. Continuing the previous paper (arXiv:0906.3800) where arbitrary solutions in AdS_2 x S^2 and homogeneous solutions were considered, we provide explicit demonstration of this conjecture at the one-loop level for several string solutions in AdS_3 x S^1 embedded into AdS_5 x S^5. Quadratic fluctuations derived in the reduced theory for inhomogeneous strings are equivalent to respective fluctuations found from the Nambu action in the original string theory. We also show the equivalence of fluctuation frequencies for homogeneous strings with both the orbital momentum and the winding on a big circle of S^5.Comment: 45 pages, references added, minor correction

Y-system for Scattering Amplitudes

We compute N=4 Super Yang Mills planar amplitudes at strong coupling by considering minimal surfaces in AdS_5 space. The surfaces end on a null polygonal contour at the boundary of AdS. We show how to compute the area of the surfaces as a function of the conformal cross ratios characterizing the polygon at the boundary. We reduce the problem to a simple set of functional equations for the cross ratios as functions of the spectral parameter. These equations have the form of Thermodynamic Bethe Ansatz equations. The area is the free energy of the TBA system. We consider any number of gluons and in any kinematic configuration.Comment: 69 pages, 19 figures, v2: references added, minor addition

Comments on operators with large spin

We consider high spin operators. We give a general argument for the logarithmic scaling of their anomalous dimensions which is based on the symmetries of the problem. By an analytic continuation we can also see the origin of the double logarithmic divergence in the Sudakov factor. We show that the cusp anomalous dimension is the energy density for a flux configuration of the gauge theory on $AdS_3 \times S^1$. We then focus on operators in ${\cal N}=4$ super Yang Mills which carry large spin and SO(6) charge and show that in a particular limit their properties are described in terms of a bosonic O(6) sigma model. This can be used to make certain all loop computations in the string theory.Comment: 33 pages, 1 figure,v2:reference to more recent work added, minor correction

Hidden Simplicity of Gauge Theory Amplitudes

These notes were given as lectures at the CERN Winter School on Supergravity, Strings and Gauge Theory 2010. We describe the structure of scattering amplitudes in gauge theories, focussing on the maximally supersymmetric theory to highlight the hidden symmetries which appear. Using the BCFW recursion relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory, and describe how it produces a sum of invariants of a large symmetry algebra. We review amplitudes in the planar theory beyond tree-level, describing the connection between amplitudes and Wilson loops, and discuss the implications of the hidden symmetries.Comment: 46 pages, 15 figures. v2 ref added, typos fixe

Correlation function of null polygonal Wilson loops with local operators

We consider the correlator of a light-like polygonal Wilson loop with n cusps with a local operator (like the dilaton or the chiral primary scalar) in planar N =4 super Yang-Mills theory. As a consequence of conformal symmetry, the main part of such correlator is a function F of 3n-11 conformal ratios. The first non-trivial case is n=4 when F depends on just one conformal ratio \zeta. This makes the corresponding correlator one of the simplest non-trivial observables that one would like to compute for generic values of the `t Hooft coupling \lambda. We compute F(\zeta,\lambda) at leading order in both the strong coupling regime (using semiclassical AdS5 x S5 string theory) and the weak coupling regime (using perturbative gauge theory). Some results are also obtained for polygonal Wilson loops with more than four edges. Furthermore, we also discuss a connection to the relation between a correlator of local operators at null-separated positions and cusped Wilson loop suggested in arXiv:1007.3243.Comment: 36 pages, 2 figure

Mass-Gaps and Spin Chains for (Super) Membranes

We present a method for computing the non-perturbative mass-gap in the theory of Bosonic membranes in flat background spacetimes with or without background fluxes. The computation of mass-gaps is carried out using a matrix regularization of the membrane Hamiltonians. The mass gap is shown to be naturally organized as an expansion in a 'hidden' parameter, which turns out to be $\frac{1}{d}$: d being the related to the dimensionality of the background space. We then proceed to develop a large $N$ perturbation theory for the membrane/matrix-model Hamiltonians around the quantum/mass corrected effective potential. The same parameter that controls the perturbation theory for the mass gap is also shown to control the Hamiltonian perturbation theory around the effective potential. The large $N$ perturbation theory is then translated into the language of quantum spin chains and the one loop spectra of various Bosonic matrix models are computed by applying the Bethe ansatz to the one-loop effective Hamiltonians for membranes in flat space times. Apart from membranes in flat spacetimes, the recently proposed matrix models (hep-th/0607005) for non-critical membranes in plane wave type spacetimes are also analyzed within the paradigm of quantum spin chains and the Bosonic sectors of all the models proposed in (hep-th/0607005) are diagonalized at the one-loop level.Comment: 36 Page

Generic multiloop methods and application to N=4 super-Yang-Mills

We review some recent additions to the tool-chest of techniques for finding compact integrand representations of multiloop gauge-theory amplitudes - including non-planar contributions - applicable for N=4 super-Yang-Mills in four and higher dimensions, as well as for theories with less supersymmetry. We discuss a general organization of amplitudes in terms of purely cubic graphs, review the method of maximal cuts, as well as some special D-dimensional recursive cuts, and conclude by describing the efficient organization of amplitudes resulting from the conjectured duality between color and kinematic structures on constituent graphs.Comment: 42 pages, 18 figures, invited review for a special issue of Journal of Physics A devoted to "Scattering Amplitudes in Gauge Theories", v2 minor corrections, v3 added reference

Large spin behavior of anomalous dimensions and short-long strings duality

We are considering the semi-classical string soliton solution of Gubser, Klebanov and Polyakov which represents highly excited states on the leading Regge trajectory, with large spin in $AdS_5$. A prescription relates this soliton solution with the corresponding field theory operators with many covariant derivatives, whose anomalous scaling dimension grows logarithmically with the space-time spin. We explicitly derive the dependence of anomalous dimension on spin for all leading and next-to-leading orders at strong coupling. We develop an iteration procedure which, in principle, allows to derive all terms in the large spin expansion of the anomalous scaling dimension of twist two operators. Our string theory results are consistent with the conjectured "reciprocity" relation, which has been verified to hold in perturbation theory up to five loops in N=4 SYM. We also derive a duality relation between long and short strings.Comment: 15 pages, 1 figure, comments and references adde