Solving N=4 super-Yang-Mills theory via Scattering Amplitudes: January 8, 2024 – March 8, 2024

Organizing by:

• Benjamin Basso (Ecole Normale Superieure),
• Lance Dixon (SLAC/Stanford U.)
• Jaroslav Trnka (UC Davis)
• Anastasia Volovich (Brown)

This program will bring together physicists and mathematicians with expertise in different facets of particle scattering in planar N=4 super-Yang-Mills theory, in order to try to solve the theory for generic values of the coupling and kinematical variables.

It has been a long-standing quest to solve strongly-interacting four-dimensional relativistic quantum field theories by analytical methods. N = 4 super-Yang-Mills theory (SYM) is the “hydrogen atom” of gauge theories. It is the archetype for the AdS/CFT correspondence, connecting it to string theory and gravity, particularly at strong coupling. Perturbatively, its gluon scattering amplitudes closely resemble those for the gauge theory of the strong interactions, quantum chromodynamics (QCD). Hence it has proven indispensable as a testing ground for powerful new computational methods for precision collider physics. In the planar limit of a large number of colors (gauge group SU(Nc) with Nc → ∞), N = 4 SYM is integrable, and its scattering amplitudes are dual to the expectation values of Wilson loops for closed light-like polygons. The perturbative expansion of planar N = 4 SYM has remarkable geometric features, centered around polytopes related to various Grassmannians and the
differential forms that live on them.

There are currently three independent descriptions of scattering amplitudes in planar N = 4 SYM, illustrated in the figure.

A weak-coupling formulation makes contact with perturbative methods involving Feynman diagrams; a “holographic” strong-coupling formulation employs minimal-area surfaces in Anti-de Sitter space; and a pentagon operator product expansion (POPE) approach exploits the two-dimensional integrability of a dual string picture at finite coupling in various kinematic limits. These formulations are all mutually consistent. (Indeed the compatibility of the POPE and perturbative approaches can be established through eight loop orders.) But they are formulated in mathematically very distinct ways, based on different underlying physical principles, and they make different properties of amplitudes manifest. In this program, the goal is to find a unified description of amplitudes in this theory that matches onto each of these formulations in the appropriate limit. Mathematically, the proposal is to search for functions that are able to interpolate between weak and strong coupling, for arbitrary kinematics, using all the knowledge we have about them in the respective limits. In this way one could gain a quantitative picture of how weak coupling excitations (gluons) evolve into strong coupling ones (strings) under a variety of physical circumstances.  The spin-off for QCD could also be very important.

This program also has an associated workshop: Mathematical Aspects of N=4 Super-Yang-Mills Theory: February 26-March 1, 2024