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

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

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

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

Black Holes as Effective Geometries

Gravitational entropy arises in string theory via coarse graining over an underlying space of microstates. In this review we would like to address the question of how the classical black hole geometry itself arises as an effective or approximate description of a pure state, in a closed string theory, which semiclassical observers are unable to distinguish from the "naive" geometry. In cases with enough supersymmetry it has been possible to explicitly construct these microstates in spacetime, and understand how coarse-graining of non-singular, horizon-free objects can lead to an effective description as an extremal black hole. We discuss how these results arise for examples in Type II string theory on AdS_5 x S^5 and on AdS_3 x S^3 x T^4 that preserve 16 and 8 supercharges respectively. For such a picture of black holes as effective geometries to extend to cases with finite horizon area the scale of quantum effects in gravity would have to extend well beyond the vicinity of the singularities in the effective theory. By studying examples in M-theory on AdS_3 x S^2 x CY that preserve 4 supersymmetries we show how this can happen.Comment: Review based on lectures of JdB at CERN RTN Winter School and of VB at PIMS Summer School. 68 pages. Added reference

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction

Quantizing N=2 Multicenter Solutions

N=2 supergravity in four dimensions, or equivalently N=1 supergravity in five dimensions, has an interesting set of BPS solutions that each correspond to a number of charged centers. This set contains black holes, black rings and their bound states, as well as many smooth solutions. Moduli spaces of such solutions carry a natural symplectic form which we determine, and which allows us to study their quantization. By counting the resulting wavefunctions we come to an independent derivation of some of the wall-crossing formulae. Knowledge of the explicit form of these wavefunctions allows us to find quantum resolutions to some apparent classical paradoxes such as solutions with barely bound centers and those with an infinitely deep throat. We show that quantum effects seem to cap off the throat at a finite depth and we give an estimate for the corresponding mass gap in the dual CFT. This is an interesting example of a system where quantum effects cannot be neglected at macroscopic scales even though the curvature is everywhere small.Comment: 49 pages + appendice

Generalized cusp in AdS_4 x CP^3 and more one-loop results from semiclassical strings

We evaluate the exact one-loop partition function for fundamental strings whose world-surface ends on a cusp at the boundary of AdS_4 and has a "jump" in CP^3. This allows us to extract the stringy prediction for the ABJM generalized cusp anomalous dimension Gamma_{cusp}^{ABJM} (phi,theta) up to NLO in sigma-model perturbation theory. With a similar analysis, we present the exact partition functions for folded closed string solutions moving in the AdS_3 parts of AdS_4 x CP^3 and AdS_3 x S^3 x S^3 x S^1 backgrounds. Results are obtained applying to the string solutions relevant for the AdS_4/CFT_3 and AdS_3/CFT_2 correspondence the tools previously developed for their AdS_5 x S^5 counterparts.Comment: 48 pages, 2 figures, version 3, corrected misprints in formulas 2.12, B.86, C.33, added comment on verification of the light-like limi