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

Expansion Trees with Cut

Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information which instances have been chosen for which quantifiers, known in the literature as expansion trees. Such a representation is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cut have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper we present a new approach that directly extends expansion trees by cuts and covers also non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps. We prove that it is weakly normalizing using methods from the epsilon-calculus

Ghost-free and Modular Invariant Spectra of a String in SL(2,R) and Three Dimensional Black Hole Geometry

Spectra of a string in $SL(2,R)$ and three dimensional (BTZ) black hole geometry are discussed. We consider a free field realization of ^sl (2,R) different from the standard ones in treatment of zero-modes. Applying this to the string model in SL(2,R), we show that the spectrum is ghost-free. The essence of the argument is the same as Bars' resolution to the ghost problem, but there are differences; for example, the currents do not contain logarithmic cuts. Moreover, we obtain a modular invariant partition function. This realization is also applicable to the analysis of the string in the three dimensional black hole geometry, the model of which is described by an orbifold of the SL(2,R) WZW model. We obtain ghost-free and modular invariant spectra for the black hole theory as well. These spectra provide examples of few sensible spectra of a string in non-trivial backgrounds with curved time and, in particular, in a black hole background with an infinite number of propagating modes.Comment: 18 pages, Latex; typos and some terminology correcte

Efficient Linear Programming Decoding of HDPC Codes

We propose several improvements for Linear Programming (LP) decoding algorithms for High Density Parity Check (HDPC) codes. First, we use the automorphism groups of a code to create parity check matrix diversity and to generate valid cuts from redundant parity checks. Second, we propose an efficient mixed integer decoder utilizing the branch and bound method. We further enhance the proposed decoders by removing inactive constraints and by adapting the parity check matrix prior to decoding according to the channel observations. Based on simulation results the proposed decoders achieve near-ML performance with reasonable complexity.Comment: Submitted to the IEEE Transactions on Communications, November 200

Bubbling Calabi-Yau geometry from matrix models

We study bubbling geometry in topological string theory. Specifically, we analyse Chern-Simons theory on both the 3-sphere and lens spaces in the presence of a Wilson loop insertion of an arbitrary representation. For each of these three manifolds we formulate a multi-matrix model whose partition function is the vev of the Wilson loop and compute the spectral curve. This spectral curve is the reduction to two dimensions of the mirror to a Calabi-Yau threefold which is the gravitational dual of the Wilson loop insertion. For lens spaces the dual geometries are new. We comment on a similar matrix model which appears in the context of Wilson loops in AdS/CFT.Comment: 30 pages; v.2 reference added, minor correction