“Solving Scattering in N=4 Super-Yang-Mills Theory” is the latest review of progress with amplitudes. The “antipodal duality” is the biggest new mystery there.
Also, a reminder that the correct ontology of this theory should also be the correct ontology of string theory, because of AdS/CFT: studying subtle changes in the operator algebra of the theory, in phases dual to an AdS black hole.
It was recently the tenth anniversary of the founding of this blog. My focus has long since shifted to the standard model. But mathematical progress here may in time affect those more practical theories too.
There is a paper today talking about “exact solvability of N=4 Yang-Mills”; but the author seems to mean only the classical theory.
For those who want to understand how interest in N=4 developed out of actual physics, “Electromagnetic duality for children” may help, by tracing a path from Maxwell’s equations, monopoles and dyons, and Montonen-Olive duality, to Yang-Mills with extended supersymmetry.
It’s almost three years since I last updated this blog, and the blur of life makes it less possible than ever, to keep up with this important sub-field. But as we approach a new decade, I thought I would mention two things. First, Arkani-Hamed and collaborators keep turning out papers with new concepts like “binary geometry” and “string canonical form”; and second, Sheppeard (vixra, my notes) keeps laboring away, in an attempt to apply similar concepts directly to the standard model.
“4gravitons” predicts future progress.
Much of the most advanced work on the N=4 theory takes place in the “planar limit” or “planar approximation”. In this limit, one only considers Feynman diagrams that are planar. This limit goes back 40 years to ‘t Hooft’s large-N expansion of gauge theories, which has also played a large part in AdS/CFT.
Now we are beginning to see hopes for progress beyond the planar limit, here, here, and here.
For context, perhaps consider the first picture from Beisert et al.
The quest for an objective physical theory that reduces in some limit to quantum mechanics is a valid and important line of inquiry, but usually it is carried out without much attention to the mathematically most advanced quantum theories. Here I’m going to mention a few ideas for how N=4 Yang-Mills might emerge from a “realist” or “objective” ontology.
1) Gerard ‘t Hooft’s cellular automata. In a previous post, I mentioned ‘t Hooft’s mapping between a particular cellular automaton and a theory of free bosonic fields in 1+1 dimensions. In a subsequent paper, he extended the mapping to fermions as well, and explicitly noted that there might be some relationship to a superstring worldsheet theory. There are limitations to the utility of what ‘t Hooft has done, but the general ideas here are definitely of interest.
2) Bohmian mechanics for the CFT. N=4 Yang-Mills contains quantum fields of spin 0, 1/2, and 1. A Bohmian mechanics for spin-0 fields was worked out long ago, Peter Holland extended it to spin-1/2 fields, and now Tatiana Seletskaia points out that he claims to have done it for spin-1 fields too. I can’t endorse Holland’s work yet, but if a Bohmian formulation of the N=4 theory exists, that seems to imply that the AdS string theory would have to be implicit in it, too; which would be remarkable.
3) Hamilton-Jacobi theory in the bulk. “On the Holographic Renormalization Group”, an early and important AdS/CFT paper, tries to analyze the duality using Hamilton-Jacobi theory on the bulk side. This is another potential connection to Bohmian mechanics, which can be described as a deformation of classical Hamilton-Jacobi theory.
4. The twistor revolution. At the end of 2012, Arkani-Hamed et al finally came out with another major installment of their series on the Grassmannian representation of the N=4 theory. This paper is not about ontology, but it is a comprehensive rewrite of the math which will surely have ontological implications.
Part of the bulk S-matrix is a product of two copies of the R-matrix of the Hubbard model. Read all about it.
AdS/CFT says that N=4 super-Yang-Mills is equal to the IIB string on AdS5 x S5, and Arkani-Hamed et al are also constructing a twistorial formulation of the theory in which “locality”, “unitarity”, “gauge symmetry” etc are all replaced by a new set of concepts.
Now I want to mention a very intriguing possibility. For many years, Gerard ‘t Hooft has been pushing the idea that you could reconstruct quantum field theory from a cellular automaton, possibly in a lower number of dimensions. In his latest paper, he actually claims a “duality” between a certain C.A. and a free bosonic field theory in 1+1 dimensions. It has something to do with combinations of eigenvalues of operators on a special lattice of points – these points are the “cells”, and the combinations are the states of the cells.
It would be an interesting project to extend this work in the direction of the IIB worldsheet theory which produces the N=4 theory.
A talk by Linda Uruchurtu (based on this paper) reviews the study of correlation functions in AdS5/CFT4.
Joseph Minahan has the best introduction to N=4 super-Yang-Mills that I’ve seen.
Slides from Marcus Spradlin at KITP touch on many developments.
Luis Alday reviews the interpretation of scattering amplitudes at strong coupling as a minimal surface in anti-de-Sitter space.
I believe Simon Caron-Huot’s latest is currently the most advanced paper on the subject: a generalization of scattering amplitudes is proposed, that would possess superconformal and dual superconformal symmetry, and which would begin to explain the simplicity recently discovered in the BDS remainder function, as discussed by Spradlin.
N=4 Super-Yang-Mills Theory 
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